Xenharmonic Wiki - User contributions [en]

User talk:Cmloegcmluin/2.11.13.17.19 subgroup

2026-06-13T15:53:45Z

Cmloegcmluin: Cmloegcmluin moved page Talk:2.11.13.17.19 subgroup to User talk:2.11.13.17.19 subgroup: nominated for deletion from main b/c arguably unnotable subgroup, but relevant to the Yer tuning

== On deletion ==

Rather than delete, if this doesn’t meet standards, I’ll move it to a user page, because it is valuable to my own [[Yer]] tuning system. --[[User:Cmloegcmluin| Cmloegcmluin]] [[User_talk:Cmloegcmluin|(talk)]], 15:22, 12 June 2026 (UTC)

: You can move it to your own user page, no problem about that. --[[User:Eufalesio|Eufalesio]] ([[User talk:Eufalesio|talk]]) 21:19, 12 June 2026 (UTC)

User:Cmloegcmluin/2.11.13.17.19 subgroup

2026-06-13T15:53:44Z

Cmloegcmluin: Cmloegcmluin moved page 2.11.13.17.19 subgroup to User:2.11.13.17.19 subgroup: nominated for deletion from main b/c arguably unnotable subgroup, but relevant to the Yer tuning

{{Delete|This is just a comma dump page, and the subgroup is arguably not notable.}}

== Commas ==
Here follows a selection of commas in the 2.11.13.17.19 [[domain basis]].

{| class="wikitable"
|-
! rowspan="2" | Name
! rowspan="2" | Prime-count vector
! rowspan="2" | Ratio
! rowspan="2" | Cents
! colspan="5" | As composition of other commas
! colspan="10" | Tempered by patent val for ed2?
|-
! yama
! bean
! Blumeyer
! frouggie
! pollar
! 13
! 24
! 33
! 37
! 57
! 70
! 80
! 113
! 124
! 137
|-
| Blumeyer
| {{vector| 7 0 0 0 -1 -1 -1 1 }}
| 2432 / 2431
| 0.7120024978
|
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| 1
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| ✓
| ✓
| ✓
| ✓
| ✓
| ✓
| ✓
| ✓
| ✓
| ✓
|-
| yama
| {{vector| -4 0 0 0 1 -1 0 1 }}
| 209 / 208
| 8.303296728
| 1
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| ✓
| ✓
| ✓
| ✓
| ✓
|
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|-
| Blume
| {{vector| -11 0 0 0 2 0 1 0 }}
| 2057 / 2048
| 7.59129423
| 1
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| −1
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| ✓
| ✓
| ✓
| ✓
| ✓
| ✓
|
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|-
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| {{vector| 3 0 0 0 0 -2 -1 2 }}
| 2888 / 2873
| 9.015299226
| 1
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| 1
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| ✓
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| ✓
| ✓
| ✓
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| eye
| {{vector| 3 0 0 0 -2 0 2 -1 }}
| 2312 / 2299
| 9.761918139
| 1
| 1
| −2
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| ✓
| ✓
| ✓
| ✓
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| ✓
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| lum
| {{vector| -1 0 0 0 -1 -1 2 0 }}
| 289 / 286
| 18.06521487
| 2
| 1
| −2
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| ✓
| ✓
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| ✓
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|-
| ume
| {{vector| -8 0 0 0 0 0 3 -1 }}
| 4913 / 4864
| 17.35321237
| 2
| 1
| −3
|
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| ✓
| ✓
| ✓
| ✓
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| mey
| {{vector| -7 0 0 0 0 3 -1 0 }}
| 2197 / 2176
| 16.62757581
| 2
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| 1
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| ✓
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| {{vector| -5 0 0 0 0 -2 2 1 }}
| 5491 / 5408
| 26.36851159
| 3
| 1
| −2
|
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| ✓
| ✓
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| ✓
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|-
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| {{vector| -5 0 0 0 -1 0 0 2 }}
| 361 / 352
| 43.7080899
| 5
| 1
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| 1
| ✓
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| {{vector| -1 0 0 0 -2 1 0 1 }}
| 247 / 242
| 35.40479317
| 4
| 1
| −1
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| 1
| ✓
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| {{vector| 6 0 0 0 -1 -4 3 0 }}
| 314432 / 314171
| 1.437639059
|
| 1
| −2
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| −1
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| ✓
| ✓
| ✓
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| ✓
| ✓
|-
|
| {{vector| 14 0 0 0 -4 0 1 -1 }}
| 278528 / 278179
| 2.170623909
|
| 1
| −1
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| ✓
| ✓
| ✓
|-
|
| {{vector| 4 0 0 0 -2 3 -2 0 }}
| 35152 / 34969
| 9.036281578
| 1
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| 1
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| ✓
| ✓
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| {{vector| 10 0 0 0 -3 -1 1 0 }}
| 17408 / 17303
| 10.47392064
| 1
| 1
| −1
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| {{vector| 0 0 0 0 -1 2 -2 1 }}
| 3211 / 3179
| 17.33957831
| 2
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| 1
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| ✓
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| {{vector| 6 0 0 0 -2 -2 1 1 }}
| 20672 / 20449
| 18.77721736
| 2
| 1
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| ✓
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| {{vector| -4 0 0 0 0 1 -2 2 }}
| 4693 / 4624
| 25.64287503
| 3
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| ✓
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| {{vector| 3 0 0 0 -3 2 0 0 }}
| 1352 / 1331
| 27.10149644
| 3
| 1
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| ✓
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| {{vector| -8 0 0 0 -1 2 1 0 }}
| 2873 / 2816
| 34.69279067
| 4
| 1
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| 1
| ✓
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| {{vector| 14 0 0 0 2 0 -2 -3 }}
| 1982464 / 1982251
| 0.1860173318
|
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| 1
| 1
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| ✓
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| ✓
| ✓
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| ✓
|-
|
| {{vector| 13 0 0 0 1 -1 0 -3 }}
| 90112 / 89167
| 18.2512322
| 2
| 1
| −2
| 1
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| ✓
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| {{vector| 7 0 0 0 -3 1 2 -2 }}
| 480896 / 480491
| 1.458621411
|
| 1
| −2
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| ✓
| ✓
| ✓
|-
| bean
| {{vector| 21 0 0 0 -5 -1 0 0 }}
| 2097152 / 2093663
| 2.882626407
|
| 1
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| ✓
| ✓
| ✓
|-
|
| {{vector| -3 0 0 0 -1 4 -1 -1 }}
| 28561 / 28424
| 8.32427908
| 1
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| 1
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| ✓
| ✓
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| {{vector| -11 0 0 0 1 2 -1 1 }}
| 35321 / 34816
| 24.93087254
| 3
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| 1
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| ✓
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| {{vector| 3 0 0 0 4 0 -1 -3 }}
| 117128 / 116603
| 7.777311562
| 1
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| −1
| 1
| 1
|
|
| ✓
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| ✓
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| ✓
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|-
|
| {{vector| -18 0 0 0 2 3 0 0 }}
| 265837 / 262144
| 24.21887004
| 3
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| −1
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| 1
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| ✓
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|
| {{vector| 9 0 0 0 2 -2 0 -2 }}
| 61952 / 61009
| 26.55452893
| 3
| 1
| −2
| 1
| 1
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| ✓
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|-
|
| {{vector| 10 0 0 0 3 -1 -2 -2 }}
| 1362944 / 1356277
| 8.48931406
| 1
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| 1
| 1
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| ✓
| ✓
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| ✓
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| ✓
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|-
|
| {{vector| 21 0 0 0 1 -1 -3 -2 }}
| 23068672 / 23056709
| 0.8980198296
|
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| 1
| 1
| 1
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| ✓
| ✓
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| ✓
|-
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| {{vector| -7 0 0 0 -3 -1 1 4 }}
| 2215457 / 2214784
| 0.525985166
|
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| 1
| −1
| −1
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| ✓
| ✓
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| ✓
|-
|
| {{vector| -10 0 0 0 2 3 -3 1 }}
| 5050903 / 5030912
| 6.865657669
| 1
| −1
| 2
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| 1
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| ✓
| ✓
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| ✓
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|-
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| {{vector| -3 0 0 0 1 2 -4 2 }}
| 671099 / 668168
| 7.577660166
| 1
| −1
| 3
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| 1
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| ✓
| ✓
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| {{vector| 5 0 0 0 3 -3 0 -1 }}
| 42592 / 41743
| 34.85782565
| 4
| 1
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| 1
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| ✓
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| {{vector| 6 0 0 0 -3 0 -1 2 }}
| 23104 / 22627
| 36.11679567
| 4
| 1
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| 1
| ✓
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|-
| pollar
| {{vector| 1 0 0 0 -2 5 -1 -2 }}
| 742586 / 742577
| 0.02098235203
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| 1
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| ✓
| ✓
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| {{vector| 17 0 0 0 2 -2 -3 -1 }}
| 15859712 / 15775643
| 9.201316557
| 1
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| 1
| 1
| 1
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| ✓
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| ✓
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| ✓
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|-
|
| {{vector| 11 0 0 0 -3 2 -3 1 }}
| 6576128 / 6539203
| 9.748284075
| 1
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| 2
|
| 1
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| ✓
| ✓
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|-
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| {{vector| 24 0 0 0 -1 -1 -1 -3 }}
| 16777216 / 16674229
| 10.65993797
| 1
| 1
| −1
| 1
| 1
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| ✓
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| ✓
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|-
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| {{vector| 17 0 0 0 -4 -2 0 1 }}
| 2490368 / 2474329
| 11.18592313
| 1
| 1
|
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| ✓
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|-
|
| {{vector| 2 0 0 0 3 -1 1 -3 }}
| 90508 / 89167
| 25.84252643
| 3
| 1
| −3
| 1
| 1
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| ✓
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| {{vector| 16 0 0 0 1 -3 -1 -1 }}
| 720896 / 709631
| 27.26653142
| 3
| 1
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| 1
| 1
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| ✓
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| {{vector| -15 0 0 0 2 1 -1 2 }}
| 567853 / 557056
| 33.23416926
| 4
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| 1
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| ✓
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| {{vector| -2 0 0 0 -3 0 2 1 }}
| 5491 / 5324
| 53.47000804
| 6
| 2
| −3
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| 1
| ✓
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| {{vector| 8 0 0 0 -3 4 -2 -1 }}
| 7311616 / 7308521
| 0.7329848499
|
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| 1
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| 1
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| ✓
| ✓
|-
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| {{vector| 28 0 0 0 -2 0 -1 -4 }}
| 268435456 / 268070297
| 2.356641241
|
| 1
| −1
| 1
| 1
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| ✓
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| ✓
|-
|
| {{vector| -10 0 0 0 -4 3 0 3 }}
| 15069223 / 14992384
| 8.850264246
| 1
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| 1
| −1
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| ✓
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| ✓
| ✓
| ✓
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|-
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| {{vector| 6 0 0 0 4 -2 -2 -1 }}
| 937024 / 927979
| 16.79261079
| 2
|
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| 1
| 1
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| ✓
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| ✓
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| {{vector| 15 0 0 0 -4 3 -3 0 }}
| 71991296 / 71931233
| 1.444987348
|
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| 2
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| 1
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| ✓
| ✓
|-
|
| {{vector| -17 0 0 0 3 4 -2 0 }}
| 38014691 / 37879808
| 6.153655171
| 1
| −1
| 1
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| 1
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| ✓
| ✓
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| {{vector| 0 0 0 0 -6 0 3 2 }}
| 1773593 / 1771561
| 1.984606577
|
| 1
| −1
| −1
| −1
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| ✓
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| ✓
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| {{vector| 14 0 0 0 -5 2 -1 0 }}
| 2768896 / 2737867
| 19.51020221
| 2
| 1
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| 1
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| ✓
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| {{vector| 18 0 0 0 0 3 -4 -3 }}
| 575930368 / 572870539
| 9.222298909
| 1
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| 1
| 1
| 2
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| ✓
| ✓
| ✓
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|-
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| {{vector| -6 0 0 0 -5 4 0 2 }}
| 10310521 / 10307264
| 0.5469675181
|
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| 1
| −1
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| ✓
|-
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| {{vector| 20 0 0 0 1 -4 1 -3 }}
| 196083712 / 195899899
| 1.623656391
|
| 1
| −2
| 1
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| ✓
|
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| {{vector| -14 0 0 0 3 2 -3 2 }}
| 81202979 / 80494592
| 15.1689544
| 2
| −1
| 2
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| 1
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| ✓
| ✓
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|-
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| {{vector| -21 0 0 0 -2 3 1 3 }}
| 256176791 / 253755392
| 16.44155848
| 2
|
|
| −1
|
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| ✓
| ✓
|
| ✓
|
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|-
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| {{vector| -1 0 0 0 4 1 -3 -1 }}
| 190333 / 186694
| 33.42018659
| 4
|
|
| 1
| 2
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| ✓
|
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| {{vector| 8 0 0 0 1 -3 2 -2 }}
| 813824 / 793117
| 44.61974379
| 5
| 2
| −4
| 1
| 1
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| ✓
|
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|-
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| {{vector| -7 0 0 0 -5 1 4 1 }}
| 20629687 / 20614528
| 1.272604079
|
| 1
| −2
| −1
| −1
|
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| ✓
|
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|-
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| {{vector| 2 0 0 0 6 -5 0 -1 }}
| 7086244 / 7054567
| 7.75632921
| 1
|
| −1
| 1
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| ✓
|
|-
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| {{vector| -17 0 0 0 -3 4 1 2 }}
| 175278857 / 174456832
| 8.138261748
| 1
|
|
| −1
|
|
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| ✓
|
| ✓
|
|
|
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|-
| frouggie
| {{vector| 13 0 0 0 4 -5 -1 -1 }}
| 119939072 / 119927639
| 0.1650349798
|
|
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| 1
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| ✓
|
| ✓
|
|
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|-
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| {{vector| 16 0 0 0 2 -5 1 -2 }}
| 134807552 / 134036773
| 9.926953119
| 1
| 1
| −2
| 1
|
|
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| ✓
|
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|-
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| {{vector| -24 0 0 0 -2 5 2 1 }}
| 2038769863 / 2030043136
| 7.42625925
| 1
|
| −1
| −1
|
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|
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| ✓
|
| ✓
|
|
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|-
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| {{vector| -4 0 0 0 -5 -1 3 3 }}
| 33698267 / 33498608
| 10.28790331
| 1
| 1
| −1
| −1
| −1
|
|
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| ✓
|
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|-
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| {{vector| -4 0 0 0 7 -1 -3 -1 }}
| 19487171 / 19416176
| 6.31869015
| 1
| −1
| 1
| 1
| 1
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| ✓
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|-
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| {{vector| 18 0 0 0 -6 3 -1 -1 }}
| 575930368 / 572214203
| 11.20690549
| 1
| 1
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| 1
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| ✓
|
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|-
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| {{vector| 0 0 0 0 5 2 -5 -1 }}
| 27217619 / 26977283
| 15.35497173
| 2
| −1
| 2
| 1
| 2
|
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| ✓
|
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|-
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| {{vector| 22 0 0 0 -1 4 -4 -4 }}
| 119793516544 / 119729942651
| 0.9190021816
|
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| 1
| 1
| 2
|
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| ✓
|-
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| {{vector| -18 0 0 0 -4 3 3 2 }}
| 3896583821 / 3838050304
| 26.20347661
| 3
| 1
| −2
| −1
|
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| ✓
|
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|-
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| {{vector| 22 0 0 0 -7 4 -1 -2 }}
| 119793516544 / 119592768427
| 2.903608759
|
| 1
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| 1
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| ✓
| ✓
|-
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| {{vector| -27 0 0 0 -1 7 -2 3 }}
| 430392078103 / 426678157312
| 15.00391942
| 2
| −1
| 2
| −1
| 1
|
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| ✓
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|}

== See also ==
* [[Yer]], a hexadecatonic [[Just intonation|JI]] [[wikipedia:Euler–Fokker_genus|EFG]] in this subgroup
* [[Gjaeck]], a [[Tridecatonic MOS|tridecatonic MOS]] of [[57edo|57ed2]] in this subgroup
* [[Hilim13]], a tridecatonic JI tuning in this subgroup

[[Category:Just intonation subgroups|#]]
[[Category:Rank-5 temperaments|#]]
[[Category:19-limit|#]]

User talk:Cmloegcmluin/2.11.13.17.19 subgroup

2026-06-12T15:22:36Z

Cmloegcmluin: Created page with "Rather than delete, if this doesn’t meet standards, I’ll move it to a user page, because it is valuable to my own Yer tuning system."

Rather than delete, if this doesn’t meet standards, I’ll move it to a user page, because it is valuable to my own [[Yer]] tuning system.

Dave Keenan & Douglas Blumeyer's guide to RTT/Tuning in nonstandard domains

2026-06-09T03:08:17Z

Cmloegcmluin:

{{breadcrumb}}{{texmap}}
This is article 9 of 9 in [[Dave Keenan]] & [[Douglas Blumeyer]]'s guide to RTT, or "[[D&D's guide]]" for short. This concluding chapter assumes much of the knowledge that comes from the earlier chapters, so we encourage readers to get up to speed with that material before digging into the stuff covered here.

At the time of this writing, not much has been done regarding the novel problems involved with optimizing tunings for [[regular temperaments]] of nonstandard [[domain basis|domains]].

If "nonstandard" is taken to mean "non-prime-limit" (as we think it's a good idea to do), then there's nothing fundamentally different about tuning a temperament of 2.3.7 than there is to tuning a temperament of 2.3.5. These prime-subgroup domains do not raise novel ''tuning'' problems, at least not conceptual ones (i.e, you may have to refactor your tuning code to accommodate the fact that the third entry may not always correspond to prime 5, but that's the extent of the problem).

At the other extreme, "nonstandard" may be taken to mean "irrational". That is, we might find a temperament whose basis includes the [[golden ratio]], φ, in order to achieve a sort of anti-JI effect. But we won't be considering these sorts of temperaments here, and that is because irrational numbers cannot be factored into primes.<ref group="note">At least not ordinary primes, and not without opening a whole other can of worms. We know about these worms all too well. See: [https://forum.sagittal.org/viewtopic.php?p=4604#p4604 Noble frequency ratios as prime-count vectors in ℚ(√5)]</ref>

The bulk of this article section will deal with nonstandard bases between these two extremes. Things get interesting when we think about possibilities for tuning temperaments with '''nonprime''' domain bases, i.e. those which are still rational (JI) but include nonprime basis elements such as 9, or 5/3. As far as we know, this topic has only been written about in the context of a single pair of all-interval tuning schemes: minimax-S and minimax-ES (see: [[Tp tuning]]). On the Tp tuning wiki page, two possible approaches are presented for tuning temperaments with nonprime bases. It calls these approaches the "subgroup notion" and the "inharmonic notion". We prefer the terminology '''prime-based''' and '''nonprime-based''', respectively, for reasons that we will explain soon. But in fact, the choice between these two approaches is relevant to ''any'' conceivable tuning scheme applied to a temperament with a nonprime basis. This is to say that there's nothing specific to these two all-interval schemes about the choice between "prime-based" and "nonprime-based".

But not only has this choice only been written about with regards to a mere two tuning schemes, it appears only to have been ''applied'' to a single one of those, in a single context. This context is Graham Breed's online tuning app x31eq, and that single tuning scheme is Graham's favored tuning scheme: minimax-ES. He did invent it after all. He calls it "Tenney-Euclidean", or "TE", and so in his app these two variations for temperaments with nonprime bases are called "subgroup TE" and "inharmonic TE" (so for us, those'd be "prime-based minimax-ES" and "nonprime-based minimax-ES", respectively).

The goal with this section of this article is to explain what this fundamental choice regarding nonprime-basis temperament-tunings is all about, and speak to why one might prefer using one approach to the other. We will also introduce the fact that one may choose to take ''neither'' of these two approaches, a possibility which does not seem to have been addressed in the practice thus far. We will discuss the topic in a way that effectively generalizes it to all nonprime-basis tuning schemes, as it should be. Finally, we will also work through the computation process for both approaches.

== The three approaches ==
One property of prime numbers (that has particular relevance for the way we do things in RTT) is being building blocks for the rational numbers; this property taps into the way we take the prime factorizations of rational numbers and put them into vector form and do linear algebra on them with transformation matrices and such. When composers or theorists use a domain basis that includes nonprimes, such as the basis 2.9.7/5,<ref group="note">"2.9.7/5" should be read as <math>\small 2.9.\frac{7}{5}</math> not <math>\small (2.9.7)/5</math>. The dots are best thought of as separators rather than multiplication operators, although they do serve to remind us that our domain consists of all possible ''products'' of integer powers of these elements.</ref> they are designating these numbers as their "basis elements" instead, thereby asking us to treat them ''as protected-against-prime-factoring''.<ref group="note">Elsewhere this has been called a "formal prime", but the coiner of that terminology, Inthar, has recommended its disuse.</ref> So when notating, or building scales and chord progressions on lattice diagrams, or whatever, these basis elements are what shape what's possible to do musically, and how.

One of our considerations is: having chosen a nonprime basis like this, how to approach optimizing its tuning? Here are three possible options:
* With the prime-based approach, we temporarily factor all of the basis elements of our nonprime basis into primes. This essentially means that we've embedded our temperament into a new domain, a larger one, i.e. a superspace of the original domain, and not just any superspace, but specifically the one with the ''simplest prime-only basis'' (or ''SPOB'' for short). Once we've achieved this, we can tune with respect to any target-intervals we want in ''that'' superspace, then convert back to the original nonprime basis as a final step. (For all-interval tunings such as minimax-ES, the assumption is that we take advantage of this by choosing the primes of the prime-only basis of this superspace as our target-intervals.)
* With the nonprime-based approach, we choose to not break our basis down into its prime elements. And when computing any interval complexities involved in damage weights, we also refuse to prime factor their basis elements.
* With the neutral approach, we simply tune as we normally would, which is to say that we don't do any manipulation of the basis (prime factoring or otherwise), but when computing interval complexities we will prime factor intervals as per usual.

Given this explanation, our nomenclature for these approaches should be clear now (there ''are'' [[Tp_tuning|explanations for the names "subgroup" and "inharmonic"]], of course, but we don't think they're nearly as clear as ours). When you're working with a temperament of a domain with any nonprime basis elements, then the "prime-based" approach factors them into actual primes and tunes with respect to those (then unfactors), whereas the "nonprime-based" approach preserves your nonprime-based intervals as they are, unbroken. While the neutral approach leads to neither of those special things happening.

And so now we can see why this choice wouldn't have made any difference for temperaments of the domain 2.3.7. What standard superspace would you embed that in? What basis element needs prime-factorization here? Its basis elements are already as prime-factored as they can go, in either case.<ref group="note">This is why, on pages of Graham's app, for temperaments of ''standard domains'', you will find the message, "This is a trivial subgroup of the rational numbers so TE is TE is TE."</ref>

=== Note on coprime ===
We note that while the [[Tp tuning]] page claims that the choice between approaches only matters when two basis elements ''share a prime factor'', or in other words, that the choice doesn't matter for ''coprime'' bases such as 2.9.5 or 2.5/3.7, this ''is'' true of "Tp tunings" such as minimax-ES as used on Graham's app and minimax-S, however, this is not true in general when the prime-based and nonprime-based approaches are extended to apply to ordinary non-all-interval tuning schemes, alternative complexities, etc.

In particular, when following a scheme whose target-interval set is itself determined by a scheme (such as [[TILT]]), it is likely to result in a different set of target-intervals depending on whether you're tuning in the original nonprime basis or the standard superspace which will support many more different target-intervals.

== How to choose your approach ==
Now we suppose you must be wondering: why would anyone have a preference among these three approaches to tuning optimisation? What would the actual effect on the tuning be? Those are both excellent questions, and we will cover both of those before this section is up.

There are valid reasons to prefer the prime-based or the nonprime-based approach, but we feel the clear default here is choosing neither one.

We'll begin with a sort of flowchart to help you choose your favored approach (of course, your preference may change from one piece of music to the next, but that's another issue):

* Do you wish to use the opportunity of having chosen a nonprime domain basis to explore experimental new possibilities of tuning optimization, essentially exploring alternate realities where your basis elements really were protected-against-prime-factoring? ''Choose the nonprime-based approach''.
* Are you not interested in (or actively against the idea of) such experimentation, not really thinking of your choice of a nonprime domain basis as a tuning choice like that, but rather thinking of it as a compositional constraint in terms of how it affects your lattice and such? ''Choose the neutral approach'', unless:
* You are also optimizing the tuning of this temperament for simultaneous (or near simultaneous) performance with other temperaments that share the same SPOB (simplest prime-only basis), and you wish for the tunings to match better across them. ''Choose the prime-based approach.''

=== Prime-based ===
The distinguishing characteristic of the prime-based approach is that it's the only one which supports tuning with respect to intervals that you can't actually reach to play in the temperament. For example, you would choose the prime-based approach to tuning a temperament of the 2.5/3.7 domain if you still want prime 5 and prime 3 to be individually optimized for, ''even while you are preventing yourself from ever using them in isolation from each other''.

Now, putting it that way, this approach seems ''not'' logical, though. But that's because we haven't included all of the reasoning yet. So you would only choose to do ''that'' if you ''also'' intended to integrate this music you're writing—with no lone 5's or lone 3's, only the 5/3—with other music (whether actually simultaneously or in quick succession) that is written in temperaments with other nonprime bases that share the same SPOB (which would be collectively tuned with respect to this shared superspace basis) and/or temperaments of this SPOB itself, which ''would'' contain the lone 5's or 3's. In other words, the prime-based approach only makes sense if you are using the subspacing to compositionally constrain yourself in one place or another at one time—one voice, one movement, etc.—but ultimately recognize that the music is still fundamentally built out of the actual primes, ''and you make use of those primes'' somewhere at some point.

Said another way, you still view the music as "truly being in" the SPOB (the one found by the prime factorization of your basis elements), but have chosen to use a subspace of it in order to restrain yourself from accessing certain intervals or chords.

Note that if you don't actually ''use'' any of the new intervals afforded to you by the change of basis to the superspace, you will not find a different tuning than you would by the neutral approach here. That's because it's only the nonprime-based approach which actually changes how complexity is measured.

The prime-based approach is the preferred approach of the tuning theorist and composer Flora Canou for these reasons.

=== Nonprime-based ===
With the nonprime-based approach, you are essentially trying to make music as if in a world where certain primes ''never existed at all''.

Now, this statement is always trivially true, since no one uses all the primes up to infinity; most people cut off before getting much further than prime 13. But what we mean here is more the effect of, say, to use the same example from the previous section, to paint a musical picture in a world where it's as if prime 3 and prime 5 never existed, and all that ever existed was this 5/3 interval.

The key defining characteristic of this approach is that when computing interval complexity, one is forbidden from thinking of the nonprime basis elements as prime factorizable. For example, in the computation of log-product complexity, if 9/7 was one of our nonprime basis elements, then its log-product complexity would be <math>\log_2{\frac{9}{7}}</math> rather than the usual <math>\log_2{9·7}</math>.

So the nonprime-based approach is for those people who wish to actually go a bit further into the promise of the experimental territory of nonprime bases.

The nonprime-based approach is the default approach used by Graham Breed's temperament app.

We note that on account of this difference in pre-scaling within the double bars of the <math>1</math>-norm, there is an argument that nonprime-based minimax-ES is not actually using the default complexity of <math>\text{lp-C}()</math>, or that what we've been considering to be log-product complexity this whole time should more accurately have been thought of exclusively in its vector-based form, never its quotient-based form, i.e. the way more directly akin to [https://en.xen.wiki/w/N2D3P9#copfr copfr] and [https://mathworld.wolfram.com/SumofPrimeFactors.html sopfr] that might more fittingly be referred to as "lopfr" for logs-of-prime-factors-with-repetition! But there's also an argument that <math>\log_2{\frac{9}{7}}</math> can still be seen as the log-product complexity of 9/7 when you realize that if 9/7 is considered a basis element, that means that it's ''not actually allowed to be broken down into a numerator and denominator''; if anything, it would be <math>\frac{9/7}{1}</math>, i.e. the numerator is 9/7 and the denominator is 1. In other words, its the nonprime-based approach which should be the thing that shoulders the burden of this sort of weirdness, not any complexity functions; the nonprime-based approach redefines reality broadly and you just have to think about these sorts of things on a case-by-case basis, how things work differently if 9/7 is now protected-against-prime-factoring.

=== Neither ===
The neutral approach is the most straightforward to understand the motivation for. This is about optimizing the tuning for the specific temperament at hand, with no considerations either made for inter-temperamental music-making such as we've seen is the motivation for the prime-based approach, nor for the experimental treatment of the nonprimes as intervals protected-against-prime-factoring for tuning purposes as we see with the nonprime approach.

This approach is for people who think of the use of a nonstandard domain basis primarily as a compositional constraint, like the prime-based folk do, however, unlike the prime-based folk, these people have no interest in tuning with respect to the SPOB in order for synergy with other temperaments to be played together with the one at hand.

Considering that the RTT community's command of ordinary tuning schemes—i.e non-all-interval ones, which ask the user to specify their target interval set—has been so stunted historically, we consider it an important warning here that if optimizing your tuning for nonprime intervals is your only consideration, there are much better ways to do it than choosing a nonprime domain basis and tuning it with this neutral approach. In that case, you are probably experiencing what might be deemed "'''target-interval-set envy'''", and should instead consider exploring how inclusion and exclusion of particular intervals from your {{subpage|tuning fundamentals|uprev|s=target-intervals|text=target-interval set}} affects your tuning.

== Illustrative example ==
Let's consider an example. We won't be working through the computation method here yet; at this point we seek only to illustrate the points made in the previous example by showing how the tuning map is affected for a specific real-world case depending on the choice of prime-based or nonprime-based tuning. This example ''will'' use minimax-ES, whose use with these two approaches has already been well-established. (In the computation section coming up next, though, we will do more to work toward generalizing this concept beyond it, though, by demoing its use with a completely different tuning scheme.) Remember, minimax-ES is an all-interval tuning scheme, so our target-intervals are only proxy targets for all the intervals, and also, these proxy targets are always our basis elements (in the standard cases we've looked up until this point, they've always been our primes).

The temperament we'll be looking at for this example is called [[Subgroup_temperaments#Marveltri|marveltri]], and it's a temperament of the 2.5.9/7 domain, with mapping <math>M</math> = {{rket|{{map|1 2 1}} {{map|0 1 -2}}}}. See http://x31eq.com/cgi-bin/rt.cgi?limit=2_5_9%2F7&ets=22_69 for details of this example. Intuitively, we should expect the nonprime-based tuning to tune the nonprimes better than the prime-based tuning would. In this case we have just the one nonprime: 9/7. Hopefully that prediction is self-explanatory.

=== Nonprime-based ===
Following the nonprime-based approach, we find our retuning map <math>𝒓 =</math> {{map|0.826 -1.781 -0.022}}. And so we're off to a good start: with only about one-fiftieth of a cent of absolute error, our nonprime 9/7 is essentially unchanged! (and tuned better than either of the other two actual primes).

=== Prime-based ===
Under the prime tuning, then, we find a retuning map <math>𝒓 =</math> {{map|0.598 -1.289 -2.144}}. Indeed, the error on the nonprime 9/7 is nearly 100 times worse now, and is the worst of any of our three basis elements. However, we may note that the tunings of both the actual primes improved slightly.

So let's dig into why exactly the prime-based approach led to this result.

We know that the prime-based approach works by embedding into the simplest prime-only superspace. With our nonprime basis 2.5.9/7, our SPOB (simplest prime-only basis) is going to:
# Disentangle the 7 from the 9, and
# Access all powers of 3, not just the even ones via 9
So that takes us to the superspace 2.3.5.7. Here is this domain basis:

<math>
\begin{array} {ccc}
\\
\begin{array} {ccc}
\\
\end{array} \\

\begin{array} {rrr}
\scriptsize{2} \\
\scriptsize{3} \\
\scriptsize{5} \\
\scriptsize{7} \\
\end{array}

\end{array}

\begin{array} {ccc}
B_{2.3.9/7} \\
\begin{array} {ccc}
\scriptsize{2} & \scriptsize{5} & \scriptsize{9/7} \\
\end{array} \\

\left[ \begin{matrix}
1 & 0 & 0 \\
0 & 0 & 2 \\
0 & 1 & 0 \\
0 & 0 & -1 \\
\end{matrix} \right]

\end{array}
</math>

This matrix can perhaps most easily be read in terms of its columns; we can see it as expressing the basis elements of our nonprime domain basis in terms of the actual underlying primes: 2 as {{vector|1 0 0 0}}, 5 as {{vector|0 0 1 0}}, and 9/7 as {{vector|0 2 0 -1}}. We can use this matrix to convert our temperament from its current nonprime basis to this SPOB. We can't convert our mapping directly from the subspace to the superspace, but what we can do is convert the corresponding [[comma basis]] <math>\mathrm{C}</math> which is a [[basis]] for the [[nullspace]] of our mapping and which also defines our temperament. That's {{bra|{{ket|-5 2 1}}}}, or in other words <math>(\frac{2}{1})^{-5} × (\frac{5}{1})^2 × (\frac{9}{7})^1 = \frac{25×9}{32×7} = \frac{225}{224}</math>, so it's a matrix containing a single column vector for the [[marvel comma]] but expressed with the 2.5.9/7 domain basis.

We can say already, then, that this example temperament we chose—"marveltri"—''is'' in fact marvel temperament, or we could say more precisely that it is a reformulation of it with an additional constraint. While in a standard domain we think of the marvel comma as <math>\frac{1}{2}·\frac{1}{2}·\frac{1}{2}·\frac{1}{2}·\frac{1}{2}·3·3·5·5·\frac{1}{7}</math>, here we group up the 3's and the 7 and define that interval to be protected-against-prime-factoring, i.e. designate it as a basis element, and so instead the comma we're tempering out becomes interpreted as <math>\frac{1}{2}·\frac{1}{2}·\frac{1}{2}·\frac{1}{2}·\frac{1}{2}·5·5·\frac{9}{7}</math>.

We can confirm this fact with our domain basis matrix, which doubles as a basis-changer. Simply convert our comma basis in the nonprime basis <math>\mathrm{C}_{2.5.9/7}</math> to the standard domain equivalent <math>\mathrm{C}</math>. Note that when moving from the subspace to the superspace, we left-multiply by the basis-changer (see [[Cross-domain_temperament_merging#Using_the_basis_change_matrix|here]] for more information). We've reversed the usual direction of data flow here by putting the calculation on the right and its result on the left, and we've indicated this with an arrow under the equals sign, because we think it will be easier to keep track of what's happening if we always put the superspace on the left and the subspace on the right.

<math>
\begin{array} {ccc}
\\
\begin{array} {rrr}
\\
\end{array} \\

\begin{array} {rrr}
\scriptsize{2} \\
\scriptsize{3} \\
\scriptsize{5} \\
\scriptsize{7} \\
\end{array}

\end{array}

\begin{array} {ccc}
\mathrm{C} \\
\begin{array} {ccc}
\\
\end{array} \\

\left[ \begin{matrix}
-5 \\
2 \\
2 \\
-1 \\
\end{matrix} \right]
\end{array}

\hspace{0.5cm}
{\huge ⭀}
\hspace{0.5cm}

\begin{array} {ccc}
\\
\begin{array} {ccc}
\\
\end{array} \\

\begin{array} {rrr}
\scriptsize{2} \\
\scriptsize{3} \\
\scriptsize{5} \\
\scriptsize{7} \\
\end{array}

\end{array}

\begin{array} {ccc}
B_{2.5.9/7} \\
\begin{array} {ccc}
\scriptsize{2} & \scriptsize{5} & \scriptsize{9/7} \\
\end{array} \\

\left[ \begin{matrix}
1 & 0 & 0 \\
0 & 0 & 2 \\
0 & 1 & 0 \\
0 & 0 & -1 \\
\end{matrix} \right]

\end{array}

\hspace{0.5cm}
{\large ×}
\hspace{0.5cm}

\begin{array} {ccc}
\\
\begin{array} {ccc}
\\
\end{array} \\

\begin{array} {rrr}
\scriptsize{2} \\
\scriptsize{5} \\
\scriptsize{9/7} \\
\end{array} \\

\begin{array} {rrr}
\\
\end{array} \\

\end{array}

\begin{array} {ccc}
\mathrm{C}_{2.5.9/7} \\
\begin{array} {ccc}
\\
\end{array} \\

\left[ \begin{array} {rrr}
-5 \\
2 \\
1 \\
\end{array} \right] \\

\begin{array} {rrr}
\\
\end{array} \\
\end{array}
</math>

Indeed, here is the more familiar standard vector for the marvel comma, {{vector|-5 2 2 -1}}. And to further confirm, when we do a column-wise nullspace on this comma, we get {{rket|{{map|1 0 0 -5}} {{bra|0 1 0 2}} {{bra|0 0 1 2}}}}, which is the mapping for 7-limit marvel temperament.

So, again, this is telling us that one way to think of this temperament {{rket|{{map|1 2 1}} {{bra|0 1 -2}}}} is that it's 7-limit marvel (the marvel comma is made to vanish) except where we don't have access to just any old interval with factors of prime 3 or 7; any factor of prime 7 is necessarily accompanied by two factors of prime 3 (and you can still manage to make the marvel comma [[vanish]] that way, because that comma contains that {{sfrac|9|7}} interval). The point is: we can think of the tuning we found with the prime-based approach as if we had simply tuned marvel, then thrown away our ability to actually use a lot of those intervals we tuned for.

We should, then, be able to confirm that this is the same tuning we would have chosen had we simply decided to compose with marvel in its standard domain, and use this to learn why {{sfrac|9|7}} ended up with −2.144¢ error here, so much more than it did with the nonprime-based approach. When we check the minimax-ES tuning of marvel (see: http://x31eq.com/cgi-bin/rt.cgi?ets=72_31_53&limit=7), which gives <math>𝒓 =</math> {{map|0.598 -0.601 -1.289 0.942}}, we can see that the absolute error for each prime rounds to 1¢; it's a pretty middle-of-the-road tuning, just the sort of thing you'd expect and want from a (pretransformed) <math>2</math>-norm, minimizing neither the max nor the total of the damages but somewhere in between, but the key thing to note is what that entails for the interval {{sfrac|9|7}}, which from marvel's perspective is just one of many many intervals you can make out of 2's, 3's, 5's, and 7's. Notice that while the absolute errors for all the primes are about the same, critically, the ''signs'' on the errors for prime 3 and 7 are ''opposite'': prime 3 is tuned narrow while prime 7 is tuned wide. So when they appear on opposite sides of the quotient bar, as they do in {{sfrac|9|7}}, the errors are going to ''compound'', and the interval will be tuned doubly-narrow. And indeed we find that summing the errors of the constituent primes -0.601 + -0.601 - 0.942 = -2.144, exactly what we were expecting.

The prime-based tuning of marveltri is {{rbra|1200.5978 383.8288}} and can be found here: http://x31eq.com/cgi-bin/rt.cgi?limit=2_5_9%2F7&ets=22_69&subgroup=on

The story for the nonprime-based approach, on the other hand, is much simpler: We tuned {{sfrac|9|7}} as best we could, given the constraints that the other basis elements, <math>2</math> and <math>5</math>, also needed to be tuned well. And we ended up with a really, really accurate tuning of {{sfrac|9|7}} doing it that way.

=== Neutral ===
Going the neutral route, we find a tuning map <math>𝒕 = \val{1200.598 & 2785.025 & 432.940}</math>, which means a retuning map of <math>𝒓 = \val{0.598 & -1.289 & -2.144}</math>. So in this case it's identical to the prime-based result. Which makes sense, because we're measuring complexity in the same way here, and our basis elements are coprime. But again, this won't always be the case, as we'll see with our example in the computation section below.

== Regarding target-interval sets ==
Now that we've gotten that example out of the way, which used the established, but all-interval, minimax-ES to illustrate the essence of the choice between these three approaches, we're ready to start focusing on ordinary tuning schemes, i.e. non-all-interval tuning schemes, and the problems involved in their target-interval set specification.

=== Quotient-based vs vector-based specification ===
When manually specifying the target-interval set for a prime-based tuning scheme, and providing the target intervals in vector form, they should all be given in the SPOB (the simplest prime-only basis), because that's the only basis where all of them could be described. When doing this for the other two approaches, though, of course as usual you should give them in the same basis that the temperament mapping is given in. This issue does not arise if you specify your target-intervals in quotient form.

For example, we might have the "prime-based {2/1, 4/3, 10/7, 3/2, 15/7, 7/5} miniaverage-U" tuning of some temperament in the 2.3.7/5 basis. If we wanted to give those target-intervals in vector form, then for the neutral and nonprime-based approaches, we would give those as {{{vector|1 0 0}}, {{vector|2 -1 0}}, {{vector|1 0 -1}}, {{vector|-1 1 0}}, {{vector|0 1 -1}}, {{vector|0 0 1}}}. However, for the prime-based approach, it would only make sense to give those as {{{vector|1 0 0 0}}, {{vector|2 -1 0 0}}, {{vector|1 0 1 -1}}, {{vector|-1 1 0 0}}, {{vector|0 1 1 -1}}, {{vector|0 0 -1 1}}}.

But even that doesn't quite make sense, because why are we using the prime-based approach here (as opposed to the neutral approach) if we're not taking advantage of the independence of primes 5 and 7? We should only do this if we're also going to include target-intervals such as 5/4, 6/5, and 7/4.

=== Scheme filtering ===
Regarding target-interval set ''schemes'', such as the truncated integer limit triangle, or [[TILT]], the most logical-seeming way to handle them is to compute them as normal, then filter out any intervals unsupported by the basis. For example, take the 8-TILT. Normally that would be {2/1, 3/1, 3/2, 4/3, 5/2, 5/3, 5/4, 6/5, 7/3, 7/4, 7/5, 7/6, 8/3, 8/5}. However, if we've asked for the TILT miniRMS-S tuning of some temperament in the 2.3.7 basis, we can't use several of those intervals; instead, TILT here should be understood to give {2/1, 3/1, 3/2, 4/3, 7/3, 7/4, 7/6, 8/3} only. This issue is pertinent not only to nonprime bases, but indeed any basis which even so much as skips a prime.

And so, when we compare "TILT minimax-ES" or "TILT nonprime-based minimax-C" with "TILT prime-based minimax-ES", there's actually more difference between the schemes than whether or not we change basis to the SPOB, or revise our complexity formulas to prevent the factorization of our nonprime basis elements; there's also the difference in results from the target-interval set scheme within the tuning scheme. This is implicit in all-interval tunings where the target-interval set scheme is not explicitly specified and is thus assumed to be the primes—at least the ''proxy'' targets are the primes, and by primes we actually mean more generally the ''basis elements''. In other words, we could say that for the "minimax-ES" tuning scheme, our target-interval set scheme is actually the "all-intervals-as-proxied-by-the-basis-elements" scheme, which like any target-interval set scheme as described above, is understood to translate into a different set of target-intervals depending on the basis.

=== Default max ===
We also need to address how the default max integer for the TILT should be computed (and the default max odd for the [[OLD]], etc.). Previously, when working only with standard (prime-limit) bases, we defined it to be the integer one less than the next prime above the prime limit. But that definition may not be the best in general.

Consider the basis 2.9.21. The current definition would give us the 10-TILT, since 7 is the prime limit of this basis, 11 is the next prime after that, and 10 is the integer one less than that. But it seems pretty clear that if someone is interested in 21/1 as a ''basis element'', then they'll certainly be interested in tuning some target-intervals which include 21's in them, but the 10-TILT won't have any of those, considering its integer limit is less than half of that!

So, we suggest that the better generalization of the definition of the max integer for TILT is actually one less than the next prime above the ''integer'' limit of the basis. Whenever the basis is prime-only (standard or otherwise, i.e. this works for 2.3.5.7 but also 2.3.7), this will come out to the same thing.

Regarding our example, then, the integer limit of that basis is 21, the next prime is 23, and so the default TILT here is the 22-TILT (filtered, of course, of all intervals with 5's, 11's, 13's, 17's, and 19's in their factorizations, or odd counts of 3's).

== Computation ==
For this example, we'll tune a temperament of the 2.7/3.11/3 domain with the TILT minimax-C tuning scheme.

=== Neutral ===
First, let's go through the neutral approach, which is the simplest, on account of it essentially being the do-nothing-different approach.

Here's our mapping, which gives us a ~{{sfrac|2|1}} period and ~{{sfrac|12|11}} generator:

<math>
\begin{array} {ccc} \begin{array} {ccc}
\\
\end{array} \\ \begin{array} {rrr}\\
\\
\\
\\
\end{array} \\ \end{array} \begin{array} {c}
M_s \\
\begin{array} {ccc}
 & \scriptsize{2} & \scriptsize{\frac{7}{3}} & \scriptsize{\frac{11}{3}} \\
\end{array} \\
\left[ \begin{matrix}
1 & 1 & 2 \\
0 & 2 & -1 \\
\end{matrix} \right] \\
\end{array}
</math>

And here's what our just tuning map looks like in this nonprime basis, just the cents per basis element for each of 2, 7/3, and 11/3:

<math>
\begin{array} {ccc} \begin{array} {ccc} \\ \end{array} \\ \begin{array} {rrr}
\\
\end{array} \\ \end{array} \begin{array} {c}
𝒋_s \\
\begin{array} {lll}
 & \scriptsize{2} & \quad\quad\quad \scriptsize{\frac{7}{3}} \quad\quad\quad & \scriptsize{\frac{11}{3}} \\
\end{array} \\
\left[ \begin{array} {rrr}
1200 & 1466.871 & 2249.363 \\
\end{array} \right] \\
\end{array}
</math>

As for our target-interval set, the [[TILT]], or truncated integer limit triangle, defaults to the highest integer limit within the prime limit, so in our case, even though we're not using a prime limit as our domain basis, our maximum prime is 11, so that makes 12 our max integer for purposes of the TILT. But we can't accept every interval in the 12-TILT, because it contains a bunch of them with factors of 5, or factors of 3 unaccompanied by factors of 7 or 11, etc. So we end up with a subset of the 12-TILT, namely <math>{ \frac{2}{1}, \frac{7}{3}, \frac{7}{6}, \frac{11}{6}, \frac{11}{7}, \frac{12}{7} }</math>. So that gives us a target-interval list in matrix form:

<math>
\begin{array} {ccc} \begin{array} {ccc} \\ \end{array} \\ \begin{array} {rrr}
\scriptsize{2} \\
\scriptsize{\frac{7}{3}} \\
\scriptsize{\frac{11}{3}}
\end{array} \end{array} \begin{array} {c}
\mathrm{T}_s \\
\left[ \begin{array} {c|c|c|c|c|c}
1 & 0 & -1 & -1 & 0 & 2 \\
0 & 1 & 1 & 0 & -1 & -1 \\
0 & 0 & 0 & 1 & 1 & 0 \\
\end{array} \right] \\
\end{array}
</math>

And here's our complexity weight matrix, using our default log-product complexity:

<math>
\begin{array} {c}
C_s \\
\left[ \begin{matrix}
\log_2{2·1} & 0 & 0 & 0 & 0 & 0 \\
0 & \log_2{7·3} & 0 & 0 & 0 & 0 \\
0 & 0 & \log_2{7·6} & 0 & 0 & 0 \\
0 & 0 & 0 & \log_2{11·6} & 0 & 0 \\
0 & 0 & 0 & 0 & \log_2{11·7} & 0 \\
0 & 0 & 0 & 0 & 0 & \log_2{12·7} \\
\end{matrix} \right]
\end{array}
</math>

This matrix is shown with <math>\text{lp-C}()</math> computed using its quotient-based form, i.e <math>\log_2{\!n·d}</math>, which requires no modification to perform correctly in this nonprime basis situation. However, we note that if one prefers the vector-based form, i.e. the pretransformed norm, one should be careful with the diagonal entries of the pretransformer. For an all-integer domain basis such as 2.9.7.11 this issue wouldn't make a difference, but it ''does'' make a difference if your basis contains rationals such as ours does. The issue is that one cannot simply set each basis element's pretransformer entry to its log, ''one must set it to its own log-product complexity''. For example, for our second basis element, its pretransformer entry should be <math>\log_2{7·3}</math>, ''not'' <math>\log_2{\frac{7}{3}}</math>. Here's our full complexity pretransformer, for clarity:

<math>
\begin{array} {ccc} \begin{array} {ccc} \\ \end{array} \\ \begin{array} {rrr}
\scriptsize{2} \\
\scriptsize{\frac{7}{3}} \\
\scriptsize{\frac{11}{3}} \\
\end{array} \\ \end{array} \begin{array} {c}
\begin{array} {lll}
X \\
\end{array} \\
\left[ \begin{matrix}
\log_2{2·1} & 0 & 0 \\
0 & \log_2{7·3} & 0 \\
0 & 0 & \log_2{11·3} \\
\end{matrix} \right] \\
\end{array}
</math>

We must add a caveat we originally missed, and it bites in ''this very example''. Setting each entry to the basis element's own log-product complexity is necessary, but it is ''not sufficient'' to make the vector-based form agree with the quotient-based form; it suffices only when the basis elements are pairwise coprime. Ours are not — both <math>\frac{7}{3}</math> and <math>\frac{11}{3}</math> carry a prime 3. That shared 3 cancels in the quotient of any target that divides one of them by the other: our <math>\frac{11}{7} = \frac{11/3}{7/3}</math> is simply <math>11·7^{-1}</math>, with no 3's remaining. But a diagonal pretransformer reading the basis-vector cannot see that cancellation; it counts the 3 once through <math>\frac{7}{3}</math> and again through <math>\frac{11}{3}</math>, and so overstates <math>\text{lp-C}\!\left(\frac{11}{7}\right)</math> as <math>\log_2{(7·3·11·3)} \approx 9.437</math> rather than the correct <math>\log_2{(11·7)} \approx 6.267</math>. So whenever the basis elements share a prime, the vector-based form must act on the interval's ''prime-count vector'' rather than its basis-vector — that is, lift the interval into the prime-only superspace (the same 2.3.7.11 we build below for the prime-based approach) and prescale it there with the ordinary log-prime pretransformer. This is exactly why we reach for the quotient-based form here: it sidesteps the issue entirely.

Okay, where to from here? Well, it's rather straightforward at this point. We just plug these objects into whatever pertinent optimization method. The result should be: <math>𝒈</math> = {{map|1191.880 133.594}}.

=== Nonprime-based ===
Everything's the same from the neutral approach to the nonprime approach, except that we may need to modify some complexity calculations. In our case, we're changing our pre-transformer matrix slightly:

<math>
\begin{array} {c}
C_s \\
\left[ \begin{matrix}
\log_2{2·1} & 0 & 0 & 0 & 0 & 0 \\
0 & \log_2{\frac{7}{3}} & 0 & 0 & 0 & 0 \\
0 & 0 & \log_2{\frac{7}{3}·2} & 0 & 0 & 0 \\
0 & 0 & 0 & \log_2{\frac{11}{3}·2} & 0 & 0 \\
0 & 0 & 0 & 0 & \log_2{\frac{11}{3}·\frac{7}{3}} & 0 \\
0 & 0 & 0 & 0 & 0 & \log_2{2^2·\frac{7}{3}} \\
\end{matrix} \right]
\end{array}
</math>

Which gives a different result, <math>𝒈 = \val{1192.399 & 133.768}</math>.

=== Prime-based ===
Finally, we'll look at the prime-based approach. We already briefly touched at how some of this looks in the example section above, in the course of explaining how the tuning of marveltri changes depending on your approach, prime-based or nonprime-based. But we're going to go through all the steps this time around. This process was inspired by the one described here: [[Generalized_Tenney_Norms_and_Tp_Interval_Space#Examples]].

At the high level, this is what we're going to do:
# Find a comma basis of our temperament, as a basis for the nullspace of our mapping.
# Prepare a matrix which we'll use to convert our temperament from its nonprime basis over to the simplest possible standard (prime-only) domain basis.
# Go ahead and change our temperament's basis in this way, doing so by multiplying the comma basis we found in step 1 with the basis change matrix we prepared in step 2, in order to compute what our comma basis looks like in the prime-only domain basis.
# Using the nullspace operation, find a mapping for this new temperament, which is the same as the old temperament except that it has this prime-only domain basis.
# Determine target-intervals and held-intervals as desired and afforded by this superspace. And determine the weights for the target-intervals.
# Optimize for a generator tuning map as usual, using the mapping from step 4 and the tuning targets from step 5.
# Convert the generator tuning map to the tempered-prime tuning map, so we can change its domain basis.
# Change domain basis from the temporary standard domain basis back into the original nonprime basis.
# Find the pseudoinverse of the mapping, in order to convert this tempered-prime tuning map into a generator tuning map with a form which matches the mapping.
# Convert to a generator tuning map using this pseudoinverse from the previous step.

And then we're done! We wish we could say it was possible in 3 easy steps, but it's more like 10 medium-hard steps.

It looks like this:

[[File:Tuning in nonstandard domains - cases.png|frameless|900x900px]]

For the duration of this section, we will be subscripting objects in the original 2.7/3.11/3 basis with <math>s</math>, for "smaller" space, and objects in the SPOB 2.3.7.11 with <math>L</math>, for "larger" space (AKA superspace).

==== 1. Find comma basis ====
We expect the reader to understand why we temporarily change basis to a basis for another domain in order to tune our temperament, and then change back; this is, after all, the entire point of this tuning approach. But we also expect there are many readers who are uncertain as to why it is the case that as part of this process it is apparently necessary to convert our mapping temporarily into a comma basis in order to change domain basis, and then convert back. Why should this be necessary—why can't we cut off the two "comma basis" nodes of the diagram above, going straight from the original mapping to the standard domain mapping?

The short answer is: because one can only ''directly'' change the basis of a mapping to a ''subspace'' of its current space, not a superspace. And vice versa, one can only directly change the basis of a comma basis (the dual manner of representing temperaments) to a superspace, not a subspace. And since it's a superspace that is our destination, we must get there via a comma basis detour.
The full explanation for this necessity is beyond the scope of this computation example, however, we encourage the reader to review the material here ([[Cross-domain_temperament_merging#Application:_determining_whether_it_is_possible_to_change_the_domain]]) if interested.

So, how to find the comma basis from the mapping? Easy enough. By using the nullspace operation on the mapping. The method of finding this is explained here in the {{subpage|exploring temperaments|uprev|s=Nullspace|text=section on null spaces]]. Here's what you should find:

<math>
\begin{array} {ccc}
\\
\begin{array} {ccc}
\\
\end{array} \\

\begin{array} {rrr}
\scriptsize{2} \\
\scriptsize{\frac{7}{3}} \\
\scriptsize{\frac{11}{3}} \\
\end{array}

\end{array}

\begin{array} {ccc}
\mathrm{C}_s \\

\begin{array} {ccc}
\end{array} \\

\left[ \begin{matrix}
5 \\
-1 \\
-2 \\
\end{matrix} \right]
\end{array}
</math>

So that's <math>2^{5}(\frac{7}{3})^{-1}\left(\frac{11}{3}\right)^{-2}</math>, or in other words, <math>\frac{32×27}{7×121} = \frac{864}{847}</math>, a comma about 34.4¢ in size.

==== 2. Prepare basis change ====
With our nonprime basis of 2.7/3.11/3, our SPOB is going to need to disentangle both the prime 7 and the prime 11 from their respective prime 3's. This will lead us to the superspace 2.3.7.11, or in other words, the 11-limit except without any prime 5.

Another way to think about this SPOB is just the set of actual primes we would need to be able to express all of our nonprime basis elements as vectors. Because that's essentially what we're going to do! When we represent our domain basis in this fashion, we actually get another powerful result for free. The resultant matrix can be used to transform intervals from the nonprime basis we're describing ''into'' that SPOB — intervals such as commas, and in particular our temperament's comma basis.

Here is this domain basis:

<math>
\begin{array} {ccc}
\\
\begin{array} {ccc}
\\
\end{array} \\

\begin{array} {rrr}
\scriptsize{2} \\
\scriptsize{3} \\
\scriptsize{7} \\
\scriptsize{11} \\
\end{array}

\end{array}

\begin{array} {ccc}
B_s \\
\begin{array} {ccc}
\scriptsize{2} & \scriptsize{\frac{7}{3}} & \scriptsize{\frac{11}{3}} \\
\end{array} \\

\left[ \begin{matrix}
1 & 0 & 0 \\
0 & -1 & -1 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{matrix} \right]
\end{array}
</math>

Note that even though we're in a sense going from our weird nonprime basis and "getting to basics" here, we still don't necessarily go all the way to the most basic thing, which would be a prime limit. That is, we don't include an all-zeros row for prime 5 even though our temperament doesn't deal with it, just for simpleness' sake we can say we're in a prime limit and so we find all the primes in their typically expected positions (so that we're not surprised when the 3rd prime here is 7 and the 4th is 11, instead of those being the 4th and 5th, respectively). It's not just a cosmetic thing either. Adding that extra row essentially means that the output intervals will have one more dimension than they did before. And remember the everpresent formula <math>d = n + r</math>, that is, dimensionality equals nullity plus rank. So any increase in dimensionality will correspond to a change in either nullity or rank. Which opens the impossible-to-answer question of: which comma do we add to this temperament on account of converting its basis, etc. Just unask the question, and don't add new rows for primes your temperament doesn't deal with.

==== 3. Change domain basis ====
To effect the basis change, we simply matrix multiply our comma basis by the basis change. We put the transformation matrix on the left, just like we would for any interval vector, or list thereof, with a temperament mapping matrix.

<math>
\begin{array} {ccc}
\\
\begin{array} {rrr}
\\
\end{array} \\

\begin{array} {rrr}
\scriptsize{2} \\
\scriptsize{3} \\
\scriptsize{7} \\
\scriptsize{11} \\
\end{array}
\end{array}

\begin{array} {ccc}
\mathrm{C} \\
\begin{array} {ccc}
\\
\end{array} \\

\left[ \begin{matrix}
-5 \\
-3 \\
1 \\
2 \\
\end{matrix} \right]

\end{array}
\hspace{0.5cm}
{\huge ⭀}
\hspace{0.5cm}

\begin{array} {ccc}
\\
\begin{array} {ccc}
\\
\end{array} \\

\begin{array} {rrr}
\scriptsize{2} \\
\scriptsize{3} \\
\scriptsize{7} \\
\scriptsize{11} \\
\end{array}
\end{array}

\begin{array} {ccc}
B_s \\
\begin{array} {ccc}
\scriptsize{2} & \scriptsize{\frac{7}{3}} & \scriptsize{\frac{11}{3}} \\
\end{array} \\

\left[ \begin{matrix}
1 & 0 & 0 \\
0 & -1 & -1 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{matrix} \right]
\end{array}

\hspace{0.5cm}
{\large ×}
\hspace{0.5cm}
\begin{array} {ccc}
\\

\begin{array} {ccc}
\\
\end{array} \\

\begin{array} {rrr}
\scriptsize{2} \\
\scriptsize{\frac{7}{3}} \\
\scriptsize{\frac{11}{3}} \\
\end{array} \\
\begin{array} {rrr}
\\
\end{array} \\

\end{array}

\begin{array} {ccc}
\mathrm{C}_s \\
\begin{array} {ccc}
\\
\end{array} \\

\left[ \begin{matrix}
5 \\
-1 \\
-2 \\
\end{matrix} \right] \\

\begin{array} {rrr}
\\
\end{array} \\
\end{array}
</math>

We can double-check that this is the same comma: <math>(2)^{-5}3^{-3}7^{1}11^{2} = \frac{7×121}{32×27}</math>. Again, yes, this is {{sfrac|864|847}}.

So this is our new comma basis. It's the same one comma, but as tempered by a temperament in the 2.3.7.11 space.

==== 4. Find mapping ====
To get us back into the form of a mapping—which we'll need in order to optimize the temperament's tuning as we've been practicing throughout this article series—we again need to use the nullspace operation (though this time we do it column-wise).

With that, we find:

<math>
\begin{array} {ccc}
\\
\begin{array} {ccc}
\\
\end{array} \\

\begin{array} {rrr}
\\
\\
\\
\end{array}

\end{array}

\begin{array} {ccc}
M_L \\
\begin{array} {ccc}
\scriptsize{2} & \scriptsize{3} & \scriptsize{7} & \scriptsize{11} \\
\end{array} \\

\left[ \begin{matrix}
1 & 0 & 1 & 2 \\
0 & 1 & 1 & 1 \\
0 & 0 & 2 & -1 \\
\end{matrix} \right]
\end{array}
</math>

So again, this is the same as the old temperament, except that it has this prime-only domain basis. (We're still labeling its rows or columns and using subscripts to indicate its domain basis, since it may be prime-only, but on account of the gap where it skips prime 5, it is still not ''standard''.)

Note that by splitting our two basis elements 7/3 and 11/3 up into three basis elements 3, 7, and 11, we have increased [[dimensionality]] of our temperament by 1. And our temperament's identity is based on its comma basis, so our [[nullity]] hasn't changed. Therefore, by the [[rank-nullity theorem]] which states that dimensionality <math>d</math> must equal rank <math>r</math> plus nullity <math>n</math>, we can see that our temperament's rank must go up by 1 in this new basis. And indeed we can see that while our mapping used to have only 2 rows, it now has 3 rows, each corresponding to a generator. Unlike the consequences we warned against of spuriously adding a blank row for the unused prime 5, this change in <math>d</math>, <math>n</math>, and <math>r</math> is very real. What this means is that when we run our optimization procedure, we will be imagining as if we have three generators in our temperament, even though we know that in our actual original basis it only has two. In the end, we'll convert our three-entry generator tuning map from this superspace basis back down into a two-entry generator tuning map, by consolidating those generators together. This is one way of looking at the fact of how the prime-based approach tunes for intervals that one won't be able to use in the original temperament that this temperament is being proxy-tuned for.

==== 5. Determine target-intervals ====
Regarding our target-interval set, we're still optimizing our tuning for the appropriate TILT here, however, what exactly that TILT includes will be slightly different in this new space. The default max integer is the same, as changing basis hasn't changed our max integer (it would have if our basis's largest integer had been composite, however, such as 33 instead of 11/3!) so we're still asking for a 12-TILT here. But it will be filtered differently. Now we've exchanged our 9 for a 3 in our basis, we're no longer limited to intervals with even counts of prime 3. So we get: {2/1, 3/1 , 3/2, 4/3, 7/3, 7/4, 7/6, 8/3, 9/4, 9/7, 11/4, 11/6, 11/7, 11/8, 11/9, 12/7}. And here's that in matrix form:

<math>
\begin{array} {ccc}
\\
\begin{array} {ccc}
\\
\end{array} \\

\begin{array} {rrr}
\scriptsize{2} \\
\scriptsize{3} \\
\scriptsize{7} \\
\scriptsize{11} \\
\end{array}

\end{array}

\begin{array} {ccc}

\mathrm{T}_L \\

\begin{array} {ccc}
\\
\end{array} \\

\left[ \begin{array} {c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c}
1 & 0 & -1 & 2 & 0 & -2 & -1 & 3 & -2 & 0 & -2 & -1 & 0 & -3 & 0 & 2 \\
0 & 1 & 1 & -1 & -1 & 0 & -1 & -1 & 2 & 2 & 0 & -1 & 0 & 0 & -2 & 1 \\
0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & -1 & 0 & 0 & -1 & 0 & 0 & -1 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 0 \\
\end{array} \right]
\end{array}
</math>

We're not bothering with any held-intervals in this tuning scheme, so that's settled already.

As for weights, because we're in a nonprime basis, we don't expect there should be too many surprises here. The actual weight matrix will be too cumbersome to spell out here, since our target-interval count <math>k</math> is 16. But we will show our <math>(4, 4)</math>-shaped complexity pretransformer, at least, for which there should be no surprises, as it's just like any other one we would have looked at before worrying about nonstandard bases, since all of its elements are primes:

<math>
\begin{array} {ccc} \begin{array} {ccc} \\ \end{array} \\ \begin{array} {rrr}
\\
\\
\\
\\
\end{array} \\ \end{array} \begin{array} {c}
\begin{array} {lll}
X \\
\end{array} \\
\left[ \begin{matrix}
\log_2{2} & 0 & 0 & 0 \\
0 & \log_2{3} & 0 & 0 \\
0 & 0 & \log_2{7} & 0 \\
0 & 0 & 0 & \log_2{11} \\
\end{matrix} \right] \\
\end{array}
</math>

==== 6. Optimize ====
Now for the part we already know how to do: optimize for our generator tuning map <math>𝒈_L</math>, using the mapping from step 4 and the target-interval set from step 5. We should find {{map|1193.102 1891.756 135.810}}. As promised, that's three generators, one for each row of our <math>M_L</math>.

==== 7. Convert to tempered-prime tuning map ====
This generator tuning map is great, but we want to get this optimized tuning information back into our original basis somehow, and we only know how to change the basis for objects that are in terms of primes, and in general, generator tuning maps are ''not'' in terms of primes, but of generators. Notice that if we were to write out <math>𝒈_L</math> in matrix form, we could neither label its row nor its columns with either our original basis or our superspace basis:

<math>
\begin{array} {ccc}
\\
\begin{array} {ccc}
\\
\end{array} \\

\begin{array} {rrr}
\\
\end{array}
\end{array}

\begin{array} {ccc}

𝒈_L \\

\begin{array} {ccc}
\\
\end{array} \\

\left[ \begin{array} {r}
1193.102 & 1891.756 & 135.810 \\
\end{array} \right]
\end{array}
</math>

And so that's why—if you were wondering, based on the diagram at the beginning of this exercise—that computing the prime-based tuning requires not just one detour (the one to the comma basis back to a mapping) but ''two'' detours: it's the same reason again, just manifesting in a slightly different way, really; we just need to be in a form temporarily where we can accomplish the basis change.

Fortunately, this is an easily solvable problem. We only need to convert our generator tuning map into a ''prime'' tuning map (which we usually shorten to simply "tuning map"). And this is as easy as multiplying it by our mapping, which essentially tells us how many of each of these generators it takes to each prime. Multiplying <math>𝒈_LM_L</math> together, then, we find our tuning map <math>𝒕_L</math>, which equals {{map|1193.102 1891.756 3356.477 4142.150}}.

==== 8. Return to original basis ====
Now that we've got our optimized tuning in the basis-change-friendly form of a tuning map <math>𝒕_L</math>, we're ready to change basis back in the other direction, back to our original basis of 2.7/3.11/3. To do this, we right-multiply it by the basis change matrix:

<math>
\begin{array} {ccc}
𝒕_L \\

\begin{array} {ccc}
\scriptsize{2} &  \scriptsize{3} & \scriptsize{7} &  \scriptsize{11} \\
\end{array} \\

\left[ \begin{matrix}
1193.102 & 1891.756 & 3356.477 & 4142.150 \\
\end{matrix} \right]
\end{array}

\hspace{0.5cm}
{\large ×}
\hspace{0.5cm}

\begin{array} {ccc}
\\
\begin{array} {ccc}
\\
\end{array} \\

\begin{array} {rrr}
\scriptsize{2} \\
\scriptsize{3} \\
\scriptsize{7} \\
\scriptsize{11} \\
\end{array}
\end{array}

\begin{array} {ccc}
B_{L↔s} \\
\begin{array} {ccc}
\scriptsize{2} & \scriptsize{7/3} & \scriptsize{11/3} \\
\end{array} \\

\left[ \begin{matrix}
1 & 0 & 0 \\
0 & -1 & -1 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{matrix} \right]
\end{array}

\hspace{0.5cm}
{\huge ⥱}
\hspace{0.5cm}

\begin{array} {ccc}
𝒕_s \\
\begin{array} {ccc}
\scriptsize{2} & \scriptsize{7/3} & \scriptsize{11/3} \\
\end{array} \\

\left[ \begin{array} {rrr}
1193.102 & 1464.722 & 2250.394 \\
\end{array} \right] \\

\begin{array} {rrr}
\\
\end{array} \\
\end{array}
</math>

==== 9. Find pseudoinverse ====
Great. So we're back home, in our original basis. However, we've got our information in the form of a ''prime'' tuning map. Maybe that's sufficient for you; if so, you may be excused. For the rest of us who wish to get our actual ''generator'' sizes in this temperament tuning, we have a little more work to do.

As you'll recall, we went from generator tuning map <math>𝒈_L</math> to tempered-prime tuning map <math>𝒕_L</math> using multiplication, specifically multiplying <math>𝒈_L</math> by our mapping <math>M_L</math>. Unfortunately, however, it's not quite so easy to go the other direction, i.e. to extract a <math>𝒈</math> from a <math>𝒕</math>; while we ''do'' happen to know our <math>M</math> here, it's not as straightforward as, say, "dividing" a <math>𝒕</math> by an <math>M</math> to extract a <math>𝒈</math>. It ''is'' possible, however, but it just requires a bit more cleverness than one might expect.

The key idea is to realize that <math>𝒕 = 𝒈M</math>, and so all we really need to do is right-multiply by something that will cancel out the <math>M</math> part of it. Such a thing would be called a ''right-inverse'' of <math>M</math>, which is to say, something that when <math>M</math> is right-multiplied by it, we end up with an identity matrix <math>I</math>. So we don't need a ''full'' inverse of <math>M</math>, which would be a matrix that cancels <math>M</math> out to an identity matrix by ''either'' right-multiplication ''or'' left-multiplication; and in fact, since any meaningful <math>M</math> will be rectangular, and rectangular matrices can never have full inverses, it couldn't be. But fortunately for us, since <math>M</math> is the ''wide'' type of rectangular matrix, not tall (i.e. it has more columns than rows), we need look no further for a right-inverse than its [[pseudoinverse]], <math>M^{+}</math>. That's right: it's the same generalized inverse we have used previously for finding tunings with [[optimization power]] <math>p = 2</math> or {{subpage|all-interval_tuning_schemes|uprev|s=Dual_norms|text=interval complexity norm power}} <math>q = 2</math>. But it has other convenient uses that have nothing to do with this power of <math>2</math>. In this case, the math works out so that for wide matrices <math>A</math>, it's the case that <math>AA^{+} = I</math> (for tall matrices, it's the other way around; <math>A^{+}A = I</math>).

The pseudoinverse has a simple formula (explained at the link above). Following it, we find:

<math>
\begin{array} {ccc}
\\
\begin{array} {ccc}
\\
\end{array} \\

\begin{array} {rrr}
\scriptsize{2} \\
\scriptsize{\frac{7}{3}} \\
\scriptsize{\frac{11}{3}} \\
\end{array}
\end{array}

\begin{array} {ccc}
M^{+}_s \\
\begin{array} {ccc}
\end{array} \\

\left[ \begin{array} {ccc}
\frac{1}{6} & 0 \\
\frac{1}{6} & \frac{2}{5} \\
\frac{1}{3} & -\frac{1}{5} \\
\end{array} \right]
\end{array}
</math>

Don't be thrown if <math>M^{+}</math> contains non-integer entries like; it'll still cancel out with <math>M</math>, and it won't make a difference. You may confirm that <math>M_sM^{+}_s = I</math> if you would like.

==== 10. Convert to generator tuning map ====
Finally, we can reach our goal of the prime-based TILT minimax-C generator tuning map <math>𝒈_{2.7/3.11/3}</math>, if we just multiply <math>𝒕_s</math> on the right by <math>M^{+}_s</math>:

<math>
\begin{array} {ccc}
𝒕_s \\
\begin{array} {ccc}
\scriptsize{2} &  \scriptsize{\frac{7}{3}} & \scriptsize{\frac{11}{3}} \\
\end{array} \\

\left[ \begin{matrix}
1193.102 & 1464.722 & 2250.394 \\
\end{matrix} \right]
\end{array}

\hspace{0.5cm}
{\large ×}
\hspace{0.5cm}

\begin{array} {ccc}
\\
\begin{array} {ccc}
\\
\end{array} \\

\begin{array} {ccc}
\scriptsize{2} \\
\scriptsize{7/3} \\
\scriptsize{11/3} \\
\end{array}

\end{array}

\begin{array} {ccc}

M^{+}_s \\

\begin{array} {ccc}
\\
\end{array} \\

\left[ \begin{matrix}
\frac{1}{6} & 0 \\
\frac{1}{6} & \frac{2}{5} \\
\frac{1}{3} & -\frac{1}{5} \\
\end{matrix} \right]

\end{array}

\hspace{0.5cm}
{\huge ⥱}
\hspace{0.5cm}

\begin{array} {ccc}
𝒈_s \\
\begin{array} {ccc}
\\
\end{array} \\

\left[ \begin{array} {ccc}
1193.102 & 135.810 \\
\end{array} \right] \\

\begin{array} {ccc}
\\
\end{array} \\
\end{array}
</math>

That 135.810¢ value certainly isn't super close to 116¢, the cents size of the purely-tuned {{sfrac|77|72}} interval for our generator, but hey, it's in the ballpark, certainly for a temperament with as low accuracy as this one.

And so we're done!

== See also ==
Well, you've reached the very end of our article series on RTT tuning. The only place left to go from here is tables. If you'd like to review the conventions for communicating about RTT that we developed over the course of this project, you can behold them in all their centralized glory on {{subpage|conventions for names, variables, units, and notations|uprev|text=this page}}. Otherwise, thanks for your attention, and we hope you got a lot out of this, and will go on to use what you learned to make some excellent music!

Dave: Catch ya later.

Douglas: Happy trails.

We hope you enjoyed reading this series as much as we enjoyed collaborating to write it. 😊😊

Hey Douglas, shouldn't one of those emoticons have hair?

== Footnotes and references ==
<references group="note" />

[[Category:Dave Keenan & Douglas Blumeyer's guide to RTT]]
[[Category:Tuning]]

Dave Keenan & Douglas Blumeyer's guide to RTT/All-interval tuning schemes

2026-01-19T19:05:32Z

Cmloegcmluin: /* Example all-interval tuning schemes */

{{breadcrumb}}{{texmap}}{{texops}}
When tuning [[regular temperaments]]—that is, choosing exact sizes for their [[generator]]s (typically in [[cents]])—one of the fundamental choices we make is which consonant musical [[interval]]s to optimize the tuning for. In other words, we choose a set of intervals whose [[damage]]s we [[target-interval list|target for minimization]]. However, a special family of tuning schemes have been developed which do not require this choice; instead, a certain kind of damage is minimized for ''every'' interval. In this article, we will be discussing such '''all-interval tuning schemes'''.

This is article 7 of 9 in [[Dave Keenan]] & [[Douglas Blumeyer]]'s guide to RTT, or "[[D&D's guide]]" for short. In order to get the most out of this article, we suggest that you first familiarize yourself with all the concepts explained in the earlier article, {{subpage|tuning fundamentals|uprev}}; we're going to build upon a lot of the concepts introduced there (and introduce more). First, we'll touch quickly upon the pros and cons of using all-interval tuning schemes, and a bit on their history. Next, we'll explain them conceptually (continuing in the vein of our fundamentals article). After that, if you're interested in such things, stick around as we work through examples of computing them, discussing various methods and their derivations (and this section is in the vein of our article 6, {{subpage|tuning computation|uprev}}, which you should read before coming here).

= Pros and cons =
All-interval tuning schemes have great value for consistently and reasonably documenting the tunings of regular temperaments, in large part because they don't require the specification of a target-interval set. Another major strength of all-interval tunings is that they are comparatively easy for computers to calculate.

On the other hand, all-interval tunings are somewhat tricky for humans to understand, as evidenced by our choice to break out an entire separate article dedicated to making sense of them. Also, they do not necessarily produce tunings which are ideal for use in real-life musical practice; when it comes to actually doing something—like building an instrument, or tuning a synth for a specific piece of music—a better approach would be to tune directly for the intervals you plan to use in the music. And as we'll see in more detail in a moment, all-interval tuning schemes require {{subpage|tuning fundamentals|uprev|s=Weight slope|text=simplicity-weighting}} of absolute error to obtain damage, which is not for everyone (for a more detailed discussion, see {{subpage|tuning fundamentals|uprev|s=Rationale for choosing your slope|text=this section of the fundamentals article}}).

We could make a loose analogy, then, between all-interval tunings and [[canonical form|canonical mappings]] on one side—where both of these are good for mass categorization, sanity checking, and automated processes—while on the other side we'd compare non-all-interval tunings to [[Normal_lists#Minimal_generator_form|mingen mappings]], both of which are immediately reasonable for musicians to make good music with.

$$ \text{all-interval tunings : canonical mappings :: non-all-interval tunings : mingen mappings} $$

So if there were to be a ''canonical tuning scheme'', then—a good compromise to avoid opinionated arguments over target-interval sets, and one whose use case might be appearing in infoboxes on wiki pages for temperaments to help give people an immediate sense of the ballpark for generator sizes—then that tuning scheme would likely be an all-interval tuning scheme. If you are working on something like that, or perhaps automated processes for searching or categorizing temperaments, then this article may be valuable to you.

But on the other hand, if you consider yourself primarily a practical musician, and you do have an opinion about which consonances are most important to get right in your music, then
this article may not be of great value to you. In that case, please just tune for your favored intervals directly. Accommodating crazily complex intervals like 1953125/1259712—the ones out there among "all intervals" beyond the ones we typically look at—may be clouding the optimization of your tuning, i.e. making it optimize with respect to a ton of junk you'll never want, rather than having it be optimized precisely and only for the stuff you do want (see the beginning of the {{subpage|tuning fundamentals|uprev|s=The importance of exclusivity|text=target-intervals section of our fundamentals article}} for a review of why it's important for target-interval sets to be exclusive).

We certainly recognize the mathematical simplicity, and the beauty of the feat of all-interval tunings. And knowing ourselves to be susceptible to seduction by such qualities, we caution our readers to be mindful not to let themselves be seduced either. It can be a little like [[Wikipedia:Streetlight_effect|streetlight effect]].

= History =
All-interval tuning schemes are a relatively recent development in the history of temperament tuning. Non-all-interval tuning schemes, however, have been used for almost 200 years.<ref group="note">At least as early as Wesley Woolhouse's proposal to use 7/26-comma meantone in 1835; Woolhouse advocated it on the basis of being the held-octave OLD miniRMS-U tuning (though he didn't use our systematic name, of course). See http://tonalsoft.com/monzo/woolhouse/essay.aspx#book. And there is also an argument that the quarter-comma meantone tuning from 1523 was understood then as the held-octave OLD minimax-U tuning.</ref>

The first proposed tuning scheme that leveraged dual norms—the key technology enabling all-interval tunings—was the [[TOP]] tuning scheme, from [[Paul Erlich]]'s ''[[A Middle Path]]'' paper, though Paul did not unpack it as such at that time. When it comes to plumbing the underlying mathematical reasons for how all-interval tunings work, we are indebted to [[Gene Ward Smith]] and [[Mike Battaglia]].<ref group="note">It was Gene who first pointed out the relevance of dual norms to TOP:
http://lumma.org/tuning/gws/top.htm
Then in 2012 Mike took the idea and ran with it, applying it to tunings in all the ways we understand today:
https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_20461#20461
https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_20929#20929
https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_20996#20996
https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_21052#21052
https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_21054#21054
https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_21082#21082
https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_21415#21415
https://www.facebook.com/groups/xenharmonic/posts/10150650778389482/
</ref>

= Concepts =
By the end of this section, you will have a deep understanding of two of the most commonly-used all-interval tuning schemes: TOP, proposed by Paul Erlich; and [[Tenney-Euclidean tuning|TE]], proposed by [[Graham Breed]] (in {{subpage|tuning fundamentals|uprev|s=Systematic tuning scheme names|text=our naming system}}, these tunings are called "minimax-S" and "minimax-ES", respectively, where the 'S' stands for "simplicity-weight damage", as explained with the introduction of the naming system in the tuning fundamentals article, and the 'E' stands for "Euclideanized", which will be explained later). You'll be able to explain how they work, how they are similar to and different from each other, and also how they compare with the more basic tuning schemes that we've explained previously.

== The two conditions ==
Being able to minimize the damage to every interval is contingent upon two conditions:
# You define "least overall damage" as the minimization of the ''maximum'' damage dealt to any one target-interval; in other words, you use a [[Target tuning#Minimax_tuning|minimax]] tuning scheme.
# You use a {{subpage|tuning fundamentals|uprev|s=Weight slope|text=simplicity-weight damage}}.

=== Condition one: Minimax ===
When we target ''no'' intervals specifically, it would be equivalent to say that we care about each interval the same as any other interval (at least in terms of the damage we're willing to let it take), or in other words that we have an ''infinite'' target-interval set, i.e. ''every'' interval in your domain, or said another way, every interval that can be generated by your primes (has only those primes as prime factors). Using the 5-limit, for example, would mean that every interval able to be generated by primes 2, 3, and 5 was in your set.

We couldn't follow the instructions for computing generator tunings that were explained in the fundamentals article of this series with <math>k = \infty</math>, that is, with our target-interval list being a <math>(d, \infty)</math>-shaped matrix. We could never find the power mean (or power sum) of the resulting target-interval damage list, or at least we could never find the power sum when the power <math>p</math> is <math>1</math>, or <math>2</math>, or any finite number. We ''could'', however, theoretically do that when <math>p=\infty</math>.

It may not be immediately obvious why <math>p=\infty</math> makes this possible. Try thinking of it this way: it would be theoretically possible to establish a ''maximum'' of an infinitely long list, if you could prove that no matter what else comes up in the infinite remaining part of the list that you haven't observed yet, no item you'd ever find there could possibly be greater than some bound you'd established by whatever means.

In our case, we can prove that no matter which interval may appear in the list, we can guarantee that its damage will not be any greater than the magnitude of the retuning map. Don't worry if you don't know what we're talking about yet —consider that a sneak preview of where we're ultimately going with this. For now it is enough to for you to understand that this sort of proof by external bounding technique is why the first of the two conditions of an infinite target-interval set tuning scheme must be that it is a minimax tuning scheme (because minimax schemes are the ones where <math>p=\infty</math>).

=== Condition two: Simplicity-weighting ===
But how exactly ''would'' we prove such a situation as that?

Imagine the target-interval set starting out as a set of simple consonances, like a tonality diamond, and then think about continuously expanding it toward including all intervals in the prime limit, by adding each next complex interval to it, one by one. Think about what each of these new interval's absolute errors must be like. As we go further and further out, eventually to intervals like 1953125/1259712 and even crazier, will their absolute errors be getting generally bigger or smaller?

Well, bigger, to be sure. That's because we can find the error of any one of these intervals by multiplying it by the retuning map <math>𝒓</math>, and while this presents opportunities for some primes' errors to counteract other primes' errors, if some are positive and others negative (e.g. if prime 2 and prime 3 are both tuned narrow, then the ~3/2 may turn out to be near just, because 2 and 3 are on opposite sides of the quotient bar), in the worst cases their errors will compound all in one direction or the other (positive or negative, wide or narrow), which means the absolute error will be large, and there will always be some worst case types as we continue to add new complex intervals.

How could we possibly offset this inevitable increase in absolute error as we spiral further and further away from unison toward infinitely complex intervals, then? Well, the answer involves ''weighting''.

Note that we haven't specified our damage weight slope in this thought experiment thus far. What if we ''simplicity-weight'' damage, then? In that case, it may be possible to establish that no matter how much absolute error an interval may be capable of incurring, any additional complexity required to achieve that higher error will offset it when we simplicity-weight the result.

And that, in fact, is exactly how we do it. This is why simplicity-weighting is the second of the two conditions of all-interval tuning schemes. So these schemes essentially have an infinitely-sized target-interval set, with no hard bound on interval complexity, rather, the set just kind of "fades out" gradually, starting with the simplest consonance, the octave.

There's not too much more left to say about the first condition, i.e. being a minimax tuning scheme. But the precise reasoning and execution of the second condition is quite the rabbit hole! It entails a fancy feat of mathematics known as a ''dual norm''. Fortunately we have been there and back for you. We think we have some good words and images to demystify how exactly it all works out, and hope you get a lot out of it.

== Power norms ==
In order to understand dual norms, we should begin by understanding ''norms'', which is to say ''power'' norms.

A '''power norm'''<ref group="note">Every other mathematical use of the Latin root "norm" (a carpenter's square) relates to perpendicularity or standardization, as in the normal to a plane or to normalize a vector, which means to standardize it by giving it a length of 1 while retaining its direction. We could think of the "norm" in "power norm" as short for "normalizer", as it is the quantity you must divide all the entries of the vector by in order to normalize it. This is a very indirect way of saying that a power norm is a kind of length.
 
For a detailed history of "norm" in mathematics see: https://math.stackexchange.com/questions/465414/who-introduced-the-term-norm-into-mathematics
</ref> is another type of statistic similar to the power mean which we covered in the fundamentals article (also similar to the power ''sum'', if you went through the computations article), just with slightly different steps.

=== Steps ===
# Take the absolute value of each entry.
# Raise each absolute entry to the <math>p</math>th power.
# Sum the powers of the absolute entries.
# Take the matching <math>p</math>th root of this sum.

So power norms are like power sums with two extra steps: the absolute value taking step at the start, and the matching root step at the end. They can also be compared with power means, with which they share the matching root step, but means don’t take the absolute value and norms don’t divide by the count.

=== Formula ===
The formula for the <math>p</math>-norm, which we notate as <math>\|\textbf{i}\|_p</math><ref group="note">It can also be notated as <math>L^p(\textbf{i})</math>. The "L" doesn't stand for "norm", of course, but it is the conventional notation for power norms. The reasons for this are beyond the scope of this article, but we will at least note that it stands for Lebesgue, a mathematician who was involved in the pioneering of this topic. We especially don't prefer the <math>L^p</math> notation due to its conventionally being notated with superscript rather than subscript. Sometimes normal size script (neither superscript nor subscript) is used, but never subscript, as with the double-bar notation.</ref>, looks like this.

<math>
\norm{\textbf{i}}_p = \sqrt[p]{\strut \sum\limits_{n=1}^d \abs{\mathrm{i}_n}^p}
</math>

Though you'll notice that instead of doing this to a damage list <math>\textbf{d}</math> as we did for power means and sums, we're taking the power norm of an interval vector <math>\textbf{i}</math> here. And instead of iterating up to <math>k</math>, the count of target-intervals, we're iterating up to <math>d</math>, the dimensionality of the temperament (which is the same as the count of entries in any interval vector).

We can expand this out like so:

<math>
\norm{\textbf{i}}_p = \sqrt[p]{\strut \abs{\mathrm{i}_1}^p + \abs{\mathrm{i}_2}^p + ... + \abs{\mathrm{i}_d}^p}
</math>

This can also be written as:

<math>
\norm{\textbf{i}}_p = \Big(\abs{\mathrm{i}_1}^p + \abs{\mathrm{i}_2}^p + ... + \abs{\mathrm{i}_d}^p\Big)^\frac{1}{p}
</math>

Expressing the <math>p</math>th root as raising to the <math>\frac{1}{p}</math> power is how you are likely to have to do it on a calculator, spreadsheet or programming language.

=== Examples ===
Consider the vector for the interval <math>\frac{27}{20}</math>, which is {{vector|-2 3 1}}.

* Its <math>1</math>-norm is <math>\sqrt[1]{\strut \abs{{-2}}^1 + \abs{3}^1 + \abs{1}^1} = \sqrt[1]{\strut 2^1 + 3^1 + 1^1} = \sqrt[1]{2 + 3 + 1} = \sqrt[1]{6} = 6</math>.
* Its <math>2</math>-norm is <math>\sqrt[2]{\strut \abs{{-2}}^2 + \abs{3}^2 + \abs{1}^2} = \sqrt[2]{\strut 2^2 + 3^2 + 1^2} = \sqrt[2]{4 + 9 + 1} = \sqrt[2]{14} \approx 3.742</math>.
* Its <math>\infty</math>-norm is <math>\sqrt[\infty]{\strut \abs{{-2}}^\infty + \abs{3}^\infty + \abs{1}^\infty} = \sqrt[\infty]{\strut 2^\infty + 3^\infty + 1^\infty} = \max(2, 3, 1) = 3</math>.

When we write <math>\sqrt[\infty]{\strut 2^\infty + 3^\infty + 1^\infty}</math> this is shorthand for the more mathematically-correct <math>\lim_{p\to\infty}\sqrt[p]{\strut 2^p + 3^p + 1^p}</math>. For a refresher, {{subpage|tuning fundamentals|uprev|s=Max|text=click here}}.

=== Relationship with distance ===
When we introduced the power mean, we presented it as a generalization of the familiar formula for, well, the mean. We can also introduce the power norm as a generalization of a familiar formula: the formula for distance. As an example, if we move 3 meters to the right, and 4 meters forward, what's our distance from our starting position? Well, with a change along the <math>x</math>-axis of 3 and along the <math>y</math>-axis of 4, many of us may already be ready to give the answer:

<math>
\sqrt{\strut x^2 + y^2} = \sqrt{\strut 3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
</math>

The answer is that we're 5 meters from where we started, and we can find this readily using the 2D version of the distance formula (which in turn is often understood as a generalization of the Pythagorean formula, the one that shows how the hypotenuse of a right triangle squared is the same area as the sum of the squares of the two other sides). And in 3D the formula stays basically the same; no changes to the structure or to the power of 2, we just add another term: <math>\sqrt{\strut x^2 + y^2 + z^2}</math>.<ref group="note">In case you were wondering, the smallest 3D case with all integers is <math>\sqrt{\strut 1^2 + 2^2 + 2^2}=3</math>, and the next is <math>\sqrt{\strut 2^2 + 3^2 + 6^2}=7</math>. But integer-valued norms are of no particular interest in RTT.</ref>

[[File:Distances2.png|frameless|400x400px]]

So the power norm is the same idea, but with a couple generalizations.
* Any power (and matching root) can be used, not only <math>2</math>, which is how we generalize the formula to be able to measure distance in other types of spaces besides those similar to the physical type we embody as humans together.
* We take the absolute value at the beginning. When dealing with triangle side lengths and damage amounts, there are no negative values, but in other cases, such as those we'll use norms for in RTT—prime retunings, and prime counts—we certainly can have negative values, and it'll be important to get those positive. When the power is <math>2</math> (or any even number) this doesn't matter because the taking of the power will enforce positivity, but it's still an important part of the general formula.

=== Comparison with power means and sums ===
So if a power sum is a type of ''total'' and a power mean is a type of ''average'', then a power norm is sort of in between, but also sort of its own thing; it's a type of ''length''.

Closely related though all three of these power statistics may be, we'd like to take this opportunity to drive home some important distinctions between this latest one—the power norm—and the two that we've looked at up to this point, the power sum and power mean. In the previous article of this guide, we showed how <math>p</math>-sums and <math>p</math>-means could be used roughly interchangeably, but certain use cases prefer one to the other. Well, norms are the odd man out here, and really shouldn't be thought of as applying in the same situations as sums and means. Norms are used on vectors (and row vectors) whose entries represent pieces of information that are in different dimensions from each other (different primes, for instance), in different units, and thus can't be directly compared; whereas sums and means are used on lists of things that are all of the same type with the same units (like lists of damages). This closer conceptual kinship between sums and means should be apparent through the coloration of this next table:

<math>
% \slant{} command approximates italics to allow slanted bold characters, including digits, in MathJax.
\def\slant#1{\style{display:inline-block;margin:-.05em;transform:skew(-14deg)translateX(.03em)}{#1}}
% Latex equivalents of the wiki templates llzigzag and rrzigzag for double zigzag brackets.
\def\llzigzag{\hspace{-1.6mu}\style{display:inline-block;transform:scale(.62,1.24)translateY(.07em);font-family:sans-serif}{ꗨ\hspace{-3mu}ꗨ}\hspace{-1.6mu}}
\def\rrzigzag{\hspace{-1.6mu}\style{display:inline-block;transform:scale(-.62,1.24)translateY(.07em);font-family:sans-serif}{ꗨ\hspace{-3mu}ꗨ}\hspace{-1.6mu}}
</math>
{| class="wikitable"
!   !! Power-sum !! Power-mean !! Power-norm
|-
| Operator:
| style="background-color: #f8f8cc;" | <math>\llzigzag·\,\rrzigzag\!_p</math>
| style="background-color: #f4cccc;" | <math>\llangle\,·\,\rrangle_p</math>
| style="background-color: #c9daf8;" | <math>\norm{ · }_q</math>
|-
| Takes the absolute value:
| style="background-color: #fff2cc;" | No
| style="background-color: #fff2cc;" | No
| style="background-color: #c9daf8;" | Yes
|-
| Raises to power:
| style="background-color: #e4e4e4;" | Yes
| style="background-color: #e4e4e4;" | Yes
| style="background-color: #e4e4e4;" | Yes
|-
| Sums:
| style="background-color: #e4e4e4;" | Yes
| style="background-color: #e4e4e4;" | Yes
| style="background-color: #e4e4e4;" | Yes
|-
| Divides by count:
| style="background-color: #d9ead3;" | No
| style="background-color: #f4cccc;" | Yes
| style="background-color: #d9ead3;" | No
|-
| Takes the root:
| style="background-color: #fff2cc;" | No
| style="background-color: #d9d2e9;" | Yes
| style="background-color: #d9d2e9;" | Yes
|-
| Input structure:
| style="background-color: #fff2cc;" | List
| style="background-color: #fff2cc;" | List
| style="background-color: #c9daf8;" | Vector
|-
| Input values referred to as:
| style="background-color: #fff2cc;" | Items
| style="background-color: #fff2cc;" | Items
| style="background-color: #c9daf8;" | Entries
|-
| Input values are in same dimension:
| style="background-color: #fff2cc;" | Yes
| style="background-color: #fff2cc;" | Yes
| style="background-color: #c9daf8;" | No
|-
| Input quantity:
| style="background-color: #fff2cc;" | Damage
| style="background-color: #fff2cc;" | Damage
| style="background-color: #c9daf8;" | Scaled interval, scaled retuning
|-
| Input units:
| style="background-color: #fff2cc;" | ¢ ({{^(}}weighting{{)^}})
| style="background-color: #fff2cc;" | ¢ ({{^(}}weighting{{)^}})
| style="background-color: #c9daf8;" | '''p''' ({{^(}}weighting{{)^}}), ¢({{^(}}weighting{{)^}})
|-
| Output structure:
| style="background-color: #e4e4e4;" | Scalar
| style="background-color: #e4e4e4;" | Scalar
| style="background-color: #e4e4e4;" | Scalar
|-
| Output quantity:
| style="background-color: #f8f8cc;" | p-sum of damages
| style="background-color: #f4cccc;" | p-mean damage
| style="background-color: #c9daf8;" | Interval complexity, retuning magnitude
|-
| Output units:
| style="background-color: #fff2cc;" | ¢ ({{^(}}weighting{{)^}})
| style="background-color: #fff2cc;" | ¢ ({{^(}}weighting{{)^}})
| style="background-color: #c9daf8;" | ({{^(}}weighting{{)^}}), ¢({{^(}}weighting{{)^}})
|}

=== Special note about the infinity norm ===
If we ignore for the moment that means do not take the absolute value—which we can ignore in our application's case of ''damage'' means, because damages are never negative—we note that the <math>\infty</math>-norm is the same as the <math>\infty</math>-mean, that is, they are both equivalent to taking the max, despite the fact that the mean divides by the count of entries or items, and the norm does not. This is because the <math>\infty</math>th root of this count is equal to <math>1</math>, and thus dividing by this count does not distinguish the <math>\infty</math>-mean from the <math>\infty</math>-norm.

We can see this by subtly rewriting our <math>p</math>-mean formula from

<math>
\llangle\textbf{d}\rrangle_p = \sqrt[p]{\strut \dfrac{\sum\limits_{n=1}^k \mathrm{d}_n^p}{k}}
</math>

to

<math>
\llangle\textbf{d}\rrangle_p = \dfrac{\sqrt[p]{\strut \sum\limits_{n=1}^k \mathrm{d}_n^p}}{\sqrt[p]{k}}
</math>

We can now see that when <math>p =\infty</math>, whatever the value of <math>k</math> is, <math>\sqrt[p]{k} = \sqrt[\infty]{k} = 1</math>, and so <math>⟪\textbf{d}⟫_\infty</math> simplifies to <math>\sqrt[\infty]{\strut \sum\limits_{n=1}^k \mathrm{d}_n^\infty} = \max\limits_{n=1}^k \mathrm{d}_n</math> which would be the same as <math>\norm{\textbf{d}}_\infty</math>, considering that the items in <math>\textbf{d}</math> are always positive.

However, we note that the <math>\infty</math>-mean and <math>\infty</math>-norm are ''not'' the same as the <math>\infty</math>-sum (which doesn't even exist). This fact is an interesting complement to the fact that (when we continue to ignore that norms take absolute values), the <math>1</math>-sum is the same as the <math>1</math>-norm (they're both the total), but not the same as the <math>1</math>-mean (which does exist, as the average).<ref group="note">A series of observations which may give insight to some readers:
* When <math>p = 1</math>, taking the root (as in a power norm or mean) makes no difference.
* When <math>p = 2</math>, taking the absolute value (as in a power norm) makes no difference, and this goes for any even <math>p</math>.
* When <math>p = \infty</math>, dividing by the count (as in a power mean) makes no difference.</ref>

== Dual norms ==
Now that we understand norms, we can start taking a look at '''dual norms'''. And we're going to switch to using <math>q</math> for the norm power instead of <math>p</math>. You may have already noticed we did that in the table above. We did so because it is important to maintain a distinction between the optimization power <math>p</math> of a tuning scheme and the norm power <math>q</math> of its complexity calculation, which we will explain four subsections from now. Remember: the only optimization power for which all-interval tuning schemes work is <math>p=\infty</math>, the one for minimax. But, as you will learn, they can work with any complexity norm <math>q \geq 1</math>.

It is still OK to refer to power norms in general as <math>p</math>-norms for short, but we'll avoid it for the rest of this article.

=== Formula relating dual powers ===
Let's begin by stating some key facts about the most commonly used norm powers that we'll be using with all-interval tunings.

* The <math>1</math>-norm is the dual norm of the <math>\infty</math>-norm, and the <math>\infty</math>-norm in turn is the dual of the <math>1</math>-norm; they are each other's dual norm. (These are the extreme norms, by the way; there's no norm with power less than <math>1</math> or greater than <math>\infty</math>.)<ref group="note">But interestingly, there are other power means. The <math>0</math>-mean is the geometric mean. The <math>{-1}</math>-mean is the harmonic mean, and the <math>{-\infty}</math>-mean is the minimum.</ref>
* The <math>2</math>-norm is ''self-dual''. It is special in this way; no other norm boasts this property. It is the pivot point right in the middle of the norm continuum from <math>1</math> to <math>\infty</math>, and as such it finds itself to be ''its own dual norm''.

In general, we can find the '''dual power''' for a norm power <math>q</math> using the following equality<ref group="note">A demonstration of this relationship is fairly involved and we won't be getting into it. It involves Hölder and Young inequalities if you want to look into it yourself. Perhaps you might begin here: https://math.stackexchange.com/questions/1839906/inequality-ab-le-fracapp-fracbqq?noredirect=1&lq=1</ref>, where <math>\text{dual}(q)</math> gives the dual power:

<math>
\dfrac{1}{q} + \dfrac{1}{\text{dual}(q)} = 1
</math>

and therefore

<math>
\text{dual}(q) = \dfrac{1}{1 - \frac{1}{q}}
</math>

With this formula we can see how the <math>1</math>-norm and <math>\infty</math>-norm relate by <math>\frac{1}{1} + \frac{1}{\infty} = 1 + 0 = 1</math>.

We can also see the self-duality of the <math>2</math>-norm by <math>\frac{1}{2} + \frac{1}{2} = 1</math>.

For one further example, the dual norm of the <math>3</math>-norm would be the <math>\frac{3}{2}</math>-norm (or <math>1.5</math>-norm), because <math>\frac{1}{3} + \dfrac{1}{\frac{3}{2}} = \frac{1}{3} + \frac{2}{3} = 1</math>.

[[File:Dual power.png|frameless|900x900px]]

So when we speak of "dual norms", we speak of a ''pair'' of norms which are in a special relationship with each other.

=== The dual norm inequality ===
We now know how the relationship between dual norms is defined. But what does this relationship ''mean'', and what can we ''use'' it for, exactly?

Well, what's special about dual norms can be articulated as a single effect: ''the absolute value of the dot product of any two vectors is always less than or equal to the norm of one vector times the dual norm of the other vector.''

That's a complete mouthful, to be sure. But this is just the sort of idea that natural language struggles to express, but mathematical notation excels at it. So let's now look at that same idea but in a new way—the mathematical way—using <math>\textbf{x}</math> and <math>\textbf{y}</math> for our two vectors:

<math>
\abs{\textbf{x}·\textbf{y}} \leq \norm{\textbf{x}}_q × \norm{\textbf{y}}_{\text{dual}(q)}
</math>

Let's do a couple examples. Suppose <math>\textbf{x} = \monzo{1 & 0 & 0}</math> and <math>\textbf{y} = \monzo{-4 & 4 & -1}</math>. If <math>p = 1</math>, we have:

<math>
\begin{align}
\left|
\left[ \begin{matrix} 1 & 0 & 0 \\ \end{matrix} \right]
·
\left[ \begin{matrix} {-4} & 4 & {-1} \\ \end{matrix} \right]
\right|
\;\; \leq& \;\;
\|
\left[ \begin{matrix} 1 & 0 & 0 \\ \end{matrix} \right]
\|_1 × \|
\left[ \begin{matrix} {-4} & 4 & {-1} \\ \end{matrix} \right]
\|_{\infty}

\\[8pt]

\left|
(1)({-4}) + (0)(4) + (0)({-1})
\right|
\;\; \leq& \;\;
\sqrt[1]{\strut |1|^1 + |0|^1 + |0|^1}
×
\sqrt[\infty]{\strut |{-4}|^\infty + |4|^\infty + |-1|^\infty}

\\[8pt]

\left|
{-4} + 0 + 0
\right|
\;\; \leq& \;\;
\sqrt[1]{\strut 1^1 + 0^1 + 0^1}
×
\sqrt[\infty]{\strut 4^\infty + 4^\infty + 1^\infty}

\\[8pt]

\left|
{-4}
\right|
\;\; \leq& \;\;
\sqrt[1]{1 + 0 + 0}
×
\max(4, 4, 1)

\\[8pt]

4
\;\; \leq& \;\;
\sqrt[1]{1}
×
4

\\[8pt]

4 \;\; \leq& \;\; 1×4

\\[8pt]

4 \;\; \leq& \;\; 4
\end{align}
</math>

So here we have the left-hand side exactly equal to the right hand side.

Or suppose <math>\textbf{x} = \monzo{0 & -3 & -5}</math> and <math>\textbf{y} = \monzo{6 & 6 & 6}</math>, and <math>q = 2</math>:

<math>
\begin{align}
\left|
\left[ \begin{matrix} 0 & {-3} & {-5} \\ \end{matrix} \right]
·
\left[ \begin{matrix} 6 & 6 & 6 \\ \end{matrix} \right]
\right|
\;\; \leq& \;\;
\left\|
\left[ \begin{matrix} 0 & {-3} & {-5} \\ \end{matrix} \right]
\right\|_2 × \left\|
\left[ \begin{matrix} 6 & 6 & 6 \\ \end{matrix} \right]
\right\|_2

\\[8pt]

\left|
(0)(6) + ({-3})(6) + ({-5})(6)
\right|
\;\; \leq& \;\;
\sqrt[2]{\strut |0|^2 + |{-3}|^2 + |{-5}|^2}
×
\sqrt[2]{\strut |6|^2 + |6|^2 + |6|^2}

\\[8pt]

\left|
0 + {-18} + {-30}
\right|
\;\; \leq& \;\;
\sqrt[2]{\strut 0^2 + 3^2 + 5^2}
×
\sqrt[2]{\strut 6^2 + 6^2 + 6^2}

\\[8pt]

\left|
{-48}
\right|
\;\; \leq& \;\;
\sqrt[2]{0 + 9 + 25}
×
\sqrt[2]{36 + 36 + 36}

\\[8pt]

48
\;\; \leq& \;\;
\sqrt[2]{34}
×
\sqrt[2]{108}

\\[8pt]

48 \;\; \leq& \;\; 5.831×10.392

\\[8pt]

48 \;\; \leq& \;\; 60.597
\end{align}
</math>

In this case, the left-hand side is less than the right-hand side.

Please feel free to run some more examples if you'd like, to convince yourself this is true. (Or see the later section of this article to develop your intuition for it.) Do not worry if the musical implications of this are not readily apparent to you yet. We have more work to do on this equation.

=== Substituting RTT objects into the formula ===
For our next step, let's substitute in some of our tuning-related objects for <math>\textbf{x}</math> and <math>\textbf{y}</math>. Specifically, we'll use the retuning map <math>𝒓</math> for <math>\textbf{x}</math>, and any old arbitrary interval <math>\textbf{i}</math> for <math>\textbf{y}</math>:

<math>
\abs{\color{red}𝒓\color{black}\color{red}\textbf{i}\color{black}} \leq \norm{\color{red}𝒓\color{black}}_{\text{dual}(q)} × \norm{\color{red}\textbf{i}\color{black}}_q
</math>

If you would like a refresher on the retuning map, please review {{subpage|tuning fundamentals|uprev|s=Alternative approach|text=this section}} of the fundamentals article. In brief, <math>𝒓 = 𝒕 - 𝒋</math>, which is to say, it is the difference between a tempered-prime tuning map and the just-prime tuning map. It is used to find the error for an interval in the tuning that is represented by <math>𝒕</math>.

If you're paying close attention, you may have noticed that we dropped the dot in the dot product between the <math>𝒓</math> and the <math>\textbf{i}</math>. That's because it's optional to write here, since we chose a row vector for the left vector and a column vector for the right vector. The dot product of two vectors gives the same result as matrix multiplication between one row vector and one column vector of the same length, in that order.

You may also have noticed that we changed the position of the <math>\text{dual}()</math>. Because duality is symmetrical, it doesn't matter which one we call <math>q</math> and which one we call <math>\text{dual}(q)</math>. We did this because the norm of the vector (the interval <math>\textbf{i}</math>) is more fundamental than the norm of the row vector (the retuning map <math>𝒓</math>), for reasons that will become clear later.

As an example, consider the interval <math>\frac{6}{5}</math> with vector {{vector|1 1 -1}} and the retuning map {{map|1.699 -2.692 3.944}}, with <math>q=2</math>:

<math>

\begin{align}

\left|
\left[ \begin{matrix} 1.699 & {-2.692} & 3.944 \\ \end{matrix} \right]
\left[ \begin{matrix} 1 \\ 1 \\ {-1} \\ \end{matrix} \right]
\right|
\;\; \leq& \;\;
\left\|
\left[ \begin{matrix} 1.699 & {-2.692} & 3.944 \\ \end{matrix} \right]
\right\|_2 × \left\|
\left[ \begin{matrix} 1 \\ 1 \\ {-1} \\ \end{matrix} \right]
\right\|_2

\\[8pt]

\left|
(1.699)(1) + ({-2.692})(1) + (3.944)({-1})
\right|
\;\; \leq& \;\;
\sqrt[2]{\strut |1.699|^2 + |{-2.692}|^2 + |3.944|^2}
×
\sqrt[2]{\strut |1|^2 + |1|^2 + |-1|^2}

\\[8pt]

\left|
1.699 + {-2.692} + {-3.944}
\right|
\;\; \leq& \;\;
\sqrt[2]{\strut 1.699^2 + 2.692^2 + 3.944^2}
×
\sqrt[2]{\strut 1^2 + 1^2 + 1^2}

\\[8pt]

\left|
{-4.937}
\right|
\;\; \leq& \;\;
\sqrt[2]{2.887 + 7.247 + 15.555}
×
\sqrt[2]{1 + 1 + 1}

\\[8pt]

4.937
\;\; \leq& \;\;
\sqrt[2]{25.689}
×
\sqrt[2]{3}

\\[8pt]

4.937
\;\; \leq& \;\;
5.068
×
1.732

\\[8pt]

4.937
\;\; \leq& \;\;
8.779
\end{align}
</math>

And if we did <math>\frac{5}{1}</math> with vector {{vector|0 0 1}} with the same retuning map but <math>q=1</math>:

<math>
\begin{align}
\left|
\left[ \begin{matrix} 1.699 & {-2.692} & 3.944 \\ \end{matrix} \right]
\left[ \begin{matrix} 0 \\ 0 \\ 1 \\ \end{matrix} \right]
\right|
\;\; \leq& \;\;
\left\|
\left[ \begin{matrix} 1.699 & {-2.692} & 3.944 \\ \end{matrix} \right]
\right\|_\infty × \left\|
\left[ \begin{matrix} 0 \\ 0 \\ 1 \\ \end{matrix} \right]
\right\|_1

\\[8pt]

\left|
(1.699)(0) + ({-2.692})(0) + (3.944)(1)
\right|
\;\; \leq& \;\;
\sqrt[\infty]{\strut |1.699|^\infty + |{-2.692}|^\infty + |3.944|^\infty}
×
\sqrt[1]{\strut |0|^1 + |0|^1 + |1|^1}

\\[8pt]

\left|
0 + 0 + {-3.944}
\right|
\;\; \leq& \;\;
\sqrt[\infty]{\strut 1.699^\infty + 2.692^\infty + 3.944^\infty}
×
\sqrt[1]{\strut 0^1 + 0^1 + 1^1}

\\[8pt]

\left|
{-3.944}
\right|
\;\; \leq& \;\;
\max(1.699, 2.692, 3.944)
×
\sqrt[1]{0 + 0 + 1}

\\[8pt]

3.944
\;\; \leq& \;\;
3.944
×
\sqrt[1]{1}

\\[8pt]

3.944
\;\; \leq& \;\;
3.944
×
1

\\[8pt]

3.944
\;\; \leq& \;\;
3.944
\end{align}
</math>

But at this point we still haven't even explained what in the world we need another power for... isn't an optimization power enough? What use do we have for a norm power, now? Well, we assure you that we'll get to that as soon as we can.

=== Isolating damage ===
Let's take the next step toward understanding how this dual norm formula applies to regular temperament tuning. That step is multiplying both sides of the equation by the reciprocal of <math>\|\textbf{i}\|_q</math>:

<math>
\abs{𝒓\textbf{i}} \color{red} × \dfrac{1}{\norm{\textbf{i}}_q} \color{black} \leq \norm{𝒓}_{\text{dual}(q)} × \norm{\textbf{i}}_q \color{red} × \dfrac{1}{\norm{\textbf{i}}_q}
</math>

This causes the <math>\norm{\textbf{i}}_q</math> on the right-hand side of the equation to cancel out.

<math>
\abs{𝒓\textbf{i}} × \dfrac{1}{\norm{\textbf{i}}_q} \leq \|𝒓\|_{\text{dual}(q)} × \cancel{\norm{\textbf{i}}_q} × \cancel{\dfrac{1}{\norm{\textbf{i}}_q}}
</math>

Leaving us with:

<math>
\abs{𝒓\textbf{i}} × \dfrac{1}{\norm{\textbf{i}}_q} \leq \norm{𝒓}_{\text{dual}(q)}
</math>

So what sense can we make of this, now? It's generally a good thing whenever one manages to isolate some value on one side of the equation, so you may think we're immediately interested in <math>\norm{𝒓}_{\text{dual}(q)}</math>. Well, we will be interested in that soon enough, but for now this value is less interesting in and of itself. It's really more of a knob we'll turn later to get what we want on the left-hand side.

So let's start contemplating what we have on the left-hand side here, then. To begin with, can we answer the question: what's <math>|𝒓\textbf{i}|</math>? Well, if you recall from the fundamentals article, <math>𝒓\textbf{i}</math> is the ''error'' of the interval. Are the "ah-ha" alarms starting to go off?<ref group="note">If not, that's alright. I (Douglas here) spent weeks at this point following a red herring, where I was convinced that the best way forward was to understand the <math>\dfrac{\textbf{i}}{\norm{\textbf{i}}_q}</math> part as the normalized vector of <math>\textbf{i}</math>, i.e. a unit vector pointing in the same direction as the original vector, notated with a hat on the variable, like <math>\hat{\textbf{i}}</math>. I keep this thought here as a footnote in case it makes anyone feel any better, or maybe—in spite of it being an anti-insight with respect to Dave’s and my pedagogical work here—it may actually help someone one day.</ref>

What if I told you that the entire left-hand side of this inequality could be understood as the ''damage'' to <math>\textbf{i}</math>? To see how this is possible, first we must recognize the <math>× \dfrac{1}{\norm{\textbf{i}}_q}</math> part of this expression as ''simplicity-weighting'': multiplying by the reciprocal of a complexity function. And if that's the case, then that tells us that the norm part must be (drumroll please) a complexity function!

For example, if <math>\textbf{i}</math> is {{vector|1 1 -1}} and <math>𝒓</math> is {{map|1.699 -2.692 3.944}} (same as we chose for another recent example), then we have the interval error <math>e</math> equal to:

<math>
\begin{align}
e
\;\; =& \;\;
𝒓\textbf{i}
\\[8pt]
\;\; =& \;\;
\left[ \begin{matrix} 1.699 & {-2.692} & 3.944 \\ \end{matrix} \right]
\left[ \begin{matrix} 1 \\ 1 \\ {-1} \\ \end{matrix} \right]
\\[8pt]
\;\; =& \;\;
(1.699)(1) + ({-2.692})(1) + (3.944)({-1})
\\[8pt]
\;\; =& \;\;
1.699 + {-2.692} + {-3.944}
\\[8pt]
\;\; =& \;\;
{-4.937}
\end{align}
</math>

So the absolute interval error <math>|e| = 4.937</math>. And the interval complexity <math>c</math>, when <math>q=1</math> is:

<math>
\begin{align}
c
\;\; =& \;\;
\norm{\textbf{i}}_1
\\[8pt]
\;\; =& \;\;
\norm{ \left[ \begin{matrix} 1 \\ 1 \\ {-1} \\ \end{matrix} \right] }_1
\\[8pt]
\;\; =& \;\;
\sqrt[1]{\strut |1|^1 + |1|^1 + |{-1}|^1}
\\[8pt]
\;\; =& \;\;
\sqrt[1]{\strut 1^1 + 1^1 + 1^1}
\\[8pt]
\;\; =& \;\;
\sqrt[1]{1 + 1 + 1}
\\[8pt]
\;\; =& \;\;
\sqrt[1]{3}
\\[8pt]
\;\; =& \;\;
3
\end{align}
</math>

And so the damage <math>d</math> is

<math>
\begin{align}
d
\;\; =& \;\;
\dfrac{\abs{𝒓\textbf{i}}}{\norm{\textbf{i}}_q}
\\[8pt]
\;\; =& \;\;
\dfrac{|e|}{c}
\\[8pt]
\;\; =& \;\;
\dfrac{4.937}{3}
\\[8pt]
\;\; =& \;\;
1.645
\end{align}
</math>

And the norm on our retuning map {{map|1.699 -2.692 3.944}}, when <math>\text{dual}(q) = \infty</math>, would be <math>\max\left(|1.699|, |{-2.692}|, |3.944|\right) = 3.944</math>, so the inequality still holds here.

=== Connecting norms and complexities ===
So we've talked about norms, and we've talked about complexities too, but we haven't yet talked about them in the same context. It's now time to bring these two concepts together.

Yes, as it turns out, there is a way to ''define a complexity as a norm'', or we might say ''normify'' a complexity. At least, there are ways to "normify" ''many of'' the complexity functions we might wish to use in RTT (not all of them). We'll look at how to do that soon enough.

For now we'd just like to end a bit of the suspense regarding the difference between the power for a tuning scheme's power ''mean'' (its optimization power, for the mean of the target-interval damage list) and the power for its power ''norm'' (its interval complexity norm's power, for the simplicity-weighting of its damage statistic itself). We'll end it by giving the norm power for the default complexity we use in our text: the log-product complexity, or <math>\text{lp-C}()</math> for short. When defined as a power norm, <math>\text{lp-C}()</math> uses a norm power of <math>1</math>. So that's certainly different than the optimization power of <math>\infty</math> required for all-interval tuning schemes (but again, even if <math>\text{lp-C}()</math>'s norm power was also <math>\infty</math>, that'd just be a coincidence; the point is that conceptually speaking, these are completely different powers.)

In the following sections, we'll unpack the right-hand side of this inequality, so that we can finally explain why the dual norm inequality is useful to tuning at all. Before moving on, though, we should be able to see at this point that if our interval complexity function is defined as a norm, then the left-hand side of this equation (with the notation slightly simplified here now) represents the simplicity-weight damage to an arbitrary interval:

<math>
\dfrac{\abs{𝒓\textbf{i}}}{\norm{\textbf{i}}_q}
</math>

=== Retuning magnitude ===
And now let's finally unpack the right-hand side of the inequality. Let's reproduce the whole thing here for convenience, along with that newly simplified left-hand side:

<math>
\dfrac{\abs{𝒓\textbf{i}}}{\norm{\textbf{i}}_q} \leq \norm{𝒓}_{\text{dual}(q)}
</math>

Inside the double bars we have our retuning map <math>𝒓</math>, and the double bars tell us to take its norm. And not just any norm: the ''dual'' norm of whichever norm that we're using for our interval complexity. So if, for example, our interval vector's norm power was <math>2</math>, then our retuning map's norm power would also be <math>2</math>. Or if our interval vector's norm power was <math>1</math>, then our retuning map's norm power would be <math>\infty</math>.

We have a special name for a norm on our interval vector <math>\textbf{i}</math>—a "complexity"—so let's give ourselves a special name for the norm on our retuning map <math>𝒓</math>, too, to help us compartmentalize these concepts (remember, a [[map]] is just another type of vector, specifically, a [[Wikipedia:Covector|row vector]]). We can refer to this norm as our '''retuning magnitude''' (or "mistuning magnitude", if you prefer). "Magnitude" is a near synonym for norm that nicely connotes size<ref group="note">Though we do recognize that it often connotes a norm with power of <math>2</math>, and that will certainly not always be the case here.</ref> (and perhaps in particular of things that are problems, like earthquakes). So by decreasing the magnitude of our retuning, we move toward a closer-to-just tuning.

And, since we're going to be using these phrases a lot moving forward, let's use "<math>\textbf{i}</math>-norm" as short for "interval complexity norm", and "<math>𝒓</math>-norm" as short for "retuning magnitude norm".

=== How to use the inequality ===
Next, let’s note which direction this inequality points. It's telling us that no matter which interval we choose—even a crazily complex one!—its simplicity-weight damage will always be less than or equal to whatever the dual norm is of our retuning map. In other words, if we can minimize the simpler right-hand side of this inequality, then we will also have thereby minimized the left-hand side, which is the side we more directly care about. This is what we meant earlier by the right-hand side being more of a knob we adjust, in order to get what we want out of the left-hand side.

And this is an extremely powerful effect here, because remember, our <math>\textbf{i}</math> variable represents ''any'' old arbitrary interval in our entire interval subspace—in other words, an infinitude of possibilities. But the <math>𝒓</math> variable, the thing we can try minimizing, represents a ''singular'' object. Put another way: any given tuning we may check on our way to minimization has infinity different <math>\textbf{i}</math>'s, but only a single <math>𝒓</math>.

And so this is what we've been looking for: a way to dismiss the infinitude of damages we don't specifically know about, in our theoretically <math>(d, k)=(d,\infty)</math>-shaped target-interval set containing every possible interval in our interval subspace, because we know for a fact that not one of them could possibly be greater than the magnitude of the errors on our primes.

What this has given us now is ''a new way to compute a minimax damage tuning''. Rather than using the method discussed already in the computations article for computing minimax tunings, we can instead minimize whichever norm we want on the retuning map, and due to the implications of the dual norm inequality, we will have thereby minimized the maximum damage across all intervals (here's where it's always the maximum! No optimization power other than <math>\infty</math> is possible here)— so long as, of course, we're satisfied with that damage being defined as a simplicity-weight damage whose interval complexity is expressible as a norm, where the norm we minimized on the retuning map is its ''dual''.

And so we can think of the less-than-or-equals sign in the middle of the dual norm equality as setting the maximum—the equivalent of our ever-present optimization power of <math>p=\infty</math>.

Finally, then, we can see why it is unnecessary to provide a target-interval set when using this minimax tuning technique: We've managed to minimize the damage to every interval in the prime limit.

== Normifying complexities ==
Time to tie up a loose end: we've established that ''some'' complexities ''can'' be norms, but ''when'' can a complexity be a norm, and ''how''?

=== Quotient-based versus vector-based formulas ===
Perhaps the best way to explain complexity normification is by example. And what better place to start than with our default complexity function: log-product complexity. We've even spoiled a couple things about it already: one, that it's one of the complexities that ''can'' indeed be a norm, and two, that when it is in norm form, its power is <math>1</math>.

So how do we get from point A to point B—how to convert <math>\text{lp-C}()</math> into a norm? Let's begin at the beginning, at point A, i.e with the formula we've been using for <math>\text{lp-C}()</math> thus far. This formula is ''quotient-based'', i.e. it takes as inputs <math>n</math> and <math>d</math>, the numerator and denominator of the JI interval expressed as a quotient:

<math>
\text{lp-C}\left(\frac{n}{d}\right) = \log_2\left(n×d\right)
</math>

We can see there are two steps to the log-product complexity. First we turn the quotient into a product. Then we take the base-2 log of that product.

<math>\text{lp-C}\left(\frac{10}{9}\right) = \log_2(10×9) = \log_2{90} \approx 6.492</math>.

Now that looks obvious enough when the interval is in quotient form, but how about when the interval is in the form of a prime-count vector? Let's convert <math>\frac{10}{9}</math> to a vector.

<math>\frac{10}{9} = \dfrac{2×5}{3×3} = \dfrac{2^1×5^1}{3^2}=2^1×3^{-2}×5^1 = \left[ \begin{matrix} 1 & {-2} & 1 \\ \end{matrix} \right]</math>

Now we convert its product, <math>10×9</math>, to a vector.

<math>10×9 = 2×5×3×3 = 2^1×3^2×5^1 = \left[ \begin{matrix} 1 & 2 & 1 \\ \end{matrix} \right]</math>

So we see that changing the vector from a quotient to a product is equivalent to taking the absolute value of all its entries, which is the first step in taking any norm.

Now we have:

<math>\text{lp} \left[ \begin{matrix} 1 & {-2} & 1 \\ \end{matrix} \right] = \log_2 \left[ \begin{matrix} |1| & |{-2}| & |1| \\ \end{matrix} \right] = \log_2\left(2^{|1|} × 3^{|{-2}|} × 5^{|1|}\right)</math>

We can now apply one of the many helpful logarithmic identities:

<math>\log(a×b) = \log(a) + \log(b)</math>

This lets us change from a single logarithm of a product of prime powers, to a sum of logarithms of individual prime powers:

<math>\log_2\left(2^{|1|}\right) + \log_2\left(3^{|{-2}|}\right) + \log_2\left(5^{|1|}\right)</math>

So we gear down from multiplication to addition. Good stuff.

But that's not all. Here's another log identity we can make use of:

<math>\log\left(a^b\right) = \log(a)×b</math>

This lets us extract the exponents from inside the logarithm parentheses to coefficients outside of them. So again we gear down one level of operational hierarchy, from exponentiation to multiplication.

<math>\log_2(2) × |1| + \log_2(3) × \abs{-2} + \log_2(5) × |1|</math>

Now because the logs of the primes are always positive (primes are all greater than 1), there's no reason we can't extend the absolute value bars to encompass the logs as well:

<math>\abs{\log_2(2) × 1} + \abs{\log_2(3) × -2} + \abs{\log_2(5) × 1}</math>

And at last we have the log-product complexity in the form of a <math>1</math>-norm: a sum of vector entries, each absolute valued (and raised to the <math>1</math>st power, then the <math>1</math>st root is taken of the whole thing, but these are no-ops so we don't need to show them).

Note that this is not the <math>1</math>-norm of the original vector, but the <math>1</math>-norm of ''a rescaled version of'' the original vector; each entry has been individually scaled by the log of its corresponding prime. One way to think about this is that we've converted each entry from a prime-count into an octave-count.

<math>\norm{ \left[ \begin{matrix} \log_2(2) × 1 & \log_2(3) × {-2} & \log_2(5) × 1 \end{matrix} \right] }_1</math>

=== Diagonal matrices ===
We have a little more work to do before we can see the original vector in the expression. We begin with a row vector <math>{\large\textbf{𝓁}}\hspace{2mu}</math>, the '''log-prime map'''.

<math>{\large\textbf{𝓁}}\hspace{2mu} = \left[ \begin{matrix} \log_2{2} & \log_2{3} & \log_2{5} \\ \end{matrix} \right]</math>

One way to think about the scaled vector inside the norm-double-bars, which was the last thing we looked at in the previous section, is as the ''entry-wise'' product of <math>{\large\textbf{𝓁}}\hspace{2mu}</math> and {{vector|1 -2 1}}. If we simply took their ''matrix'' product (AKA ''dot'' product), then all the individual products would be added together as the last step, leaving us with a scalar, which we don't want. The way to prevent them from being added together like that is to convert one of these two vectors into a matrix, specifically by putting each entry into a different row and column. In other words, we ''diagonalize'' the vector, turning it into a ''diagonal matrix'', or in other words, a matrix with all zeros except the numbers along its main diagonal.

So when we wish to achieve an entry-wise product of two vectors in linear algebra, multiplying by a diagonal matrix is how we do that (diagonal matrices like these are sometimes called "scaling matrices" for this reason, because they're the way linear algebra scales entries of vectors individually.) We can think of it this way. The diagonal matrix is a special kind of linear mapping, where the first row says: take all the entries from the incoming vector other than its ''first'' entry and throw them out (multiply them by 0), then multiply that first entry by whatever this first entry is; then for the second row, take all the entries from the incoming vector other than the ''second'' entry and throw them out, then multiply that second entry by whatever this second entry is; and so on.

In this case—since we want the result to be a ''column'' vector—its the ''row'' vector that we convert into a diagonal matrix, leaving the existing column vector alone.

<math>
\text{diag}\left({\large\textbf{𝓁}}\hspace{2mu}\right)
\left[ \begin{matrix} 1 \\ {-2} \\ 1 \\ \end{matrix} \right]

=

\left[ \begin{matrix} \log_2{2} & 0 & 0 \\ 0 & \log_2{3} & 0 \\ 0 & 0 & \log_2{5} \\ \end{matrix} \right]
\left[ \begin{matrix} 1 \\ {-2} \\ 1 \\ \end{matrix} \right]

=

\left[ \begin{matrix} \log_2(2) × 1 \\ \log_2(3) × {-2} \\ \log_2(5) × 1 \\ \end{matrix} \right]

</math>

Instead of writing <math>\text{diag}\left({\large\textbf{𝓁}}\hspace{2mu}\right)</math> we can define the variable <math>L</math> to be equal to that:

<math>L = \text{diag}\left({\large\textbf{𝓁}}\hspace{2mu}\right) = \left[ \begin{matrix} \log_2{2} & 0 & 0 \\ 0 & \log_2{3} & 0 \\ 0 & 0 & \log_2{5} \\ \end{matrix}\right] \approx
\left[ \begin{matrix}
1.000 & 0 & 0 \\
0 & 1.585 & 0 \\
0 & 0 & 2.322 \\
\end{matrix} \right]</math>

We can call this <math>L</math> the '''log-prime matrix'''.

Now we can write the log-product complexity of <math>\frac{10}{9}</math> as:

<math>\text{lp}\,\left[1\ {-2}\ \ 1\right\rangle = \norm{L\,\left[1\ {-2}\ \ 1\right\rangle}_1</math>

And in general, the log-product complexity of an interval <math>\textbf{i}</math> in vector form, can be written as:

<math>\text{lp-C}\left(\textbf{i}\right) = \norm{L\textbf{i}}_1</math> where <math>L</math> is a diagonal matrix containing the base-2 logs of the primes in our vector basis.

And we can say that the log-product complexity of an interval in vector form is its log-prime prescaled<ref group="note">Many articles on the Xenharmonic wiki at the time of writing, describe this kind of thing as a "weighted norm", but this conflicts with general mathematical usage. Although it makes no difference in the case of a <math>1</math>-norm, we found two examples online where a "weighted <math>2</math>-norm" is defined so that the weight is applied 'after' the squaring, and no examples where it was applied beforehand (see
https://math.stackexchange.com/questions/2263447/proximal-operator-of-weighted-l-2-norm, and https://www-users.cse.umn.edu/~olver/num_/lnn.pdf). Weighting after taking the powers is also standard for “weighted power-means” (see https://en.wikipedia.org/wiki/Generalized_mean#Definition). With "prescaled norm" we make it clear that the scaling occurs before any norm steps are taken.
 
We also find "weighted norms" defined as things entirely different from what we use in RTT (see https://en.wikipedia.org/wiki/Weighted_space, and https://encyclopediaofmath.org/wiki/Weighted_space).
 
As for our use of "scaled" over "weighted", we justify this choice in the main text of this article in a couple places: [[#we consciously|here]], beginning with "We consciously chose to avoid emphasizing these parallels", and [[#as for the weighting|here]], beginning with "And as for the 'weighting' vs. 'scaling' issue".</ref> <math>1</math>-norm.

Let's confirm that we get the same result for our <math>\frac{10}{9}</math> example when we do it this way. First we apply the log-prime matrix as our prescaler.

<math>
L\textbf{i} \approx

\left[ \begin{matrix}
{1.000} & 0 & 0 \\
0 & {1.585} & 0 \\
0 & 0 & {2.322} \\
\end{matrix} \right]

\left[ \begin{matrix}
{1} \\
{-2} \\
{1} \\
\end{matrix} \right]

=

\left[ \begin{matrix}
{1.000} & × & {1} \\
{1.585} & × & {-2} \\
{2.322} & × & {1} \\
\end{matrix} \right]

=

\left[ \begin{matrix}
1.000 \\
{-3.170} \\
2.322 \\
\end{matrix} \right]
</math>

Then we apply the <math>1</math>-norm.

<math>
\begin{align}
\norm{L\textbf{i}}_1
&\approx
\norm{\left[ \begin{array} {r} 1.000 \\ {-3.170} \\ 2.322 \\ \end{array} \right]}_1
\\[5pt] &=
\sqrt[1]{|1.000|^1 + |{-3.170}|^1 + |2.322|^1}
\\[5pt] &=
|1.000|^1 + |{-3.170}|^1 + |2.322|^1
\\[5pt] &=
|1.000| + |{-3.170}| + |2.322|
\\[5pt] &=
1.000 + 3.170 + 2.322
\\[5pt] &=
6.492
\\[5pt] &\approx
\log_2(10×9)
\end{align}
</math>

And so we have found the same answer with our input in vector form as we did with our input in quotient form.

=== When normifying a complexity is not possible ===
So in general, to normify a complexity function, we must find a way to express it as some power-norm of an interval's vector, that may be transformed by some scaling matrix before the norm is taken, as we managed to do above with log-product complexity. We can even allow off-diagonal entries in the scaling matrix, but in most cases the matrix must be invertible (exceptions to this will be dealt with in the advanced tuning concepts article). The reason for this will become apparent later.

We can't accomplish this with the plain old (non-logarithmic) product complexity, that is, we can’t express that complexity function as a norm.

It’s not hard to see why. Let's go back to our <math>\frac{10}{9}</math> example. To get from {{vector|1 -2 1}} to <math>2^{|1|} × 3^{|{-2}|} × 5^{|1|}</math> we need to ''exponentiate'' each entry using a different prime base, then ''scale'' those exponentials together. All we can do with prescaled 1-norms is ''scale'' each entry by a different value, then ''add'' those products together. They are one gear lower in the hierarchy of operations. Other power-norms merely insert a constant power into that sequence, then take a constant root at the end, neither of which help us here.

This illuminates one of the things that are powerful about logarithms: they can be understood as gearing down one level in the operational hierarchy, from multiplication to addition (and from exponentiation to multiplication). Using logarithms enables a factor of <math>2</math> to always be worth the same amount of complexity, from an additive perspective (in the case of log-product complexity, a factor of <math>2</math> is always worth <math>1</math> unit of complexity, which is intuitive enough). This is one key way, then, to appreciate why we typically use log-product complexity instead of (plain old) product complexity in RTT.

== Dual-norm prescalers ==
We've introduced the idea of dual norms, but so far we've only touched upon them in terms of their dual ''powers'', i.e. the <math>1</math>-norm is the dual norm of the <math>\infty</math>-norm, and vice versa. But it turns out that there's more to our dual norms than just dual powers. In the previous section we learned that a norm can have a ''prescaler'', represented by a diagonal matrix.

To kick off this part of our discussion, we pose the question: if <math>\text{lp-C}()</math> can be expressed as a <math>1</math>-norm, with a log-prime prescaler, then what does its dual norm look like? We know it will be ''some sort of'' <math>\infty</math>-norm. But will it have a prescaler? And if so, what will its matrix look like?

Before we answer that, we want to generalize our previous result for the case of log-product complexity, to allow for other kinds of complexities. So instead of <math>\text{lp-C}\left(\textbf{i}\right) = \norm{L\textbf{i}}_1</math> where <math>L</math> is a diagonal matrix containing the base-2 logs of the primes—the specific complexity prescaler we need for log-product complexity—we write, more generally <math>\text{complexity}\left(\textbf{i}\right) = \norm{X\textbf{i}}_q</math> where <math>q</math> is a norm power and <math>X</math> is whatever '''complexity prescaler''' we need at the time.

It is important to distinguish this complexity prescaler <math>X</math> from the complexity weight matrix <math>C</math>. While the complexity weight matrix <math>C</math> is always a <math>(k, k)</math>-shaped matrix—that is, with one diagonal entry for each ''targeted interval'', the complexity prescaler <math>X</math> is always a <math>(d, d)</math>-shaped matrix with just one diagonal entry for each ''prime''.

=== Prescaled norms ===
So a prescaled norm can be fully specified by these two things:

# Its power
# Its prescaler

The prescaler is a square matrix with shape <math>(d, d)</math>,<ref group="note">Except in some advanced tuning schemes, as described in the next article.</ref> so that it can take in any arbitrary interval <math>\textbf{i}</math> of our interval subspace, which has shape <math>(d, 1)</math>, and spit it out as a new vector of the same <math>(d, 1)</math> shape, but now rescaled.

At this point in our understanding of all-interval tuning schemes, we are working with two different powers, and two different multipliers. We in fact have one pair of a power and a multiplier, and another separate pair of a power and a multiplier. In order to understand their interrelations better, let's visualize them on a diagram:

[[File:Multipliers and powers.png|frameless|900x900px]]

Note, however, that reality is not actually so complicated as it may seem at first glance at this diagram. That's because analyzing the interval complexity in terms of being a norm with a prescaler and power is only of ''particular'' interest when dealing with an all-interval tuning scheme (and that's because you need to know the "duals" of each of these two things: the dual power, and the "dual" prescaler), and in that case, then everything else about the other higher-tier pair of multiplier and power—damage weight, and optimization power—are locked-in (to simplicity-weight, and <math>\infty</math>, respectively). In other words, even though there are four pieces of information on this diagram, whether you're using an all-interval tuning scheme or an ordinary one, you only ''need'' to worry about two of them at a time.

We put "dual" in scare-quotes above, in the case of the prescaler, because dual matrices have previously been defined as those where one is a null-space basis for the other, like {{subpage|exploring temperaments|uprev|s=Duality|text=mappings and comma bases}}. That is not the case here. They are simply matrix inverses.

=== Bringing it back to the dual norm inequality ===
Suppose now that we want to use log-product complexity as our interval complexity when we simplicity-weight our absolute error to obtain our damage (remember, simplicity-weight is just the reciprocal of complexity-weight). Let's plug that into our dual norm inequality (reproduced here):

<math>
\dfrac{\abs{𝒓\textbf{i}}}{\|\textbf{i}\|_q} \leq \|𝒓\|_{\text{dual}(q)}
</math>

That is, let's replace our generic and basic norm <math>\|\textbf{i}\|_q</math> with our specific prescaled one, <math>\|X\textbf{i}\|_1</math>:

<math>
\dfrac{\abs{𝒓\textbf{i}}}{\color{red}\|X\textbf{i}\|_1\color{black}} \leq \norm{𝒓}_{\text{dual}(q)}
</math>

Oh, and by the dual power equality, we know our dual power must be <math>\infty</math>, so we can specify that, too:

<math>
\dfrac{\abs{𝒓\textbf{i}}}{\|X\textbf{i}\|_1} \leq \|𝒓\|_{\color{red}\infty}
</math>

Hm. But even that's not quite right. Remember, we got here by plugging in our own special RTT objects into this dual norm inequality we got from general mathematics, the one that started out with these completely abstract <math>\textbf{x}</math> and <math>\textbf{y}</math> vector variables. We had plugged in <math>𝒓</math> for <math>\textbf{x}</math> and <math>\textbf{i}</math> for <math>\textbf{y}</math>, if you recall. So if we've just now substituted in a <math>X\textbf{i}</math> in place of one <math>\textbf{i}</math> here, then we really ought to substitute that <math>X\textbf{i}</math> in for ''every'' <math>\textbf{i}</math> here! Let's take care of that then:

<math>
\dfrac{\abs{𝒓\color{red}X\color{black}\textbf{i}}}{\norm{X\textbf{i}}_1} \leq \norm{𝒓}_\infty
</math>

Alright. But now here's the problem. What the heck is <math>𝒓X\textbf{i}</math>, the numerator on the left-hand side there? That no longer represents the error of <math>\textbf{i}</math>, which we've established is <math>𝒓\textbf{i}</math> (i.e. without the <math>X</math>). And if the inside of those absolute value bars doesn't represent the interval error, then the left-hand side of this inequality no longer represents the simplicity-weighted absolute value of the error, AKA damage.

So what do we do now?

Well, it's not the end of the world. All we have to do, actually, is cancel out that annoying <math>X</math> that's cropped up in that numerator. And we can do this easily enough. Just as we've adjusted what we substitute in for <math>\textbf{y}</math>, from <math>\textbf{i}</math> to <math>X\textbf{i}</math>, we can adjust what we substitute in for <math>\textbf{x}</math>, in this case from <math>𝒓</math> to <math>𝒓\color{red}X^{-1}</math>!

Why <math>𝒓X^{-1}</math>? Well, by including the matrix-inverse of <math>X</math> here, we'll ensure that the extra <math>X</math> that we've ended up with in the numerator there gets canceled out, just in the same way that any scalar variable <math>x</math> would cancel out when multiplied with its multiplicative inverse <math>x^{-1}</math>. So where we multiplied <math>\textbf{i}</math> one way, we multiply <math>𝒓</math> the inverse (equal and opposite) way:

<math>
\dfrac{\abs{𝒓\color{red}X^{-1}\color{black}X\textbf{i}}}{\norm{X\textbf{i}}_1} \leq \norm{𝒓\color{red}X^{-1}\color{black}}_\infty
</math>

Cancelling out:

<math>
\dfrac{\abs{𝒓\cancel{X^{-1}}\cancel{X}\textbf{i}}}{\norm{X\textbf{i}}_1} \leq \norm{𝒓X^{-1}}_\infty
</math>

And we're left with:

<math>
\dfrac{\abs{𝒓\textbf{i}}}{\norm{X\textbf{i}}_1} \leq \norm{𝒓X^{-1}}_\infty
</math>

So now we're back to a representation of a simplicity-weight damage on the left-hand side, and as a byproduct of achieving this, the right-hand side has changed a bit. Specifically, just as our interval complexity <math>\text{fn-C}\left(\textbf{i}\right) = \norm{X\textbf{i}}_1</math> is a prescaled norm—the <math>1</math>-norm prescaled by <math>X</math>—so is our retuning magnitude a prescaled norm: the <math>\infty</math>-norm prescaled by <math>X^{\color{red}-1}</math>. So our "dual" prescaler <math>X^{-1}</math> is really our '''inverse prescaler.'''<ref group="note">You may be tempted to think that a complexity prescaler's matrix-inverse could be called a ''simplicity'' prescaler, but we note that in the case of target intervals, a simplicity prescaler is not defined as, and is not in general, the matrix-inverse of a complexity prescaler, but rather its entry-wise reciprocal. So this would only lead to confusion.</ref>

<math>
\dfrac{\abs{𝒓\textbf{i}}}{\norm{X\textbf{i}}_1} \leq \norm{𝒓\color{red} X^{-1} \color{black}}_\infty
</math>

To wrap up here, we can say that if we want to minimize the maximum log-product-simplicity-weight damage across all intervals, we must minimize the <math>\infty</math>-norm of the retuning map prescaled by the inverse of the ''complexity'' prescaler that the intervals are prescaled by. And the <math>\infty</math>-norm, as we've seen earlier, just grabs whichever entry is the maximum out of all the entries in the given vector.

We note that unlike the situation with the dual powers, there's nothing inherent to the dual norm inequality about inverse prescalers, which is to say that using inverse prescalers for <math>\textbf{i}</math> and <math>𝒓</math> is not at all ''necessary'' according to this general mathematical inequality. Doing so is simply the only ''useful'' thing for us to do here given our use case, since we wish to end up with <math>𝒓\textbf{i}</math> in the numerator on the left-hand side.

=== Inverse prescaler for log-product complexity ===
So, then, what is this <math>X^{-1}</math>? Finding the inverse of a matrix is a basic linear algebra operation you'll find in any math software package, or spreadsheet. But in the case of a diagonal matrix, as we have here, it's particularly simple. It's the same matrix but with each entry along the diagonal replaced with its reciprocal—AKA its inverse. So to review, if <math>X</math> is this:

<math>
\begin{align}
X
&=
\left[ \begin{matrix}
\log_2{2} & 0 & 0 \\
0 & \log_2{3} & 0 \\
0 & 0 & \log_2{5} \\
\end{matrix} \right]

\\[10pt] &\approx

\left[ \begin{matrix}
1.000 & 0 & 0 \\
0 & 1.585 & 0 \\
0 & 0 & 2.322 \\
\end{matrix} \right]
\end{align}
</math>

Then <math>X^{-1}</math> is this:

<math>
\begin{align}

X^{-1} &=

\left[ \begin{matrix}
\log_2{2} & 0 & 0 \\
0 & \log_2{3} & 0 \\
0 & 0 & \log_2{5} \\
\end{matrix} \right]

^{\Large -1} \normalsize

\\[10pt] &=

\left[ \begin{matrix}
(\log_2{2})^{-1} & 0 & 0 \\
0 & (\log_2{3})^{-1} & 0 \\
0 & 0 & (\log_2{5})^{-1} \\
\end{matrix} \right]

\\[10pt] &=

\left[ \begin{matrix}
\frac{1}{\log_2{2}} & 0 & 0 \\
0 & \frac{1}{\log_2{3}} & 0 \\
0 & 0 & \frac{1}{\log_2{5}} \\
\end{matrix} \right]

\\[10pt] & \approx

\left[ \begin{matrix}
1.000 & 0 & 0 \\
0 & 0.631 & 0 \\
0 & 0 & 0.431 \\
\end{matrix} \right]
\end{align}
</math>

And note that because, in this case, the inverse happens to equal the entry-wise reciprocal, <math>X^{-1} = \dfrac{1}{X}</math><ref group="note">The inverse prescaler is not defined as, and is not in general, the entry-wise reciprocal of the complexity prescaler, but rather its matrix-inverse. The complexity prescaler is not always a diagonal matrix, as in some advanced tuning schemes, as described in the next article.</ref> we could also rewrite our inequality as:

<math>
\dfrac{\abs{𝒓\textbf{i}}}{\norm{{\color{red} X}\textbf{i}}_1} \leq \norm{\dfrac{𝒓}{\color{red} X}}_\infty
</math>

This way of writing it illuminates how both sets of logs-of-primes, that have not canceled out with each other here, are now on the denominator side of the fraction bar (both occurrences of <math>\color{red}X</math> have been highlighted in red text above to drive this point home). They ended up on this side for two different reasons, but this side they've ended up on nonetheless.

We'll present one last way of looking at this inequality, which uses a common mathematical notation for duals of functions: a superscript asterisk. So if <math>\text{fn-C}()</math> is our complexity function, which we call on our interval <math>\textbf{i}</math>, then <math>\text{fn-C}^{*}\!()</math> is that function's dual, which we call on our retuning map <math>𝒓</math>:

<math>
\dfrac{\abs{𝒓\textbf{i}}}{{\color{red}\text{fn-C}(}\textbf{i}{\color{red}t)}} \leq {\color{red} \text{fn-C}^{*}\!(}\color{black}𝒓{\color{red})}
</math>

But how can we actually minimize the right-hand side of this inequality? Well, in short, you can plug it into a computer; please give our RTT Library in Wolfram Language a shot. If you want to understand its inner workings, however, it uses specialized methods depending on the norm power, and we'll get into all that detail in the computation section below.

=== Sanity-check example ===
Let's replay an example from earlier, but this time using a prescaled norm, to make sure the inequality still holds as expected. So we have our interval <math>\textbf{i}</math> being <math>\frac{5}{1}</math> with vector {{vector|0 0 1}}, our retuning map <math>𝒓</math> being {{map|1.699 -2.692 3.944}}, and our <math>q</math> being <math>1</math>:

<math>
\begin{align}
\dfrac{
\left|
\left[ \begin{matrix} 1.699 & {-2.692} & 3.944 \\ \end{matrix} \right]
\left[ \begin{matrix} 0 \\ 0 \\ 1 \\ \end{matrix} \right]
\right|
}
{
\left\|
\left[ \begin{matrix} \log_2{2} & 0 & 0 \\ 0 & \log_2{3} & 0 \\ 0 & 0 & \log_2{5} \\ \end{matrix} \right]
\left[ \begin{matrix} 0 \\ 0 \\ 1 \\ \end{matrix} \right]
\right\|_1
}
\;\; \leq& \;\;
\left\|
\left[ \begin{matrix} 1.699 & {-2.692} & 3.944 \\ \end{matrix} \right]
\left[ \begin{matrix} \frac{1}{\log_2{2}} & 0 & 0 \\ 0 & \frac{1}{\log_2{3}} & 0 \\ 0 & 0 & \frac{1}{\log_2{5}} \\ \end{matrix} \right]
\right\|_\infty

\\[8pt]

\dfrac{
\left|
(1.699)(0) + ({-2.692})(0) + (3.944)(1)
\right|
}
{
\left\|
\left[ \begin{matrix} (\log_2{2})(0) + (0)(0) + (0)(0) \\ (0)(0) + (\log_2{3})(0) + (0)(0) \\ (0)(0) + (0)(0) + (\log_2{5})(1) \\ \end{matrix} \right]
\right\|_1
}
\;\; \leq& \;\;
\left\|
\left[ \begin{matrix} (1.699)\left(\frac{1}{\log_2{2}}\right) + ({-2.692})(0) + (3.944)(0) & (1.699)(0) + ({-2.692})\left(\frac{1}{\log_2{3}}\right) + (3.944)(0) & (1.699)(0) + ({-2.692})(0) + (3.944)\left(\frac{1}{\log_2{5}}\right) \\ \end{matrix} \right]
\right\|_\infty

\\[8pt]

\dfrac{
\left|
0 + 0 + 3.944
\right|
}
{
\left\|
\left[ \begin{matrix} 0 + 0 + 0 \\ 0 + 0 + 0 \\ 0 + 0 + \log_2{5} \\ \end{matrix} \right]
\right\|_1
}
\;\; \leq& \;\;
\left\|
\left[ \begin{matrix} \frac{1.699}{\log_2{2}} + 0 + 0 & 0 + \frac{{-2.692}}{\log_2{3}} + 0 & 0 + 0 + \frac{3.944}{\log_2{5}} \\ \end{matrix} \right]
\right\|_\infty

\\[8pt]

\dfrac{
\left|
3.944
\right|
}
{
\left\|
\left[ \begin{matrix} 0 \\ 0 \\ \log_2{5} \\ \end{matrix} \right]
\right\|_1
}
\;\; \leq& \;\;
\left\|
\left[ \begin{matrix} \frac{1.699}{\log_2{2}} & \frac{{-2.692}}{\log_2{3}} & \frac{3.944}{\log_2{5}} \\ \end{matrix} \right]
\right\|_\infty

\\[8pt]

\dfrac{
3.944
}
{
\sqrt[1]{\strut |0|^1 + |0|^1 + |\log_2{5}|^1}
}
\;\; \leq& \;\;
\max\left(\abs{\frac{1.699}{\log_2{2}}}, \abs{\frac{{-2.692}}{\log_2{3}}}, \abs{\frac{3.944}{\log_2{5}}}\right)

\\[8pt]

\dfrac{
3.944
}
{
\sqrt[1]{\strut 0^1 + 0^1 + (\log_2{5})^1}
}
\;\; \leq& \;\;
\abs{\frac{3.944}{\log_2{5}}}

\\[8pt]

\dfrac{
3.944
}
{
\sqrt[1]{0 + 0 + \log_2{5}}
}
\;\; \leq& \;\;
\frac{3.944}{\log_2{5}}

\\[8pt]

\dfrac{
3.944
}
{
\sqrt[1]{\log_2{5}}
}
\;\; \leq& \;\;
1.699

\\[8pt]

\dfrac{
3.944
}
{
\log_2{5}
}
\;\; \leq& \;\;
1.699

\\[8pt]

1.699
\;\; \leq& \;\;
1.699
\end{align}
</math>

== Example all-interval tuning schemes ==
At the beginning of this Concepts section, you were promised that by the end of it, you'd have a deep understanding of two of the most commonly-used all-interval tuning schemes: minimax-S and minimax-ES. We claimed you'd be able to explain how they work, how they are similar to and different from each other, and also how they compare with the more basic tuning schemes that we've explained previously. Well, we've got great news: you're closer to the end than you may think!

=== Minimax-S ===
For starters, at this point, attaining a complete understanding of the minimax-S tuning scheme is a freebie. That's because it's the example we've been working through this entire article already. Yes, that's right, minimax-S is the scheme which gives us—for any given temperament—the tuning which minimizes the log-prime-simplicity-weight damage to all intervals in its subspace, i.e. where we choose our target-interval set to be literally every possible interval.

It's the tuning that uses this take on the dual norm inequality:

<math>
\dfrac{\abs{𝒓\textbf{i}}}{\norm{L\textbf{i}}_1} \leq \norm{𝒓L^{-1}}_\infty
</math>

If you recall from the fundamentals article when we introduced our naming system for tuning schemes, by leaving off the target-interval set (as we have with the name "minimax-S"), we assume that all intervals are being targeted. And while all our talk about dual norms and normifying complexities in this article certainly might have distracted you—giving you a peek into the Pandora's box of the variety of complexities we could choose to use when weighting absolute error to obtain damage—if you recall, we made it all the way through the fundamentals article without needing any other complexity besides log-product complexity, and that's our default interval complexity, so it shouldn't surprise you in retrospect that it's the interval complexity minimax-S uses, either. If you like, you could imagine that the full name, without the default applied, would be "minimax-lp-S".

As stated earlier, "minimax-S" is just our systematic name for the tuning introduced by Paul Erlich in his paper ''A Middle Path'', where he named it "TOP".<ref group="note">"TOP" is a double acronym. It stands either for "Tempered Octaves, Please" or for "Tenney OPtimal".
 
At the time this tuning scheme was proposed, tempering octaves was a novel prospect. According to Paul, "almost all the types of optimal tuning my colleagues and I had considered until this year had pure octaves" (p173 of https://dkeenan.com/Music/MiddlePath.pdf). One of those examples—predating ''A Middle Path'' by 9 months—was the "What is a linear microtemperament?" section of Dave's article ''Optimising JI guitar designs using linear microtemperaments (or: If it aint Baroque don’t waste your lute fixing it)'' (p2-6 of https://www.dkeenan.com/Music/MicroGuitar.pdf); it mentions tempered octaves, though doesn't give them. Nowadays, however, tempering octaves is ubiquitous, so naming a tuning scheme for the practice is not nearly specific enough.
 
And regarding the second name, since Tenney refers to the Tenney lattice—which is to say, it only refers to the combination of scaling prime factors by the logs of the respective primes, then moving along the rungs only (using the taxicab norm, i.e. norm power <math>1</math>)—then ''any'' tuning scheme which weights absolute error to obtain damage using an interval complexity which uses the log-prime matrix <math>L</math> and norm power <math>1</math>, could be considered "optimal" with respect to "Tenney" no matter whether that's simplicity-weight damage or complexity-weight damage, or whether the target-interval set contains all intervals or not, so this name is not nearly specific enough anymore either.
 
This lack of specificity, on both accounts, is what led to Graham adopting the alternative name of TOP-max (see https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_88292#88470) for it, while what we now know as TE tuning he called TOP-RMS at that time (see https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_88292#88375). Since then, a generalized naming was developed whereby "TOP" is "T1" and "TE" is "T2", but we think this doesn't improve the situation much.
 
The first issue is that since Tenney implies norm power <math>1</math>, Euclideanization of Tenney is already self-contradictory, or at best, Euclideanization involves a wasteful overriding of part of the meaning of Tenney where instead something referring only to log-prime prescaling should be used (such as we do in our naming system, by adding "lp-", though this is the default, so it is rarely shown).
 
But the main problem with this numeric naming scheme is that it's too easy to get confused about what the number refers to. Is it <math>p</math>, <math>q</math> or <math>\text{dual}(q)</math>? In fact, it refers to <math>q</math>, the norm power for the interval complexity. There's an argument that this makes sense because the user wants to know which norm power the interval complexity uses, i.e. the complexity which simplicity-weights the absolute errors in their target-intervals to obtain their damages, and that it doesn't matter what you have to do to achieve this minimization. But there's also an argument that the user of an all-interval tuning scheme tends to know too much about the tool they're using, and would expect to be told the power used for the retuning map norm, <math>\text{dual}(q)</math>, which is what they directly minimize to perform the minimax optimization of all intervals (this is how Flora Canou's temperament utilities library handles things, and we can also see that this is the thinking Graham used when he changed "TOP" to "TOP-max"). Our systematic name disambiguates what we refer to through context, because everything past the mini-<math>p</math>-mean name is part of the description of the damage minimized. So if an "E" for "Euclideanized" appears there, it is simply part of the name of the interval complexity used in weighting the absolute error to obtain damage.
 
The name "TOP-RMS", by the way, is a great example of the inherent danger of conflating optimization powers <math>p</math> and norm powers <math>q</math>. Remember, our systematic name for this scheme is "minimax-ES", which definitively shows it to be a tuning which minimizes the ''max'' (AKA (<math>p\!=\!\infty</math>)-mean), not the ''RMS'' (AKA (<math>p\!=\!2</math>)-mean) damage. What TOP-RMS really involves is not a <math>(p\!=\!2)</math>-''mean'', but a <math>(q\!=\!2)</math>-''norm'' (as the interval complexity function). (One might argue that because minimax-ES tuning is equivalent to primes miniRMS-S tuning, i.e. it is equivalent to minimizing the <math>2</math>-mean over a target-interval set consisting only of the primes, but this principle doesn't hold in general, and the argument is a bit of a stretch.)
 
Furthermore, the use of "Tenney" in the name of this tuning scheme seems to have set the stage for a procession of eponymous tuning scheme namings, tapping Benedetti, Weil, Kees, Wilson, and possibly more names we don't even know about yet; eponyms are no good because they don't convey any meaning unless you're already familiar with the history of the information, and so our naming system has stuck entirely to descriptive naming, with the notable exception of Euclid who shows up in "Euclideanized", but we consider this ancient Greek thinker's name to have [[Wikipedia:Euclid's Elements|transcended eponymity]], at least in the context of Euclidean space, distance, length, and geometry, which is where we, and other microtonal theorists before us, have applied it.
 
We have one final piece to this note regarding the naming of TOP. Historically, people have sometimes distinguished tuning schemes which find the true (unique) optimum tuning from the rest of the set of tunings that are tied for minimax or miniaverage damages (distinguished them, that is, from the tuning schemes that can return any or all of the tied tunings) by prefixing the tuning scheme name with "TIP", coined by [[Keenan Pepper]] to stand for "Tiebreaker in Polytope" (see: https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_20405.html#20412). This was intended to distinguish the "TIPTOP" tuning scheme from the "TOP" tuning scheme, but in fact Paul always intended his TOP tunings to be "TIPTOP", and all tunings given in the current version of ''A Middle Path'' are "TIPTOP" (there was a small error in one tuning out of the 55 in the first version). So this prefix is no longer necessary, now that the community has widely recognized that there is no use for tuning schemes which merely return an arbitrary value from a range of near-optimum tunings when the ability to acquire the true optimum tuning is readily available (the optimum in the limit as <math>\text{dual}(q)→\infty</math>), so we may as well use "TOP" and our equivalent "minimax-S" to refer to the scheme which returns the true optimum tuning. If ever necessary, we may call tunings which tie with the true optimum for basic minimax damage "tunings with the same maximum damage as the minimax tuning" or "same max as minimax" for short; we'd rather avoid dignifying them with formal naming.
 
No wait: one more gripe about the naming of "TOP" variants. Unfortunately, people decided to name a pure-octave version of it "POTOP", which is silly because one of the acronyms of TOP is "tempered octaves please", so that's self-contradictory: "pure octave tempered octaves please." (By the way, "POTT" is just short for "POTIPTOP", so you already know what we think of that.) POTE, which is pure-octave TE, is less bad; that is only interpretable as "pure-octave Tenney-Euclidean". But both of these PO-tunings were unfortunately defined to use the destretched-interval style rather than the held-intervals approach to unchanged octaves. For more information, see {{subpage|tuning fundamentals|uprev|s=Destretching vs. holding|text=destretching vs. holding}} and [[#Destretched-octave minimax-(E)S]].</ref> The equivalence between minimax-S and TOP may not be obvious, even if you are familiar with Paul's paper. Paul explains the concept in a very different way than we have (no mention of dual norms at all), and using different terminology, e.g. while we use the general mathematical terminology "log-product complexity", Paul used the terminology "harmonic distance". This is the terminology of [[James Tenney]], the first person to apply this function to microtonality.

=== Minimax-ES ===
The first thing you may notice about the minimax-ES tuning scheme is that its name appears very similar to that of minimax-S. The only difference is the insertion of that "E" in there. So let's start with that. What does it stand for, and how does it change our scheme?

This 'E' stands for "'''Euclideanized'''", and so it is calling for us to "Euclideanize" whichever interval complexity function we use to simplicity-weight the absolute error to obtain the damage. The full name of this scheme might be read as "the minimized maximum of Euclideanized-simplicity-weight damage (to all intervals)".

How, then, might we Euclideanize a complexity function? Maybe it's an alternative to normifying it? Well, not quite. Euclideanizing a complexity function is something you do ''after'' you already have a complexity function's formula in norm form (by normifying it from quotient form, if necessary). '''Euclideanization''' is quite simple: take what you have already, and change the norm power to <math>2</math>. (Usually the power changes ''from'' <math>1</math>, as it does when we Euclideanize <math>\text{lp-C}()</math>, but that part's not critical.) To be clear, leave the norm prescaler (if any) alone.

So if this is the summation form of <math>\text{lp-C}()</math>, as we found earlier:

<math>
\text{lp-C}\left(\textbf{i}\right) = \sum\limits_{n=1}^d \log_2{p_n}\abs{\mathrm{i}_n}
</math>

And this is how that looks like before we eliminate the no-op <math>1</math>st power and <math>1</math>st root:

<math>
\text{lp-C}\left(\textbf{i}\right) =\color{red} \sqrt[ 1 ]{\strut \color{black} \sum\limits_{n=1}^d \color{red}\left(\color{black}\log_2{p_n}\abs{\mathrm{i}_n} \color{red}\right)^{1} }
</math>

Then this is the summation form of Euclideanized log-product complexity, <math>\text{E-lp-C}()</math>. We keep the norm prescaler <math>\log_2{p_n}</math> as is, and swap out all powers and roots of <math>1</math> for <math>2</math>'s:

<math>
\text{lp-C}\left(\textbf{i}\right) = \sqrt[\color{red} 2 \color{black} ]{\strut \sum\limits_{n=1}^d \left(\log_2{p_n}\abs{\mathrm{i}_n}\right)^{\color{red}2} }
</math>

We could expand that out like this:

<math>
\text{lp-C}\left(\textbf{i}\right) = \sqrt[2]{\strut \left(\log_2{p_1} × \abs{\mathrm{i}_1}\right)^2 + \left(\log_2{p_2} × \abs{\mathrm{i}_2}\right)^2 + ... + \left(\log_2{p_d} × \abs{\mathrm{i}_d}\right)^2 }
</math>

Note that even if a complexity has a quotient form, its Euclideanized version will not. At least, it won't have a meaningfully distinct quotient form, i.e. one that works any way other than by unpacking the rational's prime factors, in which case it would merely be an extraneously complicated formulation of the same ideas which would be better expressed through a summation or norm form.

So why do we call this "Euclideanizing"? It's because a common name for the basic <math>2</math>-norm is the "Euclidean norm". And that, in turn, is because the Euclidean norm is how to find Euclidean distance (as we explained in the earlier section [[#Relationship with distance|Power norms: Relationship with distance]]), or in other words, distance in Euclidean space, which is just another way of saying the basic geometric space we understand as representing the way space works in our everyday reality.

So to be absolutely clear, minimax-ES is the all-interval tuning that uses <math>\text{E-lp-C}()</math> as its interval complexity function. As stated earlier, the original name for this tuning, per its inventor Graham Breed, is "TE", which stands for Tenney-Euclidean, in recognition of the fact that it's a Euclideanized version of the Tenney-style TOP tuning that Paul had innovated.<ref group="note">https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_18357#18357</ref>

And now that we have <math>\text{E-lp-C}()</math> defined as a type of <math>2</math>-norm, and understand that its prescaler (same as it is with <math>\text{lp-C}()</math>) is the logs of the primes, then we can know that if we want to minimize the log-product-simplicity-weight damage to all intervals in our subspace, we need to minimize the <math>2</math>-norm of our retuning map <math>𝒓</math>, where that map has been prescaled by the inverse of the logs of the primes:

<math>
\dfrac{\abs{𝒓\textbf{i}}}{\norm{X\textbf{i}}_{\color{red}2 \color{black}}} \leq \norm{𝒓X^{-1}}_{\color{red}2}
</math>

Note in particular how our dual norm, the one we're minimizing on <math>𝒓</math>, has a power of <math>2</math>. (You didn’t think we’d teach you all that stuff about the dual power continuum only to end the article only using a single norm power, did you?)

Now why in the world would we use minimax-ES when we could use minimax-S? Well, the short answer is: not because it gives better tunings. It gives ''worse'' tunings, actually. The advantage here is that minimax-ES is easier to compute, because there's a special way to solve for it.

Regarding it being a worse tuning, this can be quickly addressed by noting that unlike <math>\text{lp-C}()</math>, <math>\text{E-lp-C}()</math> is ''not'' monotonic over the integers. We'll save a full audit of various complexity functions used as interval complexities until the advanced tuning concepts article, but for now we'll just note that from 5 to 6 to 7 we get complexities of 2.322, dipping down to 1.874, and back up to 2.807 (whereas for <math>\text{lp-C}()</math> we get the same values for 5 and 7 but for 6 we get 2.585, in between them). While there's an argument that 6 is lower complexity than 5 or 7—being that it's lower prime limit than either of them—in general this sort of irregularity leads to strangenesses like <math>\frac{9}{8}</math> being ranked more complex than <math>\frac{10}{9}</math>.<ref group="note">We note that there's nothing inextricably linking Euclideanized complexity functions to all-interval tuning schemes (or minimax tuning schemes, or simplicity-weight damage tuning schemes). For example, TILT minimax-ES, TILT minimax-EC, TILT miniaverage-ES, TILT miniaverage-EC, TILT miniRMS-ES, and TILT miniRMS-EC are all possible tuning schemes. Since it has less psychoacoustic plausibility than <math>\text{lp-C}()</math> and offers no computational benefits in these cases, we see no particular reason to use these schemes, but nothing is stopping you if you really want to.</ref> That said, <math>\text{E-lp-C}()</math> isn't complete garbage; it's close enough to <math>\text{lp-C}()</math> that the computational simplicity may be of interest to some people.

Regarding minimax-ES being easier to compute, well, if you went through the computations article, then you may already have guessed: it's because we have the pseudoinverse to compute it with.

Again, as with minimizing an <math>\infty</math>-norm, we do have specialized techniques for actually computing the answer, which will be discussed in the computations section below. Otherwise, you can just plug it into a library such as ours in Wolfram.

Here is minimax-ES's take on the dual norm inequality. It's almost identical to minimax-S, except the dual powers of <math>1</math> and <math>\infty</math> have been replaced with dual powers of <math>2</math> and <math>2</math>:

<math>
\dfrac{\abs{𝒓\textbf{i}}}{\norm{L\textbf{i}}_2} \leq \norm{𝒓L^{-1}}_2
</math>

And tell you what, we'll throw in a third example all-interval tuning scheme for free. The [[CTE]] tuning scheme, the initialism for "Constrained Tenney-Euclidean", is just held-octave minimax-ES. In other words, by "constrained" it means a specific constraint: namely, on the octave, and that it is held unchanged.

=== Others ===
Most of the tunings that have been named and described on the wiki at the time of this writing are all-interval tunings. As you know, they all are minimax tuning schemes, and they all use simplicity-weight damage. The main trait that distinguishes them, then, is which interval complexity function they use. The relationship between these tunings is much clearer, of course, when using our systematic naming. For examples, "BOP" is just "minimax-p-S", using product complexity (the non-logarithmic version), and "Weil" is just "minimax-lil-S", using log-integer-limit complexity. The other half of these schemes are just Euclideanized versions, e.g. "BE" is just "minimax-E-p-S" and "WE" is just "minimax-E-lil-S". We also see tunings with held-intervals (like CTE, which is held-octave minimax-ES), or destretched intervals (POTE, which is destretched-octave minimax-ES), but anyway. If you're eager to learn more about other all-interval tuning schemes, you can then continue your studies here on our article about {{subpage|alternative complexities|uprev}}.

== A geometric demonstration of dual norms ==
So now we've learned basically everything we need to know to get cracking with all-interval tuning schemes. But maybe you're still a bit bothered. Our logic and equations check out, but you still just don't feel it in your bones. It doesn't really ''feel'' yet like minimizing the retuning magnitude should cap the max damage on all intervals. If you're lacking intuition for this effect (as we certainly did when we were learning it), then perhaps one or the other of the following two demonstrations will solidify things for you in a new and helpful way.

Here we'll give a nice little demonstration of how <math>\abs{\textbf{x}·\textbf{y}} \leq \norm{\textbf{x}}_q × \norm{\textbf{y}}_{\text{dual}(q)}</math> using some geometry.

=== Setup ===
We'll represent vectors as arrows, and we'll use the double-bar notation <math>\norm{\mathbf{x}}</math> (with no subscript) for the ordinary geometrical length of the arrow representing the vector <math>\mathbf{x}</math>. Single bars <math>|x|</math> take the absolute value of a scalar.

When you scale the length of a vector—that is, when you multiply all the vector's entries by the same factor—its norm scales by that factor too, no matter what kind of norm it is. In fact, this is one requirement for a function to be considered a norm. This applies to both norms on the right hand side of the inequality. Also when you scale a vector, its dot product with another vector scales by that factor too. This applies to both x and y in the dot product on the left side of the inequality. So we could simplify our demonstration by considering <math>\textbf{x}</math> and <math>\textbf{y}</math> to be of unit length, because multiplying both sides of the inequality by the same factors leaves the inequality unchanged. This is true even for negative scale factors, thanks to the absolute values being taken on both sides.

However, we think that going all the way to unit vectors would obscure some of what is going on, so we will merely make <math>\textbf{x}</math> and <math>\textbf{y}</math> have the same length. This will be simplification enough.

If we fix the lengths of two vectors, their dot product is a measure of the degree to which they point in the same direction. The dot product is a maximum when they point in ''exactly'' the same direction. So if we're trying to show that <math>\abs{\textbf{x} · \textbf{y}}</math> is less than or equal to something, we only need to check its maximum value, and so not only can <math>\textbf{x}</math> and <math>\textbf{y}</math> be of the same length, they can be the ''same vector'', simplifying the demonstration still further.
Let's use some arbitrary interval <math>\textbf{i}</math> as our <math>\textbf{x}</math> and <math>\textbf{y}</math>, then.

[[File:Dual_norm_geometric_demo_1.png|frameless|220x220px]]

So our inequality now looks like:

<math>
\abs{\textbf{i} · \textbf{i}} \leq \norm{\textbf{i}}_q × \norm{\textbf{i}}_{\text{dual}(q)}
</math>

The dot product of a vector with itself is a simple scalar that corresponds numerically to the area of a square whose sides are the length of its arrow. So we can substitute <math>\norm{\textbf{i}}^2</math> in for <math>\abs{\textbf{i} · \textbf{i}}</math>:

<math>
\norm{\textbf{i}}^2 \leq \norm{\textbf{i}}_q × \norm{\textbf{i}}_{\text{dual}(q)}
</math>

We can visualize this <math>\|\textbf{i}\|^2</math> area as a square lying along the length of <math>\textbf{i}</math>.

[[File:Dual_norm_geometric_demo_2.png|frameless|300x300px]]

The first norm we'll check is the <math>2</math>-norm. It is self-dual, so our inequality looks like:

<math>
\norm{\textbf{i}}^2 \leq \norm{\textbf{i}}_2 × \norm{\textbf{i}}_2
</math>

This says that the square of the length of the vector is less than or equal to its <math>2</math>-norm times its <math>2</math>-norm. But, based on [[Pythagoras of Samos|Pythagoras]]' theorem, the <math>2</math>-norm of a vector is simply its ordinary length, which is why it's also called the Euclidean norm. Euclidean geometry is ordinary everyday geometry. So we have:

<math>
\norm{\textbf{i}}^2 \leq \norm{\textbf{i}}× \norm{\textbf{i}}
</math>

Which simplifies to:

<math>
\norm{\textbf{i}}^2 \leq \norm{\textbf{i}}^2
</math>

This inequality is therefore true, because the included equality is true, being this identity.

<math>
\norm{\textbf{i}}^2 = \norm{\textbf{i}}^2
</math>

[[File:Dual norm geometric demo 3.png|frameless|300x300px]]

It's slightly trickier to demonstrate this inequality for the <math>1</math>-norm and its dual, the <math>\infty</math>-norm, but it's doable. We may begin with our arbitrary interval <math>\textbf{i}</math> and its dot product with itself equal to its length squared.

But we'll need to find suitable substitutes for <math>\norm{\textbf{i}}_1</math> and <math>\norm{\textbf{i}}_\infty</math> in the inequality:

<math>
\norm{\textbf{i}}^2 \leq \norm{\textbf{i}}_1 × \norm{\textbf{i}}_\infty
</math>

To do this, we'll need to look at the entries of <math>\textbf{i}</math>.

Let's make this example as simple as possible to illustrate the concept. Let's give our vector only two entries, which is enough entries that we can't treat it as a scalar, but no more entries than that. It could be a 3-limit vector, with its two entries corresponding to primes 2 and 3. But for our geometrical demonstration we will refer to them as <math>\mathrm{i}_{\text{h}}</math> and <math>\mathrm{i}_{\text{v}}</math> for horizontal and vertical. And we can visualize them as the legs (the two sides at right angles) of a right triangle with <math>\textbf{i}</math> as the hypotenuse. And let's also assume that <math>|\mathrm{i}_{\text{h}}| > |\mathrm{i}_{\text{v}}|</math>, that is, that <math>\mathrm{i}_{\text{h}}</math> is the longer side.

[[File:Dual norm geometric demo 4.png|frameless|300x300px]]

We know that the <math>1</math>-norm of an interval is simply the sum of the absolute values of its entries. So:

<math>
\norm{\textbf{i}}_1 = \abs{\mathrm{i}_{\text{h}}} + \abs{\mathrm{i}_{\text{v}}}
</math>

And we know that the <math>\infty</math>-norm of an interval is simply the maximum of the absolute value of its entries. And since we've assumed for this demonstration that <math>\abs{\mathrm{i}_{\text{h}}} > \abs{\mathrm{i}_{\text{v}}}</math>, we have:

<math>
\norm{\textbf{i}}_\infty = \abs{\mathrm{i}_{\text{h}}}
</math>

After substituting both of those in for our norms, our inequality now looks like:

<math>
\norm{\textbf{i}}^2 \leq \left(\abs{\mathrm{i}_{\text{h}}} + \abs{\mathrm{i}_{\text{v}}}\right) × \left(\abs{\mathrm{i}_{\text{h}}}\right)
</math>

We can distribute the <math>\abs{\mathrm{i}_{\text{h}}}</math>:

<math>
\norm{\textbf{i}}^2 \leq \abs{\mathrm{i}_{\text{h}}}^2 + \abs{\mathrm{i}_{\text{h}} × \mathrm{i}_{\text{v}}}
</math>

=== The key visualization ===
Now for a particularly cool visualization! We can show <math>|\mathrm{i}_{\text{h}}|^2</math> as a square positioned along <math>\mathrm{i}_{\text{h}}</math>, and we can visualize <math>\abs{\mathrm{i}_{\text{h}} × \mathrm{i}_{\text{v}}}</math> as a rectangle positioned along <math>\mathrm{i}_{\text{v}}</math> extending <math>\mathrm{i}_{\text{h}}</math> outwards from the triangle.

[[File:Dual norm geometric demo 5.png|frameless|450x450px]]

Let's set up another similar diagram to compare the previous one with.

[[File:Dual norm geometric demo 6.png|frameless|450x450px]]

By the Pythagorean theorem, the square of a hypotenuse is equal to the sum of the squares of the legs. So in this case:

<math>
\norm{\textbf{i}}^2 = \abs{\mathrm{i}_{\text{h}}}^2 + \abs{\mathrm{i}_{\text{v}}}^2
</math>

We can visualize this in a similar way.

[[File:Dual norm geometric demo 7.png|frameless|525x525px]]

Comparing this with the previous diagram, we can see how the area of the square on the hypotenuse must always be less than the sum of the areas of the square and the rectangle positioned on the legs: because the <math>\abs{\mathrm{i}_{\text{h}} × \mathrm{i}_{\text{v}}}</math> rectangle will by definition always be at least the size of the <math>\abs{\mathrm{i}_{\text{v}}}^2 = \abs{\mathrm{i}_{\text{v}} × \mathrm{i}_{\text{v}}}</math> square that would make their sum equal, because <math>\abs{\mathrm{i}_{\text{h}}} \geq \abs{\mathrm{i}_{\text{v}}}</math>.

=== Edge cases ===
Now let's check some edge cases.

At one extreme, where <math>\abs{\mathrm{i}_{\text{v}}}</math> is as large as possible, that is where <math>\abs{\mathrm{i}_{\text{v}}} = \abs{\mathrm{i}_{\text{h}}}</math>, then the rectangle becomes a square—<math>\abs{\mathrm{i}_{\text{h}} × \mathrm{i}_{\text{v}}}</math> becomes <math>|\mathrm{i}_{\text{v}}|^2</math>—and so the diagram becomes an instance of the Pythagorean theorem, where the right triangle happens to be isosceles.

And so, here the dual norm product is equal to the vector dot product, which satisfies the less-than-or-equal-to inequality.

[[File:Dual norm geometric demo 8.png|frameless|450x450px]]

At the other extreme, <math>\abs{\mathrm{i}_{\text{v}}}</math> is as ''small'' as possible. It certainly could be <math>0</math>, but we're showing it on this diagram as a value very close to <math>0</math>, so the associated rectangle can still be visualized.

[[File:Dual norm geometric demo 9.png|frameless|450x450px]]

If <math>\abs{\mathrm{i}_{\text{v}}} \approx 0</math>, then the rectangle's area is also <math>\approx 0</math>, and so the area of the hypotenuse's square simplifies to <math>\approx \abs{\mathrm{i}_{\text{h}}}^2</math>. At this point, it is approximately equal to the area of the other leg's square <math>\abs{\mathrm{i}_{\text{h}}}^2</math>, so again the inequality holds.

We have thus demonstrated the dual norm inequality for the <math>2</math>-norm with itself and the <math>1</math>-norm with the <math>\infty</math>-norm. Because these are the worst cases, and it works for them, it must also work for all other pairs of dual norms in between these extremes.<ref group="note">For an interesting take on a similar idea, see Mathologer's animation here: https://www.youtube.com/watch?v=Y5wiWCR9Axc&t=1307</ref>

== Unit shapes ==
Here's another handy geometric way to think of the <math>1</math>, <math>2</math>, and <math>\infty</math> -norms: by their '''unit shapes'''. What is meant by "unit shape" is this: given a central point, what is the shape you get from drawing a line through all points that are exactly one unit away from that point, given the present definition of distance.

=== Shape: Circle, Distance: Crow ===
Let's first consider the case of <math>q = 2</math>, because in terms of unit shapes, this is actually the power that gives the most familiar results: a unit circle. As mentioned earlier, <math>q = 2</math> is related to the distance formula. If you remember learning the Pythagorean formula—the formula that gives the length of the hypotenuse of a right triangle—this is that. One side of the triangle represents the coordinate in one dimension, and the other side of the triangle represents the coordinate in the other of the two dimensions. And so the hypotenuse is the shortest distance from the point you started at to the point you arrive at by moving by each side of the triangle. You could imagine a procession of right triangles which all have a hypotenuse of length 1, starting with a degenerate triangle where one side is length 1 and the other is length 0 (so the hypotenuse simply is the side of length 1), immediately transitioning into a really long flat triangle, then to an isosceles one in the middle, and finally a tall skinny one (and ultimately another degenerate triangle). If you locked one of the vertices with an acute angle in place, you'd see the other angle trace out a quarter of a circle. Repeating four copies of this triangle gives the unit circle. This is just a restatement of the definition of a circle, which is the set of all points that are the same distance from a shared center point. Also recall that the formula for a circle is <math>x^2 + y^2 = r</math>, where <math>r</math> here is not the temperament rank but rather the circle's radius. And this is generalizable to higher dimensions; a sphere is the set of points in three dimensions that are equidistant from a center point, and so on.

This is distance "as the crow flies", or in other words, with no constraints, just as straight as possible from point A to point B.

[[File:Unit shapes - circle.png|frameless|450x450px]]

=== Shape: Diamond, Distance: Cab ===
Next let's look at the case of <math>q = 1</math>. This unit shape is a diamond. Again, this means that this is the shape of the set of points that are all 1 away from the center point. Think of it this way. If you go straight up and down or straight right or left, the coordinates whose absolute values sum to 1 will be (1, 0), (-1, 0), (0, 1), and (0, -1). But "distance" works differently in this space based on the <math>1</math>-norm. Think about how far we can go exactly diagonally here, that is, where we go the same distance along both the x and y axes. In physical space, the kind modeled by the <math>2</math>-norm, we could move by <math>\sqrt{\frac{1}{2}} \approx 0.707</math> in each dimension, because those each get squared before being summed, and <math>\sqrt{\frac{1}{2}}^2 = \frac{1}{2}</math>, and <math>\frac{1}{2} + \frac{1}{2} = 1</math>. But in <math>1</math>-normed space, we can only move by 0.5 in the x and y axes before we've moved a total of 1 between the two dimensions. Any amount extra we move in one direction has to come out exactly as much from how far we move in the other dimension. And so we trace out a sharp-cornered diamond.

This is "taxicab distance", as it corresponds to the distance it would take a cab to get from point A to point B, constrained to a square grid of roads.

[[File:Unit shapes - diamond.png|frameless|450x450px]]

=== Shape: Square, Distance: Max ===
Finally, let's look at the case of <math>q = \infty</math>. Remember, this essentially gives us the max of the two coordinates' absolute values. So if we go straight left, right, up, or down, the coordinates (1, 0), (-1, 0), (0, 1), and (0, -1) all have a norm value of 1, just as with the other two norms. But notice what happens when we go exactly diagonally here. We can actually go all the way to the opposite corner, to (1, 1), (1, -1), (-1, 1), and (-1, -1), and the norm values of these points are all still just 1. So the unit shape for <math>q = \infty</math> is a ''square''.

We can call this the "maximum-leg distance". This is an even shorter distance than "as the crow flies". So, to continue the taxicab versus crow analogy we need "'''Max the magician'''" who can teleport through all the dimensions except the longest one.

[[File:Unit shapes - square.png|frameless|450x450px]]

=== How it helps ===
So now we bet you're wondering how we can use these unit shapes to visualize the dual norm relationship. Well, just choose a pair of dual norms, and then pick a direction away from the center. For each of the two chosen norms, calculate the actual distance (yes, the <math>2</math>-norm) of the line segment from the center to its intersection with the unit shape. If you multiply these two distances together, you will always get 1. This is easy to see if the direction chosen is straight right, left, up, or down, since those distances will always be 1, and 1 × 1 = 1. But how about exactly diagonal? In the case of the <math>2</math>-norm, that distance is also exactly 1, so 1 × 1 = 1. In the case of the pair of <math>1</math>-norm and <math>\infty</math>-norm, those distances are <math>\frac{\sqrt{2}}{2}</math> and <math>\sqrt{2}</math>, respectively, and <math>\frac{\sqrt{2}}{2} × \sqrt{2}</math> also equals 1.

So yet again we find <math>q = 2</math> as a curved entity halfway between two blocky entities for <math>q = 1</math> and <math>q = \infty</math>. And if we were to check the unit shapes of other powers between <math>1</math> and <math>2</math>, and <math>2</math> and <math>\infty</math>, we would find a series of shapes, like the diamond bulging outward until it's the shape of a circle, and then the circle spiking outwards until it's the shape of a square.

All this is to say: we can see that pairs of vectors whose distances are measured by dual norms balance each other out.

= Units analysis =
In this section we're going to perform a units analysis of the dual norm inequality, in the vein of {{subpage|units analysis|uprev|text=article 5 of this guide}}:

<math>
\dfrac{\abs{𝒓\textbf{i}}}{\norm{X\textbf{i}}_q} \leq \norm{𝒓X^{-1}}_{\text{dual}(q)}
</math>

Let's break this problem down into three parts:
# The left-hand side's numerator
# The left-hand side's denominator
# The right-hand side

== Left-hand side's numerator ==
Here's what we're working with:

<math>
\abs{𝒓\textbf{i}}
</math>

Our arbitrary interval vector <math>\textbf{i}</math> has units of primes <math>\small 𝗽</math>. And our [[retuning map]] <math>𝒓</math> has units of <math>\mathsf{¢}</math>/<math>\small 𝗽</math>. The absolute value bars have no effect on units. And so we have: (<math>\mathsf{¢}</math>/<math>\small 𝗽</math>)<math>\small ·𝗽</math>, the primes cancel, and the end result is cents <math>\mathsf{¢}</math>. This is unsurprising because we know the retuning map gives us the error for a given interval under a temperament, and so this is just that interval's absolute error here.

== Left-hand side's denominator ==
Here's the denominator of the left-hand side:

<math>
\norm{X\textbf{i}}_q
</math>

We'll be using the default complexity of [[log-product complexity]] here for our complexity prescaler, so let's substitute its log-prime matrix <math>L</math> in for the prescaler. And let's choose a norm power of <math>q=1</math>. (So we're doing the minimax-S tuning scheme here):

<math>
\norm{L\textbf{i}}_1
</math>

So again, the units of our arbitrary interval vector <math>\textbf{i}</math> are the vectorized unit <math>\small 𝗽</math>, for primes. So if we take a units-only view, this is what we have:

<math>
\norm{{\large\mathsf{𝟙}}\mathsf{(C)}·𝗽}_1
</math>

So our annotation has something visible to annotate now, so we could rewrite this as:

<math>
\norm{𝗽\mathsf{(C)}}_1
</math>

Let's suppose this is a 5-limit vector, and so we have:

<math>
\norm{[ \; \mathsf{p_1} \; \mathsf{p_2} \; \mathsf{p_3} \; ⟩\mathsf{(C)}}_1
</math>

Then we could distribute that annotation. Essentially, each of the entries in this vector is a complexity-annotated prime:

<math>
\norm{\left[ \; \mathsf{p_1}\mathsf{(C)} \; \mathsf{p_2}\mathsf{(C)} \; \mathsf{p_3}\mathsf{(C)} \; \right\rangle}_1
</math>

The formula for this <math>1</math>-norm is very simple. We sum the absolute values of each of the entries:

<math>
\abs{\mathsf{p_1}\mathsf{(C)}} + \abs{\mathsf{p_2}\mathsf{(C)}} + \abs{\mathsf{p_3}\mathsf{(C)}}
</math>

(Again, absolute value bars don't affect units, only quantity, so we can pretty much ignore them here too.) So the formula is simple, but what this means for our units analysis is not so simple! We're now summing quantities with different units! Sure, they're all primes, but they're all different primes, corresponding to completely different dimensions in the JI lattice. Your first reaction might be to think that this is only about as offensive as being asked to sum meters, feet, and furlongs; we just need to convert to the same unit and ''then'' we can sum them properly. But no! The idea behind these primes is deeper than that. Meters, feet, and furlongs are all units of ''length'', where ''length'' is their dimension; they're all measurements of the same dimension. Whereas our primes are meant to be interpreted as completely different dimensions. So what we're being asked to do here is actually more like being asked to sum meters, seconds, and kilograms! Can't do!

So what ''do'' we do, then? Our intuition on this has been: drop the part of this that is nonsensical, and keep the part that still makes arguable sense. In other words, our annotation ''does'' appear in each term, so it makes some sense that it's still valid to keep around for the final result. But the primes don't. They get junked. And so our final units for this chunk of the expression are:

<math>
\mathsf{(C)}
</math>
<math>
% \slant{} command approximates italics to allow slanted bold characters, including digits, in MathJax.
\def\slant#1{\style{display:inline-block;margin:-.05em;transform:skew(-14deg)translateX(.03em)}{#1}}
</math>

Now, there is a good argument (which we considered for many months) that since the just tuning map can be broken down into <math>1200×\slant{\mathbf{1}}L</math>, where <math>L</math> is the log-prime matrix doing a units conversion, taking all of our temperament information from its original units of the various prime harmonics, and consolidating it all into one shared unit type, that shared unit being octaves, in which case we think of it as having units of <math>\small\mathsf{oct}</math>/<math>\small 𝗽</math>. If this is taken to be the case, then all units would be in units of octaves before we take the norm, and therefore—being consistent between entries—the units would be preserved in the end by the norm. Further alternatively, we could have it ''both'' ways, i.e. convert each individual prime unit to a shared unit of octaves and ''also'' annotate. Like so:

<math>
\begin{align}
\norm{(\mathsf{oct}/𝗽\mathsf{(C))}(𝗽)}_1 &= \\
\norm{(\mathsf{oct}/\cancel{𝗽}\mathsf{(C))})(\cancel{𝗽})}_1 &= \\
\norm{\mathsf{oct}\mathsf{(C)}}_1 &= \\
\norm{\left[ \; \mathsf{oct}\mathsf{(C)} \; \mathsf{oct}\mathsf{(C)} \; \mathsf{oct}\mathsf{(C)} \; \right\rangle}_1 &= \\
\abs{\mathsf{oct}\mathsf{(C)}} + \abs{\mathsf{oct}\mathsf{(C)}} + \abs{\mathsf{oct}\mathsf{(C)}} &= \\
\mathsf{oct}\mathsf{(C)}
\end{align}
</math>

and thus the units of the complexity could be interpreted as "weighted octaves", in a way we can't interpret results from complexities that use prescalers other than <math>L</math>. And so, we could say that prescaling by the log-prime matrix gives a complexity function a "badge of honor".

But we finally decided against this interpretation, after reflecting on the quotient-based form of the formula, <math>n·d</math>, numerator times denominator. What would the units of those be? They're not logarithmic pitch; they're more like frequency, or frequency multipliers against some base pitch. So they'd both be <math>\small\mathsf{Hz}</math> for units of <math>\small\mathsf{Hz}/\mathsf{Hz}</math> or equivalently dimensionless. Or perhaps it should be interpreted as ''squaring'' the two separate <math>\small\mathsf{Hz}</math> values? Who's to say:

<math>
\log_2{\!(n·d)} → \log_2{\!({\small\mathsf{Hz\!·\!Hz}})} → \log_2{\small\mathsf{Hz^2}}
</math>

On account of these two interpretations of the same value not agreeing on units, we decided that we couldn't accept any interpretation of a norm that preserves actual units in any such way.

There's also the intuition that a complexity is an abstract measurement of an object, and no longer a real physical property of it, so it makes sense for it to be dimensionless.

Alright, but we're actually not ''quite'' done with this chunk yet, because there's another effect to recognize: the influence of the ''power'' of the norm we chose. The <math>1</math>-norm is sometimes also called the "taxicab" norm, and so we say that a complexity computed via a taxicab norm like this may include a 't' in its annotation symbol.<ref group="note">Although we normally use uppercase letters in annotations, we use lowercase 't' for taxicab to avoid possible confusion with "T" for "Tenney", which we don't use, but has been used by other authors to refer to (log-product) simplicity weighting, for which we use "S".</ref> So not only does the <math>\norm{·}_1</math> preserve whatever consistent units and annotations exist among the entries it is called on, it furthermore ''augments'' any existing annotation with this taxicab 't' element. Think of it this way: the annotation doesn't change the fact that a quantity with units is in those units or not; it's more like the annotation is there to give us a little extra background information about the context of these units—where they came from, and where they're going. So our end result is actually:

<math>
\mathsf{(tC)}
</math>

Except for the fact that since the taxicab norm (the <math>1</math>-norm) is our default norm for computing complexity (and simplicity), we don't have to show the 't', so this was fine as it was:

<math>
\mathsf{(C)}
</math>

But this step, of including the norm's effect on the units, will be important in the next step.

== Right-hand side ==
Here's what we have over here:

<math>\norm{𝒓X^{-1}}_{\text{dual}(q)}</math>

Again, the retuning map <math>𝒓</math> has units of <math>\mathsf{¢}</math>/<math>\small 𝗽</math>. But what about our inverse prescaler? Well, this is always supposed to be the inverse of our complexity prescaler. So if our complexity prescaler had units of <math>\small\mathsf{(C)}</math>, i.e. unitless but with a complexity annotation, then this has units of <math>\small\mathsf{(C^{-1})}</math>. Finally, we used <math>1</math> as our norm power for our interval complexity, so we must use the dual norm power here for our retuning magnitude's norm, that being <math>\infty</math>. So, we've got:

<math>
\norm{{\large\mathsf{¢}}/𝗽\mathsf{\left(C^{-1}\right)}}_\infty
</math>

(Note that when we write <math>\mathsf{¢}</math>/<math>\small 𝗽\mathsf{\left(C^{-1}\right)}</math>, we're not saying that the <math>\small\mathsf{\left(C^{-1}\right)}</math> annotation is in the denominator; the annotation is understood to apply to the unit as a whole, so it's sort of floating out to the conceptual side, here.)

Let's evaluate the norm (remember, the <math>\infty</math> is equivalent to taking the max of the absolute values):

<math>
\max\left(\abs{{\large\mathsf{¢}}/\mathsf{p_1}\mathsf{\left(C^{-1}\right)}}, \abs{{\large\mathsf{¢}}/\mathsf{p_2}\mathsf{\left(C^{-1}\right)}}, \abs{{\large\mathsf{¢}}/\mathsf{p_3}\mathsf{\left(C^{-1}\right)}}\right)
</math>

Like with our left-hand side denominator, the primes are disparate here and are just going to have be thrown away (though it's a bit headier here; technically the max function returns exactly one of these options and throws away the others, so individual max calls could be said to preserve units, and yet in the general case sometimes it will be <math>\small\mathsf{p_1}</math>, sometimes <math>\small\mathsf{p_2}</math>, etc. so we can't really say).

But the other two things—the cents, and the annotation—are perfectly consistent across every entry. So those can stay. And our end result is:

<math>{\large\mathsf{¢}}\mathsf{(C^{-1})}</math>

Except, again, that's not quite it, because we still haven't applied the effect from the norm power. The letter we use for the <math>\infty</math>-norm is "M" for "Max", and this one is ''not'' our default, so we do have to show it:

<math>{\large\mathsf{¢}}\mathsf{(MC^{-1})}</math>

And ''now'' we're done here.

== Putting it all back together ==
Reassembling the dual norm inequality with the units we've found, we get

<math>\dfrac{{\large\mathsf{¢}}}{\mathsf{(C)}} \leq {\large\mathsf{¢}}\mathsf{(MC^{-1})}</math>

And remember, <math>\dfrac{1}{\mathsf{(C)}} = \mathsf{(S)}</math>, so we can swap that out on the left, and we've got:

<math>{\large\mathsf{¢}}\mathsf{(S)} \leq {\large\mathsf{¢}}\mathsf{(MC^{-1})}</math>

And there's nowhere really further we can go with this from here. Our end result tells us that when we leverage the dual norm inequality, we say that the simplicity weight damage is less than or equal to the retuning magnitude as measured using the inverse of the complexity prescaler and using the dual power. The annotations on either side do not exactly match, but it's okay because this is an inequality, not an equality. Cool!

= Computation =
To get the most out of this section, we strongly suggest that you have read the previous article in this series, on [[Dave Keenan & Douglas Blumeyer's guide to RTT/Tuning computation|tuning computation]]. The material here builds upon it.

Minimizing the power norm of a (possibly prescaled) retuning map is—for better or worse—a problem remarkably similar to minimizing the power mean of a target-interval damage list. We say this is "for better or worse" because while computationally speaking it means we can reuse much of the processes and computer code we developed already, it also means that the problem space is rife for confusion in human minds.

Conceptually speaking, all-interval tuning schemes are very different from ordinary tuning schemes, i.e. non-all-interval ones, the type where a finite set of target-intervals are specified, as were covered in the fundamentals article of this series. That is to say: leveraging the dual norm inequality to minimize damage via a proxy—the retuning (or prime-error) magnitude—is very different conceptually from simply minimizing the damage to a list of target-intervals. This is the same distinction we noted earlier when we described all-interval tuning schemes as representing an entirely other way of finding minimax tunings.

Computationally speaking, however, all-interval tuning schemes turn out not to be that different after all. The computation process for an all-interval tuning scheme strongly parallels the process for computing an ordinary tuning. You'll see that there's actually not much new to learn here; the methodology is barely more than an alternative version of what we already taught in the computation article, when you swap out the optimization power for the norm power. Even more mercifully, these alternative versions are actually ''simpler'' to compute than the ones we worked through in the computations article; as we hinted at in the introduction of this article; the computational simplicity of all-interval tuning schemes, in fact, is the primary benefit of using them at all.

Some readers of this article up to this point may already have been receiving parallelism alerts from the backs of their minds. We consciously chose to avoid emphasizing these parallels during the concepts section of this article, and sometimes we even suppressed them, using our terminological choices to compartmentalize them. This is because we too struggled a lot with disentangling optimization powers and norm powers (quite different!), and between damage weights and norm prescalers (also quite different!). We've seen some of even the sharpest of xen theorists we know get stuck in the web of conflations here, too. So we figured it was a better choice, pedagogically, to avoid drawing attention to their similarities when introducing all-interval tuning schemes conceptually. But now that we're in the computations section, it's time to wade into the dangerously murky waters of parallelism between these ideas.

== Visualizing the problem ==
Before we dig into the various methods, we're going to review the shared problem between them all, just as we did in the computations article.

=== The analogous objects ===
We can find a nearly one-to-one correspondence between tuning objects in the all-interval case and the ordinary case:

{| class="wikitable"
! colspan="2" | Ordinary tuning schemes
! colspan="2" | All-interval tuning schemes
|-
| <math>\mathrm{T}</math>
| Target-interval list
| <math>\mathrm{T}_{\text{p}} = \mathrm{I}</math>
| Prime proxy target-interval list (an identity matrix)
|-
| <math>\textbf{e}</math>
| Target-interval error list
| <math>\textbf{e}_{\text{p}}</math>
| Prime proxy target-interval error list
|-
| <math>p</math>
| Optimization power
| <math>\text{dual}(q)</math>
| Dual norm power (dual to the <math>\textbf{i}</math>-norm power)
|-
| <math>S</math>
| Target-interval simplicity weight matrix
| <math>X^{-1}</math>
| Inverse prescaler (inverse of the <math>\textbf{i}</math>-norm prescaler)
|}

It's important not to confuse <math>S</math> and <math>X^{-1}</math>. There's only one case where the <math>S</math> for an ordinary tuning scheme would look the same as <math>X^{-1}</math> for an all-interval tuning scheme (applied to the same temperament), and that's if we ({{subpage|tuning fundamentals|uprev|s=Only the primes|text=ill-advisedly}}) used only the primes as our target-intervals.

Another way to look at the difference is to imagine what <math>S</math> would look like for the all-interval tuning scheme. Being the ''target-interval'' simplicity weight matrix, <math>S</math> is a <math>(k, k)</math>-shaped matrix. But remember, for all-interval tunings, <math>k = \infty</math>; that's ''why'' they're called "all-interval" tuning schemes: because they target ''all'' intervals! So we can't see the entirety of <math>S</math> at once, because it's an infinitely-large matrix. But we can take a look at its top-left corner to get a sense for what's inside:

<math>
S =
\text{diag}(𝒔) =

\left[ \begin{matrix}
s_1 & 0 & 0 & \cdots \\
0 & s_2 & 0 & \cdots \\
0 & 0 & s_3 & \cdots \\
\vdots & \vdots & \vdots & \ddots \\
\end{matrix} \right] =

\left[ \begin{matrix}
\dfrac{1}{c_1} & 0 & 0 & \cdots \\
0 & \dfrac{1}{c_2} & 0 & \cdots \\
0 & 0 & \dfrac{1}{c_3} & \cdots \\
\vdots & \vdots & \vdots & \ddots \\
\end{matrix} \right] =

\left[ \begin{matrix}
\dfrac{1}{\norm{X\textbf{t}_1}_{q}} & 0 & 0 & \cdots \\
0 & \dfrac{1}{\norm{X\textbf{t}_2}_{q}} & 0 & \cdots \\
0 & 0 & \dfrac{1}{\norm{X\textbf{t}_3}_{q}} & \cdots \\
\vdots & \vdots & \vdots & \ddots \\
\end{matrix} \right]
</math>

The simplicity weight matrix for an all-interval tuning is a diagonalized version of the list of target-interval simplicities <math>𝒔</math>. Each element of this list <math>s_i</math> is the reciprocal of the corresponding complexity <math>c_i</math> of that target-interval <math>\textbf{t}_i</math>. And each of these interval complexities is a norm-ified complexity <math>\norm{X\textbf{t}_i}_{q}</math>, with complexity prescaler <math>X</math> and norm power <math>q</math>.

You'll note above that the analogous object in the table above to the "target-interval" list <math>\mathrm{T}</math> is essentially "the primes": the '''prime proxy target-interval list'''. We've denoted this as <math>\mathrm{T}_{\text{p}}</math>, using the subscript <math>\text{p}</math> as short for "primes", meaning that this the same concept as before but with only members corresponding to the primes. We can give the <math>\text{p}</math> of <math>\mathrm{T}_{\text{p}}</math> a secondary meaning, as well: short for "proxy", as this matrix no longer truly represents our target-intervals (remember, all-interval tunings minimize damage across ''all'' intervals!), but actually just our ''proxy'' target-intervals, the primes, the things we use as a sort of intermediary targeting mechanism.<ref group="note">A minor note is that "target" here can refer both to the minimization procedure's consideration of an interval as well as our human choice to include the interval in a set, and for all-interval tunings, we have only the former property.</ref>

You'll also notice that <math>\mathrm{T}_{\text{p}}</math> is equivalent to <math>\mathrm{I}</math>, an identity matrix with units of primes <math>\small 𝗽</math>. This is because if you take the vector for each prime interval and assemble them into a matrix, that's just what you get: an identity matrix. Like so, for the 5-limit anyway:

<math>
\begin{array}{c}
\frac{2}{1} \\
\left[ \begin{matrix}
1 \\
0 \\
0 \\
\end{matrix} \right]
\end{array}

\begin{array}{c}
\\
\Huge | \normalsize
\end{array}

\begin{array}{c}
\frac{3}{1} \\
\left[ \begin{matrix}
0 \\
1 \\
0 \\
\end{matrix} \right]
\end{array}

\begin{array}{c}
\\
\Huge | \normalsize
\end{array}

\begin{array}{c}
\frac{5}{1} \\
\left[ \begin{matrix}
0 \\
0 \\
1 \\
\end{matrix} \right]
\end{array}

\begin{array}{c}
\\
=
\end{array}

\begin{array}{c}
\mathrm{I} \\
\left[ \begin{matrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1\\
\end{matrix} \right]
\end{array}
</math>

This identity matrix is, in fact, the crux of the computational simplicity of all-interval tuning schemes: wherever <math>\mathrm{T}</math> figured in computations for ordinary schemes, we can replace it with <math>\mathrm{I}</math>. And since an identity matrix is what it sounds like—it leaves things identical to how they started—we might as well just leave it out entirely, then! It has no effect on our computations. This is a particularly big win, considering that <math>\mathrm{T}</math> was typically the largest matrix we dealt with (tied with <math>W</math>, anyway, but then <math>W</math> is always just a diagonal matrix with shape matching those of our choice of <math>\mathrm{T}</math>).

We've chosen to keep this matrix around anyway, in the form of <math>\mathrm{T}_{\text{p}}</math>, at least when setting up the overall problem, because we find that it aids in grasping the rationale behind the computations. It eliminates a potential for confusion that some other articles on the Xenharmonic wiki contain. Specifically, they speak about "weighted mappings", and they use this term because with all-interval tunings we seem to find <math>M</math> multiplied directly by the inverse prescaler which is the inverse of the <math>\textbf{i}</math>-norm's complexity prescaler (which they consider "weighting"; fine for now, but more on that in a second). But this is pretty confusing; there's no direct way to reason about what a "weighted mapping" would be or mean. However, when we recognize that between the mapping and inverse prescaler matrices we find an invisible identity matrix representing the primes, it all makes sense; we simply have <math>M\mathrm{T}_\text{p}X^{-1} = M\mathrm{I}X^{-1}</math> in place of where for ordinary tuning schemes we had our <math>M\mathrm{T}W</math> formation. And as for the "weighting" vs. "prescaling" issue, we have avoided referring to the effect of multiplying retunings by an inverse prescaler as "weighting"; we've found that it's best to restrict the use of that word "weighting" exclusively to weighting absolute error to obtain ''damage'',<ref group="note">Though we certainly recognize that anyone familiar with the meaning of weighting from statistics will understand these multiplications as acts of weighting, we prefer to restrict our usage to cases where "weight" has the everyday meaning as in "these are weighty matters", i.e. of placing additional importance on things.</ref> and we restrict the use of "damage" to possibly-multiplied/weighted absolute error to (specific finite sets of) ''target-intervals'', while the primes here are only ''proxy'' target-intervals. That is, we restrain ourselves from defining <math>\textbf{d}_{\text{p}}</math> as "proxy damages" or "damages to the primes", since it's more confusion than it's worth. And we find it's valuable to use the specialized term "prescaled" in this case, so that both are distinct from generic multiplication, and where "prescaled" carries the helpful and important information that it occurs ''before'' the norm is taken.

=== Substituting the all-interval objects into the expression ===
To review, for ordinary tuning schemes, we seek to minimize the <math>p</math>-mean of the items in the target-interval damage list <math>\textbf{d}</math>, which is:

<math>
\textbf{d} = \abs{\textbf{e}}\phantom{_{\text{p}}}W\phantom{^{-\,}} = \abs{𝒓}\mathrm{T}\phantom{_{\text{p}}}W\phantom{^{-\,}} = \abs{𝒕 - 𝒋}\mathrm{T}\phantom{_{\text{p}}}W\phantom{^{-\,}} = \abs{𝒈M\mathrm{T}\phantom{_{\text{p}}}W\phantom{^{-\,}} - 𝒋M_{\text{j}}\mathrm{T}\phantom{_{\text{p}}}W\phantom{^{-\,}}}
</math>

Whereas for all-interval tuning schemes, we seek to minimize the <math>\color{red}\text{dual}(q)</math>-norm of the entries in the absolute errors of the primes prescaled by the inverse prescaler <math>\color{red}\abs{𝒓}X^{-1}</math>, which is similar looking:

<math>
\phantom{\textbf{d}} = \abs{\color{red}\textbf{e}_{\text{p}}\color{black}}\color{red}X^{-1}\color{black} = \abs{𝒓}\color{red}\mathrm{T}_{\text{p}}\color{red}X^{-1}\color{black} = \abs{𝒕 - 𝒋}\color{red}\mathrm{T}_{\text{p}}\color{red}X^{-1}\color{black} = \abs{𝒈M_{\text{j}}\color{red}\mathrm{T}_{\text{p}}\color{red}X^{-1}\color{black} - 𝒋M_{\text{j}}\color{red}\mathrm{T}_{\text{p}}\color{red}X^{-1}\color{black}}
</math>

Both <math>\mathrm{T}_{\text{p}}</math> and <math>M_{\text{j}}</math> are identity matrices.

=== Power sum simplification ===
In the computations article, we saw that we can simplify computation of a tuning per an ordinary tuning scheme by substituting a power sum for our power mean, i.e. skipping the steps of division-by-count and taking-the-matching-root-at-the-end, neither of which make a difference when comparing one candidate tuning to another. Well, it turns out we can also simplify computation of a tuning per an all-interval tuning scheme by substituting a power sum for our power norm, i.e. skipping the steps of taking-the-matching-root-at-the-end and taking-the-absolute-values-of-the-entries. The first of these two step skippings is already explained in the same way it is for substituting a power sum for a power mean. The second of these two step skippings is accounted for by the fact that our retunings have already had their absolute values taken by the definition of our process, so there's no need for the power statistic itself to do any absolute-value-taking.

So this may be confusing in light of the earlier section [[#Comparison with power means and sums|Power norms: Comparison with power means and sums]] where we showed that sums have a much closer conceptual kinship with means in general. But so it goes.

== General method ==
The [[Dave Keenan & Douglas Blumeyer's guide to RTT/Tuning computation#General method|general method of optimizing tunings]] may be adapted to finding all-interval tunings. In fact, it is already discussed on the [[Tp_tuning]] page where it says:
"...we can choose a TOP tuning canonically by setting it to the limit as ''p'' tends to 1 of the T''p'' tuning, thereby defining a unique tuning T''p''...". Though please note that this author is using ''p'' to refer to the power of the ''interval complexity norm'', not its dual, the ''retuning magnitude norm'', which is analogous (computation-wise) to the optimization power used for ordinary tunings, which ''we'' call <math>p</math>.

So here's the original pseudocode for ordinary tunings:

: <code>Minimize(Sum(((g.M - j).T.W)^p), byChanging: g);</code>

Note that we omit the absolute value for efficiency reasons when p is even, which includes the p→\infty case.

And here's the revised version, swapping our <code>T</code> out for <code>Tp</code> (that is, our proxy prime target-interval list <math>\mathrm{T}_{\text{p}}</math>), our <code>W</code> out for <code>Inverse(X)</code> (that is, our retuning magnitude norm prescaler <math>X^{-1}</math>), and our <code>p</code> out for <code>dual(q)</code> (that is, our retuning magnitude norm power <math>\text{dual}(q)</math>).

: <code>dual(q) := 1/(1-1/q);</code>
: <code>Minimize(Sum(((g.M - j).Tp.Inverse(X))^dual(q)), byChanging: g);</code>

Note that <code>g.M - j</code> is the same thing as <code>r</code>, our retuning map, and <code>Tp</code> is an identity matrix, so this may be as simple as:

: <code>Minimize(Sum((r.Inverse(X))^dual(q)), byChanging: g);</code>

That assumes you are comfortable with the <code>byChanging:</code> parameter not explicitly appearing in the expression whose value is to be <code>Minimize</code>d.

== Paul's method for nullity-1 minimax-S ==
In Paul's ''A Middle Path'' paper, he gives an alternative means of computing a minimax-S tuning. And this one can be done by hand! However, it only works when the nullity of the temperament <math>n</math> equals 1, or in other words, when only a single comma is made to vanish. To be clear, this is irrespective of the rank of the temperament; this trick works for rank-1, -2, -3, etc. temperaments as long as only a single comma vanishes.

Basically, Paul's trick works by distributing the scaled absolute error equally across the tunings of the primes. (To be clear, this means one equal serving of scaled error for each basis element prime, ''not'' one equal serving of scaled error for each occurrence of a prime in the comma's prime factorization).

One way to understand how Paul's trick works is hinted at by the end result of our example above. When <math>r + 1</math> (proxy) target-intervals can be tied for the same minimum absolute scaled error, and <math>n = 1</math>, and <math>d = r + n</math>, we can see that <math>d</math> i.e. ''all'' of our (proxy) target-intervals can be tied, because <math>r + 1 = (d - n) + 1 = d - 1 + 1 = d</math>.

For comma <math>a/b</math>, this minimum scaled absolute error amount will be <math>1200 \dfrac{\log_2\left(\frac{a}{b}\right)}{\log_2{ab}}</math> ¢/oct.

In meantone's case, that's <math>1200 \dfrac{\log_2\left(\frac{81}{80}\right)}{\log_2{81·80}} \approx 1200 \frac{0.018}{12.662} \approx 1.699</math> ¢/oct.

If we want to know the actual tuning that causes these tied-across-the-board values, though the first step is to get from the scaled absolute error we have already to the ''not''-scaled ''not''-absolute error:
* To un-scale-ify, multiply by the log of the prime.
* To un-absolute-ify, i.e. to recover the sign, just look at the sign of the entries in the vector of the comma. If the entry is positive, the corresponding prime's error is negative; if it is negative, the error is positive.

So for example, the meantone comma is {{vector|-4 4 -1}}, so the errors for prime 2 and 5 will be positive (primes tuned wide) and the error for prime 3 will be negative (tuned narrow). And prime 2's error will be unchanged by the log of the prime step, i.e. it's still 1.699, but prime 5's error will be <math>1.699 × \log_2{5} = 3.945</math>, and indeed when we take the {{map|1201.699 697.564}} generator tuning map and convert it to the tuning map by multiplying by meantone's mapping {{rket|{{map|1 1 0}} {{map|{0 1 4}}}}, we get {{map|1201.699 1899.260 2790.258}} and since a purely-tuned prime 5 is 2786.314{{c}}, that's indeed {{nowrap|2790.258 − 2786.313 {{=}} 3.945{{cent}}}} error.

= Unchanged-octave variants =
== Destretched-octave minimax-(E)S ==
This section is here on account of the historical popularity of tuning schemes called "POTOP", "POTT", and "POTE". The first two are the same; that's just "pure octave TOP" (where "TOP" is "Tenney OPtimal"), and "pure octave TIP-TOP" (where "TIPTOP is nowadays an extraneously complicated name for "TOP"; see the footnote about naming issues in the previous section: [[#Minimax-S|Minimax-S]]. And the latter is just "pure octave TE" (where "TE" is "Tenney Euclidean"). Both of these "PO"-tunings were unfortunately defined to use the dumb destretched-interval approach rather than the smart held-intervals optimization approach to achieving unchanged octaves.

To compute the destretched-octave minimax-S tuning of meantone, we begin with the minimax-S tuning we found above: {{map|1201.699 697.564}}. For a refresher on computing destretched-interval tunings, see {{subpage|tuning fundamentals|uprev|s=Destretching vs. holding}}. Basically we just multiply the thing by the ratio between its tuning of the interval in question and its pure size: {{map|1201.699 697.564}} <math>× \frac{1200}{1201.699}</math> = {{map|1200.000 696.578}}. And for the destretched-octave minimax-ES tuning, we take the minimax-ES tuning from above, {{map|1201.397 697.049}}, and destretch that by <math>\frac{1200}{1201.397}</math> to {{map|1200 696.239}}.

== Held-octave minimax-(E)S ==
Fortunately we do have some community momentum around shifting from the dumb destretched-octave variants of minimax-S and minimax-ES toward the constrained optimization variants, which are known as "CTOP" and "CTE": prefixing "TOP" and "TE", respectively, with a 'C' for "constrained." We feel it's not appropriate to assume both the fact that the constraint is the octave and that the constraint is that it's pure. We prefer our nomenclature here which prefixes the tuning scheme name with "held-octave" (this also works for any other interval, or set of intervals, one might wish to hold unchanged).

Because the computation of held-interval optimizations is more complex, we will instead refer you to {{subpage|tuning computation|uprev|s=with held-intervals|text=this section on held-intervals}}. You should find {{map|1200.000 696.578}} for held-octave minimax-S and {{map|1200.000 697.214}} for held-octave minimax-ES.

= See also =
Thus concludes our deep dive into all-interval tuning schemes! At this point, if you'd like to continue with our series of articles on this topic, don't miss the units analysis article in this intermediate section:
* 5. {{subpage|Units analysis|prev}}: To look at temperament and tuning in a new way, think about the units of the values in frequently used matrices

Or you might be interested in more advanced stuff that largely builds on all-interval tuning schemes:
* 8. {{subpage|Alternative complexities|prev}}: For tuning optimizations with error weighted by something other than log-product complexity
* 9. {{subpage|Tuning in nonstandard domains|prev}}: For temperaments of domains other than prime limits, and in particular nonprime domains

= Footnotes and references =
<references group="note" />

[[Category:Dave Keenan & Douglas Blumeyer's guide to RTT]]
[[Category:Tuning]]

Talk:72edo

2025-10-19T18:51:02Z

Cmloegcmluin: /* mobile format edit breaks links in SVGs */ new section

{{WSArchiveLink}}

== Comma table ==

Has anyone an idea if such tables make sense at all? Also a hint about sensible criteria what to take in what to leave out would be helpful. --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 18:56, 21 December 2020 (UTC)

: I do in fact think that comma tables make sense, but I'd recommend choosing individual commas based on their size, their significance, and their p-limit. For example, commas with p-limits above 31 should not be included at all- the numbers that result from such primes get too complicated too easily. At the same time, commas that are of significant size relative to the EDO's Step size should also be included on the list. Commas like the Nexus comma should also be included on these lists due to their significance- it's not often that two pure prime chains get so close to one another that the difference between them is unnoticeable. I hope I'm being at least somewhat reasonable. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 21:50, 21 December 2020 (UTC)

:: I like the way you guys didn't write out ratios of many many digits. I would go further and only include ratios of 3 or 4 digits or less. --[[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 00:40, 22 December 2020 (UTC)

::: Are you talking about the number of digits in total- as in throughout the whole ratio? Sorry, but most commas that are well known have ratios of at least three digits per single side of the ratio, so that would be unreasonable. --[[User:Aura|Aura]] ([[User talk:Aura|talk]]) 01:12, 22 December 2020 (UTC)

:::: Not the total number of digits, just the number of digits in the numerator. --[[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 20:00, 22 December 2020 (UTC)

::::: In my opinion also a 8-digit (or <abbr title="9-digits would be better, in this case the numerator would start with 1 anyway">9-digit</abbr>) limit would look good in the comma table. See [[41edo#Commas]] for an example. The 10-digit limit was also in use for naming interval pages and redirect lemmas to them. Maybe also this could be changed? BTW: I think for finding a new consensus, we should move this discussion to a better place. --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 20:50, 22 December 2020 (UTC)

:::::: I like what you did with the 41edo table, --[[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 22:17, 24 December 2020 (UTC)

: I heard these tables are generated by scala, aren't they? [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 05:54, 22 December 2020 (UTC)

:: Yes it seems so to me, I also darkly remember reading that somewhere. But I'd like to understand the criteria without looking at scale. Do you see chances to get there? BTW: I have scala installed, I tried to get this list but with no success so far. If anyone knows scala, please help! --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 09:16, 22 December 2020 (UTC)

::: You read that in [[50edo]]: ''This list is not all-inclusive, and is based on the interval table from Scala version 2.2. '' Seems the criteria is simply the presence in the interval table. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 13:29, 22 December 2020 (UTC)

== mobile format edit breaks links in SVGs ==

[https://en.xen.wiki/index.php?title=72edo&curid=982&diff=210275&oldid=210254 This edit] kills all the links in Sagittal images in favor of fitting images to mobile screen size. Shrinking the links in the imagemap would happen automatically if Mediawiki would just use the svg file instead of converting it to a png. Is there a way to make an imagemap shrink to the display width? Someone [https://www.reddit.com/r/mediawiki/comments/hsa10y/imagemap_resizing_of_original_image/ had the same question 5 years ago]; sadly, it remains unanswered. Does anyone have any ideas?

Isoharmonic series

2025-09-29T16:56:07Z

Cmloegcmluin: hopefully clarify and include Mike's feedback from the Talk page

An '''isoharmonic series''' is a variation on the [[harmonic series]], where every pitch has been linearly shifted by a rational number:

<math>f(n) = c + n</math> where <math>c</math> is rational

So for a:b:c:d:... you have b-a = c-b = d-c = etc.

It is synonymous with the term [[OS]], otonal sequence, which is part of a system of [[arithmetic tuning|arithmetic]] and [[harmonotonic tuning|harmonotonic]] tunings. It is also essentially the series form of an [[isoharmonic chord]].

==See also==

* [[Xenharmonic series]], for other variations on the harmonic series

[[Category:Otonality]]
[[Category:Harmonic]]
[[Category:Harmonic series‏‎]]
[[Category:Xenharmonic series]]

OS

2025-09-29T16:51:30Z

Cmloegcmluin: /* Vs. OS */

An '''OS''', or '''otonal sequence''', is a kind of [[Arithmetic tunings|arithmetic]] and [[Harmonotonic tunings|harmonotonic]] tuning.

== Specification ==

Its full specification is (n-)OSp: (n pitches of an) [[otonal]] sequence adding by rational interval p. The "n" is optional. If unspecified, you describe an open-ended sequence.

== Formula ==

The formula for step <math>k</math> of an OSp is:

<math>
f(k) = 1 + k⋅p
</math>

== Tips ==

The OSp could be read as "1 out of every p harmonics of the harmonic series" (starting with harmonic 1). So OS2 would give the odd harmonics: 1, 3, 5, 7...

And OS(1/p) could be read as "every harmonic but over p" (again, always starting with harmonic 1). For example, OS(1/5) gives <math>\frac 55, \frac 65, \frac 75, \frac 85, etc.</math>

For an example combining specifying the numerator and denominator: if you say OS3/4, in other words vary the overtone series to have a step size of 3/4 instead of 1, then you get the tuning <math>1, 1\frac 34, 2\frac 24, 3\frac14</math>, which is equivalent to <math>\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}</math>, or in other words, a class iii [[isoharmonic_chords|isoharmonic]] tuning with starting position of 4. We call this the otonal sequence of 3 over 4, or OS3/4.

== Relationship to other tunings ==

=== Vs. AFS ===

An OS is a specific (rational) type of [[AFS]]; the only difference is that the p for an n-OSp must be rational.

=== As shifted overtone series ===

Both OS and AFS are equivalent to taking an overtone series and adding (or subtracting) a constant amount of frequency (for OS it is rational, for AFS it is probably irrational). By doing this, the step sizes remain equal in frequency, but their relationship in pitch changes. For a detailed explanation of this, see the later section on the [[OS#Derivation|derivation]].

Yet another term for this structure is an [[isoharmonic series]].

=== Vs. OD ===

By specifying n, your OS will be equivalent to some [[OD|OD (otonal division)]]. E.g. 8-OS3/4 = 8-OD7, because 8(3/4) = 6, so you will have traveled 6 away from the root of 1, and reached 7.

=== Vs. US ===

The analogous undertone equivalent of an OS is a [[US]].

== Examples ==

{| class="wikitable"
|+example: 8-OS(3/4)
|-
! quantity
! (0)
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
|-
! frequency (''f'', ratio)
|(4/4)
|7/4
|10/4
|13/4
|16/4
|19/4
|22/4
|25/4
|28/4
|-
! pitch (log₂''f'', octaves)
|(0)
|0.81
|1.32
|1.70
|2.00
|2.25
|2.46
|2.64
|2.81
|-
! length (1/''f'', ratio)
|(1/1)
|4/7
|2/5
|4/13
|1/4
|4/19
|2/11
|4/25
|1/7
|}

== Derivation ==

The tuning OS3/4 is the sequence <math>\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}...</math> and so on. Any OS is equivalent to shifting the overtone series by a constant amount of frequency. In the case of OS3/4, it is a shift by <math>\frac 13</math>. Let's show how.

Begin with the overtone series:

<math>1, 2, 3, 4...</math>

Shift it by <math>\frac 13</math>:

<math>
1\frac 13, 2\frac 13, 3\frac 13, 4\frac 13... \\
</math>

Convert to improper fractions by first expanding the whole number:

<math>
\frac 33 + \frac 13, \frac 63 + \frac 13, \frac 93 + \frac 13, \frac {12}{3} + \frac 13... \\
</math>

...then consolidating numerators:

<math>
\frac 43, \frac 73, \frac{10}{3}, \frac{13}{3}...
</math>

Resize to start at <math>\frac 11</math> by multiplying every term by the reciprocal of the first term, <math>\frac 43</math>, which is <math>\frac 34</math>:

<math>
\frac 43 \cdot \frac 34, \frac 73 \cdot \frac 34, \frac{10}{3} \cdot \frac 34, \frac{13}{3} \cdot \frac 34...
</math>

Cancel out:

<math>
\frac{4}{\cancel{3}} \cdot \frac{\cancel{3}}{4}, \frac{7}{\cancel{3}} \cdot \frac{\cancel{3}}{4}, \frac{10}{\cancel{3}} \cdot \frac{\cancel{3}}{4}, \frac{13}{\cancel{3}} \cdot \frac{\cancel{3}}{4}...
</math>

And we've arrived:

<math>
\frac 44, \frac 74, \frac{10}{4}, \frac{13}{4}...
</math>

So we can see that <math>\frac 13</math> was the right amount to shift by because it is the delta from the starting position <math>1</math> to <math>\frac 43</math>, the latter of which is the reciprocal of the target step size <math>\frac 34</math> and therefore the value that we need the starting position to equal in order to be sent ''back'' to <math>1</math> when we resize all steps from 1 to the target step size by multiplying everything by it.

[[Category:Otonality]]
[[Category:Harmonic]]
[[Category:Harmonic series‏‎]]
[[Category:Xenharmonic series]]

User talk:Sintel

2025-06-16T14:30:56Z

Cmloegcmluin: /* Plucker rewrite */

== Welcome ==

Hi Sintel, welcome to the xenharmonic wiki! I'd like to add your native language to the categories. But I'm not sure if this should be <code>nl</code>? BTW, I heard a lot of your music and sounds, which I found very interesting. Best regards --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 10:31, 16 December 2021 (UTC)

:Thank you! <code>nl</code> is correct, yes. [[User:Sintel|Sintel]] ([[User talk:Sintel|talk]]) 12:53, 16 December 2021 (UTC)

== Some deletion requests ==

Hi Sintel,
I tried to fix (some of) the problems you addressed, so also your original request has been moved (see [[User talk:Moremajorthanmajor/Ed7/3]]). If I did something wrong, please let me know. Best regards --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 14:27, 26 February 2022 (UTC)

: This seems like a good solution to me! [[User:Sintel|Sintel]] ([[User talk:Sintel|talk]]) 14:46, 26 February 2022 (UTC)

== Pathological scales ==

I saw that you deleted some "Pathological" scales from several EDO pages, so I figured maybe you could tell me what that means in the first place, since attempting to search for pages on the subject just directed me to pages where this term appears. (Naively, I would have thought "Pathological" would imply a negative step size or something like that — does this have a different term?) And why delete them anyway?

: Well, nobody knows what it means, since they were added by [[User:Moremajorthanmajor]], who never explained his terms, and has since been banned.
: – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 08:18, 16 April 2025 (UTC)
:: So could the term be repurposed for something more useful (like scales with negative or blown-out steps)? [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 10:04, 16 April 2025 (UTC)
::: Common examples like collapsed and equalized scales are more accurately described as {{w|Degeneracy (mathematics)|degenerate}} than {{w|Pathological (mathematics)|pathological}}. If you look at examples of pathological objects in mathematics from the Wikipedia article, you'll notice that they're not just "exaggerated" versions of common stuff, they really behave strangely and unintuitively. --[[User:Fredg999|Fredg999]] ([[User talk:Fredg999|talk]]) 15:10, 16 April 2025 (UTC)
:::: I thought about collapsed and equalized, but decided those don't really belong to "pathological" — but I am still willing to propose that something like a diatonic scale with a negative value for ''s'' is pathological, since you end up with with D sharper than E in the same octave, and B of one octave sharper than C of the next octave. An example on the other end of the Meantone spectrum would be 5L 2s with a fifth flatter than 7edo, but continuing down the Meantone spectrum instead of switching to Mavila   here the sharp of one note is flatter than the flat of the same note, and thirds which sound major are actually minor and vice versa, and you have to redefine how the circle of fifths works or redefine major and minor. [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 22:17, 16 April 2025 (UTC)

== Plucker rewrite ==

Please try to use musical terminology first, and describe how you're modelling it mathematically second. Remember that you are not Battaglia.
Context: [[Plücker coordinates]]

-- [[User:VectorGraphics|VectorGraphics]] ([[User talk:VectorGraphics|talk]]) 21:31, 11 June 2025 (UTC)

: There's barely any musical application, I can't magically make it more relevant than it is. Do you think the current [[wedgies and multivals]] page does a better job? If so in what way? – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 21:53, 11 June 2025 (UTC)

:: Well, why are we not retiring wedgies entirely, then, if they have no musical relevance? -- [[User:VectorGraphics|VectorGraphics]] ([[User talk:VectorGraphics|talk]]) 02:53, 12 June 2025 (UTC)

::: Could someone explain to me what a wedgie even is? I went to the page for that and never could figure it out. It says "This page or section may be difficult to understand to those unfamiliar with the mathematical concepts involved." No kidding. [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 05:29, 12 June 2025 (UTC)

:::: It appears to be some kind of matrix thing called a "multival" that represents a temperament. Most people on the Discord agree that they are effectively useless, hence I have been working on largely removing them from the wiki as part of [[User:VectorGraphics/Operation_Loosen_Underpants|Operation Loosen Underpants]]. Check [[Plucker coordinates]] for Sintel's explanation. -- [[User:VectorGraphics|VectorGraphics]] ([[User talk:VectorGraphics|talk]]) 05:54, 12 June 2025 (UTC)

::: As you might know I've been the one to remove most references to exterior algebra on the wiki. I want to document the techniques properly though, since they're actually useful for people who want some deeper understanding. If by "retiring wedgies entirely" you mean removing them from the temperament sections, I agree. Also why is this on my user talk? If you have any comments specifically on [[Plücker coordinates]], please state them there for future reference.

::: @Lucius Chiaraviglio 'wedgies' are called 'Plücker coordinates' in the real world, they're a way to assign natural coordinates to matrices, which have some interesting geometrical properties that makes them useful. Too much to explain here, but I'll refer you to [[Plücker coordinates]] and [[Hodge dual]], which has some example computations. Let me know if there's any way I can make the content more accessible. – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 11:44, 12 June 2025 (UTC)

:::: I followed those links — still looks like an awful lot of stuff that requires some mathematics that I don't have. I guess I'll have to follow the lead of what seems to be the current direction on the Xenharmonic Wiki and just make do without. On the bright side, if I understand correctly, if I ever need to describe a temperament that isn't described on here, this removal will make the work a bit easier. [[User:Lucius Chiaraviglio|Lucius Chiaraviglio]] ([[User talk:Lucius Chiaraviglio|talk]]) 23:03, 12 June 2025 (UTC)

::::: It's not really worth your time right now unless you already have some math prerequisites. And yeah, just now they were removed from all temperament data, so you don't have to worry about it. – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 10:16, 13 June 2025 (UTC)

:::::: Since there's so much buzz about multimaps these days and I don't see anyone referencing the excellent resource Dave and I assembled a few years back, here it is, specifically the section where we attempt to give any reason to use them (this part was difficult, haha): https://en.xen.wiki/w/Dave_Keenan_%26_Douglas_Blumeyer%27s_guide_to_EA_for_RTT#Advantages_of_EA We basically just did this in an attempt to prove to the people who used them that we understood them well enough to show that nobody needed them. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 14:30, 16 June 2025 (UTC)

Talk:Normal forms

2025-06-15T20:25:17Z

Cmloegcmluin: /* Title */

{{WSArchiveLink}}

== Smith normal form ==

I noticed "Smith normal form" was added to this page as a name for the form of the normal val list. I've done a couple examples and I'm pretty sure that [https://en.wikipedia.org/wiki/Smith_normal_form Smith normal form] is not equivalent to this process; e.g. meantone's normal val list would be [⟨1 0 -4] ⟨0 1 4]] while I suppose you could say meantone's val list in Smith normal form (taking the first k rows only, as is demonstrated in the penultimate paragraph of the page on [[saturation]]) would be [⟨1 0 4] ⟨0 1 -4]]. If this is an attempt to name this other form for Gene Ward Smith, I think it's not a great idea, because of the preexistence of the linked Smith normal form which was named for Henry John Stephen Smith. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 18:24, 23 June 2021 (UTC)

: I saw it somewhere and thought this was what it referred to. I was really sorry about that. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 07:31, 24 June 2021 (UTC)

:: Oh, no big deal at all. Thanks for updating the page. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 15:17, 24 June 2021 (UTC)

== How best to handle the new canonical form for RTT matrices w/r/t normal form ==

I recently published [[Canonical form|a page on the xen wiki proposing a canonical form for RTT mappings (or comma bases)]]. This new page of mine solves a problem that this page seems to have set out to solve but not finished the job: establishing a form for RTT matrices which ''uniquely identifies them'', for a definition of uniqueness that is appropriate to the RTT domain. And so my page refers to this page in several places, mostly critiquing it.

I note that this page doesn't even present a unified front. It presents both HNF and IRREF as ''potential'' normal forms. But it is not clear whether a given matrix on a temperament page marked as "normal" is its HNF or IRREF (sometimes they are the same, other times different).

But in particular I note that neither the HNF nor IRREF methods discussed here reliably defactor matrices (or in other words, saturate them, or remove contorsion, though those are confusing terms for the issue that my new page sets out to eliminate). This is what I'm referring to when I speak of a definition of uniqueness that is appropriate to the RTT domain. The HNF of a matrix is unique, yes, as is the IRREF. But these definitions of uniqueness treat e.g. {{map|12 19 28}} and {{map|24 38 56}} as distinct, when from a strict RTT perspective the latter is not distinct ''insofar as how it tempers JI'' from the former. At least when enfactoring is found in mappings, it has musical reality, but in comma bases it's meaningless and confusing (I can still play music in 24-ET that sounds different than 12-ET, but I can't pump the comma {{vector|-8 8 -2}} any differently than the comma {{vector|-4 4 -1}})). So: enfactored matrices are pathological. (If you're interested in this issue, my new page discusses it in detail.)

My concern is that the normal forms for RTT matrices which are discussed here have proliferated widely, but I believe that now that a canonical form has been developed (by [[Dave Keenan]], in collaboration with myself, [[Douglas Blumeyer]], inspired in no small part by many insights from [[Gene Ward Smith]]) should be the primary form of RTT matrices used throughout the wiki. Of course I don't plan to do this myself immediately, for numerous reasons. For starters, that'd be a Herculean task. But mostly I wouldn't do something that impactful without soliciting input from the community first.

And as for this page itself: I am not saying that it is harmful or that there is nothing of value on it. Far from it! There's some good thinking here. But I do wonder what people here think about what the best approach should be for recognizing its relationship with the new canonical form. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 18:10, 27 September 2021 (UTC)

: First of all we must make clear what purposes these pages serve. Imo this page should always be kept up to date and used as the canonical reference. I'm not sure about your view of your ''canonical form'' article. To me it acts more like a development note thereof.

: Currently the form in all temp pages is the form defined in the ''normal val list'' section, and they are in canonical form already since they are not enfactored to start with. (More precisely, there is not a start but only the end and the end is correct.) So they are in canonical form and we don't need to do anything about them.

: The important part is the amendment of the definition in the ''normal val list'' section so that we'll convert the form to the canonical form, to ensure the correct end even if we start with an enfactored map. It can be a one-liner amendment, as simple as "defactor it", or rewritten to focus on the new reduction method.

: Finally, the irref form should be removed as it's almost never used. We cover it in another page i.e. ''generator size manipulation'', just like other forms. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 15:07, 28 September 2021 (UTC)

:: My view of my new canonical form article is that it is mostly about defactoring, or in other words, the main element that is missing from the existing normal form and its wiki page. It is only partially about a canonical form for matrices which uses defactoring. I certainly agree with you that the extreme level of detail I included on my page, and in particular the documentation of failed experiments and tangential information, gives it a "development notes" character. Totally fair. :) You may have noticed that I also initiated similar discussion on the page for saturation/contorsion here: https://en.xen.wiki/w/Talk:Saturation, i.e. discussion re: how best to integrate this material about defactoring and canonical form in with what already exists on the wiki.

:: I think that at the end of the day, only two pages are necessary: one page about defactoring, and one page about the matrix form which uses defactoring along with Hermite normalization in order to achieve a unique identifier for a temperament. Due to recent freaky experiences proposing changes to the wiki on Facebook, my confidence about making major contributions directly to existing pages has been shaken. That's why I added all my new material to one new page: perhaps only as a staging ground. If I were to have felt more confident, I would have directly:
# added most of that new material to the page for saturation/contortion, added redirect pages to it for "enfactored", "defactoring", etc., and then in the Talk page drawn people's attention to the parts of the new material explaining the major flaws in the existing terminology for the concept and recommending that the primary name of the page be updated to "defactoring".
# added a small amount of material to this existing normal lists page, added a redirect page to it for "canonical form", and then in the Talk page drawn people's attention to the parts of the new material explaining the preference for "canonical" over "normal" as the term for this form and recommend that the primary name of the page be updated to "canonical form".

:: I believe that there are some people who will never accept proposals to rename concepts like "saturation" and "contorsion", nor would they appreciate the reworking and relegation of most of the existing material on that page to a "mathematical theory" subsection of it as I would prefer. So I think I'll never accomplish the consolidation of the defactoring material into the saturation/contorsion page. However, I do think we can migrate relevant information from my page into the existing normal lists page. In other words, extract all the information from my page about canonical form into the normal lists until I can rename my new page to be focused exclusively on "defactoring" independent of the canonical form it's used for, and then rename the normal lists page to "canonical form". The amount of migrated material to accomplish that may indeed literally be a one-liner, as you suggest. :)

:: You make an excellent point that while people exploring on their own are likely to encounter enfactored temperaments, probably most of those that have managed to get documented here are not enfactored. And also if you think that the IRREF form is almost never used, then there's not that risk of mismatch either. Therefore I agree that we probably ''could'' simply override the existing normal form with the new canonical form, or in other words, conflate the two while keeping the name "normal form", without causing a ton of inaccuracies across the wiki.

:: However, I don't think that's a good idea. While I personally agree that there may be little value in maintaining "normal form" as a term which refers to an RTT matrix which has been normalized but not also defactored, I think it is smarter and safer to allow for the possibility that there are people for whom the existing normal form w/o defactoring does hold some importance which we don't see ourselves at this time. Just for backwards compatibility's sake, I mean. There's also an argument that switching to the new term "canonical form" would be important because it helps signal to the community that the form has changed conceptually.

:: I agree that the IRREF form should be removed from this page. It's not mentioned in my new page for "generator size manipulation" though, so I'm not sure what you mean by that. Perhaps you mean that we should extract it to its own dedicated page? --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 17:42, 28 September 2021 (UTC)

::: So we've basically reached agreement that this page should be updated first. We definitely want to keep the original normal form (for both val list and monzo list), with the new canonical form added probably as a separate section. I'll try taking care of this.

::: Hmmm I remember at one point seeing irref in the page ''generator size manipulation''. If not, let's move it there or somewhere else (like ''Mathematical theory of regular temperaments''). [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 00:50, 29 September 2021 (UTC)

:::: That sounds good to me so far. Re: IRREF, I know it's not anywhere on the page [[generator size manipulation]], because I created that page myself only a week or two ago, and it doesn't really have anything to do with IRREF. Around the same time as I was creating that page, though, I was working on the canonical form page, and I do have a section about IRREF there: [[canonical form#IRREF]] I believe my section contains every bit of information re: IRREF and its relationship to HNF that is presently on the normal list page, and supplements it with visual diagrams that make comparison easy. So I think we could simply remove the IRREF stuff from this page. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 02:21, 29 September 2021 (UTC)

:::: Thanks for making your latest changes to the page. I'm glad to have learned about the Databox template and syntax highlighting! I've gone ahead and used it myself on other pages now.

:::: I think the technique to link out to the canonical form page works fine. Maybe I'll keep it the way it is, i.e. not work to rename it to simply "defactoring" (I did add a redirect page for that, though).

:::: Re: the beep example. I recently fixed that example myself. But I've since noticed it's not quite perfect. Normal form (and canonical form) require the pivots to be positive. And you find the pivots for comma bases by anti-transposing them, i.e. flipping them across the anti-diagonal, between top-right and bottom-left, so that when the HNF tries to put all the zeros in the bottom-left corner, it gravitates them toward where we want them: the higher primes, and commas earlier in the list. Technically, then, the canonical commas for beep should be 25/27 and 35/36, even though with n < d those are negative in pitch and that's not the typical way we write commas. It looks less off-putting when the canonical form is presented as a matrix, i.e. {{map|{{vector|0 -3 2 0}} {{vector|2 2 -1 -1}}}}, so I suggest we write them like that. Or I'm open to other suggestions.

:::: Speaking of lists vs. matrices, I would like to rename the page from "normal lists" to "normal form". I see that this reflects my preference to think of RTT structures as matrices rather than lists of vectors or covectors. Because we are using linear algebra extensively here, I think this is the natural and appropriate way to think of them. What do you think?

:::: Re: the new "Tenney minimal" section. I think it's an interesting idea to present a comma list in a different normal form than HNF (or defactored + HNF = canonical form), namely, some definition of the simplest possible ratios. However, I have several questions.

::::# You state that this is already the case that temperament pages use this form. I have no reason to believe they're not. But I didn't know that was the case. How do you know this?
::::# Do you have a definition for this normal form somewhere? If you don't yet, I would recommend excluding it from this page until it's better developed. I can easily see how the product complexity of ratios can be easily calculated individually, but minimizing the simplicity of multiple ratios may be somewhat subtle.
::::# Why name it "Tenney minimal"? I do not see that this term has wide use on the wiki or Discord already. It seems like an unnecessary eponym. If it is related to the Tenney height of the ratios, wouldn't it be equivalent and simpler to just refer to their product complexity?

:::: That's all for now. Thanks for helping with this. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 18:54, 29 September 2021 (UTC)

:::: I would also like to see across the wiki places where "normal list", "normal comma list", "normal interval list", "normal val list", "normal list basis", etc. standardized to "canonical mapping" or "canonical comma basis", linking here. What do you think? --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 19:54, 29 September 2021 (UTC)

::::: Sounds like a lot of work to be done!

::::: Re: the beep example. We've adopted an additional step for positive generator in Hermite normal form. Analogously, we can add one to flip the monzo if it turns out negative. So [{{monzo| 0 3 -2 0 }}, {{monzo| 2 2 -1 -1 }}] instead.

::::: Re: rename and standardization. I don't think renaming this page and/or "standardize" the terms in other pages is a priority. Until some data from other users are collected, I can't exclude that what we project as standardization may actually be prescriptivism.

::::: Re: Tenney minimal. 1. By examining some comma lists. For septimal meantone, the normal form would be {81/80, 59049/57344}. Yet the comma list shown in the temp page is {81/80, 126/125}. 2. I've worked out a lot of comma bases and haven't encountered a single example where the definition shown in this page leads to ambiguous results. Now I definitely can't assert there's no exceptions, I'm afraid it'd better be there since all the comma bases shown in the temp pages need an explanation. 3. It's attested in the ''Genesisplus'' page. We can call it ''ratio-product simplest form'', but that's longer and still needs explanation. You know, you can also remove ''Benedetti height'' in favor of ''ratio-product'', and remove ''Tenney height'' in favor of ''logarithm of ratio-product''. I just don't think that's how human language works. Benedetti height is one of many types of heights. In the topic of heights, each type is equally distinct despite one of them being simpler in formula, so each type equally deserves a name. From there is derived ''Tenney-minimal'' as one of many possible minimal forms. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 23:55, 29 September 2021 (UTC)

:::::: Re: flipping the sign of a row for a positive generator or comma. I hadn't thought about that last step given there in a while. You're right; the normal form as defined here ensures positive generators. So it would be consistent with that for the commas to also be flipped to be positive using a similar process. That said, as discussed on the canonical form page, Dave and I iterated on and ultimately decided to explicitly reject any stipulations of that sort from our definition. So in that case, canonical form is ''not'' necessarily equivalent to normal form in all defactored cases. This a perfect example of why it was a good decision for us not to conflate the two or override one with the other! Certainly some people may prefer to normalize to positive commas and generators, while Dave and I prefer the simplicity and purity of our canonicalization method. For instance, I have no intention or desire to complicate the functions I implemented in my RTT library for Wolfram Language by incorporating this sort of comma or generator positivity. So, this leads me to think that we should keep the two pages more separate than we originally planned.

:::::: Re: rename and standardization. Good point. Okay. The paint's still wet. Let's give it some time. Because even this seemingly innocuous issue that cropped up for beep is throwing me for a loop, so I am no position to propose such widespread changes yet. Never mind.

:::::: Re: Tenney minimal. Yes, I am aware that "Benedetti height" is xen jargon for "product complexity" and "Tenney height" is xen jargon for "log product complexity". In fact I recently added notes about that to their wiki pages here. I prefer descriptive names over eponymous ones, and established ones over new coinages, whenever possible. I agree that there are many distinct types of height that deserve names and that some are simpler than others, but I'm not sure what your point by that is. Anyway, because "simple" and "complex" are antonyms, we might be able to get away with "product-simplest form" then. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 01:07, 30 September 2021 (UTC)

::::::: Re: positive generator. If you don't want the canonical form normalized to positive generators, many mappings shown in the temp pages aren't in canonical form. I, for one, expect that the canonical form should be normalized to positive generators. After all, we've been using this form for so many years, and it wouldn't be there in the first place if it's undesirable. Anyway, we seem to be in a really awkward position now :).

::::::: Re: Tenney-minimal. My point is that some heights being simpler in formula doesn't imply they don't deserve a name – if you accept ''Wilson height'' and ''Kees height'' you may well accept ''Benedetti height'' and ''Tenney height'', even tho these are much simpler. And since ''Tenney-minimal'' derives direct from ''Tenney height'', and since it's been attested, I think it's a quite standard way to convey the idea of ratio-product simplicity. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 02:53, 30 September 2021 (UTC)

:::::::: Re: positive generator. Here's a bit more color on Dave and I's experience working on canonical form. We thought that the way it's done in Graham Breed's temperament finder was probably the most familiar and well-liked form in the community, i.e. that generators should always be positive and less than half the size of the previous generator (that's the "mingen" form that I ended up documenting somewhere on that new page I made re: generator size manipulation). But we found it problematic to extend this constraint past rank-2, and then heard from Graham that he'd never managed to find a way to do it himself either. Dave and I decided that probably preferences about generator forms would come and go over time like flavors of the month, and that if we wanted our proposal to be taken most seriously, we should laser-focus it on the main element of importance — defactoring — and leave the rest to well-established mathematical precedents like HNF that are already implemented in many code libraries. And it was right about this time I discovered the chroma-positive generator form that some folks on Discord seem to be using now, which to me was perfect evidence of changing preferences for generator form, and therefore the nail in the coffin for involving generator size considerations in the form we advocate for the express and primary purpose of uniquely identifying temperaments AKA canonicalization.

:::::::: I note that it's misleading to include the final steps in the "normal val list" and "normal interval list" in sections titled "Hermite Normal Form" because those are not part of the definition of HNF. That's part of the reason why I forgot those steps were there; the last time I read those lists, many months ago, I suppose I had written them off as tediously worded but probably ultimately accurate descriptions of how to reach HNF. But those final steps deviate them from HNF. So I think we should extract the "For any number q < 1 on this list, replace q with 1/q" and the "Find the Moore–Penrose pseudoinverse..." steps to another section, which we might call "positive comma form" and "positive generator form". This has the other benefit of allowing us to maintain the integration of canonical form into this page, as the canonical form Dave and I defined will be in a state where it extends the HNF as defined here. If you agree this is a reasonable solution, I am happy to implement it (esp. since you did the work for the previous edit).

:::::::: As a result of this change, there would now be several options for normal forms documented on this page: Hermite normal form, positive comma/generator form, Tenney-minimal form, and canonical form. At this point we may even want to tilt the other way, i.e. rather than toward a consolidated effort into a single form, a survey of all relevant forms, and therefore additionally include the chroma-positive form and mingen form, and any others you're aware of. Across the wiki many temperaments have a normal form (or "normal lists" or whatever) documented; it may be to our advantage that most of these places are given generically. Surely whatever's given in most of those places is at least one of these normal forms. I suppose over time people may revise individual pages to be more specific about which one it is, and possibly include more than one.

:::::::: Re: Tenney-minimal. I think Benedetti, Tenney, Wilson, and Kees height are all ideas that deserve names. Maybe we have a different definition of "name" here? I think "product complexity" is a name. It's not a person's name, but I prefer that it's not eponymous. Of these four heights, only Kees does not have a preexisting name in mathematics. Just now I had to look up what Wilson height was again, but if you had just used "sum of prime factors with repetition" which is often abbreviated "sopfr" I would have immediately known what you're talking about. "Tenney" is used in xen jargon as a synonym for "log", and because logarithms are so basic to xen, "Tenney" ends up in an excessive number of things' names. That it is attested in a .scl file that was uploaded to the wiki is not very persuasive to me. The old Scala file archives have tons of junk in them, as I can attest after doing an audit of them for some Sagittal-related reason recently: https://forum.sagittal.org/viewtopic.php?p=1515#p1515 I would be looking for a much stronger precedent than a single mention in a .scl file to justify "Tenney-minimal form" over "product-simplest form".

:::::::: I see that you commented out the information about IRREF. Do you mind if I simply delete it? I just noticed during a search across the wiki for misinformation re: torsion that it is one of the few places that contains it. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 17:20, 30 September 2021 (UTC)

::::::::: Re: positive generator. Good to hear that you're willing to have all the recognized forms included in this page. That seems like the best solution we currently have.

::::::::: Down to some details. 1. first of all, I must inform you that the chroma-positive form isn't a specification of generator in a temperament, but in a MOS. For example, meantone[7] (diatonic) and meantone[12] (m-chromatic) have different chroma-positive generators since they have different chromas. 2. I feel like mentioning again the musician's form – I suggest supplying it with an alternative name: ''equave-reduced generator form'', and accordingly have the definition revised to reflect that. In this take, meantone's generator is ~3/2 and porcupine's generator is ~10/9 (not ~9/5). 3. I can't expect the mappings in the temp pages are to be changed to another form, but we do need to find out a form to be used consistently in the ''POTE generator(s)'' lines.

::::::::: So if you accept my proposal, we'll include Hermite normal form, canonical form, positive generator form, equave-reduced generator form / musician's form, and minimal generator form.

::::::::: Re: Tenney-minimal. I'd rather not see ''sopfr'' since it requires the same amount, if not more, of explanation as ''Wilson height''. As I said earlier, the Benedetti height, whatever the formula is, manifests itself as a type of height. ''Height'' is an immediately recognizable tag which carries not only the meaning of a measure of interval complexity but also implies all the properties. So we acquire the relevant knowledge without checking the formula. On the other hand, ''product complexity'' only demonstrates the formula, that is, it emphasises the computation aspect, but omits the aspects of properties and significance. That to me is penny wise and pound foolish.

::::::::: In this case, however, I admit that ''Tenney-minimal form'' is not the best term since it's equivalent to ''Benedetti-minimal form''. I guess ''minimal'' is still good. I hope to drop ''product'' as both the numerator and the denominator correlates strongly with the product in any good commas. So how about ''minimal comma form'' or ''minimal ratio form''? You coined ''minimal generator form'' and ''mingen'' after all. (I guess I won't be using ''mingen'' if there isn't ''mincom'' as well.) [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 18:53, 30 September 2021 (UTC)

:::::::::: Re: chroma-positive generators. Thank you for correcting me. Yes, that makes complete sense why they're a different matter.

:::::::::: Re: "musician's form": Sorry, but I haven't heard back from Dave yet about what he means when he uses that term. I searched the old tuning list archives and didn't find anything there either.

:::::::::: Re: "equave-reduced generator form", could you define it please? I can't guess at what that means from its name, association w/ the name musician's form, and the examples you give.

:::::::::: Re: the POTE generator lines: those aren't the only places on temperament pages where a specific form of a generator is chosen which may or may not be the one in the mappings shown, are they? Or is there a specific concern you have re: POTE generators that I'm not ascertaining?

:::::::::: Re: your proposal of which forms to include. That list looks fine though we still need to define equave-reduced and musician's forms.

:::::::::: Re: height. You make an interesting point about "height" as a tag, of membership in a suite of similarly useful measurements. In fact, for what it's worth, I had started typing something to that effect myself, anticipating your likely reply, but I couldn't figure out how to say it without muddying the points I was already making. So I'm glad to see that you went ahead and made the point, and I think you put it a lot more clearly than I was going to. :) So while I personally disagree that sopfr requires any explanation, I concur that there is value in a domain such as xenharmonics rebranding established mathematical operations in order to underline important interrelations such as this. I would still prefer that these rebrands be descriptive. Why not "product height", "log product height", and "factor-sum height", I mean?

:::::::::: Re: product. When you say, "I hope to drop ''product'' as both the numerator and the denominator correlates strongly with the product in any good commas," what do you mean exactly? The "product" here as I understand is the product (as in multiplication) of the numerator and denominator. So of course both the numerator and the denominator correlate strongly with the product. And why would this be a reason to drop the word "product"? Sorry if I missed something obvious here.

:::::::::: Re: mingen coinage. For the record that was Dave's coinage, not mine, and I have some qualms about it, but I have been promulgating it, so your point stands. Of your suggestions, I like "minimal ratio form" the best; "minimal comma form" leaves open for me the interpretation that the comma be smallest in ''cents'', which is more akin to the sense in which "minimal generator form" uses "minimal", but I think it is more appropriate to deal with commas as ratios but generators as cents values. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 20:27, 30 September 2021 (UTC)

::::::::::: Re: equave-reduced generator form. This is defined as follows. The mapping is normalized such that each generator is reduced by the formal prime represented by the first column (which is formally regarded as the equave). It's usually the octave but can be others, depending on the subgroup. That's my understanding of ''musician's form'' but I'm aware it's up to Dave to determine what musician's form is.

::::::::::: Re: height and product. Look, I presented you ''equave-reduced generator form'' which I reckon is straightforward enough, yet you still asked me about the definition. I think that's an exemplar case to show that a name isn't anything but a sign, and the relatability between the signifier and the signified can't be expected due to semiotic arbitrariness. Specifically, even if we go with ''product height'', the reader will wonder it's the product of what. The answer "numerator and denominator" isn't emergent, with so many existing measures of intervals. Therefore, I reserve that it requires about the same amount of explanation.

::::::::::: Since "product of what" isn't emergent, it's questionable whether there's any substance in the word ''product''. I'd rather use ''ratio'' or ''comma'' instead in our name of the matrix form because that contains some stuff at least. So I'll go with ''minimal ratio form''. Ok?

::::::::::: Re: POTE generator(s) line. Problem is the generators in this line in all the temp pages don't consistently come from a certain form. My plan is to regulate them to the equave-reduced generator form. Not related but I'll also revise this line to ''Optimal tuning (POTE)''. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 16:59, 1 October 2021 (UTC)

:::::::::::: Re: equave-reduced generator form. You had said "In this take ... porcupine's generator is ~10/9 (not ~9/5)." But both 10/9 and 9/5 are octave-reduced (the equave is the octave here). In what way does your definition select 10/9 over 9/5? That's what I was confused about earlier, but this definition you've now shared has not resolved my confusion. Because this is the definition of a normal form, not a canonical form, I understand that it's not necessary that the results be unique; i.e. it might be the case that both 10/9 and 9/5 were options for a equave-reduced generator (normal) form. But you specifically claimed that 9/5 did not qualify, and I expect you do wish for this form to uniquely identify. So maybe there's something about your definition that I'm missing, that should be included explicitly.

:::::::::::: Re: names. The job of a name is not to unambiguously and completely specify the thing it names; sure, "equave-reduced" wasn't enough to convey the entire definition of that form to me, but I wouldn't expect it to have done that. So a name shouldn't contain ''all'' the information about a thing, but a name does a good job if it contains ''some'' of the most important info about it. For a newcomer, "product complexity" vs "Benedetti height" have the same cost: there's not enough information in either name to understand the thing, so you must look it up to learn it. But in the case of people like me who have already learned the thing but, say, use it only once every couple months or something, "product complexity" would have a clear advantage because of its descriptiveness: it doesn't contain the entire definition, but it's the right amount to jog my memory and let me feel confident about using it without requiring another look up. On the other hand, I have to look up "Benedetti height" every freaking time just to remind myself which person's name got associated with which complexity metric. I agree that the term "product complexity" leaves out the information about what exactly gets its product taken, and so while for me that's plenty to remember that it's the numerator and denominator, other people might not be so certain based on that name. But that's an acceptable cost. No one name could be perfect for everyone. By pushing for "product complexity" I'm pushing for the name I think should work the best for the most people, is all.

:::::::::::: That all said! Yes, I think "minimal ratio form" is good and I would be happy if you used it. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 20:59, 1 October 2021 (UTC)

::::::::::::: Re: equave-reduced generator form. You're aware that 3/1 octave-reduces to 3/2 and not 4/3? Porcupine's generator, in the positive generator form, is 10/9. 10/9 octave-reduces to 10/9 and not 9/5. I think the part we both missed is that the positive generator form should be used as a starting point to derive this form.

::::::::::::: Re: names. You're totally right that including ''product'' in the name is a helpful mnemonic. Good point. However, by using descriptive names its distinctiveness is lost in return. Here's an extension of the "product of what" problem: what if there's another "product complexity"? I just came up with one, which is a product of numerator, denominator, and interval size, in the hope of punishing larger intervals so that 77/1 is measured more complex than 11/7. Not to seriously use it, what I mean to show is how easy for the phrase to be neutralized and turned into an empty signifier.

::::::::::::: Anyway, I'mma work on this page. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 01:04, 2 October 2021 (UTC)

:::::::::::::: Re: equave-reduced generator form. Good stuff, totally got it now. Thanks.

:::::::::::::: Re: names. You've done it again, i.e. in my previous comment I began to anticipate a point I thought you might make, but ended up cutting it. In this case the point was about how "Benedetti" is at least a word you don't hear every day. So I could imagine for people with other learning styles, who can do a one-time match of "Benedetti height" with the concept and retain it strongly, then the distinctiveness of "Benedetti" could be superior. Anyway. Naming is hard. I expect you're familiar with the adage about the two hardest things in computer science being cache invalidation, naming things, and off-by-one errors. Heh. Looking forward to your edits. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 01:20, 2 October 2021 (UTC)

::::::::::::::: I've reworked it and pls give it a review. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 14:39, 2 October 2021 (UTC)

:::::::::::::::: W00t! That's basically exactly how I was going to do it: for the mapping side, build them cumulatively on top of each other: HNF, canonical, positive generator, equave-reduced, mingen. Then a similar progression for the comma basis side. Great work.

:::::::::::::::: I just layered on an edit myself which came out bigger than I was expecting as I added some context that I think is helpful and some further explanations and examples. Let me know what you think. I didn't disagree with anything you had put on the page, so if in my edit I screwed up anything you cared about, please go right ahead and rework/redo. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 00:51, 4 October 2021 (UTC)

::::::::::::::::: I like these additions of contexts and details. Hopefully they will give readers easier time. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 11:06, 4 October 2021 (UTC)

:::::::::::::::::: Thank you. I did a little more work on it this morning based on random things that popped in my head last night. The main thing being that there were some inaccuracies re: canonical form (defactoring removes rows of zeros, so the third step to do so again after HNF was pointless). Also, I thought it was better to separate the issue re: converting intervals between ratio and vector format from discussions of the normal forms, since the format they're written in is independent from the different normal forms. Again, let me know if there's any problem with it and feel free to further adjust yourself. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 18:17, 4 October 2021 (UTC)

== IRREF ==

The footnote criticizes IRREF for making enfactored mappings. But I originally proposed IRREF for the comma list, not the mapping. I agree that IRREF mappings are a horrible idea! From my last edit of this page: "For a monzo list, it has the advantage of limiting the appearance of the N highest primes to only one comma each (where N is the codimension), isolating each prime's effect on the pergen, but has the disadvantage that the commas tend to have high odd limits, and the comma list may have torsion." I think IRREF is valuable as a sort of secondary comma list. It would be nice if x31eq listed the IRREF comma list as well as the usual one, so that one could see all the various restrictions at a glance. It also helps compare two different temperaments to see what they have in common. For example consider (81/80 36/35) and (2048/2025 64/63), two comma lists defined by ascending prime limit, least odd limit and no enfactoring. Their IRREF forms are (81/80 64/63) and (2048/2025 64/63). The IRREFs show that they have 64/63 in common. --[[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 22:55, 13 October 2021 (UTC)

: Thanks for this Kite. I'm sorry that I misrepresented your writing. I didn't do it intentionally; I think I just wasn't being careful to mind possible importance that you may have placed on distinguishing between mappings and comma lists. As far as I was concerned IRREF was no good for either mappings or comma lists, because I was concerned about enfactoring (what you refer to here as "torsion", but as Dave and I determined in our recent research, torsion is the name for a related problem which only pertains to periodicity blocks, not temperaments; I can refer to you our findings again if you like).

: I've simply removed that part of the footnote, because I realized not only did it misrepresent your writing, what it says otherwise is no longer really relevant at all.

: Okay. So I see your point that IRREF can provide a different sort of value here. And at this point I can't see any reason why it wouldn't qualify as a "normal form". So if you want, feel free to add it back to the page. Flora politely commented it out but I was brash and straight up deleted your stuff. So here it is, dug up from the change logs:

:: Another important normalized form for integral matrices is what [[Kite Giedraitis]] has dubbed the IRREF, the '''integral reduced row echelon form'''. It is the [[Wikipedia: Row echelon form|reduced row echelon form]] made integral by multiplying each row of the matrix by the least common multiple of all denominators in that row. It differs from the Hermite normal form in that each pivot is the only nonzero entry in its column. For a monzo list, it has the advantage of limiting the appearance of the ''N'' highest primes to only one comma each (where ''N'' is the codimension), isolating each prime's effect on the [[pergen]], but has the disadvantage that the commas tend to have high odd limits, and the comma list may have torsion. Sometimes the IRREF is identical to the Hermite normal form.

: I had forgotten that the name IRREF was coined by you! Well, I think Dave and I did find one or two other academic sites which came up with the same term. But I believe you did so independently :)

: Right! So, however, if you want to add it back to the page, there is a slight issue. As you can see, at the moment, HNF is up top, because it's part of every normal form presently on the page. So if you want to add IRREF back, you would have to rework the page a bit to accommodate that. I got us into this mess so I'm happy to help with that. Perhaps we could do it together next week. Let me know. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 01:04, 14 October 2021 (UTC)

:: Yes, I would like it back in the article. BTW I thought of another use for IRREFs. Makes it easy to compute meets and joins if you have the IRREF for both temperaments. I don't know if it always makes it easier but it certainly makes it easier some times. --[[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 20:59, 16 October 2021 (UTC)

== Revision to the equave-reduced generator form ==

Actually, the equave reduction need not start with the positive generator form, but may start with the canonical form, which means a step of manipulation is saved. The result of course can be different, but sometimes perferable.

My proposed revision changes the old "equave-reduced generator form" to the "positive equave-reduced generator form", which signifies that its starting point is the positive generator form. And it has the new "equave-reduced generator form" derived from the canonical form.

Sensi is a great example to show how all these will differ from each other:

# Canonical form: {{mapping| 1 6 8 11 | 0 7 9 13 }} with generators ~2, ~9/14
# Positive generator form: {{mapping| 1 6 8 11 | 0 -7 -9 -13 }} with generators ~2, ~14/9
# Equave-reduced generator form: {{mapping| 1 -1 -1 -2 | 0 7 9 13 }} with generators ~2, ~9/7
# Positive equave-reduced generator form: {{mapping| 1 6 8 11 | 0 -7 -9 -13 }} with generators ~2, ~14/9
# Minimal generator form: {{mapping| 1 -1 -1 -2 | 0 7 9 13 }} with generators ~2, ~9/7

My expectation is that the positive equave-reduced generator form should be left largely obsolete since musicians will work with the new equave-reduced generator form. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 08:00, 16 January 2023 (UTC)

: I like it. Well-motivated and well-executed. Yes, I agree that positive-ization and equave-reduction should not have been bound together as they were, and that canonical form is the baseline form. They are independent interests. And I also agree with you that their combination will likely be less popular than simple equave-reduction (or simple positive-ization). I expect after your revision, the combo form "Positive equave-reduced generator form" will be left with a very brief section, saying only that you equave-reduce the generator from the positive gen form, and that's all. Again, good thinking and good work; thanks for looking into this. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 16:51, 16 January 2023 (UTC)

:: Great! I'll work on it soon. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 18:43, 16 January 2023 (UTC)

== Title ==
The article discusses normal forms mostly, so I feel like it should be titled 'normal forms' or something like that.
I don't see many places calling these 'normal lists' at all, though for a list of commas I suppose this makes more sense.

– [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 12:41, 15 June 2025 (UTC)

: Makes sense. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 13:09, 15 June 2025 (UTC)

: yes I’ve always wished for that myself. —[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 20:25, 15 June 2025 (UTC)

== I propose an even better form/rule ==
A combo of the positive generator form and equave-reduced generator form (there should technically be a hyphen between ''positive''/''equave-reduced'' and ''generator'', right?): it makes sure the generators are positive not by flipping but by octave-reducing. Iow it's the canonical form if the canonical gens are positive, but octave-reduced form if the canonical gens are negative. This also makes sure the first nonzero entry of each row in the mapping is always positive; for rank-2 temps with primes 2 and 3 this means prime 3 is always a positive number of gensteps.
* For meantone this gives {{mapping| 1 0 -4 -13 | 0 1 4 10 }} with gens ~2, ~3, same as canonical and positive generator forms.
* For sensi this gives {{mapping| 1 -1 -1 -2 | 0 7 9 13 }} with gens ~2, ~9/7, same as equave-reduced and minimal generator forms.
* For würschmidt this gives {{mapping| 1 -1 2 | 0 8 1 }} with gens ~2, ~5/4(!), same as equave-reduced and minimal generator forms.

But that is still not good enough, for it gives {{mapping| 1 -1 -2 | 0 3 5 }} for porcupine, with gens ~2, ~9/5, yet we're generally more familiar with ~10/9 as the gen. So here's an additional rule based on the omega extension of ploidacot: if the temperament is omega-''n''-cot i.e. splits ~4/3 into three or more steps, we should stick to the positive generator form, so that porcupine remains {{mapping| 1 2 3 | 0 -3 -5 }}. I'm still contemplating how this should be extended to no-2 and/or no-3 temps.

I think it's very humanized and fits well for the wiki, tho perhaps it's more like a rule for choosing forms than an individual form.

[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 13:12, 15 June 2025 (UTC)

User talk:Cmloegcmluin/APS

2025-05-18T19:01:42Z

Cmloegcmluin: /* Merge */

== Periodicity and infiniteness? ==
By "if not provided, the sequence is open-ended" it seems to suggest that systems in this category are finite and aperiodic when ''n'' is provided. By "''n''-EPD-''x'' = ''n''-APS-(''x''/''n'')" it seems to suggest that this tuning system is infinite and periodic just as equal multiplications. Which one is intended? [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 11:17, 4 March 2023 (UTC)

: Great catch. The intention was that they are finite and aperiodic when ''n'' is provided. So they are only equivalent to a single period of a EPD. I have corrected accordingly (and the similar statements on the ALS and AFS pages). Thank you. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 18:28, 28 March 2023 (UTC)

== Vagueness in the spec ==

> "The pitch of the ''k''-th step of an APS-''p'' is quite simply ''k''⋅''p''."

That implies ''p'' must be a ''pitch relation'' (i.e. log-frequency relation), such as cents or octaves.

> ''n''-EPD-''p'' = APS(''p''/''n'')

That is also true only if ''p'' is a pitch relation. I've tried to clarify this matter in the lastest changes.

However, in "APS⁴√2" and "APS1.189" the numbers are frequency ratios. Using pitches it should be APS(300 ¢), APS(1/4 oct), or APS(1\4). I think there are two ways to specify the tuning, one by frequency ratio, the other by pitch relations. It can work without confusion, cuz obviously frequency ratio is dimensionless, whereas pitch relations never go without a unit.

[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 10:45, 14 October 2023 (UTC)

: Oh dang, good catch! I clearly wasn't paying careful enough attention to what I was doing when I wrote this page back then. I suppose if even I found myself specifying APS by frequency ratios, I should imagine that others in the wild will do this too. And I see your point that because of the presence or absence of units, there is no ambiguity caused by allowing both. But yes, we'd need to acknowledge this in the page. I support it. Feel free to make the changes yourself, or if you'd rather I take care of it, I'm happy to. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 17:20, 17 October 2023 (UTC)

:: I'll make the changes myself, then. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 07:40, 18 October 2023 (UTC)

== How to use the example table ==

I'm much confused by the table in the "examples" section. The first row is labeled "frequency (''f'')", whose entries are 1, 1.19, 1.41, 1.68, 2, and that's perfectly clear to me. Now the second row is labeled "pitch (log2''f'')", and the entries, in terms of contents, are identical to those in the first row. So I wondered, maybe these are the ''f'' which should be plugged to log2''f''? That gives you the correct pitch relations. But interpreting them this way leads to inconsistent results in the third row. The entries in this row are 1, 0.84, 0.71, 0.59, 0.5, which are clearly lengths, and not meant to be the ''f'' as is plugged to 1/''f''.

I suggest the following changes:
* frequency → frequency ratio
* pitch → size
* length → length ratio
And then the "size" row can show the pitch relations, that is, 0/4 oct, 1/4 oct, 2/4 oct, 3/4 oct, 4/4 oct.

[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 11:00, 14 October 2023 (UTC)

: Yikes. I screwed up the middle row of this table. If I had done it correctly and consistently with how I did the tables on all the other arithmetic tuning pages (OD, EFD, OS, AFS, EPD, AS, UD, ELD, US, ALS), that row would simply be 0/4, 1/4, 2/4, 3/4, 4/4. So I've made that change already. Thanks for catching that.
: I prefer the simplicity of frequency/pitch/length which are fundamentals of this naming system, i.e. I'd rather not introduce the term "size" here. But the point you've raised about how pitch requires units while frequency and length do not (being ratios of whichever unit to the same unit, canceling out) is very compelling. So I suggest something like this:
:* frequency → frequency (ratio)
:* pitch → pitch (octaves)
:* length → length (ratio)
: One advantage of this is that we consolidate the units into the row headers, rather than putting 'oct' into each cell of that row, which would clutter it. What do you think? If you agree, I volunteer to make the changes across all the tables, since that will be pretty tedious, and it's a problem I created in the first place. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 17:20, 17 October 2023 (UTC)
:: I agree with the proposed changes. [[User:Fredg999|Fredg999]] ([[User talk:Fredg999|talk]]) 21:16, 17 October 2023 (UTC)

::: Sounds great. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 07:40, 18 October 2023 (UTC)

:::: Wunderbar. It's done. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 20:37, 19 October 2023 (UTC)

== Merge ==

This page can be merged with [[equal-step tuning]]. – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 17:05, 18 May 2025 (UTC)

: I've made the same point on several discussion pages now. These pages are part of a family of pages in a system of arithmetic tunings. Yes, they are all equal step. I am fine with merging if it's done like this: https://en.xen.wiki/index.php?title=Talk:EPD&action=edit&section=1 --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 19:01, 18 May 2025 (UTC)

User talk:Cmloegcmluin/AS

2025-05-18T19:00:08Z

Cmloegcmluin:

== Merge ==

This page can be merged with [[equal-step tuning]]. – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 17:05, 18 May 2025 (UTC)

: Could be, but it's part of a family of pages with identical structures (see https://en.xen.wiki/w/Arithmetic_tuning#Types) --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 18:54, 18 May 2025 (UTC)

: Ah, though if the merge is handled similarly to how it's discussed here, I could be okay with it: https://en.xen.wiki/index.php?title=Talk:EPD&action=edit&section=1 --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 19:00, 18 May 2025 (UTC)

Talk:EDO

2025-05-18T18:58:14Z

Cmloegcmluin: /* Daniel Anthony Stearns */

{{WSArchiveLink}}
{{High priority}}

== Daniel Anthony Stearns ==

He is credited with having coined the term but there is no reference for this? Is it true at all? --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 20:39, 22 November 2020 (UTC)

: There are two "sources" now, both of them posts on the tuning list with zero replies. I find they make very little sense so it's hard to consider these real sources. – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 09:15, 18 May 2025 (UTC)

:: Not sure how you will find better in-writing documentation, but if you ask anyone who was on the tuning list at that time, they'll tell you that it was Daniel who coined it. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 18:58, 18 May 2025 (UTC)

User talk:Cmloegcmluin/AS

2025-05-18T18:54:26Z

Cmloegcmluin: /* Merge */

16edo

2025-05-10T06:57:41Z

Cmloegcmluin: /* Theory */ remove "Logarithmic divisions of intervals" table for same reasons it was removed from 15edo: https://en.xen.wiki/w/Talk:15edo#Table%20of%20divisors

{{interwiki
| de = 16-EDO
| en = 16edo
| es = 16 EDO
| ja = 16平均律
}}
{{Infobox ET}}
{{ED intro}}

16edo's step size is sometimes called an '''eka''', a term proposed by [[Luca Attanasio]], from Sanskrit [[wikt:%E0%A4%8F%E0%A4%95#Sanskrit|एक]] (''éka'', "one", "unit"),<ref>[http://www.armodue.com/risorse.htm Armodue: le risorse di un nuovo sistema musicale]</ref> when used as an [[interval size unit]], especially in the context of [[Armodue]] theory.

== Theory ==
In general, 16edo tends to better approximate the differences between odd [[harmonic]]s than odd harmonics themselves, though there are exceptions: it has a [[7/4|7/1]] which is only six cents sharp, and a [[5/4|5/1]] which is only eleven cents flat. Most low harmonics are tuned very flat, but some such as [[21/16|21]]:[[11/8|22]]:[[23/16|23]]:[[3/2|24]]:[[25/16|25]]:[[13/8|26]] are well in tune with each other. Having a flat tendency, 16et is best tuned with [[stretched octave]]s, which improve the accuracy of wide-voiced JI chords and [[rooted]] harmonics especially on inharmonic timbres such as bells and gamelans, with [[25edt]], [[41ed6]], and [[57ed12]] being good options.

Four steps of 16edo gives the 300{{c}} minor third interval shared by [[12edo]] (and other multiples of [[4edo]]), and thus the familiar [[diminished seventh chord]] may be built on any scale step with 4 unique tetrads up to [[octave equivalence]].

=== Odd harmonics ===
{{Harmonics in equal|16}}

=== Subsets and supersets ===
Since 16 factors into primes as 24, 16edo has subset edos {{EDOs| 2, 4, and 8 }}.

== Intervals ==
16edo can be notated with conventional notation, including the staff, note names, relative notation, etc. in two ways.

The first and most common defines sharp/flat, major/minor and aug/dim in terms of the native antidiatonic scale, such that sharp is higher pitched than flat, and major/aug is wider than minor/dim, as would be expected. Because it does not follow diatonic conventions, conventional interval arithmetic no longer works, e.g. {{nowrap|M2 + M2}} isn't M3, and {{nowrap|D + M2}} isn't E. Because antidiatonic is the sister scale to diatonic, you can solve this by swapping major and minor in interval arithmetic rules (see [[16edo#Interval_arithmetic_examples]]). Note that the notes that form chords are different from in diatonic: for example, a major chord, {{dash|P1, M3, P5|med}}, is approximately 4:5:6 as would be expected, but is notated C-E#-G on C. (But see below in "Chord Names".)

Alternatively, one can essentially pretend 16edo's antidiatonic scale is a normal diatonic, meaning that sharp is lower in pitch than flat (since the "S" step is larger than the "L" step) and major/aug is narrower than minor/dim. The primary purpose of doing this is to allow music notated in 12edo or another diatonic system to be directly translated to 16edo "on the fly" (or to allow support for 16edo in tools that only allow chain-of-fifths notation), and it carries over the way interval arithmetic works from diatonic notation, at the cost of notating the sizes of intervals and the shapes of chords incorrectly: that is, a major chord, P1-M3-P5, is notated C-E-G on C, but is no longer ~4:5:6 (since the third is closer to a minor third).

For the sake of clarity, the first notation is commonly called "melodic notation", and the second is called "harmonic notation", but this is a bit of a misnomer as both preserve different features of the notation of harmony.
{| class="wikitable"
|+
!
!P1-M3-P5 ~ 4:5:6
!P1-M3-P5 = C-E-G on C
|-
!Diatonic notation
|NO
|YES
|-
!Antidiatonic notation
|YES
|NO
|}
Alternatively, one can use Armodue nine-nominal notation; see [[Armodue theory]]

{| class="wikitable center-all"
|-
! rowspan="2" | Degree
! rowspan="2" | [[Cent]]s
! rowspan="2" | Approximate ratios*
! colspan="6" | Names
|-
! colspan="2" | Antidiatonic
! colspan="2" | Diatonic
! Just
! Simplified
|-
| 0
| 0
| 1/1
| unison
| D
| unison
| D
| unison
| unison
|-
| 1
| 75
| 28/27, 27/26
| aug 1, dim 2nd
| D♯, E♭
| dim 1, aug 2nd
| D♭, E♯
| subminor 2nd
| min 2nd
|-
| 2
| 150
| 35/32
| minor 2nd
| E
| major 2nd
| E
| neutral 2nd
| maj 2nd
|-
| 3
| 225
| 8/7
| major 2nd
| E♯
| minor 2nd
| E♭
| supermajor 2nd, septimal whole-tone
| perf 2nd
|-
| 4
| 300
| 19/16, 32/27
| minor 3rd
| F♭
| major 3rd
| F♯
| minor 3rd
| min 3rd
|-
| 5
| 375
| 5/4, 16/13, 26/21
| major 3rd
| F
| minor 3rd
| F
| major 3rd
| maj 3rd
|-
| 6
| 450
| 13/10, 35/27
| aug 3rd, dim 4th
| F♯, G♭
| dim 3rd, aug 4th
| F♭, G♯
| sub-4th, supermajor 3rd
| min 4th
|-
| 7
| 525
| 19/14, 27/20, 35/26, 256/189
| perfect 4th
| G
| perfect 4th
| G
| wide 4th
| maj 4th
|-
| 8
| 600
| 7/5, 10/7
| aug 4th, dim 5th
| G♯, A♭
| dim 4th, aug 5th
| G♭, A♯
| tritone
| aug 4th, dim 5th
|-
| 9
| 675
| 28/19, 40/27, 52/35, 189/128
| perfect 5th
| A
| perfect 5th
| A
| narrow 5th
| min 5th
|-
| 10
| 750
| 20/13, 54/35
| aug 5th, dim 6th
| A♯, B♭
| dim 5th, aug 6th
| A♭, B♯
| super-5th, subminor 6th
| maj 5th
|-
| 11
| 825
| 8/5, 13/8, 21/13
| minor 6th
| B
| major 6th
| B
| minor 6th
| min 6th
|-
| 12
| 900
| 27/16, 32/19
| major 6th
| B♯
| minor 6th
| B♭
| major 6th
| maj 6th
|-
| 13
| 975
| 7/4
| minor 7th
| C♭
| major 7th
| C♯
| subminor 7th, septimal minor 7th
| perf 7th
|-
| 14
| 1050
| 64/35
| major 7th
| C
| minor 7th
| C
| neutral 7th
| min 7th
|-
| 15
| 1125
| 27/14, 52/27
| aug 7th, dim 8ve
| C♯, D♭
| dim 7th, aug 8ve
| C♭, D♯
| supermajor 7th
| maj 7th
|-
| 16
| 1200
| 2/1
| 8ve
| D
| 8ve
| D
| octave
| octave
|}
<nowiki />* Based on treating 16edo as a 2.5.7.13.19.27 subgroup temperament; other approaches are possible.

== Notation ==
16edo notation can be easy utilizing [[Goldsmith's Circle]] of keys, nominals, and respective notation{{clarify}}. The nominals for a 6 line staff can be switched for [[Erv Wilson]]'s Beta and Epsilon additions to A–G. The Armodue model uses a 4-line staff for 16edo.

Mos scales like Mavila[7] (or "inverse/anti-diatonic" which reverses step sizes of diatonic from LLsLLLs to ssLsssL in the heptatonic variation) can work as an alternative to the traditional diatonic scale, while maintaining conventional A–G ♯/♭ notation as described above. Alternatively, one can utilize the Mavila[9] mos, for a sort of "hyper-diatonic" scale of 7 large steps and 2 small steps. [[Armodue theory|Armodue notation]] of 16edo "Mavila[9] Staff" does just this, and places the arrangement (222122221) on nine white "natural" keys of the 16edo keyboard. If the 9-note (enneatonic) mos is adopted as a notational basis for 16edo, then we need an entirely different set of interval classes than any of the heptatonic classes described above; perhaps it even makes sense to refer to the octave ([[2/1]]) as the "[[decave]]". This is identical to the KISS notation for this scale when using numbers.

{| class="wikitable center-all"
|-
! Degree
! Cents
! colspan="2" | Mavila[9] notation
|-
| 0
| 0
| unison
| 1
|-
| 1
| 75
| aug unison, minor 2nd
| 1♯, 2♭
|-
| 2
| 150
| major 2nd
| 2
|-
| 3
| 225
| aug 2nd, minor 3rd
| 2♯, 3♭
|-
| 4
| 300
| major 3rd, dim 4th
| 3, 4𝄫
|-
| 5
| 375
| minor 4th
| 4♭
|-
| 6
| 450
| major 4th, dim 5th
| 4, 5♭
|-
| 7
| 525
| aug 4th, minor 5th
| 4♯, 5
|-
| 8
| 600
| aug 5th, dim 6th
| 5♯, 6♭
|-
| 9
| 675
| perfect 6th, dim 7th
| 6, 7𝄫
|-
| 10
| 750
| aug 6th, minor 7th
| 6♯, 7♭
|-
| 11
| 825
| major 7th
| 7
|-
| 12
| 900
| aug 7th, minor 8th
| 7♯, 8♭
|-
| 13
| 975
| major 8th, dim 9th
| 8, 9𝄫
|-
| 14
| 1050
| minor 9th
| 9
|-
| 15
| 1125
| major 9th, dim 10ve
| 9♯, 1♭
|-
| 16
| 1200
| 10ve (Decave)
| 1
|}

=== Sagittal notation ===
This notation uses the same sagittal sequence as [[21edo #Sagittal notation|21edo]].

<imagemap>
File:16-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 471 0 631 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 471 106 [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation | limma-fraction notation]]
default [[File:16-EDO_Sagittal.svg]]
</imagemap>

=== Armodue notation (4-line staff) ===
[http://www.armodue.com/ricerche.htm Armodue]: Pierpaolo Beretta's website for his Armodue theory for 16edo (esadekaphonic), including compositions.

For resources on the Armodue theory, see the [[Armodue]] on this wiki

== Chord names ==
16edo chords can be named using ups and downs. Using diatonic interval names, chord names bear little relationship to the sound: a minor chord (spelled {{dash|A, C, E|med}}) sounds like [[4:5:6]], the classical major triad, and a major chord (spelled {{dash|C, E, G|med}}) sounds like [[10:12:15]], a classical minor triad! Instead, using antidiatonic names, the chord names will match the sound—but finding the name from the spelling follows the rules of antidiatonic rather than diatonic interval arithmetic.

{| class="wikitable center-all"
|-
! rowspan="2" | Chord
! rowspan="2" | JI ratios
! colspan="6" | Name
|-
! colspan="3" | Diatonic
! colspan="3" | Antidiatonic
|-
| {{dash|0, 5, 9|med}}
| 4:5:6
| D F A
| Dm
| D minor
| D F A
| D
| D major
|-
| {{dash|0, 4, 9|med}}
| 10:12:15
| D F♯ A
| D
| D major
| D F♭ A
| Dm
| D minor
|-
| {{dash|0, 4, 8|med}}
| 5:6:7
| D F♯ A♯
| Daug
| D augmented
| D F♭ A♭
| Ddim
| D diminished
|-
| {{dash|0, 5, 10|med}}
|
| D F A♭
| Ddim
| D diminished
| D F A♯
| Daug
| D augmented
|-
| {{dash|0, 5, 9, 13|med}}
| 4:5:6:7
| D F A C♯
| Dm(M7)
| D minor-major
| D F A C♭
| D7
| D seven
|-
| {{dash|0, 5, 9, 12|med}}
|
| D F A Bb
| Dm(♭6)
| D minor flat-six
| D F A B♯
| D6
| D six
|-
| {{dash|0, 5, 9, 14|med}}
|
| D F A C
| Dm7
| D minor seven
| D F A C
| DM7
| D major seven
|-
| {{dash|0, 4, 9, 13|med}}
|
| D F♯ A C♯
| DM7
| D major seven
| D F♭ A C♭
| DM7
| D minor seven
|}

Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord {{dash|6, 1, 3, 5, 7, 9, 11, 13}}). See [[Ups and downs notation #Chords and chord progressions]] for more examples.

Using antidiatonic names, if you're used to diatonic interval arithmetic, you can do antidiatonic interval arithmetic by following the simple guideline that qualities are '''reversed''' from standard diatonic. As in, just as adding two major seconds gives you a major third in 12edo, adding two minor seconds gives a minor third in 16edo.

That is, reversing means exchanging major for minor, aug for dim, and sharp for flat. Perfect and natural are unaffected.

Examples can be found at the bottom of the page.

== Approximation to JI ==
=== Selected just intervals by error ===
{{Q-odd-limit intervals|16}}

It's worth noting that the 525{{c}} interval is almost exactly halfway in between 4/3 and 11/8, making it very discordant, although playing this in the context of a larger chord, and with specialized timbres, can make this less noticeable.

[[File:16ed2-001.svg|alt=alt : Your browser has no SVG support.]]

[[:File:16ed2-001.svg|16ed2-001.svg]]

=== Zeta peak index ===
{{ZPI
| zpi = 51
| steps = 15.9443732426877
| step size = 75.2616601314409
| tempered height = 4.191572
| pure height = 3.476281
| integral = 0.812082
| gap = 13.070433
| octave = 1204.18656210305
| consistent = 6
| distinct = 6
}}

== Octave theory ==
The scale supports the diminished temperament with its 1/4 octave period, though its generator size, equal to its step size of 75{{c}}, is smaller than ideal. Its very flat 3/2 of 675{{c}} [[support]]s Mavila temperament, where the mapping of major and minor is reversed. The temperament could be popular for its 150{{c}} "3/4-tone" equal division of the traditional 300{{c}} minor third.

16edo is also a tuning for the [[Jubilismic clan|no-threes 7-limit temperament tempering out 50/49]]. This has a period of a half-octave (600{{c}}), and a generator of a flat septimal major 2nd, for which 16edo uses 3\16. For this, there are mos scales of sizes 4, 6, and 10; extending this temperament to the full 7-limit can produce either Lemba or Astrology (16edo supports both, but is not a very accurate tuning of either).

16edo is also a tuning for the no-threes 7-limit temperament tempering out [http://x31eq.com/cgi-bin/uv.cgi?uvs=%5B-19%2C7%2C1%3E&limit=2_5_7 546875:524288], which has a flat major third as generator, for which 16-EDO provides 5\16 octaves. For this, there are MOS of sizes 7, 10, and 13; these are shown below under "'''Magic family of scales'''".

[[Easley Blackwood Jr]] writes of 16edo:

"''16 notes: This tuning is best thought of as a combination of four intertwined diminished seventh chords. Since 12-note tuning can be regarded as a combination of three diminished seventh chords, it is plain that the two tunings have elements in common. The most obvious difference in the way the two tunings sound and work is that triads in 16-note tuning, although recognizable, are too discordant to serve as the final harmony in cadences. Keys can still be established by successions of altered subdominant and dominant harmonies, however, and the Etude is based mainly upon this property. The fundamental consonant harmony employed is a minor triad with an added minor seventh.''"

From a harmonic series perspective, if we take 13\16 as a 7/4 ratio approximation, sharp by 6.174{{c}}, and take the 300{{c}} minor third as an approximation of the harmonic 19th ([[19/16]], approximately 297.5{{c}}), that can combine with the approximation of the harmonic seventh to form a 16:19:28 triad .

The interval between the 28th & 19th harmonics, 28:19, measures approximately 671.3{{c}}, which is 3.7{{c}} away from 16edo's "narrow fifth". Another voicing for this chord is 14:16:19, which features 19:14 as the outer interval (528.7{{c}} just, 525.0{{c}} in 16edo). A perhaps more consonant open voicing is 7:16:19

== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|edo=16}}

=== Commas ===
16et [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes [[val]] {{val| 16 25 37 45 55 59 }}.)

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime limit]]
! [[Ratio]]<ref group=note>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name
|-
| 5
| [[135/128]]
| {{monzo| -7 3 1 }}
| 92.18
| Layobi
| Mavila comma, major chroma
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, major diesis
|-
| 5
| [[3125/3072]]
| {{monzo| -10 -1 5 }}
| 29.61
| Laquinyo
| Magic comma
|-
| 5
| [[6115295232/6103515625|(20 digits)]]
| {{monzo| 23 6 -14 }}
| 3.34
| Sasepbiru
| [[Vishnuzma]]
|-
| 7
| [[36/35]]
| {{monzo| 2 2 -1 -1 }}
| 48.77
| Rugu
| Mint comma, septimal quartertone
|-
| 7
| [[525/512]]
| {{monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma
|-
| 7
| [[50/49]]
| {{monzo| 1 0 2 -2 }}
| 34.98
| Biruyo
| Jubilisma
|-
| 7
| [[64827/64000]]
| {{monzo| -9 3 -3 4 }}
| 22.23
| Laquadzo-atrigu
| Squalentine comma
|-
| 7
| [[3125/3087]]
| {{monzo| 0 -2 5 -3 }}
| 21.18
| Triru-aquinyo
| Gariboh comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[1029/1024]]
| {{monzo| -10 1 0 3 }}
| 8.43
| Latrizo
| Gamelisma
|-
| 7
| [[6144/6125]]
| {{monzo| 11 1 -3 -2 }}
| 5.36
| Sarurutrigu
| Porwell comma
|-
| 11
| [[121/120]]
| {{monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[385/384]]
| {{monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
| 3.93
| Luzozogu
| Werckisma
|-
| 11
| [[3025/3024]]
| {{monzo| -4 -3 2 -1 2 }}
| 0.57
| Loloruyoyo
| Lehmerisma
|}

=== Rank-2 temperaments ===
* [[List of 16et rank two temperaments by badness]]

{| class="wikitable center-1 center-2"
|+ Table of temperaments by generator
|-
! Periods per 8ve
! Generator
! Temperaments
|-
| 1
| 1\16
| [[Valentine]], [[slurpee]]
|-
| 1
| 3\16
| [[Gorgo]]
|-
| 1
| 5\16
| [[Magic]]/[[muggles]]
|-
| 1
| 7\16
| [[Mavila]]/[[armodue]]
|-
| 2
| 1\16
| [[Bipelog]]
|-
| 2
| 3\16
| [[Lemba]], [[astrology]]
|-
| 4
| 1\16
| [[Diminished (temperament)|Diminished]]/[[demolished]]
|-
| 8
| 1\16
| [[Semidim]]
|}

== Scales ==
* {{Main|List of MOS scales in {{PAGENAME}}}}
Important mosses include:
* [[magic]] anti-diatonic 3L4s 1414141 (5\16, 1\1)
* [[magic]] superdiatonic 3L7s 1311311311 (5\16, 1\1)
* [[magic]] chromatic 11121121112 3L10s (5\16, 1\1)
* [[mavila]] anti-diatonic 2L5s 2223223 (9\16, 1\1)
* [[mavila]] superdiatonic 7L2s 222212221 (9\16, 1\1)
* [[gorgo]] 5L1s 333331 (3\16, 1\1)
* [[lemba]] 4L2s 332332 (3\16, 1\2)

'''Mavila'''

{| class="wikitable"
|-
| [5]:
| 5 2 5 2 2
|
|-
| [7]:
| 3 2 2 3 2 2 2
|[[File:MavilaAntidiatonic16edo.mp3]]
|-
| [9]:
| 1 2 2 2 1 2 2 2 2
|[[File:MavilaSuperdiatonic16edo.mp3]]
|}
See also [[Mavila Temperament Modal Harmony]].

'''Diminished'''

{| class="wikitable"
|-
| [8]:
| 1 3 1 3 1 3 1 3
|[[File:htgt16edo.mp3]]
|-
| [12]:
| 1 1 2 1 1 2 1 1 2 1 1 2
|
|}

'''Magic'''

[7]: 1 4 1 4 1 4 1

[10]: 1 3 1 1 3 1 1 1 3 1

[13]: 1 1 2 1 1 1 2 1 1 1 2 1 1

'''Cynder/Gorgo'''

[5]: 3 3 4 3 3

[6]: 3 3 1 3 3 3

[11]: 1 2 1 2 1 2 1 2 1 2 1

'''Lemba/Astrology'''

[4]: 3 5 3 5

[6]: 3 2 3 3 2 3

[10]: 2 1 2 1 2 2 1 2 1 2

== Metallic harmony ==
In 16edo, triadic harmony can be based on on heptatonic sevenths (or seconds) rather than thirds. For instance, 16edo approximates 7/4 well enough to use

it in place of the usual 3/2, and in Mavila[7] this 7/4 approximation shares an interval class with a well-approximated 11/6 (at 1050{{c}}). Stacking these two intervals reaches 2025{{c}}, or a minor 6th plus an octave. Thus the out-of-tune 675{{c}} interval is bypassed, and all the dyads in the triad are consonant.

Depending on whether the Mavila[7] major 7th or minor 7th is used, one of two triads is produced: a small one, {{nowrap|{{dash|0, 975, 2025{{c}}}}}}, and a large one, {{nowrap|{{dash|0, 1050, 2025{{c}}}}}}. William Lynch, a major proponent of this style of harmony, calls these two triads "hard" and "soft", respectively. In addition, two other "symmetrical" triads are also obvious possible chords: a narrow symmetrical triad at {{nowrap|{{dash|0, 975, 1950{{c}}}}}}, and a wide symmetrical triad at {{nowrap|{{dash|0, 1050, 2100{{c}}}}}}. These are sort of analogous to "diminished" and "augmented" triads. The characteristic buzzy/metallic sound of these seventh-based triads inspired William Lynch to call them "Metallic triads".

=== MOS scales supporting metallic harmony in 16edo ===
The ssLsssL mode of Mavila[7] contains two hard triads on degrees 1 and 4 and two soft triads on degrees 2 and 6. The other three chords are wide symmetrical triads 0-1050-2025{{c}}. In Mavila[9], hard and soft triads cease to share a triad class, as 975{{c}} is a major 8th, while 1050{{c}} is a minor 9th; the triads may still be used, but parallel harmonic motion will function differently.

Another possible MOS scales for this approach would be Lemba[6], which gives two each of the soft, hard, and narrow symmetric triads.

''See: [[Metallic Harmony]].''

== Diagrams ==
'''16-tone piano layout based on the mavila[7]/antidiatonic scale'''

This Layout places mavila[7] on the black keys and mavila[9] on the white keys, according to antidiatonic notation.

[[File:16-EDO-PIano-Diagram.png|alt=16-EDO-PIano-Diagram.png|748x293px|16-EDO-PIano-Diagram.png]]

'''Un-annotated diagram'''

Please explain this image. {{todo|annotate}}

[[File:16edo_wheel_01.png|alt=16edo wheel 01.png|325x325px|16edo wheel 01.png]]

'''Lumatone mapping'''

See: [[Lumatone mapping for 16edo]]

== Interval arithmetic examples ==
These examples show the correspondence between interval arithmetic using diatonic and antidiatonic notation.
{| class="wikitable" style="text-align: center;"
! colspan="2" |Diatonic (i.e. 12edo)
! colspan="2" |Antidiatonic (i.e. 16edo)
|-
! Question
! Result
! Question
! Result
|-
| M2 + M2
| aug3
| m2 + m2
| dim3
|-
| D to F♯
| aug3
| D to F♭
| dim3
|-
| D to F
| M3
| D to F
| m3
|-
| E♭ + m3
| Gbb
| E♯ + M3
| G♯♯
|-
| E♭ + P5
| B♭
| E♯ + P5
| B♯
|-
| A minor chord
| A C♭ E
| A major chord
| A C♯ E
|-
| E♭ major chord
| E♭ G♭ D♭
| E♯ minor chord
| E♯ G♯ B♯
|-
| Gm7 = G + m3 + P5 + m7
| G B D F♭
| G + M3 + P5 + M7
| G B D F♯
|-
| A♭7aug = A♭ + M3 + A5 + m7
| A♭ C♭ E Gbb
| A♯ + m3 + d5 + M7
| A♯ C♯ E G♯♯
|-
| what chord is D F A♯?
| D + M3 + A5 = Daug
| D F A♭
| D + m3 + d5
|-
| what chord is C E G♭ B♭?
| C + m3 + d5 + d7 = Cdim7
| C E G♯ B♯
| C + M3 + A5 + A7
|-
| C major scale = C + M2 + M3 + P4 + P5 + M6 + M7 + P8
| C D♯ E♯ F G A♯ B♯ C
| C + m2 + m3 + P4 + P5 + m6 + m7 + P8
| C D♭ E♭ F G A♭ B♭ C
|-
| C minor scale = C + M2 + m3 + P4 + P5 + m6 + m7 + P8
| C D♯ E F G A B C
| C + m2 + M3 + P4 + P5 + M6 + M7 + P8
| C D♭ E F G A B C
|-
| what scale is A B♯ C♭ D E F G♭ A?
| A + M2 + m3 + P4 + P5 + M6 + m7 = A dorian
| A B♭ C♯ D E F G♯ A
| A + m2 + M3 + P4 + P5 + m6 + M7
|}

== Music ==
{{Catrel| 16edo tracks }}

; [[Abnormality]]
* [https://www.youtube.com/watch?v=zao6E8GdQh0 ''it's not not opposite day''] (2023)
* [https://www.youtube.com/watch?v=1pa3dztk8o0 ''nightfall''] (2024)

; [[Beheld]]
* [https://www.youtube.com/watch?v=kzPeVB2mncc ''Nebulous vibe'']

; [[City of the Asleep]]
* [https://cityoftheasleep.bandcamp.com/track/huckleberry-regional-preserve ''Huckleberry Regional Preserve'']
* [https://cityoftheasleep.bandcamp.com/track/illegible-red-ink ''Illegible Red Ink'']

; [[Bryan Deister]]
* [https://www.youtube.com/shorts/IfVvjoRqqNk ''16edo jam''] (2025)

; [[E8 Heterotic]]
* [https://youtu.be/a8Jgb_XIj7c "Hexed"]

; [[Fabrizio Fiale]]
* [https://www.soundclick.com/music/songInfo.cfm?songID=12370649 ''Prenestyna Highway'']
* [https://www.soundclick.com/music/songInfo.cfm?songID=7715803 ''Palestrina Morta, fantasia quasi una sonata'']
* [https://soundcloud.com/fff-fiale/in-sospensione-neutra ''In Sospensione Neutra'']

; [[Aaron Andrew Hunt]]
* [https://soundcloud.com/uz1kt3k/fuga-a3-in-16et ''Fuga a3 in 16ET'']

; [[Last Sacrament]]
* [http://lastsacrament.bandcamp.com/album/enantiodromia ''Enantiodromia''] (album) (from 2013)
* [https://lastsacrament.bandcamp.com/album/maniacal-meditations-ep ''Maniacal Meditations''] (EP) (2013 EP)

; [[William Lynch]]
* [[:File:Mavila_Jazz_Rhodes_1.mp3|''Mavila Jazz Groove'']]
* [[:File:mavila4.mp3|''Cold, Dark Night for a Dance'']]

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=vIWxP_C0aUM ''Mavila Fugue'']
* [https://www.youtube.com/watch?v=KYkmT46oGhw ''Canon at the Semitone on The Mother's Malison Theme'', for Cor Anglais and Violin] ([https://www.youtube.com/watch?v=I6BUauD8EaE for Organ])
* [https://www.youtube.com/watch?v=P7LUSRd1kMg ''Canon on Twinkle Twinkle Little Star'', for Organ] (2023) ([https://www.youtube.com/watch?v=QHJYyqge_JQ for Baroque Oboe and Viola])

; [[Herman Miller]]
* [http://www.io.com/%7Ehmiller/midi/16tet.mid ''Etude in 16-tone equal tuning'']{{dead link}} [http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/16tet.mp3 play]{{dead link}} ([http://soonlabel.com/xenharmonic/archives/2604 organ version]{{dead link}})

; [[Nae Ayy]]
* [https://www.youtube.com/watch?v=H74psBvdeT4 ''Rambling'']
* [https://www.youtube.com/watch?v=OAhV8ol2Hbw ''a n g e r y'']
* [https://www.youtube.com/watch?v=-MboZelse90 ''Maundering'']

; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=LXsZIbT6wpM ''Edolian - Seventhic''] (2020)
* [https://www.youtube.com/watch?v=UrQPr7V9feA ''Finality''] (2021)

; [[Jean-Pierre Poulin]]
* [http://www.jeanpierrepoulin.com/mp3/Armodue78.mp3 ''Armodue78'']

; [[Ron Sword]]
* [https://soundcloud.com/ron-sword/mavila-fog ''The Foggy Road from Pasadena'']{{dead link}}

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/16-ET/20120527-16-malathion.mp3 ''Malathion''] - [http://chrisvaisvil.com/?p=2358 details]
* [http://micro.soonlabel.com/16-ET/20130216_16edo_vesta.mp3 ''Being of Vesta''] - [http://chrisvaisvil.com/?p=3061 details]
* [http://micro.soonlabel.com/simultaneous-tunings/20130607_thin_ice_christiane.mp3 ''Thin Ice''] - [http://chrisvaisvil.com/?p=3354 details]

; [[Stephen Weigel]]
* [https://www.youtube.com/watch?v=0t7ZmlmrE0Q ''Shot Fades the Sum Of'']
* [https://www.youtube.com/watch?v=2y01AlgOPvk ''When the Saints go Marching'']

; [[Randy Winchester]]
* [http://micro.soonlabel.com/gene_ward_smith/Others/Winchester/05%20-%205.%2016%20octave.mp3 Comets Over Flatland 5]{{dead link}}

; [[Woyten]]
* [https://www.youtube.com/watch?v=LLgClI8pyNw ''Don't Take Five''] (2021)

; [[Xotla]]
* "Robotic Dialogue" from ''Microtones & Garden Gnomes'' (2017) [https://xotla.bandcamp.com/track/robotic-dialogue-16edo Bandcamp] | [https://youtu.be/sFxny2JNGpo?si=8MKPuIMCR_Xx1DTi YouTube]
* "Cognitive Climate" from Science Fraction (2022) [https://open.spotify.com/track/52v382I0OUotQjHo0pPoXs Spotify] | [https://xotla.bandcamp.com/track/cognitive-climate-16edo Bandcamp] | [https://youtu.be/dNBDG4wymN8?si=XGbpNkRp3qUo0Xgb YouTube]

; [[User:Nick_Vuci|Nick Vuci]]
* [https://en.xen.wiki/images/4/44/NickVuci-20220206-16edo-Prelude.mp3 ''Prelude'']
* [https://en.xen.wiki/images/9/9a/NickVuci-20231102-16edo-SofterForJ.mp3 ''Softer for J'']
* [https://en.xen.wiki/images/4/48/NickVuci-20220306-16edo-Invention.mp3 ''2-Part Invention'']
* [https://en.xen.wiki/w/User:Nick_Vuci#Modal_Studies ''Mavila Modal Studies'']
* [https://en.xen.wiki/images/c/c6/NV-20210526-16NEJI128-SerialismDubstepSketch.mp3 ''EDM based on a tone row'']

; [[Zewen Senpai]]
* [https://www.youtube.com/watch?v=QOzBGd64Pi4 ''Simple Ambient Study No. 1'']

== Notes ==
<references group=note/>

== See also ==
* [[57ed12]] - octave stretched version of 16edo; 57ed12 improves 3.5.11.13.17 but damages 2.7

=== Approaches ===
* [[User:VectorGraphics/16edo theory|Vector's approach]]
* [[Armodue theory]]
** [[Armodue armonia]]

== References ==
<references />

== Further reading ==
* [[Sword, Ron]]. ''[https://ronsword.bigcartel.com/product/esadekaphonic-scales-for-guitar Hexadecaphonic Scales for Guitar: A Microtonal Guitar Method Book, for Theory, Scales, and Information on the Sixteen Equal Division Octave System]''. 2009. (semi-diminished fourth tuning)
* Sword, Ron. ''[http://www.metatonalmusic.com/books.html Hexadecaphonic Scales for Guitar: Theory, Scales and Information on the Sixteen Equal Division Octave system]''. 2010? (superfourth tuning)
* Sword, Ron. "Thesaurus of Melodic Patterns and Intervals for 16-Tones" IAAA Press, USA. First Ed: August, 2011{{citation needed}}

[[Category:Teentuning]]
[[Category:Listen]]
[[Category:Mavila]]
[[Category:Guitar]]
[[Category:Pages with internal sound examples]]

{{Todo|cleanup}}

Talk:15edo

2025-05-02T18:37:19Z

Cmloegcmluin: /* Table of divisors */

{{WSArchiveLink}}
==Draft at 15edo-a==
Is there anyone who opposes replacing this with [[15edo-a]]? If so, speak up.

In the future note that drafts aren't really necessary, since we have a complete history of all versions. If you still want a draft use a user subpage, for example [[User:Keenan Pepper/15edo]]. [[User:Keenan Pepper|Keenan Pepper]] ([[User talk:Keenan Pepper|talk]]) 04:11, 30 September 2018 (UTC)

== Table of divisors ==

The table "Logarithmic divisions of intervals in 15edo" is just marking divisors.
I don't really need a table to know 6 steps is divisible by 2 and 3. It takes up quite some space, and actually it took me a bit to even figure out what the point of it was. I'd rather remove it.
– [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 11:42, 2 May 2025 (UTC)
: Agreed. --[[User:Fredg999|Fredg999]] ([[User talk:Fredg999|talk]]) 18:31, 2 May 2025 (UTC)
:: Killed it. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 18:37, 2 May 2025 (UTC)

15edo

2025-05-02T18:37:05Z

Cmloegcmluin: /* Tuning theory */ remove unhelpful table

{{interwiki
| de = 15-EDO
| en = 15edo
| es =
| ja = 15平均律
}}
{{Infobox ET}}
{{Wikipedia|15 equal temperament}}
{{ED intro}}

== Theory ==

=== Composition theory ===

* [[User:Unque/15edo Composition Theory|Unque's approach]] - covers scales, chords, intervals, and functional harmony.
* [[15edo/Vector's compositional guides|Vector's guides]] - covers the construction of scales, the kinds of chords found in 15edo, and a possible notation system.
* [[Metallic harmony]] - harmony involving stacking sevenths instead of thirds; 15edo is one of the systems it is intended for.

=== Tuning theory ===
15edo can be thought of as three sets of [[5edo]] which do not connect by [[3/2|fifths]]. The fifth at 720 cents is quite wide yet still usable as a perfect fifth. Some would describe the fifth as more shimmery and pungent than anything closer to a just 3/2. The perfect fifth of 15edo returns to the octave if stacked five times. In regular temperament terms, this means the [[Pythagorean limma]] is tempered out, which is radically different than a meantone system. This has a variety of ramifications for chord progressions based on {{w|Function (music)|functional harmony}}, because if you use the equipentatonic as your "diatonic scale", the same interval can have multiple functions. Additionally, 15 being equal to {{nowrap|3 × 5}} also implies that 15edo contains five sets of [[3edo]].

15edo can be seen as a [[7-limit]] temperament because of its ability to approximate some septimal intervals, but it also contains some fairly obvious approximations to [[11-limit]] intervals, so it can reasonably be described as an 11-limit temperament, and is generally considered to be the first edo to work as an 11-limit system; however, due to its rather distant approximation of the 3rd harmonic (and therefore the 9th harmonic as well), those seeking to represent JI with 15edo would be best advised to avoid chords requiring those harmonics (or to at least treat them with sensitivity, for instance, only using 9/8 when it is being made up of two 3/2s to make its identity clear). 15edo is also notable for being the smallest edo with recognizable, distinct representations of 5-odd limit intervals (3/2, 5/4, 6/5, and their octave inverses) that has a positive [[syntonic comma]].

In the 15edo system, major thirds cannot be divided perfectly into two, while minor thirds, fourths, wide tritones, subminor sevenths, and supermajor sevenths can. Similarly, supermajor seconds, fourths, fifths, and subminor sevenths can all be divided into 3 equal parts, while minor thirds, tritones, and major sixths cannot.

This gives 15edo a whole new set of pitch symmetries and modes of limited transposition. Coupled with the lack of a [[5L 2s|5L 2s diatonic scale]] and of a standard tritone, this tuning can be disorienting at first. Nonetheless, 15edo is notable for being the next-smallest edo after 9edo, 12edo and 14edo that contains recognizable major and minor triads. Under a stricter definition excluding 9edo and 14edo, this is a property noted in the works of theorists like [[Ivor Darreg]] and [[Easley Blackwood]]. In addition, because the guitar can be tuned symmetrically, from E to e (6th to 1st strings) unlike the 12-tone system on guitars, the learning curve is very manageable. All chords look the same modulated anywhere, and minor arpeggios are vertically stacked, making them very easy to play. 15-tone may be a promising start for anyone interested in xenharmony, due to its manageable number of tones and for containing the relatively popular 5edo.

A possible analogue to the diatonic scale in 15edo is the [[Zarlino]] diatonic, which flattens one fifth to a large tritone in order to make all 7 notes distinct (and close to corresponding JI intervals, especially if you use the left-handed version). The fact that 15edo supports [[porcupine]] temperament is equivalent to the fact that both accidentals generally required to notate zarlino collapse to a single chromatic step. For a moment-of-symmetry scale, the [[1L 6s]] (onyx) and [[5L 5s]] (blackwood) scales are also an option.

15edo is also the second-smallest edo (after [[10edo]]) that maintains [[minimal consistent EDOs|25% or lower relative error]] on all of the first eight harmonics of the [[harmonic series]].

==== Prime harmonics ====
{{Harmonics in equal|15}}

==Intervals==
{{See also|15edo-interval names}}
Relative to 12edo, 15edo maintains some categorically-similar intervals, particularly the 3rds, 4ths, 5ths, and 6ths, but is quite different in the categories of 2nds and 7ths. The closest intervals it has to a 12edo [[whole tone]] are both 40 cents sharp or flat of the 200-cent 12edo whole tone. This makes it rather difficult to translate traditional diatonic melodic approaches into 15edo, and also means that things like 7th, 9th, and 11th chords will behave very differently, even though major and minor triads are still relatively familiar-sounding. One step of 15edo almost exactly equals the reduced 67th harmonic, [[67/64]].

{| class="wikitable center-all left-8"
|-
!Degree
!Cents
!Approximate Ratios<ref group="note">{{rd|limit=11-limit}}</ref>
![[Solfege]] (porcupine-based)
!Porcupine[7] (traditional)
!Porcupine[8] (Greek)
!Zarlino diatonic notation
! Blackwood "guitar notation"
!Blackwood Decimal
!Audio
|-
| 0
|0
|1/1
|do
| D
|α
|C
|E
|1
|[[File:piano_0_1edo.mp3]]
|-
|1
| 80
|25/24, 21/20, 16/15, 22/21
|di
|D# / Eb
| α/ β\
|Db / C#
| E#
|1# / 2b
|[[File:piano_1_15edo.mp3]]
|-
|2
|160
|11/10, 12/11, 10/9
|ru
|E
|β
|D
| Gb
|2
|[[File:piano_2_15edo.mp3]]
|-
|3
| 240
|8/7, 7/6, 9/8
| re
| E# / Fb
|β/ χ\
|D#
| G
|3
|[[File:piano_1_5edo.mp3]]
|-
|4
|320
|6/5, 11/9
|me
|F
| χ
|Eb
|G#
|3# / 4b
|[[File:piano_4_15edo.mp3]]
|-
|5
| 400
|5/4, 14/11
|mi
|F# / Gb
|χ/ δ\
|E
|Ab
|4
|[[File:piano_1_3edo.mp3]]
|-
| 6
|480
|4/3, 9/7, 21/16
|fa
|G
|δ
|F
|A
|5
|[[File:piano_2_5edo.mp3]]
|-
|7
|560
|11/8, 7/5
|fu
|G#
|δ/ ε\
|F#
|A#
|5# / 6b
|[[File:piano_7_15edo.mp3]]
|-
|8
|640
|16/11, 10/7
|su
| Ab
|ε
|Gb
|Bb
|6
|[[File:piano_8_15edo.mp3]]
|-
|9
|720
|3/2, 14/9, 32/21
|sol
|A
|ε/ φ\
|G
|B
|7
|[[File:piano_3_5edo.mp3]]
|-
|10
|800
| 8/5, 11/7
|le
|A# / Bb
|φ
|Ab / G#
|B#
|7# / 8b
|[[File:piano_2_3edo.mp3]]
|-
|11
|880
|5/3, 18/11
|la
|B
|φ/ γ\
|A
|Db
|8
|[[File:piano_11_15edo.mp3]]
|-
|12
| 960
|7/4, 12/7, 16/9
|ta
|B# / Cb
|γ
|A# / Bbb
|D
| 9
|[[File:piano_4_5edo.mp3]]
|-
|13
| 1040
|20/11, 11/6, 9/5
|tu
|C
|γ/ η\
|Bb
|D#
|9# / 0b
|[[File:piano_13_15edo.mp3]]
|-
|14
| 1120
|48/25, 40/21, 15/8, 21/11
|ti
|C# / Db
|η
|B
|Eb
|0
|[[File:piano_14_15edo.mp3]]
|-
| 15
|1200
|2/1
|do
|D
| α
|C
| E
| 1
|[[File:piano_1_1edo.mp3]]
|}

===Alternative interval names===
{| class="wikitable center-all"
|-
! Step
! Cents
! colspan="3" | [[Ups and downs notation]] ([[Enharmonic unisons in ups and downs notation|EUs]]: v3A1 and m2) (partial list, e.g. M2/m3 is also A1 and d4)
! colspan="2" | Porcupine notation ([[Enharmonic unison|EU]]: dd2)
|-
| 0
| 0
| P1, m2
| unison, min 2nd
| C# / D / Eb
| unison
| D
|-
| 1
| 80
| ^1, ^m2
| up-unison, upminor 2nd
| ^C# / ^D / ^Eb
| aug unison, dim 2nd
| D# / Eb
|-
| 2
| 160
| vM2
| downmajor 2nd
| vD# / vE / vF / vGb
| perfect 2nd
| E
|-
| 3
| 240
| M2, m3
| major 2nd, minor 3rd
| D# / E / F / Gb
| aug 2nd, dim 3rd
| E# / Fb
|-
| 4
| 320
| ^m3
| upminor 3rd
| ^D# / ^E / ^F / ^Gb
| minor 3rd
| F
|-
| 5
| 400
| vM3
| downmajor 3rd
| vF# / vG / vAb
| major 3rd, dim 4th
| F# / Gb
|-
| 6
| 480
| M3, P4, d5
| major 3rd, perfect 4th, dim 5th
| F# / G / Ab
| aug 3rd, minor 4th
| Fx / G
|-
| 7
| 560
| ^4, ^d5
| up 4th, updim 5th
| ^F# / ^G / ^Ab
| major 4th, dim 5th
| G# / Abb
|-
| 8
| 640
| vA4, v5
| downaug 4th, down 5th
| vG# / vA / vBb
| aug 4th, minor 5th
| Gx / Ab
|-
| 9
| 720
| A4, P5, m6
| aug 4th, perfect 5th, minor 6th
| G# / A / Bb
| major 5th, dim 6th
| A / Bbb
|-
| 10
| 800
| ^5, ^m6
| up 5th, upminor 6th
| ^G# / ^A / ^Bb
| aug 5th, minor 6th
| A# / Bb
|-
| 11
| 880
| vA5, vM6
| downaug 5th, downmajor 6th
| vA# / vB / vC / vDb
| major 6th
| B
|-
| 12
| 960
| M6, m7
| major 6th, minor 7th
| A# / B / C / Db
| aug 6th, dim 7th
| B# / Cb
|-
| 13
| 1040
| ^m7
| upminor 7th
| ^A# / ^B / ^C / ^Db
| perfect 7th
| C
|-
| 14
| 1120
| vM7, v8
| downmajor 7th, down octave
| vC# / vD / vEb
| aug 7th, dim 8ve
| C# / Db
|-
| 15
| 1200
| M7, P8
| major 7th, octave
| C# / D / Eb
| 8ve
| D
|}

The 15edo porcupine genchain in both absolute and relative notation:

* …{{dash|Fx, Gx, A#, B#, C#, D#, E#, F#, G#, A, B, C, D, E, F, G, Ab, Bb, Cb, Db, Eb, Fb, Gb, Abb, Bbb|long}}…
* …{{dash|A3, A4, A5, A6, A7, A1, A2, M3, M4, M5, M6, P7, P1, P2, m3, m4, m5, m6, d7, d8, d2, d3, d4, d5, d6|long}}…

All 15edo chords can be named using ups and downs. Because many intervals have several names, many chords do too. Alterations are always enclosed in parentheses, additions never are. An up or down immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).

0-3-9 = D E A = D2 = "D sus 2", or D F A = Dm = "D minor" (approximate 6:7:9)

0-4-9 = D ^F A = D^m = "D upminor" (approximate 10:12:15)

0-5-9 = D vF# A = Dv = "D down" or "D downmajor" (approximate 4:5:6)

0-6-9 = D G A = D4, or D F# A = D = "D" or "D major" (approximate 14:18:21)

0-3-9-12 = D F A C = Dm7 = "D minor seven", or D F A B = Dm6 = "D minor six"

0-4-9-12 = D ^F A C = D^m,7 = "D upminor, add seven", or D ^F A B = D^m,6 = "D upminor add-six"

0-5-9-12 = D vF# A C = Dv,7 = "D down add-seven", or D vF# A B = Dv,6 = "D down add-six"

0-6-9-12 = D F# A C = D7 = "D seven", or D F# A B = D6 = "D six"

0-5-9-14 = D vF# A vC# = DvM7 = "D downmajor seven"

0-4-9-13 = D ^F A ^C = D^m7 = "D upminor-seven", or D ^F A ^B = D^m6 = "D upminor-six"

For a more complete list, see [[Ups and downs notation#Chords and Chord Progressions]].
==Notation ==
There are many ways to notate 15edo, and the choice of notation depends heavily on which temperament or scale one wishes to focus on.

=== Notations generated by the fifth ===
In these notations, the nominals form a circle of perfect fifths. The other notes are notated using accidentals that raise or lower by one edostep.

==== Ups and downs notation (heptatonic) ====
15edo can be notated with [[ups and downs]], spoken as up, dup, downsharp, sharp, upsharp etc. and down, dud, upflat etc. Note that downsharp is equivalent to dup (double-up) and upflat is equivalent to dud (double-down).{{Sharpness-sharp3a}}[[Alternative symbols for ups and downs notation]] uses sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:{{Sharpness-sharp3}}

==== "Eef" notation (pentatonic) ====
[[Kite Giedraitis]] proposes pentatonic (as opposed to heptatonic) note names that omit B and merge E and F into a new letter "eef" that rhymes with "leaf". Eef, like E, is a 5th above A. Eef, like F, is a 4th above C. Eef is written like an E, but with the bottom horizontal line going not right but left from the vertical line. Eef can be typed as ꘙ (unicode A619) or ⊧ (unicode 22A7) or 𐐆 (unicode 10406). The circle of 5ths is C G D A ꘙ C. All intervals are either perfect, upperfect or dowperfect (never major or minor). This is similar to heptatonic interval names in 7edo, 14edo, 21edo, etc.
{| class="wikitable"
|C
|^C
|vD
|D
|^D
|vꘙ
|ꘙ
|^ꘙ
|vG
|G
|^G
|vA
|A
|^A
|vC
|C
|-
|P1
|^1
|v2
|P2
|^2
|v3
|P3
|^3
|v4
|P4
|^4
|v5
|P5
|^5
|v6
|P6
|}

==== Sagittal notation (heptatonic)====
This notation uses the same sagittal sequence as EDOs [[22edo#Sagittal notation|22]] and [[29edo#Sagittal notation|29]], is a subset of the notation for [[30edo#Sagittal notation|30-EDO]], and is a superset of the notation for [[5edo#Sagittal notation|5-EDO]].<imagemap>
File:15-EDO_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 439 0 599 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 439 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
default [[File:15-EDO_Sagittal.svg]]
</imagemap>

==== Blackwood guitar notation ====
On a 15edo guitar, because the "perfect fourth" comes from 5edo, all of the open strings can be tuned a perfect fourth apart and still span exactly two octaves. If one starts the [[circle of fourths]] on B — B-E-A-D-G-(B) — then the open strings of the guitar can be notated as usual (E-A-D-G-B-E). However, because the circle of fourths closes at five, and does not continue to circulate through the other 10 notes of 15edo, it is necessary to use accidentals to notate intervals on the other two chains of 5edo. This notation is not particularly ideal as a basis for a staff notation (as it requires all non-5edo chords to be notated with accidentals). It is nevertheless useful because it reflects an intuitive approach to 15edo on the guitar, since 5edo provides a useful set of 3-limit landmarks (or "perfect fourths" and "perfect fifths") that can be used to navigate the fretboard. It's especially convenient for writing chord charts, where the funky accidental-laden spellings can be more or less ignored.

=== Blackwood decatonic notation ===
Using the nominals 1-0 (with 0 representing "10"), one of the three circles of 5edo is represented by the odd numbers, the second by the even numbers, and the third by numbers with accidentals (either odd numbers with sharps, or even numbers with flats).

One could name the nominals with letters instead of numbers, such as ABC... or JKL...

=== Notations generated by the second ===
In these notations, the nominals form a chain of perfect 2nds, each of which are two edosteps wide. From the last note of the chain up to the first there is an augmented 2nd of three edosteps. Accidentals raise or lower by one edostep.

====Porcupine notation (heptatonic) ====
Porcupine notation can be based on the Porcupine[7] Lssssss scale. By representing the 3|3 mode (sssLsss) with a chain of seconds (D E F G A B C D) and using sharps and flats (#/b) to denote an edostep up or down respectively, 15edo can be notated using standard notation. Its intervals are here named with respect to diatonic intervals, i.e., as if fifth-generated. Thus the 4th and 5th are called perfect even though they are not generators, and the 2nd and 7th are not called perfect even though they are generators.
{| class="wikitable"
!Cents
!Interval Name(s)
!Note name(s)
!Diamond-mos (on symmetric mode)
|-
|0
|Unison
|D
|J
|-
|80
|Augmented Unison / Minor Second
|D# / Eb
|J&/K@
|-
|160
|Major Second
|E
|K
|-
|240
|Augmented Second / Diminished Third
|E# / Fb
|K&/L@
|-
|320
|Minor Third
|F
|L
|-
|400
|Major Third / Diminished Fourth
|F# / Gb
|L&/M@
|-
|480
|Perfect Fourth
|G
|M
|-
|560
|Augmented Fourth
|G#
|M&
|-
|640
|Diminished Fifth
|Ab
|N@
|-
|720
|Perfect Fifth
|A
|N
|-
|800
|Augmented Fifth / Minor Sixth
|A# / Bb
|N&/O@
|-
|880
|Major Sixth
|B
|O
|-
|960
|Augmented Sixth / Diminished Seventh
|B# / Cb
|O&/P@
|-
|1040
|Minor Seventh
|C
|P
|-
|1120
|Major Seventh / Diminished Octave
|C# / Db
|P&/J@
|-
|1200
|Octave
|D
|J
|}
One advantage of this notation is that its notated D major scale, D E F# G A B C# D, directly corresponds to 15edo’s zarlino LH Ionian scale. However, this only holds true for the key of D. Furthermore, the perfect 4th and/or 5th of most other keys is notated the same way a diminished or augmented fourth or fifth is in standard diatonic. For example, in the key of A the perfect fifth is E#.

==== Zarlino notation (heptatonic) ====
15edo's zarlino scale can also be treated as the primary scale, analogously to diatonic.
{| class="wikitable"
!Cents
!Note name(s)
|-
|0
|D
|-
|80
|D#
|-
|160
|Eb
|-
|240
|E
|-
|320
|F
|-
|400
|F#
|-
|480
|Gb
|-
|560
|G
|-
|640
|G# / Ab
|-
|720
|A
|-
|800
|A#
|-
|880
|Bb
|-
|960
|B
|-
|1040
|C
|-
|1120
|C# / Db
|-
|1200
|D
|}

==== Porcupine "quill" notation (octatonic) ====
Porcupine notation can also be based on the Porcupine[8] LLLLLLLs scale using eight nominals: either α β χ δ ε φ γ η or A B C D E F G H. Latin letters are easier to type and more generalizable, but they have the downside of conflicts with standard notation. Thus, Greek letters can be used in their place with a close resemblance to the spelling of ABCDEFGHA. The letters are not in greek alphabetic order.

The eight nominals form the base diatonic scale. In the "quill name" column, the "quill" is the name given to the two-edostep interval (160¢) of 15edo while the "small quill" (80¢) is the chroma of 15edo. This produces a very consistent notation for both Porcupine[8] and Blackwood[10], moreso than putting 15edo into a 5L 2s framework.
{| class="wikitable"
!Cents
!Quill Name
!MOSstep Name
!Note names (Greek)
!Note names (Latin)
|-
|0
|Zeroquill
|Perfect 0-step
|α - α
|A - A
|-
|80
|Small Quill / Half Quill
|Diminished 1-step
|α - β\
|A - Bb
|-
|160
|Quill
|Perfect 1-step
|α - β
|A - B
|-
|240
|Small Diquill
|Minor 2-step
|α - χ\
|A - Cb
|-
|320
|Large Diquill
|Major 2-step
|α - χ
|A - C
|-
|400
|Small Triquill
|Minor 3-step
|α - δ\
|A - Db
|-
|480
|Large Triquill
|Major 3-step
|α - δ
|A - D
|-
|560
|Small Fourquill
|Minor 4-step
|α - ε\
|A - Eb
|-
|640
|Large Fourquill
|Major 4-step
|α - ε
|A - E
|-
|720
|Small Fivequill
|Minor 5-step
|α - φ\
|A - Fb
|-
|800
|Large Fivequill
|Major 5-step
|α - φ
|A - F
|-
|880
|Small Sixquill
|Minor 6-step
|α - γ\
|A - Gb
|-
|960
|Large Sixquill
|Major 6-step
|α - γ
|A - G
|-
|1040
|Small Sevenquill
|Perfect 7-step
|α - η\
|A - Hb
|-
|1120
|Large Sevenquill
|Augmented 7-step
|α - η
|A - H
|-
|1200
|Octoquill
|Perfect 8-step
|α - α
|A - A
|}
A regular keyboard can be designed using this system by placing 7 black keys as Porcupine[7] and 8 whites as Porcupine[8]. In fact, [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel] has already done this with his pink Halberstadt keyboard.

==Approximation to JI==
[[File:15ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 13-limit intervals]]
15edo offers some minor improvements over 12et in ratios of 5 (particularly in 6/5 and 5/3), and has a much better approximation to the 7th and 11th harmonics, but its approximation to the 3rd harmonic is rather off. However, the particular way in which this approximation is off is as much a feature as it is a bug, for it allows the construction of a [[5L 5s]] [[MOS scale]] wherein every note of the scale can serve as a root for a 7-limit otonal or utonal tetrad, as well as either a 5-limit major or minor 7th chord. This is known as the [[blackwood]] temperament, named after [[Easley Blackwood Jr.]], who is the first to document its existence. It has also been written on extensively by [[Igliashon Jones]] in the paper [http://www.cityoftheasleep.com/etc/5nEDOs.pdf ''Five is Not an Odd Number'']. For an in-depth treatment of harmony in 15edo based on this temperament (and its 7- and 11-limit extensions), see [[Blacksmith temperament modal harmony (in 15edo)]].

===15-odd-limit interval mappings===
{{Q-odd-limit intervals}}

===Zeta peak index===
{{ZPI
| zpi = 47
| steps = 15.0534898676781
| step size = 79.7157343943591
| tempered height = 5.050324
| pure height = 4.390681
| integral = 1.104057
| gap = 14.918297
| octave = 1195.73601591539
| consistent = 8
| distinct = 7
}}

==Regular temperament properties==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" |[[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal 8ve stretch (¢)
! colspan="2" |Tuning error
|-
!
[[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|2.3.5
| 128/125, 250/243
|{{mapping| 15 24 35 }}
| −5.75
|4.63
|5.81
|-
|2.3.5.7
|28/27, 49/48, 126/125
|
{{mapping| 15 24 35 42 }}
|−3.55
|5.56
|6.97
|-
|2.3.5.7.11
|28/27, 49/48, 55/54, 77/75
|{{mapping| 15 24 35 42 52 }}
|−3.34
|4.99
| 6.25
|}

=== Errors by subgroup ===
{| class="wikitable center-3 mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" |Errors by subgroup
|-
! Subgroup
! Mapping
! Adjusted error (¢)
|-
| 2.3
| {{mapping| 15 24 }}
| 8.979801
|-
| 2.5
| {{mapping| 15 35 }}
| 6.826357
|-
| 2.7
| {{mapping| 15 42 }}
| 4.418738
|-
| 2.11
| {{mapping| 15 52 }}
| 4.336492
|-
| 2.3.5
|
{{mapping| 15 24 35 }}
| 10.742841
|-
| 2.3.7
| {{mapping| 15 24 42 }}
| 17.481581
|-
| 2.3.11
|
{{mapping| 15 24 52 }}
| 16.831238
|-
| 2.5.7
| {{mapping| 15 35 42 }}
| 10.509269
|-
| 2.5.11
| {{mapping| 15 35 52 }}
| 8.335693
|-
| 2.7.11
| {{mapping| 15 42 52 }}
| 8.002641
|-
| 2.3.5.7
| {{mapping| 15 24 35 42 }}
| 15.603114
|-
| 2.3.5.11
| {{mapping| 15 24 35 52 }}
| 14.693746
|-
| 2.3.7.11
| {{mapping| 15 24 42 52 }}
| 18.660367
|-
| 2.5.7.11
| {{mapping| 15 35 42 52 }}
| 11.462127
|-
| 2.3.5.7.11
| {{mapping| 15 24 35 42 52 }}
| 17.258371
|}

===Uniform maps===
{{Uniform map|edo=15}}

===Rank-2 temperaments===
* [[List of 15et rank two temperaments by badness]]
*[[List of edo-distinct 15et rank two temperaments]]

{| class="wikitable center-all left-4 left-5"
|-
! Periods per 8ve
!Generator
! Associated ratio
!Temperaments
!Mos scales
|-
| 1
|1\15
|21/20
|[[Nautilus]] [[Valentine]]
|
|-
|1
|2\15
|11/10
|[[Porcupine]] / [[opossum]]
|[[1L 6s]], [[7L 1s]]
|-
|1
|4\15
|6/5 77/64
|[[Cata]] / [[keemun]] / [[catalan]] [[Orgone]] / [[superkleismic]]
| [[3L 1s]], [[4L 3s]], [[4L 7s]]
|-
|1
|7\15
|7/5
|[[Progress]] [[Parakangaroo]]
|[[2L 1s]], [[2L 3s]], [[2L 5s]], [[2L 7s]] [[2L 9s]], [[2L 11s]]
|-
| 3
| 1\15
| 16/15
| [[Augmented]] / [[augene]]
|[[3L 3s]], [[3L 6s]], [[3L 9s]]
|-
|3
|2\15
|7/6
|[[Triforce]]
|[[3L 3s]], [[6L 3s]]
|-
|5
| 1\15
|16/15
|[[Blackwood]] / [[blacksmith]]
| [[5L 5s]]
|}

===Commas===
15et [[tempering out|tempers out]] the following [[comma]]s using the [[patent val]] {{val| 15 24 35 42 52 56 }}.

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime limit]]
! [[Ratio]]<ref>{{rd}}</ref>
![[Monzo]]
![[Cent]]s
![[Color name]]
!Name
|-
|3
|[[256/243]]
| {{monzo| 8 -5 }}
|90.225
|Sawa
|Blackwood comma, Pythagorean limma
|-
|5
|<abbr title="254803968/244140625">(18 digits)</abbr>
|{{monzo| 20 5 -12 }}
|74.01
|Saquadtrigu
| [[Hypovishnuzma]]
|-
|5
|[[250/243]]
|{{monzo| 1 -5 3 }}
| 49.166
|Triyo
|Porcupine comma, maximal
|-
|5
|[[128/125]]
|{{monzo| 7 0 -3 }}
|41.059
|Trigu
| Augmented comma, lesser diesis
|-
|5
|[[15625/15552]]
|{{monzo| -6 -5 6 }}
|8.107
|Tribiyo
|Kleisma
|-
|7
|[[28/27]]
|{{monzo| 2 -3 0 1 }}
|62.961
|Zo
| Septimal third-tone, trienstonic comma
|-
|7
|[[1029/1000]]
|{{monzo| -3 1 -3 3 }}
|49.492
| Trizogu
|Keega
|-
|7
|[[49/48]]
|{{monzo| -4 -1 0 2 }}
|35.697
|Zozo
|Semaphoresma, Slendro diesis
|-
|7
|[[64/63]]
|{{monzo| 6 -2 0 -1 }}
|27.264
|Ru
|Archytas' comma, septimal comma
|-
|7
| [[64827/64000]]
|{{monzo| -9 3 -3 4 }}
|22.227
|Laquadzo-atrigu
|Squalentine comma
|-
|7
|
[[875/864]]
|{{monzo| -5 -3 3 1 }}
|21.902
|Zotriyo
|Keema
|-
|7
|[[126/125]]
|{{monzo| 1 2 -3 1 }}
| 13.795
| Zotrigu
| Starling comma
|-
|7
|[[4000/3969]]
|
{{monzo| 5 -4 3 -2 }}
|13.469
| Rurutriyo
|Octagar comma
|-
|7
| [[1029/1024]]
|{{monzo| -10 1 0 3 }}
|8.433
|Latrizo
|Gamelisma
|-
|7
|[[6144/6125]]
|{{monzo| 11 1 -3 -2 }}
|5.362
|Saruru-atrigu
|Porwell comma
|-
|7
|<abbr title="250047/250000">(12 digits)</abbr>
|{{monzo| -4 6 -6 3 }}
|0.325
| Trizogugu
|[[Landscape comma]]
|-
|11
|[[100/99]]
|{{monzo| 2 -2 2 0 -1 }}
| 17.399
|Luyoyo
|Ptolemisma
|-
| 11
|[[121/120]]
|
{{monzo| -3 -1 -1 0 2 }}
|14.367
| Lologu
|Biyatisma
|-
|11
|[[176/175]]
|{{monzo| 4 0 -2 -1 1 }}
| 9.865
|Lorugugu
| Valinorsma
|-
|11
|[[65536/65219]]
|{{monzo| 16 0 0 -2 -3 }}
|8.394
|Satrilu-aruru
|Orgonisma
|-
| 11
|[[385/384]]
|{{monzo| -7 -1 1 1 1 }}
| 4.503
|Lozoyo
| Keenanisma
|-
|11
|[[441/440]]
| {{monzo| -3 2 -1 2 -1 }}
|3.930
|Luzozogu
|Werckisma
|-
|11
|[[4000/3993]]
|{{monzo| 5 -1 3 0 -3 }}
|3.032
|Triluyo
| Wizardharry
|-
|11
|
[[3025/3024]]
|{{monzo| -4 -3 2 -1 2 }}
|0.572
|Loloruyoyo
| Lehmerisma
|-
|13
|[[91/90]]
|{{monzo| -1 -2 -1 1 0 1 }}
|19.130
| Thozogu
|Superleap comma, biome comma
|-
|13
|[[676/675]]
|{{monzo| 2 -3 -2 0 0 2 }}
|2.563
|Bithogu
|Island comma
|}
<references />

== Scales==
Some scales commonly used in 15edo, written in a common mode, in steps of 15edo:

=== MOS scales===
* Augene[6] [[3L 3s]] (period = 5\15, gen = 1\15): 4 1 4 1 4 1
*Augene[9] [[3L 6s]] (period = 5\15, gen = 1\15): 3 1 1 3 1 1 3 1 1
*Augene[12] [[3L 9s]] (period = 5\15, gen = 1\15): 2 1 1 1 2 1 1 1 2 1 1 1
*Triforce[6] [[3L 3s]] (period = 5\15, gen = 2\15): 3 2 3 2 3 2
*Triforce[9] [[6L 3s]] (period = 5\15, gen = 2\15): 2 1 2 2 1 2 2 1 2
*Porcupine[7] [[1L 6s]] (gen = 2\15): 3 2 2 2 2 2 2
*Porcupine[8] [[7L 1s]] (gen = 2\15): 2 1 2 2 2 2 2 2
*Hanson/Keemun/Orgone[7] [[4L 3s]] (gen = 4\15): 1 3 1 3 1 3 3
*Hanson/Keemun/Orgone[11] [[4L 7s]] (gen = 4\15): 1 2 1 1 2 1 1 2 1 2 1
*Blackwood[10] [[5L 5s]] (period = 3\15, gen = 1\15): 2 1 2 1 2 1 2 1 2 1 (Blackwood Decatonic)

[[File:BlackwoodMajor 15edo.mp3]] [[:BlackwoodMajor 15edo.mp3|BlackwoodMajor 15edo.mp3]]

Blackwood decatonic, major mode, in 15edo

===Other scales===
*[[Zarlino]]/Ptolemy diatonic, "just" major (Porcupine[7] 6|0 b4 #7): 3 2 1 3 2 3 1
*inverse of [[Zarlino]]/Ptolemy diatonic, natural minor (Porcupine[7] 3|3 #2 b6): 3 1 2 3 1 3 2
*tetrachordal major: 3 2 1 3 3 2 1
*inverse of tetrachordal major, tetrachordal minor: 3 1 2 3 1 2 3
*"just"/[[The Pinetone System#Pinetone pentatonic|Pinetone major pentatonic]] (subset of Porcupine[7]): 3 2 4 2 4
* "just"/[[The Pinetone System#Pinetone pentatonic|Pinetone minor pentatonic]] (inverse of "just" major pentatonic, subset of Porcupine[7]): 4 2 3 4 2
*Porcupine bright major #7 (Porcupine harmonic major) - Porcupine[7] 6|0 #7: 3 2 2 2 2 3 1
*Porcupine bright major #6 #7 (Porcupine melodic major) - Porcupine[7] 6|0 #7: 3 2 2 2 3 2 1
*Porcupine bright minor #2 (Porcupine harmonic minor) - Porcupine[7] 4|2 #2: 3 1 3 2 2 2 2 (mode of bright major #7)
*Porcupine dark minor #2 (Porcupine melodic minor) - Porcupine[7] 3|3 #2: 3 1 2 3 2 2 2 (inverse of bright major #6 #7)
*Porcupine bright harmonic 11th mode - Porcupine[7] 6|0 b7: 3 2 2 2 2 1 3
* [[The Pinetone System#Pinetone harmonic diminished octatonic|Pinetone diminished octatonic]] / Porcupine[8] bright minor #2 - Porcupine[8] 2|5 #5: 2 2 1 3 1 2 2 2
*"just" harmonic minor: 3 1 2 3 1 4 1
*"just" harmonic major: 3 2 1 3 1 4 1
* "just" melodic minor ascending: 3 1 2 3 2 3 1
* Marvel double harmonic hexatonic (Augene[6] [[4M]]): 1 4 1 4 4 1, 1 4 4 1 4 1
*[[Marvel double harmonic major]]: 1 4 1 3 1 4 1
*Marvel double harmonic nonatonic (Augene[9] [[4M]]): 1 3 1 1 3 1 3 1 1, 1 1 3 1 3 1 1 3 1
*Marvel double harmonic decatonic: 1 3 1 1 2 1 1 3 1 1
*enharmonic trichord octave species: 1 5 3 1 5 , 5 1 3 5 1
*chromatic tetrachord octave species: 1 1 4 3 1 1 4, 4 1 1 3 4 1 1, 1 4 1 3 1 4 1
*[[Chopsticks]] double octave scale: 4 2 4 2 4 2 4 2 4 2
*[[5- to 10-tone scales in 47zpi]] (slightly stretched 15edo)

=== Horagrams===
[[File:Screen Shot 2020-04-24 at 12.04.03 AM.png|none|thumb|986x986px|2\15 MOS with 1L 1s, 1L 2s, 1L 3s, 1L 4s, 1L 5s, 1L 6s, 7L 1s]]
[[File:Screen Shot 2020-04-24 at 12.04.45 AM.png|none|thumb|735x735px|4\15 MOS using 1L 1s, 1L 2s, 3L 1s, 4L 3s, 4L 7s]]
[[File:Screen Shot 2020-04-24 at 12.05.29 AM.png|none|thumb|927x927px|7\15 MOS using 1L 1s, 2L 1s, 2L 3s, 2L 5s, 2L 7s, 2L 9s, 2L 11s ]]

==Diagrams==
[[File:15edo_wheel.png|alt=15edo wheel.png|225x225px|15edo wheel.png]] [[File:15edo_wheel_02.png|alt=15edo wheel 02.png|250x250px|15edo wheel 02.png]] [[File:15edo_wheel_03.png|alt=15edo wheel 03.png|220x220px|15edo wheel 03.png]]

=== Keyboard===
[[File:Porcupine keyboard major triad shapes.png|none|thumb|500x500px|Major chord shapes for a Porcupine keyboard in 15edo, using Blackwood logic (i.e. native fifth notation) for the letters.]]

===Lumatone ===
''See [[Lumatone mapping for 15edo]]''

==Music==
{{Main| 15edo/Music }}
{{Catrel|15edo tracks}}
; [http://micro.soonlabel.com/15-ET/ XA 15-ET Directory]

==Other info==
==== Keyboard layouts ====
<gallery widths="300px">
File:15_tone_keyboard.png|Porcupine layout for 15edo
File:Screen Shot 2020-04-23 at 11.59.17 PM.png|Hanson layout for 15edo
File:15edo kb3.png|Zarlino layout for 15edo
</gallery>
==Further reading==
=== Theory===
* Carson, Brent. [http://web.archive.org/web/20121025054304/http://home.comcast.net/~brentishere/15noteequaltempermenttutorial.html Fifteen Note Equal Temperment]
*[[Darreg, Ivor]]. ''[http://tonalsoft.com/sonic-arts/darreg/dar35.htm 15-Tone Scale System]''. 1991. (Originally at [http://sonic-arts.org/darreg/dar35.htm], now broken)
*InTeAS. ''[https://web.archive.org/web/20110713044141/http://www.inteas.com/Penta01.htm The Pentadecaphonic System]'' (2001, archived)

===Guitar===
*[[Sword, Ron]]. [http://www.metatonalmusic.com/books.html Pentadecaphonic Scales for Guitar: Scales, Chord-Scales, Notation, and Theory for Fifteen Equal Divisions of the Octave]''. (A large repository of all known scales and temperament families in the 15edo system. 300+ examples with chord-scale progressions.)''

==Notes==
<references group="note" />

[[Category:15edo| ]] 
[[Category:Equal divisions of the octave|##]] 
[[Category:Guitar]]
[[Category:Teentuning]]

Talk:Tenney norm

2025-04-30T19:46:58Z

Cmloegcmluin: /* Move to Tenney norm */

{{WSArchiveLink}}

== Move to Tenney norm ==

This complexity is defined as a norm, which is (mostly) a stronger condition than being a height function.
In fact, the whole reason it is useful is because it is a norm, not a height.
For this reason, the title should be either "Tenney norm", or "Tenney harmonic distance", because that's what it was called originally.
– [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 18:18, 30 April 2025 (UTC)

: Agreed. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 19:46, 30 April 2025 (UTC)

Talk:Height

2025-04-30T02:38:55Z

Cmloegcmluin: /* Counterexample */

{{WSArchiveLink}}

== Height is not dissonance ==
Not a big fan of this opening:
: The height is a tool to measure the dissonance of JI intervals.

It measures the complexity. This might be related to dissonance, but not in an obvious way.

-[[User:Sintel|Sintel]] ([[User talk:Sintel|talk]]) 14:20, 11 April 2022 (UTC)

: Agreed. Feel free to improve it. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 15:44, 11 April 2022 (UTC)

:: Agreed. As far as I can tell, "height" and "complexity" are synonymous. If they're not, then it may be a good idea to explain the difference on the page. I don't understand what the motivation is for using the term "height" when we already have the descriptive and in-common-use term "complexity". --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 15:56, 11 April 2022 (UTC)

::: Maybe that's naive, but I'd say, it's not obvious that complexity (in a more common sense) needs to be measured in one dimension. Height seems to be a one-dimensional measure. --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 06:29, 12 April 2022 (UTC)

:::: Sorry, but I'm a little confused by your previous post; I don't understand whether you're saying that what I had just said was naive or that what you say next is naive. Because I'm confused by that, I'm not sure whether you agree with me or not. In either case, what you say next reads to me as a defense of "complexity" over "height", because many of the "heights" we use in xen are indeed measurements of multidimensional objects: prime-count vectors representing JI intervals. So that's another good point, and one that I hadn't considered before. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 14:56, 12 April 2022 (UTC)

::: 'I don't understand what the motivation is for using the term "height" when we already have the descriptive and in-common-use term "complexity".'
::: It's borrowed from mathematics: [[Wikipedia: Height function]] [[User:ResonantFrequencies|ResonantFrequencies]] ([[User talk:ResonantFrequencies|talk]]) 19:22, 30 December 2022 (UTC)

:::: Right. Complexity is a subjective concept. Height is a rigorously defined mathematical object. So complexity isn't necessarily measured by height. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 07:49, 31 December 2022 (UTC)

== gradus suavitatis ==

Does Euler's ''gradus suavitatis'' count? What about the other metrics listed in Scala? (DEPTH, ENTROPY, GRADUS, HARMON, HEIGHT, MANN, MAX, PROOIJEN, RHSM, RECTANGULAR, TENNEY, TE_NORM, TRIANGLE, TR_LOG, VOGEL, WEIL, WILSON) [[User:ResonantFrequencies|ResonantFrequencies]] ([[User talk:ResonantFrequencies|talk]]) 19:30, 30 December 2022 (UTC)

: Could you provide some materials on what it is? [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 07:49, 31 December 2022 (UTC)

: The gradus suavitatis is indeed a proper height function. It might be worth adding because of its historical significance. Not sure about the other ones, though a lot of those (tenney, te_norm, weil, wilson) are already described as height functions on the wiki. – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 13:43, 25 April 2025 (UTC)

== Counterexample ==

Not sure how relevant this is, but I was wondering why we have "height" and "complexity" as different things. A counterexample might clear up the difference. Consider the total number of prime factors of a number. This is some kind of complexity measure. It's really just the unweighted l_1 norm when expressed in vector form, so it seems quite sensible. For example 5/4 = 2^-2 * 5^1, so h(5/4) = 3. Similarly we have h(81/80) = 9. This satisfies all of the criteria except finiteness, all prime numbers p have h(p/1) = 1. So there are infinitely many rationals for which h(x) <= C.

So this defines a complexity which is not a height.

– [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 13:41, 25 April 2025 (UTC)

: Agreed. The [[Complexity]] page will also require an update, as it literally states that the complexity of an interval is called "height", whereas this counterexample shows that heights are a subset of interval complexity measures. --[[User:Fredg999|Fredg999]] ([[User talk:Fredg999|talk]]) 20:45, 25 April 2025 (UTC)

:: Actually, I'm not really sure if we should be talking about height functions at all, since those properties aren't really used anywhere! The only place heights actuall show up is in the proof for Dirichlet (logflat) badness, where it's actually a height on temperaments, not intervals! And then the actual height functions we do end up using are actually just norms on some vector space. – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 11:19, 26 April 2025 (UTC)

::: I'd love to do away with heights and stick to complexities. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 02:38, 30 April 2025 (UTC)

Talk:Generator embedding optimization

2025-04-20T18:02:15Z

Cmloegcmluin:

== Consider splitting page ==

This page is rather long and uses a lot of latex, which takes some time to typeset properly.

On my old laptop, the page takes 12 seconds (!) to lay out.

– [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 21:52, 19 April 2025 (UTC)

: lol. Okay, I will do this soonish. Should be straightforward enough, but I want to make sure to update all the other pages that link here. And I am 'extremely' busy for the next couple weeks, but I have bookmarked it to do after I get an apartment, move, and settle into my new role at work. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 18:02, 20 April 2025 (UTC)

Sagittal notation

2025-03-29T16:38:27Z

Cmloegcmluin: /* The Athenian symbol set */ tweaks for clarity (feel free to revert anything you don't like, Dave)

[[de:Sagittalnotation]]
[[file:sagittal_sample.gif]]

'''Sagittal notation''' is a [[musical notation]] system capable of notating almost any conceivable tuning. It was developed by [[Dave Keenan]] and [[George Secor]] with significant contributions from numerous others.

== Flavors of Sagittal notation ==
Sagittal notation comes in two mutually compatible ''flavors''.

=== Evo ===
The '''Evo''' flavor (short for "evolutionary", previously called "mixed") uses only single-shaft Sagittal symbols, e.g. {{sagittal| /| }} {{sagittal| \! }} {{sagittal| |) }} {{sagittal| !) }}, alone or in combination with conventional sharps and flats and their doubles. Only the large variant of the double sharp {{sagittal| x }} (U+E47D) is considered to be stylistically-compatible with Sagittal symbols. Evo is much easier to learn, but it results in a greater number of symbols on the sheet, which can give it a more cluttered appearance, particularly with chords, and it may be confusing when two symbols alter the same note in opposite directions.

 A sub-flavor of Evo is '''Evo-SZ''' (Evo with Stein–Zimmermann). This is where any sagittals that are notating exactly half the alteration of a sharp or flat (most often {{sagittal| /|\ }} {{sagittal| \!/ }}) are replaced by the Stein–Zimmermann semisharp {{sagittal| > }} and narrow reversed flat {{sagittal| < }}, and the corresponding combinations (most often {{sagittal| /|\ }}{{sagittal| # }} and {{sagittal| \!/ }}{{sagittal| b }}) are replaced by {{sagittal| ># }} and {{sagittal| <b }}. The narrow variants of the fractional flats {{sagittal| < }} (U+E284) and {{sagittal| <b }} (U+E285) are preferred because they preserve the Sagittal principle that the visual size of a symbol should indicate the relative size of its alteration and they reduce left-right confusion.

=== Revo ===
The '''Revo''' flavor (short for "revolutionary", previously called "pure") only requires one accidental per note. Revo therefore takes up less space on the sheet and presents a cleaner appearance, and it clearly indicates the direction of the overall alteration. It discards the conventional sharps and flats and their doubles and replaces them with these multi-shaft arrow-like symbols: {{sagittal| /||\ }} {{sagittal| \!!/ }} {{sagittal| /X\ }} {{sagittal| \Y/ }}. Adding a sharp or flat to a Sagittal is achieved by adding two more shafts, e.g. {{sagittal| /| }}{{sagittal| # }} becomes {{sagittal| /||| }} and {{sagittal| !) }}{{sagittal| b }} becomes {{sagittal| !!!) }}. When the Sagittal part alters in the opposite direction to the sharp or flat part, the rules are not so simple, e.g. {{sagittal| \! }}{{sagittal| # }} becomes {{sagittal| ||\ }} and {{sagittal| |) }}{{sagittal| b }} becomes {{sagittal| !!) }}; one must simply learn these ''[[2187/2048#Notation|apotome]] complements''.

== Resources ==
* [http://sagittal.org/ Official site]
* [http://forum.sagittal.org Sagittal Forum]
* [http://sagittal.org/sagittal.pdf The original Xenharmonikon article (updated)]
* [http://sagittal.org/gift/GiftOfTheGods.htm Gift of the Gods: a Mythical introduction to Sagittal notation]
* [http://sagittal.org/Sagittal%20Standard%20JI%20Notation%20Calculator%20Spreadsheet.xlsx spreadsheet-based calculator for Sagittal JI notation]
* [http://sagittal.org/Sagittal-SMuFL-Map.pdf Sagittal-SMuFL-Map, a table of every Sagittal symbol]
* [[Pain free guide to Sagittal]] by [[William Lynch]]
* [[:File:24 Edo.pdf]] – Sagittal notation guide for 24edo by William Lynch (download: [{{filepath:24 Edo.pdf}} ''24_Edo.pdf''])
* [[Introductory_examples_in_Sagittal_notation|Introductory examples]] by [[Hans Straub]]
* [https://andrewmeronek.com/music-resources/sagittal-chord-lists/ Sagittal chord chart] by [[Andrew Meronek]]

== Notation software support ==
=== Sibelius ===
Sagibelius 2.0 – plugins for using Sagittal notation in Sibelius 4 and up. By [http://www.jacobbarton.net/2011/10/sagibelius-2-0-released/ Jacob Barton]. Hosted on this wiki. Donationware.

[[:File:Sagibelius_2.0.zip|Sagibelius_2.0.zip]]

=== Lilypond ===
[http://x31eq.com/lilypond/ Plugin for Sagittal notation in Lilypond] by Graham Breed

=== MuseScore ===
Sagittal accidentals are available in MuseScore via the [https://www.smufl.org/fonts/ Bravura font which implements the SMuFL standard]. They can be accessed by opening the Master Palette and finding them in the Symbols section at the end.

=== Scala ===
Sagittal notation is available in Scala.

=== Dorico ===
Because Dorico is built by Steinberg Media, the same company that maintains the SMuFL standard, it supports Sagittal.

== Scores in Sagittal notation ==
* [[The Sagittal Songbook]]
** [http://oneforall.ytmnd.com One for All]
** [[land_urchin|Land Urchin]]
** [[Clouds_(Andrew_Heathwaite)|Clouds]]
** [[Prayer of Thanks]]

* [[:File:sunday3.pdf|Sunday Pipes]] in [[22edo]] by [[Mats Öljare]]
* Tibia in [[22edo]] by [[Paul Erlich]] ([http://www.youtube.com/watch?v=d44Lfp9lAG8 Listen]). Sagittal score [[:File:TIBIA.pdf|in F||\]] or [[:File:tibia_in_g.pdf|in G]] (contains errors in measures 9, 19 and 20)
* [[:File:Ivor_Darreg,_Suite.pdf|On the Enharmonic Tetrachord (from Suite, Op. 62)]], in [[22edo]], by [[Ivor Darreg]]. Originally printed in the Spring 1975 issue of Xenharmonikon in quarter-tone notation. Transcribed to Sagittal by [[Juhani Nuorvala]].([https://www.youtube.com/watch?v=DvHvza1vtfo Listen])

== The Athenian symbol set ==
Early in the design of the Sagittal notation system, Secor and Keenan found that an economical JI notation system could be defined by dividing the apotome (Pythagorean sharp or flat) into 21 almost-equal divisions. This required only five pairs of up and down microtonal accidentals, although a few others were added for convenience in alternative spellings. This 10-symbol set ([[Sagittal notation#Athenian extension single-shaft|shown below]]), in combination with the Spartan symbol set which it extends, is called the Athenian symbol set. It exactly notates many common 11-limit ratios and the 17th harmonic, while also approximating many common 13-limit ratios within ±0.4{{c}}. If the divisions were made exactly equal, this would be an example of [[Brahmagupta]] temperament.

== Gallery ==
=== Spartan single-shaft ===
<div style="margin-bottom: 20px;">
{{sge| ¦( | !( | n | 5120/5103 | 7/5 kleisma }}
{{sge| /¦ | \! | p | 81/80 | 5 comma }}
{{sge| ¦) | !) | t | 64/63| 7 comma }}
{{sge|//¦ |\\! | ph | 6561/6400 | 25 small diesis }}
{{sge| /¦) | \!) | pat | 36/35 | 35 medium diesis }}
{{sge| /¦\ | \!/ | pak | 33/32 | 11 medium diesis }}
{{sge| (¦) | (!) | jat | 729/704 | 11 large diesis }}
{{sge| (¦\ | (!/ | jak | 8505/8192 | 35 large diesis }}
</div>

=== Spartan multi-shaft ===
Multi-shaft sagittals are only used in the [[#Revo|Revo]] flavor of Sagittal.
<div style="margin-bottom: 20px;">
{{sge| )¦¦( | )!!( | ph | 6561/6400 | 25S }}
{{sge| ¦¦) | !!) | t | 64/63 | 7C }}
{{sge| ¦¦\ | !!/ | p | 81/80 | 5C }}
{{sge| /¦¦) | \!!) | n | 5120/5103 | 7/5k }}
{{sge| /¦¦\ | \!!/ }}
{{sge| ¦¦¦( | !!!( | n | 5120/5103 | 7/5k }}
{{sge| /¦¦¦ | \!!! | p | 81/80 | 5C }}
{{sge| ¦¦¦) | !!!) | t | 64/63 | 7C }}
{{sge|//¦¦¦ |\\!!! | ph | 6561/6400 | 25S }}
{{sge| /¦¦¦) | \!!!) | pat | 36/35 | 35M }}
{{sge| /¦¦¦\ | \!!!/ | pak | 33/32 | 11M }}
{{sge| (¦¦¦) | (!!!) | jat | 729/704 | 11L }}
{{sge| (¦¦¦\ | (!!!/ | jak | 8505/8192 | 35L }}
{{sge| )X( | )Y( | ph | 6561/6400 | 25S | mid=48px }}
{{sge| X) | Y) | t | 64/63 | 7C | mid=48px }}
{{sge| X\ | Y/ | p | 81/80 | 5C | mid=48px }}
{{sge| /X) | \Y) | n | 5120/5103 | 7/5k | mid=48px }}
{{sge| /X\ | \Y/ | mid=48px }}
</div>

=== Athenian extension single-shaft ===
<div style="margin-bottom: 20px;">
{{sge| )¦( | )!( | ran | 896/891 | 11/7 kleisma }}
{{sge| ~¦( | ~!( | san | 4131/4096 | 17 comma }}
{{sge| ¦\ | !/ | k | 55/54 | 55 comma }}
{{sge| (¦ | (! | j |45927/45056| 11/7 comma }}
{{sge| (¦( | (!( | jan | 45/44 | 11/5 small diesis }}
</div>

=== Athenian extension multi-shaft ===
Multi-shaft sagittals are only used in the [[#Revo|Revo]] flavor of Sagittal.
<div style="margin-bottom: 20px;">
{{sge| ~¦¦( | ~!!( | jan | 45/44 | 11/5S }}
{{sge| )¦¦~ | )!!~ | j |45927/45056| 11/7C }}
{{sge| /¦¦ | \!! | k | 55/54 | 55C }}
{{sge| (¦¦( | (!!( | san | 4131/4096 | 17C }}
{{sge|//¦¦ |\\!! | ran | 896/891 | 11/7k }}
{{sge| )¦¦¦( | )!!!( | ran | 896/891 | 11/7k }}
{{sge| ~¦¦¦( | ~!!!( | san | 4131/4096 | 17C }}
{{sge| ¦¦¦\ | !!!/ | k | 55/54 | 55C }}
{{sge| (¦¦¦ | (!!! | j |45927/45056| 11/7C }}
{{sge| (¦¦¦( | (!!!( | jan | 45/44 | 11/5S }}
{{sge| ~X( | ~Y( | jan | 45/44 | 11/5S | mid=48px }}
{{sge| )X~ | )Y~ | j |45927/45056| 11/7C | mid=48px }}
{{sge| /X | \Y | k | 55/54 | 55C | mid=48px }}
{{sge| (X( | (Y( | san | 4131/4096 | 17C | mid=48px }}
{{sge|//X |\\Y | ran | 896/891 | 11/7k | mid=48px }}
</div>

== See also ==
* [[Sagitype]]

{{Navbox notation}}

[[Category:Notation]]
[[Category:Sagittal notation| ]]

Talk:Detempering

2025-03-27T02:02:15Z

Cmloegcmluin:

== Compact lattice ==

What is a "compact lattice"? I don't find a definition for this concept on the wiki. Does this relate to a specific mathematical concept of compactness, e.g. https://en.wikipedia.org/wiki/Compact_space? I'm embarrassed to note that the phrase has been on the page since the day I created it, so I'm responsible for this. Often I add stuff to the wiki at the behest of old-school RTT friends who don't participate in the wiki, and this has the air of something someone might have emailed, messaged, or texted me on a recommendation, but I can't find a record of it. It's possible this is just an informal usage, but I feel like I do see people sometimes use this "compact" term in a technical-seeming way, so if it is an objectively-defined property of importance to xen, I'd like to see it explicitly defined at least here and possibly in another centralized place on the wiki. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 22:51, 24 January 2022 (UTC)

== Incorrect usage of "transversal"? ==

I set out to edit this page for clarity just now, and as I did so, I ended up finding what appeared to be a mistake, and so I removed it. But then I noticed this mistake in another place, which got me thinking that perhaps this was an intentional use of the word in a way I was unfamiliar with. (For now, it has still been removed from that page, but not declared to have been a mistake.)

Here's what it is. [https://en.xen.wiki/index.php?title=Detempering&diff=67398&oldid=65268 Inthar modified the page] to read "'''detempering''' is the process of taking a tempered pitch system and replacing each of its pitches with one or more [[JI]] pitches that the pitch represents (called a [[transversal]])}}" (my highlighting). I thought this was a mistake because I figured "called a transversal" was referring to the entire phrase, a "[replacement] of each of [a tempered pitch system]'s pitches with one ore more JI pitches that the pitch represents". And a transversal is defined to select only ''exactly one pitch'' that the tempered pitch represents, never more than one.

But then I noticed the following sentence: "The first tval maps the first generator (for which 2/1 is a transversal) to 7 steps and the second generator (for which 10/9 is a transversal) to 1 step, and similarly with the second tval." It is found on this page, toward the bottom: https://en.xen.wiki/w/Temperament_mapping_matrices So here "transversal" is being used apparently to mean "member of its preimage". (This edit was from Mike Battaglia back in 2012, but it looks like he may have just imported it in an automated move along with a ton of other stuff wholesale from the previous version of the wiki, so I'm not attributing any personal intentionality to him here: https://en.xen.wiki/index.php?title=Temperament_mapping_matrices&diff=27089&oldid=27090)

And when you come back to the Detempering page and review Inthar's edit, it now is clear that he may also have intended "called a transversal" not to refer to the entire resulting detempering, but actually just to the phrase "JI [pitch] that the pitch represents," or in other words "member of its preimage."

So to me, this usage is quite confusing. It no longer has anything to do with the actual meaning of "transversal". A transversal transverses a list of sets. Such as a transversal of preimages of generators for a temperament, or of pitches in a tempered scale. It seems that "transversal", via its typical association in RTT with lists of preimages, has come to be used simply for a member of a preimage, out of any context of transversing. I don't think we should use "transversal" in this way, but I'd like to hear what other people think. Maybe I'm all wet on this, or maybe there's something else I don't know yet.

--[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 22:57, 24 January 2022 (UTC)

Also I note that this usage of "transversal" does not appear on the present xen wiki page for [[transversal]]. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 16:48, 26 January 2022 (UTC)

I now understand that "transversal" can be used as short for "transversal scale", but in contexts where it can be this confusing, I suggest including "scale" should be done. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 18:25, 31 January 2023 (UTC)

It appears that my concerns have been addressed by now, by removing references to "transversal" on this page, except for one at the bottom where it is merely acknowledged that people have (confusingly!) called these that. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 23:11, 26 March 2025 (UTC)

: Imo the word transversal is just confusing. All existing detemperings just pick one pitch of the preimage, so we don't really have to worry about it. If you ask me just get rid of any mention of transversal. – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 23:49, 26 March 2025 (UTC)

:: Word. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 23:46, 26 March 2025 (UTC)

::: Oh, and there is also [[preimage]] which we may at some point merge into here? Although it is sort of a generic concept. I added 'detempering' to the 'see also' section on there since there are zero references to it otherwise. – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 23:49, 26 March 2025 (UTC)

:::: Yeah I saw you did that, good stuff; I supplemented it with a few-word description of the connection between the two ideas. Re: going further and actually merging the "preimage" material into this "detempering" page, I could see that working — perhaps by simply yoinking the entirety of the current preimage page and making it the first subsection of the page, before the "One-to-one detemperings of equal temperaments" subsection? --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 02:02, 27 March 2025 (UTC)

Talk:Detempering

2025-03-26T23:46:45Z

Cmloegcmluin:

== Compact lattice ==

What is a "compact lattice"? I don't find a definition for this concept on the wiki. Does this relate to a specific mathematical concept of compactness, e.g. https://en.wikipedia.org/wiki/Compact_space? I'm embarrassed to note that the phrase has been on the page since the day I created it, so I'm responsible for this. Often I add stuff to the wiki at the behest of old-school RTT friends who don't participate in the wiki, and this has the air of something someone might have emailed, messaged, or texted me on a recommendation, but I can't find a record of it. It's possible this is just an informal usage, but I feel like I do see people sometimes use this "compact" term in a technical-seeming way, so if it is an objectively-defined property of importance to xen, I'd like to see it explicitly defined at least here and possibly in another centralized place on the wiki. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 22:51, 24 January 2022 (UTC)

== Incorrect usage of "transversal"? ==

I set out to edit this page for clarity just now, and as I did so, I ended up finding what appeared to be a mistake, and so I removed it. But then I noticed this mistake in another place, which got me thinking that perhaps this was an intentional use of the word in a way I was unfamiliar with. (For now, it has still been removed from that page, but not declared to have been a mistake.)

Here's what it is. [https://en.xen.wiki/index.php?title=Detempering&diff=67398&oldid=65268 Inthar modified the page] to read "'''detempering''' is the process of taking a tempered pitch system and replacing each of its pitches with one or more [[JI]] pitches that the pitch represents (called a [[transversal]])}}" (my highlighting). I thought this was a mistake because I figured "called a transversal" was referring to the entire phrase, a "[replacement] of each of [a tempered pitch system]'s pitches with one ore more JI pitches that the pitch represents". And a transversal is defined to select only ''exactly one pitch'' that the tempered pitch represents, never more than one.

But then I noticed the following sentence: "The first tval maps the first generator (for which 2/1 is a transversal) to 7 steps and the second generator (for which 10/9 is a transversal) to 1 step, and similarly with the second tval." It is found on this page, toward the bottom: https://en.xen.wiki/w/Temperament_mapping_matrices So here "transversal" is being used apparently to mean "member of its preimage". (This edit was from Mike Battaglia back in 2012, but it looks like he may have just imported it in an automated move along with a ton of other stuff wholesale from the previous version of the wiki, so I'm not attributing any personal intentionality to him here: https://en.xen.wiki/index.php?title=Temperament_mapping_matrices&diff=27089&oldid=27090)

And when you come back to the Detempering page and review Inthar's edit, it now is clear that he may also have intended "called a transversal" not to refer to the entire resulting detempering, but actually just to the phrase "JI [pitch] that the pitch represents," or in other words "member of its preimage."

So to me, this usage is quite confusing. It no longer has anything to do with the actual meaning of "transversal". A transversal transverses a list of sets. Such as a transversal of preimages of generators for a temperament, or of pitches in a tempered scale. It seems that "transversal", via its typical association in RTT with lists of preimages, has come to be used simply for a member of a preimage, out of any context of transversing. I don't think we should use "transversal" in this way, but I'd like to hear what other people think. Maybe I'm all wet on this, or maybe there's something else I don't know yet.

--[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 22:57, 24 January 2022 (UTC)

Also I note that this usage of "transversal" does not appear on the present xen wiki page for [[transversal]]. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 16:48, 26 January 2022 (UTC)

I now understand that "transversal" can be used as short for "transversal scale", but in contexts where it can be this confusing, I suggest including "scale" should be done. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 18:25, 31 January 2023 (UTC)

It appears that my concerns have been addressed by now, by removing references to "transversal" on this page, except for one at the bottom where it is merely acknowledged that people have (confusingly!) called these that. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 23:11, 26 March 2025 (UTC)

: Imo the word transversal is just confusing. All existing detemperings just pick one pitch of the preimage, so we don't really have to worry about it. If you ask me just get rid of any mention of transversal. 23:41, 26 March 2025 (UTC)

:: Word. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 23:46, 26 March 2025 (UTC)

Talk:Transversal

2025-03-26T23:45:59Z

Cmloegcmluin: /* See also */

== See also ==
https://en.xen.wiki/w/Transversal_generators

- [[User:Sintel|Sintel]] ([[User talk:Sintel|talk]]) 02:23, 21 December 2021 (UTC)

: Indeed, this page makes no mention of the second major application of transversals in RTT: transversals of the preimages of generators, per the link Sintel shared above. That should be added to this page, at least as a "See also" link, but probably briefly conceptually introduced as well as linked. I cannot do this myself, though, and neither can Sintel, because we are not admins, and the page along with other mathematically-advanced pages has recently been locked. So would an admin please add something like this for us.

: Furthermore, this page defines transversal in general, which is fine. But it is primarily about ''transversal scales'', and at this time, it is the main place where those are defined. For now I've created the page Transversal scale to redirect here. But I suggest one of two possible things should be done next:
# Option 1: Most of the material from this page should be extracted to turn this new Transversal scale page into a page in its own right, leaving behind here only information that's about transversals in and of themselves (and links out to the two existing applications of them: scales and generators).
# Option 2: The page about generators preimage transversals should be merged in with this page, and so the resulting page will cover all three topics — transversals in general, scales, and generators.

: --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 22:56, 24 January 2022 (UTC)

: I still think one of these two options I've presented should be done. I do not have a strong preference between them. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 19:51, 5 December 2022 (UTC)

:: I see that we have taken an Option #3, which is to simply remove this material entirely, rather than move most of it into "transversal scale". That's fine I guess. And it redirects to "detempering" rather than "generator detempering", the latter of which was the one I suggested could be merged in with this. That's fine. I still think we should merge the "generator detempering" page in with the "detempering" page, since it's really just a special case of it. Maybe I'll try that out and see how it fits. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 23:09, 26 March 2025 (UTC)

::: I think detempering should just be about scales that are detempers, while the transversal generator page is about finding out what JI intervals generate a certain temperament. Their uses are not really overlapping that much so I'm fine with keeping them seperate pages. EDIT: to add to this: I decided to make this a redirect since most of the references to this page are literally about detempering, and the contents of this page weren't really adding much relevant information on that topic. – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 23:43, 26 March 2025 (UTC)

:::: I came to the same conclusion when I started actually trying to make it work. Sweet. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 23:45, 26 March 2025 (UTC)

== Suggested ASCII diagram to illustrate concept ==

Dave and I have been discussing this topic over email and suggest that it could be a nice idea to supplement the definition of transversal here with a bare-bones illustration of the concept:

<nowiki>
|
{.........|...}
{...|.........}
{......|......}
|</nowiki>

visualizing how a transversal "transverses" a list of sets, sampling one from each. Again, I can't add it to the page myself, since it has been locked recently for only admins to edit. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 22:57, 24 January 2022 (UTC)

: I still think this should be added. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 19:51, 5 December 2022 (UTC)

: I'm just posting here as a reminder to admins about these two requests, in case they all missed them the first time. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 22:58, 6 May 2023 (UTC)

Talk:Detempering

2025-03-26T23:11:39Z

Cmloegcmluin: /* Incorrect usage of "transversal"? */

Talk:Transversal

2025-03-26T23:09:17Z

Cmloegcmluin:

== See also ==
https://en.xen.wiki/w/Transversal_generators

- [[User:Sintel|Sintel]] ([[User talk:Sintel|talk]]) 02:23, 21 December 2021 (UTC)

: Indeed, this page makes no mention of the second major application of transversals in RTT: transversals of the preimages of generators, per the link Sintel shared above. That should be added to this page, at least as a "See also" link, but probably briefly conceptually introduced as well as linked. I cannot do this myself, though, and neither can Sintel, because we are not admins, and the page along with other mathematically-advanced pages has recently been locked. So would an admin please add something like this for us.

: Furthermore, this page defines transversal in general, which is fine. But it is primarily about ''transversal scales'', and at this time, it is the main place where those are defined. For now I've created the page Transversal scale to redirect here. But I suggest one of two possible things should be done next:
# Option 1: Most of the material from this page should be extracted to turn this new Transversal scale page into a page in its own right, leaving behind here only information that's about transversals in and of themselves (and links out to the two existing applications of them: scales and generators).
# Option 2: The page about generators preimage transversals should be merged in with this page, and so the resulting page will cover all three topics — transversals in general, scales, and generators.

: --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 22:56, 24 January 2022 (UTC)

: I still think one of these two options I've presented should be done. I do not have a strong preference between them. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 19:51, 5 December 2022 (UTC)

:: I see that we have taken an Option #3, which is to simply remove this material entirely, rather than move most of it into "transversal scale". That's fine I guess. And it redirects to "detempering" rather than "generator detempering", the latter of which was the one I suggested could be merged in with this. That's fine. I still think we should merge the "generator detempering" page in with the "detempering" page, since it's really just a special case of it. Maybe I'll try that out and see how it fits. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 23:09, 26 March 2025 (UTC)

== Suggested ASCII diagram to illustrate concept ==

Dave and I have been discussing this topic over email and suggest that it could be a nice idea to supplement the definition of transversal here with a bare-bones illustration of the concept:

<nowiki>
|
{.........|...}
{...|.........}
{......|......}
|</nowiki>

visualizing how a transversal "transverses" a list of sets, sampling one from each. Again, I can't add it to the page myself, since it has been locked recently for only admins to edit. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 22:57, 24 January 2022 (UTC)

: I still think this should be added. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 19:51, 5 December 2022 (UTC)

: I'm just posting here as a reminder to admins about these two requests, in case they all missed them the first time. --[[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 22:58, 6 May 2023 (UTC)

Generator preimage

2025-03-26T19:28:23Z

Cmloegcmluin: I agree it's no good that this page doesn't link to Detempering, but isn't it better to give the explanatory link in the opening, since it's half of the name for this concept?

Also known as a '''generator preimage transversal''' or a '''generator [[detempering]]'''. Every [[generator]] of a [[regular temperament]] has a [[preimage]], which is an infinite set of JI intervals that map to it. A [[transversal]] means a selection of one representative element from each of a list of sets. So if for each generator in our temperament's list of generators we choose one JI interval that maps to it, then we have a generator preimage transversal for that temperament.

=Technical Definition=
Given a reduced list of [[Harmonic_Limit|p-limit]] vals V, we may define a set of transversal generators for V as a set of p-limit intervals q such that one of the vals of V maps q to 1 and the rest map it to 0. By ''reduced'' is meant that the GCD of the elements of each of the vals is 1--or in other words, none of the vals are [[contorted]]--and that they are [[linearly independent]], so that if there are r vals, the rank of V as a matrix is r.

If v1, v2, ... vr are the vals of V and t1, t2, ... tr are the corresponding transversal generators, then for any p-limit q we have, modulo the regular temperament defined by V

q ≅ t1^v1(q) * t2^v2(q) * ... * tr^vr(q)

In this way the transversal generators provide a transversal of the p-limit, and hence the name.

=Examples=
Suppose V consists of the 7-limit patent vals for 12 and 19; that is, V = [<12 19 28 34|, <19 30 44 53|]. Then a corresponding list of transversal generators is [49/48, 36/35]. 49/48 corresponds to one step of 12et, and zero steps of 19et, whereas 36/35 is zero steps of 12et, and one step of 19et. This gives us a septimal meantone transversal of the 7-limit where 3/2 is represented by (49/48)^7 * (36/35)^11, and 2 is represented by (49/48)^12 * (36/35)^19. A more familiar septimal meantone transversal starts from the normal val list, [<1 0 -4 -13|, <0 1 4 10|], which corresponds to the transversal generators [2, 3].

Given a list of transversal generators, we may append a [[comma basis]] for V and obtain a basis for the entire p-limit. For instance, we may extend [49/48, 36/35] to [49/48, 36/35, 81/80, 126/125]. Taking the corresponding matrix of monzos, whose rows are monzos for this list, inverting it and then transposing, we obtain

[<12 19 28 34|, <19 30 44 53|, <-4 -6 -9 -11|, <-5 -8 -12 -14|]

This is a [http://en.wikipedia.org/wiki/Unimodular_matrix unimodular matrix] defining a change of basis for the p-limit.

For another example, consider [<1 1 1 2|, <0 2 1 1|, <0 0 2 1|] which is the [[Normal_lists|normal val list]] for breed temperament, the temperament tempering out 2401/2400. A corresponding list of transversal generators is [2, 49/40, 10/7].

=Finding the generator preimage transversal=
{{todo|cleanup|inline=1|text=Add simpler algorithm}}

Two methods for finding the generator preimage transversal have been developed. The first was developed by [[Gene Ward Smith]] sometime in or before June 2011, which uses the [[Hermite normal form]]. The second was developed by [[User:Sintel|Sintel]] in December 2021, which uses the [[Smith normal form]].

== Method using the Smith Normal Form ==

So we want to find a generator preimage transversal <math>T</math> for a mapping <math>M</math> where:

<math>MT = I,</math>

and where <math>I</math> is the identity matrix. When this is the case, then for each generator of the temperament represented by <math>M</math>, a different column of <math>T</math> as a [[prime-count vector]] represents an interval that <math>M</math> maps to that generator. And when this <math>T</math> has all integer entries, then these generators are all JI.

Essentially we need a way to do:

<math>
M^{-1}MT = M^{-1}I \\
\cancel{M^{-1}M}T = M^{-1}I \\
T = M^{-1}
</math>

But there's two major problems with this simplistic approach:
# <math>M</math> is rectangular and therefore cannot be inverted (it has no inverse).
# The pseudoinverse we use in lieu of a true inverse in situations like this doesn't generally give integer results, and so it won't give us the JI generators we seek.
So the pseudoinverse won't solve this problem for us, but the Smith decomposition can help! For a given <math>M</math>, it finds invertible matrices <math>L</math> and <math>R</math> for which

<math>
LMR = D,
</math>

where <math>D</math> is the Smith Normal Form of <math>M</math>, which is a rectangular diagonal matrix (the name 'D' is for "diagonal"). Furthermore, if <math>M</math> was defactored, then not only is <math>D</math> diagonal (only has nonzero values along its main diagonal) but the numbers along its main diagonal are all equal to 1. This type of matrix is called an orthonormal matrix, and has the property that <math>D^{T}D = I</math>, which we're going to take advantage of.

So first let's solve the decomposition equation for <math>M</math>. First, left-multiply by <math>L^{-1}</math>:

<math>
L^{-1}LMR = L^{-1}D \\
\cancel{L^{-1}L}MR = L^{-1}D \\
MR = L^{-1}D
</math>

And then right-multiply by <math>R^{-1}</math>:

<math>
MRR^{-1} = L^{-1}DR^{-1} \\
M\cancel{RR^{-1}} = L^{-1}DR^{-1} \\
M = L^{-1}DR^{-1}
</math>

Now we can substitute this result in for <math>M</math> in our original equation:

<math>
(L^{-1}DR^{-1})T = I
</math>

And we can proceed to solve this for our target <math>T</math>. The rest is busywork. First, left-multiply by <math>L</math>:

<math>
LL^{-1}DR^{-1}T = LI \\
\cancel{LL^{-1}}DR^{-1}T = LI \\
DR^{-1}T = L
</math>

Then left-multiply by <math>D^{T}</math>:

<math>
D^{T}DR^{-1}T = D^{T}L \\
\cancel{D^{T}D}R^{-1}T = D^{T}L \\
R^{-1}T = D^{T}L
</math>

And finally, left-multiply by <math>R</math>:

<math>
RR^{-1}T = RD^{T}L \\
\cancel{RR^{-1}}T = RD^{T}L \\
T = RD^{T}L
</math>

And there's our answer! It's still not necessarily giving the simplest or best JI generators, but arriving at those is an independent problem (perhaps by choosing a complexity metric and minimizing it though linear combinations with the commas and the other generators, or perhaps using LLL).

=== Example ===

Our mapping is 5-limit meantone, in [[equave-reduced generator form]]:

<math>
\left[ \begin{array} {rrr}
1 & 1 & 0 \\
0 & 1 & 4 \\
\end{array} \right]
</math>

Its Smith Decomposition <math>L, D, R</math> is:

<math>
\begin{array} {lll}

\left[ \begin{array} {rrr}
1 & -1 \\
0 & 1 \\
\end{array} \right]

, &

\left[ \begin{array} {rrr}
1 & 0 & 0 \\
0 & 1 & 0 \\
\end{array} \right]

, &

\left[ \begin{array} {rrr}
1 & 0 & 4 \\
0 & 1 & -4 \\
0 & 0 & 1 \\
\end{array} \right]

\end{array}
</math>

And so a transversal of the preimages of its generators are <math>RD^{T}L</math>:

<math>
\left[ \begin{matrix}
1 & 0 & 4 \\
0 & 1 & -4 \\
0 & 0 & 1 \\
\end{matrix} \right]
\left[ \begin{matrix}
1 & 0 \\
0 & 1 \\
0 & 0 \\
\end{matrix} \right]
\left[ \begin{matrix}
1 & -1 \\
0 & 1 \\
\end{matrix} \right]
=
\left[ \begin{matrix}
1 & -1 \\
0 & 1 \\
0 & 0 \\
\end{matrix} \right]
</math>

That is, 2/1 and 3/2.

=== Wolfram Language implementation ===

{{Databox|getGpt[]|
<syntaxhighlight lang="mathematica">
getGpt[m_] := Module[{decomp, left, snf, right},
decomp = SmithDecomposition[m];
left = Part[decomp, 1];
snf = Part[decomp, 2];
right = Part[decomp, 3];

right.Transpose[snf].left
];
</syntaxhighlight>}}

== Method using the Hermite Normal Form ==

We can find a generator preimage transversal for V by the following procedure:

<ul><li>Take the transpose of the [[Tenney-Euclidean_Tuning#The pseudoinverse|pseudoinverse]] of V, call that U</li><li>Find a basis for the commas of V</li><li>For each row U[i] of U, clear denominators and append the monzos of the comma basis for V</li><li>[[Saturation|Saturate]] the result to a list of monzos, call that S</li><li>Apply the ith val V[i] (dot product) to each element of S</li><li>Insert V[i].S[j] in front of the elements of S[j] as the first element, obtaining the jth element T[j] of a modified list T</li><li>Hermite reduce the modified list T, take the first row, and remove the first element (which should be a 1.)</li><li>Consider the rest to be a monzo and convert it to a rational number</li><li>This is a corresponding transveral generator to the ith val V[i] of V; it may be reduced to an equivalent generator of minimal [[Tenney_Height|Tenney height]] by multiplying by the commas of V</li></ul>

=== Example ===

Note: I've followed the algorithm as described above. But clearly I am transposing things here way more often than is necessary than if the algorithm was superficially revised.

For 5-limit meantone, <math>V</math> is

<math>
\begin{bmatrix}
1 & 2 & 4 \\
0 & -1 & -4 \\
\end{bmatrix}
</math>

The pseudoinverse <math>V⁺</math> is

<math>
\frac{1}{33}
\begin{bmatrix}
17 & 18 \\
16 & 15 \\
-4 & -12 \\
\end{bmatrix}
</math>

And that transposed is <math>(V⁺)ᵀ</math>, which we'll call <math>U</math>:

<math>
\frac{1}{33}
\begin{bmatrix}
17 & 16 & -4 \\
18 & 15 & -12 \\
\end{bmatrix}
</math>

And here's the comma basis for <math>V</math>:

<math>
\begin{bmatrix}
-4 \\
4 \\
-1 \\
\end{bmatrix}
</math>

Beginning with <math>i = 1</math>, we'll create a matrix out of <math>U[i]</math> with denominators cleared and the comma basis appended (transposed so each comma is a row):

<math>
\begin{bmatrix}
17 & 16 & -4 \\
-4 & 4 & -1 \\
\end{bmatrix}
</math>

Defactor and call this <math>S</math>:

<math>
\begin{bmatrix}
1 & 32 & -8 \\
0 & 4 & -1 \\
\end{bmatrix}
</math>

Now take the dot product of <math>V[1]</math> to each element of <math>S</math>, or in other words, left multiply <math>Sᵀ</math> by <math>V[1]</math>:

<math>
\begin{bmatrix}
1 & 2 & 4 \\
\end{bmatrix}
\begin{bmatrix}
1 & 0 \\
32 & 4 \\
-8 & -1 \\
\end{bmatrix} =
\begin{bmatrix}
33 & 4 \\
\end{bmatrix}
</math>

Now prepend that result, transposed, to <math>S</math>:

<math>
\begin{bmatrix}
33 & 1 & 32 & -8 \\
4 & 0 & 4 & -1 \\
\end{bmatrix}
</math>

Take the Hermite Normal Form:

<math>
\begin{bmatrix}
1 & 1 & 0 & 0 \\
0 & 4 & -4 & 1 \\
\end{bmatrix}
</math>

Take the first row:

<math>
\begin{bmatrix}
1 & 1 & 0 & 0 \\
\end{bmatrix}
</math>

Remove the first element (which should and indeed is a 1):

<math>
\begin{bmatrix}
1 & 0 & 0 \\
\end{bmatrix}
</math>

So indeed that gives the period for meantone, 2/1.

Let's repeat the latter steps but now with <math>i = 2</math>. Here's <math>U[2]</math> with denominators cleared and the comma basis appended:

<math>
\begin{bmatrix}
18 & 15 & -12 \\
-4 & 4 & -1 \\
\end{bmatrix}
</math>

Defactor to get our new <math>S</math>:

<math>
\begin{bmatrix}
2 & 31 & -16 \\
0 & 2 & -1 \\
\end{bmatrix}
</math>

Now take the dot product of <math>V[2]</math> to each element of <math>S</math>, or in other words, left multiply <math>Sᵀ</math> by <math>V[2]</math>:

<math>
\begin{bmatrix}
0 & -1 & -4 \\
\end{bmatrix}
\begin{bmatrix}
2 & 0 \\
31 & 2 \\
-16 & -1 \\
\end{bmatrix} =
\begin{bmatrix}
33 & 2 \\
\end{bmatrix}
</math>

Now prepend that result, transposed, to <math>S</math>:

<math>
\begin{bmatrix}
33 & 2 & 31 & -16 \\
2 & 0 & 2 & -1 \\
\end{bmatrix}
</math>

Take the Hermite Normal Form:

<math>
\begin{bmatrix}
1 & 2 & -1 & 0 \\
0 & 4 & -4 & 1 \\
\end{bmatrix}
</math>

Take the first row:

<math>
\begin{bmatrix}
1 & 2 & -1 & 0 \\
\end{bmatrix}
</math>

Remove the first element (which should and indeed is a 1):

<math>
\begin{bmatrix}
2 & -1 & 0 \\
\end{bmatrix}
</math>

So indeed that gives the generator for meantone, 4/3. We're done!

=== Wolfram Language implementation ===

{{Databox|getGpt[]|
<syntaxhighlight lang="mathematica">
getGptEntry[u_, v_, c_] := Module[{base},
base = Transpose[columnHermiteDefactor[Join[{u}, Transpose[c]]]];

Drop[First[Take[hnf[Transpose[Join[{v}.base,base]]],1]],1]
];

getGptSmithMethod[m_] := Module[{c},
c = nullSpaceBasis[m];
Transpose[MapThread[getGptEntry[#1, #1, c]&, {Map[multByLcd,Transpose[PseudoInverse[m]]],m}]]
];

getGptSmithMethod[{{1,2,4},{0,-1,-4}}] (* {{1,2},{0,-1},{0,0}} = 2/1 and 4/3 as expected *)
</syntaxhighlight>}}

[[Category:generator]]
[[Category:Regular temperament theory]]
[[Category:todo:reduce_mathslang]]

Preimage

2025-03-26T19:26:19Z

Cmloegcmluin: /* See also */ briefly explicitly explain the reason for the connection between this page and the new one linked to in its See Also

For a [[regular temperament]], the '''preimage''' of a mapped interval is the set of all (typically [[JI|justly intoned]]) intervals that map to it.

For any interval that's a member of such a preimage, altering it by any one of the commas that the temperament [[vanish]]es finds another interval that's also a member of the same preimage. Preimages thereby technically contain an infinite number of such intervals, but usually only the simplest ones are of any interest.

== Examples ==

This section uses curly brackets for [[tmonzos and tvals|generator-count vector]]s representing mapped intervals. This helps distinguish them from the angle brackets used for [[prime-count vector]]s representing (unmapped) intervals.

=== A temperament with only one comma ===

For the meantone mapping {{rket|{{map|1 1 0}} {{map|0 1 4}}}}, let's find the preimage of the mapped interval [-1 2}. One member of the preimage is 10/9 AKA {{vector|1 -2 1}} because {{rket|{{map|1 1 0}} {{map|0 1 4}}}}{{vector|1 -2 1}} = [-1 2}. Another member of the preimage is 9/8 AKA {{vector|-3 2 0}} because {{rket|{{map|1 1 0}} {{map|0 1 4}}}}{{vector|-3 2 0}} = [-1 2} as well. Notice that these two intervals are one meantone comma (that's 81/80, or {{vector|-4 4 -1}}) apart: {{vector|1 -2 1}} + {{vector|-4 4 -1}} = {{vector|-3 2 0}}.

The meantone comma is the only comma that meantone makes vanish (meantone is a [[nullity]]-1 temperament, which means its [[comma basis]] or nullspace contains only a single interval), so any mapped interval's preimage here is going to be simply a series of intervals off from each other by yet another meantone comma. So more members can be found by repeatedly adding this one comma.

First we find {{vector|-3 2 0}} + {{vector|-4 4 -1}} = {{vector|-7 6 -1}} which is 729/640; then from there we find {{vector|-7 6 -1}} + {{vector|-4 4 -1}} = {{vector|-11 10 -2}} = 59049/51200; then we find {{vector|-11 10 -2}} + {{vector|-4 4 -1}} = {{vector|-15 14 -3}} = 4782969/4096000. We could go on like this literally forever, but clearly these intervals are quickly getting wildly complex. Furthermore, with each additional comma we add, the difference between the original interval's size (such as measured in cents) and its tempered size grows greater, so they are of less interest in that respect as well.

We could also proceed along this series of meantone-comma-separated members of [-1 2}'s preimage in the other direction, by repeatedly ''subtracting'' meantone commas from 9/8. But we won't work through that because we know a similar effect will happen: the intervals found will grow steadily more complex and with greater error under the temperament.

=== A temperament with multiple commas ===

We could repeat this experiment but with septimal meantone, {{ket|{{val|1 0 -4 -13}} {{val|0 1 4 10}}}}, which is still [[rank]]-2 but due to the one extra [[dimensionality]] according to the [[rank-nullity theorem]] is nullity-2. The additional comma it makes to vanish is [[126/125]] {{vector|1 2 -3 1}}, the starling comma. Here, 10/9 {{vector|1 -2 1 0}} and 9/8 {{vector|-3 2 0 0}} both still map to [-1 2}, so they are still members of [-1 2}'s preimage, along with all the intervals listed in the previous section. But here we have some additional members which are off by multiples of the second comma as well, such as {{vector|1 -2 1 0}} + {{vector|1 2 -3 1}} = {{vector|2 0 -2 1}}, which is 28/25, or {{vector|-3 2 0 0}} - {{vector|1 2 -3 1}} = {{vector|-4 0 3 -1}}, which is 125/112.

So in this case the full member set of the preimage for any mapped interval of this temperament such as [-1 2} could be arranged as a 2D grid, with members differing by a meantone comma along one axis and members differing by a starling comma along the other axis.

== Image ==

The "preimage" is literally "that which came before the image", or in other words, the set of possible things that the image might have been projected from.

A [[mapping]] in RTT always maps any given interval to only a single other interval. The '''image''' for an interval is that one interval that it's mapped to by a (regular) temperament.

== Terminology ==

The terms "[[Wikipedia:Image_(mathematics)|image]]" and "[https://mathworld.wolfram.com/Preimage.html preimage]" come from general mathematics where they are used regarding mathematical functions and their inputs (their domain) and outputs (their range). They are used in the same sense in regular temperament theory; in our application, [[mappings]] are the functions, intervals are the inputs, and mapped intervals are the outputs.

== See also ==
* [[Detempering]], replacing tempered pitches with pitches from their preimages

[[Category:Regular temperament theory]]
[[Category:Terms]]
[[Category:Math]]

Pathology of enfactoring

2025-03-20T23:39:50Z

Cmloegcmluin: /* Enfactored comma bases vs. periodicity blocks with torsion */ clarifications per Kite's feedback

In this article that relates to [[regular temperaments]], we will use lattices to demonstrate the musical implications of [[mappings]] that contain common factors, and the lack of musical implications of [[comma basis|comma bases]] that contain common factors.

== Defactored case ==

[[File:Unenfactored mapping.png|365px|thumb|right|A 3-limit tempered lattice, superimposed on the JI lattice]]

First, let's look at a [[defactoring|defactored]] mapping. This example temperament is so simple that it is not of practical musical interest. It was chosen because it's basically the numerically simplest possible example, where this type of simplicity empowers us to visualize the problem at a practical scale as clearly as possible. Please consider the diagram at right.

This is a representation of 2-ET, a 3-limit, rank-1 (equal) temperament, with mapping {{rket|{{map|2 3}}}}, meaning it has a single generator which takes two steps to reach the octave, and three steps to reach the tritave. This temperament makes a single comma [[vanish]], a comma whose vector representation looks similar to the mapping: {{vector|-3 2}}, AKA 9/8, the 3-limit major 2nd. And so the comma basis for this temperament is [{{vector|-3 2}}]. The generator is both 4/3 and 3/2. In musical terms, both the fourth and the fifth are so heavily tempered that they each become a half-octave.

We can imagine that we started out with a JI lattice, where movement up and down correspond to prime 2 (the octave) and movements right and left correspond to prime 3 (the tritave). We have tempered JI here, and so we've faded the JI lattice out to a faint grey color in the background. What we've done specifically is made the comma {{vector|-3 2}} vanish so that any nodes in this lattice which are 2 over and 3 up from each other are equivalent. Therefore we only need to consider a thin swath of the lattice anymore, specifically, a swath which connects the origin {{vector|0 0}}, AKA 1/1, to {{vector|-3 2}}, and then runs perpendicularly to infinity in either direction.

There's a couple good ways to interpret this situation:
# We've turned on a teleportation field for every point outside this swath, so that it moves by these (-3,2) intervals until it finds its way inside the swath.
# We've rolled up space, so that the line from 1/1 to 9/8 — the width of our swath — is like the circumference of a tube. In this case, since we're only working with 3-limit JI, "space" was only ever 2D, so we can just think of it as paper that we've rolled up, so the pair of dotted lines visualized here are touching along their entire lengths (and if you wanted to imagine further copies of these dotted lines at every (-3,2) interval farther out in either direction, and the paper is infinitely thin, just stack all the dotted lines on top of each other forever).

Either way, then we just superimpose the new tempered lattice on top. It's drawn in blue.

You can see that in the grey lattice underneath, coordinates have 2 values, e.g. {{vector|-1 2}}. That's because the JI lattice essentially had two generators: the octave and the tritave. But tempering has helped us simplify things by reducing us to a single generator, so here, in the new blue-colored tempered lattice, the coordinates have only 1 value, e.g. {{vector|8}}, and they simply indicate how many iterations of the single generator here that we've taken to reach the given point.

We've superimposed the tempered lattice atop the former JI lattice so we can see which JI intervals map to which tempered intervals. For example, {{vector|-1 2}}, AKA 9/2, maps to {{vector|8}}. That tells us that if we want to use the temperament's approximation of the JI interval 9/2, then we want the tempered pitch arrived at by moving by the generator 8 times.

Note that whenever the path the generator takes leaves the main swath, it wraps around to the point on the opposite side. Again, you can think of this either way you prefer: the world is still flat, and you've just warped over there; or, the world has been curled up, and in reality you've been looping back around toward that point the whole time, and the dotted line in this flat representation just represents the point you cut the tube and unrolled it so it could be better visualized on a screen.

The tempered vector for the generator of this temperament is of course {{vector|1}}, but the simplest JI pitch that maps to this generator is {{vector|-1 1}}, or 3/2. It's visually clear why this is the generator. We need to choose an interval which if we repeatedly move by it, while wrapping around the swath, we'll visit every node inside the swath. The best way to do that is to move from the origin to the node that's nearest to the dotted line labelled "tube circumference". We want this node because we want to move away from the tube circumference as little as possible each time, so we avoid skipping any nodes. If it's not obvious, you may want to experiment with drawing the generator line as if it had gone to any other point inside the swath; if you repeated that movement, would you visit every node? No.

All of this so far is actually only just explaining the basic setup for any tempered lattice. But we've got to lay the basics down first in order to discuss the effect of enfactoring. We'll do that now!

== Enfactored mapping ==

[[File:2-enfactored mapping.png|365px|thumb|right|A 2-enfactored mapping represents a temperoid for which every other step of its generator lands on a pitch which no JI interval would ever temper to.]]

We are now comparing the previous diagram, which had the mapping {{rket|{{map|2 3}}}}, with the 2-enfactored version of it, i.e. the mapping 2×{{rket|{{map|2 3}}}} = {{rket|{{map|4 6}}}}, AKA 4-ET.

If you compare this lattice of an enfactored mapping with the previous lattice for a healthy, defactored mapping, they should look almost the same. They have the same comma and tube circumference. And the generator follows the same path through that tube/swath. The key difference is how far the generator moves with each step along that path.

Starting from the origin, we can see that it takes us 2 moves of the generator to reach the approximation of {{vector|-1 1}}, AKA 3/2, where before we made that step in one go. Then another 2 moves to reach the approximation of {{vector|-2 2}}, AKA 9/4, for a total of 4 moves, where before it only took us 2 steps. As you keep going, you'll see that each node it has taken us 2x as many steps as before to reach it.

And for what? What happens in the steps that are halfway between nodes that were on the JI lattice? These are shown with hollow blue circles instead of filled blue circles, to indicate that there's no JI lattice node underneath them. In other words, while these are legitimate musical intervals, there is no JI interval which would be said to temper to them. In still other words, since this is 4-ET, that first generator step is to a node {{vector|1}} that's about 300¢. But {{vector|0 0}} tempers to {{vector|0}} and {{vector|-1 1}} tempers to {{vector|2}}; nothing tempers to {{vector|1}}. It's an interval that can certainly at least be heard and understood musically, but it has no meaning with respect to tempering JI, or said another way, it has no RTT purpose.

And so this 4-ET doesn't bring anything to the table that isn't already brought by 2-ET. And so it is fitting to consider it only a temperoid, rather than a true temperament. Were this as bad as things got, it might not be worth pushing for distinguishing temperoids from temperaments. But once we look at enfactored comma bases, we'll see why things get pretty pathological.

== Enfactored comma bases ==

[[File:2-enfactored comma-basis.png|365px|thumb|left|enfactored comma bases are garbage]]

Here's where things get kind of nuts. Most recently we experimented with enfactoring our healthy temperament's mapping. Now let's experiment with enfactoring its comma basis. In the defactored situation, if our comma basis was [{{vector|-3 2}}] = 9/8, then 2-enfactoring it produces 2×[{{vector|-3 2}}] = [{{vector|-6 4}}] = 81/64, the 3-limit major 3rd.

We know that in the original diagram, the large-labelled {{vector|-3 2}} represented our comma, and this was the point that our dotted line ran through, the one that represented our boundary of warp/wrap. So our first thought should be: we must alter our diagram so that now {{vector|-6 4}} is that point instead. Fine.

But here's the problem. It simply doesn't make sense to double the width of our swath/tube! If {{vector|-6 4}} is made to vanish, then so is {{vector|-3 2}}. That is, while nothing would stop you from drawing a diagram with a double-width swath/tube, the musical reality is that it is impossible to make {{vector|-6 4}} vanish without also making {{vector|-3 2}} vanish. In musical terms, the 3-limit major 3rd is the sum of two 3-limit major 2nds. If the major 3rd vanishes, the major 2nd must also. And so there is no meaning or purpose to the comma basis {{vector|-6 4}}, whether RTT-wise or musically in general. It is garbage.

And so our lattice for an enfactored comma basis looks almost identical to the original defactored lattice. The only difference here is that we've drawn a "supposed (but false)" tube circumference out to {{vector|-6 4}}, while the half of this length which is real is now labelled the "true" circumference.

== Enfactored comma bases vs. periodicity blocks with torsion ==

[[File:Torsion.png|400px|thumb|right|a reworking of the classic torsion example from Tonalsoft to reveal the twinned generator paths]]

And now we're prepared to confront the key difference between the enfactored comma basis of a temperament, and torsion of a periodicity block.

What they have in common is that both take the form of a common factor found somewhere in linear combinations of entries in a list of commas defining a pitch structure, and that these commas can be visualized by slicing the JI lattice into swaths of "periodicity" (that's just a fancy word for the effect we've already been observing for temperaments, where nodes outside the swath related by the size of that comma are considered equivalent and therefore redundant, or repetitions of the same pitch class).

The key difference is that the former (temperaments) tempers the commas out, while the latter (periodicity blocks) does not.

Why is this the key difference? Well, remember how in the previous section, the reason we couldn't actually extend the width of the swath/tube to {{vector|-6 4}} was because the tempering: if {{vector|-6 4}} is made to vanish, then {{vector|-3 2}} is as well, so the swath/tube cannot legitimately be extended. Since there is no tempering in the case of periodicity blocks, however, the width ''can'' legitimately be extended in this way.

Let's take a look at the example given in Tonalsoft's page for torsion. The diagram there has been reworked here to help clarify things. The origin, 1/1, has been placed in the corner of this parallelogram-shaped block, and the two commas that define it are in two of the other corners: [[2048/2025]] ({{vector|11 -4 -2}}) and 625/324 ({{vector|-2 -4 4}}). The value at the fourth corner, 12800/6561, has vector {{vector|9 -8 2}}. You may recognize this as the meantone comma (its inverse, anyway) squared, (80/81)^2 = {{vector|4 -4 1}}×2, but with an extra factor of 2 introduced in order to octave-reduce (well, in this case, "octave-increase") the ratio to between 1 and 2. The most important part is that the vector is 2-enfactored. You can see that the node at the very center of this block is 160/81 = {{vector|5 -4 1}}, which is unsurprisingly half of the ratio at the fourth corner, i.e. the ''non-enfactored'' version of the meantone comma (though, again, inverted and octave-adjusted to between 1 and 2).

The red and blue lines that wrap around this block are two different generator paths. The point here is to show that by doubling the size of this periodicity block, we have made it impossible to choose a node to travel to from the origin, i.e. a generator, such that you can reach every node in the block. Instead, the best you can do is reach half of the nodes; that's the red path from the origin 1/1. The blue path is an exact copy of the red path, but offset.

So we can see how tempting the duality can be here. In the case of a 2-enfactored mapping, the generator path reaches ''twice'' as many nodes as there were JI nodes. But in the case of a 2-enfactored comma basis — if we could legitimately extend the width of the block, as we do in untempered periodicity blocks! — we would reach ''half'' as many nodes. But this duality just is not musically, audibly real.

== Enfactored mappings vs. enfactored comma bases ==

One may pose the question: is there any relationship between an enfactored mapping and an enfactored comma basis? Does one imply the other. Does one imply the absence of the other? Or are they completely independent?

We note that there is no such thing as an enfactored temperament, only enfactored matrices. And these essentially result from an arithmetic oversight, namely failing to defactor, and as such they are completely independent<ref>as observed on the page [[Color_notation/Temperament_Names|color notation]], which reads "it's possible that there is both torsion and contorsion"</ref>. When we use matrix math for RTT we defactor as a matter of course. For example, defactoring is built in to the NullSpace matrix operation in the Wolfram Language.

However, if you have an artistic or other reason to generate scales using generators derived directly from enfactored mappings, and therefore to treat enfactored mappings as representing different musical objects, which we call temperoids, not temperaments, then you might choose to encode the same common factor into the comma basis representation of the temperoid with the understanding that this is a mere bookkeeping exercise and has no mathematical basis.

=Footnotes=
<references/>

[[Category:Regular temperament theory]]
[[Category:Terms]]
[[Category:Math]]

Pathology of enfactoring

2025-03-18T13:10:01Z

Cmloegcmluin: improvements suggested (verbatim) by Dave

In this article that relates to [[regular temperaments]], we will use lattices to demonstrate the musical implications of [[mappings]] that contain common factors, and the lack of musical implications of [[comma basis|comma bases]] that contain common factors.

== Defactored case ==

[[File:Unenfactored mapping.png|365px|thumb|right|A 3-limit tempered lattice, superimposed on the JI lattice]]

First, let's look at a [[defactoring|defactored]] mapping. This example temperament is so simple that it is not of practical musical interest. It was chosen because it's basically the numerically simplest possible example, where this type of simplicity empowers us to visualize the problem at a practical scale as clearly as possible. Please consider the diagram at right.

This is a representation of 2-ET, a 3-limit, rank-1 (equal) temperament, with mapping {{rket|{{map|2 3}}}}, meaning it has a single generator which takes two steps to reach the octave, and three steps to reach the tritave. This temperament makes a single comma [[vanish]], a comma whose vector representation looks similar to the mapping: {{vector|-3 2}}, AKA 9/8. And so the comma basis for this temperament is [{{vector|-3 2}}].

We can imagine that we started out with a JI lattice, where movement up and down correspond to prime 2 (the octave) and movements right and left correspond to prime 3 (the tritave). We have tempered JI here, and so we've faded the JI lattice out to a faint grey color in the background. What we've done specifically is made the comma {{vector|-3 2}} vanish so that any nodes in this lattice which are 2 over and 3 up from each other are equivalent. Therefore we only need to consider a thin swath of the lattice anymore, specifically, a swath which connects the origin {{vector|0 0}}, AKA 1/1, to {{vector|-3 2}}, and then runs perpendicularly to infinity in either direction.

There's a couple good ways to interpret this situation:
# We've turned on a teleportation field for every point outside this swath, so that it moves by these (-3,2) intervals until it finds its way inside the swath.
# We've rolled up space, so that the line from 1/1 to 9/8 — the width of our swath — is like the circumference of a tube. In this case, since we're only working with 3-limit JI, "space" was only ever 2D, so we can just think of it as paper that we've rolled up, so the pair of dotted lines visualized here are touching along their entire lengths (and if you wanted to imagine further copies of these dotted lines at every (-3,2) interval farther out in either direction, and the paper is infinitely thin, just stack all the dotted lines on top of each other forever).

Either way, then we just superimpose the new tempered lattice on top. It's drawn in blue.

You can see that in the grey lattice underneath, coordinates have 2 values, e.g. {{vector|-1 2}}. That's because the JI lattice essentially had two generators: the octave and the tritave. But tempering has helped us simplify things by reducing us to a single generator, so here, in the new blue-colored tempered lattice, the coordinates have only 1 value, e.g. {{vector|8}}, and they simply indicate how many iterations of the single generator here that we've taken to reach the given point.

We've superimposed the tempered lattice atop the former JI lattice so we can see which JI intervals map to which tempered intervals. For example, {{vector|-1 2}}, AKA 9/2, maps to {{vector|8}}. That tells us that if we want to use the temperament's approximation of the JI interval 9/2, then we want the tempered pitch arrived at by moving by the generator 8 times.

Note that whenever the path the generator takes leaves the main swath, it wraps around to the point on the opposite side. Again, you can think of this either way you prefer: the world is still flat, and you've just warped over there; or, the world has been curled up, and in reality you've been looping back around toward that point the whole time, and the dotted line in this flat representation just represents the point you cut the tube and unrolled it so it could be better visualized on a screen.

The tempered vector for the generator of this temperament is of course {{vector|1}}, but the simplest JI pitch that maps to this generator is {{vector|-1 1}}, or 3/2. It's visually clear why this is the generator. We need to choose an interval which if we repeatedly move by it, while wrapping around the swath, we'll visit every node inside the swath. The best way to do that is to move from the origin to the node that's nearest to the dotted line labelled "tube circumference". We want this node because we want to move away from the tube circumference as little as possible each time, so we avoid skipping any nodes. If it's not obvious, you may want to experiment with drawing the generator line as if it had gone to any other point inside the swath; if you repeated that movement, would you visit every node? No.

All of this so far is actually only just explaining the basic setup for any tempered lattice. But we've got to lay the basics down first in order to discuss the effect of enfactoring. We'll do that now!

== Enfactored mapping ==

[[File:2-enfactored mapping.png|365px|thumb|right|A 2-enfactored mapping represents a temperoid for which every other step of its generator lands on a pitch which no JI interval would ever temper to.]]

We are now comparing the previous diagram, which had the mapping {{rket|{{map|2 3}}}}, with the 2-enfactored version of it, i.e. the mapping 2×{{rket|{{map|2 3}}}} = {{rket|{{map|4 6}}}}, AKA 4-ET.

If you compare this lattice of an enfactored mapping with the previous lattice for a healthy, defactored mapping, they should look almost the same. They have the same comma and tube circumference. And the generator follows the same path through that tube/swath. The key difference is how far the generator moves with each step along that path.

Starting from the origin, we can see that it takes us 2 moves of the generator to reach the approximation of {{vector|-1 1}}, AKA 3/2, where before we made that step in one go. Then another 2 moves to reach the approximation of {{vector|-2 2}}, AKA 9/4, for a total of 4 moves, where before it only took us 2 steps. As you keep going, you'll see that each node it has taken us 2x as many steps as before to reach it.

And for what? What happens in the steps that are halfway between nodes that were on the JI lattice? These are shown with hollow blue circles instead of filled blue circles, to indicate that there's no JI lattice node underneath them. In other words, while these are legitimate musical intervals, there is no JI interval which would be said to temper to them. In still other words, since this is 4-ET, that first generator step is to a node {{vector|1}} that's about 300¢. But {{vector|0 0}} tempers to {{vector|0}} and {{vector|-1 1}} tempers to {{vector|2}}; nothing tempers to {{vector|1}}. It's an interval that can certainly at least be heard and understood musically, but it has no meaning with respect to tempering JI, or said another way, it has no RTT purpose.

And so this 4-ET doesn't bring anything to the table that isn't already brought by 2-ET. And so it is fitting to consider it only a temperoid, rather than a true temperament. Were this as bad as things got, it might not be worth pushing for distinguishing temperoids from temperaments. But once we look at enfactored comma bases, we'll see why things get pretty pathological.

== Enfactored comma bases ==

[[File:2-enfactored comma-basis.png|365px|thumb|left|enfactored comma bases are garbage]]

Here's where things get kind of nuts. Most recently we experimented with enfactoring our healthy temperament's mapping. Now let's experiment with enfactoring its comma basis. In the defactored situation, if our comma basis was [{{vector|-3 2}}], then 2-enfactoring it produces 2×[{{vector|-3 2}}] = [{{vector|-6 4}}].

We know that in the original diagram, the large-labelled {{vector|-3 2}} represented our comma, and this was the point that our dotted line ran through, the one that represented our boundary of warp/wrap. So our first thought should be: we must alter our diagram so that now {{vector|-6 4}} is that point instead. Fine.

But here's the problem. It simply doesn't make sense to double the width of our swath/tube! If {{vector|-6 4}} is made to vanish, then so is {{vector|-3 2}}. That is, while nothing would stop you from drawing a diagram with a double-width swath/tube, the musical reality is that it is impossible to make {{vector|-6 4}} vanish without also making {{vector|-3 2}} vanish. And so there is no meaning or purpose to the comma basis {{vector|-6 4}}, whether RTT-wise or musically in general. It is garbage.

And so our lattice for an enfactored comma basis looks almost identical to the original defactored lattice. The only difference here is that we've drawn a "supposed (but false)" tube circumference out to {{vector|-6 4}}, while the half of this length which is real is now labelled the "true" circumference.

== Enfactored comma bases vs. periodicity blocks with torsion ==

[[File:Torsion.png|400px|thumb|right|a reworking of the classic torsion example from Tonalsoft to reveal the twinned generator paths]]

And now we're prepared to confront the key difference between the enfactored comma basis of a temperament, and torsion of a periodicity block.

What they have in common is that both take the form of a common factor found somewhere in linear combinations of entries in a list of commas defining a pitch structure, and that these commas can be visualized by slicing the JI lattice into swaths of "periodicity" (that's just a fancy word for the effect we've already been observing for temperaments, where nodes outside the swath related by the size of that comma are considered equivalent and therefore redundant, or repetitions of the same pitch class).

The key difference is that the former (temperaments) tempers the commas out, while the latter (periodicity blocks) does not.

Why is this the key difference? Well, remember how in the previous section, the reason we couldn't actually extend the width of the swath/tube to {{vector|-6 4}} was because the tempering: if {{vector|-6 4}} is made to vanish, then {{vector|-3 2}} is as well, so the swath/tube cannot legitimately be extended. Since there is no tempering in the case of periodicity blocks, however, the width ''can'' legitimately be extended in this way.

Let's take a look at the example given in Tonalsoft's page for torsion. The diagram there has been reworked here to help clarify things. The origin, 1/1, has been placed in the corner of this parallelogram-shaped block, and the two commas that define it are in two of the other corners: [[2048/2025]] ({{vector|11 -4 -4}}) and 625/324 ({{vector|-2 -4 4}}). The value at the fourth corner, 12800/6561, has vector 2×{{vector|-8 8 -2}}. The first 2 is just to octave-reduce it to being positive, but you may recognize the actual vector part as 2 times the meantone comma. The most important part is that the vector is 2-enfactored. You can see that the node at the very center of this block is 160/81, which again is 2×{{vector|-4 4 -1}}, or the octave-reduced non-enfactored version of that same comma.

The red and blue lines that wrap around this block are two different generator paths. The point here is to show that by doubling the size of this periodicity block, we have made it impossible to choose a node to travel to from the origin, i.e. a generator, such that you can reach every node in the block. Instead, the best you can do is reach half of the nodes; that's the red path from the origin 1/1. The blue path is an exact copy of the red path, but offset.

So we can see how tempting the duality can be here. In the case of a 2-enfactored mapping, the generator path reaches ''twice'' as many nodes as there were JI nodes. But in the case of a 2-enfactored comma basis — if we could legitimately extend the width of the block, as we do in untempered periodicity blocks! — we would reach ''half'' as many nodes. But this duality just is not musically, audibly real.

== Enfactored mappings vs. enfactored comma bases ==

One may pose the question: is there any relationship between an enfactored mapping and an enfactored comma basis? Does one imply the other. Does one imply the absence of the other? Or are they completely independent?

We note that there is no such thing as an enfactored temperament, only enfactored matrices. And these essentially result from an arithmetic oversight, namely failing to defactor, and as such they are completely independent<ref>as observed on the page [[Color_notation/Temperament_Names|color notation]], which reads "it's possible that there is both torsion and contorsion"</ref>. When we use matrix math for RTT we defactor as a matter of course. For example, defactoring is built in to the NullSpace matrix operation in the Wolfram Language.

However, if you have an artistic or other reason to generate scales using generators derived directly from enfactored mappings, and therefore to treat enfactored mappings as representing different musical objects, which we call temperoids, not temperaments, then you might choose to encode the same common factor into the comma basis representation of the temperoid with the understanding that this is a mere bookkeeping exercise and has no mathematical basis.

=Footnotes=
<references/>

[[Category:Regular temperament theory]]
[[Category:Terms]]
[[Category:Math]]

Saturation, torsion, and contorsion

2025-03-18T13:01:53Z

Cmloegcmluin: /* History and terminology */ remove not really correct statement; per the linked terminology proposal page, we suggest that torsion is not a good word for the periodicity block concept either

: ''This is a general introduction to this concept; for a more mathematical take on this, see [[Mathematical theory of saturation]].''

[[Category:Regular temperament theory]]
[[Category:Terms]]
[[Category:Math]]
In [[regular temperament theory]], a [[temperament]] is saturated if its set of available intervals matches what is suggested by its mapping or comma basis. A temperament's mapping can fail with respect to saturation by being contorted, and its comma basis can fail through torsion.

This article briefly explains these issues; for lattice-based visualizations and intuitive explanations, see [[Pathology of enfactoring|Pathology of saturation]].

==Contorsion==
A temperament (more specifically, its [[mapping]]) displays '''contorsion''' if there is some generatable interval which no [[just intonation]] interval maps to. This generatable interval is a '''contorted generator''', which has the property that every just interval's mapping has a multiple of c of that generator, where c, the '''contorsion order''', is greater than one. In a contorted temperament, all generator bases will contain at least one contorted generator. The largest contorsion order is called the '''greatest factor'''<ref>This term is inspired by H. J. S. Smith's [https://www.jstor.org/stable/pdf/108738.pdf ''On Systems of Linear Indeterminate Equations and Congruences''], where Smith describes the GCD of a matrix's minor determinants as its "greatest divisor". "Divisor" and "factor" are synonyms and they prefer "factor" for its connection with the term "defactor".</ref>.

For example, [[5-limit]] [[36edo|36et]] (with mapping [36 57 84]) uses 12 of its pitches per octave (the ones within [[12edo|12et]]) to map the entire 5-limit gamut. As a result, no 5-limit just intonation interval maps to any of the other 24 pitches, making 36et contorted in the 5-limit. Therefore there is a contorted generator; since there is only one generator of 36et (namely, the 36th-octave), that generator must be contorted. Every pitch is mapped to a number of [[Generator|generators]] that is a multiple of 3 (where the generator of 36et is a 36th of the octave), so this generator has contorsion order 3. For a higher-rank example, the 13-limit 87&111 temperament Hemimist, with mapping [⟨3 0 26 56 8], ⟨0 2 -8 -20 1]], when restricted to the 2.5.7.11.13 subgroup, has no just intonation interval corresponding to the period or the square of the period, although there is a just intonation interval (namely, 2/1) corresponding to the cube of the period. Therefore, this a contorted generator with contorsion order 3.

If a temperament has a subgroup which is contorted, especially a subgroup with small primes (for example, the 11-limit subgroup of 23-limit [[44edo|44et]]), that temperament will likely be easier to traverse than the number of generators required according to the mapping would suggest.
==Torsion in temperaments==
A temperament (more specifically, its [[comma basis]]) displays '''torsion''' if there is some interval mapped to zero which is not formable by multiplying commas in the basis. This interval is a '''comma with torsion''', which has the property that commas in the basis can be multiplied to form the ''c''th power of this ratio, but not that ratio itself or any smaller power, where c is the '''torsion order'''.

For instance, in a temperament with comma basis {[[6561/6250]], [[128/125]]}, there is an interval 81/80 which is not formable by multiplying commas in the basis, but is nevertheless forced to be mapped to zero because (81/80)^2 = (6561/6250)/(128/125) is part of the basis. Thus, 81/80 displays torsion with torsion order 2.
==Torsion in periodicity blocks==
A comma basis in the context of periodicity blocks displays torsion if it displays torsion as a temperament—precisely when there is some comma with torsion where commas in the basis can be multiplied to form the ''c''th power of this ratio, but not that ratio itself or any smaller power, where c is the torsion order.

Within periodicity blocks, no mapping needs to be defined from a comma basis, so comma bases with torsion are able to form periodicity blocks where the smallest comma with torsion is not tempered out.
==Saturation algorithms ==
An unsaturated mapping or comma basis can be made saturated, ensuring our ability to most simply—and thereby uniquely—identify temperaments using only matrices. The simplest and fastest algorithm for saturating matrices is called [[column Hermite defactoring]]. For more information on such algorithms, see [[Defactoring algorithms|Saturation algorithms]].
==History and terminology==
The term ''saturation'' was coined by {{w|Nicolas Bourbaki}} in 1972<ref>[https://pdfcoffee.com/commutative-algebra-bourbaki-pdf-free.html Nicolas Bourbaki. ''Commutative Algebra'']</ref>, working in the field of commutative algebra. It came to RTT via [[Gene Ward Smith]] and [[Graham Breed]]'s observations of the work of the mathematician {{w|William A. Stein|William Stein}} and his {{w|SageMath|Sage}} software<ref>It may also have come through PARI/GT.</ref>. The earliest identified terminology for this concept was in 1861 by {{w|Henry John Stephen Smith|H. J. S. Smith}}<ref>H. J. S. Smith is the creator of the {{w|Smith normal form}} used in [[Defactoring algorithms #Precedent: Smith defactoring|Gene Ward Smith's saturation algorithm]].</ref> who called saturated matrices "prime matrices"<ref>Also from ''On Systems of Linear Indeterminate Equations and Congruences'', linked above. Neither ''prime matrix'' nor ''greatest divisor'' seems to have caught on in the mathematical community.</ref>.

The term ''torsion'' has been used since at least as early as 1932<ref>[https://scholar.google.com/scholar?q=%22torsion+group%22&hl=en&as_sdt=0%2C5&as_ylo=1900&as_yhi=1940 Google Scholar: Torsion group]</ref><ref>[https://math.stackexchange.com/questions/300586/where-does-the-word-torsion-in-algebra-come-from Stack Exchange | ''Where does the word "torsion" in algebra come from?'']</ref> and came to RTT from the mathematical field of group theory. Historically, a group-theory formalism was used to analyze comma bases with torsion, where the smallest comma displaying torsion was not made to vanish although a power of that comma was, which is musically impossible; using a linear algebra formalism as is preferred now, no such impossibility is suggested. The term ''contorsion'' was invented for RTT in 2002 by [[Paul Erlich]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2033.html#2456 Yahoo! Tuning Group | ''My top 5--for Paul'']</ref>, as a play on the word "co-torsion", being dual to the situation with "torsion" above.

In the case of temperaments, [[Dave Keenan]] and [[Douglas Blumeyer]] have proposed<ref>See [[Defactoring terminology proposal]] for details.</ref> and used '''defactoring''' as a replacement for ''saturation'' and '''enfactoring''' as a replacement for both ''torsion'' and ''contorsion''. So, a mapping or comma basis of a temperament is either defactored (saturated) or enfactored (unsaturated, having torsion/contorsion).

==See also==
* [http://www.tonalsoft.com/enc/t/torsion.aspx Tonalsoft's page on torsion]
* [http://www.tonalsoft.com/enc/c/contortion.aspx Tonalsoft's page on contorsion]

==References and footnotes==
<references />

Saturation, torsion, and contorsion

2025-03-18T12:59:03Z

Cmloegcmluin: add links to Tonalsoft

: ''This is a general introduction to this concept; for a more mathematical take on this, see [[Mathematical theory of saturation]].''

[[Category:Regular temperament theory]]
[[Category:Terms]]
[[Category:Math]]
In [[regular temperament theory]], a [[temperament]] is saturated if its set of available intervals matches what is suggested by its mapping or comma basis. A temperament's mapping can fail with respect to saturation by being contorted, and its comma basis can fail through torsion.

This article briefly explains these issues; for lattice-based visualizations and intuitive explanations, see [[Pathology of enfactoring|Pathology of saturation]].

==Contorsion==
A temperament (more specifically, its [[mapping]]) displays '''contorsion''' if there is some generatable interval which no [[just intonation]] interval maps to. This generatable interval is a '''contorted generator''', which has the property that every just interval's mapping has a multiple of c of that generator, where c, the '''contorsion order''', is greater than one. In a contorted temperament, all generator bases will contain at least one contorted generator. The largest contorsion order is called the '''greatest factor'''<ref>This term is inspired by H. J. S. Smith's [https://www.jstor.org/stable/pdf/108738.pdf ''On Systems of Linear Indeterminate Equations and Congruences''], where Smith describes the GCD of a matrix's minor determinants as its "greatest divisor". "Divisor" and "factor" are synonyms and they prefer "factor" for its connection with the term "defactor".</ref>.

For example, [[5-limit]] [[36edo|36et]] (with mapping [36 57 84]) uses 12 of its pitches per octave (the ones within [[12edo|12et]]) to map the entire 5-limit gamut. As a result, no 5-limit just intonation interval maps to any of the other 24 pitches, making 36et contorted in the 5-limit. Therefore there is a contorted generator; since there is only one generator of 36et (namely, the 36th-octave), that generator must be contorted. Every pitch is mapped to a number of [[Generator|generators]] that is a multiple of 3 (where the generator of 36et is a 36th of the octave), so this generator has contorsion order 3. For a higher-rank example, the 13-limit 87&111 temperament Hemimist, with mapping [⟨3 0 26 56 8], ⟨0 2 -8 -20 1]], when restricted to the 2.5.7.11.13 subgroup, has no just intonation interval corresponding to the period or the square of the period, although there is a just intonation interval (namely, 2/1) corresponding to the cube of the period. Therefore, this a contorted generator with contorsion order 3.

If a temperament has a subgroup which is contorted, especially a subgroup with small primes (for example, the 11-limit subgroup of 23-limit [[44edo|44et]]), that temperament will likely be easier to traverse than the number of generators required according to the mapping would suggest.
==Torsion in temperaments==
A temperament (more specifically, its [[comma basis]]) displays '''torsion''' if there is some interval mapped to zero which is not formable by multiplying commas in the basis. This interval is a '''comma with torsion''', which has the property that commas in the basis can be multiplied to form the ''c''th power of this ratio, but not that ratio itself or any smaller power, where c is the '''torsion order'''.

For instance, in a temperament with comma basis {[[6561/6250]], [[128/125]]}, there is an interval 81/80 which is not formable by multiplying commas in the basis, but is nevertheless forced to be mapped to zero because (81/80)^2 = (6561/6250)/(128/125) is part of the basis. Thus, 81/80 displays torsion with torsion order 2.
==Torsion in periodicity blocks==
A comma basis in the context of periodicity blocks displays torsion if it displays torsion as a temperament—precisely when there is some comma with torsion where commas in the basis can be multiplied to form the ''c''th power of this ratio, but not that ratio itself or any smaller power, where c is the torsion order.

Within periodicity blocks, no mapping needs to be defined from a comma basis, so comma bases with torsion are able to form periodicity blocks where the smallest comma with torsion is not tempered out.
==Saturation algorithms ==
An unsaturated mapping or comma basis can be made saturated, ensuring our ability to most simply—and thereby uniquely—identify temperaments using only matrices. The simplest and fastest algorithm for saturating matrices is called [[column Hermite defactoring]]. For more information on such algorithms, see [[Defactoring algorithms|Saturation algorithms]].
==History and terminology==
The term ''saturation'' was coined by {{w|Nicolas Bourbaki}} in 1972<ref>[https://pdfcoffee.com/commutative-algebra-bourbaki-pdf-free.html Nicolas Bourbaki. ''Commutative Algebra'']</ref>, working in the field of commutative algebra. It came to RTT via [[Gene Ward Smith]] and [[Graham Breed]]'s observations of the work of the mathematician {{w|William A. Stein|William Stein}} and his {{w|SageMath|Sage}} software<ref>It may also have come through PARI/GT.</ref>. The earliest identified terminology for this concept was in 1861 by {{w|Henry John Stephen Smith|H. J. S. Smith}}<ref>H. J. S. Smith is the creator of the {{w|Smith normal form}} used in [[Defactoring algorithms #Precedent: Smith defactoring|Gene Ward Smith's saturation algorithm]].</ref> who called saturated matrices "prime matrices"<ref>Also from ''On Systems of Linear Indeterminate Equations and Congruences'', linked above. Neither ''prime matrix'' nor ''greatest divisor'' seems to have caught on in the mathematical community.</ref>.

The term ''torsion'' has been used since at least as early as 1932<ref>[https://scholar.google.com/scholar?q=%22torsion+group%22&hl=en&as_sdt=0%2C5&as_ylo=1900&as_yhi=1940 Google Scholar: Torsion group]</ref><ref>[https://math.stackexchange.com/questions/300586/where-does-the-word-torsion-in-algebra-come-from Stack Exchange | ''Where does the word "torsion" in algebra come from?'']</ref> and came to RTT from the mathematical field of group theory. Historically, a group-theory formalism was used to analyze comma bases with torsion, where the smallest comma displaying torsion was not made to vanish although a power of that comma was, which is musically impossible; using a linear algebra formalism as is preferred now, no such impossibility is suggested. The term ''contorsion'' was invented for RTT in 2002 by [[Paul Erlich]]<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_2033.html#2456 Yahoo! Tuning Group | ''My top 5--for Paul'']</ref>, as a play on the word "co-torsion", being dual to the situation with "torsion" above.

In the case of temperaments, [[Dave Keenan]] and [[Douglas Blumeyer]] have proposed and used '''defactoring''' as a replacement for ''saturation'' and '''enfactoring''' as a replacement for both ''torsion'' and ''contorsion''. So, a mapping or comma basis of a temperament is either defactored (saturated) or enfactored (unsaturated, having torsion/contorsion).<ref>See [[Defactoring terminology proposal]] for details.</ref> Keenan and Blumeyer reserve the word "torsion" for the case of periodicity blocks.

==See also==
* [http://www.tonalsoft.com/enc/t/torsion.aspx Tonalsoft's page on torsion]
* [http://www.tonalsoft.com/enc/c/contortion.aspx Tonalsoft's page on contorsion]

==References and footnotes==
<references />

Pathology of enfactoring

2025-03-18T12:56:00Z

Cmloegcmluin: /* Enfactored comma bases vs. periodicity blocks with torsion */

In this article, we will use lattices to visualize [[enfactoring|enfactored]] temperaments, to demonstrate the musical implications of mappings with common factors, and the lack of musical implications of comma bases with common factors.

== Defactored case ==

[[File:Unenfactored mapping.png|365px|thumb|right|A 3-limit tempered lattice, superimposed on the JI lattice]]

First, let's look at a defactored mapping. This example temperament is so simple that it is not of practical musical interest. It was chosen because it's basically the numerically simplest possible example, where this type of simplicity empowers us to visualize the problem at a practical scale as clearly as possible. Please consider the diagram at right.

This is a representation of 2-ET, a 3-limit, rank-1 (equal) temperament, with mapping {{rket|{{map|2 3}}}}, meaning it has a single generator which takes two steps to reach the octave, and three steps to reach the tritave. This temperament makes a single comma [[vanish]], a comma whose vector representation looks similar to the mapping: {{vector|-3 2}}, AKA 9/8. And so the comma basis for this temperament is [{{vector|-3 2}}].

We can imagine that we started out with a JI lattice, where movement up and down correspond to prime 2 (the octave) and movements right and left correspond to prime 3 (the tritave). We have tempered JI here, and so we've faded the JI lattice out to a faint grey color in the background. What we've done specifically is made the comma {{vector|-3 2}} vanish so that any nodes in this lattice which are 2 over and 3 up from each other are equivalent. Therefore we only need to consider a thin swath of the lattice anymore, specifically, a swath which connects the origin {{vector|0 0}}, AKA 1/1, to {{vector|-3 2}}, and then runs perpendicularly to infinity in either direction.

There's a couple good ways to interpret this situation:
# We've turned on a teleportation field for every point outside this swath, so that it moves by these (-3,2) intervals until it finds its way inside the swath.
# We've rolled up space, so that the line from 1/1 to 9/8 — the width of our swath — is like the circumference of a tube. In this case, since we're only working with 3-limit JI, "space" was only ever 2D, so we can just think of it as paper that we've rolled up, so the pair of dotted lines visualized here are touching along their entire lengths (and if you wanted to imagine further copies of these dotted lines at every (-3,2) interval farther out in either direction, and the paper is infinitely thin, just stack all the dotted lines on top of each other forever).

Either way, then we just superimpose the new tempered lattice on top. It's drawn in blue.

You can see that in the grey lattice underneath, coordinates have 2 values, e.g. {{vector|-1 2}}. That's because the JI lattice essentially had two generators: the octave and the tritave. But tempering has helped us simplify things by reducing us to a single generator, so here, in the new blue-colored tempered lattice, the coordinates have only 1 value, e.g. {{vector|8}}, and they simply indicate how many iterations of the single generator here that we've taken to reach the given point.

We've superimposed the tempered lattice atop the former JI lattice so we can see which JI intervals map to which tempered intervals. For example, {{vector|-1 2}}, AKA 9/2, maps to {{vector|8}}. That tells us that if we want to use the temperament's approximation of the JI interval 9/2, then we want the tempered pitch arrived at by moving by the generator 8 times.

Note that whenever the path the generator takes leaves the main swath, it wraps around to the point on the opposite side. Again, you can think of this either way you prefer: the world is still flat, and you've just warped over there; or, the world has been curled up, and in reality you've been looping back around toward that point the whole time, and the dotted line in this flat representation just represents the point you cut the tube and unrolled it so it could be better visualized on a screen.

The tempered vector for the generator of this temperament is of course {{vector|1}}, but the simplest JI pitch that maps to this generator is {{vector|-1 1}}, or 3/2. It's visually clear why this is the generator. We need to choose an interval which if we repeatedly move by it, while wrapping around the swath, we'll visit every node inside the swath. The best way to do that is to move from the origin to the node that's nearest to the dotted line labelled "tube circumference". We want this node because we want to move away from the tube circumference as little as possible each time, so we avoid skipping any nodes. If it's not obvious, you may want to experiment with drawing the generator line as if it had gone to any other point inside the swath; if you repeated that movement, would you visit every node? No.

All of this so far is actually only just explaining the basic setup for any tempered lattice. But we've got to lay the basics down first in order to discuss the effect of enfactoring. We'll do that now!

== Enfactored mapping ==

[[File:2-enfactored mapping.png|365px|thumb|right|A 2-enfactored mapping represents a temperoid for which every other step of its generator lands on a pitch which no JI interval would ever temper to.]]

We are now comparing the previous diagram, which had the mapping {{rket|{{map|2 3}}}}, with the 2-enfactored version of it, i.e. the mapping 2×{{rket|{{map|2 3}}}} = {{rket|{{map|4 6}}}}, AKA 4-ET.

If you compare this lattice of an enfactored mapping with the previous lattice for a healthy, defactored mapping, they should look almost the same. They have the same comma and tube circumference. And the generator follows the same path through that tube/swath. The key difference is how far the generator moves with each step along that path.

Starting from the origin, we can see that it takes us 2 moves of the generator to reach the approximation of {{vector|-1 1}}, AKA 3/2, where before we made that step in one go. Then another 2 moves to reach the approximation of {{vector|-2 2}}, AKA 9/4, for a total of 4 moves, where before it only took us 2 steps. As you keep going, you'll see that each node it has taken us 2x as many steps as before to reach it.

And for what? What happens in the steps that are halfway between nodes that were on the JI lattice? These are shown with hollow blue circles instead of filled blue circles, to indicate that there's no JI lattice node underneath them. In other words, while these are legitimate musical intervals, there is no JI interval which would be said to temper to them. In still other words, since this is 4-ET, that first generator step is to a node {{vector|1}} that's about 300¢. But {{vector|0 0}} tempers to {{vector|0}} and {{vector|-1 1}} tempers to {{vector|2}}; nothing tempers to {{vector|1}}. It's an interval that can certainly at least be heard and understood musically, but it has no meaning with respect to tempering JI, or said another way, it has no RTT purpose.

And so this 4-ET doesn't bring anything to the table that isn't already brought by 2-ET. And so it is fitting to consider it only a temperoid, rather than a true temperament. Were this as bad as things got, it might not be worth pushing for distinguishing temperoids from temperaments. But once we look at enfactored comma bases, we'll see why things get pretty pathological.

== Enfactored comma bases ==

[[File:2-enfactored comma-basis.png|365px|thumb|left|enfactored comma bases are garbage]]

Here's where things get kind of nuts. Most recently we experimented with enfactoring our healthy temperament's mapping. Now let's experiment with enfactoring its comma basis. In the defactored situation, if our comma basis was [{{vector|-3 2}}], then 2-enfactoring it produces 2×[{{vector|-3 2}}] = [{{vector|-6 4}}].

We know that in the original diagram, the large-labelled {{vector|-3 2}} represented our comma, and this was the point that our dotted line ran through, the one that represented our boundary of warp/wrap. So our first thought should be: we must alter our diagram so that now {{vector|-6 4}} is that point instead. Fine.

But here's the problem. It simply doesn't make sense to double the width of our swath/tube! If {{vector|-6 4}} is made to vanish, then so is {{vector|-3 2}}. That is, while nothing would stop you from drawing a diagram with a double-width swath/tube, the musical reality is that it is impossible to make {{vector|-6 4}} vanish without also making {{vector|-3 2}} vanish. And so there is no meaning or purpose to the comma basis {{vector|-6 4}}, whether RTT-wise or musically in general. It is garbage.

And so our lattice for an enfactored comma basis looks almost identical to the original defactored lattice. The only difference here is that we've drawn a "supposed (but false)" tube circumference out to {{vector|-6 4}}, while the half of this length which is real is now labelled the "true" circumference.

== Enfactored comma bases vs. periodicity blocks with torsion ==

[[File:Torsion.png|400px|thumb|right|a reworking of the classic torsion example from Tonalsoft to reveal the twinned generator paths]]

And now we're prepared to confront the key difference between the enfactored comma basis of a temperament, and torsion of a periodicity block.

What they have in common is that both take the form of a common factor found somewhere in linear combinations of entries in a list of commas defining a pitch structure, and that these commas can be visualized by slicing the JI lattice into swaths of "periodicity" (that's just a fancy word for the effect we've already been observing for temperaments, where nodes outside the swath related by the size of that comma are considered equivalent and therefore redundant, or repetitions of the same pitch class).

The key difference is that the former (temperaments) tempers the commas out, while the latter (periodicity blocks) does not.

Why is this the key difference? Well, remember how in the previous section, the reason we couldn't actually extend the width of the swath/tube to {{vector|-6 4}} was because the tempering: if {{vector|-6 4}} is made to vanish, then {{vector|-3 2}} is as well, so the swath/tube cannot legitimately be extended. Since there is no tempering in the case of periodicity blocks, however, the width ''can'' legitimately be extended in this way.

Let's take a look at the example given in Tonalsoft's page for torsion. The diagram there has been reworked here to help clarify things. The origin, 1/1, has been placed in the corner of this parallelogram-shaped block, and the two commas that define it are in two of the other corners: [[2048/2025]] ({{vector|11 -4 -4}}) and 625/324 ({{vector|-2 -4 4}}). The value at the fourth corner, 12800/6561, has vector 2×{{vector|-8 8 -2}}. The first 2 is just to octave-reduce it to being positive, but you may recognize the actual vector part as 2 times the meantone comma. The most important part is that the vector is 2-enfactored. You can see that the node at the very center of this block is 160/81, which again is 2×{{vector|-4 4 -1}}, or the octave-reduced non-enfactored version of that same comma.

The red and blue lines that wrap around this block are two different generator paths. The point here is to show that by doubling the size of this periodicity block, we have made it impossible to choose a node to travel to from the origin, i.e. a generator, such that you can reach every node in the block. Instead, the best you can do is reach half of the nodes; that's the red path from the origin 1/1. The blue path is an exact copy of the red path, but offset.

So we can see how tempting the duality can be here. In the case of a 2-enfactored mapping, the generator path reaches ''twice'' as many nodes as there were JI nodes. But in the case of a 2-enfactored comma basis — if we could legitimately extend the width of the block, as we do in untempered periodicity blocks! — we would reach ''half'' as many nodes. But this duality just is not musically, audibly real.

== Enfactored mappings vs. enfactored comma bases ==

One may pose the question: what is the relationship between an enfactored mapping and an enfactored comma basis? Can you have one but not the other? Must you? Or must you not? Or does the question even make sense? Certainly at least some have suggested these cases are meaningfully independent<ref>such as the page [[Color_notation/Temperament_Names|color notation]], which reads "it's possible that there is both torsion and contorsion"</ref>.

The conclusion we arrive at here is that because enfactored comma bases don't make any sense, or at least don't represent any legitimately new musical information of any kind that their defactored version doesn't already represent, it is not generally useful to think of enfactored mappings and enfactored comma bases as independent phenomena. It only makes sense to speak of enfactored temperaments. Of course, one will often use the term "enfactored mapping" because enfactored mappings are the kind which do have some musical purpose, and often the enfactored mapping will be being used to represent the enfactored temperament — or temperoid, that is.

=Footnotes=
<references/>

[[Category:Regular temperament theory]]
[[Category:Terms]]
[[Category:Math]]

128afdo

2025-03-07T06:33:09Z

Cmloegcmluin: /* Music */

{{Infobox AFDO|steps=128}}

'''128afdo''' ([[AFDO|arithmetic frequency division of the octave]]), or '''128odo''' ([[otonal division]] of the octave), divides the octave into 128 parts of 1/128 each. It is a superset of [[127afdo]] and a subset of [[129afdo]]. As a scale it may be known as [[harmonic mode|mode 128 of the harmonic series]] or the [[overtone scale #Over-n scales|Over-128]] scale.

The '''8th Octave Overtone Tuning''', sometimes known as '''128 Tuning''', is a tuning developed by [[Johnny Reinhard]]. It is equivalent to 128afdo, except that it has a fixed root and cannot be rotated. It consists of harmonics of the [[harmonic series]], numbers 128 (27, hence 8th octave) through 255. It is an Over-1 scale, specifically mode 128 of the harmonic series. Scales can be selected as subsets of these 128 pitches, or the entire set can be used.

A key benefit of using pitches exclusively from the same harmonic series is that they share a fundamental. By using the 8th octave of a harmonic series, said fundamental will almost certainly be [https://www.merriam-webster.com/dictionary/infrasonic infrasonic], but it will still have a [[psychoacoustic]] presence.

An illustratively surprising result of this higher harmonic tuning is that, since a just [[4/3]] does not have a power of 2 in the denominator and thus does not exist in the (octave-reduced) harmonic series, it will not be used in this tuning. Instead, when the inverse of the [[3/2]] ratio is needed, one may use [[43/32]] (511.517706¢) or [[171/128]] (501.423018¢).

Due to having only one prime factor (2), yet also being a higher octave of a prime mode (mode 2), it is a very strong tuning for [[primodality]], providing a large gamut of intervals without compromising their clear prime identity.

== Music ==
; [[Georg Friedrich Haas]]
* [https://www.youtube.com/watch?v=TxGcveURI-I ''For Johnny Reinhard''] (2015)

; [[La Monte Young]]
* [https://www.nicovideo.jp/watch/sm7119661 ''The Well-Tuned Piano''] (1964) – actually up to the 11th octave harmonics, but the same idea

; [[Johnny Reinhard]]
* [https://open.spotify.com/album/7jtoRTNK2Pm7vxkq5PH12b ''True''] (2014)

; [[Glenn Branca]]
* [https://www.youtube.com/watch?v=t4re9tjY5es ''Symphony #3 "Gloria"''] (1983) – actually only the 7th octave harmonics, but the same idea

; [[Philipp Gerschlauer]]
* [https://www.youtube.com/watch?v=lGa66qHzKME ''128 notes per octave on Alto Saxophone''] (2015)

; [[Juhani Nuorvala]]
* ''Toivo 128'' (2017) [https://soundcloud.com/juhani-nuorvala/toivo-128 recording] [https://nuotisto.s3-eu-west-1.amazonaws.com/store/e6fc131f958d13f87f3ea56b0d57beab50473c79bbc5a705b0dd6878214a.pdf score]

Composers John Eaton, Anton Rovner, Peter Alexander Thoegersen, Monroe Golden, and others have also worked with 8th Octave Overtone Tuning.{{citation needed}}

== External links ==
* [https://stereosociety.com/20/jpg/Johnny-Reinhard/8th-Octave-Overtone-Tuning.pdf Johnny Reinhard's original paper].
* [https://www.cassgb.org/features/post/128-note-octave/ 128 NOTES PER OCTAVE ON THE SAXOPHONE: HOW I DID IT AND WHY!: Saxophonist Philipp Gerschlauer on how he went about devising a 128-note per octave fingering chart]
* [https://books.google.com/books/about/8th_Octave_Overtone_Tuning_and_Bassoon_F.html?id=YE9gAQAACAAJ Johnny Reinhard - 8th Octave Overtone Tuning and Bassoon Fingerings in 128]
* [https://www.kylegann.com/13th-Harmonic.html The tuning for Nursery Tunes for Demented Children by Kyle Gann] is a subset of 8th Octave Overtone Tuning.

[[Category:Harmonic series]]
[[Category:Primodality]]
[[Category:Listen]]

Eigenmonzo basis

2025-02-22T16:13:14Z

Cmloegcmluin: /* See also */

An [[eigenmonzo|eigenmonzo or unchanged-interval]] is a rational interval tuned justly by a [[regular temperament]] tuning. In other words, if a tuning is ''T'', then an eigenmonzo ''q'' satisfies {{nowrap| ''T''(''q'') {{=}} ''q'' }}. The eigenmonzos of ''T'' define a [[just intonation subgroup]], the eigenmonzo subgroup, whose basis is an '''eigenmonzo basis''' or '''unchanged-interval basis'''.

One sort of example is provided by any equal division of the octave, where 2 (the octave) is always an eigenmonzo and the group {2''n''} of powers of 2 is the eigenmonzo subgroup.

The idea is most useful in connection to the [[Target tuning #Minimax tuning|minimax tunings]] of regular temperaments, where for a rank-''r'' regular temperament, the eigenmonzo subgroup is a rank-''r'' JI subgroup whose generators, together with generators for the commas of the subgroup, can be used to define the [[projection matrix]] of the minimax tuning and hence define the tuning.

== See also ==
* [[Projection #The unchanged-interval basis]], for a discussion of this concept in the context of other related temperament tuning objects

[[Category:Regular temperament theory]]
[[Category:Terms]]
[[Category:Math]]
[[Category:Monzo]]

Eigenmonzo basis

2025-02-20T00:34:57Z

Cmloegcmluin: Undo revision 181519 by Lériendil (talk) - I think the link was only unintuitive because the anchor within the destination page had been broken by careless edits to that page

Given a [[regular temperament]] tuning T, an [[eigenmonzo]] (unchanged-interval) is a rational interval q such that T(q) = q; that is, T tunes q justly. The eigenmonzos of T define a [[just intonation subgroup]], the eigenmonzo subgroup.

One sort of example is provided by any equal division of the octave, where 2 (the octave) is always an eigenmonzo and the group {2''n''} of powers of 2 is the eigenmonzo subgroup. The idea is most useful in connection to the [[Target tuning #Minimax tuning|minimax tunings]] of regular temperaments, where for a rank-''r'' regular temperament, the eigenmonzo subgroup is a rank-''r'' JI subgroup whose generators, together with generators for the commas of the subgroup, can be used to define the [[projection matrix]] of the minimax tuning and hence define the tuning.

== See also ==

* [[unchanged-interval basis]]

[[Category:Regular temperament theory]]
[[Category:Terms]]
[[Category:Math]]
[[Category:Monzo]]

Projection

2025-02-20T00:33:54Z

Cmloegcmluin: /* The unchanged-interval basisv */ fix broken links introduced by Arrowhead

{{Beginner|Projection matrices}}

A '''projection matrix''', or '''projection''' for short<ref group="note">If an alternative word to "mapping" is sought that does not generically refer to the mathematical structure in the way that "matrix" does, the noun "operator" could be used, according to this: https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_12787#12787</ref>, is an object in [[regular temperament theory]] (RTT) that uniquely identifies a specific tuning of a specific regular temperament.<ref group="note">All temperaments discussed on this page hereafter will be assumed to be "regular".</ref>

== Shape ==
A projection is typically a square matrix with shape <math>(d, d)</math>, where <math>d</math> is the [[dimensionality]] of the temperament (for exceptions to this, see [[Projection#Projecting to other spaces]]).

== With respect to other RTT objects ==
[[File:Temperament and tuning matrices 2.png|frameless|1200x1200px]]

Projections are perhaps best understood in comparison with other more frequently used RTT objects:

=== The mapping ===
Like [[mapping]]s, projections:
# Represent regular temperaments,
# Transform [[JI]] intervals, and
# Accept these JI intervals as inputs in the form of [[prime-count vector]]s.

Perhaps the simplest way to explain the difference would be to say that a mapping outputs intervals in the form of [[generator-count vector]]s, whereas a projection outputs intervals that are also in the form of prime-count vectors.

The key reason for this difference is that mappings represent temperaments in the abstract, that is, how intervals are approximated but without any specific information about how to embed them into tuning space; to find the cents value<ref group="note">Any logarithmic pitch unit—cents, octaves, millioctaves, etc.—may be used, but this article has chosen to consistently use cents.</ref> of a ''mapped'' interval—one that has been mapped by a mapping—one must further map it by a [[generator tuning map]]. On the other hand, a ''projected'' interval—one that has been mapped by a projection, or "projected"—already includes the embedding information, and so their cents value can be obtained by mapping them with the generic [[just tuning map]] for the primes. In other words, the projection has applied tuning to the mapped intervals in a particular way, by embedding them back into the original JI space, where the tuning is known, so all we're really doing at that point is sizing the interval.

While a projection maps one prime-count vector to another prime-count vector, the output vector is usually quite different from the input vector. Most notably, the input interval is justly intoned, and therefore the entries of its vector are integers, while the output interval is tempered, and therefore the entries of its vector may be non-integers. Some temperament tunings are chosen so that certain JI intervals remain unchanged by the temperament; in such cases, if the input interval is one of the unchanged-intervals, then its output will exactly match the input.

=== The tuning map ===
Like a [[tuning map]], a projection transforms a JI interval into a new interval that is both mapped and tuned. One key difference is that a tuning map sends the input interval straight to its cents value, whereas the projection sends the interval to an intermediate form as a vector with typically non-integer entries, which must be further mapped by the just tuning map to find its cents value. This difference in behavior is explained by the fact that the tuning map is the projection left-multiplied by the just tuning map, or in other words, that the tuning map projects the input interval and then sizes it to cents all in one go.

At a glance, tuning maps may seem more convenient, then. But the advantage of a projection is that it still identifies the tuning of a temperament, whereas the tuning map, due to being injected with and collapsed down with the just tuning map, has obscured that information and thereby lost the ability to serve as a unique identifier. It only serves the function of mapping intervals to cents values.

=== The generator embedding ===
A projection matrix may be found as the combination of a mapping with a generator embedding, through matrix multiplication. The mapping represents the approximation information in the abstract, i.e. abstracted from any specific embedding, while the generator embedding specifies such a embedding. And so together, the projection matrix represents the specific embedding of the given approximation, or in other words, specific tuning of the given temperament, de-abstracting it.

Multiplying the mapping and generator embedding together in the opposite order, <math>MG</math>, instead gives an identity matrix, <math>I</math>. Using the example of 1/4-comma meantone again, {{rket|{{map|1 1 0}} {{map|0 1 4}}}}{{rbra|{{vector|1 0 0}} {{vector|0 0 1/4}}}} = {{rbra|{{rket|1 0}} {{rket|0 1}}}}. This simply proves that the generator embedding is, in fact, a matrix of generators.

Think about it like any other interval mapping situation: if an interval is mapped to the generator-count vector {{rket|0 1}}, that tells us that the interval maps to exactly one of the second generator and nothing else; in cases like this, we can say it that it is a member of the [[preimage]] for that second generator, or in other words, that it is one of the many possible JI intervals which is approximated by exactly one of that generator. And similarly, if an interval maps to a generator-count vector of {{rket|1 0}}, that would mean that whatever prime-count vector we put in was a member of the preimage for the ''other'' generator.

So, if an entire matrix is mapped by a temperament's mapping matrix to an identity matrix, then that is a very special case; it tells us that each of this matrix's columns can be thought of as a vector that maps to a different one of each of that same temperament's generators. It is, in other words, a [[generator detempering]].

== Examples ==
The generator of meantone temperament is the fifth. A justly intoned fifth is the interval <math>\frac32</math> at about 701.955 ¢, which as a 5-limit prime-count vector looks like {{vector|-1 1 0}}. But in the [[quarter comma meantone|quarter-comma tuning of meantone]], the fifth is flattened. Since 1 fifth is a quarter comma flat, 4 fifths are a full comma flat. 4 just fifths equals <math>\frac{81}{16}</math>, and 4 fifths minus a comma works out to exactly <math>\frac51</math>. Thus the tuning of the fifth is one-quarter of <math>\frac51</math>, which is <math>5^\frac14 = \sqrt[4]5</math> at about 696.578 ¢, which as a vector looks like {{vector|0 0 1/4}}. JI ratios have prime counts that contain only integers, but this one has fractions in it; <math>\sqrt[4]5</math> is an irrational number, so it is not JI.

So, by combining this vector for the tuned fifth with the vector {{vector|1 0 0}} for a purely-tuned octave <math>\frac21</math> as the period, we produce the full generator embedding <math>G</math> for quarter-comma meantone as {{rbra|{{vector|1 0 0}} {{vector|0 0 1/4}}}}:

<math>
\left[ \begin{array} {rrr}
1 & 0 \\
0 & 0 \\
0 & \frac14 \\
\end{array} \right]
</math>

And so here is the projection matrix for quarter-comma meantone, shown as the product of that generator embedding with the meantone mapping:

<math>

\begin{array} {c}
P \\
\left[ \begin{array} {r}
1 & 1 & 0 \\
0 & 0 & 0 \\
0 & \frac14 & 1 \\
\end{array} \right]
\end{array}

=

\begin{array} {c}
G \\
\left[ \begin{array} {r}
1 & 0 \\
0 & 0 \\
0 & \frac14 \\
\end{array} \right]
\end{array}

\begin{array} {c}
M \\
\left[ \begin{array} {r}
1 & 1 & 0 \\
0 & 1 & 4 \\
\end{array} \right]
\end{array}

</math>

The columns of <math>P</math> are vectors, one for each prime. The 1st column of <math>P</math> tells us that prime 2 is projected to {{vector|1 0 0}} = <math>\frac21</math>. Thus <math>\frac21</math> is projected to itself, and is an unchanged-interval. The 2nd column tells us that prime 3 is projected to {{vector|1 0 1/4}} = an octave plus the tempered fifth. The 3rd column tells us that prime 5 is projected to {{vector|0 0 1}} = <math>\frac51</math>. Thus <math>\frac51</math> is also an unchanged-interval, as is any combination of our two unchanged-intervals <math>\frac21</math> and <math>\frac21</math>, such as <math>\frac54</math>, <math>\frac85</math>, <math>\frac{25}{16}</math>, etc.

We can use this matrix to determine what a JI interval <math>\textbf{i}</math> is projected to. Multiply <math>P</math> by <math>\textbf{i}</math> to get <math>P\textbf{i}</math>. Let's start with <math>\textbf{i}</math> = <math>\frac43</math>:

<math>

\begin{array} {c}
P \\
\left[ \begin{array} {r}
1 & 1 & 0 \\
0 & 0 & 0 \\
0 & \frac14 & 1 \\
\end{array} \right]
\end{array}

\begin{array} {c}
\textbf{i} \\
\left[ \begin{array} {r}
2 \\
-1 \\
0 \\
\end{array} \right]
\end{array}

=

\begin{array} {c}
P\textbf{i} \\
\left[ \begin{array} {r}
1 \\
0 \\
-\frac14 \\
\end{array} \right]
\end{array}

</math>

Thus <math>\frac43</math> is projected to {{vector|1 0 -1/4}}. Note that {{vector|1 0 -1/4}} is one quarter of an exact untempered JI ratio, <math>\frac{16}{5}</math>, and thus four tempered fourths equals a pure 5-limit minor thirteenth.

Now let <math>\textbf{i}</math> = <math>\frac65</math>.

<math>

\begin{array} {c}
P \\
\left[ \begin{array} {r}
1 & 1 & 0 \\
0 & 0 & 0 \\
0 & \frac14 & 1 \\
\end{array} \right]
\end{array}

\begin{array} {c}
\textbf{i} \\
\left[ \begin{array} {r}
1 \\
1 \\
-1 \\
\end{array} \right]
\end{array}

=

\begin{array} {c}
P\textbf{i} \\
\left[ \begin{array} {r}
2 \\
0 \\
-\frac34 \\
\end{array} \right]
\end{array}

</math>

Thus <math>\frac65</math> becomes {{vector|2 0 -3/4}}, and four of them equals an untempered <math>\frac{256}{125}</math>. The reader can do similar calculations to verify that <math>\frac21</math> is projected to <math>\frac21</math>, <math>\frac54</math> is projected to <math>\frac54</math>, and <math>\frac32</math> is projected to {{vector|0 0 1/4}}.

For quarter-comma meantone, it's plain to see from the projection matrix that it's a rank-2 temperament: one of the rows is already all-zeros, so clearly it's not full-rank (rank-3)! This will not in general be true of projection matrices, however. For example, we can take a look at ''third''-comma meantone's projection.

For third-comma meantone, three fifths (<math>\frac{27}{8}</math>) are a full comma flat. That works out to <math>\frac{10}{3}</math>. Thus the generator is {{vector|1/3 -1/3 1/3}}. Again, the period is a pure octave. This gives us our generator embedding <math>G</math>. Multiply it by the same mapping <math>M</math> to find the projection matrix <math>P</math>:

<math>

\begin{array} {c}
P \\
\left[ \begin{array} {r}
1 & \frac43 & \frac43 \\
0 & -\frac43 & -\frac13 \\
0 & \frac43 & \frac13 \\
\end{array} \right]
\end{array}

=

\begin{array} {c}
G \\
\left[ \begin{array} {r}
1 & \frac13 \\
0 & -\frac13 \\
0 & \frac13 \\
\end{array} \right]
\end{array}

\begin{array} {c}
M \\
\left[ \begin{array} {r}
1 & 1 & 0 \\
0 & 1 & 4 \\
\end{array} \right]
\end{array}

</math>

Let's use <math>P</math> to find out what <math>\frac65</math> gets projected to:

<math>

\begin{array} {c}
P \\
\left[ \begin{array} {r}
1 & \frac43 & \frac43 \\
0 & -\frac43 & -\frac13 \\
0 & \frac43 & \frac13 \\
\end{array} \right]
\end{array}

\begin{array} {c}
\textbf{i} \\
\left[ \begin{array} {r}
1 \\
1 \\
-1 \\
\end{array} \right]
\end{array}

=

\begin{array} {c}
P\textbf{i} \\
\left[ \begin{array} {r}
1 \\
1 \\
-1 \\
\end{array} \right]
\end{array}

</math>

Thus <math>\frac65</math> is an unchanged-interval of third-comma meantone, instead of <math>\frac54</math>. The reader can verify that <math>\frac21</math> is still an unchanged-interval.

Nearly two-dozen further examples of generator embeddings may be found throughout the article [[Generator embedding optimization]], including some with tempered octaves.

== Generator embedding ==
In order to understand projections, it is critical to understand the lesser-used and lesser-understood half of them: the generator embedding. So let's briefly cover this object next.

A '''generator embedding''' is an object that represents the ''embedding'' of a [[regular temperament]] from the tempered lattice back into tuning space. It could be thought of as representing the "tuning" information of a temperament, if one leaves out the actual "sizing" part of that (the conversion of prime factors to their logarithmic pitch size). It has one column for each of the temperament's [[generators]]. Each of these columns represents its generator's tuning in the form of a vector.

=== With respect to the generator tuning map ===
A more common way to view the tuning of a temperament than as a generator ''embedding'' is as a [[generator tuning map|generator ''tuning map'']]. In cases where tuning is thought of as approximation followed by embedding, the generator tuning map <math>𝒈</math> is closely related to the generator embedding <math>G</math>; it is simply <math>G</math> left-multiplied by the [[just tuning map]] <math>𝒋</math><ref group="note">Similarly, the projection matrix, when left-multiplied by <math>𝒋</math>, gives the ''temperament'' [[tuning map]] <math>𝒕</math>, usually referred to simply as the "tuning map" for short. 1/4-comma meantone's <math>𝒕</math> is {{map|1.000 1.585 2.232}}·{{ket|{{map|1 1 0}} {{map|0 0 0}} {{map|0 1/4 1}}}} = {{map|1.000 1.580 2.232}}. This is clearly closely related to the just tuning map, which represents the tuning of JI.</ref> (see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Units analysis#Just tuning map, generator embedding: generator tuning map]]). For example, since meantone is 5-limit, its just tuning map is {{map|log₂2 log₂3 log₂5}} ≈ {{map|1.000 1.585 2.232}}, so 1/4-comma meantone's <math>𝒈</math> is {{map|1.000 1.585 2.232}}·{{rbra|{{vector|1 0 0}} {{vector|0 0 1/4}}}} = {{map|1.000 0.580}}, or in cents instead of octaves, that's {{rbra|1200.000 696.578}}.

Many popular regular temperament tuning schemes work by optimizing for the entries of <math>𝒈</math> directly, and many times it's not helpful or insightful to view the generators in non-integer vector form, which are reasons for <math>𝒈</math>'s popularity over <math>G</math>. Some practitioners may not even view tuning as an optimization problem and will simply choose values for <math>𝒈</math> on gut feeling. This is all to say that this idea of approximating and then re-embedding, AKA projecting, is not an inherently necessary feature of RTT; it is only one way to look at it which may be valuable to some musicians and theoreticians but completely bonkers-seeming and convoluted to others.

=== Units ===
The units of a prime-count vector are typically understood to be "primes", which is natural enough given their name. But the units of the generator embedding <math>G</math> are better taken to be '''p'''/'''g''', read "primes ''per generator''." This makes sense because their job is to translate temperament generators back into terms of primes.

Here is an example generator embedding for a [[5-limit]], [[Tour_of_Regular_Temperaments#Rank-2_temperaments|rank-2 temperament]], with units given for each entry:

<math>
\left[ \begin{array} {rrr}
1\;{}^{\text{p}_1}{\mskip -5mu/\mskip -3mu}_{\text{g}_1} & \frac13\;{}^{\text{p}_1}{\mskip -5mu/\mskip -3mu}_{\text{g}_2} \\
0\;{}^{\text{p}_2}{\mskip -5mu/\mskip -3mu}_{\text{g}_1} & {-\frac13}\;{}^{\text{p}_2}{\mskip -5mu/\mskip -3mu}_{\text{g}_2} \\
0\;{}^{\text{p}_3}{\mskip -5mu/\mskip -3mu}_{\text{g}_1} & \frac13\;{}^{\text{p}_3}{\mskip -5mu/\mskip -3mu}_{\text{g}_2} \\
\end{array} \right]
</math>

The subscripts indicate which primes and which generators are related. So the columns, as previously stated, correspond to the two generators of the temperament, g₁ and g₂, while the rows correspond to the three primes for this temperament, p₁, p₂, and p₃, which are primes 2, 3, and 5, respectively.

See also [[Dave Keenan & Douglas Blumeyer's guide to RTT/Units analysis]], and/or the [[#Units|Units]] section later in this article for more details.

== Uniqueness ==
As just mentioned, projection matrices represent specific tunings of abstract temperaments, being the matrix product of a generator embedding which provides the embedding information, and a mapping which provides the approximation information. Notably, not only does the projection matrix represent the tuning of a temperament, it does so ''uniquely''. We can say that mappings and generator embeddings contain not only approximation and embedding information, but also ''generator form'' information, and it is this generator form information which causes them to be non-unique; however, when they are combined into a projection matrix, their generator form information ''cancels out'', and so no matter which combination of matching mapping and generator embedding we choose for a given temperament tuning, we will end up with the same exact projection.<ref group="note">The present author is not sure if any combination of mapping and generator embedding should lead to a projection matrix, and what the conditions on this would be. If anyone can figure this out, please add it to the article.</ref>

=== Mapping non-uniqueness===
To be clear, a mapping matrix does ''not'' uniquely represent approximation information. Multiple mappings can be found that describe the same temperament, in the sense that the same set of [[commas]] [[Tempering_out|vanish]]. This non-uniqueness is the reason why a [[canonical form]] for mappings was developed, which can be understood as a function which takes any equivalent mapping and converts it to the same exact mapping.

What distinguishes these equivalent mappings from each other is the sizes of the generators they use.

This concept is best demonstrated by example. Consider the following mapping :

<math>
M_1 = \left[ \begin{array} {r}
1 & 1 & 0 \\
0 & 1 & 4 \\
\end{array} \right]
</math>

This is an example of a mapping which represents meantone. This mapping represents it in a form where the first generator (the [[period]]) is an octave and the second generator (or simply ''the'' generator) is a perfect fifth. But this mapping ''also'' represents meantone:

<math>
M_2 = \left[ \begin{array} {r}
1 & 2 & 4 \\
0 & {-1} & {-4} \\
\end{array} \right]
</math>

This mapping represents it in a form where the period is an octave still but the generator is a perfect fourth. We can also have:

<math>
M_3 = \left[ \begin{array} {r}
1 & 0 & {-4} \\
0 & 1 & 4 \\
\end{array} \right]
</math>

Where the generator is a perfect twelfth, or even something like:

<math>
M_4 = \left[ \begin{array} {r}
12 & 19 & 28 \\
7 & 11 & 16 \\
\end{array} \right]
</math>

which technically makes the [[meantone comma]] <math>\frac{81}{80}</math> vanish—the main requirement of being a meantone temperament—but it has a period of about 76 ¢ and generator of about 41 ¢, which is pretty strange indeed (we're not being specific about tuning here, just giving the ballpark sizes).

What we can say is that generator form information differentiates these forms of the meantone mapping.<ref group="note">There is a way to represent approximation information without generator form information, which therefore means a data structure which inherently uniquely represents temperaments, and that is a [[multimap]].</ref>

=== Matching generator embeddings ===
For a given temperament tuning, such as [[quarter-comma meantone]], each possible form of the mapping will be matched with a generator embedding which it multiplies together with to find the unique quarter-comma meantone projection matrix. For example, for the {{ket|{{bra|1 1 0}} {{bra|0 1 4}}}} version we gave above which describes meantone in terms of an octave and a fifth, the matching generator embedding is:

<math>
G_1 = \left[ \begin{array} {r}
1 & 0 \\
0 & 0 \\
0 & \frac14 \\
\end{array} \right]
</math>

We can see here in the first column that the period is given by the integer vector {{vector|1 0 0}}, representing <math>2^1</math>, and the fifth is given by the non-integer vector {{vector|0 0 <math>\frac14</math>}}, representing <math>\sqrt[4]{5} \approx 1.495 \approx 1.5 = \frac32</math>.

For the second version we gave above, then, {{ket|{{bra|1 2 4}} {{bra|0 -1 -4}}}}, which describes meantone in terms of an octave and a ''fourth'', the matching generator embedding is:

<math>
G_ 2 = \left[ \begin{array} {r}
1 & 1 \\
0 & 0 \\
0 & {-\frac14} \\
\end{array} \right]
</math>

This has the same period, but now the generator is {{vector|1 0 <math>{-\frac14}</math>}}, representing <math>\frac{2}{\sqrt[4]{5}} \approx 1.337 \approx 1.\overline{3} \approx \frac43</math>.

=== Converging to the same projection ===
Now check out what happens when we find both <math>G_{1}M_{1}</math> and <math>G_{2}M_{2}</math>:

<math>

\begin{array} {c}
P \\
\left[ \begin{array} {r}
1 & 1 & 0 \\
0 & 0 & 0 \\
0 & \frac14 & 1 \\
\end{array} \right]
\end{array}

=

\begin{array} {c}
G_1 \\
\left[ \begin{array} {r}
1 & 0 \\
0 & 0 \\
0 & \frac14 \\
\end{array} \right]
\end{array}

\begin{array} {c}
M_1 \\
\left[ \begin{array} {r}
1 & 1 & 0 \\
0 & 1 & 4 \\
\end{array} \right]
\end{array}

\\ \text{ } \\ \text{ } \\

\begin{array} {c}
P \\
\left[ \begin{array} {r}
1 & 1 & 0 \\
0 & 0 & 0 \\
0 & \frac14 & 1 \\
\end{array} \right]
\end{array}

=

\begin{array} {c}
G_2 \\
\left[ \begin{array} {r}
1 & 1 \\
0 & 0 \\
0 & {-\frac14} \\
\end{array} \right]
\end{array}

\begin{array} {c}
M_2 \\
\left[ \begin{array} {r}
1 & 2 & 4 \\
0 & {-1} & {-4} \\
\end{array} \right]
\end{array}

\\

</math>

This shows us that there is no <math>P_1</math> or <math>P_2</math> or otherwise; we have only a single shared projection matrix <math>P</math> for any possible combination of mapping and generator embeddings representing this temperament tuning (which, again, in this case is quarter-comma meantone).

=== Keeping the mapping and generator embedding in sync ===
The way to transform from one mapping form <math>M_1</math> to another equivalent mapping form <math>M_2</math> is to perform elementary row operations, the most common of which is to add some multiple of one row to another (or subtract some multiple of one row from another). For more information on this, please see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Mappings#Adding and subtracting rows|the detailed explanation here]]. Similarly, we can transform from one generator embedding <math>G_1</math> to another equivalent generator embedding <math>G_2</math> by performing elementary ''column'' operations.

Supposing one desires to transform from a pair of <math>M_1</math> and <math>G_1</math> to another pair of <math>M_2</math> and <math>G_2</math> where both pairs multiply to the same <math>P</math>, or—said another way—you wish to keep your <math>M</math> and <math>G</math> ''in sync'', the simplest approach would be to—for each elementary row operation you apply to <math>M</math> you must apply the opposite elementary column operation to <math>G</math>, e.g. if you add three times the second row to the first row of <math>M</math>, then you must ''subtract'' three times the second ''column'' from the first ''column'' of <math>G</math>. This is along the same lines as the explanations provided for manipulating generator form by changing forms of <math>M</math>, which you can find here: [[Generator form manipulation]].

For example, if we have <math>M_1</math> = {{rket|{{bra|1 1 0}} {{bra|0 1 4}}}} and <math>G_1</math> = {{rbra|{{ket|1 0 0}} {{ket|0 0 <math>\frac14</math>}}}}, then <math>M_1</math> and <math>G_1</math> are in sync because they're both in the form where <math>g_1</math> is ~2 and <math>g_2</math> is ~3/2. Or if we have <math>M_2</math> = {{rket|{{bra|1 0 -4}} {{bra|0 1 4}}}} and <math>G_2</math> = {{rbra|{{ket|1 0 0}} {{ket|1 0 <math>\frac14</math>}}}} then they're still in sync because they're both <math>g_1</math> ~2 and <math>g_2</math> ~3 here. But if we mismatched those, they'd be out of sync. Those are both <math>M</math>'s for meantone, and both <math>G</math>'s that can work for quarter-comma meantone, but if you mismatch them with respect to the generator form information, you won't find the same <math>P</math> by matrix multiplication <math>GM</math>.

(This notion of "sync" is the same idea pointed out in the diagram at the start of the "Obtaining objects from the projection" section below, with the note on <math>G</math> reading "(the one matching M)". And for more information on generator form information, see the "Generator information types" below.)

We note in particular that putting <math>M</math> and <math>G</math> into their canonical forms independently is not a guarantee that they will remain in sync; canonicalization will not necessarily arrive at the same generator form information in <math>G</math> as it does in <math>M</math>.

=== Form matrix ===
When performing these elementary row and column operations, we can actually keep track of them in a way, by applying them in parallel to an identity matrix.

This is a special type of matrix called a unimodular matrix; that is, unimodularity is simply a property of a matrix which says that its determinant is ±1, which in itself is not particularly important except that it means the determinant is never 0 in which case the matrix would be uninvertible, and we do need it to be invertible. And this matrix will stay unimodular so long as we only apply elementary operations to it; that's what special about elementary operations.

Let's look at an example of this sort of form change tracking. Let's go from the octave-twelfth form of meantone to its octave-fifth form. We'll consider the octave-twelfth form to be our "home base" of sorts, and let it be "plain" <math>M</math>, and mark the octave-fifth form with subscripts:

<math>
\begin{align}

\begin{array} {c}
F \\
\left[ \begin{array} {c}
1 & 0 \\
{\color{blue}0} & {\color{blue}1} \\
\end{array} \right]
\end{array}

&\cdots

\begin{array} {c}
M \\
\left[ \begin{array} {c}
1 & 0 & -4 \\
{\color{blue}0} & {\color{blue}1} & {\color{blue}4} \\
\end{array} \right]
\end{array}

\\[24pt]

\begin{array} {c}
F_{\text{8ave,5th}} \\
\left[ \begin{array} {c}
1{\color{blue}+0} & 0{\color{blue}+1} \\
0 & 1 \\
\end{array} \right]
\end{array}

&\cdots

\begin{array} {c}
M_{\text{8ave,5th}} \\
\left[ \begin{array} {c}
1{\color{blue}+0} & 0{\color{blue}+1} & -4{\color{blue}+4} \\
0 & 1 & 4 \\
\end{array} \right]
\end{array}

\\[24pt]

\begin{array} {c}
F_{\text{8ave,5th}} \\
\left[ \begin{array} {c}
1 & 1 \\
0 & 1 \\
\end{array} \right]
\end{array}

&\cdots

\begin{array} {c}
M_{\text{8ave,5th}} \\
\left[ \begin{array} {c}
1 & 1 & 0 \\
0 & 1 & 4 \\
\end{array} \right]
\end{array}

\end{align}

</math>

Now that was only just one step, so it's not much of a feat. But you can keep going and going with your parallel changes to your <math>M</math> and <math>F</math> so long as you always stick to elementary operations, and everything we're about to discuss that you can do with trick will work just the same.

So end up with this as our unimodular matrix:

<math>
\begin{array} {c}
F_{\text{8ave,5th}} \\
\left[ \begin{array} {c}
1 & 1 \\
0 & 1 \\
\end{array} \right]
\end{array}
</math>

What's powerful about this <math>F_{\text{8ave,5th}}</math> matrix is that we can now use it ''as a transformation'' on the original <math>M</math> to change it directly into the <math>M_{\text{8ave,5th}}</math> we arrived at via the elementary row operations:

<math>

\begin{array} {c}
M_{\text{8ave,5th}} \\
\left[ \begin{array} {c}
1 & 1 & 0 \\
0 & 1 & 4 \\
\end{array} \right]
\end{array}

=

\begin{array} {c}
F_{\text{8ave,5th}} \\
\left[ \begin{array} {c}
1 & 1 \\
0 & 1 \\
\end{array} \right]
\end{array}

\begin{array} {c}
M \\
\left[ \begin{array} {c}
1 & 0 & -4 \\
0 & 1 & 4 \\
\end{array} \right]
\end{array}

</math>

So with one act of matrix multiplication, we can replace an arbitrarily large number of elementary row operations. Again, this may not seem so impressive considering in this case we only manipulated it by one step to begin with, but in general this is a very powerful effect. And, it synergizes with what happens to the generator embedding matrix simultaneously. That's what we'll look at next.

So, all of the same effects are there for the generator embedding matrix, but with respect to its columns:

<math>

\begin{array} {c}

\begin{array} {c}
G \\
\left[ \begin{array} {c}
{\color{blue}1} & 1 \\
{\color{blue}0} & 0\\
{\color{blue}0} & \frac14 \\
\end{array} \right]
\end{array}

& &

\begin{array} {c}
G_{\text{8ave,5th}} \\
\left[ \begin{array} {c}
1 & 1{\color{blue}-1} \\
0 & 0{\color{blue}-0}\\
0 & \frac14{\color{blue}-0} \\
\end{array} \right]
\end{array}

& &

\begin{array} {c}
G_{\text{8ave,5th}} \\
\left[ \begin{array} {c}
1 & 0 \\
0 & 0 \\
0 & \frac14 \\
\end{array} \right]
\end{array}

\\ \vdots & & \vdots & & \vdots \\

\begin{array} {c}
F^{-1} \\
\left[ \begin{array} {c}
{\color{blue}1} & 0 \\
{\color{blue}0} & 1 \\
\end{array} \right]
\end{array}

& &

\begin{array} {c}
F_{\text{8ave,5th}}^{-1} \\
\left[ \begin{array} {c}
1 & 0{\color{blue}-1} \\
0 & 1{\color{blue}-0} \\
\end{array} \right]
\end{array}

& &

\begin{array} {c}
F_{\text{8ave,5th}}^{-1} \\
\left[ \begin{array} {c}
1 & -1 \\
0 & 1 \\
\end{array} \right]
\end{array}

\end{array}
</math>

Notice that we have <math>F^{-1}</math> here, the inverse of <math>F</math>. You may recall that previously we had stated that the changes we make to the generator embedding matrix to keep it in sync with the mapping were the inverses of the ones we were applying to the mapping. Well, now we can see that they are inverses not just in an informal sense, but a very literal mathematical way! In the beginning, when both <math>F</math> and <math>F^{-1}</math> were identity matrices, the fact that they are inverses is apparent, in the same way that 1⁻¹ or 1/1 is 1. But if you wish to double-check the following:

<math>

\left[ \begin{array} {c}
1 & 1 \\
0 & 1 \\
\end{array} \right]
^{-1}

=

\left[ \begin{array} {c}
1 & -1 \\
0 & 1 \\
\end{array} \right]
</math>

Please go ahead and do so in your favored math software.

What this tells us is that anywhere we could write <math>P = GM</math> in our RTT equations, we can now write <math>P = GM = GF^{-1}FM</math>.

The injection of these <math>F</math> matrices doesn't affect the temperament or tuning in any way. Think about it this way. 9/8 always goes to 193.157 cents in quarter comma meantone, whether you're using the octave-and-fifth form or the octave-and-fourth form or any other form. All the form does is tell you what your generator sizes themselves are. They generate (span) the same space regardless. <math>F</math> is only good for keeping the embedding and approximating parts of your projection in sync when you're changing the basis, that's all. The point is for it to have no effect on the general intervals.

In other words, the service it provides is rather to ''give us a way to speak of the generator form of a temperament with respect to a particular mapping and a particular generator embedding.'' In other words, if we possess a scheme for unambiguously determining a "home base" <math>M</math>—such as a [[canonical form]], which we ''do'' have—and we possess a scheme for unambiguously determining a particular embedding—such as any tuning scheme which gives tunings expressible as embeddings (e.g. miniaverage, miniRMS)—then we now also have a way to describe with cold, hard matrices (i.e. not fuzzier instructions like "octave-fifth") what exact form we wish our generators to be in. We can speak of a pair of a linked bases&mdash"
;the generator embedding treated as a basis, and the mapping treated as a mapping-row basis—as the (generator) '''form''' of a temperament. And so a tuning system can be more fully specified than it could previously, leveraging this conceit.

It may be helpful to define <math>M_{\text{c}}</math> as the canonical mapping, the one in canonical form, and <math>G_{\text{c}}</math> as the one in "corresponding form" form to the canonical form of the mapping, i.e. per whichever tuning scheme is being used, it's the <math>G</math> you get from <math>M_{\text{c}}</math>. In this case, then with respect to <math>P = G_{\text{c}}M_{\text{c}}</math>, any <math>M_{\text{f}}</math> and <math>G_{\text{f}}</math> are viable so long as <math>M_{\text{c}} = F_{\text{f}}M_{\text{f}}</math> and <math>G_{\text{c}} = G_{\text{f}}F_{\text{f}}^{-1}</math> for some generator form <math>F_{\text{f}}</math>.

So, for a concrete example, we can say that for meantone temperament, given that its canonical form is the octave-twelfth form (that was no accident that we chose it as our "home base" earlier!) this generator form matrix <math>F_{\text{8ave,4th}}</math> represents the octave-fourth form:

<math>
\begin{array} {c}
F_{\text{8ave,4th}} \\
\left[ \begin{array} {c}
1 & 2 \\
0 & -1 \\
\end{array} \right]
\end{array}
</math>

Because we have the quarter-comma tuning of meantone as our tuning, then we have this as our canonical <math>M</math> and corresponding <math>G</math>:

<math>
\begin{array} {c}
G_{\text{c}} \\
\left[ \begin{array} {c}
1 & 1 \\
0 & 0\\
0 & \frac14 \\
\end{array} \right]
\end{array}

\begin{array} {c}
M_{\text{c}} \\
\left[ \begin{array} {c}
1 & 0 & -4 \\
0 & 1 & 4 \\
\end{array} \right]
\end{array}
</math>

And injecting <math>F_{\text{8ave,4th}}</math> and its inverse <math>F_{\text{8ave,4th}}^{-1}</math> (yes, it happens to be its own inverse):

<math>
\begin{array} {c}
G_{\text{c}} \\
\left[ \begin{array} {c}
1 & 1 \\
0 & 0\\
0 & \frac14 \\
\end{array} \right]
\end{array}

\begin{array} {c}
F_{\text{8ave,4th}}^{-1} \\
\left[ \begin{array} {c}
1 & 2 \\
0 & -1 \\
\end{array} \right]
\end{array}

\begin{array} {c}
F_{\text{8ave,4th}} \\
\left[ \begin{array} {c}
1 & 2 \\
0 & -1 \\
\end{array} \right]
\end{array}

\begin{array} {c}
M_{\text{c}} \\
\left[ \begin{array} {c}
1 & 0 & -4 \\
0 & 1 & 4 \\
\end{array} \right]
\end{array}
</math>

We can then find <math>G_{\text{8ave,4th}}</math> and <math>M_{\text{8ave,4th}}</math> if we like, as <math>G_{\text{c}}F_{\text{8ave,4th}}^{-1} </math> and <math>F_{\text{8ave,4th}}M_{\text{c}}</math>, respectively:

<math>
\begin{array} {c}
G_{\text{8ave,4th}} \\
\left[ \begin{array} {c}
1 & 1 \\
0 & 0\\
0 & -\frac14 \\
\end{array} \right]
\end{array}

\begin{array} {c}
M_{\text{8ave,4th}} \\
\left[ \begin{array} {c}
1 & 2 & 4 \\
0 & -1 & -4 \\
\end{array} \right]
\end{array}
</math>

== Units ==
The units of a projection matrix are a unique case: '''p'''/'''p''', read "primes per prime". At first glance it may appear that this expression should cancel out. The reason why the primes in the numerator and the primes in the denominator do not cancel is that the former series of primes progresses across the matrix by rows while the latter series of primes progresses across the matrix by columns:

<math>
\Large
\left[ \begin{array} {rrr}
\frac{\color{ForestGreen}\text{p}_1}{\color{ForestGreen}\text{p}_1} & \frac{\color{ForestGreen}\text{p}_1}{\color{NavyBlue}\text{p}_2} & \frac{\color{ForestGreen}\text{p}_1}{\color{Plum}\text{p}_3} \\[10pt]
\frac{\color{NavyBlue}\text{p}_2}{\color{ForestGreen}\text{p}_1} & \frac{\color{NavyBlue}\text{p}_2}{\color{NavyBlue}\text{p}_2} & \frac{\color{NavyBlue}\text{p}_2}{\color{Plum}\text{p}_3} \\[10pt]
\frac{\color{Plum}\text{p}_3}{\color{ForestGreen}\text{p}_1} & \frac{\color{Plum}\text{p}_3}{\color{NavyBlue}\text{p}_2} & \frac{\color{Plum}\text{p}_3}{\color{Plum}\text{p}_3} \\[10pt]
\end{array} \right]
</math>

So the primes only match along the main diagonal. There, they could be considered to cancel out. But we see no value to this. We may as well keep those as '''p'''/'''p''' like all the other entries.

Here's an example (again, of quarter-comma meantone) of a projection matrix with both its amounts and units:

<math>
\left[ \begin{array} {rrr}
1\;{}^{\text{p}_1}{\mskip -5mu/\mskip -3mu}_{\text{g}_1} & 1\;{}^{\text{p}_1}{\mskip -5mu/\mskip -3mu}_{\text{g}_2} & 0\;{}^{\text{p}_1}{\mskip -5mu/\mskip -3mu}_{\text{g}_3} \\
0\;{}^{\text{p}_2}{\mskip -5mu/\mskip -3mu}_{\text{g}_1} & 0\;{}^{\text{p}_2}{\mskip -5mu/\mskip -3mu}_{\text{g}_2} & 0\;{}^{\text{p}_2}{\mskip -5mu/\mskip -3mu}_{\text{g}_3} \\
0\;{}^{\text{p}_3}{\mskip -5mu/\mskip -3mu}_{\text{g}_1} & \frac14\;{}^{\text{p}_3}{\mskip -5mu/\mskip -3mu}_{\text{g}_2} & 1\;{}^{\text{p}_3}{\mskip -5mu/\mskip -3mu}_{\text{g}_3}\\
\end{array} \right]
</math>

In the first several of the following subsections, we examine units-only analyses of some RTT objects; for simpler of examples of these, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Units analysis#Units-only analyses]]. Then the last few sections are more general units analyses.

=== Generator embedding, mapping: Projection matrix ===
A <math>\small 𝗽</math>/<math>\small 𝗴</math> generator embedding and a <math>\small 𝗴</math>/<math>\small 𝗽</math> mapping combine to make a <math>\small 𝗽</math>/<math>\small 𝗽</math> projection matrix.

<math>
\begin{align}

\begin{array} {c} G \\[4pt]
\left[\begin{array} {rrr}
\frac{\color{ForestGreen}\mathsf{p_1}}{\color{BurntOrange}\mathsf{g_1}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{OrangeRed}\mathsf{g_2}} \\[6pt]
\frac{\color{NavyBlue}\mathsf{p_2}}{\color{BurntOrange}\mathsf{g_1}} & \frac{\color{NavyBlue}\mathsf{p_2}}{\color{OrangeRed}\mathsf{g_2}} \\[6pt]
\frac{\color{Plum}\mathsf{p_3}}{\color{BurntOrange}\mathsf{g_1}} & \frac{\color{Plum}\mathsf{p_3}}{\color{OrangeRed}\mathsf{g_2}} \\[6pt]
\end{array} \right]
\end{array}

\begin{array} {c} M \\[4pt]
\left[ \begin{array} {rrr}
\frac{\color{BurntOrange}\mathsf{g_1}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{BurntOrange}\mathsf{g_1}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{BurntOrange}\mathsf{g_1}}{\color{Plum}\mathsf{p_3}} \\[6pt]
\frac{\color{OrangeRed}\mathsf{g_2}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{OrangeRed}\mathsf{g_2}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{OrangeRed}\mathsf{g_2}}{\color{Plum}\mathsf{p_3}} \\[6pt]
\end{array} \right]
\end{array}

&\begin{array}{c}\\[4pt]=\end{array}

\begin{array} {c} GM \\[4pt]
\left[ \begin{array} {rrr}

(\frac{\color{ForestGreen}\mathsf{p_1}}{\cancel{\color{BurntOrange}\mathsf{g_1}}})
(\frac{\cancel{\color{BurntOrange}\mathsf{g_1}}}{\color{ForestGreen}\mathsf{p_1}})
+
(\frac{\color{ForestGreen}\mathsf{p_1}}{\cancel{\color{OrangeRed}\mathsf{g_2}}})
(\frac{\cancel{\color{OrangeRed}\mathsf{g_2}}}{\color{ForestGreen}\mathsf{p_1}})

&

(\frac{\color{ForestGreen}\mathsf{p_1}}{\cancel{\color{BurntOrange}\mathsf{g_1}}})
(\frac{\cancel{\color{BurntOrange}\mathsf{g_1}}}{\color{NavyBlue}\mathsf{p_2}})
+
(\frac{\color{ForestGreen}\mathsf{p_1}}{\cancel{\color{OrangeRed}\mathsf{g_2}}})
(\frac{\cancel{\color{OrangeRed}\mathsf{g_2}}}{\color{NavyBlue}\mathsf{p_2}})

&

(\frac{\color{ForestGreen}\mathsf{p_1}}{\cancel{\color{BurntOrange}\mathsf{g_1}}})
(\frac{\cancel{\color{BurntOrange}\mathsf{g_1}}}{\color{Plum}\mathsf{p_3}})
+
(\frac{\color{ForestGreen}\mathsf{p_1}}{\cancel{\color{OrangeRed}\mathsf{g_2}}})
(\frac{\cancel{\color{OrangeRed}\mathsf{g_2}}}{\color{Plum}\mathsf{p_3}})

\\[6pt]

(\frac{\color{NavyBlue}\mathsf{p_2}}{\cancel{\color{BurntOrange}\mathsf{g_1}}})
(\frac{\cancel{\color{BurntOrange}\mathsf{g_1}}}{\color{ForestGreen}\mathsf{p_1}})
+
(\frac{\color{NavyBlue}\mathsf{p_2}}{\cancel{\color{OrangeRed}\mathsf{g_2}}})
(\frac{\cancel{\color{OrangeRed}\mathsf{g_2}}}{\color{ForestGreen}\mathsf{p_1}})

&

(\frac{\color{NavyBlue}\mathsf{p_2}}{\cancel{\color{BurntOrange}\mathsf{g_1}}})
(\frac{\cancel{\color{BurntOrange}\mathsf{g_1}}}{\color{NavyBlue}\mathsf{p_2}})
+
(\frac{\color{NavyBlue}\mathsf{p_2}}{\cancel{\color{OrangeRed}\mathsf{g_2}}})
(\frac{\cancel{\color{OrangeRed}\mathsf{g_2}}}{\color{NavyBlue}\mathsf{p_2}})

&

(\frac{\color{NavyBlue}\mathsf{p_2}}{\cancel{\color{BurntOrange}\mathsf{g_1}}})
(\frac{\cancel{\color{BurntOrange}\mathsf{g_1}}}{\color{Plum}\mathsf{p_3}})
+
(\frac{\color{NavyBlue}\mathsf{p_2}}{\cancel{\color{OrangeRed}\mathsf{g_2}}})
(\frac{\cancel{\color{OrangeRed}\mathsf{g_2}}}{\color{Plum}\mathsf{p_3}})

\\[6pt]

(\frac{\color{Plum}\mathsf{p_3}}{\cancel{\color{BurntOrange}\mathsf{g_1}}})
(\frac{\cancel{\color{BurntOrange}\mathsf{g_1}}}{\color{ForestGreen}\mathsf{p_1}})
+
(\frac{\color{Plum}\mathsf{p_3}}{\cancel{\color{OrangeRed}\mathsf{g_2}}})
(\frac{\cancel{\color{OrangeRed}\mathsf{g_2}}}{\color{ForestGreen}\mathsf{p_1}})

&

(\frac{\color{Plum}\mathsf{p_3}}{\cancel{\color{BurntOrange}\mathsf{g_1}}})
(\frac{\cancel{\color{BurntOrange}\mathsf{g_1}}}{\color{NavyBlue}\mathsf{p_2}})
+
(\frac{\color{Plum}\mathsf{p_3}}{\cancel{\color{OrangeRed}\mathsf{g_2}}})
(\frac{\cancel{\color{OrangeRed}\mathsf{g_2}}}{\color{NavyBlue}\mathsf{p_2}})

&

(\frac{\color{Plum}\mathsf{p_3}}{\cancel{\color{BurntOrange}\mathsf{g_1}}})
(\frac{\cancel{\color{BurntOrange}\mathsf{g_1}}}{\color{Plum}\mathsf{p_3}})
+
(\frac{\color{Plum}\mathsf{p_3}}{\cancel{\color{OrangeRed}\mathsf{g_2}}})
(\frac{\cancel{\color{OrangeRed}\mathsf{g_2}}}{\color{Plum}\mathsf{p_3}})

\\[6pt]

\end{array} \right]
\end{array}

\\[20pt] &\begin{array}{c}\\[4pt]=\end{array}

\begin{array} {c} P \\[4pt]
\left[ \begin{array} {rrr}
\frac{\color{ForestGreen}\mathsf{p_1}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{Plum}\mathsf{p_3}} \\[6pt]
\frac{\color{NavyBlue}\mathsf{p_2}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{NavyBlue}\mathsf{p_2}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{NavyBlue}\mathsf{p_2}}{\color{Plum}\mathsf{p_3}} \\[6pt]
\frac{\color{Plum}\mathsf{p_3}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{Plum}\mathsf{p_3}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{Plum}\mathsf{p_3}}{\color{Plum}\mathsf{p_3}} \\[6pt]
\end{array} \right]
\end{array}

\end{align}
</math>

=== Just tuning map, generator embedding: Generator tuning map ===
Here, a <math>\mathsf{¢}</math>/<math>\small 𝗽</math> just tuning map and a <math>\small 𝗽</math>/<math>\small 𝗴</math> generator embedding combine to make a <math>\mathsf{¢}</math>/<math>\small 𝗴</math> generator tuning map.

<math>
\begin{align}

\begin{array} {c} 𝒋 \\[4pt]
\left[ \begin{array} {rrr}
\frac{{\large\mathsf{¢}}}{\color{ForestGreen}\mathsf{p_1}} & \frac{{\large\mathsf{¢}}}{\color{NavyBlue}\mathsf{p_2}} & \frac{{\large\mathsf{¢}}}{\color{Plum}\mathsf{p_3}} \\
\end{array} \right]
\end{array}

\begin{array} {c} G \\[4pt]
\left[\begin{array} {rrr}
\frac{\color{ForestGreen}\mathsf{p_1}}{\color{BurntOrange}\mathsf{g_1}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{OrangeRed}\mathsf{g_2}} \\[6pt]
\frac{\color{NavyBlue}\mathsf{p_2}}{\color{BurntOrange}\mathsf{g_1}} & \frac{\color{NavyBlue}\mathsf{p_2}}{\color{OrangeRed}\mathsf{g_2}} \\[6pt]
\frac{\color{Plum}\mathsf{p_3}}{\color{BurntOrange}\mathsf{g_1}} & \frac{\color{Plum}\mathsf{p_3}}{\color{OrangeRed}\mathsf{g_2}} \\[6pt]
\end{array} \right]
\end{array}

&\begin{array}{c}\\[4pt]=\end{array}

\begin{array} {c} 𝒋G \\[4pt]
\left[ \begin{array} {rrr}

(\frac{{\large\mathsf{¢}}}{\cancel{\color{ForestGreen}\mathsf{p_1}}})(\frac{\cancel{\color{ForestGreen}\mathsf{p_1}}}{\color{BurntOrange}\mathsf{g_1}}) + (\frac{{\large\mathsf{¢}}}{\cancel{\color{NavyBlue}\mathsf{p_2}}})(\frac{\cancel{\color{NavyBlue}\mathsf{p_2}}}{\color{BurntOrange}\mathsf{g_1}}) + (\frac{{\large\mathsf{¢}}}{\cancel{\color{Plum}\mathsf{p_3}}})(\frac{\cancel{\color{Plum}\mathsf{p_3}}}{\color{BurntOrange}\mathsf{g_1}})
&
(\frac{{\large\mathsf{¢}}}{\cancel{\color{ForestGreen}\mathsf{p_1}}})(\frac{\cancel{\color{ForestGreen}\mathsf{p_1}}}{\color{OrangeRed}\mathsf{g_2}}) + (\frac{{\large\mathsf{¢}}}{\cancel{\color{NavyBlue}\mathsf{p_2}}})(\frac{\cancel{\color{NavyBlue}\mathsf{p_2}}}{\color{OrangeRed}\mathsf{g_2}}) + (\frac{{\large\mathsf{¢}}}{\cancel{\color{Plum}\mathsf{p_3}}})(\frac{\cancel{\color{Plum}\mathsf{p_3}}}{\color{OrangeRed}\mathsf{g_2}})
\\[6pt]

\end{array} \right]
\end{array}

\\ &\begin{array}{c}\\[4pt]=\end{array}

\begin{array} {c} 𝒈 \\[4pt]
\left[ \begin{array} {rrr}
\frac{{\large\mathsf{¢}}}{\color{BurntOrange}\mathsf{g_1}} & \frac{{\large\mathsf{¢}}}{\color{OrangeRed}\mathsf{g_2}}
\end{array} \right]
\end{array}

\end{align}
</math>

=== Just tuning map, projection matrix: Tuning map ===
A <math>\mathsf{¢}</math>/<math>\small 𝗽</math> just tuning map and a <math>\small 𝗽</math>/<math>\small 𝗽</math> projection matrix combine to make a <math>\mathsf{¢}</math>/<math>\small 𝗽</math> tuning map.

<math>
\begin{align}

\begin{array} {c} 𝒋 \\[4pt]
\left[ \begin{array} {rrr}
\frac{{\large\mathsf{¢}}}{\color{ForestGreen}\mathsf{p_1}} & \frac{{\large\mathsf{¢}}}{\color{NavyBlue}\mathsf{p_2}} & \frac{{\large\mathsf{¢}}}{\color{Plum}\mathsf{p_3}} \\
\end{array} \right]
\end{array}

\begin{array} {c} P \\[4pt]
\left[ \begin{array} {rrr}
\frac{\color{ForestGreen}\mathsf{p_1}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{Plum}\mathsf{p_3}} \\[6pt]
\frac{\color{NavyBlue}\mathsf{p_2}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{NavyBlue}\mathsf{p_2}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{NavyBlue}\mathsf{p_2}}{\color{Plum}\mathsf{p_3}} \\[6pt]
\frac{\color{Plum}\mathsf{p_3}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{Plum}\mathsf{p_3}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{Plum}\mathsf{p_3}}{\color{Plum}\mathsf{p_3}} \\[6pt]
\end{array} \right]
\end{array}

&\begin{array}{c}\\[4pt]=\end{array}

\begin{array} {c} 𝒋P \\[4pt]
\left[ \begin{array} {rrr}

(\frac{{\large\mathsf{¢}}}{\cancel{\color{ForestGreen}\mathsf{p_1}}})
(\frac{\cancel{\color{ForestGreen}\mathsf{p_1}}}{\cancel{\color{ForestGreen}\mathsf{p_1}}})
+
(\frac{{\large\mathsf{¢}}}{\cancel{\color{NavyBlue}\mathsf{p_2}}})
(\frac{\cancel{\color{NavyBlue}\mathsf{p_2}}}{\cancel{\color{ForestGreen}\mathsf{p_1}}})
+
(\frac{{\large\mathsf{¢}}}{\cancel{\color{Plum}\mathsf{p_3}}})
(\frac{\cancel{\color{Plum}\mathsf{p_3}}}{\cancel{\color{ForestGreen}\mathsf{p_1}}})

&

(\frac{{\large\mathsf{¢}}}{\cancel{\color{ForestGreen}\mathsf{p_1}}})
(\frac{\cancel{\color{ForestGreen}\mathsf{p_1}}}{\cancel{\color{NavyBlue}\mathsf{p_2}}})
+
(\frac{{\large\mathsf{¢}}}{\cancel{\color{NavyBlue}\mathsf{p_2}}})
(\frac{\cancel{\color{NavyBlue}\mathsf{p_2}}}{\cancel{\color{NavyBlue}\mathsf{p_2}}})
+
(\frac{{\large\mathsf{¢}}}{\cancel{\color{Plum}\mathsf{p_3}}})
(\frac{\cancel{\color{Plum}\mathsf{p_3}}}{\cancel{\color{NavyBlue}\mathsf{p_2}}})

&

(\frac{{\large\mathsf{¢}}}{\cancel{\color{ForestGreen}\mathsf{p_1}}})
(\frac{\cancel{\color{ForestGreen}\mathsf{p_1}}}{\cancel{\color{Plum}\mathsf{p_3}}})
+
(\frac{{\large\mathsf{¢}}}{\cancel{\color{NavyBlue}\mathsf{p_2}}})
(\frac{\cancel{\color{NavyBlue}\mathsf{p_2}}}{\cancel{\color{Plum}\mathsf{p_3}}})
+
(\frac{{\large\mathsf{¢}}}{\cancel{\color{Plum}\mathsf{p_3}}})
(\frac{\cancel{\color{Plum}\mathsf{p_3}}}{\cancel{\color{Plum}\mathsf{p_3}}})

\end{array} \right]
\end{array}

\\ &\begin{array}{c}\\[4pt]=\end{array}

\begin{array} {c} 𝒕 \\[4pt]
\left[ \begin{array} {rrr}
\frac{{\large\mathsf{¢}}}{\color{ForestGreen}\mathsf{p_1}} & \frac{{\large\mathsf{¢}}}{\color{NavyBlue}\mathsf{p_2}} & \frac{{\large\mathsf{¢}}}{\color{Plum}\mathsf{p_3}} \\
\end{array} \right]
\end{array}

\end{align}
</math>

=== Just tuning map, projected interval: Tempered interval size ===
A <math>\mathsf{¢}</math>/<math>\small 𝗽</math> just tuning map and a vector representing a projected interval combine to give the interval's size in <math>\mathsf{¢}</math>.

<math>
\begin{align}

\begin{array} {c} 𝒋 \\[4pt]
\left[ \begin{array} {rrr}
\frac{{\large\mathsf{¢}}}{\color{ForestGreen}\mathsf{p_1}} & \frac{{\large\mathsf{¢}}}{\color{NavyBlue}\mathsf{p_2}} & \frac{{\large\mathsf{¢}}}{\color{Plum}\mathsf{p_3}} \\
\end{array} \right]
\end{array}

\begin{array} {c} P\textbf{i} \\[4pt]
\left[ \begin{array} {rrr}
\color{ForestGreen}\mathsf{p_1} \\[6pt]
\color{NavyBlue}\mathsf{p_2} \\[6pt]
\color{Plum}\mathsf{p_3} \\[6pt]
\end{array} \right]
\end{array}

&\begin{array}{c}\\[4pt]=\end{array}

\begin{array} {c} 𝒋P\textbf{i} \\[4pt]
\left[ \begin{array} {rrr}
(\frac{{\large\mathsf{¢}}}{\cancel{\color{ForestGreen}\mathsf{p_1}}})(\cancel{\color{ForestGreen}\mathsf{p_1}}) + (\frac{{\large\mathsf{¢}}}{\cancel{\color{NavyBlue}\mathsf{p_2}}})(\cancel{\color{NavyBlue}\mathsf{p_2}}) + (\frac{{\large\mathsf{¢}}}{\cancel{\color{Plum}\mathsf{p_3}}})(\cancel{\color{Plum}\mathsf{p_3}})
\end{array} \right]
\end{array}

\\ &\begin{array}{c}\\[4pt]=\end{array}

\begin{array} {c} 𝒕\textbf{i} \\[4pt]
\left[ \begin{array} {rrr}
{\large\mathsf{¢}}
\end{array} \right]
\end{array}

\end{align}
</math>

=== Generator embedding, mapped interval: Projected interval ===
A <math>\small 𝗽</math>/<math>\small 𝗴</math> generator embedding and a generator-count vector (units of <math>\small 𝗴</math>) combine to make a vector (units of <math>\small 𝗽</math>) representing the projected interval.

<math>
\begin{align}

\begin{array} {c} G \\[4pt]
\left[\begin{array} {rrr}
\frac{\color{ForestGreen}\mathsf{p_1}}{\color{BurntOrange}\mathsf{g_1}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{OrangeRed}\mathsf{g_2}} \\[6pt]
\frac{\color{NavyBlue}\mathsf{p_2}}{\color{BurntOrange}\mathsf{g_1}} & \frac{\color{NavyBlue}\mathsf{p_2}}{\color{OrangeRed}\mathsf{g_2}} \\[6pt]
\frac{\color{Plum}\mathsf{p_3}}{\color{BurntOrange}\mathsf{g_1}} & \frac{\color{Plum}\mathsf{p_3}}{\color{OrangeRed}\mathsf{g_2}} \\[6pt]
\end{array} \right]
\end{array}

\begin{array} {c} M\textbf{i} \\[4pt]
\left[ \begin{array} {rrr}
\color{BurntOrange}\mathsf{g_1} \\[6pt]
\color{OrangeRed}\mathsf{g_2} \\[6pt]
\end{array} \right]
\end{array}

&\begin{array}{c}\\[4pt]=\end{array}

\begin{array} {c} GM\textbf{i} \\[4pt]
\left[ \begin{array} {rrr}
(\frac{\color{ForestGreen}\mathsf{p_1}}{\cancel{\color{BurntOrange}\mathsf{g_1}}})(\cancel{\color{BurntOrange}\mathsf{g_1}}) + (\frac{\color{ForestGreen}\mathsf{p_1}}{\cancel{\color{OrangeRed}\mathsf{g_2}}})(\cancel{\color{OrangeRed}\mathsf{g_2}}) \\[6pt]
(\frac{\color{NavyBlue}\mathsf{p_2}}{\cancel{\color{BurntOrange}\mathsf{g_1}}})(\cancel{\color{BurntOrange}\mathsf{g_1}}) + (\frac{\color{NavyBlue}\mathsf{p_2}}{\cancel{\color{OrangeRed}\mathsf{g_2}}})(\cancel{\color{OrangeRed}\mathsf{g_2}}) \\[6pt]
(\frac{\color{Plum}\mathsf{p_3}}{\cancel{\color{BurntOrange}\mathsf{g_1}}})(\cancel{\color{BurntOrange}\mathsf{g_1}}) + (\frac{\color{Plum}\mathsf{p_3}}{\cancel{\color{OrangeRed}\mathsf{g_2}}})(\cancel{\color{OrangeRed}\mathsf{g_2}}) \\[6pt]
\end{array} \right]
\end{array}

\\[20pt] &\begin{array}{c}\\[4pt]=\end{array}

\begin{array} {c} P\textbf{i} \\[4pt]
\left[ \begin{array} {rrr}
\color{ForestGreen}\mathsf{p_1} \\[6pt]
\color{NavyBlue}\mathsf{p_2} \\[6pt]
\color{Plum}\mathsf{p_3} \\[6pt]
\end{array} \right]
\end{array}

\end{align}
</math>

=== Projection matrix, interval: Projected interval ===
A <math>\small 𝗽</math>/<math>\small 𝗽</math> projection matrix and a vector (units of <math>\small 𝗽</math>) representing an interval combine to make a new vector (still with units of <math>\small 𝗽</math>) representing the projected interval.

<math>
\begin{align}

\begin{array} {c} P \\[4pt]
\left[ \begin{array} {rrr}
\frac{\color{ForestGreen}\mathsf{p_1}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{Plum}\mathsf{p_3}} \\[6pt]
\frac{\color{NavyBlue}\mathsf{p_2}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{NavyBlue}\mathsf{p_2}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{NavyBlue}\mathsf{p_2}}{\color{Plum}\mathsf{p_3}} \\[6pt]
\frac{\color{Plum}\mathsf{p_3}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{Plum}\mathsf{p_3}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{Plum}\mathsf{p_3}}{\color{Plum}\mathsf{p_3}} \\[6pt]
\end{array} \right]
\end{array}

\begin{array} {c} \textbf{i} \\[4pt]
\left[ \begin{array} {rrr}
\color{ForestGreen}\mathsf{p_1} \\[6pt]
\color{NavyBlue}\mathsf{p_2} \\[6pt]
\color{Plum}\mathsf{p_3} \\[6pt]
\end{array} \right]
\end{array}

&\begin{array}{c}\\[4pt]=\end{array}

\begin{array} {c} P\textbf{i} \\[4pt]
\left[ \begin{array} {rrr}
(\frac{\color{ForestGreen}\mathsf{p_1}}{\cancel{\color{ForestGreen}\mathsf{p_1}}})(\cancel{\color{ForestGreen}\mathsf{p_1}}) + (\frac{\color{ForestGreen}\mathsf{p_1}}{\cancel{\color{NavyBlue}\mathsf{p_2}}})(\cancel{\color{NavyBlue}\mathsf{p_2}}) + (\frac{\color{ForestGreen}\mathsf{p_1}}{\cancel{\color{Plum}\mathsf{p_3}}})(\cancel{\color{Plum}\mathsf{p_3}}) \\[6pt]
(\frac{\color{NavyBlue}\mathsf{p_2}}{\cancel{\color{ForestGreen}\mathsf{p_1}}})(\cancel{\color{ForestGreen}\mathsf{p_1}}) + (\frac{\color{NavyBlue}\mathsf{p_2}}{\cancel{\color{NavyBlue}\mathsf{p_2}}})(\cancel{\color{NavyBlue}\mathsf{p_2}}) + (\frac{\color{NavyBlue}\mathsf{p_2}}{\cancel{\color{Plum}\mathsf{p_3}}})(\cancel{\color{Plum}\mathsf{p_3}}) \\[6pt]
(\frac{\color{Plum}\mathsf{p_3}}{\cancel{\color{ForestGreen}\mathsf{p_1}}})(\cancel{\color{ForestGreen}\mathsf{p_1}}) + (\frac{\color{Plum}\mathsf{p_3}}{\cancel{\color{NavyBlue}\mathsf{p_2}}})(\cancel{\color{NavyBlue}\mathsf{p_2}}) + (\frac{\color{Plum}\mathsf{p_3}}{\cancel{\color{Plum}\mathsf{p_3}}})(\cancel{\color{Plum}\mathsf{p_3}}) \\[6pt]
\end{array} \right]
\end{array}

\\[20pt] &\begin{array}{c}\\[4pt]=\end{array}

\begin{array} {c} P\textbf{i} \\[4pt]
\left[ \begin{array} {rrr}
\color{ForestGreen}\mathsf{p_1} \\[6pt]
\color{NavyBlue}\mathsf{p_2} \\[6pt]
\color{Plum}\mathsf{p_3} \\[6pt]
\end{array} \right]
\end{array}

\end{align}
</math>

=== The JI mapping times the JI generator embedding ===
This situation is a variation upon the situation [[Dave Keenan & Douglas Blumeyer's guide to RTT/Units analysis#Cancelation|described here]], where the projection matrix <math>P</math> is derived as the matrix product of the generator embedding matrix <math>G</math> and the temperament mapping matrix <math>M</math>. What we're going to do here is work through the variation where it's the ''JI'' embedding matrix <math>G_{\text{j}}</math> and the ''JI'' mapping matrix <math>M_{\text{j}}</math>.

The key difference here is that both of these matrices are [[identity matrix|identity matrices]]. Thus, upon multiplying them, the result is ''also'' an identity matrix.

As you'll recall, the units of a projection matrix are <math>\small 𝗽</math>/<math>\small 𝗽</math>, where these vectorized units do not cancel, because one increments along rows and the other along columns, so we get different combinations of primes in different entries. However, ''along the diagonal'', the indices match, and cancel out. Like so:

<math>
\left[ \begin{array} {c}
\frac{\color{ForestGreen}\mathsf{p_1}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{Plum}\mathsf{p_3}} \\[6pt]
\frac{\color{NavyBlue}\mathsf{p_2}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{NavyBlue}\mathsf{p_2}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{NavyBlue}\mathsf{p_2}}{\color{Plum}\mathsf{p_3}} \\[6pt]
\frac{\color{Plum}\mathsf{p_3}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{Plum}\mathsf{p_3}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{Plum}\mathsf{p_3}}{\color{Plum}\mathsf{p_3}} \\[6pt]
\end{array} \right]

→

\left[ \begin{array} {c}
\frac{\cancel{\color{ForestGreen}\mathsf{p_1}}}{\cancel{\color{ForestGreen}\mathsf{p_1}}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{Plum}\mathsf{p_3}} \\
\frac{\color{NavyBlue}\mathsf{p_2}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\cancel{\color{NavyBlue}\mathsf{p_2}}}{\cancel{\color{NavyBlue}\mathsf{p_2}}} & \frac{\color{NavyBlue}\mathsf{p_2}}{\color{Plum}\mathsf{p_3}} \\
\frac{\color{Plum}\mathsf{p_3}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{Plum}\mathsf{p_3}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\cancel{\color{Plum}\mathsf{p_3}}}{\cancel{\color{Plum}\mathsf{p_3}}} \\
\end{array} \right]

→

\left[ \begin{array} {c}
{\large\mathsf{𝟙}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{NavyBlue}\mathsf{p_2}} & \frac{\color{ForestGreen}\mathsf{p_1}}{\color{Plum}\mathsf{p_3}} \\[6pt]
\frac{\color{NavyBlue}\mathsf{p_2}}{\color{ForestGreen}\mathsf{p_1}} & {\large\mathsf{𝟙}} & \frac{\color{NavyBlue}\mathsf{p_2}}{\color{Plum}\mathsf{p_3}} \\[6pt]
\frac{\color{Plum}\mathsf{p_3}}{\color{ForestGreen}\mathsf{p_1}} & \frac{\color{Plum}\mathsf{p_3}}{\color{NavyBlue}\mathsf{p_2}} & {\large\mathsf{𝟙}} \\[6pt]
\end{array} \right]
</math>

So we keep the <math>\small 𝗽</math>/<math>\small 𝗽</math> unit because most of the entries still relate one prime to another.

However, if this units pattern were applied ''to an identity matrix'', like so:

<math>
\left[ \begin{array} {c}
1\,{\large\mathsf{𝟙}} & 0\,\frac{\color{ForestGreen}\mathsf{p_1}}{\color{NavyBlue}\mathsf{p_2}} & 0\,\color{ForestGreen}\frac{\mathsf{p_1}}{\color{Plum}\mathsf{p_3}} \\[6pt]
0\,\frac{\color{NavyBlue}\mathsf{p_2}}{\color{ForestGreen}\mathsf{p_1}} & 1\,{\large\mathsf{𝟙}} & 0\,\frac{\color{NavyBlue}\mathsf{p_2}}{\color{Plum}\mathsf{p_3}} \\[6pt]
0\,\frac{\color{Plum}\mathsf{p_3}}{\color{ForestGreen}\mathsf{p_1}} & 0\,\frac{\color{Plum}\mathsf{p_3}}{\color{NavyBlue}\mathsf{p_2}} & 1\,{\large\mathsf{𝟙}} \\[6pt]
\end{array} \right]
</math>

And then we eliminate units for all the entries whose quantities are zero:

<math>
\left[ \begin{array} {c}
1\,{\large\mathsf{𝟙}} & 0 & 0 \\
0 & 1\,{\large\mathsf{𝟙}} & 0 \\
0 & 0 & 1\,{\large\mathsf{𝟙}} \\
\end{array} \right]
</math>

Having zeroed out all of the entries who had any dimensionful units, we can now rightly say that this matrix as a whole is unitless (has dimensionless units).

Now we can see why, despite the fact that <math>GM = P</math> in a meaningful sense, it is not the case that <math>G_{\text{j}}M_{\text{j}} = P_{\text{j}}</math> in any meaningful sense. Instead <math>G_{\text{j}}M_{\text{j}} = I</math>, a completely generic, unitless identity matrix.

=== Derivation of generator embedding matrix from unchanged-interval basis and mapping ===
The generator embedding matrix <math>G</math> is related to the mapping <math>M</math> and unchanged-interval basis <math>\mathrm{U}</math> by the following formula:

<math>
G=\mathrm{U}(M\mathrm{U})^{-1}
</math>

This formula is used, among other places, in the [[Generator_embedding_optimization#Zero-damage_method|zero-damage method]] for computing miniaverage tunings of regular temperaments (see here: [[Generator embedding optimization#Convert to generators]]).

And it turns out this formula works fine when subjected to a units analysis. <math>\mathrm{U}</math> has units of primes. And <math>M</math> has units of generators per primes. So we have <math>\small 𝗽\!·\!((𝗴</math>/<math>\small 𝗽)\!·\!𝗽)^{-1}</math>. The <math>\small 𝗽</math>'s on the inside of the parens cancel, and we're left with <math>\small 𝗽𝗴^{-1}</math>, or in other words, <math>\small 𝗽</math>/<math>\small 𝗴</math>, which are indeed the units of <math>G</math>.

=== Derivation of projection matrix from unrotated vector list and scaling factor matrix ===
In a similar fashion we can show that the formula for the projection matrix <math>P</math> from the [[unrotated vector list]] <math>\mathrm{V}</math> and the [[scaling factor matrix]] <math>\textit{Λ}</math> holds up to units analysis. You can read about these objects and their relationships below: [[Projection#The unrotated vectors and scaling factors]]. The formula is:

<math>
\mathrm{V}\textit{Λ}\mathrm{V}^{-1}
</math>

With <math>\mathrm{V}</math> having units of <math>\small 𝗽</math> and <math>\textit{Λ}</math> having no units (having dimensionless units), we find: <math>{\small 𝗽}\!·\!\mathsf{𝟙}\!·\!{\small 𝗽^{-1}}</math>. This simplifies to <math>\small 𝗽</math>/<math>\small 𝗽</math>, which matches our projection matrix's units.

== Projection properties ==
=== Unrotated vectors and scaling factors ===
A <math>(d, d)</math>-shaped projection matrix represents both all of the [[comma]]s of a temperament as well as all of the [[unchanged-interval]]s of the tuning. It does so via <math>d</math> '''unrotated vectors''', which is to say, this projection only ''scales'' the vector. Each one of these unrotated vectors, then, is paired with a corresponding '''scaling factor''', which tells us by how much the projection scales it. Each of these scaling factors—which we'll represent with the Greek letter lambda <math>λ</math>—is equal to either 1 or 0, and no other value is possible (more on this in the next section).

* When a scaling factor is equal to 1, this essentially means "no scaling", since multiplying something by 1 has no effect; so, these unrotated vectors are ''also'' unscaled vectors, which means they are completely unchanged. In other words, these are the unchanged-intervals of this tuning of this temperament.
* When a scaling factor is equal to 0, this means the projection scales it down to nothing. These, then, are the commas of the temperament. These are technically unrotated vectors, in the sense that rotation is no longer meaningful for something that has been vanished.

For readers familiar with the linear algebra concepts of [https://mathworld.wolfram.com/Eigenvector.html eigenvector]s and eigenvalues, this all may sound familiar: eigenvector is technical math speak for an "unrotated vector", and eigenvalue is technical math speak for its "scaling factor"; the prefix "eigen-" comes from the German word for "own", which we can think of as referring to how it projects onto its own span, or in other words, its projection falls along the infinite line through space we get if we extend the original vector in both directions forever, which is just another way of saying that the projection doesn't rotate the vector off of its original span like it does for most other vectors (in actuality, though, the "eigen" part of "eigenvector" means "own" in a different, but related way: it refers to the fact that the vector is the projection's "own" vector, or in other words, that it ''characterizes'' the projection, which is why another commonly used term for a vector like this is a "characteristic vector"). For those readers unfamiliar with these ideas, we recognize that this may be a lot to process at once.

It may be helpful to visualize the projection as a distortion field across tuning space, with curvy vortices something like how we might see warm and cold fronts on a weather map, or specks of iron patterned by a magnetic field. In these sorts of visualizations, we could imagine the unrotated vectors as the arrows that points along the paths where the distortion pattern happens to come out to be perfectly straight.

Now what's mathematically powerful about the idea of unrotated vectors is this: because the projection is only scaling things along these vectors, not rotating them, then we no longer require a full-blown matrix to represent that change. We can dramatically simply our formula. We can represent such a change using nothing but a single number, called a ''scalar''. This scalar is its corresponding scaling factor.

=== Idempotence ===
The only possible scaling factors of a projection matrix are 1 and 0. That is because projections are ''idempotent''<ref group="note">Etymologically, the roots of this word are "same" and "power", meaning it has only the same power each time it is applied.</ref>, meaning that if we interpret them as functions, then repeatedly applying the function has no effect beyond the first application. In other words, if <math\textbf{i}<nowiki></math></nowiki> is an interval vector:

<math>
P\textbf{i} = PP\textbf{i} = PPP\textbf{i} \text{...}
</math>

We can see this by example with quarter-comma meantone:

<math>

\begin{array} {c}
P \\
\left[ \begin{array} {r}
1 & 1 & 0 \\
0 & 0 & 0 \\
0 & \frac14 & 1 \\
\end{array} \right]
\end{array}

\begin{array} {c}
\textbf{i} \\
\left[ \begin{array} {r}
{-1} \\
1 \\
0 \\
\end{array} \right]
\end{array}

=

\begin{array} {c}
P\textbf{i} \\
\left[ \begin{array} {r}
0 \\
0 \\
\frac14 \\
\end{array} \right]
\end{array}

\\ \text{ } \\ \text{ } \\

\begin{array} {c}
P \\
\left[ \begin{array} {r}
1 & 1 & 0 \\
0 & 0 & 0 \\
0 & \frac14 & 1 \\
\end{array} \right]
\end{array}

\begin{array} {c}
P\textbf{i} \\
\left[ \begin{array} {r}
0 \\
0 \\
\frac14 \\
\end{array} \right]
\end{array}

=

\begin{array} {c}
P\textbf{i} \\
\left[ \begin{array} {r}
0 \\
0 \\
\frac14 \\
\end{array} \right]
\end{array}

</math>

In other words, once we map a JI perfect fifth <math>\frac32 = 701.955\,¢</math> with quarter-comma meantone, we get <math>\sqrt[4]{5} = 696.578\,¢</math>. And if quarter-comma meantone's fifth goes through its own portal, it comes out the other side still as quarter-comma meantone's fifth.

So we can see why the only possible values a projection could scale its eigenvectors by would be 1 and 0, because these are the only values one can repeatedly scale things by without changing them past the first scaling.<ref group="note">Here is a helpful proof, giving another way to look at this fact: https://math.stackexchange.com/questions/1393656/show-that-if-%ce%bb-is-an-eigenvalue-of-a-projection-matrix-p-then-%ce%bb-1-or-%ce%bb-0/1393661#1393661</ref>

=== Flattening ===
A good way to understand the idempotence of projections is geometrically. This also agrees with our natural language intuitions about projections, such as projecting shadows of objects onto a wall. A projection is any transformation that reduces the dimensionality. What we've done is taken information in <math>d</math>-dimensional space and projected it down by one dimension, into <math>(d-1)</math>-dimensional space. For example we might take some 3D object and project it down to its silhouette. If we take the projection we just made and try to project it again, nothing is left to change.

But how to connect this to RTT? Well, when we make commas vanish, we reduce dimensionality. For example, 5-limit JI is three-dimensional, or ''rank-3'', because it is represented by a mapping matrix with three rows:

<math>
\left[ \begin{array} {r}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{array} \right]
</math>

In general, it is an <math>r × d</math> matrix where <math>r = d</math>, or in other words, a <math>d × d</math> matrix. Whereas a rank-2 temperament in 5-limit still boasts three columns (those correspond to the 3 primes of the 5-limit), but only has two rows, corresponding to how it has reduced the three generators of pure JI down to two generators approximating JI:

<math>
\left[ \begin{array} {r}
1 & 1 & 0 \\
0 & 1 & 4 \\
\end{array} \right]
</math>

So in general this is an <math>r × d</math> matrix where <math>r < d</math>.

So it may be confusing at first to realize that a projection matrix represents a lower-dimensional object, given that, like JI, it is always a <math>d × d</math> matrix! For example, the quarter-comma meantone projection we've been looking at:

<math>
\left[ \begin{array} {r}
1 & 1 & 0 \\
0 & 0 & 0 \\
0 & \frac14 & 1 \\
\end{array} \right]
</math>

projects all the vectors of the 3-dimensional space of 5-limit JI onto a 2-dimensional plane. On one hand, how could it not—it is, after all, an object representing a tuning of meantone, which is definitely a rank-2 temperament. But on the other hand, this projection matrix has three rows. So how can we reconcile this situation?

Here's one way to think about it. When we look at a mapping matrix like meantone's {{ket|{{bra|1 1 0}} {{bra|0 1 4}}}} above, we're basically plotting vectors in a brand-new 2D space, that is, one completely distinct from the 3D space we started with in JI. So while the <math>x</math>, <math>y</math>, and <math>z</math> axes for JI corresponded to our primes <math>\text{p}_1</math>, <math>\text{p}_2</math>, and <math>\text{p}_3</math>, our <math>x</math> and <math>y</math> axes (no <math>z</math> here!) of our new meantone space correspond to its generators <math>\text{g}_1</math> and <math>\text{p}_2</math>. That's how ''its'' 2D plane exists. However, as for a tuning of meantone's projection matrix, such as quarter-comma's, it remains in the original 3D space with the <math>\text{p}_1</math>, <math>\text{p}_2</math>, and <math>\text{p}_3</math> axes. What this means is that everything in that space gets projected—or smooshed down, you could think of it—into a single plane. So this plane is 2D, certainly, but importantly, it still requires three coordinates to describe ''because it exists titled at some angle through the original JI space''. It is a 2D object occupying 3D space, just like any 2D document you might have in your real-life 3D physical space sitting up on a bookshelf at some slight angle.

=== Visualization of simpler problem ===
It's rather tricky to visualize planes tilted within volumes, as it turns out. But perhaps a simpler example will be welcome. So let's gear down to the 3-limit, which is 2D. Consider 3-limit [[5edo|5-ET]], which is a rank-1, nullity-1 temperament. The one comma that it makes vanish is the [[Blackwood|blackwood comma]]. Here is the projection matrix for an arbitrary tuning of this temperament:

<math>
\begin{array} {c}
P \\
\left[ \begin{array} {r}
-\frac53 & -\frac83 & \\
\frac53 & \frac83 & \\
\end{array} \right]
\end{array}

=

\begin{array} {c}
G \\
\left[ \begin{array} {r}
-\frac13 \\
\frac13 \\
\end{array} \right]
\end{array}

\begin{array} {c}
M \\
\left[ \begin{array} {r}
5 & 8 \\
\end{array} \right]
\end{array}
</math>

In particular, this is the tuning where <math>\frac32</math> is unchanged (as are all of its multiples). So our unchanged-interval basis contains a single column vector. This describes a line we could draw across the 3-limit lattice, which we could call our "unchanged-interval line". The idea is that every pitch (i.e., every point in this 2D 3-limit space) will get projected onto this line. Every pitch that is already on this line therefore won't be moved by this tuning; that's why it's unchanged!

Note that this unchanged-interval line is also our tempered lattice. Normally, when we draw the tempered lattice separately from the JI lattice, we wouldn't draw it at an angle like this. But it's important that it's at this angle here, since that's the angle at which it has been re-embedded into our original JI space.

And since we have only a single comma, we know the angle at which every point in space that's off the unchanged-interval line will be projected onto it. We can figure it out by drawing a line from our comma, {{Vector|-8 5}}, to the origin, {{Vector|0 0}}. Every other projection line will be parallel to this line.
[[File:Projection demo - arbitrary tuning case.png|none|thumb|900x900px|The purple grid is the original 2D JI lattice. The green line is the 1D tempered lattice for this unchanged-3/2 tuning of 5-ET, which we can also think of as the unchanged-interval line. We've only shown 5 points on it (other than the origin), i.e. we iterated the generator until we reached the next octave. For each of these green points on the tempered lattice, we show one purple point on the JI lattice which projects to it, via a blue arrow.

In particular, we show the blackwood comma, the vanishing comma of this temperament, projecting to the unison [0}.

Note that each next iteration of the generator adds a new green dot 1/3 of the way to the next JI lattice node which is diagonally up and to the left; this corresponds to the fact that 3/2 is our unchanged-interval, and 3/2 [-1 1⟩ maps to ~3/2 [3} in 5-ET. In other words, each generator is 2^(-1/3)·3^(1/3) so that (2^(-1/3)·3^(1/3))^3 = 2^(-1)·3^(1) = 3/2.

Notice that all of the blue projection lines are exactly parallel, but that the green unchanged-interval line is not parallel or perpendicular to them.]]
For comparison's sake, here's the tuning of 5-ET we get by taking the pseudoinverse of its mapping (i.e. the [[minimax-E-copfr-S]] tuning). Note that here, the projection lines are exactly perpendicular to the unchanged-interval line.
[[File:Projection demo - pseudoinverse case.png|none|thumb|900x900px|The projection line from the blackwood comma to the origin (its vanishing) is identical to the previous diagram, and we're showing the same set of JI pitches projecting to the generator steps of 5-ET up to the octave, and all projection lines are still parallel. What's different in this diagram is the angle of the unchanged-interval line, which is now exactly perpendicular to the projection lines (note the 90° square angle indicators).

The reason why the pseudoinverse gives this perpendicular line is that in a sense this tuning is the "best fit line" to this temperament; it minimizes the overall lengths of all projection lines. Tuning tempered intervals is much trickier than this, however; this simplistic approach is deceptive in that it doesn't how 1) harmonic space isn't based on Euclidean distances and 2) primes 2 and 3 are not necessarily of equal importance.

The green dots are much more closely spaced here than in the previous diagram, but the generator size (in cents) is very similar; it's just that depending on how you're angled in this space, the combination of powers of 2 and 3 that define the generator size will be different, which can manifest as very different spacings visually.]]

And so from here we can try to generalize these insights to higher dimensions. Meantone temperament, again, would exist in 3D. The meantone comma at the point {{vector|-4 4 -1}} would make a blue line to unison at the origin {{vector|0 0 0}}. Also through the origin we'd have an green "unchanged-interval plane". This could be tilted at any angle; the tilt would depend only the tuning of the temperament. No matter the tuning, every interval will have a parallel blue line through the space projecting it onto this plane. And this plane would also therefore represent the tempered lattice, and would have a regularly-spaced 2D grid of points on it.

=== Tunings and commas ===
Any projection will occupy some line/plane/space/etc. that is perpendicular to all of the temperaments commas' vectors, no matter what the tuning happens to be. What differentiates one tuning from another is what path all the other intervals take onto the projection. Remember how we described projections earlier as distortion fields with curvy vortices something like how we might see warm and cold fronts on a weather map, and among those waves of distortion one will occasionally find paths or spots where the distortion works out perfectly straight or perfectly still. So any tuning of a temperament will have the same such straight paths for the commas mapping to the origin (the point with all zeros). But the tunings will all be set aside from each other by which unchanged-intervals they have, that is, the "eyes of the storm" if you will, the points in space that don't budge at all by the distortion. And whatever these are will be part of the complicated distortion field that leads to all the other non-eigenvector intervals landing somewhere or another on that set projection line/plane/space (its dimensionality depends on the rank of the temperament).

== Obtaining objects from the projection ==
From a projection <math>P</math> it is possible to obtain many other useful objects. See the below diagram.

[[File:Projection relations.png|frameless|800x800px]]

All you need is purple to get anything on here (a purple bit, or one blue bit and one red bit). Meaning that if you have any of these:
*<math>P</math>
*<math>\textit{Λ}</math> and <math>\mathrm{V}</math>
*<math>M</math> and <math>G</math>
*<math>M</math> and <math>\textrm{U}</math>
*<math>\textrm{C}</math> and <math>G</math>
*<math>\textrm{C}</math> and <math>\textrm{U}</math>
then you can get everything else using the methods described below.

=== The comma basis ===
To obtain the comma basis from <math>P</math>, simply take the nullspace as you would take it of the mapping (see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Exploring temperaments#Nullspace]] for more information).

Remember, the mapping represents the temperament, and the projection represents a particular tuning of this temperament, so no matter which projection we use, while they will each have their own unchanged-intervals, they will share the same commas: the commas of the temperament.

<math>\textrm{C} = \text{nullspace}(P)</math>

An alternative method for finding <math>\textrm{C}</math> is discussed in the "Alternative method for the comma and unchanged-interval bases" section below.

=== The unchanged-interval basis ===
The '''unchanged-interval basis''' of a tuning is the [[basis]] for all of its [[unchanged-interval]]s.

Obtaining the unchanged-interval basis <math>\textrm{U}</math> means to obtain the unchanged-intervals <math>\textbf{u}_1, \textbf{u}_2, …</math> of <math>P</math> or in other words any <math>\textbf{u}_i</math> for which <math>P\textbf{u}_i = \textbf{u}_i</math>. There are many ways to find these, but one way stands out for its clarity. We can rewrite this equation by subtracting <math>\textbf{u}_i</math> from both sides to get:

<math>
P\textbf{u}_i - \textbf{u}_i = 0
</math>

Next, we can factor out the <math>\textbf{u}</math> from both terms:

<math>
(P - I)\textbf{u}_i = 0
</math>

To be clear, that's an identity matrix <math>I</math>, the matrix equivalent of a 1.

So now what have we gained? We've given ourselves a way to think of this as a nullspace problem, in the same way we found the commas of the projection!

Let's review the commas problem but in another way. If <math>\textbf{c}_1</math> is a comma of the temperament, then <math>M\textbf{c}_1 = 0</math> and <math>P\textbf{c}_1 = 0</math>, which tells us that <math>\text{nullspace}(M)</math> or <math>\text{nullspace}(P)</math> will give us the comma basis <math>\textrm{C}</math>, or in other words a basis for all such commas <math>\textbf{c}_1, \textbf{c}_2, …</math>.

So, if <math>(P - I)\textbf{u}_i = 0</math>, then <math>\text{nullspace}(P - I)</math> should similarly give us a basis for all the <math>\textbf{u}_i</math> which satisfy that equation.

<math>\textrm{U} = \text{nullspace}(P - I)</math>

An alternative method for finding <math>\textrm{U}</math> is discussed in the "Alternative method for the comma and unchanged-interval bases" section below.

=== The mapping ===
To obtain (some form of) the mapping from a projection, find its comma basis per the above, then take the nullspace of that comma basis to get the mapping. For more information, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Exploring temperaments#Nullspace]].

<math>M = \text{nullspace}(\text{nullspace}(P))</math>

In other words, do a double nullspace.

=== The generator embedding ===
To obtain (some form of) a generator embedding for a projection, find the unchanged-interval basis per the above, and then use <math>G = \textrm{U}(M\textrm{U})^{-1}</math>. Let's unpack why this is so.

If the projection matrix is <math>P</math>, and a matrix whose columns are vectors representing the unchanged-intervals of a tuning is <math>\mathrm{U}</math>, then by this definition of unrotated (only-scaled) vectors:

<math>
P\mathrm{U} = λ\mathrm{U}
</math>

Again, this is because the intervals in <math>\mathrm{U}</math> are merely scaled, so we can represent their change with something simpler than a matrix like <math>P</math>, namely, a mere scalar like <math>λ</math>.

Furthermore, if <math>λ = 1</math>, then we have:

<math>
P\mathrm{U} = (1)\mathrm{U}
</math>

Or even more simply:

<math>
P\mathrm{U} = \mathrm{U}
</math>

Again, this shows how the projection matrix maps—or more specifically, we can say it ''projects''—the interval onto itself, or in other words, that it is unchanged by the tuning.

Because we know what <math>\mathrm{U}</math> is—we've specifically decided which intervals we wish to be unchanged here—we could solve for <math>P</math> now. But we don't actually care about <math>P</math> directly; what we want to find are the generators.

Fortunately, <math>P</math> is defined ''in terms of'' our desired generators, specifically our generator embedding <math>G</math>, as well as our mapping, <math>M</math>. The formula is this:

<math>
P = GM
</math>

So we can substitute <math>GM</math> in for <math>P</math>, and our equation will now be:

<math>
(GM)\mathrm{U} = \mathrm{U}
</math>

Since we're solving for our generators <math>G</math>, we right multiply both sides by the inverse of <math>M\mathrm{U}</math>, in order to cancel out the <math>M\mathrm{U}</math> on the left-hand side, and thereby isolate <math>G</math>:

<math>
GM\mathrm{U}(M\mathrm{U})^{-1} = \mathrm{U}(M\mathrm{U})^{-1} \\
G\cancel{M\mathrm{U}}\cancel{(M\mathrm{U})^{-1}} = \mathrm{U}(M\mathrm{U})^{-1} \\
G = \mathrm{U}(M\mathrm{U})^{-1}
</math>

For a simpler take on this idea which doesn't involve the projection matrix, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Tuning computation#Solving for the generators]].

=== The unrotated vectors and scaling factors ===
To obtain the unrotated vectors and scaling factors, we can find these in matrix form via a mathematical process known as "eigendecomposition", which can be handled by most math software. For example, in Wolfram Language, we can perform an eigendecomposition using the function <code>Eigensystem[]</code>. Here's an example of quarter-comma meantone:
<nowiki>In: P = {{1,1,0},{0,0,0},{0,1/4,1}}
In: Eigensystem[P] // MatrixForm
Out: {{1,1,0},{{0,0,1},{1,0,0},{4,-4,1}}}</nowiki>

The answer comes in the form of a tuple. The first element is a list of our scaling factors (eigenvalues), and the second element is a list of our unrotated vectors (eigenvectors). So, this result is telling us that for scaling factors 1, 1, and 0, respectively, we have corresponding unrotated vectors {{vector|0 0 1}}, {{vector|1 0 0}}, and {{vector|4 -4 1}}. The set of eigenvectors with eigenvalue 0 constitute a comma basis, while the set of eigenvectors with eigenvalue 1 constitute an unchanged-interval basis (so, quarter-comma meantone is characterized by an unchanged <math>\frac21</math> and <math>\frac51</math>).

The '''unrotated vector list''' may be treated as a matrix <math>\mathrm{V}</math>, and the list of scaling factors may be diagonalized (placed along the main diagonal of an otherwise all-zeros matrix) as the '''scaling factor matrix''' <math>\textit{Λ}</math> (that's a capital lambda, the same Greek letter we use the lowercase version of for the individual scaling factors).

As a bonus, we can get back to the projection via <math>P = \mathrm{V}\textit{Λ}\mathrm{V}^{-1}</math>.

By the way, the general method to find scaling factors is to solve the "characteristic equation" <math>\det(P - λI) = 0</math> for <math>λ</math>; that's lamdba times an identity matrix, or in other words, subtract lambda from each entry along <math>P</math>'s diagonal. But this shouldn't be necessary if one follows the other suggestions provided here.

=== Alternative method for the comma and unchanged-interval bases ===
This section assumes you've reviewed the immediately previous section.

The pair of <math>\textit{Λ}</math> and <math>\mathrm{V}</math> also provide us an alternative way to find <math>\textrm{C}</math> and <math>\textrm{U}</math>. If the 1's come first in <math>\textit{Λ}</math> and the 0's afterwards, then <math>\mathrm{V}</math> is simply the concatenation of <math>\textrm{C}</math> and <math>\textrm{U}</math>. To continue the above example, we have:

<math>
\begin{array} {c}
\textit{Λ} \\
\left[ \begin{array} {r}
\style{background-color:#98CC70;padding:5px}{1} & 0 & 0 \\
0 & \style{background-color:#98CC70;padding:5px}{1} & 0 \\
0 & 0 & \style{background-color:#F2B2B4;padding:5px}{0} \\
\end{array} \right]
\end{array}

\text{&}

\begin{array} {c}
\mathrm{V} \\
\left[ \begin{array} {rr|r}
\style{background-color:#98CC70;padding:5px}{0} & \style{background-color:#98CC70;padding:5px}{1} & \style{background-color:#F2B2B4;padding:5px}{4} \\
\style{background-color:#98CC70;padding:5px}{0} & \style{background-color:#98CC70;padding:5px}{0} & \style{background-color:#F2B2B4;padding:5px}{{-4}} \\
\style{background-color:#98CC70;padding:5px}{1} & \style{background-color:#98CC70;padding:5px}{0} & \style{background-color:#F2B2B4;padding:5px}{1} \\
\end{array} \right]
\end{array}

→

\begin{array} {c}
\textrm{U} \\
\left[ \begin{array} {r}
\style{background-color:#98CC70;padding:5px}{0} & \style{background-color:#98CC70;padding:5px}{1} \\
\style{background-color:#98CC70;padding:5px}{0} & \style{background-color:#98CC70;padding:5px}{0} \\
\style{background-color:#98CC70;padding:5px}{1} & \style{background-color:#98CC70;padding:5px}{0} \\
\end{array} \right]
\end{array}

\text{&}

\begin{array} {c}
\textrm{C} \\
\left[ \begin{array} {r}
\style{background-color:#F2B2B4;padding:5px}{4} \\
\style{background-color:#F2B2B4;padding:5px}{{-4}} \\
\style{background-color:#F2B2B4;padding:5px}{1} \\
\end{array} \right]
\end{array}

</math>

On the left, we've highlighted the diagonalized scaling factors with green if they are for unchanged-intervals, and with red if they are for vanishing commas. Then on the right, we've colored the entire corresponding vector columns, and placed a vertical line between the two green columns corresponding to the two green scaling factors of 1 and the one red column corresponding to the one red scaling factor of 0. And so we can see that the green-colored part of <math>\mathrm{V}</math> is <math>\textrm{U}</math>, and the red-colored part of <math>V</math> is <math>\textrm{C}</math>.

== Unrotated vector lists are not bases ==
As seen above, the unrotated vector list <math>V</math> is the concatenation of the unchanged-interval basis <math>\mathrm{U}</math> and the comma basis <math>\mathrm{C}</math>. Yet <math>V</math> itself is not a basis; it is merely a ''list'' of vectors. Why is this? Perhaps it is best visualized in the diagram below:

[[File:V is not basis.png|frameless|900x900px]]

The basic idea is that any two commas' projections are the zero vector, so an indefinite number of these may be combined with each other. And any unchanged-interval's projection is equal to itself, so an indefinite number of these may be combined as well. But any interval that is a combination of some number of unchanged-intervals and some number of commas will have the comma part projected to zero but the unchanged-interval part left alone, and thus be rotated by the projection. In other words, only intervals with the same scaling factor can be combined and still remain unrotated, which is to say that the commas form one type of unrotated vector basis and the unchanged-intervals form another type of unrotated vector basis, but these bases do not combine into one basis together.

That said, <math>V</math> must still be full-rank, meaning there is some relationship between the comma vectors and the unchanged-interval vectors. Specifically, it means that no unchanged-interval can be a linear combination of other unchanged-intervals and commas. If any were, we'd find an impossible situation, such as two unchanged-intervals off by a comma, so neither interval can change, yet they must both project to the same thing. For another take on this idea, see [[Dave Keenan & Douglas Blumeyer's guide to RTT/Tuning computation#Edge cases]].

== Generator information types ==
[[File:Info types - flow chart.png|thumb|602x602px|A diagram showing how the three information types (approximation, embedding, and form) break down across the tuning map, projection, mapping, generator embedding, multimap, and form matrices. ]]
One way to think about what's happening in this vicinity of RTT is that we have three different generator information types:
# approximation
#embedding
# form

Mappings combine types (1) and (3). Generator embeddings combine types (2) and (3). Projections combine types (1) and (2). So each possible subset of two of these pieces of information is accounted for by these three objects.

One advantage of using exterior algebra for RTT, i.e. representing a temperament with a multimap rather than a mapping matrix, is that it isolates the approximation information (1) from the form information (3), i.e. that any equivalent mapping is sent to the same multimap (largest minors list). For more information, see [[Douglas Blumeyer and Dave Keenan's Intro to exterior algebra for RTT#Pure representation of temperament information]].

In a similar way, when you combine a mapping with a generator embedding into a projection, the generator form information goes away from both, and you're left with just pure approximation and embedding information. We've used color to help convey this idea in the diagram to the right, with type (1) red, type (2) blue, type (3) green.

So, when you compress the multi-row projection matrix into a single-row tuning map by multiplying it by the just tuning map <math>𝒋</math>, the two types of information are still there, but blended together such that they are unrecoverable, or in other words, it's now ambiguous how we arrived at this <math>𝒕</math> and could have arrived to it from a different combination of <math>M</math> and <math>G</math>.

So again, while the mapping represents approximation information abstracted from any embedding, and the generator embedding represents embedding information that could be applied to any suitable approximation, they are each impure in the sense that they bind their respective generator information types to a particular form. This is the nature of how they must match to multiply together to give a certain projection. But no matter which two <math>G</math> and <math>M</math> you choose, as long as they do match, then their form information cancels out and we end up with just the approximation and embedding information.

(The top-left object is something no one has ever spoken about, as far as we know, and we see no use for it. We can't even say what "pure embedding information" would mean, independent of a temperament, or what it would mean to explore that space, in the way theorists have explored multimap space using temperament addition, etc. So we can see that the "multituning", perhaps we could call it, of quarter-comma meantone, is {{multivector|0 ¼ 0}})

And here's a series of tables that show various parts of the tempering process color-coded according to the above diagram:

[[File:Info types - checkers.png|frameless|800x800px]]

== Mapping projected intervals ==
For any interval vector <math>\textbf{i}</math> that has already been projected by the projection matrix <math>P</math>, if you then map it by the temperament mapping <math>M</math>, you'll get the same thing as you would have if you hadn't projected it. In other words:

<math>
MP\textbf{i} = M\textbf{i}
</math>

that is, so long as <math>P</math> and <math>M</math> are for the same temperament. Said yet another way, even though temperament mappings are primarily designed to map vectors with ''all integer'' entries (therefore representing JI intervals), if you happen to try mapping one of the projected vectors which typically have ''non-integer but at least rational'' entries, it will nonetheless find itself mapped to the same generator-count vector as whatever all-integer JI vector it came from would have been mapped to.

== Projecting to other spaces ==
The typical use case for a projection matrix is to re-embed a temperament lattice back into the original JI space from which it was tempered. But it is also possible to project to a completely different JI space after tempering, even a higher-dimensional one than the original space, or a lower-dimensional one than the tempered lattice. As long as you know how to translate vectors in the destination space into size, it's fair game. In fact, there's nothing stopping you from taking vectors in that space and projecting again, and again, and again. Though the present author fails to see if there are any meaningful musical purposes for this much temperamental distortion.

== Comma bases should always have integer entries ==
While projections, generator embeddings, and unchanged-interval bases may have non-integer entries, comma bases should not. Rather than temper out a comma with rational entries, clear its denominators. And a comma with irrational entries is departing from a scenario where RTT has much practical value.

== Applications ==
Projections and generator embeddings come up in some methods for calculating tunings according to commonly-used schemes:
# Those that use the pseudoinverse method, such as miniRMS and minimax-ES tunings: see [[Generator embedding optimization#Pseudoinverse method]].
# Those that use the zero-damage method, such as miniaverage tunings: see [[Generator embedding optimization#Zero-damage method]].
# Tunings based on unchanged-intervals: see [[Generator embedding optimization#Unchanged-interval method]].

== See also ==
* [[Projection matrices]]: a more mathematical discussion of these ideas
* [[Eigenmonzo basis]]: an alternative conceptualization for the unchanged-interval basis

Note that projection matrices do not have anything deeply to do with [[projective tuning space]] or [[projective tone space]], other than the fact that they both use the mathematical operation of projecting something from a higher dimension to a lower one.

== Footnotes ==
<references group="note" />

[[Category:Regular temperament theory]]
[[Category:Tuning]]

Wedgie/Archived version

2025-02-14T00:21:29Z

Cmloegcmluin: Undo revision 180619 by Domin (talk) - these concepts may have been given silly names, and are advanced and therefore difficult to understand, but they are not novelties

{{texmap}}
{{inaccessible}}

An alternating [[Wikipedia: Multilinear map|multilinear map]] which is a multilinear function taking a certain number ''n'' of [[monzos]] as arguments and returning an integer as a value we may call an '''''n''-map'''. This definition is quite a mouthful, and we will attempt to unpack it in more comprehensible language and explain why these things are valuable in tuning theory.

The simplest kind of ''n''-map is the 1-map, or [[val]]. This takes p-limit rational numbers, which may be written as monzos, and returns an integer, and may be called both a [[Wikipedia: Group homomorphism|group homomorphism]] and a [http://mathworld.wolfram.com/ModuleHomomorphism.html module homomorphism]. Vals are [[Wikipedia: Linear map|linear]]: If you take the product of two ''p''-limit rationals (or equivalently, add the corresponding monzos) then the val applied to the product/sum is the sum of the val applied to each separately, and so forth. Next come the 2-maps. These are linear functions {{nowrap|f(''u'', ''v'')}}, linear for ''u'' fixing ''v'', and linear for ''v'' fixing ''u'', and alternating. meaning that {{nowrap|f(''u'', ''u'') {{=}} 0}} and {{nowrap|f(''u'', ''v'') {{=}} −f(''v'', ''u'')}}.

One use for such things is as "machines" for measuring complexity. If we consider the 1-map which is the val for 11-limit 31et, we find we have <math>\tval{31 & 49 & 72 & 87 & 107}</math>. This tells us that it takes 72 steps of 31 equal to get to the approximate 5, which therefore has a complexity of 72 in this system. Now consider a 2-map {{nowrap|"meantone(''u'', ''v'')"}} which tells us, roughly speaking, how many generator steps it takes to get to ''v'' assuming ''u'' is being used as a period in septimal meantone. Using 2 as a period we can take (the approximate) 3/2 as a generator, in which case we have {{nowrap|meantone(2, 3) {{=}} 1}}, {{nowrap|meantone(2, 5) {{=}} 4}}, {{nowrap|meantone(2, 7) {{=}} 10}}. With 3 as a period and 3/2 as a generator, we get {{nowrap|meantone(3, 5) {{=}} 4}} and {{nowrap|meantone(3, 7) {{=}} 13}}. Finally, with if we take 5 as a period we find that four 3/2s give 5, so 5{{frac|1|4}} (or equivalently, {{frac|3|2}}) is the basic period. Using {{frac|3|2}} as a period and {{frac|9|8}} as a generator we get three generator steps to 7, and multiplying by four to be using 5 and not 5{{frac|1|4}} gives us {{nowrap|meantone(5, 7) {{=}} 12}}. This description does not make clear where the signs come from, which will emerge from the discussion of the wedge product, but it may help to elucidate how these things are connected to complexity.

Given an ''n''-map ''f'' and an ''m''-map ''g'' we may define a new ({{nowrap|''n'' + ''m''}})-map, the [[Wikipedia: Exterior algebra|wedge product]] of ''f'' and ''g'', written {{nowrap|''f'' ∧ ''g''}}, as follows:

<math>\displaystyle f\wedge g = \sum_s \operatorname{sgn}\left(s\right)f\left(x_s(1), x_s(2), \ldots, x_s\left(n\right)\right)g\left(x_s\left(n +1\right), \ldots, x_s\left(n + m\right)\right)</math>

where the sum is taken over {{nowrap|S(''n'', ''m'')}}, the set of all [[Wikipedia:Permutation|permutations]] of the first {{nowrap|''n'' + ''m''}} integers which are an [[Wikipedia:(p,q)_shuffle|{{nowrap|(''n'', ''m'')}} shuffle]], and sgn(''t'') is the [[Wikipedia: Parity of a permutation|parity of the permutation]] ''t'', which is +1 if ''t'' is even meaning an even number of transpositions of two numbers will get to ''t'', and −1 if ''t'' is odd.

If ''f'' and ''g'' are both vals (1-maps) then this becomes especially easy: {{nowrap|(''f'' ∧ ''g'')(''u'', ''v'') {{=}} ''f''(''u'')''g''(''v'') − ''f''(''v'')''g''(''u'')}}. Let's consider a specific example. Suppose <math>E_{19} = \val{19 & 30 & 44 & 53}</math> is the equal temperament val for septimal 19et, and <math>E_{31} = \val{31 & 49 & 72 & 87}</math> is the val for septimal 31et. Then writing intervals multiplicatively, we have

<math>\left(E_{19}\wedge E_{31}\right)\left(2,3\right) = E_{19}\left(2\right)E_{31}\left(3\right) - E_{19}\left(3\right)E_{31}\left(2\right) = 19*49 - 31*30 = 1.</math>

We may continue in this way to consider (2,5), (2,7), (3,5), (3,7), and (5,7), and writing them in this alphabetical order yields <math>\bitval{1 & 4 & 10 & 4 & 13 & 12}</math>. Here, the double angle braces are to indicate that the object is a 2-map. In fact, it is a special kind of 2-map in that it is the result of taking a wedge product rather than being, eg, the sum of two wedge products and is called a '''bival'''. In the same way, triple wedge products yield trivals which we depict with three angle braces, and so forth. Just as vals as associatd to rank one (equal) temperaments, bivals are associated to [[rank two temperament]]s such as [[meantone]], trivals to [[rank three temperament]]s, and so forth. In tuning theory the necessity to look at any n-maps aside from vals, bivals and trivals seldom arises, so this notation, which is not standardly mathematical but which has been adopted for convenience by tuning theorists, is quite practical. As we can see by comparing the numbers, {{nowrap|E19 ∧ E31}} is the same object we were calling {{nowrap|"meantone(''u'', ''v'')"}} which gives us complexity measurements for meantone.

This particular bival has the properties that the first nonzero coordinate (1, in this case) is positive, and that the [[Wikipedia: Greatest common divisor|GCD]] of all of the coordinates is 1. An n-map with these properties we may call ''reduced'', and reduced n-vals can be used to give unique names to [[regular temperament]]s.

These reduced ''n''-vals, and particularly reduced bivals, are called '''wedgies''' (or [[Plücker coordinates]]), and the fact that they are reduced both makes the name unique and tells us that wedgies are [[Wikipedia: Projective space|projective]], and hence the definition of regular temperaments in terms of them is projective. Thus, <math>E_{24} = \tval{24 & 38 & 56}</math> is a perfectly valid val, but since it is not reduced, it does not define a 1-wedgie and hence there is no 5-limit 24et temperament to go with it. Sometimes such a temperament, where more than one set of notes exists in it each of which is unreachable from the others via intervals with defined prime mappings is called ''contorted''. Wedgies do not name or signify contorted temperaments.

== Computing the previous example in Maple ==
In fact one can directly do many computations in Maple. Let us associate to the i'th prime the variable [math] x_i [/math]. So for example 7 corresponds to [math] x_4 [/math]. Then we introduce a basis vector [math] dx_i [/math] associated to the variable [math] x_i [/math]. Then to a pair of primes, for example [math] (3,7) [/math], we associate a basis vector [math] dx_2 \wedge dx_4 [/math]. Similarly if we have 3 or more primes. Expressions where there are [math] dx_i [/math] can be called 1 forms, [math] dx_i \wedge dx_j [/math] 2 forms etc.

In this way let's write <math>E_{19} = \val{19 & 30 & 44 & 53}</math> and <math>E_{31} = \val{31 & 49 & 72 & 87}</math> as [math] e_{19}=19dx_1+30dx_2+44dx_3+53dx_4 [/math] and [math] e_{31}=31dx_1+49dx_2+72dx_3+87dx_4 [/math]. Then we simply compute the exterior product

[math] \displaystyle \alpha =e_{19} \wedge e_{31}=dx_1\wedge dx_2+4dx_1\wedge dx_3+10dx_1\wedge dx_4+4dx_2\wedge dx_3 +13dx_2\wedge dx_4+12dx_3\wedge dx_4[/math].

A form is said to be decomposable if it can be written as an exterior product of 1 forms. So given above [math] \alpha [/math] how do we know if it is decomposable or not? Let us introduce a linear map [math] L (b)=b\wedge \alpha [/math]. This is a map from 1 forms to 3 forms. Now a kernel or nullspace of this map are all 1 forms such that [math] L(b)=0 [/math]. A basis for this nullspace in the present case is

[math] \displaystyle b_1=dx_1-4dx_3-13dx_4 \quad, \quad b_2=dx_2+4dx_3+10dx_4 [/math].

Now one can check that [math] \alpha=b_1\wedge b_2 [/math]. All these computations can be done easily in Maple when the things are properly set up. But is this useful to anyone?

The original [math] (e_{19}, e_{31}) [/math] is a different basis of the nullspace. In matrix terms the connection between them is as follows. If

[math] \displaystyle A=\begin{pmatrix} 19 & 30 & 44 & 53 \\ 31 & 49 & 72 & 87 \end{pmatrix}[/math]

then its Hermite (normal) form is

[math] \displaystyle H=\begin{pmatrix} 1 & 0 & -4 & -13 \\ 0 & 1 & 4 & 10 \end{pmatrix}[/math]

Let us take another [[Dave Keenan & Douglas Blumeyer's guide to EA for RTT|example]]. Suppose we have [math] \alpha_0=dx_1\wedge dx_2\wedge dx_3+2dx_1\wedge dx_2\wedge dx_4-2dx_1\wedge dx_3\wedge dx_4 -5dx_2\wedge dx_3\wedge dx_4[/math]. Now we have [math] L_0 (b)=b\wedge \alpha_0 [/math] and the basis for nullspace is

[math] \displaystyle b_1=dx_1+5dx_4 \quad, \quad b_2=dx_2+2dx_4 \quad, \quad b_3=dx_3+2dx_4 [/math].

and one can check that [math] \alpha=b_1\wedge b_2 \wedge b_3[/math]. Note by the way that {{nowrap|''n'' − 1}} forms are always decomposable (here n=4 and we computed the decomposition of 3 form).

== Truncation of wedgies ==
A useful operation to perform on any multivector, including wedgies, is truncation of the wedgie to a lower prime limit. This in effect sets all the basis vectors of a ''p''-limit wedgie which are greater than ''q'', the prime limit being truncated to zero. An algorithm to produce the truncation is to list the ''r''-subsets of the primes to ''p'' in alphabetical order, and add the corresponding coefficient to the list of the ''q''-limit truncation if and only if the maximum prime in the ''r''-subet is less than or equal to ''q''. Truncating a wedgie can lead to a non-wedgie if the GCD of the coefficients is greater than one; this means that in the lower limit, [[Wedgies_and_Multivals|contortion]] has appeared.

== Conditions on being a wedgie ==
If we take any three integers <math>\bitval{a & b & c}</math> such that {{nowrap|GCD(''a'', ''b'', ''c'') {{=}} 1}} and {{nowrap|''a'' ≥ 1}} the result is always a wedgie, the wedgie tempering out the [[The_dual|dual]] [[monzos|monzo]] <math>\tmonzo{c & -b & a}</math>. Since three such integers chosen at random are unlikely to produce a suitably small comma, the temperament will probably not be worth much, but at least it can be defined.

However, this is no longer the case in higher limits. There, not everything which looks like a wedgie will be one; for instance the wedgies must also satisfy the condition, for any wedgie W, that {{nowrap|W ∧ W {{=}} 0}}, where the "0" means the multival of rank 2''r'' obtained by wedging W with W. For prime limits 7 and 11 this condition suffices for rank two, but in general we need to check, for every prime {{nowrap|''q'' ≤ ''p''}} and every basis val ''v'' sending ''q'' to 1 and everything else to 0, that {{nowrap|(W ∨ ''q'') ∧ W {{=}} 0}} and {{nowrap|(W ∧ ''v'')º ∧ Wº {{=}} 0}}, where "∨" denotes the [[interior product]]. These conditions, the complete set along with the basic reduction conditions for being a wedgie, are known as the [[Wikipedia:Plücker embedding|Plücker relations]]. Note that the Plücker relations must be satisfied, since for a rank-''r'' multival, {{nowrap|W ∨ ''q''}} is a rank-({{nowrap|''r'' − 1}}) multival corresponding to tempering out all the commas of W, as well as ''q''.

In the 7-limit case, if we wedge a prospective rank two multival <math>W = \bitval{a & b & c & d & e & f}</math> with itself, we obtain <math>W \wedge W = 2\left(af - be + cd\right)</math>. The quantity {{nowrap|''af'' − ''be'' + ''cd''}} is the [[Wikipedia:Pfaffian|Pfaffian]] of the wedgie, and that the Pfaffian is zero tells us that in the five-dimensional projective space '''P⁵''' in which wedgies live, the wedgie lies on a (four-dimensional) [[Wikipedia:Hypersurface|hypersurface]], known as the [[Mathematical theory of regular temperaments#Geometry_of_regular_temperaments|Grassmannian]] {{nowrap|'''Gr'''(2, 4)}}. For an 11-limit rank-two wedgie <math>W = \bitval{w_1 & w_2 & w_3 & w_4 & w_5 & w_6 & w_7 & w_8 & w_9 & w_{10}}</math> we have that <math>W \wedge W = 2\quadtval{w_1 w_8 - w_2 w_6 + w_3 w_5 & w_1 w_9 - w_2 w_7 + w_4 w_5 & w_1 w_{10} - w_3 w_7 + w_4 w_6 & w_2 w_{10} - w_3 w_9 + w_4 w_8 & w_5 w_{10} - w_6 w_9 + w_7 w_8}</math> is zero. These conditions allow us to solve for three of the coefficients in terms of the other seven, and so that '''Gr'''(2, 5), the Grassmannian of rank-two 11-limit temperaments, is a six-dimensional projective [[Wikipedia: Algebraic variety|algebraic variety]] in nine-dimensional projective space '''P⁹'''. Wedgies correspond to rational points on this variety. For 11-limit rank three temperaments, we have {{nowrap|''w''6''w''1 − ''w''5''w''2 + ''w''4''w''3}} = {{nowrap|''w''9''w''1 − ''w''8''w''2 + ''w''7''w''3}} = {{nowrap|''w''10''w''1 − ''w''8''w''4 + ''w''7''w''5}} = {{nowrap|''w''10''w''2 − ''w''9''w''4 + ''w''7''w''6}} = {{nowrap|''w''10''w''3 − ''w''9''w''5 + ''w''8''w''6 {{=}} 0}}; again, this leads to a six-dimensional variety, this time {{nowrap|'''Gr'''(3, 5)}}.

== Constrained wedgies ==
Most of the wedgies which are legitimate according to the previous section do not represent temperaments which are in any way reasonable. To get temperaments which are, we need to constrain the relevant metrics--complexity should not be too high, error should not be too high, and badness should not be so high that competing temperaments are much better. Let us consider how bounding [[Tenney-Euclidean_temperament_measures#TE simple badness|relative error]] E, aka simple badness, constrains a 7-limit rank-w wedgie <math>W = \bitval{a & b & c & d & e & f}</math>.

By definition, <math>E = \left\|J \wedge Z\right\|</math>, where Z is the weighted version of W; if ''q''3, ''q''5, and ''q''7 are the logarithms base two of 3, 5, and 7, then <math>Z = \bival{\frac{a}{q_3} & \frac{b}{q_5} & \frac{c}{q_7} & \frac{d}{q_3 q_5} & \frac{e}{q_3 q_7} & \frac{f}{q_5 q_7}}</math>. We now have

<math>\left(\frac{d}{q_3q_5}-\frac{b}{q_5}+\frac{a}{q_3}\right)^2+\left(\frac{e}{q_3q_7}-\frac{c}{q_7}+\frac{a}{q_3}\right)^2+\left(\frac{f}{q_5q_7}-\frac{c}{q_7}+\frac{b}{q_5}\right)^2+\left(\frac{f}{q_5q_7}-\frac{e}{q_3q_7}+\frac{d}{q_3q_5}\right)^2 = 4 E^2</math>

From this we can conclude that ''d'', ''e'', and ''f'' satisfy <math>\left|d - q_3b + q_5a\right| \leqslant 2Eq_3q_5</math>, <math>\left|e - q_3c + q_7a\right| \leqslant 2Eq_3q_7</math>, and <math>\left|f - q_5c + q_7d\right| \leqslant 2Eq_5q_7</math>. This has an interesting interpretation: since <math>\tval{1 & q_3 & q_5 & q_7} \wedge \tval{0 & a & b & c} =</math> <math>\bitval{a & b & c & q_3b - q_5a & q_3c - q_7a & q_5c - q_7b}</math>, if {{nowrap|E ≤ {{frac|1|4''q''5''q''7}}}}, then the full wedgie can be recovered from the octave equivalent (OE) portion of the wedgie simply by wedging it with <math>\tval{1 & q_3 & q_5 & q_7}</math> and rounding to the nearest integer. This is not a very serious constraint to place on relative error; it seems unlikely anyone would be interested in a temperament which did not fall well under this low standard. Hence we may compile lists of reasonable temperaments by presuming "reasonable" requires this bound to be met, searching through triples <math>\bitval{a & b & c & \ldots}</math> (note that if all of these are zero, 2 is being tempered out) up to some complexity bound, wedging with <math>\tmonzo{1 & q_3 & q_5 & q_7}</math>, and rounding, then checking if the GCD is one and the Pfaffian is zero (i.e. {{nowrap|''af'' − ''be'' + ''cd'' {{=}} 0}}). Then we may toss everthing which does not meet the bound on relative error; however, for a reasonable list we will want a tighter bound.

If {{nowrap|C {{=}} {{!!}}W{{!!}}}} is the TE complexity, then the formula for the [[Tenney-Euclidean_metrics#Logflat TE badness|logflat badness]] B in the 7-limit rank-two case is particularly simple: {{nowrap|B {{=}} CE}}. If complexity is bounded by, for example, 20 (which allows for some quite complex temperaments) then since {{nowrap|E ≤ {{frac|1|4''q''5''q''7}}}}, {{nowrap|B ≤ {{frac|20|4''q''5''q''7}} {{=}} 0.767}}. This badness figure is easily met. While simply bounding complexity will lead to a finite list, the list would be enormous. An alternative is also to bound badness; for instance, we might produce a list of 7-limit rank-two temperaments with complexity less than 20 and a more reasonable badness limit, such as 0.05 or 0.06.

== Reconstituting wedgies in general ==
Essentially the same situation obtains for rank two temperaments in higher limits. The rule then is that if <math>E \leqslant \frac{1}{\binom{n}{3}\log_2\left(q\right)\log_2\left(p\right)}</math> then wedging <math>K = \tval{1 & \log_2\left(3\right) & \log_2\left(5\right) & \ldots & \log_2\left(p\right)}</math> with the val consisting of 0 followed by the first {{nowrap|''n'' − 1}} coefficients of the wedgie and rounding will give the wedgie, where ''p'' and ''q'' are the largest and second largest primes in the prime limit.

More generally, we can reconstitute W by rounding {{nowrap|Y {{=}} (W ∨ 2) ∧ K}} to the nearest integer coefficients, where K is the JI point <math>\tval{1 & \log_2\left(3\right) & \log_2\left(5\right) & \ldots & \log_2\left(p\right)}</math> in unweighted coordinates. Then we have <math>\left\|\left(W - Y\right) + Y\right\| \leqslant \left\|W-Y\right\| + \left\|Y\right\|</math> by the triangle inequality, and since {{nowrap|{{!!}}W − Y{{!!}}}} is bounded by the fact that W has been obtained by rounding, complexity, which is {{nowrap|{{!!}}(W − Y) + Y{{!!}} {{=}} {{!!}}W{{!!}}}}, can be bounded by ||Y||; which means it can be bounded by the coefficients of Y, which are those coefficients of W which can be found in W∨2 and over which we could be conducting a search. Moreover, we have from {{nowrap|Y ∧ K {{=}} ((W ∨ 2) ∧ K) ∧ K {{=}} 0}} that relative error, which is {{nowrap|{{!!}}W ∧ K{{!!}}}}, is {{nowrap|{{!!}}((W − Y) + Y) ∧ K{{!!}} {{=}} {{!!}}(W − Y) ∧ K{{!!}}}}, hence relative error is also bounded by the fact that {{nowrap|{{!!}}W − Y{{!!}}}} is bounded. This means that unless relative error is large, W can be recovered by rounding Y, and hence all wedgies within such a bound, which we may call ''recoverable'', can be found by a search on only some prospective coefficients. Temperaments which are not recoverable seem of little interest and may be ruled out of consideration. Search spaces for complexity measures such as [[Tenney-Euclidean_temperament_measures#TE Complexity|TE complexity]] which are defined in terms of the wedgie can be obtained by assuming all wedgie coefficients which are not being used to recover a wedgie are zero, which gives a minimum value for the complexity. In the case of rank two temperaments, an especially efficient complexity measure for such searches, and one with some other desirable properties, is [[generator complexity]].

In the particular case of the 11-limit in rank three, we have that {{nowrap|(W ∨ 2) ∧ K}} gives the full wedgie, which has ten coefficents, in terms of the first six upon rounding off. Using this for a search is less difficult than it sounds, since the complexity numbers for rank three are so much lower. If the relative error E satisifes {{nowrap|E ≤ {{frac|1|2√(5)''q''5''q''7''q''11}}}}, then the rounding off is guaranteed to lead to the correct result. This amount, 0.0099, is again easily met.

== See also ==
* [[Intro to exterior algebra for RTT]]: for detailed background about and further explanations for how to work with multivectors and multicovectors such as wedgies

[[Category:Math]]
[[Category:Exterior algebra]]
{{todo| improve readability }}

== Notes ==
<references />

Xenharmonic Wiki:Referendum

2025-02-09T15:11:31Z

Cmloegcmluin:

This page was created on Sun 9 Feb 2025 to '''enable the community to directly vote on major existential concerns''' the wiki is facing at this time.

'''Every editor who wishes to vote is encouraged to vote'''.

'''This page will be divided into two sections, "Yes/no questions", and "Multiple choice questions". Each section will begin with an explanation of how the voting method works'''.

'''You may vote on as many or as few questions as you wish'''. You may not vote more than once on the same question, but you may edit your vote after posting it if you wish.

; Authorship and responsibility for implementation

This page was mostly written by [[Budjarn Lambeth]], who copied the wording of some questions from [[Mike Battaglia]]. The results of the referendum will be implemented by [[Tyler Henthorn]].

(''Do note, however, that Henthorn is neither for nor against this referendum itself, nor is he for nor against any particular outcome. He is simply acting as a neutral party to facilitate it. It is his intention to allow the community (all of you) to take the lead, and he will simply implement whatever you decide.''

''Also note that Battaglia is neither for nor against the referendum either, Lambeth just simply copied Battaglia’s wording from an unrelated post Battaglia made.'')
[[Category:Xenharmonic Wiki]]

== Yes/no questions ==

=== How to vote ===
<nowiki>Edit the page and write either "* YES. ~~~~" or "* NO. ~~~~" in a new line under the question.

The "~~~~" will automatically tag the vote with a name and date to make counting easier.</nowiki>

=== How votes will be counted ===
After the third vote is cast on a question, a 7 day timer begins counting down. After 7 days have passed, whichever option has more votes - YES or NO - is the one that wins.

If there is a tie after 7 days, then the poll stays open and the very next vote cast acts as the tie breaker.

 

=== Question 1. Should FloraC be granted bureaucrat permissions? YES or NO? ===
* YES. [[User:BudjarnLambeth|BudjarnLambeth]] ([[User talk:BudjarnLambeth|talk]]) 07:22, 9 February 2025 (UTC)
* YES. [[User:Fitzgerald Lee|Fitzgerald Lee]] ([[User talk:Fitzgerald Lee|talk]]) 07:49, 9 February 2025 (UTC)
* YES. [[User:Xenoindex|Xenoindex]] ([[User talk:Xenoindex|talk]]) 07:57, 9 February 2025 (UTC)
* YES. We are in dire need of more bureaucrats, and FloraC has been a great admin. [[User:Sintel|Sintel]] ([[User talk:Sintel|talk]]) 12:43, 9 February 2025 (UTC)
* YES. [[User:Francium|Francium]] ([[User talk:Francium|talk]]) 13:01, 9 February 2025 (UTC)
* YES. [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 13:14, 9 February 2025 (UTC)
* YES. [[User:Inthar|Inthar]] ([[User talk:Inthar|talk]]) 15:04, 9 February 2025 (UTC)
* YES. [[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 15:11, 9 February 2025 (UTC)
 

=== Question 2. Should Fredg999 be granted sysop permissions? YES or NO? ===
* YES. [[User:BudjarnLambeth|BudjarnLambeth]] ([[User talk:BudjarnLambeth|talk]]) 07:22, 9 February 2025 (UTC)
* YES. [[User:Fitzgerald Lee|Fitzgerald Lee]] ([[User talk:Fitzgerald Lee|talk]]) 07:49, 9 February 2025 (UTC)
* YES. [[User:Xenoindex|Xenoindex]] ([[User talk:Xenoindex|talk]]) 07:57, 9 February 2025 (UTC)
* YES. Fredg999 has a long history of good edits and is a very level-headed person who is good at conflict resolution. [[User:Sintel|Sintel]] ([[User talk:Sintel|talk]]) 12:46, 9 February 2025 (UTC)
* YES. [[User:Francium|Francium]] ([[User talk:Francium|talk]]) 13:01, 9 February 2025 (UTC)
* YES. [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 13:14, 9 February 2025 (UTC)
* YES. [[User:Inthar|Inthar]] ([[User talk:Inthar|talk]]) 15:04, 9 February 2025 (UTC)
* YES. [[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 15:11, 9 February 2025 (UTC)
 

=== Question 3. Should Fredg999 be granted interface admin permissions? YES or NO? ===
* YES. [[User:BudjarnLambeth|BudjarnLambeth]] ([[User talk:BudjarnLambeth|talk]]) 07:22, 9 February 2025 (UTC)
* YES. [[User:Fitzgerald Lee|Fitzgerald Lee]] ([[User talk:Fitzgerald Lee|talk]]) 07:49, 9 February 2025 (UTC)
* YES. [[User:Xenoindex|Xenoindex]] ([[User talk:Xenoindex|talk]]) 07:57, 9 February 2025 (UTC)
* YES. (see above) [[User:Sintel|Sintel]] ([[User talk:Sintel|talk]]) 12:46, 9 February 2025 (UTC)
* YES. [[User:Francium|Francium]] ([[User talk:Francium|talk]]) 13:01, 9 February 2025 (UTC)
* YES. [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 13:14, 9 February 2025 (UTC)
* YES. [[User:Inthar|Inthar]] ([[User talk:Inthar|talk]]) 15:04, 9 February 2025 (UTC)
* YES. [[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 15:11, 9 February 2025 (UTC)
 

=== Question 4. Should Lériendil be granted sysop permissions? YES or NO? ===
* YES. [[User:BudjarnLambeth|BudjarnLambeth]] ([[User talk:BudjarnLambeth|talk]]) 07:22, 9 February 2025 (UTC)
* YES. [[User:Fitzgerald Lee|Fitzgerald Lee]] ([[User talk:Fitzgerald Lee|talk]]) 07:49, 9 February 2025 (UTC)
* YES. [[User:Xenoindex|Xenoindex]] ([[User talk:Xenoindex|talk]]) 07:57, 9 February 2025 (UTC)
* Neutral. Lériendil has not been active for very long, and seems mostly concerned with specific RTT projects rather than the wiki as a whole. But we need more admins and I don't think they'd do a bad job. [[User:Sintel|Sintel]] ([[User talk:Sintel|talk]]) 12:48, 9 February 2025 (UTC)
* YES. [[User:Francium|Francium]] ([[User talk:Francium|talk]]) 13:01, 9 February 2025 (UTC)
* YES. [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 13:14, 9 February 2025 (UTC)
* YES. [[User:Inthar|Inthar]] ([[User talk:Inthar|talk]]) 15:04, 9 February 2025 (UTC)
* YES. [[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 15:11, 9 February 2025 (UTC)
 

=== Question 5. Should Lériendil be granted interface admin permissions? YES or NO? ===
* YES. [[User:BudjarnLambeth|BudjarnLambeth]] ([[User talk:BudjarnLambeth|talk]]) 07:22, 9 February 2025 (UTC)
* YES. [[User:Fitzgerald Lee|Fitzgerald Lee]] ([[User talk:Fitzgerald Lee|talk]]) 07:49, 9 February 2025 (UTC)
* YES. [[User:Xenoindex|Xenoindex]] ([[User talk:Xenoindex|talk]]) 07:57, 9 February 2025 (UTC)
* Neutral. (see above) [[User:Sintel|Sintel]] ([[User talk:Sintel|talk]]) 12:49, 9 February 2025 (UTC)
* YES. [[User:Francium|Francium]] ([[User talk:Francium|talk]]) 13:01, 9 February 2025 (UTC)
* YES. [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 13:14, 9 February 2025 (UTC)
* YES. [[User:Inthar|Inthar]] ([[User talk:Inthar|talk]]) 15:04, 9 February 2025 (UTC)
* YES. [[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 15:11, 9 February 2025 (UTC)
 

== Multiple choice questions ==

=== How to vote ===
<nowiki>Edit the page and write "* The options I'm okay with are (whatever options you choose). ~~~~" in a new line under the question.

For example, "* The options I'm okay with are A, C, D, F. ~~~~" is a valid vote.</nowiki>

=== How votes are counted ===
After the third vote is cast on a question, a 7 day timer begins counting down. After 7 days have passed, whichever choice appeared the most times in peoples' votes is the one that wins.

If there is a tie after 7 days, then the poll stays open and the very next vote cast acts as the tie breaker.

 
=== Question 6. Who has the last say on the RTT pages, particularly Gene's stuff? ===
; Options
* A. They should be treated just like any other page, anyone can edit them.
* B. Any proposed edit must be first suggested on the talk page, and recieve more replies in favour than against, as of exactly 7 days after the first reply is posted.
* C. Any proposed edit must be first suggested in both a Discord poll in #wiki, and a Facebook poll in Xenwiki Work Group. It must recieve more yes votes than no votes on BOTH platforms, as of exactly 7 days after the poll is posted. If nobody votes at all on one of the platforms, that is considered a "yes" vote.
* D. Any proposed edit must be suggested to a bureaucrat on that bureaucrat's talk page, and the bureaucrat must say yes in order for it to be implemented.
* E. Any proposed edit must be suggested to any admin on that admin's talk page, and the admin must say yes in order for it to be implemented.
* F. None of these options are acceptable, the community should be asked to invent a different option which can then be voted on.

; Votes
* The options I'm okay with are A, B, E. [[User:BudjarnLambeth|BudjarnLambeth]] ([[User talk:BudjarnLambeth|talk]]) 07:22, 9 February 2025 (UTC)
* The options I'm okay with are A, B. [[User:Fitzgerald Lee|Fitzgerald Lee]] ([[User talk:Fitzgerald Lee|talk]]) 07:54, 9 February 2025 (UTC)
* AB [[User:Xenoindex|Xenoindex]] ([[User talk:Xenoindex|talk]]) 07:57, 9 February 2025 (UTC)
* A. As far as I'm concerned there is no good reason to treat these pages in a different way. Use the talk page and build consensus. [[User:Sintel|Sintel]] ([[User talk:Sintel|talk]]) 12:52, 9 February 2025 (UTC)
* The options I'm okay with are A, B, E. [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 13:17, 9 February 2025 (UTC)
* The options I'm okay with are A, B, E. [[User:Francium|Francium]] ([[User talk:Francium|talk]]) 13:20, 9 February 2025 (UTC)
* A. See Sintel's comment above. [[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 15:11, 9 February 2025 (UTC)

 

=== Question 7. Who is overseeing the mathematical content, and is responsible for making sure it is correct? ===
; Options
* A. It should be treated just like any other content, anyone can edit it.
* B. The bureaucrats should be responsible for overseeing it directly.
* C. The admins in general should be responsible for overseeing it directly.
* D. We should hold a one-off election to elect a team of five "math experts" with the responsibility and authority to correct all math on the wiki.
* E. We should hold a recurring annual election to elect a team of five "math experts" with the responsibility and authority to correct all math on the wiki.
* F. There should be a team of "math experts" which can be appointed by, or dismissed by, any bureaucrat.
* G. There should be a team of "math experts" which can be appointed by, or dismissed by, any admin in general.
* H. None of these options are acceptable, the community should be asked to invent a different option which can then be voted on.

; Votes
* The options I'm okay with are A, C, F, G. [[User:BudjarnLambeth|BudjarnLambeth]] ([[User talk:BudjarnLambeth|talk]]) 07:22, 9 February 2025 (UTC)
* The options I'm okay with are A, B, C. [[User:Fitzgerald Lee|Fitzgerald Lee]] ([[User talk:Fitzgerald Lee|talk]]) 07:54, 9 February 2025 (UTC)
* ACFG [[User:Xenoindex|Xenoindex]] ([[User talk:Xenoindex|talk]]) 07:57, 9 February 2025 (UTC)
* A. Having some kind of wiki-wide math project / task force is not a bad idea, but it should be on a voluntary basis and their edits/opinions should not be given special priority. [[User:Sintel|Sintel]] ([[User talk:Sintel|talk]]) 12:55, 9 February 2025 (UTC)
* The options I'm okay with are A, D, E, G. [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 13:17, 9 February 2025 (UTC)
* The options I'm okay with are A, C, E, G. Although Sintel has a good point with the voluntary basis and the priority. [[User:Francium|Francium]] ([[User talk:Francium|talk]]) 13:23, 9 February 2025 (UTC)
* A. See Sintel's comment above. [[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 15:11, 9 February 2025 (UTC)
 

=== Question 8. In future, who gets to add or remove admins? ===
; Options
* A. Any bureaucrat can add or remove admins at their own discretion (that's how it works now).
* B. A page is created where anyone can nominate someone to be added or removed as admin. A yes/no vote is then held using the same rules as the yes/no votes on this page, and the bureaucrats must enforce the results of the vote.
* C. None of these options are acceptable, the community should be asked to invent a different option which can then be voted on.

; Votes
* The options I'm okay with are A, B. [[User:BudjarnLambeth|BudjarnLambeth]] ([[User talk:BudjarnLambeth|talk]]) 07:22, 9 February 2025 (UTC)
* The options I'm okay with are A, B. [[User:Fitzgerald Lee|Fitzgerald Lee]] ([[User talk:Fitzgerald Lee|talk]]) 07:54, 9 February 2025 (UTC)
* AB [[User:Xenoindex|Xenoindex]] ([[User talk:Xenoindex|talk]]) 07:57, 9 February 2025 (UTC)
* C. Admin nominations should be discussed with everyone giving their opinions freely. After a set period the discussion closes and a bureaucrat reviews all of the arguments, and makes the final decision. The outcome should be based on community consensus (near unanimity) and not majority vote. [[User:Sintel|Sintel]] ([[User talk:Sintel|talk]]) 13:15, 9 February 2025 (UTC)
* The options I'm okay with are A, C. [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 13:17, 9 February 2025 (UTC)
* The options I'm okay with are A, C. [[User:Francium|Francium]] ([[User talk:Francium|talk]]) 13:24, 9 February 2025 (UTC)
* The options I'm okay with are A, B, C. [[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 15:11, 9 February 2025 (UTC)
 

=== Question 9. In future, who gets to add or remove bureaucrats? ===
; Options
* A. A page is created where anyone can nominate someone to be added or removed as bureaucrat. A yes/no vote is then held using the same rules as the yes/no votes on this page, except that YES requires a supermajority of at least 2/3 of the vote to pass. The bureaucrats must enforce the results of the vote. If the wiki has only one bureaucrat, the community cannot vote to remove them until a new bureaucrat has been added first.
* B. A protected page is created where any admin can nominate someone to be added or removed as bureaucrat. A yes/no vote is then held using the same rules as the yes/no votes on this page, except that YES requires a supermajority of at least 2/3 of the vote to pass, and only admins are allowed to vote. The bureaucrats must enforce the results of the vote. If the wiki has only one bureaucrat, the admins cannot vote to remove them until a new bureaucrat has been added first.
* C. None of these options are acceptable, the community should be asked to invent a different option which can then be voted on.

; Votes
* The options I'm okay with are A, B. [[User:BudjarnLambeth|BudjarnLambeth]] ([[User talk:BudjarnLambeth|talk]]) 07:22, 9 February 2025 (UTC)
* The options I'm okay with are A. [[User:Fitzgerald Lee|Fitzgerald Lee]] ([[User talk:Fitzgerald Lee|talk]]) 07:54, 9 February 2025 (UTC)
* AB [[User:Xenoindex|Xenoindex]] ([[User talk:Xenoindex|talk]]) 07:57, 9 February 2025 (UTC)
* C. (same process as I outlined above) [[User:Sintel|Sintel]] ([[User talk:Sintel|talk]]) 12:59, 9 February 2025 (UTC)
* The options I'm okay with are C. [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 13:17, 9 February 2025 (UTC)
* The options I'm okay with are C. [[User:Francium|Francium]] ([[User talk:Francium|talk]]) 13:25, 9 February 2025 (UTC)
* The options I'm okay with are A, B, C. [[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 15:11, 9 February 2025 (UTC)
 

=== Question 10. Who gets to determine what pages are locked or unlocked? ===
; Options
* A. Any admin can lock or unlock a page at their discretion (that's how it works now).
* B. Only bureaucrats are allowed to lock or unlock pages.
* C. A page is created where anyone can nominate a page to be locked or unlocked. A yes/no vote is then held using the same rules as the yes/no votes on this page. The admins must enforce the results of the vote.
* D. A page is created where anyone can nominate a page to be locked or unlocked. A yes/no vote is then held using the same rules as the yes/no votes on this page, except that ONLY admins can vote. The admins must enforce the results of the vote.
* E. Any proposed lock or unlock must be first suggested in both a Discord poll in #wiki, and a Facebook poll in Xenwiki Work Group. It must recieve more yes votes than no votes on BOTH platforms, as of exactly 7 days after the poll is posted. If nobody votes at all on one of the platforms, that is considered a "yes" vote. The admins must enforce the results of the vote.
* F. None of these options are acceptable, the community should be asked to invent a different option which can then be voted on.

; Votes
* The options I'm okay with are A, B, C, D. [[User:BudjarnLambeth|BudjarnLambeth]] ([[User talk:BudjarnLambeth|talk]]) 07:22, 9 February 2025 (UTC)
* The options I'm okay with are A, B, C. [[User:Fitzgerald Lee|Fitzgerald Lee]] ([[User talk:Fitzgerald Lee|talk]]) 07:54, 9 February 2025 (UTC)
* ABCD [[User:Xenoindex|Xenoindex]] ([[User talk:Xenoindex|talk]]) 07:57, 9 February 2025 (UTC)
* F. There should be a clear protection policy (analogous to [https://en.wikipedia.org/wiki/Wikipedia:Protection_policy Wikipedia's protection policy]). Protection can be done at the discretion of an admin, but the reasoning and duration should be clearly outlined on the talk page. To quote the Wikipedia policy:
:: Applying page protection solely as a preemptive measure is contrary to the open nature of Wikipedia and is generally not allowed. Instead, protection is used when vandalism, disruption, or abuse by multiple users is occurring at a frequency that warrants protection.
: Protection should be used sparingly and mostly as a last resort. Leaving pages protected indefinitely without leaving any path to unprotect them (as is happening now!) is unacceptable on a wiki. [[User:Sintel|Sintel]] ([[User talk:Sintel|talk]]) 13:09, 9 February 2025 (UTC)
* The options I'm okay with are F. [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 13:17, 9 February 2025 (UTC)
* The options I'm okay with are F. [[User:Francium|Francium]] ([[User talk:Francium|talk]]) 13:26, 9 February 2025 (UTC)
* The options I'm okay with are F. See Sintel's comment above. [[User:Cmloegcmluin|Cmloegcmluin]] ([[User talk:Cmloegcmluin|talk]]) 15:11, 9 February 2025 (UTC)

Unnoticeable comma

2025-01-30T04:30:34Z

Cmloegcmluin: /* 19-limit commas */

'''Unnoticeable commas''' are very small intervals. These [[comma]]s are called "unnoticeable" because, being equal to or less than 3.5 cents, they are smaller than the average peak [http://musictheory.zentral.zone/huntsystem2.html#2 JND] ([[just-noticeable difference]]) of human pitch perception, as illustrated by the research of [[Aaron Andrew Hunt]]. It is improbable that even a trained listener would be able to notice these intervals, and as such they a prime target for psychoacoustically informed [[Microtempering|microtempering]]. (However, a considerably larger comma can be unnoticeable in an [[adaptive just intonation|adaptive]] tuning context. Instead of one large pitch shift of the entire comma, there can be many small pitch shifts of a fraction of a comma, one per chord change. Given this, a noticeable 3-limit comma that arguably deserves inclusion is the [[mercator comma]], corresponding to using [[53edo]] for the circle of fifths.)

For commas over 100 cents in size, see [[Large comma]]; for commas in between 30 and 100 cents in size, see [[Medium comma]]; and for commas over 3.5 cents in size, see [[Small comma]].

Due to the wide range of sizes, cents values here and on the other commas pages are listed to 5 significant digits, instead of to a fixed number of decimal places as is the [[conventions|convention]] elsewhere on the wiki. Except for the 3-limit commas, the [[color notation|color name]] refers to both the comma and the temperament created when it is tempered out.

== List of commas, by prime limit ==
=== 3-limit commas ===
{{See also| Table of 3-limit commas }}

{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name|Color Name]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[Qian's large comma]]
| Wa-359
| w-359
| {{monzo| -569 359 }}
| 1.8453
| Chen Yingshi (2009)
|-
| [[Qian's small comma]], sasktel comma
| Wa-306
| w-306
| {{monzo| 485 -306 }}
| 1.7697
| Chen Yingshi (2009) for ''Qian's small comma''
|-
| [[Satanic comma]]
| Wa-665
| w-665
| {{monzo| -1054 665 }}
| 0.075575
| [[Marc Jones]] (1990)
|-
| 15601-comma
| Wa-15601
| w-15601
| {{monzo| 24727 -15601 }}
| 0.031499
|
|-
| 31867-comma
| Wa-31867
| w-31867
| {{monzo| -50508 31867 }}
| 0.012577
|
|-
| [[Archangelic comma]]
| Wa-190537
| w-190537
| {{monzo| 301994 -190537 }}
| 0.00011162
| [[Dawson Berry]] (2022)
|}

=== 5-limit commas ===
{{See also| Table of 5-limit commas }}

{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name|Color Name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[Dodifo comma]]
| Quadla-sepquinyo
| L4y35-9
|
| {{Monzo| -67 -9 35 }}
| 3.3850
| [[Petr Pařízek]] (2011)
|-
| [[Vishnuzma]], semisuper comma
| Sasepbigu
| sg144
| 6115295232 / 6103515625
| {{Monzo| 23 6 -14 }}
| 3.3380
| [[Gene Ward Smith]] (2002)
|-
| [[Enneadeca]], 19-tone-comma
| Neyo
| y19-4
| 19073486328125 / 19042491875328
| {{Monzo| -14 -19 19 }}
| 2.8155
| [[Gene Ward Smith]] (2002)
|-
| [[Vavoom comma]]
| Quinla-seyo
| L5y17-7
|
| {{Monzo| -68 18 17 }}
| 2.5232
| [[Paul Erlich]] (2002)
|-
| [[Tricot comma]]
| Quadsa-triyo
| s4y33
| 68719476736000 / 68630377364883
| {{Monzo| 39 -29 3 }}
| 2.2461
| [[Paul Erlich]] (2002)
|-
| [[Schisma]]
| Layo
| Ly-2
| 32805 / 32768
| {{Monzo| -15 8 1 }}
| 1.9537
| Hermann von Helmholtz, Alexander Ellis (1875)
|-
| [[Aluminium comma]]
| Sepsa-thegu
| s7g138
|
| {{Monzo| 92 -39 -13 }}
| 1.6767
| [[Eliora]] (2023)
|-
| [[Counterschisma]]
| Tribilagu
| L6g-5
|
| {{Monzo| -69 45 -1 }}
| 1.6613
| [[Gene Ward Smith]] (2003)
|-
| [[Neon comma]]
| Laquinquinbigu
|
|
| {{monzo| 21 60 -50 }}
| 1.6144
| [[Eliora]] (2024)
|-
| [[Septendecima]]
| Lala-sebiyo
| LLy34-8
|
| {{Monzo| -52 -17 34 }}
| 1.4313
|
|-
| [[Luna comma]], hemithirds comma
| Sasa-quintrigu
| ssg155
| 274877906944 / 274658203125
| {{Monzo| 38 -2 -15 }}
| 1.3843
| [[Gene Ward Smith]] (2002)
|-
| [[Minortone comma]], minortonma
| Trila-segu
| L3g172
|
| {{Monzo| -16 35 -17 }}
| 1.0919
| [[Gene Ward Smith]] (2002)
|-
| [[Ennealimma]]
| Satritribiyo
| sy18-3
| 7629394531250 / 7625597484987
| {{Monzo| 1 -27 18 }}
| 0.86183
| [[Paul Erlich]], [[Gene Ward Smith]] (2001)
|-
| Astro comma
| Tribisa-thiwegu
| s6g3110
|
| {{Monzo| 91 -12 -31 }}
| 0.81486
| [[Paul Erlich]] (2002)
|-
| [[Gaster comma]]
| Quadbila-negu
| L8g19-3
|
| {{Monzo| -70 72 -19 }}
| 0.79950
| [[User:Xenllium|Xenllium]] (2022)
|-
| [[Niobium comma]]
|
|
|
| {{Monzo| -875 492 41 }}
| 0.72269
| [[Eliora]] (2023)
|-
| [[Kwazy comma]]
| Quadla-quadquadyo
| L4y16-6
|
| {{Monzo| -53 10 16 }}
| 0.56943
| [[Paul Erlich]], [[Gene Ward Smith]] (2002)
|-
| Whoosh
| Saletrigu
| sg337
|
| {{Monzo| 37 25 -33 }}
| 0.52246
| [[Paul Erlich]] (2002)
|-
| Egads
| Setriyo
| y51-9
|
| {{Monzo| -36 -52 51 }}
| 0.33936
| [[Paul Erlich]] (2002)
|-
| [[Monzisma]]
| Quinsa-yoyo
| s5yy4
|
| {{Monzo| 54 -37 2 }}
| 0.29240
| [[Margo Schulter]] (2001)
|-
| Fortune
| Tritrila-sepbiyo
| L9y14-9
|
| {{Monzo| -107 47 14 }}
| 0.27703
| [[Paul Erlich]] (2002)
|-
| Gross
| Quinbisa-fosegu
| s10g4715
|
| {{Monzo| 144 -22 -47 }}
| 0.24543
| [[Paul Erlich]] (2002)
|-
| Senior
| Quadla-sepquingu
| L4g354
|
| {{Monzo| -17 62 -35 }}
| 0.23007
| [[Paul Erlich]] (2002)
|-
| [[Counterquectisma]], deltapion
| Quintritrilayo
| L45y-31
|
| {{Monzo| -500 314 1 }}
| 0.18399
| [[Eliora]] (2023)
|-
| [[Quectisma]]
|
|
|
| {{Monzo| 554 -351 1 }}
| 0.10841
| [[User:CompactStar|CompactStar]] (2023)
|-
| Raider
| Tritrisa-thiseyo
| s9y371
|
| {{Monzo| 71 -99 37 }}
| 0.062327
| [[Gene Ward Smith]] (2002)
|-
| Pirate
| Quinla-sepsepyo
| L5y49-12
|
| {{Monzo| -90 -15 49 }}
| 0.046966
| [[Paul Erlich]] (2002)
|-
| Viking
| Nela-siweyo
| L19y61-23
|
| {{Monzo| -251 69 61 }}
| 0.031605
| [[Gene Ward Smith]] (2002)
|-
| [[Kirnberger's atom]]
| Sepbisa-quadtrigu
| s14g1212
|
| {{Monzo| 161 -84 -12 }}
| 0.015361
|
|-
| Selenia
|
|
|
| {{Monzo| -433 -137 280 }}
| 0.0047636
|
|-
| Titania
|
|
|
| {{Monzo| 2746 -521 -827 }}
| 0.0031829
|
|-
| Quark
| Twethebila-sequinyo
|
|
| {{Monzo| -573 237 85 }}
| 8.8361 × 10-4
|
|-
| Scamp
|
|
|
| {{Monzo| -5836 4293 -417 }}
| 3.3472 × 10-5
|
|-
| Rover
|
|
|
| {{Monzo| 1634 1502 -1729 }}
| 2.7513 × 10-5
|
|-
| Rascal
|
|
|
| {{Monzo| -7470 2791 1312 }}
| 5.9596 × 10-6
|
|}

=== 7-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name|Color Name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[Decovulture comma]]
| Sasa-birugugu
| ssrrg42
| 67108864 / 66976875
| {{Monzo| 26 -7 -4 -2 }}
| 3.4083
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Rainy comma]]
| Laquinzo-atriyo
| Lz5y32
| 2100875/2097152
| {{Monzo| -21 0 3 5 }}
| 3.0706
| [[User:Flirora|+merlan #flirora]] (2020)
|-
| [[Pontiqak comma]]
| Lazozotritriyo
| Lzzy9-2
| 95703125 / 95551488
| {{Monzo| -17 -6 9 2 }}
| 2.7452
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Pessoalisma]]
| Sasa-tribiru-agugu
| ssr6gg-2
| 2147483648 / 2144153025
| {{Monzo| 31 -6 -2 -6 }}
| 2.6871
| [[Dawson Berry]] (2023)
|-
| [[Mitonisma]]
| Laquadzo-agu
| Lz4g2
| 5250987/5242880
| {{Monzo| -20 7 -1 4 }}
| 2.6749
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Horwell comma]]
| Lazoquinyo
| Lzy5-2
| 65625/65536
| {{Monzo| -16 1 5 1 }}
| 2.3495
| [[Gene Ward Smith]] (2005)
|-
| [[Forge comma]]
| Lala-trizo-aquingu
| LLz3g52
| 1640558367 / 1638400000
| {{Monzo| -19 14 -5 3 }}
| 2.2792
| [[Dawson Berry]] (2022)
|-
| [[109-7-comma]]
|
| L16z10939
|
| {{Monzo| -306 0 0 109 }}
| 2.0238
|
|-
| [[Neptunisma]]
| Laruruleyo
| Lrry11-4
| 48828125 / 48771072
| {{monzo| -12 -5 11 -2 }}
| 2.0240
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Slendrisma]], slendric schisma
| Sasa-quinbiru
| ssr10-4
| 68719476736 / 68641485507
| {{monzo| 36 -5 0 -10 }}
| 1.9659
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[Septimal ennealimma]]
| Tritrizo
| z95
| 40353607 / 40310784
| {{Monzo| -11 -9 0 9 }}
| 1.8382
| [[Eliora]] (2021)
|-
| [[Meter]]
| Latriru-asepyo
| Lr3y7-4
| 703125/702464
| {{Monzo| -11 2 7 -3 }}
| 1.6283
|
|-
| [[Skeetsma]]
| Lala-rutrigu
| LLrg3-2
| 14348907 / 14336000
| {{Monzo| -14 15 -3 -1}}
| 1.5580
| [[User:Jerdle|Jerdle]] (2021)
|-
| [[Breeze comma]]
| Laquadru-atriyo
| Lr4y3-4
| 2460375 / 2458624
| {{Monzo| -10 9 3 -4 }}
| 1.2325
| [[Tristan Bay]] (2024)
|-
| [[Wizma]]
| Quinzo-ayoyo
| z5yy3
| 420175/419904
| {{Monzo| -6 -8 2 5 }}
| 1.1170
|
|-
| [[Supermatertisma]]
| Lasepru-atritriyo
| Lr7y9-6
| 52734375 / 52706752
| {{Monzo| -6 3 9 -7 }}
| 0.90708
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[Trienstonisma]]
| Laquinru-agu
| Lr5g-4
| 43046721 / 43025920
| {{monzo| -9 16 -1 -5 }}
| 0.83677
| [[User:Xenllium|Xenllium]] (2021)
|-
| [[2401/2400|Breedsma]]
| Bizozogu
| z4gg3
| 2401/2400
| {{monzo| -5 -1 -2 4 }}
| 0.72120
| [[Gene Ward Smith]] (2004)
|-
| [[Gariatom]]
| Quintrila-tribizo
| 15z6-8
|
| {{monzo| -169 96 0 6 }}
| 0.63552
| [[Tristan Bay]] (2023)
|-
| 171-9/7-comma
| Quadtribisa-netritrizo
|
|
| {{monzo| 62 -342 0 171 }}
| 0.61971
|
|-
| [[Supermasesquartisma]]
| Laquadbiru-aquinyo
| Lr8y5-6
| 184528125 / 184473632
| {{monzo| -5 10 5 -8 }}
| 0.51133
| [[User:Xenllium|Xenllium]] (2021)
|-
| 571-7-comma
|
|
|
| {{monzo| 1603 0 0 -571 }}
| 0.40741
|
|-
| [[Ragisma]]
| Zoquadyo
| zy41
| 4375/4374
| {{Monzo| -1 -7 4 1 }}
| 0.39576
|
|-
| [[Septiruthenia]], septimal ruthenia
| Nela-lequadzo
| L19z448
|
| {{Monzo| -263 88 0 44 }}
| 0.37996
| [[Eliora]] (2022)
|-
| [[Akjaysma]]
| Trisa-seprugu
| s3r7g71
| 140737488355328 / 140710042265625
| {{Monzo| 47 -7 -7 -7 }}
| 0.33765
| [[Aaron Krister Johnson]] (2011)
|-
| [[Landscape comma]]
| Trizogugu
| z3g63
| 250047/250000
| {{Monzo| -4 6 -6 3 }}
| 0.32544
| [[Yahya Abdal-Aziz]] (2005)
|-
| [[Izar comma]], bapbo schismina
| Saquadtrizo-asepgu
| sz12g78
| 13841287201 / 13839609375
| {{Monzo| 0 -11 -7 12 }}
| 0.20987
|
|-
| [[Nanisma]]
| Quinbisaru
| s10r7
|
| {{Monzo| 109 -67 0 -1 }}
| 0.18904
| [[Margo Schulter]] (2002)
|-
| laleruyo (171&1547&3125)
| Laleruyo
| Lr11y11-8
| 3955078125 / 3954653486
| {{Monzo| -1 4 11 -11 }}
| 0.18588
|
|-
| 87-fold starling comma
| Twenetri-zotrigu
|
|
| {{Monzo| 86 174 -261 87 }}
| 0.14469
|
|-
| [[Revopentisma]]
| Sasa-neru
| ssr19-8
|
| {{Monzo| 47 4 0 -19 }}
| 0.12778
| [[Eliora]] (2023)
|-
| [[Starscape comma]]
| Latritriru-ayo
| Lr9y-6
| 645700815 / 645657712
| {{Monzo| -4 17 1 -9 }}
| 0.11557
|
|-
| [[Nommisma]]
| Quinla-zoyoyo
| L5zzy-4
|
| {{Monzo| -55 30 2 1 }}
| 0.10336
| [[Flora Canou]] (2023)
|-
| [[Euzenius comma]]
| Sabiruquinyo
| srry10-3
| 78125000 / 78121827
| {{Monzo| 3 -13 10 -2 }}
| 0.070314
|
|-
| 171&1547&4973 comma
| Satwethezo-atritribigu
| sz23g1815
|
| {{Monzo| 1 -15 -18 23 }}
| 0.023986
|
|-
| [[Technologisma]]
| Trisa-quinbiru-agu
| s3r10g-3
|
| {{Monzo| 51 -13 -1 -10 }}
| 0.012210
| [[User:Godtone|Godtone]] (2024)
|-
| Termite
| Satritribiru-athiseyo
| sr18y37-14
|
| {{Monzo| 9 -28 37 -18 }}
| 0.0010723
|
|-
| Neutrino
|
|
|
| {{Monzo| 1889 -2145 138 424 }}
| 1.6361 × 10-10
|
|}

=== 11-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name|Color Name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[Lifthrasirsma]]
| Sasa-bilugu
| ss1uugg3
| 536870912 / 535869675
| {{Monzo| 29 -11 -2 0 -2 }}
| 3.2317
| [[Flora Canou]] (2023)
|-
| [[540/539|Swetisma]]
| Lururuyo
| 1urry-2
| 540/539
| {{Monzo| 2 3 1 -2 -1 }}
| 3.2090
| [[Manuel Op de Coul]] (2004)
|-
| [[Anthill comma]]
| Satrilo-ayoyo
| s1o3yy1
| 5532400/531441
| {{Monzo| 4 -12 2 0 3 }}
| 3.1212
| [[Budjarn Lambeth]] (2024)
|-
| [[4000/3993|Wizardharry comma]], pine comma
| Triluyo
| 1u3y31
| 4000/3993
| {{Monzo| 5 -1 3 0 -3 }}
| 3.0323
| [[User:Godtone|Godtone]] (2023) for ''pine comma''
|-
| [[Symbiotic comma]]
| Salozo
| s1oz2
| 19712/19683
| {{Monzo| 8 -9 0 1 1 }}
| 2.5488
| [[Dawson Berry]] (2020)
|-
| [[5632/5625|Vishdel comma]]
| Saloquadgu
| s1og42
| 5632/5625
| {{Monzo| 9 -2 -4 0 1 }}
| 2.1531
|
|-
| [[Nexus comma]], nexisma
| Tribilo
| 1o6-2
| 1771561/1769472
| {{Monzo| -16 -3 0 0 6 }}
| 2.0427
| [[Dawson Berry]] (2020)
|-
| [[Reef comma]]
| Salubizogu
| s1uzzgg3
| 200704/200475
| {{Monzo| 12 -6 -2 2 -1 }}
| 1.9764
| [[Dawson Berry]] (2022)
|-
| [[41503/41472|Argyria]], tinge
| Lolotrizo
| 1ooz32
| 41503/41472
| {{Monzo| -9 -4 0 3 2 }}
| 1.2936
| [[Gayle Young]] (2018) and [[Todd Harrop]] (2020) for ''tinge'' [[User:Lériendil|Lériendil]] (2024) for ''argyria''
|-
| [[Alpharabian schisma]]
|
|
| 618121839509504 / 617673396283947
| {{Monzo| 18 -31 0 0 9 }}
| 1.2565
| [[Dawson Berry]] (2024)
|-
| [[Olympia]]
| Salururu
| s1urr1
| 131072/130977
| {{Monzo| 17 -5 0 -2 -1 }}
| 1.2552
| [[Flora Canou]] (2021)
|-
| 11-cycle schisma, 37-11-comma
| Quinsa-thiselu
| s51u379
|
| {{Monzo| 128 0 0 0 -37 }}
| 1.2361
|
|-
| [[Seascape comma]], undecimal hemifourths comma
| Bilozogugu
| 1oozzg42
| 160083/160000
| {{Monzo| -8 3 -4 2 2 }}
| 0.89784
| [[User:Tristanbay|Tristan Bay]] (2024) for ''seascape comma''
|-
| [[Sesdecal comma]]
| Laquadlu-asepyo
| L1u4y7-2
| 234375/234256
| {{Monzo| -4 1 7 0 -4 }}
| 0.87923
|
|-
| [[Triagnoshenisma]]
| Trila-trilo-agu
| L31o3g-3
| 171885556953/171798691840
| {{Monzo| -35 17 -1 0 3 }}
| 0.87513
| [[Dawson Berry]], [[User:Frostburn|Frostburn]] (2024)
|-
| [[Frameshift comma]]
| Quadla-trilu
| L41u3-3
| 22876792454961 / 22866405883904
| {{Monzo| -34 28 0 0 -3 }}
| 0.78620
| [[Dawson Berry]] (2021)
|-
| Hemigail
| Quadlo-atriru
| 1o4r3-3
| 43923/43904
| {{Monzo| -7 1 0 -3 4 }}
| 0.74905
|
|-
| [[Sossmarvel comma]]
| Trila-lusepruyoyo
| L31ur7y14-8
| 9730975341796875 / 9726998192586752
| {{Monzo| -30 13 14 -7 -1 }}
| 0.70772
| [[Dawson Berry]] (2022)
|-
| [[3025/3024|Lehmerisma]]
| Loloruyoyo
| 1ooryy-2
| 3025/3024
| {{Monzo| -4 -3 2 -1 2 }}
| 0.57240
| [[Gene Ward Smith]] (2004)
|-
| [[Ptolemi-nicema]]
| Quinbisa-twethetriluyoyo
| s101u69y138-8
|
| {{Monzo| 137 -138 138 0 -69 }}
| 0.56437
| [[User:Eliora|Eliora]] (2023)
|-
| [[Elysia]]
| Bilutrizo
| 1uuz64
| 117649/117612
| {{Monzo| -2 -5 0 6 -2 }}
| 0.54455
| [[User:Lériendil|Lériendil]] (2024)
|-
| [[Quartisma]]
| Saquinlu-azo
| s1u5z3
| 117440512 / 117406179
| {{Monzo| 24 -6 0 1 -5 }}
| 0.50619
| [[Dawson Berry]] (2020)
|-
| [[9801/9800|Kalisma]], Gauss' comma
| Bilorugu
| 1oorrgg-2
| 9801/9800
| {{Monzo| -3 4 -2 -2 2 }}
| 0.17665
| [[Margo Schulter]] (2000) [[Gene Ward Smith]] (2004) for ''Gauss' comma''
|-
| [[151263/151250|Odiheim]]
| Luluquinzo-aquadgu
| 1uuz5g44
| 151263/151250
| {{Monzo| -1 2 -4 5 -2 }}
| 0.14879
|
|-
| [[Countercentisma]]
|
|
|
| {{Monzo| -1 -3300 2700 0 -300 }}
| 0.14187
| [[User:CompactStar|CompactStar]] (2024)
|-
| [[Spoob]]
| Tribiluzozogu
| 1u6z12g68
| 27682574402 / 27680640625
| {{Monzo|1 0 -6 12 -6 }}
| 0.12094
| [[User:Eliora|Eliora]] (2023)
|-
| [[Syntonisma]]
| Trisa-lusepyo
| s31uy72
| 83886080000000 / 83881572334857
| {{Monzo| 30 -27 7 0 -1 }}
| 0.093031
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Parimo]]
| Satribilo-agu
| s1o6g1
| 1771561/1771470
| {{Monzo| -1 -11 -1 0 6 }}
| 0.088931
|
|-
| Parisma
| Laquadlu-arurugu
| L1u4rrg-2
| 14348907 / 14348180
| {{Monzo| -2 15 -1 -2 -4 }}
| 0.087717
|
|-
| [[Blare comma]]
| Laquadquadlo-aquadtrizo
| L1o16z124
| 636003407850068828189211361 / 635974777627126753067532288
| {{Monzo| -51 -24 0 12 16 }}
| 0.077935
| [[User:Akselai|Akselai]] (2024)
|-
| Ultimo
| Quadlo-asepru-ayoyo
| 1o4r7yy-5
| 3294225/3294172
| {{Monzo| -2 2 2 -7 4 }}
| 0.027854
|
|-
| Parismina
| Sasa-quinbilo-azozo
| ss1o10zz2
| 2541867610898 / 2541865828329
| {{Monzo| 1 -26 0 2 10 }}
| 0.0012141
|
|}

=== 13-limit commas ===
{| class="wikitable center-3"
|-
! Name(s)
! colspan="2" | [[Color name|Color Name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[Wilschisma]]
| Sathoyo
| s3oy2
| 532480/531441
| {{Monzo| 13 -12 1 0 0 1 }}
| 3.3814
| [[Flora Canou]] (2021)
|-
| Bean
| Sathuquinlu
| s3u1u52
| 2097152/2093663
| {{Monzo| 21 0 0 0 -5 -1 }}
| 2.8826
|
|-
| [[625/624|Tunbarsma]]
| Thuquadyo
| 3uy4-2
| 625/624
| {{Monzo| -4 -1 4 0 0 -1 }}
| 2.7722
|
|-
| [[676/675|Island comma]]
| Bithogu
| 3oogg2
| 676/675
| {{Monzo| 2 -3 -2 0 0 2 }}
| 2.5629
| [[Mike Battaglia]] (2011)
|-
| [[729/728|Squbema]]
| Lathuru
| L3ur-2
| 729/728
| {{Monzo| -3 6 0 -1 0 -1 }}
| 2.3764
|
|-
| [[2200/2197|Petrma]]
| Trithu-aloyoyo
| 3u31oyy-2
| 2200/2197
| {{Monzo| 3 0 2 0 1 -3 }}
| 2.3624
|
|-
| [[Tridecimal eighth-octave comma]]
| Thotrilo-agu
| 3o1o3g1
| 17303/17280
| {{Monzo| -7 -3 -1 0 3 1 }}
| 2.3028
| [[Budjarn Lambeth]] (2024)
|-
| [[Sinarabian comma]]
| Lathotrilu
|
| 85293/85184
| {{Monzo| -6 8 0 0 -3 1 }}
| 2.2138
| [[Dawson Berry]] (2025)
|-
| [[1575/1573|Nicola]]
| Thululuzoyoyo
| 3u1uuzyy1
| 1575/1573
| {{Monzo| 0 2 2 1 -2 -1 }}
| 2.1998
|
|-
| [[1001/1000|Sinbadma]]
| Tholozotrigu
| 3o1ozg32
| 1001/1000
| {{Monzo| -3 0 -3 1 1 1 }}
| 1.7303
|
|-
| [[4459/4455|Tristanisma]]
| Tholutrizo-agu
| 3o1uz3g3
| 4459/4455
| {{Monzo| 0 -4 -1 3 -1 1}}
| 1.5537
| [[User:Tristanbay|Tristan Bay]] (2024)
|-
| [[Tridecapyth comma]]
| Trisatho
| s33o3
| 3489660928/3486784401
| {{Monzo| 28 -20 0 0 0 1 }}
| 1.4276
| [[Dawson Berry]] (2021)
|-
| [[Cantonisma]]
| Trithoru-ayo
| 3o3r3y-2
| 10985/10976
| {{Monzo| -5 0 1 -3 0 3 }}
| 1.4190
| [[Margo Schulter]] (2013)
|-
| [[Punctisma]]
| Sathutrizogu
| s3uz3g33
| 43904/43875
| {{Monzo| 7 -3 -3 3 0 -1 }}
| 1.1439
| [[User:Jerdle|Jerdle]], [[Dawson Berry]] (2023)
|-
| [[Neguschisma]]
| Lala-thulozo
| LL3u1oz-2
| 13640319/13631488
| {{Monzo| -20 11 0 1 1 -1 }}
| 1.1212
| [[User:Tristanbay|Tristan Bay]] (2024)
|-
| [[1716/1715|Lummic comma]]
| Tholotriru-agu
| 3o1or3g-2
| 1716/1715
| {{Monzo| 2 1 -1 -3 1 1 }}
| 1.0092
|
|-
| [[Pseudovishnuzma]]
| Sasa-thozosepbigu
| ss3ozg145
| 6106906624 / 6103515625
| {{Monzo| 26 0 -14 1 0 1 }}
| 0.96157
| [[User:Eliora|Eliora]] (2023)
|-
| [[Sercloreminisma]]
| Bithuthuzo-agu
| 3u4zzg1
| 142884/142805
| {{Monzo| 2 6 -1 2 0 -4 }}
| 0.95746
| [[Francium]] (2024)
|-
| [[2080/2079|Ibnsinma, sinaisma]]
| Tholuruyo
| 3o1ury1
| 2080/2079
| {{Monzo| 5 -3 1 -1 -1 1 }}
| 0.83252
| [[Margo Schulter]], [[Gene Ward Smith]] (2012) [[Flora Canou]] (2023)
|-
| [[Phaotic comma]], phaotisma
| Sathotriyo
| s3uy31
| 256000/255879
| {{Monzo| 11 -9 3 0 0 -1 }}
| 0.81847
| [[Dawson Berry]] (2021)
|-
| [[4096/4095|Schismina]]
| Sathurugu
| s3urg1
| 4096/4095
| {{Monzo| 12 -2 -1 -1 0 -1 }}
| 0.42272
|
|-
| [[4225/4224|Leprechaun comma]]
| Thotholuyoyo
| 3oo1uyy1
| 4225/4224
| {{Monzo| -7 -1 2 0 -1 2 }}
| 0.40981
| [[Margo Schulter]] (2012)
|-
| [[6656/6655|Jacobin comma]]
| Thotrilu-agu
| 3o1u3g2
| 6656/6655
| {{Monzo| 9 0 -1 0 -3 1 }}
| 0.26012
| [[Gene Ward Smith]] (2014)
|-
| [[Catasma]]
| Latrithuyoyo
| L3u3y6-3
| 140625/140608
| {{Monzo| -6 2 6 0 0 -3 }}
| 0.20930
| [[Tristan Bay]] (2023)
|-
| [[492128/492075|13^3⋅7/25 schismina]]
| Satritho-azogugu
| s33zgg3
| 492128/492075
| {{Monzo| 5 -9 -2 1 0 3 }}
| 0.18646
| [[User:Recentlymaterialized|Recentlymaterialized]] (2024)
|-
| [[10648/10647|Harmonisma]]
| Thuthutrilo-aru
| 3uu1o3r-2
| 10648/10647
| {{Monzo| 3 -2 0 -1 3 -2 }}
| 0.16260
| [[Margo Schulter]] (2002)
|-
| [[Pentonisma]]
| Saquinthuzogu
| s3u5z5g54
| 281974669312 / 281950621875
| {{Monzo| 24 -5 -5 5 0 -5 }}
| 0.14765
| [[User:Xenllium|Xenllium]] (2022)
|-
| [[Pontigailimma]]
| Thururuquingu
| 3urrg51
| 1990656/1990625
| {{Monzo| 13 5 -5 -2 0 -1 }}
| 0.027
| [[User:Recentlymaterialized|Recentlymaterialized]]
|-
| [[Grossmisma]]
| septholo-azogu
| 3o71o7zg2
|
| {{Monzo| -30 -13 -1 1 7 7 }}
| 0.016410
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Chalmersia]]
| Lathotholurugugu
| L3oo1urgg1
| 123201/123200
| {{Monzo| -6 6 -2 -1 -1 2 }}
| 0.01405
| [[Gene Ward Smith]] (2003)
|-
| [[Lanasma]]
| Trila-septrithu-aquinquadbizo
| L33u21z4013
| 5522323710216635 / 5522322559508318
| {{Monzo| -33 -1 0 40 0 -21 }}
| 3.6074 × 10-4
| [[User:Angekallikoita|Ange]] (2025)
|}

=== 17-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name|Color Name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[561/560|Monardisma]], tsaharuk comma
| Solorugu
| 17o1org1
| 561/560
| {{Monzo| -4 1 -1 -1 1 0 1 }}
| 3.0887
| [[Scott Dakota]] (2023) for ''monardisma''
|-
| [[595/594|Dakotisma]]
| Soluzoyo
| 17o1uzy2
| 595/594
| {{Monzo| -1 -3 1 1 -1 0 1 }}
| 2.9121
| [[Praveen Venkataramana]] (2022)
|-
| [[715/714|September comma]]
| Sutholoruyo
| 17u3o1ory-2
| 715/714
| {{Monzo| -1 -1 1 -1 1 1 -1 }}
| 2.4230
| [[Scott Dakota]] (2021)
|-
| [[833/832|Horizma, horizon comma]]
| Sothuzozo
| 17o3uzz2
| 833/832
| {{Monzo| -6 0 0 2 0 -1 1 }}
| 2.0796
| [[User:Jerdle|Jerdle]] (2021)
|-
| [[936/935|Ainisma, ainic comma]]
| Sutholugu
| 17u3o1ug1
| 936/935
| {{Monzo| 3 2 -1 0 -1 1 -1 }}
| 1.8506
| [[Dawson Berry]] (2020)
|-
| [[2025/2023|Fidesma]]
| Susuruyoyo
| 17uuryy-3
| 2025/2023
| {{Monzo| 0 4 2 -1 0 0 -2 }}
| 1.7107
| [[Dawson Berry]] (2023)
|-
| [[1089/1088|Twosquare comma]]
| Sulolo
| 17u1oo-2
| 1089/1088
| {{Monzo| -6 2 0 0 2 0 -1 }}
| 1.5905
| [[User:Kaiveran|Kaiveran]] (2018)
|-
| [[1156/1155|Quadrantonisma]]
| Sosolurugu
| 17oo1urg2
| 1156/1155
| {{Monzo| 2 -1 -1 -1 -1 0 2 }}
| 1.4983
| [[Flora Canou]] (2023)
|-
| [[1225/1224|Noellisma]]
| Subizoyo
| 17uzzyy1
| 1225/1224
| {{Monzo| -3 -2 2 2 0 0 -1 }}
| 1.4138
| [[Flora Canou]] (2022)
|-
| [[1275/1274|Cimbrisma]]
| Sothubiruyo
| 17o3urryy-2
| 1275/1274
| {{Monzo| -1 1 2 -2 0 -1 1 }}
| 1.3584
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[1701/1700|Palingenetic comma, palingenesis]]
| Suzogugu
| 17uzgg1
| 1701/1700
| {{Monzo| -2 5 -2 1 0 0 -1 }}
| 1.0181
| [[Dawson Berry]] (2021)
|-
| [[Laser comma]]
| Lasorutriyo
| L17ory3-2
| 57375/57344
| {{Monzo| -13 3 3 -1 0 0 1 }}
| 0.93564
| [[User:Jerdle|Jerdle]] (2023)
|-
| [[2058/2057|Xenisma]]
| Sululutrizo
| 17u1uuz32
| 2058/2057
| {{Monzo| 1 1 0 3 -2 0 -1 }}
| 0.84143
| [[Margo Schulter]] (2000)
|-
| [[11016/11011|Cyclops comma]]
| Sothululuru
| 17o3u1uur1
| 11016/11011
| {{Monzo| 3 4 0 -1 -2 -1 1 }}
| 0.78596
| [[Eliora]] (2022)
|-
| [[24576/24565|Mavka comma]], archagallisma
| Trisu-agu
| 17u3g-2
| 24576/24565
| {{Monzo| 13 1 -1 0 0 0 -3 }}
| 0.77506
| [[Eliora]] (2022) for ''mavka comma''
|-
| [[2431/2430|Heptacircle comma]]
| Sothologu
| 17o3o1og2
| 2431/2430
| {{Monzo| -1 -5 -1 0 1 1 1 }}
| 0.71230
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[2500/2499|Sperasma]]
| Subiruyoyo
| 17urry4-3
| 2500/2499
| {{Monzo| 2 -1 4 -2 0 0 -1 }}
| 0.69263
| [[Dawson Berry]] (2023)
|-
| [[2601/2600|Sextantonisma]]
| Sosothugugu
| 17oo3ugg2
| 2601/2600
| {{Monzo| -3 2 -2 0 0 1 3 }}
| 0.66573
| [[Flora Canou]] (2023)
|-
| [[Semisixthmisma]]
| Trisu-athutrilo
| 17u33u1o3-3
| 63888/63869
| {{Monzo| 4 1 0 0 3 -1 -3 }}
| 0.51494
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[4914/4913|Baladisma]]
| Trisu-athozo
| 17u33oz-2
| 4914/4913
| {{Monzo| 1 3 0 1 0 1 -3 }}
| 0.35234
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[5832/5831|Chlorisma]]
| Sutriru
| 17ur3-3
| 5832/5831
| {{Monzo| 3 6 0 -3 0 0 -1 }}
| 0.29688
| [[User:Jerdle|Jerdle]] (2021)
|-
| [[Galileisma]]
| Lalesu-agu
| L17u11g-7
| 171382426877952 / 171359481538165
| {{Monzo| 14 21 -1 0 0 0 -11 }}
| 0.23180
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Centisma]]
| quinquinquadquadso
| 17o400194
|
| 2.3.17 {{Monzo| -1001 -400 400 }}
| 0.16345
| [[CompactStar]] (2023)
|-
| [[Flashma]]
| Sotholuzotrigu
| 17o3o1uzg33
| 12376/12375
| {{Monzo| 3 -2 -3 1 -1 1 1 }}
| 0.13989
| [[User:Kaiveran|Kaiveran]] (2016)
|-
| [[Sparkisma]]
| Sululuruyoyo
| 17u1uuryy-2
| 14400/14399
| {{Monzo| 6 2 2 -1 -2 0 -1 }}
| 0.12023
| [[User:Kaiveran|Kaiveran]] (2016)
|-
| [[Insanobromisma]]
| Sepquinsuyoyo
| 17u35y70-29
|
| {{Monzo| 36 -35 70 0 0 0 -35 }}
| 0.095608
| [[Eliora]] (2023)
|-
| [[28561/28560|Pisanoisma]]
| Suquadtho-arugu
| 17u3o4rg1
| 28561/28560
| {{Monzo| -4 -1 -1 -1 0 4 -1 }}
| 0.060616
| [[Budjarn Lambeth]] (2024)
|-
| [[E-shaped comma]]
| Susuthoquadzo
| 17uu3oz42
| 31213/31212
| {{Monzo| -2 -3 0 4 0 1 -2 }}
| 0.055466
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Lateral comma]]
| Sasuthotholoyo
| s17u3oo1oy1
| 37180/37179
| {{Monzo| 2 -7 1 0 1 2 -1 }}
| 0.046564
| [[User:Akselai|Akselai]] (2024)
|-
| [[Clevelandisma]]
|
|
| 2000033/2000000
| {{Monzo| -7 0 -6 6 0 0 1 }}
| 0.028565
| [[Lériendil]] (2024)
|-
| [[Scintillisma]]
| Lasuthuluquadzo-agu
| L17u3u1uz4g2
| 194481/194480
| {{Monzo| -4 4 -1 4 -1 -1 -1 }}
| 0.0089018
| [[Margo Schulter]] (2012)
|-
| [[Aksial comma]]
| Sotritho-aquinru-agu
| 17o3o3r5g-2
| 336141/336140
| {{Monzo| -2 2 -1 -5 0 3 1 }}
| 0.0051503
| [[User:Akselai|Akselai]] (2024)
|}

=== 19-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name|Color Name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[513/512|Undevicesimal comma, undevicesimal schisma]], Boethius' comma, Hunt 19-minor mediant comma
| Lano
| L19o1
| 513/512
| {{Monzo| -9 3 0 0 0 0 0 1 }}
| 3.3780
| Plainsound Music Edition (2020)<ref>[https://marsbat.space/pdfs/HEJI2legend+series.pdf The Helmholtz-Ellis JI Pitch Notation (HEJI)]</ref> for ''undevicesimal schisma''
|-
| [[969/968|Kingfisher comma]]
| Nosolulu
| 19o17o1uu2
| 969/968
| {{Monzo| -3 1 0 0 -2 0 1 1 }}
| 1.7875
| [[Budjarn Lambeth]] (2025)
|-
| [[Mercurial comma]]
| Quinnosu-abiruyo
| 19o517u5rryy-3
| 557122275 / 556583944
| {{Monzo| -3 2 2 -2 0 0 -5 5 }}
| 1.6736
| [[User:Yourmusic Productions|Yourmusic Productions]] (2020)
|-
| [[1216/1215|Password, Eratosthenes' comma]]
| Sanogu
| s19og2
| 1216/1215
| {{Monzo| 6 -5 -1 0 0 0 0 1 }}
| 1.4243
|
|-
| [[1331/1330|Solvejgsma]]
| Nutrilo-arugu
| 19u1o3rg-2
| 1331/1330
| {{Monzo| -1 0 -1 -1 3 0 0 -1 }}
| 1.3012
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[1445/1444|Aureusma]]
| Nunusosoyo
| 19uu17ooy1
| 1445/1444
| {{Monzo| -2 0 1 0 0 0 2 -2 }}
| 1.1985
| [[Flora Canou]] (2021)
|-
| [[1521/1520|Pinkanberry]]
| Nuthothogu
| 19u3o3og1
| 1521/1520
| {{Monzo| -4 2 -1 0 0 2 0 -1 }}
| 1.1386
| [[User:Kaiveran|Kaiveran]] (2021)
|-
| [[1540/1539|Kevolisma]]
| Nulozoyo
| 19u1ozy1
| 1540/1539
| {{Monzo| 2 -4 1 1 1 0 0 -1 }}
| 1.1245
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Cobaltomenisma]]
| Nusothuthuzo
| 19u17o3uuz1
| 3213/3211
| {{Monzo| 0 3 0 1 0 -2 1 -1 }}
| 1.0780
| [[Francium]] (2025)
|-
| [[1729/1728|Ramanujanisma]]
| Nothozo
| 19o3oz2
| 1729/1728
| {{Monzo| -6 -3 0 1 0 1 0 1 }}
| 1.0016
| [[Frédéric Gagné]] (2021)
|-
| [[3971/3969|Heartlandisma]]
| Nonoloruru
| 19oo1orr1
| 3971/3969
| {{Monzo| 0 -4 0 -2 1 0 0 2 }}
| 0.87216
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[830297/829939|Minthtone schismina]]
| Trinuso-abitholu
| 19u317o33oo1uu2
| 830297/829939
| {{Monzo| 0 0 0 0 -2 2 3 -3 }}
| 0.74662
| [[User:Recentlymaterialized|Recentlymaterialized]] (2024)
|-
| [[2376/2375|Trichthonisma]]
| Nulotrigu
| 19u1og31
| 2376/2375
| {{Monzo| 3 3 -3 0 1 0 0 -1 }}
| 0.72879
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[83521/83486|Crawma]]
| Nuquadso-atrithu
| 19u17o43u32
| 83521/83486
| {{Monzo| -1 0 0 0 0 -3 4 -1 }}
| 0.72564
| [[User:Ks26|groundfault]] (2023)
|-
| [[2432/2431|Blumeyer comma]]
| Nosuthulu
| 19o17u3u1u1
| 2432/2431
| {{Monzo| 7 0 0 0 -1 -1 -1 1 }}
| 0.71200
| [[Douglas Blumeyer]] (2015)
|-
| [[2926/2925|Neovulture comma, neovulturisma]]
| Nothulozogugu
| 19o3u1ozgg2
| 2926/2925
| {{Monzo| 1 -2 -2 1 1 -1 0 1 }}
| 0.59177
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[3136/3135|Neomirkwai comma, neomirkwaisma]]
| Nuluzozogu
| 19u1uzzg2
| 3136/3135
| {{Monzo| 6 -1 -1 2 -1 0 0 -1 }}
| 0.55214
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[3250/3249|Martebisma]]
| Nunuthotriyo
| 19uu3oy3-2
| 3250/3249
| {{Monzo| 1 -2 3 0 0 1 0 -2 }}
| 0.53277
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[4200/4199|Neosatanisma]]
| Nusuthuzoyoyo
| 19u17u3uzyy-2
| 4200/4199
| {{Monzo| 3 1 2 1 0 -1 -1 -1 }}
| 0.41225
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[5776/5775|Neovish comma, neovishma]]
| Nonolurugugu
| 19oo1urgg2
| 5776/5775
| {{Monzo| 4 -1 -2 -1 -1 0 0 2 }}
| 0.29975
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[5985/5984|Neogrendel comma, neogrendelisma]]
| Nosuluzoyo
| 19o17u1uzy1
| 5985/5984
| {{Monzo| -5 2 1 1 -1 0 -1 1 }}
| 0.28929
| [[User:Xenllium|Xenllium]] (2023)
|-
| BMO schismina
| Sabinothu
| s19oo3uu2
| 369664/369603
| {{Monzo| 10 -7 0 0 0 -2 0 2 }}
| 0.28570
|
|-
| [[6175/6174|Neonewt comma, neonewtisma]]
| Nothotriru-ayoyo
| 19o3or3yy-2
| 6175/6174
| {{Monzo| -1 -2 2 -3 0 1 0 1 }}
| 0.28038
| [[User:Xenllium|Xenllium]] (2023)
|-
| Frouggie comma
| Nusuquinthu-aquadlo
| 19u17u3u51o4-3
| 119939072 / 119927639
| {{Monzo| 13 0 0 0 4 -5 -1 -1 }}
| 0.16503
| [[Douglas Blumeyer]] (2020)
|-
| [[Lakisma]]
| Saquadnoso-agu
| s19o417o4g5
| 10884540241 / 10883911680
| {{Monzo| -12 -12 -1 0 0 0 4 4 }}
| 0.09998
| [[User:Flirora|+merlan #flirora]] (2021)
|-
| [[Aubertisma]]
| Nosothutrilu-arutriyo
| 19o17o3u1u3ry31
| 121125/121121
| {{monzo| 0 1 3 -1 -3 -1 1 1 }}
| 0.057173
| [[Francium]] (2025)
|-
| Pollar comma
| Nunusuquintho-alulu
| 19uu17u3o51uu1
| 742586/742577
| {{Monzo| 1 0 0 0 -2 5 -1 -2 }}
| 0.020982
| [[Douglas Blumeyer]] (2020)
|-
| [[Decimillisma]], 19-limit decimill
| Sanosorurutrigu
| s19o17orrg32
| 165376/165375
| {{Monzo| 9 -3 -3 -2 0 0 1 1 }}
| 0.010469
| [[Flora Canou]] (2021), for ''decimillisma''
|-
| [[65/19 atom]]
| sasa-nuthoyo
| ss19u3oy2
| 272629760 / 272629233
| {{Monzo| 22 -15 1 0 0 1 0 -1 }}
| 0.0033465
| [[User:Recentlymaterialized|Recentlymaterialized]] (2024)
|-
| [[Devicisma]]
| Nunusothutrilo-azogu
| 19uu17o3u1o3zg1
| 633556/633555
| {{Monzo| 2 -3 -1 1 3 -1 1 -2 }}
| 0.0027326
| [[User:Xenllium|Xenllium]] (2023)
|}

=== 23-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name|Color Name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[507/506|Laodicisma]]
| Twethuthotholu
| 23u3oo1u1
| 507/506
| 2.3.11.13.23 {{Monzo| -1 1 -1 2 -1 }}
| 3.4180
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[529/528|Preziosisma]]
| Bitwetho-alu
| 23oo1u2
| 529/528
| 2.3.11.23 {{Monzo| -4 -1 -1 2 }}
| 3.2758
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[576/575|Worcester comma]]
| Twethugugu
| 23ugg1
| 576/575
| 2.3.5.23 {{Monzo| 6 2 -2 -1 }}
| 3.0082
| [[User:Kaiveran|Kaiveran]] (2023)
|-
| [[9765625/9750528]]
| Labitwethuquinyo-a
| L23uuy10-4
| 9765625 / 9750528
| 2.3.5.23 {{Monzo| -11 -2 10 -2 }}
| 2.6784
|
|-
| [[736/735|Harvardisma]]
| Twethorurugu
| 23orrg1
| 736/735
| 2.3.5.7.23 {{Monzo| 5 -1 -1 -2 1 }}
| 2.3538
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[760/759|Squadronisma]]
| Twethunoluyo
| 23u19o1uy1
| 760/759
| {{Monzo| 3 -1 1 0 -1 0 0 1 -1 }}
| 2.2794
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[875/874|Nymphisma]]
| Twethunuzotriyo
| 23u19uzy3-2
| 875/874
| 2.5.7.19.23 {{Monzo| -1 3 1 -1 -1 }}
| 1.9797
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[897/896|Lysistratisma]]
| Twethothoru
| 23o3or1
| 897/896
| 2.3.7.13.23 {{Monzo| -7 1 -1 1 1 }}
| 1.9311
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[1105/1104|Fragarisma]]
| Twethusothoyo
| 23u17o3oy1
| 1105/1104
| {{Monzo| -4 -1 1 0 0 1 1 0 -1 }}
| 1.5674
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[1197/1196|Rhodesisma]]
| Twethunothuzo
| 23u19o3uz1
| 1197/1196
| {{Monzo| -2 2 0 1 0 -1 0 1 -1 }}
| 1.4469
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[1288/1287|Triaphonisma]], santisma
| Twethothuluzo
| 23o3u1uz2
| 1288/1287
| {{Monzo| 3 -2 0 1 -1 -1 0 0 1 }}
| 1.3446
| [[User:Xenllium|Xenllium]] (2023) for ''santisma''
|-
| [[1496/1495|Turkisma]]
| Twethusothulogu
| 23u17o3u1og1
| 1496/1495
| {{Monzo| 3 0 -1 0 1 -1 1 0 -1 }}
| 1.1576
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Gordaitisma]]
| Twethonosolugugu
| 23o19o17o1ugg3
| 7429/7425
| {{monzo| 0 -3 -2 0 -1 0 1 1 1 }}
| 0.93240
| [[Francium]] (2025)
|-
| [[1863/1862|Antinousisma]]
| Twethonururu
| 23o19urr-2
| 1863/1862
| 2.3.7.19.23 {{Monzo| -1 4 -2 -1 1 }}
| 0.92952
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Kaifsma]]
| Twethunosutholuzozogu
| 23u19o17u3o1uzzg2
| 193648/193545
| {{Monzo| 4 -2 -1 2 -1 1 -1 1 -1 }}
| 0.92108
| [[User:Kaiveran|Kaiveran]] (2022)
|-
| [[2024/2023|Artifisma]], insincere comma
| Twethosusuloru
| 23o17uu1or-2
| 2024/2023
| 2.7.11.17.23 {{Monzo| 3 -1 1 -2 1 }}
| 0.85556
| [[User:Xenllium|Xenllium]] (2023) for ''insincere comma''
|-
| [[2025/2024|Cupcake comma]], cupcakesma
| Latwethuluyoyo
| L23u1uyy-2
| 2025/2024
| 2.3.5.11.23 {{Monzo| -3 4 2 -1 -1 }}
| 0.85514
| See the page.
|-
| [[2185/2184|Guangdongisma]]
| Twethonothuruyo
| 23o19o3ury1
| 2185/2184
| {{Monzo| -3 -1 1 -1 0 -1 0 1 1 }}
| 0.79251
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[2300/2299|Travellisma]]
| Twethonubiluyo
| 23o19u1uuyy1
| 2300/2299
| 2.5.11.19.23 {{Monzo| 2 2 -2 -1 1 }}
| 0.75287
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[2646/2645|Biyativice comma]]
| Bitwethuzo-agu
| 23uuzzg1
| 2646/2645
| 2.3.5.7.23 {{Monzo| 1 3 -1 2 -2 }}
| 0.65441
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[2737/2736|Kotkisma]]
| Twethonusozo
| 23o19u17oz2
| 2737/2736
| {{Monzo| -4 -2 0 1 0 0 1 -1 1 }}
| 0.63265
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Kwadransma]]
| Laquadtwethu
|
| 279936/279841
| {{Monzo| 7 7 0 0 0 0 0 0 -4 }}
| 0.58762
| [[Lériendil]] (2024)
|-
| [[3060/3059|Vicious comma]], viciousma
| Twethunusoruyo
| 23u19u17ory-2
| 3060/3059
| {{Monzo| 2 2 1 -1 0 0 1 -1 -1 }}
| 0.56586
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[3381/3380|Mikkolisma]], Seminaiadvice comma
| Twethothuthuzozogu
| 23o3uuzzg2
| 3381/3380
| {{Monzo| -2 1 -1 2 0 -2 0 0 1 }}
| 0.51212
| See the page.
|-
| [[3520/3519|Vicedim comma]]
| Twethusuloyo
| 23u17u1oy-2
| 3520/3519
| {{Monzo| 6 -2 1 0 1 0 -1 0 -1 }}
| 0.49190
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[3888/3887|Vicetride comma]]
| Twethuthuthu
| 23u3uu-2
| 3888/3887
| 2.3.13.23 {{Monzo| 4 5 -2 -1 }}
| 0.44533
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Hagendorfisma]]
| Twethunosothuyoyo
| 23u19o17o3uyy1
| 8075/8073
| {{monzo| 0 -3 2 0 0 -1 1 1 -1 }}
| 0.42884
| [[Francium]] (2025)
|-
| [[4693/4692|Viceaug comma]]
| Twethunonosutho
| 23u19oo17u3o1
| 4693/4692
| {{Monzo| -2 -1 0 0 0 1 -1 2 -1 }}
| 0.36894
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[4761/4760|Demiquartervice comma]]
| Bitwetho-asurugu
| 23oo17urg1
| 4761/4760
| {{Monzo| -3 2 -1 -1 0 0 -1 0 2 }}
| 0.36367
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[5083/5082|Broadviewsma]]
| Twethosotholuluru
| 23o17o3o1uur2
| 5083/5082
| {{Monzo| -1 -1 0 -1 -2 1 1 0 1 }}
| 0.34063
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Galeaclolusisma]]
| Twethususutholuquadyo
| 23u17uu3o1uy4-3
| 73125/73117
| {{Monzo| 0 2 4 0 -1 1 -2 0 -1 }}
| 0.18941
| [[Francium]] (2025)
|-
| [[Vicetertisma]]
| Tritwethu-athotho
| 23u33oo-2
| 12168/12167
| 2.3.13.23 {{Monzo| 3 2 2 -3 }}
| 0.14228
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Joshuavoisma]]
| Twethusutholozoyoyo
| 23u17u3o1ozyy-2
| 25025/25024
| 2.5.7.11.13.17.23 {{monzo| -6 2 1 1 1 -1 -1 }}
| 0.06918
| [[Francium]] (2024)
|-
| [[Diarithmedia]]
| Bitwethozo-agu
| 23oozzg3
| 25921/25920
| 2.3.5.7.23 {{monzo| 6 -4 -1 2 2 }}
| 0.066790
| [[Flora Canou]] (2023), modified by [[Lériendil]] (2024)
|-
| [[Jeffbenisma]]
| Labitwethu-anutholuzoyo
| L23uu19u3o1uzy-2
| 110565/110561
| {{monzo| 0 5 1 1 -1 1 0 -1 -2 }}
| 0.062633
| [[Francium]] (2025)
|}

=== Higher-limit commas ===
{| class="wikitable"
|-
! Name(s)
! colspan="2" | [[Color name|Color Name]]
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
! Named by
|-
| [[7936/7921|Lily comma]]
|
| 89uu31o1
| 7936/7921
| 2.31.89 {{monzo| 8 1 -2 }}
| 3.2753
| [[Budjarn Lambeth]] (2023)
|-
| [[551/550|Minor chthonovinema]]
| Twenonolugugu
| 29o19o1ugg2
| 551/550
| 2.5.11.19.29 {{monzo| -1 -2 -1 1 1 }}
| 3.1448
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[552/551]]
| Twenutwethonu
| 29u23o19u1
| 552/551
| 2.3.19.23.29 {{monzo| 3 1 -1 1 -1 }}
| 3.1391
|
|-
| [[609/608|Vineyard comma]]
| Twenonuzo
| 29o19uz1
| 609/608
| 2.3.7.19.29 {{monzo| -5 1 1 -1 1 }}
| 2.8451
| [[User:Xenllium|Xenllium]] (2024)
|-
| [[Ellisma]]
| Fowulozogu
| 41u1ozg2
| 616/615
| 2.3.5.7.11.41 {{monzo| 3 -1 -1 1 1 -1 }}
| 2.8127
| [[Francium]] (2024)
|-
| [[621/620|Owowhatsthisma]]
| Thiwutwethogu
| 31u23og2
| 621/620
| 2.3.5.23.31 {{monzo| -2 3 -1 1 -1 }}
| 2.7901
| [[HEHEHE I AM A SUPAHSTAR SAGA]] (2021)
|-
| [[704/703|Minimyna]]
| Thisunulo
| 37u19u1o-2
| 704/703
| 2.11.19.37 {{monzo| 6 1 -1 -1 }}
| 2.4609
| [[User:Kaiveran|Kaiveran]] (2023)
|-
| [[704969/704000|Molar comma]]
|
| 89o31ug31
| 704969/704000
| 2.5.11.89 {{monzo| -9 -3 -1 3 }}
| 2.3813
| [[Budjarn Lambeth]] (2023)
|-
| [[Botolphisma]]
| Thisunothogu
| 37u19o3og1
| 741/740
| 2.3.5.13.19.37 {{monzo| -2 1 -1 1 1 -1 }}
| 2.3379
| [[Francium]] (2025)
|-
| [[750/749|Ancient Chinese tempering comma]]{{Clarify}}
|
| 107ury3-2
| 750/749
| 2.3.5.7.107 {{monzo| 1 1 3 -1 -1 }}
| 2.3099
|
|-
| [[784/783|Biminorisma]], spoogalactic comma
| Twenuzozo
| 29uzz2
| 784/783
| 2.3.7.29 {{monzo| 4 -3 2 -1 }}
| 2.2096
| [[Scott Dakota]] for ''biminorisma''
|-
| [[841/840|Arabellisma]]
| Bitweno-arugu
| 29oorg1
| 841/840
| 2.3.5.7.29 {{monzo| -3 -1 -1 -1 2 }}
| 2.0598
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Alphonsinisma]]
| Thisoluruyoyo
| 37o1uryy1
| 925/924
| 2.3.5.7.11.37 {{monzo| -2 -1 2 -1 -1 1 }}
| 1.8726
| [[Francium]] (2025)
|-
| [[961/960|Trricesimoprimal quarter-tones comma]]
| Bithiwo-agu
|
| 961/960
| 2.3.5.31 [-6 -1 -1 2⟩
| 1.8024
| [[Godtone]] (2024)
|-
| [[Yerkesisma]]
| Fothutwethuloyo
| 43u23u1oy-2
| 990/989
| 2.3.5.11.23.43 {{monzo| 1 2 1 1 -1 -1 }}
| 1.7496
| [[Francium]] (2025)
|-
| [[1024/1023|Kibisma]]
| Thiwulu
| 31u1u2
| 1024/1023
| 2.3.11.31 {{Monzo| 10 -1 -1 -1 }}
| 1.6915
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[1025/1024|Kilobytisma]]
| Fowoyoyo
| 41oyy-2
| 1025/1024
| 2.5.41 [-10 2 1⟩
| 1.6898
| [[CompactStar]] (2024)
|-
| [[Marconisma]]
| Thisotrilu
| 37o1u32
| 1332/1331
| 2.3.11.37 {{monzo| 2 2 -3 1 }}
| 1.3002
| [[Francium]] (2024)
|-
| [[Nordenmarkisma]]
| Fothunosulozo
| 43u19o17uoz1
| 1463/1462
| 2.7.11.17.19.43 {{monzo| -1 1 1 -1 1 -1 }}
| 1.1838
| [[Francium]] (2025)
|-
| [[Aunusisma]]
| Thisotwenusuyo
| 37o29u17uy1
| 1480/1479
| 2.3.5.17.29.37 {{monzo| 3 -1 1 -1 -1 1 }}
| 1.1702
| [[Francium]] (2024)
|-
| [[Gabisma]]
| Thisothuyo
| 37o3uy1
| 1665/1664
| 2.3.5.13.37 {{monzo| -7 2 1 -1 1 }}
| 1.0401
| [[Francium]] (2025)
|-
| [[Gentisma]]
| Thisusozozogu
| 37u17ozzg2
| 1666/1665
| 2.3.5.7.17.37 {{monzo| 1 -2 -1 2 1 -1 }}
| 1.0395
| [[Francium]] (2025)
|-
| [[1682/1681|Shaftesburisma]]
| Bifowutweno
| 41uu29oo2
| 1682/1681
| 2.29.41 {{monzo| 1 2 -2 }}
| 1.0296
| [[Budjarn Lambeth]] (2023)
|-
| [[1925/1924|Misericorde]]
| Thisuthulozoyoyo
| 37u3u1ozyy-2
| 1925/1924
| 2.5.7.11.13.37 {{monzo| -2 2 1 1 -1 -1 }}
| 0.89958
| [[User:Kaiveran|Kaiveran]] (2023)
|-
| [[Magnetisma]]
| Tritrila-quinquadtrifo thutweno
| L943u6029o60-8
|
| 2.3.29.43 {{monzo| -61 60 60 -60 }}
| 0.86936
| [[User:Eliora|Eliora]] (2022)
|-
| [[Stentorisma]]
| Thisotwenothulugu
| 37o29o3u1ug2
| 2146/2145
| 2.3.5.11.13.29.37 {{monzo| 1 -1 -1 -1 -1 1 1 }}
| 0.80691
| [[Francium]] (2024)
|-
| [[Kuznetsovisma]]
| Thiwotwenolozo
| 31u29o1oz2
| 2233/2232
| 2.3.7.11.29.31 {{monzo| -3 -2 1 1 1 -1 }}
| 0.77547
| [[Francium]] (2024)
|-
| [[2520/2519|Platonisma]]
|
|
| 2520/2519
| 2.3.5.7.11.229 {{monzo| 3 2 1 1 -1 -1 }}
| 0.68713
| [[User:Eliora|Eliora]] (2022)
|-
| [[Viljevisma]]
| Thisotwenutwetholu
| 37o29u23o1u2
| 2553/2552
| 2.3.11.23.29.37 {{monzo| -3 1 -1 1 -1 1 }}
| 0.67825
| [[Francium]] (2024)
|-
| [[Beattisma]]
| Fothusuthoyoyo
| 43u17u3oyy-2
| 2925/2924
| 2.3.5.13.17.43 {{monzo| -2 2 2 1 -1 -1 }}
| 0.59198
| [[Francium]] (2025)
|-
| [[9251/9248|Helevenisma]]
| Bitwenosu-alo
| 29oo17uu1o-2
| 9251/9248
| 2.11.17.29 {{monzo| -5 1 -2 2 }}
| 0.56151
| [[Zhea Erose]] (2023)
|-
| [[Pedersenisma]]
| Fothutwetholuru
| 43u23o1ur1
| 3312/3311
| 2.3.7.11.23.43 {{monzo| 4 2 -1 -1 1 -1 }}
| 0.52279
| [[Francium]] (2025)
|-
| [[Alexisma]]
| Thisosutholuzo
| 37o17u3o1uz2
| 3367/3366
| 2.3.7.11.13.17.37 {{monzo| -1 -2 1 -1 1 -1 1 }}
| 0.51425
| [[Francium]] (2025)
|-
| [[Bronxisma]]
| Twenununusolozo
| 29u19uu17o1oz1
| 10472/10469
| 2.7.11.17.19.29 {{monzo| 3 1 1 1 -2 -1 }}
| 0.49603
| [[Francium]] (2024)
|-
| [[Veederisma]]
| Twenutholuluyo
| 29u3o1uuy1
| 3510/3509
| 2.3.5.11.13.29 {{monzo| 1 3 1 -2 1 -1 }}
| 0.49330
| [[Francium]] (2024)
|-
| [[Meranisma]]
| Thisunosolo
| 37u19o17o1o1
| 3553/3552
| 2.3.11.17.19.37 {{monzo| -5 -1 1 1 1 -1 }}
| 0.48733
| [[Francium]] (2024)
|-
| [[Smithsonianisma]]
| Fowutwethulotrizo
| 41u23u1oz32
| 3773/3772
| 2.7.11.23.41 {{monzo| -2 3 1 -1 -1 }}
| 0.45891
| [[Francium]] (2025)
|-
| [[Megumisma]]
| Thisosolutriru
| 37o17o1ur31
| 3774/3773
| 2.3.7.11.17.37 {{monzo| 1 1 -3 -1 1 1 }}
| 0.45879
| [[Francium]] (2025)
|-
| [[3969/3968|Yunzee comma]]
| Lathiwuzozo
| L31uzz2
| 3969/3968
| 2.3.7.31 {{monzo| -7 4 2 -1 }}
| 0.43624
| [[User:Kaiveran|Kaiveran]] (2021)
|-
| [[Tamashimisma]]
| Thiwutwethothozogu
| 31u23o3ozg3
| 4186/4185
| 2.3.5.7.13.23.31 {{monzo| 1 -3 -1 1 1 1 -1 }}
| 0.41363
| [[Francium]] (2025)
|-
| [[Shchedrinisma]]
| Thisosusutriyo
| 37o17uuy3-2
| 4625/4624
| 2.5.17.37 {{monzo| -4 3 -2 1 }}
| 0.37436
| [[Francium]] (2024)
|-
| [[4901/4900|Large grapevine]]
| Twenothothobirugu
| 29o3oorrgg1
| 4901/4900
| 2.5.7.13.29 {{monzo| -2 -2 -2 2 1 }}
| 0.35328
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Kalmanisma]]
| Thiwutwethuthoru
| 31u23u3or1
| 4992/4991
| 2.3.7.13.23.31 {{monzo| 7 1 -1 1 -1 -1 }}
| 0.34684
| [[Francium]] (2024)
|-
| [[5041/5040|Third brown pair comma]], 19th highly compositema
|
| 71oorg1
| 5041/5040
| 2.3.5.7.71 {{monzo| -4 -2 -1 -1 2 }}
| 0.34347
| [[Eliora]] (2022)
|-
| [[5292/5291|Bullionisma]]
| Thisuthuluzozo
| 37u3u1uzz1
| 5292/5291
| 2.3.7.11.13.37 {{monzo| 2 3 2 -1 -1 -1 }}
| 0.32717
| [[Eliora]] (2023)
|-
| [[Scarlattisma]]
| Thiwunuluyo
| 31u19u1uy1
| 6480/6479
| 2.3.5.11.19.31 {{monzo| 4 4 1 -1 -1 -1 }}
| 0.26719
| [[Francium]] (2024)
|-
| [[Kunijisma]]
| Fowutwethothorugugu
| 41u23o3orgg2
| 7176/7175
| 2.3.5.7.13.23.41 {{monzo| 3 1 -2 -1 1 1 -1 }}
| 0.24127
| [[Francium]] (2025)
|-
| [[7425/7424|Small grapevine]]
| Latwenuloyoyo
| L29u1oyy-2
| 7425/7424
| 2.3.5.11.29 {{monzo| -8 3 2 1 -1 }}
| 0.23318
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Liebisma]]
| Sathisonuthogu
| S37o19u3og2
| 7696/7695
| 2.3.5.13.19.37 {{monzo| 4 -4 -1 1 -1 1 }}
| 0.22497
| [[Francium]] (2025)
|-
| [[7777/7776|Pulsar comma]]
|
| 101o1oz2
| 7777/7776
| 2.3.7.11.101 {{monzo| -5 -5 1 1 1 }}
| 0.22262
| [[Eliora]] (2022)
|-
| [[Polishookisma]]
| Thiwubitwetho-athuru
| 31u23oo3ur2
| 8464/8463
| 2.3.7.13.23.31 {{monzo| 4 -1 -1 -1 2 -1 }}
| 0.20455
| [[Francium]] (2024)
|-
| [[Friesachisma]]
| Thisotwethothulogu
| 37o23o3u1og2
| 9361/9360
| 2.3.5.11.13.23.37 {{monzo| -4 -2 -1 1 -1 1 1}}
| 0.18495
| [[Francium]] (2024)
|-
| [[Etampesisma]]
| Fowutwethunotholuzo
| 41u23u19o3o1uz2
| 10374/10373
| 2.3.7.11.13.19.23.41 {{monzo| 1 1 1 -1 1 1 -1 -1 }}
| 0.16689
| [[Francium]] (2024)
|-
| [[Girardisma]]
| Fothunoloyoyo
| 43u19o1oyy1
| 10450/10449
| 2.3.5.11.19.43 {{monzo| 1 -5 2 1 1 -1 }}
| 0.16567
| [[Francium]] (2024)
|-
| [[Zangarisma]]
| Thisososolu
| 37o17oo1u3
| 10693/10692
| 2.3.11.17.37 {{monzo| -2 -5 -1 2 1 }}
| 0.16191
| [[Francium]] (2024)
|-
| [[Kaguyisma]]
| Fothutwethusoluyo
| 43u23u17o1uy1
| 10880/10879
| 2.5.11.17.23.43 {{monzo| 7 1 -1 1 -1 -1 }}
| 0.15913
| [[Francium]] (2024)
|-
| [[Sinachopoulosisma]]
| Thisotwethonunuluzo
| 37o23o19uu1uz2
| 11914/11913
| 2.3.7.11.19.23.37 {{monzo| 1 -1 1 -1 -2 1 1 }}
| 0.14532
| [[Francium]] (2024)
|-
| [[Berylisma]]
| Quadthisolu
| 37o41u44
| 1874161 / 1874048
| 2.11.37 {{monzo| -7 -4 4 }}
| 0.10439
| [[User:Jerdle|Jerdle]] (2023)
|-
| [[Manheimisma]]
| Fothutwenosuloloyo
| 43u29o17u1ooy-2
| 17545/17544
| 2.3.5.11.17.29.43 {{monzo| -3 -1 1 2 -1 1 -1 }}
| 0.098677
| [[Francium]] (2024)
|-
| [[Canupisma]]
| Fowutwenuthotrizo-agu
| 41u29u3oz3g3
| 17836/17835
| 2.3.5.7.13.29.41 {{monzo| 2 -1 -1 3 1 -1 -1 }}
| 0.097067
| [[Francium]] (2024)
|-
| [[Genzelisma]]
| Thisotwenonusogu
| 37o29o19u17og2
| 18241/18240
| 2.3.5.17.19.29.37 {{monzo| -6 -1 -1 1 -1 1 1 }}
| 0.094912
| [[Francium]] (2024)
|-
| [[Totziensisma]]
| Thiwotwetholurutrigu
| 31o23o1urg31
| 19251/19250
| 2.3.5.7.11.23.31 {{monzo| -1 3 -3 -1 -1 1 1 }}
| 0.089932
| [[Francium]] (2024)
|-
| [[Honeybrookisma]]
| Thiwobitwenu-atwethutho
| 31o29uu23u3o-2
| 19344/19343
| 2.3.13.23.29.31 {{monzo| 4 1 1 -1 -2 1 }}
| 0.089500
| [[Francium]] (2024)
|-
| [[3359375/3359232|Dimigenes comma]]
|
| 43oy7-2
| 3359375 / 3359232
| 2.3.5.43 {{monzo| -9 -8 7 1 }}
| 0.073696
| [[User:Akselai|Akselai]] (2024)
|-
| [[Vuillafansisma]]
| Twenunosoluyo
| 29u19o17o1uy2
| 25840/25839
| 2.3.5.11.17.19.29 {{monzo| 4 -4 1 -1 1 1 -1 }}
| 0.067000
| [[Francium]] (2024)
|-
| Palimilli
|
| 1003001o23o1u1
| 23069023 / 23068672
| 2.11.23.1003001 {{monzo| -21 -1 1 1 }}
| 0.026341
| [[User:Kaiveran|Kaiveran]] (2021)
|-
| [[Jouvisma]]
| Thisotwethotholuluzogu
| 37o23o3o1uuzg3
| 77441/77440
| 2.5.7.11.13.23.37 {{monzo| -7 -1 1 -2 1 1 1 }}
| 0.022356
| [[Francium]] (2024)
|-
| [[Sulbasutrisma]]
|
|
| 332929/332928
| 2.3.17.577 {{monzo| -7 -2 -2 2 }}
| 0.0052000
| [[User:2^67-1|Cole]] (2024)
|-
| [[Mebisma]]
| Safowuthiwulugugu
| s41u31u1ugg3
| 1048576/1048575
| 2.3.5.11.31.41 {{Monzo| 20 -1 -2 -1 -1 -1 }}
| 0.0016510
| [[User:Xenllium|Xenllium]] (2023)
|-
| [[Zudilisma]]
|
| L4397u23ur-5
| 68 630 377 364 883 / 68 630 356 164 608
| 2.3.7.23.397 {{monzo| -30 29 -1 -1 -1 }}
| 0.00053479
| [[Budjarn Lambeth]] (2023)
|-
| [[Odyssey comma]]
| Bitwenotwetho-athulurutrigu
| 29oo23oo3u1urg32
| 4004001/4004000
| 2.3.5.7.11.13.23.29 {{monzo| -5 2 -3 -1 -1 -1 0 0 2 2 }}
| 0.00043238
| [[Tristan Bay]] (2024)
|-
| [[Borcherdsma]]
|
|
| 160 561 400 000 / 160 561 399 999
| 2.5.7.11.13.19.29.31.47.59.71 {{monzo| 6 5 -1 -2 -1 1 1 1 1 -3 -1 }}
| 1.0783 × 10-8
| [[User:Akselai|Akselai]] (2024)
|}

== Notes ==

[[Category:Unnoticeable commas| ]] 
[[Category:Lists of commas]]
{{todo|complete table|add color name}}

User:Cmloegcmluin/2.11.13.17.19 subgroup

2025-01-30T04:25:48Z

Cmloegcmluin:

==Commas==
Here follows a selection of commas in the 2.11.13.17.19 [[domain basis]].
{| class="wikitable"
|+
! rowspan="2" |name
! rowspan="2" |prime-count vector
! rowspan="2" |ratio
! rowspan="2" |cents
! colspan="5" |as composition of other commas
! colspan="10" |tempered by patent val for ed2?
|-
!yama
!bean
!Blumeyer
!frouggie
!pollar
!13
!24
!33
!37
!57
!70
!80
!113
!124
!137
|-
|Blumeyer
|{{vector| 7 0 0 0 -1 -1 -1 1 }}
|2432 / 2431
|0.7120024978
|
|
|1
|
|
|✓
|✓
|✓
|✓
|✓
|✓
|✓
|✓
|✓
|✓
|-
|yama
|{{vector| -4 0 0 0 1 -1 0 1 }}
|209 / 208
|8.303296728
|1
|
|
|
|
|✓
|✓
|✓
|✓
|✓
|✓
|
|
|
|
|-
|Blume
|{{vector| -11 0 0 0 2 0 1 0 }}
|2057 / 2048
|7.59129423
|1
|
| -1
|
|
|✓
|✓
|✓
|✓
|✓
|✓
|
|
|
|
|-
|
|{{vector| 3 0 0 0 0 -2 -1 2 }}
|2888 / 2873
|9.015299226
|1
|
|1
|
|
|✓
|✓
|✓
|✓
|✓
|✓
|
|
|
|
|-
|eye
|{{vector| 3 0 0 0 -2 0 2 -1 }}
|2312 / 2299
|9.761918139
|1
|1
| -2
|
|
|✓
|✓
|✓
|✓
|
|
|✓
|
|
|
|-
|lum
|{{vector| -1 0 0 0 -1 -1 2 0 }}
|289 / 286
|18.06521487
|2
|1
| -2
|
|
|✓
|✓
|
|✓
|
|
|
|
|
|
|-
|ume
|{{vector| -8 0 0 0 0 0 3 -1 }}
|4913 / 4864
|17.35321237
|2
|1
| -3
|
|
|✓
|✓
|✓
|✓
|
|
|
|
|
|
|-
|mey
|{{vector| -7 0 0 0 0 3 -1 0 }}
|2197 / 2176
|16.62757581
|2
|
|
|
|1
|
|
|✓
|
|
|
|
|
|
|
|-
|
|{{vector| -5 0 0 0 0 -2 2 1 }}
|5491 / 5408
|26.36851159
|3
|1
| -2
|
|
|✓
|✓
|
|✓
|
|
|
|
|
|
|-
|
|{{vector| -5 0 0 0 -1 0 0 2 }}
|361 / 352
|43.7080899
|5
|1
| -1
|
|1
|✓
|
|
|
|
|
|
|
|
|
|-
|
|{{vector| -1 0 0 0 -2 1 0 1 }}
|247 / 242
|35.40479317
|4
|1
| -1
|
|1
|✓
|
|
|
|
|
|
|
|
|
|-
|
|{{vector| 6 0 0 0 -1 -4 3 0 }}
|314432 / 314171
|1.437639059
|
|1
| -2
|
| -1
|
|
|
|
|✓
|✓
|✓
|
|✓
|✓
|-
|
|{{vector| 14 0 0 0 -4 0 1 -1 }}
|278528 / 278179
|2.170623909
|
|1
| -1
|
|
|
|
|
|
|
|
|
|✓
|✓
|✓
|-
|
|{{vector| 4 0 0 0 -2 3 -2 0 }}
|35152 / 34969
|9.036281578
|1
|
|1
|
|1
|
|
|
|
|
|
|✓
|✓
|
|
|-
|
|{{vector| 10 0 0 0 -3 -1 1 0 }}
|17408 / 17303
|10.47392064
|1
|1
| -1
|
|
|
|
|
|
|
|
|✓
|
|
|
|-
|
|{{vector| 0 0 0 0 -1 2 -2 1 }}
|3211 / 3179
|17.33957831
|2
|
|1
|
|1
|✓
|
|
|
|
|
|
|
|
|
|-
|
|{{vector| 6 0 0 0 -2 -2 1 1 }}
|20672 / 20449
|18.77721736
|2
|1
| -1
|
|
|✓
|✓
|
|✓
|
|
|
|
|
|
|-
|
|{{vector| -4 0 0 0 0 1 -2 2 }}
|4693 / 4624
|25.64287503
|3
|
|1
|
|1
|
|
|✓
|
|
|
|
|
|
|
|-
|
|{{vector| 3 0 0 0 -3 2 0 0 }}
|1352 / 1331
|27.10149644
|3
|1
| -1
|
|1
|✓
|
|
|
|
|
|
|
|
|
|-
|
|{{vector| -8 0 0 0 -1 2 1 0 }}
|2873 / 2816
|34.69279067
|4
|1
| -2
|
|1
|✓
|
|
|
|
|
|
|
|
|
|-
|
|{{vector| 14 0 0 0 2 0 -2 -3 }}
|1982464 / 1982251
|0.1860173318
|
|
|
|1
|1
|
|
|✓
|
|
|
|✓
|✓
|
|✓
|-
|
|{{vector| 13 0 0 0 1 -1 0 -3 }}
|90112 / 89167
|18.2512322
|2
|1
| -2
|1
|1
|
|✓
|
|
|
|
|
|
|
|
|-
|
|{{vector| 7 0 0 0 -3 1 2 -2 }}
|480896 / 480491
|1.458621411
|
|1
| -2
|
|
|
|
|
|
|
|
|
|✓
|✓
|✓
|-
|bean
|{{vector| 21 0 0 0 -5 -1 0 0 }}
|2097152 / 2093663
|2.882626407
|
|1
|
|
|
|
|
|
|
|
|
|
|✓
|✓
|✓
|-
|
|{{vector| -3 0 0 0 -1 4 -1 -1 }}
|28561 / 28424
|8.32427908
|1
|
|
|
|1
|
|
|
|
|
|
|✓
|✓
|
|
|-
|
|{{vector| -11 0 0 0 1 2 -1 1 }}
|35321 / 34816
|24.93087254
|3
|
|
|
|1
|
|
|✓
|
|
|
|
|
|
|
|-
|
|{{vector| 3 0 0 0 4 0 -1 -3 }}
|117128 / 116603
|7.777311562
|1
|
| -1
|1
|1
|
|
|✓
|
|✓
|
|
|
|✓
|
|-
|
|{{vector| -18 0 0 0 2 3 0 0 }}
|265837 / 262144
|24.21887004
|3
|
| -1
|
|1
|
|
|✓
|
|
|
|
|
|
|
|-
|
|{{vector| 9 0 0 0 2 -2 0 -2 }}
|61952 / 61009
|26.55452893
|3
|1
| -2
|1
|1
|
|✓
|
|
|
|
|
|
|
|
|-
|
|{{vector| 10 0 0 0 3 -1 -2 -2 }}
|1362944 / 1356277
|8.48931406
|1
|
|
|1
|1
|
|✓
|✓
|
|✓
|
|
|
|✓
|
|-
|
|{{vector| 21 0 0 0 1 -1 -3 -2 }}
|23068672 / 23056709
|0.8980198296
|
|
|1
|1
|1
|
|
|
|
|
|
|✓
|✓
|
|✓
|-
|
|{{vector| -7 0 0 0 -3 -1 1 4 }}
|2215457 / 2214784
|0.525985166
|
|
|1
| -1
| -1
|
|
|
|
|
|
|✓
|✓
|
|✓
|-
|
|{{vector| -10 0 0 0 2 3 -3 1 }}
|5050903 / 5030912
|6.865657669
|1
| -1
|2
|
|1
|
|
|
|
|✓
|✓
|
|✓
|
|
|-
|
|{{vector| -3 0 0 0 1 2 -4 2 }}
|671099 / 668168
|7.577660166
|1
| -1
|3
|
|1
|
|
|
|
|✓
|✓
|
|
|
|
|-
|
|{{vector| 5 0 0 0 3 -3 0 -1 }}
|42592 / 41743
|34.85782565
|4
|1
| -2
|1
|1
|
|✓
|
|
|
|
|
|
|
|
|-
|
|{{vector| 6 0 0 0 -3 0 -1 2 }}
|23104 / 22627
|36.11679567
|4
|1
|
|
|1
|✓
|
|
|
|
|
|
|
|
|
|-
|pollar
|{{vector| 1 0 0 0 -2 5 -1 -2 }}
|742586 / 742577
|0.02098235203
|
|
|
|
|1
|
|
|
|
|
|
|
|
|✓
|✓
|-
|
|{{vector| 17 0 0 0 2 -2 -3 -1 }}
|15859712 / 15775643
|9.201316557
|1
|
|1
|1
|1
|
|
|✓
|
|✓
|
|
|
|✓
|
|-
|
|{{vector| 11 0 0 0 -3 2 -3 1 }}
|6576128 / 6539203
|9.748284075
|1
|
|2
|
|1
|
|
|
|
|
|
|✓
|✓
|
|
|-
|
|{{vector| 24 0 0 0 -1 -1 -1 -3 }}
|16777216 / 16674229
|10.65993797
|1
|1
| -1
|1
|1
|
|
|
|
|
|
|✓
|
|✓
|
|-
|
|{{vector| 17 0 0 0 -4 -2 0 1 }}
|2490368 / 2474329
|11.18592313
|1
|1
|
|
|
|
|
|
|
|
|
|✓
|
|
|
|-
|
|{{vector| 2 0 0 0 3 -1 1 -3 }}
|90508 / 89167
|25.84252643
|3
|1
| -3
|1
|1
|
|✓
|
|
|
|
|
|
|
|
|-
|
|{{vector| 16 0 0 0 1 -3 -1 -1 }}
|720896 / 709631
|27.26653142
|3
|1
| -1
|1
|1
|
|✓
|
|
|
|
|
|
|
|
|-
|
|{{vector| -15 0 0 0 2 1 -1 2 }}
|567853 / 557056
|33.23416926
|4
|
|
|
|1
|
|
|✓
|
|
|
|
|
|
|
|-
|
|{{vector| -2 0 0 0 -3 0 2 1 }}
|5491 / 5324
|53.47000804
|6
|2
| -3
|
|1
|✓
|
|
|
|
|
|
|
|
|
|-
|
|{{vector| 8 0 0 0 -3 4 -2 -1 }}
|7311616 / 7308521
|0.7329848499
|
|
|1
|
|1
|
|
|
|
|
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==See also==

[[Yer]], a hexadecatonic [[Just intonation|JI]] [[wikipedia:Euler–Fokker_genus|EFG]] in this subgroup

[[Gjaeck]], a [[Tridecatonic MOS|tridecatonic MOS]] of [[57edo|57ed2]] in this subgroup

[[Hilim13]], a tridecatonic JI tuning in this subgroup
[[Category:Subgroup]]

Interval size measure

2025-01-23T17:03:49Z

Cmloegcmluin: /* Octave-based fine measures */ undo incorrect change of spelling

'''Interval size measure''' or '''interval size unit''' means the ''distance'' between pitches. Intervals can be measured [[#logarithmic|logarithmic]] or by frequency [[#ratio|ratios]].

== Logarithmic ==
All logarithmic measures can be combined by adding and subtracting them.

=== Backslash notation ===
A common shorthand in use in the microtonal community is ''k''\''N'', written with a backslash (\) instead of a forwardslash (/), to refer to an interval with a frequency ratio of 2''k''/''N''. ''k''\''N'' is pronounced "''k'' steps of ''N'' [[edo]]", and can be derived from the meaning of "[[step]]s" in the context of edos (unless talking about steps of specific subsets/scales of some edo).

Steps are linear in the log-frequency domain, so expressions like {{nowrap|11\19 − 6\19 {{=}} 5\19}} hold. In general, we have
: {{nowrap|''a''\''N'' + ''b''\''N'' {{=}} (''a'' + ''b'')\''N''}}

which expresses the same thing as {{nowrap|2''a''/''N'' × 2''b''/''N'' {{=}} 2(''a'' + ''b'')/''N''.}}

Or equivalently, for subtraction/division:

: {{nowrap|''a''\''N'' − ''b''\''N'' {{=}} (''a'' − ''b'')\''N''}}

which expresses the same thing as {{nowrap|2''a''/''N'' / 2''b''/''N'' {{=}} 2(''a'' - ''b'')/''N''.}}

Backslash notation can be extended to support [[nonoctave]] [[equal tuning]]s by writing the tuning in full after the backslash. For example, 11\13edt means 11 steps of [[13edt]], 14\9edf means 14 steps of [[9edf]], and 7\12ed12/5 means 7 steps of [[12ed12/5]].

=== Gross ===
The [[octave]] and the [[decade]] are common coarse units for interval sizes. The {{w|decibel}}, being a relative logarithmic-scale unit for power or root-power quantities, is inappropriate for measuring intervals; the decade is used instead. Similarly, the {{w|neper}} (Np) and the dineper (dNp), like the decibel, should not be used. However, in the absence of a substitute, dinepers have an application in [[logarithmic approximants]].

Intervals are sometimes expressed in the number of scale steps between them. These steps can be of different size, compare for example the names of the major scale in the classic music. An early unit for measuring intervals is the "[[tone]]" which dates back to classic Greece.

In serial music, all intervals were measured by the number of 12edo [[semitone]]s. In analogy, the '''relative interval measure''' is the number of steps between two pitches of an [[equal tuning]], sometimes called "[[degree]]s". These measures can be written using [[#Backslash notation|backslash notation]] if the degree itself isn't sufficiently clear in context.

=== Fine ===
The [[cent]] (¢), [[1200edo|1\1200 octave]], is the classic measure for intervals when more precision than 12edo is required. Some people object to it on the grounds that it is too (obviously) closely related to 12 equal.

==== Octave-based fine measures ====
The following table demonstrates a list of measures derived from the logarithmic division of the octave: {{todo|complete table|research|comment=Add all missing citations.}}

{| class="wikitable sortable"
|+ style="font-size: 105%;" | List of octave-based fine measures (logarithmic)
|-
! Unit name (symbol):
! Divisions of octave
! Prime factors
! Origin/significance
|-
| [[Eka]]
| [[16edo|16]]
| 24
| From Sanskrit ''eka'': one, unit; chromatic unit of Armodue 16edo Theory<ref>[http://www.armodue.com/risorse.htm Armodue: le risorse di un nuovo sistema musicale]</ref>.
|-
| [[Normal diesis]]
| [[31edo|31]]
| 31 (prime)
| See the dedicated page.
|-
| [[Méride]]
| [[43edo|43]]
| 43 (prime)
| Proposed by [[Joseph Sauveur]], as 7 heptaméride units<ref name="measure">[http://www.huygens-fokker.org/docs/measures.html Stichting Huygens–Fokker: Logarithmic Interval Measures]</ref><ref>[http://tonalsoft.com/enc/m/meride.aspx Tonalsoft | ''Méride / 43-ed2 / 43-edo / 43-ET / 43-tone equal-temperament'']</ref>.
|-
| [[Holdrian comma]]
| [[53edo|53]]
| 53 (prime)
| See the dedicated page.
|-
| [[Holdrian comma|Mercator’s old comma]]
| [[55edo|55]]
| 5 x 11
| Not to be confused with [[Mercator's comma]].
|-
| [[Decitone]]
| [[60edo|60]]
| 22 × 3 × 5
|
|-
| [[Morion]]
| [[72edo|72]]
| 23 × 32
| See the dedicated page.
|-
| [[Farab]]
| [[144edo|144]]
| 24 × 32
| 1/12 of [[12edo]] semitone; Proposed by [[al-Farabi]] in 10th century<ref name="measure"/><ref>[http://tonalsoft.com/enc/f/farab.aspx Tonalsoft | ''Farab''].</ref>.
|-
| [[Mem]]
| [[205edo|205]]
| 5 × 41
| Unit used by H-Pi Instruments<ref name="measure"/><ref>[http://musictheory.zentral.zone/huntsystem1.html H-Pi Instruments | Hunt Theoretical System]</ref><ref>[http://tonalsoft.com/enc/m/mem.aspx Tonalsoft | ''Mem, 205-edo'']</ref>.
|-
| [[Tredek]]
| [[270edo|270]]
| 2 × 33 × 5
| Proposed by [[Joseph Monzo]] (2013)<ref>[http://tonalsoft.com/enc/t/tredek.aspx Tonalsoft | ''Tredek, 270-edo'']</ref>.
|-
| [[Savart]]*
| [[300edo|300]]
| 22 × 3 × 52
| [[Alexander Wood]]'s definition of the Savart<ref>''[https://books.google.com.au/books?id=NWZ8CgAAQBAJ&lpg=PT50&vq=savart&pg=PT51 The Physics of Music]'', Alexander Wood, 1944.</ref>, containing [[12edo]].
|-
| [[Heptaméride]] / [[eptaméride]] / [[savart]]*
| [[301edo|301]]
| 7 × 43
| 301 ≃ 1,000 × log102; 1/7 of Méride unit; proposed by Joseph Sauveur (1701), advocated by [[Félix Savart]]<ref name="measure"/><ref>[http://tonalsoft.com/enc/h/heptameride.aspx Tonalsoft | ''Heptaméride'']</ref>.
|-
| [[Gene]]
| [[311edo|311]]
| 311 (prime)
| Proposed by Joseph Monzo (2007)<ref>[http://tonalsoft.com/enc/g/gene.aspx Tonalsoft | ''Gene, 311-edo'']</ref>.
|-
| [[Dröbisch Angle]]
| [[360edo|360]]
| 23 × 32 × 5
| Proposed as ''angle'' by [[Moritz Dröbisch]] in the 19th century, later by [[Andrew Pikler]] as the current name in ''Logarithmic Frequency Systems'' (1966)<ref name="measure"/>.
|-
| [[Squb]]
| [[494edo|494]]
| 2 × 13 × 19
| {{Citation needed}}
|-
| Great [[iring]] / [[centitone]]
| [[500edo|500]]
| 22 × 53
| {{Citation needed}}
|-
| Dexl
| [[540edo|540]]
| 22 × 33 × 5
| Proposed by Joseph Monzo (2023)<ref>[http://tonalsoft.com/enc/d/dexl.aspx Tonalsoft | ''Dexl, 540-edo'']</ref>.
|-
| [[Iring]] / [[centitone]]
| [[600edo|600]]
| 23 × 3 × 52
| [[Relative cent]] of [[6edo]] ([[12edo]] tone); Proposed by [[Widogast Iring]] (1898), later by [[Joseph Yasser]] as a "centitone" (1932)<ref name="measure"/><ref>[http://www.tonalsoft.com/enc/c/centitone.aspx Tonalsoft | ''Centitone, iring'']</ref>.
|-
| [[Skisma]] (Sk)
| [[612edo|612]]
| 22 × 32 × 17
| Edo representation of [[Sagittal notation|Sagittal]]'s Ultra (Herculean) precision level JI notation (58eda), where it is known as an "ultrina"<ref name="measure"/><ref>[http://tonalsoft.com/enc/s/sk.aspx Tonalsoft | ''Sk, 612-edo'']</ref>.
|-
| [[Delfi]]
| [[665edo|665]]
| 5 × 7 × 19
| <ref name="measure"/>
|-
| Small [[iring]] / [[centitone]]
| [[700edo|700]]
| 22 × 52 x 7
| {{Citation needed}}
|-
| [[Woolhouse]]
| [[730edo|730]]
| 2 × 5 × 73
| Proposed by [[Wesley S.B. Woolhouse]] (1835)<ref>[https://archive.org/details/essayonmusicali00woolgoog/page/n34/mode/2up ''Essay on musical intervals, harmonics, and the temperament of the musical scale, &c''], Wesley S.B. Woolhouse. </ref>.
|-
| [[Millioctave]] (moct)
| [[1000edo|1000]]
| 23 × 53
| See the dedicated page.
|-
| [[Cent]] (¢)
| 1200
| 24 × 3 × 52
| See the dedicated page.
|-
| Greater muon
| [[1224edo|1224]]
| 23 × 32 × 17
| {{Citation needed}}
|-
| Triangular cent
| [[1260edo|1260]]
| 22 × 32 × 5 × 7
| {{Citation needed}}
|-
| Pion
| [[1272edo|1272]]
| 23 × 3 × 53
| {{Citation needed}}
|-
| Pound
| [[1344edo|1344]]
| 26 × 3 × 7
| {{Citation needed}}
|-
| Neutron
| [[1392edo|1392]]
| 24 × 3 × 29
| {{Citation needed}}
|-
| Lesser muon
| [[1428edo|1428]]
| 22 × 3 × 7 × 17
| {{Citation needed}}
|-
| Decifarab
| [[1440edo|1440]]
| 25 × 32 × 5
| 1/10 of [[Farab]] unit<ref name="measure"/>.
|-
| Quadratic cent
| [[1452edo|1452]]
| 22 × 3 × 112
| {{Citation needed}}
|-
| Ksion
| [[1476edo|1476]]
| 22 × 32 × 41
| {{Citation needed}}
|-
| Cubic cent
| [[1500edo|1500]]
| 22 × 3 × 53
| {{Citation needed}}
|-
| Heptamu (7mu)
| [[1536edo|1536]]
| 29 × 3
| Seventh MIDI-resolution unit, 1/128 (1/(27)) of [[12edo]] semitone<ref>[http://tonalsoft.com/enc/number/7mu.aspx Tonalsoft | ''7mu / heptamu'']</ref>
|-
| Rhoon
| [[1560edo|1560]]
| 23 × 3 × 5 × 13
| {{Citation needed}}
|-
| śata
| [[1600edo|1600]]
| 26 × 52
| From Sanskrit ''śatam'': hundred; [[Relative cent]] of Armodue 16edo Theory{{Citation needed}}
|-
| Tile
| [[1632edo|1632]]
| 25 × 3 × 17
| {{Citation needed}}
|-
| [[Iota]]
| [[1700edo|1700]]
| 22 × 52 × 17
| [[Relative cent]] of [[17edo]]; proposed by [[Margo Schulter]] (2002) and [[George Secor]]<ref name="measure"/>.
|-
| [[Harmos]]
| [[1728edo|1728]]
| 26 × 33
| 1728 = 123; 1/144 of [[12edo]] semitone; Proposed by [[Paul Beaver]]<ref name="measure"/><ref name="equal">[http://tonalsoft.com/enc/e/equal-temperament.aspx Tonalsoft | ''Equal temperaments'']</ref>.
|-
| Hind śat / Indian cent
| 2200
| 23 × 11 × 52
| {{Citation needed}}
|-
| [[Mina]]
| [[2460edo|2460]]
| 22 × 3 × 5 × 41
| Abbreviation of "schismina", edo representation of [[Sagittal notation|Sagittal]]'s Extreme (Olympian) precision level JI notation (233eda)<ref name="measure"/><ref>[http://tonalsoft.com/enc/m/mina.aspx Tonalsoft | ''Mina'']</ref>.
|-
| Centidiesis
| 3100
| 22 × 52 x 31
| {{Citation needed}}
|-
| Centiméride
| 4300
| 22 × 52 x 43
| {{Citation needed}}
|-
| [[Major tina]]
| [[8269edo|8269]]
| 8269 (prime)
| Proposed by [[Flora Canou]] (2021)<ref>[https://forum.sagittal.org/viewtopic.php?f=4&t=515 The Sagittal Forum | ''Definition of the tina reviewed'']</ref>.
|-
| [[Tina]]
| [[8539edo|8539]]
| 8539 (prime)
| Provides good approximations for 41-limit primes except 37; named by [[Dave Keenan]] and [[George Secor]]; edo representation of [[Sagittal notation|Sagittal]]'s Insane (Magrathean) precision level JI notation (809eda)<ref name="measure"/><ref>[http://tonalsoft.com/enc/t/tina.aspx Tonalsoft | ''Tina'']</ref>.
|-
| [[Purdal]]
| [[9900edo|9900]]
| 22 × 32 × 52 × 11
| [[Relative cent]] of [[99edo]]; Suggested by [[Osmiorisbendi]], advocated by [[Tútim Dennsuul Wafiil]]. See the dedicated page.
|-
| [[Türk sent]] / [[Turkish cent]]
| [[10600edo|10600]]
| 23 × 52 × 53
| [[Relative cent]] of [[106edo]], 1/200 of [[53edo]]; invented by [[M. Ekrem Karadeniz]] (1965), influenced by [[Abdülkadir Töre]]<ref name="measure"/><ref>[http://www.tonalsoft.com/enc/t/turk-sent.aspx Tonalsoft | ''Türk-sent'']</ref><ref>[http://www.ozanyarman.com/files/doctorate_thesis.pdf ''79-Tone Tuning & Theory for Turkish Maqam Music''], Ozan Yarman. </ref>.
|-
| [[Prima]]
| [[12276edo|12276]]
| 22 × 32 × 11 × 31
| Proposed by [[Erv Wilson]], [[Gene Ward Smith]] and [[Gavin Putland]]<ref name="measure"/>.
|-
| [[Jinn]]
| [[16808edo|16808]]
| 23 × 11 × 191
| See the dedicated page.
|-
| [[Jot]]
| [[30103edo|30103]]
| 30103 (prime)
| 30103 ≃ 100,000 × log102; Proposed by [[Augustus de Morgan]] (1864)<ref name="measure"/><ref>[http://www.tonalsoft.com/enc/j/jot.aspx Tonalsoft | ''Jot'']</ref><ref name="equal"/>.
|-
| [[Imp]]
| [[31920edo|31920]]
| 24 × 3 × 5 × 7 × 19
| <ref name="measure"/>
|-
| [[Flu]]
| [[46032edo|46032]]
| 24 × 3 × 7 × 137
| Proposed by Gene Ward Smith (2005)<ref name="measure"/><ref>[http://tonalsoft.com/enc/f/flu.aspx Tonalsoft | ''Flu'']</ref>.
|-
| [[Normal atom]]
| [[78005edo|78005]]
| 5 × 15601
| Name proposed by Tristan Bay in 2023; 78005edo consistently maps Kirnberger's atom to 1 edostep and is a very strong 5-limit system. {{Citation needed}}
|-
| [[MIDI Tuning Standard unit]] (14mu)
| [[196608edo|196608]]
| 216 × 3
| Fourteenth MIDI-resolution unit, 1/16384 (1/(214)) of [[12edo]] semitone<ref name="measure"/>.
|}
<nowiki />* More to be added regarding the Heptaméride/Savart units

==== Non-octave fine measures ====
There are other fine measurements based upon the logarithmic division of other intervals (e.g. 3/1 (twelfth)), a few of which are listed below:

{| class="wikitable sortable"
|+ style="font-size: 105%;" List of non-octave fine measures (logarithmic)
|-
! Unit name (symbol)
! Base interval
! Divisions of base interval
! Origin/significance
|-
| Hekt
| 3/1 (twelfth)
| 1300
| 1/100 of 13edt (Bohlen–Pierce) scale step
|-
| Euhekt
| 3/1 (twelfth)
| 1900
| 1/100 of 19edt (OnlyPure) scale step
|-
| Grad
| [[Pythagorean comma|531441/524288]] (Pythagorean comma)
| 12
| [[12edo]] flattens [[3/2]] by this amount
|-
| Tuning unit
| [[531441/524288]] (Pythagorean comma)
| 720
|
|}

To ''convert hekts'', which is quite common in EDT systems, ''into cents'', use following formula: <code> c = h*12/13*math.log(3)/math.log(2) </code>

=== Relative measures ===
Within a given [[Equal-step tuning|equal-stepped]] tonal system, the [[relative cent]] (rct, r¢) can be used to describe properties of pitches (for instance the approximation of [[Just intonation|JI]] intervals). It is defined as on 100th (or 1 percent) of the interval between two neighbouring pitches in the used equal tuning.

== Ratio ==
Intervals can be measured also giving their [[ratio]]. For instance the major third as [[5/4]] or the pure fifth [[3/2]]. When combining sizes given in ratios, you have to multiply or divide:

a pure fifth increased by a major third gives the major seventh {{nowrap|3/2 × 5/4 {{=}} [[15/8]]}},

which is a diatonic semitone below an octave {{nowrap|([[2/1]]) / (15/8) {{=}} 2/1 × 8/15 {{=}} [[16/15]]}}.

Another notation for ratios is a vector of prime factor exponents, often called a [[monzo]], such as {{monzo| -4 4 -1 }} (for the syntonic comma, {{nowrap|81/80 = 2−4 × 34 × 5−1}}), which builds a bridge back to the logarithmic measure: intervals can be combined by component-wise addition or subtraction of their vectors.

== See also ==
* [[Interval span]]

== Articles ==
* [http://arxiv.org/abs/0907.5249 ''Why the Kirnberger Kernel Is So Small''] by [[Don N. Page]]

== References ==
<references />

[[Category:Interval]]

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2025-01-05T15:49:25Z

Cmloegcmluin: fix typo

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'''Note:''' If you are notating for [[val]]s, use [[Template:Val]] instead.

{{documentation}}

[[Category:Formatting templates]]
</noinclude>

Sagittal notation

2025-01-04T16:49:23Z

Cmloegcmluin: /* Spartan single-shaft */ fix likely copy-pasta

[[de:Sagittalnotation]]
[[file:sagittal_sample.gif]]

'''Sagittal notation''' is a [[musical notation]] system capable of notating almost any conceivable tuning. It was developed by [[Dave Keenan]] and [[George Secor]] with significant contributions from numerous others.

== Flavors of Sagittal notation ==
Sagittal notation comes in two mutually compatible ''flavors''.

=== Evo ===
The '''Evo''' flavor (short for "evolutionary", previously called "mixed") uses only single-shaft Sagittal symbols, e.g. {{sagittal| /| }} {{sagittal| \! }} {{sagittal| |) }} {{sagittal| !) }}, alone or in combination with conventional sharps and flats and their doubles. Only the large variant of the double sharp {{sagittal| x }} (U+E47D) is considered to be stylistically-compatible with Sagittal symbols. Evo is much easier to learn, but it results in a greater number of symbols on the sheet, which can give it a more cluttered appearance, particularly with chords, and it may be confusing when two symbols alter the same note in opposite directions.

 A sub-flavor of Evo is '''Evo-SZ''' (Evo with Stein-Zimmermann). This is where any sagittals that are notating exactly half the alteration of a sharp or flat (most often {{sagittal| /|\ }} {{sagittal| \!/ }}) are replaced by the Stein-Zimmermann semisharp {{sagittal| > }} and narrow reversed flat {{sagittal| < }}, and the corresponding combinations (most often {{sagittal| /|\ }}{{sagittal| # }} and {{sagittal| \!/ }}{{sagittal| b }}) are replaced by {{sagittal| ># }} and {{sagittal| <b }}. The narrow variants of the fractional flats {{sagittal| < }} (U+E284) and {{sagittal| <b }} (U+E285) are preferred because they preserve the Sagittal principle that the visual size of a symbol should indicate the relative size of its alteration and they reduce left-right confusion.

=== Revo ===
The '''Revo''' flavor (short for "revolutionary", previously called "pure") only requires one accidental per note. Revo therefore takes up less space on the sheet and presents a cleaner appearance, and it clearly indicates the direction of the overall alteration. It discards the conventional sharps and flats and their doubles and replaces them with these multi-shaft arrow-like symbols: {{sagittal| /||\ }} {{sagittal| \!!/ }} {{sagittal| /X\ }} {{sagittal| \Y/ }}. Adding a sharp or flat to a Sagittal is achieved by adding two more shafts, e.g. {{sagittal| /| }}{{sagittal| # }} becomes {{sagittal| /||| }} and {{sagittal| !) }}{{sagittal| b }} becomes {{sagittal| !!!) }}. When the Sagittal part alters in the opposite direction to the sharp or flat part, the rules are not so simple, e.g. {{sagittal| \! }}{{sagittal| # }} becomes {{sagittal| ||\ }} and {{sagittal| |) }}{{sagittal| b }} becomes {{sagittal| !!) }}; one must simply learn these ''[[2187/2048#Notation|apotome]] complements''.

== Resources ==
* [http://sagittal.org/ Official site]
* [http://forum.sagittal.org Sagittal Forum]
* [http://sagittal.org/sagittal.pdf The original Xenharmonikon article (updated)]
* [http://sagittal.org/gift/GiftOfTheGods.htm Gift of the Gods: a Mythical introduction to Sagittal notation]
* [http://sagittal.org/Sagittal%20Standard%20JI%20Notation%20Calculator%20Spreadsheet.xlsx spreadsheet-based calculator for Sagittal JI notation]
* [http://sagittal.org/Sagittal-SMuFL-Map.pdf Sagittal-SMuFL-Map, a table of every Sagittal symbol]
* [[Pain free guide to Sagittal]] by [[William Lynch]]
* [[:File:24 Edo.pdf]] – Sagittal notation guide for 24edo by William Lynch (download: [{{filepath:24 Edo.pdf}} ''24_Edo.pdf''])
* [[Introductory_examples_in_Sagittal_notation|Introductory examples]] by [[Hans Straub]]
* [https://andrewmeronek.com/music-resources/sagittal-chord-lists/ Sagittal chord chart] by [[Andrew Meronek]]

== Notation software support ==
=== Sibelius ===
Sagibelius 2.0 - plugins for using Sagittal notation in Sibelius 4 and up. By [http://www.jacobbarton.net/2011/10/sagibelius-2-0-released/ Jacob Barton]. Hosted on this wiki. Donationware.

[[:File:Sagibelius_2.0.zip|Sagibelius_2.0.zip]]

=== Lilypond ===
[http://x31eq.com/lilypond/ Plugin for Sagittal notation in Lilypond] by Graham Breed

=== MuseScore ===
Sagittal accidentals are available in MuseScore via the [https://www.smufl.org/fonts/ Bravura font which implements the SMuFL standard]. They can be accessed by opening the Master Palette and finding them in the Symbols section at the end.

=== Scala ===
Sagittal notation is available in Scala.

=== Dorico ===
Because Dorico is built by Steinberg Media, the same company that maintains the SMuFL standard, it supports Sagittal.

== Scores in Sagittal notation ==
* [[The Sagittal Songbook]]
** [http://oneforall.ytmnd.com One for All]
** [[land_urchin|Land Urchin]]
** [[Clouds_(Andrew_Heathwaite)|Clouds]]
** [[Prayer of Thanks]]

* [[:File:sunday3.pdf|Sunday Pipes]] in [[22edo|22tET]] by [[Mats Öljare]]
* Tibia in [[22edo|22tET]] by [[Paul Erlich]] ([http://www.youtube.com/watch?v=d44Lfp9lAG8 Listen]). Sagittal score [[:File:TIBIA.pdf|in F||\]] or [[:File:tibia_in_g.pdf|in G]] (contains errors in measures 9, 19 and 20)
* [[:File:Ivor_Darreg,_Suite.pdf|On the Enharmonic Tetrachord (from Suite, Op. 62)]], in [[22edo|22tET]], by [[Ivor Darreg]]. Originally printed in the Spring 1975 issue of Xenharmonikon in quarter-tone notation. Transcribed to Sagittal by [[Juhani Nuorvala]].([https://www.youtube.com/watch?v=DvHvza1vtfo Listen])

== Gallery ==
=== Spartan single-shaft ===
{{sagittal-gallery-entry| ¦( | !( | n | 5120/5103 | 7/5 kleisma }}
{{sagittal-gallery-entry| /¦ | \! | p | 81/80 | 5 comma }}
{{sagittal-gallery-entry| ¦) | !) | t | 64/63| 7 comma }}
{{sagittal-gallery-entry|//¦ |\\! | ph | 6561/6400 | 25 small diesis }}
{{sagittal-gallery-entry| /¦) | \!) | pat | 36/35 | 35 medium diesis }}
{{sagittal-gallery-entry| /¦\ | \!/ | pak | 33/32 | 11 medium diesis }}
{{sagittal-gallery-entry| (¦) | (!) | jat | 729/704 | 11 large diesis }}
{{sagittal-gallery-entry| (¦\ | (!/ | jak | 8505/8192 | 35 large diesis }}
<div style="margin-bottom: 16px;"></div>

=== Spartan multi-shaft ===
Multi-shaft sagittals are only used in the [[#Revo|Revo]] flavor of Sagittal.
{{sagittal-gallery-entry| )¦¦( | )!!( | ph | 6561/6400 | 25S }}
{{sagittal-gallery-entry| ¦¦) | !!) | t | 64/63 | 7C }}
{{sagittal-gallery-entry| ¦¦\ | !!/ | p | 81/80 | 5C }}
{{sagittal-gallery-entry| /¦¦) | \!!) | n | 5120/5103 | 7/5k }}
{{sagittal-gallery-entry| /¦¦\ | \!!/ }}
{{sagittal-gallery-entry| ¦¦¦( | !!!( | n | 5120/5103 | 7/5k }}
{{sagittal-gallery-entry| /¦¦¦ | \!!! | p | 81/80 | 5C }}
{{sagittal-gallery-entry| ¦¦¦) | !!!) | t | 64/63 | 7C }}
{{sagittal-gallery-entry|//¦¦¦ |\\!!! | ph | 6561/6400 | 25S }}
{{sagittal-gallery-entry| /¦¦¦) | \!!!) | pat | 36/35 | 35M }}
{{sagittal-gallery-entry| /¦¦¦\ | \!!!/ | pak | 33/32 | 11M }}
{{sagittal-gallery-entry| (¦¦¦) | (!!!) | jat | 729/704 | 11L }}
{{sagittal-gallery-entry| (¦¦¦\ | (!!!/ | jak | 8505/8192 | 35L }}
{{sagittal-gallery-entry| )X( | )Y( | ph | 6561/6400 | 25S | mid=48px }}
{{sagittal-gallery-entry| X) | Y) | t | 64/63 | 7C | mid=48px }}
{{sagittal-gallery-entry| X\ | Y/ | p | 81/80 | 5C | mid=48px }}
{{sagittal-gallery-entry| /X) | \Y) | n | 5120/5103 | 7/5k | mid=48px }}
{{sagittal-gallery-entry| /X\ | \Y/ | mid=48px }}
<div style="margin-bottom: 16px;"></div>

=== Athenian extension single-shaft ===
{{sagittal-gallery-entry| )¦( | )!( | ran | 896/891 | 11/7 kleisma }}
{{sagittal-gallery-entry| ~¦( | ~!( | san | 4131/4096 | 17 comma }}
{{sagittal-gallery-entry| ¦\ | !/ | k | 55/54 | 55 comma }}
{{sagittal-gallery-entry| (¦ | (! | j |45927/45056| 11/7 comma }}
{{sagittal-gallery-entry| (¦( | (!( | jan | 45/44 | 11/5 small diesis }}
<div style="margin-bottom: 16px;"></div>

=== Athenian extension multi-shaft ===
Multi-shaft sagittals are only used in the [[#Revo|Revo]] flavor of Sagittal.
{{sagittal-gallery-entry| ~¦¦( | ~!!( | jan | 45/44 | 11/5S }}
{{sagittal-gallery-entry| )¦¦~ | )!!~ | j |45927/45056| 11/7C }}
{{sagittal-gallery-entry| /¦¦ | \!! | k | 55/54 | 55C }}
{{sagittal-gallery-entry| (¦¦( | (!!( | san | 4131/4096 | 17C }}
{{sagittal-gallery-entry|//¦¦ |\\!! | ran | 896/891 | 11/7k }}
{{sagittal-gallery-entry| )¦¦¦( | )!!!( | ran | 896/891 | 11/7k }}
{{sagittal-gallery-entry| ~¦¦¦( | ~!!!( | san | 4131/4096 | 17C }}
{{sagittal-gallery-entry| ¦¦¦\ | !!!/ | k | 55/54 | 55C }}
{{sagittal-gallery-entry| (¦¦¦ | (!!! | j |45927/45056| 11/7C }}
{{sagittal-gallery-entry| (¦¦¦( | (!!!( | jan | 45/44 | 11/5S }}
{{sagittal-gallery-entry| ~X( | ~Y( | jan | 45/44 | 11/5S | mid=48px }}
{{sagittal-gallery-entry| )X~ | )Y~ | j |45927/45056| 11/7C | mid=48px }}
{{sagittal-gallery-entry| /X | \Y | k | 55/54 | 55C | mid=48px }}
{{sagittal-gallery-entry| (X( | (Y( | san | 4131/4096 | 17C | mid=48px }}
{{sagittal-gallery-entry|//X |\\Y | ran | 896/891 | 11/7k | mid=48px }}
<div style="margin-bottom: 16px;"></div>

{{Notation navbox}}

[[Category:Notation]]
[[Category:Sagittal notation| ]]

77edo

2024-12-29T05:41:44Z

Cmloegcmluin: add Notation section, and Sagittal notation

{{Infobox ET}}
{{EDO intro|77}}

== Theory ==
With [[3/1|harmonic 3]] less than a cent flat, [[5/1|harmonic 5]] a bit over three cents sharp and [[7/1|7]]'s less flat than that, 77edo represents an excellent tuning choice for both [[valentine]], the 31 & 46 temperament, and [[starling]], the [[126/125]] [[planar temperament]], giving the [[optimal patent val]] for [[11-limit]] valentine and its [[13-limit]] extensions dwynwen and valentino, as well as 11-limit starling and [[oxpecker]] temperaments. It also gives the optimal patent val for [[grackle]] and various members of the [[unicorn family]], with a [[generator]] of 4\77 instead of the 5\77 (which gives valentine). These are 7-limit [[Unicorn family #Alicorn|alicorn]] and 11- and 13-limit [[Unicorn family #Camahueto|camahueto]].

77et tempers out [[32805/32768]] in the [[5-limit]], [[126/125]], [[1029/1024]] and [[6144/6125]] in the 7-limit, [[121/120]], [[176/175]], [[385/384]] and [[441/440]] in the 11-limit, and [[196/195]], [[351/350]], [[352/351]], [[676/675]] and [[729/728]] in the 13-limit.

The [[17/1|17]] and [[19/1|19]] are tuned fairly well, making it [[consistent]] to the no-11 [[21-odd-limit]]. The equal temperament tempers out [[256/255]] in the 17-limit; and [[171/170]], [[361/360]], [[513/512]], and [[1216/1215]] in the 19-limit.

77edo is an excellent edo for [[Carlos Alpha]], since the difference between 5 steps of 77edo and 1 step of Carlos Alpha is only −0.042912 cents.

=== Prime harmonics ===
{{Harmonics in equal|77|columns=12}}
{{Harmonics in equal|77|columns=12|start=13|collapsed=true|title=Approximation of prime harmonics in 77edo (continued)}}

== Intervals ==
{| class="wikitable center-all right-2 left-3"
|-
! Degree
! Cents
! Approximate Ratios*
|-
| 0
| 0.000
| 1/1
|-
| 1
| 15.584
| 81/80, 91/90, 99/98, 105/104
|-
| 2
| 31.169
| 49/48, 55/54, 64/63, 65/64, ''100/99''
|-
| 3
| 46.753
| 33/32, 36/35, 40/39, ''45/44'', ''50/49''
|-
| 4
| 62.338
| 26/25, 27/26, 28/27
|-
| 5
| 77.922
| 21/20, 22/21, 25/24
|-
| 6
| 93.506
| 18/17, 19/18, 20/19
|-
| 7
| 109.091
| 16/15, 17/16
|-
| 8
| 124.675
| 14/13, 15/14
|-
| 9
| 140.260
| 13/12
|-
| 10
| 155.844
| ''11/10'', 12/11
|-
| 11
| 171.429
| 21/19
|-
| 12
| 187.013
| 10/9
|-
| 13
| 202.597
| 9/8
|-
| 14
| 218.182
| 17/15
|-
| 15
| 233.766
| 8/7
|-
| 16
| 249.351
| 15/13, 22/19
|-
| 17
| 264.935
| 7/6
|-
| 18
| 280.519
| 20/17
|-
| 19
| 296.104
| 13/11, 19/16, 32/27
|-
| 20
| 311.688
| 6/5
|-
| 21
| 327.273
| 98/81
|-
| 22
| 342.857
| 11/9, 17/14
|-
| 23
| 358.442
| 16/13, 21/17
|-
| 24
| 374.026
| 26/21, 56/45
|-
| 25
| 389.610
| 5/4
|-
| 26
| 405.195
| 19/15, 24/19, 33/26
|-
| 27
| 420.779
| 14/11, 32/25
|-
| 28
| 436.364
| 9/7
|-
| 29
| 451.948
| 13/10
|-
| 30
| 467.532
| 17/13, 21/16
|-
| 31
| 483.117
| 120/91
|-
| 32
| 498.701
| 4/3
|-
| 33
| 514.286
| 27/20
|-
| 34
| 529.870
| 19/14
|-
| 35
| 545.455
| 11/8, ''15/11'', 26/19
|-
| 36
| 561.039
| 18/13
|-
| 37
| 576.623
| 7/5
|-
| 38
| 592.208
| 24/17, 38/27, 45/32
|-
| …
| …
| …
|}
<nowiki />* As a 19-limit temperament

== Notation ==

===Sagittal notation===
====Evo flavor====

<imagemap>
File:77-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 591 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 220 106 [[64/63]]
rect 220 80 340 106 [[33/32]]
default [[File:77-EDO_Evo_Sagittal.svg]]
</imagemap>

====Revo flavor====

<imagemap>
File:77-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 543 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[81/80]]
rect 120 80 220 106 [[64/63]]
rect 220 80 340 106 [[33/32]]
default [[File:77-EDO_Revo_Sagittal.svg]]
</imagemap>

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -122 77 }}
| {{mapping| 77 122 }}
| +0.207
| 0.207
| 1.33
|-
| 2.3.5
| 32805/32768, 1594323/1562500
| {{mapping| 77 122 179 }}
| −0.336
| 0.785
| 5.04
|-
| 2.3.5.7
| 126/125, 1029/1024, 10976/10935
| {{mapping| 77 122 179 216 }}
| −0.021
| 0.872
| 5.59
|-
| 2.3.5.7.11
| 121/120, 126/125, 176/175, 10976/10935
| {{mapping| 77 122 179 216 266 }}
| +0.322
| 1.039
| 6.66
|-
| 2.3.5.7.11.13
| 121/120, 126/125, 176/175, 196/195, 676/675
| {{mapping| 77 122 179 216 266 285 }}
| +0.222
| 0.974
| 6.25
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods per 8ve
! Generator*
! Cents*
! Associated ratio*
! Temperament
|-
| 1
| 4\77
| 62.34
| 28/27
| [[Unicorn]] / alicorn (77e) / camahueto (77) / qilin (77)
|-
| 1
| 5\77
| 77.92
| 21/20
| [[Valentine]]
|-
| 1
| 9\77
| 140.26
| 13/12
| [[Tsaharuk]]
|-
| 1
| 15\77
| 233.77
| 8/7
| [[Guiron]]
|-
| 1
| 16\77
| 249.35
| 15/13
| [[Hemischis]] (77e)
|-
| 1
| 20\77
| 311.69
| 6/5
| [[Oolong]]
|-
| 1
| 23\77
| 358.44
| 16/13
| [[Restles]]
|-
| 1
| 31\77
| 483.12
| 45/34
| [[Hemiseven]]
|-
| 1
| 32\77
| 498.70
| 4/3
| [[Grackle]]
|-
| 1
| 34\77
| 529.87
| 512/375
| [[Tuskaloosa]] [[Muscogee]]
|-
| 7
| 32\77 (1\77)
| 498.70 (15.58)
| 4/3 (81/80)
| [[Absurdity]]
|-
| 11
| 32\77 (3\77)
| 498.70 (46.75)
| 4/3 (36/35)
| [[Hendecatonic]]
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Music ==
; [[Jake Freivald]]
* [http://soonlabel.com/xenharmonic/wp-content/uploads/2013/10/Freivald-J.-A-Seed-Planted-2nd-Version-77edo.mp3 ''A Seed Planted'']{{dead link}}, in an [https://web.archive.org/web/20190412162407/http://soonlabel.com/xenharmonic/archives/1391 organ version] of [[Claudi Meneghin]].

; [[Joel Grant Taylor]]
* [https://web.archive.org/web/20201127015546/http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/star_1-GrimaldiA+Bmod.mp3 ''Star 1-GrimaldiA+Bmod'']

; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/star/20120830-77et-star.mp3 ''77et Star'']

[[Category:Listen]]
[[Category:Star]]
[[Category:Starling]]
[[Category:Valentine]]

75edo

2024-12-29T05:40:36Z

Cmloegcmluin: add Notation section, and Sagittal notation

{{Infobox ET}}
{{EDO intro}}

== Theory ==
75et [[tempering out|tempers out]] 20000/19683 ([[tetracot comma]]) and 2109375/2097152 ([[semicomma]]) in the [[5-limit]], and provides a good tuning for the [[tetracot]] temperament. It tempers out [[225/224]] and [[1728/1715]] in the [[7-limit]], [[support]]ing [[bunya]] and [[orwell]], and providing the [[optimal patent val]] for [[fog]].

In the [[11-limit]], 75e [[val]] {{val| 75 119 174 211 '''260''' }} (corresponding to [[#Riemann zeta function|401zpi]]) scores lower in [[TE error|error]], and tempers [[100/99]] and [[243/242]], whereas the [[patent val]] {{val| 75 119 174 211 '''259''' }} tempers [[99/98]] and [[121/120]]. It tempers out [[325/324]] and [[512/507]] in the [[13-limit]], [[120/119]] and [[256/255]] in the [[17-limit]], and [[190/189]] and 250/247 in the 19-limit.

Since 75 is part of the {{w|Fibonacci sequence}} beginning with 5 and 12, it closely approximates the [[peppermint]] temperament. The size of its fifth is exactly 704 cents, which is very close to the peppermint fifth of 704.096 cents. This makes it suitable for neo-Gothic tunings. It also approximates the [[Carlos Beta]] scale well (4\75 ≈ 1\Carlos Beta).

=== Odd harmonics ===
{{Harmonics in equal|75}}

=== Riemann zeta function ===

The [[The_Riemann_zeta_function_and_tuning|Riemann zeta function]] includes two peaks of similar magnitude around 75edo: '''400zpi''' and '''401zpi''', corresponding to the 75dfghk and 75eij vals, with differing mappings for all primes above 5. 400zpi tempers out [[686/675]], [[875/864]], and [[5120/5103]] in the [[7-limit]], [[121/120]] and [[441/440]] in the [[11-limit]], [[91/90]], [[352/351]], and [[2080/2079]] in the [[13-limit]], [[136/135]] in the [[17-limit]], [[190/189]] in the [[19-limit]], and [[161/160]] in the [[23-limit]]. 401zpi tempers out [[20000/19683]], [[1728/1715]], and [[225/224]] in the 7-limit, [[100/99]] and [[2200/2187]] in the 11-limit, [[144/143]] and [[275/273]] in the 13-limit, [[120/119]] and [[1225/1224]] in the 17-limit, [[190/189]] in the 19-limit, and [[162/161]] in the 23-limit. Its step is mapped to [[49/48]] (the slendro diesis) in 400zpi, but [[64/63]] (Archytas' comma) in 401zpi and 75p.
[[File:401zpi.png|200px|thumb|right|The Riemann zeta function around 75edo, showing 400zpi and 401zpi]]
Compare how prime harmonics are mapped in each zeta peak:
{{Harmonics in cet|16.0211986487005|title=Approximation of harmonics in 400zpi|intervals=prime|columns=11}}
{{Harmonics in cet|15.9805820697015|title=Approximation of harmonics in 401zpi|intervals=prime|columns=11}}

== Intervals ==
{{Interval table}}

== Notation ==

===Sagittal notation===
In the following diagrams, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.

This notation uses the same sagittal sequence as [[68edo#Sagittal notation|68-EDO]].
====Evo flavor====

<imagemap>
File:75-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 735 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[64/63]]
rect 120 80 220 106 [[81/80]]
rect 220 80 340 106 [[33/32]]
rect 340 80 460 106 [[27/26]]
default [[File:75-EDO_Evo_Sagittal.svg]]
</imagemap>

====Revo flavor====

<imagemap>
File:75-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 703 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[64/63]]
rect 120 80 220 106 [[81/80]]
rect 220 80 340 106 [[33/32]]
rect 340 80 460 106 [[27/26]]
default [[File:75-EDO_Revo_Sagittal.svg]]
</imagemap>

====Evo-SZ flavor====

<imagemap>
File:75-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 727 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[64/63]]
rect 120 80 220 106 [[81/80]]
rect 220 80 340 106 [[33/32]]
rect 340 80 460 106 [[27/26]]
default [[File:75-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 119 -75 }}
| {{mapping| 75 119 }}
| -0.645
| 0.645
| 4.03
|-
| 2.3.5
| 20000/19683, 2109375/2097152
| {{mapping| 75 119 174 }}
| -0.099
| 0.936
| 5.85
|-
| 2.3.5.7
| 225/224, 1728/1715, 15625/15309
| {{mapping| 75 119 174 211 }}
| -0.713
| 1.337
| 8.36
|}

== Music ==
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=oL6K6O4FBxc ''Fugue on The Lick''] (2019)

[[Category:Listen]]

73edo

2024-12-29T05:39:27Z

Cmloegcmluin: add Notation section, and Sagittal notation

{{Infobox ET}}
{{EDO intro|73}}

== Theory ==
73edo has a very sharp tendency, with the approximations of [[3/1|3]], [[5/1|5]], [[7/1|7]], [[11/1|11]] all sharp. The equal temperament [[tempering out|tempers out]] [[78732/78125]] and [[262144/253125]] in the [[5-limit]]; [[126/125]] and [[245/243]] in the [[7-limit]]; [[176/175]], [[441/440]] and [[4000/3993]] in the [[11-limit]]; [[91/90]], [[169/168]], [[196/195]], [[325/324]], [[351/350]] and [[352/351]] in the [[13-limit]]. It provides the [[optimal patent val]] for the [[marrakesh]] temperament, though [[104edo]] and [[135edo]] tunes it better.

73edo fits in [[mavila]] scale, by the 9;5 relation in the [[7L 2s|superdiatonic]] scheme.

=== Prime harmonics ===
{{Harmonics in equal|73|intervals=prime}}

=== Subsets and supersets ===
73edo is the 21st [[prime edo]], past [[71edo]] and before [[79edo]].

== Intervals ==
{{Interval table}}

== Notation ==

===Sagittal notation===
This notation uses the same sagittal sequence as [[80edo#Sagittal notation|80-EDO]].
====Evo flavor====

<imagemap>
File:73-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 719 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[64/63]]
rect 120 80 220 106 [[81/80]]
rect 220 80 350 106 [[45/44]]
rect 350 80 470 106 [[33/32]]
default [[File:73-EDO_Evo_Sagittal.svg]]
</imagemap>

====Revo flavor====

<imagemap>
File:73-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 679 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 120 106 [[64/63]]
rect 120 80 220 106 [[81/80]]
rect 220 80 350 106 [[45/44]]
rect 350 80 470 106 [[33/32]]
default [[File:73-EDO_Revo_Sagittal.svg]]
</imagemap>

== Scales ==
* Porky[7]: 10 10 10 13 10 10 10 ((10, 20, 30, 43, 53, 63, 73)\73)

== Music ==
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=NuCnLVijULo ''Little Fugue on Happy Birthday''] (2020)

[[Category:Marrakesh]]
[[Category:Listen]]

71edo

2024-12-29T05:38:31Z

Cmloegcmluin: add Notation section, and Sagittal notation

{{Infobox ET}}
{{EDO intro|71}}

== Theory ==
71edo is a [[dual-fifth]] system, with the flat fifth (which is near the fifths of [[26edo]] and [[45edo]]) [[support]]ing [[flattone]] temperament, and the sharp fifth (which is near [[22edo]]'s fifth) supporting [[superpyth]]. Unlike small dual-fifth systems such as [[18edo]], both fifths are close approximations of 3/2.

Using the [[patent val]], the equal temperament [[tempering out|tempers out]] 20480/19683 and [[393216/390625]] in the [[5-limit]], [[875/864]], [[1029/1024]] and [[4000/3969]] in the [[7-limit]], [[100/99]] and [[245/242]] in the [[11-limit]], and [[91/90]] in the [[13-limit]]. In the 13-limit it supplies the optimal [[patent val]] for the 29 & 71 and 34 & 37 temperaments.

=== Odd harmonics ===
{{Harmonics in equal|71}}

=== Subsets and supersets ===
71edo is the 20th [[prime edo]], following [[67edo]] and before [[73edo]]. [[142edo]], which doubles it, provides correction for the harmonic 3.

== Intervals ==
{{Interval table}}

== Notation ==

===Sagittal notation===
In the following diagrams, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.
====Best fifth notation====
=====Evo flavor=====

<imagemap>
File:71-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 772 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[55/54]]
rect 130 80 260 106 [[144/143]]
rect 260 80 370 106 [[81/80]]
rect 370 80 490 106 [[33/32]]
rect 490 80 600 106 [[27/26]]
default [[File:71-EDO_Evo_Sagittal.svg]]
</imagemap>

=====Revo flavor=====

<imagemap>
File:71-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 772 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[55/54]]
rect 130 80 260 106 [[144/143]]
rect 260 80 370 106 [[81/80]]
rect 370 80 490 106 [[33/32]]
rect 490 80 600 106 [[27/26]]
default [[File:71-EDO_Revo_Sagittal.svg]]
</imagemap>

=====Evo-SZ flavor=====

<imagemap>
File:71-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 605 0 765 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[55/54]]
rect 130 80 260 106 [[144/143]]
rect 260 80 370 106 [[81/80]]
rect 370 80 490 106 [[33/32]]
rect 490 80 600 106 [[27/26]]
default [[File:71-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>

====Second best fifth notation====
This notation uses the same sagittal sequence as EDOs [[50edo#Sagittal notation|50]], [[57edo#Sagittal notation|57]], and [[64edo#Sagittal notation|64]].
=====Evo flavor=====

<imagemap>
File:71b_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 551 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 160 106 [[1053/1024]]
default [[File:71b_Evo_Sagittal.svg]]
</imagemap>

=====Revo flavor=====

<imagemap>
File:71b_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 520 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 160 106 [[1053/1024]]
default [[File:71b_Revo_Sagittal.svg]]
</imagemap>

== Music ==
; [[Francium]]
* [https://www.youtube.com/watch?v=_FPTUlO6jNI ''Dancing in the Mosh Pit''] (2023)

[[Category:Listen]]

69edo

2024-12-29T05:37:19Z

Cmloegcmluin: add Notation section, and Sagittal notation (thanks for un-deleting those actually-used Intervals sections, Fredg999, sorry about that!)

{{Infobox ET}}
{{EDO intro|69}} Nice.

== Theory ==
69edo has been called "the love-child of [[23edo]] and [[quarter-comma meantone]]". As a meantone system, it is on the flat side, with a fifth of 695.652 cents. Such a fifth is closer to [[2/7-comma meantone]] than 1/4-comma, and is nearly identical to that of "Synch-Meantone", or Wilson's equal beating meantone, wherein the perfect fifth and the major third beat at equal rates. Therefore 69edo can be treated as a closed system of Synch-Meantone for most purposes.

69edo offers two kinds of meantone 12-tone scales. One is the raw meantone scale, which has a 7:4 step ratio, and other is period-3 [[Meantone family#Lithium|lithium]] scale, which has a 6:5 step ratio and stems from a temperament tempering out 3125/3087 along with 81/80. It should be noted that while the lithium scale has a meantone fifth, it produces a [[3L 6s|tcherepnin]] scale instead of traditional diatonic.

In the [[7-limit]] it is a [[mohajira]] system, tempering out 6144/6125, but not a septimal meantone system, as [[126/125]] maps to one step. In the 11-limit it tempers out [[99/98]], and supports the 31&69 variant of mohajira, identical to the standard 11-limit mohajira in [[31edo]] but not in 69.

The [[concoctic scale]] for 69edo is 22\69, and the corresponding rank two temperament is 22 & 69, defined by tempering out the [-41, 1, 17⟩ comma in the 5-limit.

=== Odd harmonics ===
{{Harmonics in equal|69}}

== Intervals ==
{{Interval table}}

=== Proposed names ===
{| class="wikitable mw-collapsible mw-collapsed collapsible center-1 right-3"
|-
! Degree
! Carmen's naming system
! Cents
! Approximate Ratios*
! Error (abs, [[cent|¢]])
|-
|0
|Natural Unison, 1
|0.000
|[[1/1]]
|0.000
|-
|1
|Ptolemy's comma
|17.391
|[[100/99]]
| -0.008
|-
|2
|Jubilisma, lesser septimal sixth tone
|34.783
|[[50/49]], [[101/99]]
| -0.193, 0.157
|-
|3
|lesser septendecimal quartertone, _____
|52.174
|[[34/33]], [[101/98]]
| 0.491, -0.028
|-
|4
|_____
|69.565
|[[76/73]]
| -0.158
|-
|5
|Small undevicesimal semitone
|86.957
|[[20/19]]
| -1.844
|-
|6
|Large septendecimal semitone
|104.348
|[[17/16]]
| -0.608
|-
|7
|Septimal diatonic semitone
|121.739
|[[15/14]]
|2.296
|-
|8
|Tridecimal neutral second
|139.130
|[[13/12]]
|0.558
|-
|9
|Vicesimotertial neutral second
|156.522
|[[23/21]]
| -0.972
|-
|10
| Undevicesimal large neutral second, undevicesimal whole tone
|173.913
|[[21/19]]
|0.645
|-
|11
|Quasi-meantone
|191.304
|[[19/17]]
| -1.253
|-
|12
|Whole tone
|208.696
|[[9/8]]
|4.786
|-
|13
|Septimal whole tone
|226.087
|[[8/7]]
| -5.087
|-
|14
|Vicesimotertial semifourth
|243.478
|[[23/20]]
|1.518
|-
|15
|Subminor third, undetricesimal subminor third
|260.870
|[[7/6]], [[29/25]]
| -6.001, 3.920
|-
|16
| Vicesimotertial subminor third
|278.261
|[[27/23]]
|0.670
|-
|17
|Pythagorean minor third
|295.652
|[[32/27]]
|1.517
|-
|18
|Classic minor third
|313.043
|[[6/5]]
| -2.598
|-
|19
|Vicesimotertial supraminor third
|330.435
|[[23/19]]
| -0.327
|-
|20
|Undecimal neutral third
|347.826
|[[11/9]]
|0.418
|-
|21
|Septendecimal submajor third
|365.217
|[[21/17]]
| -0.608
|-
|22
|Classic major third
|382.609
|[[5/4]]
| -3.705
|-
|23
| Undetricesimal major third, Septendecimal major third
|400.000
|[[29/23]], [[34/27]]
| -1.303, 0.910
|-
|24
|Undecimal major third
|417.391
|[[14/11]]
| -0.117
|-
|25
|Supermajor third
|434.783
|[[9/7]]
| -0.301
|-
|26
|Barbados third
|452.174
|[[13/10]]
| -2.040
|-
|27
|Septimal sub-fourth
|469.565
|[[21/16]]
| -1.216
|-
|28
|_____
|486.957
|[[53/40]]
| -0.234
|-
|29
|Just perfect fourth
|504.348
|[[4/3]]
|6.303
|-
|30
|Vicesimotertial acute fourth
|521.739
|[[23/17]]
| -1.580
|-
|31
|Undecimal augmented fourth
|539.130
|[[15/11]]
|2.180
|-
|32
|Undecimal superfourth, undetricesimal superfourth
|556.522
|[[11/8]], [[29/21]]
|5.204, -2.275
|-
|33
|Narrow tritone, classic augmented fourth
|573.913
|[[7/5]], [[25/18]]
| -8.600, 5.196
|-
|34
|_____
|591.304
|[[31/22]]
| -2.413
|-
|35
|High tritone, undevicesimal tritone
|608.696
|[[10/7]], [[27/19]]
| -8.792, 0.344
|-
|36
|_____
|626.087
|[[33/23]]
|1.088
|-
|37
| Undetricesimal tritone
|643.478
|[[29/20]]
|0.215
|-
|38
| Undevicesimal diminished fifth, undecimal diminished fifth
|660.870
|[[19/13]], [[22/15]]
|3.884, -2.180
|-
|39
|Vicesimotertial grave fifth, _____
|678.261
|[[34/23]], [[37/25]]
|1.580, -0.456
|-
|40
|Just perfect fifth
|695.652
|[[3/2]]
| -6.303
|-
|41
|_____
|713.043
|[[80/53]]
|0.234
|-
|42
|Super-fifth, undetricesimal super-fifth
|730.435
|[[32/21]], [[29/19]]
|1.216, -1.630
|-
|43
|Septendecimal subminor sixth
|747.826
|[[17/11]]
| -5.811
|-
|44
|Subminor sixth
|765.217
|[[14/9]]
|0.301
|-
|45
|Undecimal minor sixth
|782.609
|[[11/7]]
|0.117
|-
|46
| Septendecimal subminor sixth
|800.000
|[[27/17]]
| -0.910
|-
|47
|Classic minor sixth
|817.391
|[[8/5]]
|3.705
|-
|48
|Septendecimal supraminor sixth
|834.783
|[[34/21]]
|0.608
|-
|49
|Undecimal neutral sixth
|852.174
|[[18/11]]
| -0.418
|-
|50
|Vicesimotertial submajor sixth
|869.565
|[[38/23]]
|0.327
|-
|51
|Classic major sixth
|886.957
|[[5/3]]
|2.598
|-
|52
|Pythagorean major sixth
|904.348
|[[27/16]]
| -1.517
|-
|53
|Septendecimal major sixth, undetricesimal major sixth
|921.739
|[[17/10]], [[29/17]]
|3.097, -2.883
|-
|54
|Supermajor sixth, undetricesimal supermajor sixth
|939.130
|[[12/7]], [[50/29]]
|6.001, -3.920
|-
|55
|Vicesimotertial supermajor sixth
|956.522
|[[40/23]]
| -1.518
|-
|56
|Harmonic seventh
|973.913
|[[7/4]]
|5.087
|-
|57
|Pythagorean minor seventh
|991.304
|[[16/9]]
| -4.786
|-
|58
|Quasi-meantone minor seventh
|1008.696
|[[34/19]]
|1.253
|-
|59
|Minor neutral undevicesimal seventh
|1026.087
|[[38/21]]
| -0.645
|-
|60
|Vicesimotertial neutral seventh
|1043.478
|[[42/23]]
|0.972
|-
|61
|Tridecimal neutral seventh
|1060.870
|[[24/13]]
| -0.558
|-
|62
|Septimal diatonic major seventh
|1078.261
|[[28/15]]
| -2.296
|-
|63
|Small septendecimal major seventh
|1095.652
|[[32/17]]
|0.608
|-
|64
|Small undevicesimal semitone
|1113.043
|[[20/19]]
|1.844
|-
|65
|_____
|1130.435
|[[73/38]]
|0.158
|-
|66
| Septendecimal supermajor seventh
|1147.826
|[[33/17]]
| -0.491
|-
|67
|_____
|1165.217
|[[49/25]]
| -0.193
|-
|68
|_____
|1182.609
|[[99/50]]
|0.008
|-
|69
|Octave, 8
|1200.000
|[[2/1]]
|0.000
|}
<nowiki>*</nowiki>some simpler ratios listed

== Notation ==

===Sagittal notation===
In the following diagrams, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.

This notation uses the same sagittal sequence as EDOs [[62edo#Sagittal notation|62]] and [[76edo#Sagittal notation|76]].
====Evo flavor====

<imagemap>
File:69-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 783 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 170 106 [[1053/1024]]
rect 170 80 290 106 [[33/32]]
default [[File:69-EDO_Evo_Sagittal.svg]]
</imagemap>

====Revo flavor====

<imagemap>
File:69-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 751 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 170 106 [[1053/1024]]
rect 170 80 290 106 [[33/32]]
default [[File:69-EDO_Revo_Sagittal.svg]]
</imagemap>

====Evo-SZ flavor====

<imagemap>
File:69-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 759 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 170 106 [[1053/1024]]
rect 170 80 290 106 [[33/32]]
default [[File:69-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -109 69 }}
| {{mapping| 69 109 }}
| +1.99
| 1.99
| 11.43
|-
| 2.3.5
| 81/80, {{monzo| -41 1 17 }}
| {{mapping| 69 109 160 }}
| +1.86
| 1.64
| 9.40
|-
| 2.3.5.7
| 81/80, 126/125, 4117715/3981312
| {{mapping| 69 109 160 193 }} (69d)
| +2.49
| 1.79
| 10.28
|-
| 2.3.5.7
| 81/80, 3125/3087, 6144/6125
| {{mapping| 69 109 160 194 }} (69)
| +0.94
| 2.13
| 12.23
|}

=== Rank 2 temperaments ===
{| class="wikitable center-1 center-2"
|-
! Periods per 8ve
! Generator
! Temperaments
|-
| 1
| 2\69
| [[Gammy]] (69de)
|-
| 1
| 19\69
| [[Rarity]]
|-
| 1
| 20\69
| [[Mohaha]] (69e)
|-
| 1
| 22\69
| [[Caleb]] (69) [[marveltri]] (69)
|-
| 1
| 29\69
| [[Meantone]] (69d)
|-
| 3
| 5\69
| [[Augmented family #Ogene|Ogene]] (69bceef)
|-
| 3
| 6\69
| [[August]] (7-limit, 69cdd) [[Lithium]] (69)
|-
| 3
| 9\69
| [[Nessafof]] (69e)
|}

== Scales ==
* Supermajor[11], [[3L 8s]] – 6 6 6 7 6 6 6 7 6 6 7
* Meantone[7], [[5L 2s]] (gen = 40\69) – 11 11 7 11 11 11 7
* Meantone[12], [[7L 5s]] (gen = 40\69) – 7 4 7 4 7 4 7 7 4 7 4 7
* Lithium[9], [[3L 6s]] – 11 6 6 11 6 6 11 6 6
* Lithium[12], [[9L 3s]] – 5 6 6 6 5 6 6 6 5 6 6 6

== Music ==
; [[Eliora]]
* [https://www.youtube.com/watch?v=a4vNlDU6Vkw ''Hypergiant Sakura''] (2021)

; [[Francium]]
* [https://www.youtube.com/watch?v=Z3m4KqpuKPw ''69 hours before''] (2023)

[[Category:Meantone]]
[[Category:Listen]]

{{Todo| review }}

67edo

2024-12-29T05:33:56Z

Cmloegcmluin: add Notation section, and Sagittal notation (and remove unused Intervals section)

{{Infobox ET}}
{{EDO intro}}

== Theory ==
67edo [[tempering out|tempers out]] [[81/80]], [[support]]ing [[meantone]], with a tuning which is slightly sharp of [[1/6-comma meantone|1/6-comma]] (the tuning favored by {{w|Wolfgang Amadeus Mozart|Mozart}} and contemporaries, though they suggested the flatter & composite [[55edo]] as an approximation). It is indistinguishable from 4/25=0.16-comma meantone. In the 7-limit the [[patent val]] tempers out [[1029/1024]] and [[1728/1715]], so that it supports [[mothra]]. In the 11-limit it tempers out [[176/175]] and [[540/539]], supporting [[mosura]], an alternative 11-limit mothra. In the 13-limit it tempers out [[144/143]] and [[196/195]], supporting 13-limit mosura. It tempers out the [[orgonisma]], and on the 2.7.11 subgroup it supports the [[orgone]] temperament.

It is a promising tuning which has, as many relatively large equal temperaments do, a variety of tonal resources: it is the first edo to have both meantone and an orgone temperament ([[26edo]] could be called meantone, but it is more of a [[flattone]]). It has relatively good approximations of the [[3/1|3rd]], [[7/1|7th]], [[11/1|11th]], [[13/1|13th]], [[15/1|15th]], [[17/1|17th]] [[harmonic]]s, although the [[5/1|5th]], [[9/1|9th]], and [[19/1|19th]] as well as certain higher ones are workable as well. 33 + 34 can be used to construct this temperament explaining some of its properties. It does well on the 2.3.7.11.13.17.23.31.37.41 [[subgroup]].

=== Prime harmonics ===
{{Harmonics in equal|67|columns=13}}

=== Subsets and supersets ===
67edo is the 19th [[prime edo]], following [[61edo]] and before [[71edo]].

== Notation ==

===Sagittal notation===
In the following diagrams, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.
====Evo flavor====

<imagemap>
File:67-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 735 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 160 106 [[896/891]]
rect 160 80 280 106 [[36/35]]
rect 280 80 440 106 [[1053/1024]]
default [[File:67-EDO_Evo_Sagittal.svg]]
</imagemap>

====Revo flavor====

<imagemap>
File:67-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 759 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 160 106 [[896/891]]
rect 160 80 280 106 [[36/35]]
rect 280 80 440 106 [[1053/1024]]
default [[File:67-EDO_Revo_Sagittal.svg]]
</imagemap>

== Scales ==
=== Mos scales ===
* Meantone[5]: 11 11 17 11 17
* Meantone[7]: 11 11 6 11 11 11 6
* Barbados[5], Bustling Docks (original/default tuning): 14 14 11 14 14

=== Modmos scales ===
* Cavernous (original/default tuning): 14 14 11 21 7
* Formicarium (original/default tuning): 14 7 18 14 14
* Negri Blues (original/default tuning): 14 14 3 8 14 14
* Negri Blues Septatonic (original/default tuning): 14 14 3 8 11 3 14
* Negri Blues Octatonic (original/default tuning): 7 14 7 11 7 11 3 7
* Understory (original/default tuning): 14 7 18 7 21
* Meantone Ionian Pentatonic: 22 6 11 22 6
* Meantone Minor Melodic: 11 6 11 11 11 11 6
* Meantone Minor Harmonic: 11 6 11 11 6 16 6
* Meantone Minor Hexatonic: 11 6 11 11 17 11
* Meantone Dorian Harmonic: 11 6 16 6 11 6 11
* Meantone Mixolydian Pentatonic: 22 6 11 17 11
* Meantone Phrygian Dominant: 6 16 6 11 6 11 11
* Meantone Phrygian Dominant Hexatonic: 6 16 6 11 6 22
* Meantone Phrygian Dominant Pentatonic: 22 6 11 6 22
* Meantone Phrygian Pentatonic: 6 11 22 6 22
* Meantone Double Harmonic: 6 16 6 11 6 16 6

=== Blues scales ===
* [[Lost spirit]] (approximated from [[31edo]]): 17 11 6 5 13 4 11
* [[Blackened skies]] (approximated from [[72edo]]): 18 10 5 6 5 18 5
* Blues Aeolian Hexatonic: 17 11 6 5 6 22
* Blues Aeolian Pentatonic I: 17 11 11 6 22
* Blues Aeolian Pentatonic II: 17 22 6 11 11
* Blues Bright Double Harmonic: 6 16 6 11 6 11 6 5
* Blues Dark Double Harmonic: 11 6 11 6 5 6 16 6
* Blues Dorian Hexatonic: 17 11 11 11 6 11
* Blues Dorian Pentatonic: 17 22 11 6 11
* Blues Dorian Septatonic: 17 11 6 5 11 6 11
* Blues Harmonic Hexatonic: 11 6 11 11 22 6
* Blues Harmonic Septatonic: 17 11 6 5 6 11 5 6
* Blues Leading: 17 11 6 5 17 6 5
* Blues Minor: 17 11 6 5 17 11
* Blues Minor Maj7: 17 11 6 5 22 6
* Blues Pentachordal: 11 6 11 5 6 28
* Greyed Skies (approximated from [[91edo]]): 17 11 5 6 6 17 5
* Akebono I: 11 6 11 11 17
* Augmented: 17 6 16 6 16 6
* Dominant Pentatonic: 11 11 17 17 11
* Hirajoshi: 11 6 12 6 22
* Javanese Pentachordal: 6 11 17 4 29

=== Others ===
* Approximation of [[Pelog]] lima: 6 10 22 7 22
* Arcade (approximated from [[32afdo]]): 22 4 13 15 13
* Cosmic (approximated from [[32afdo]]): 29 10 6 11 11
* Mechanical (approximated from [[16afdo]]): 17 5 17 15 13
* Moonbeam (approximated from [[16afdo]]): 11 6 12 22 6
* Springwater (approximated from [[8afdo]]): 11 11 17 15 13
* Volcanic (approximated from [[16afdo]]): 6 16 17 15 13
* Deja Vu (approximated from [[101afdo]]): 18 21 6 12 10
* Freeway (approximated from [[6afdo]]): 15 12 11 11 9 8
* Mushroom (approximated from [[30afdo]]): 15 12 11 4 24
* Underpass (approximated from [[10afdo]]): 18 21 12 6 10
* Sourgummy (approximated from [[51afdo]]): 14 12 14 14 13
* Bubblegum/Cola (approximated from [[60afdo]]/[[99afdo]]): 14 13 13 13 14
* Spearmint/Whitechocolate (approximated from [[62afdo]]/[[90afdo]]): 13 14 13 14 13
* Lemonade (approximated from [[79afdo]]): 14 13 13 14 13
* Candycorn (approximated from [[91afdo]]): 11 12 11 10 12 11
* Trailmix (approximated from [[97afdo]]): 11 11 11 12 11 11
* Liquorice (approximated from [[101afdo]]): 11 11 12 10 12 11
* Apple Mint (approximated from [[80afdo]]): 9 11 9 9 10 9 10

==Music==
* [http://soonlabel.com/xenharmonic/wp-content/uploads/2011/11/67-edo.mp3 Beginning of a piece in 67 tone], [[Peter Kosmorsky]] {{dead link}}
* [https://youtu.be/xeOjzyXJl_M 67edo Negri8 MODMOS Improvisation], [[Budjarn Lambeth]]

[[Category:Equal divisions of the octave|##]] 
[[Category:Meantone]]
[[Category:Listen]]