Semaphoresmic clan: Difference between revisions

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+missing CTE tunings. +error maps
m Spell out 1\1 in decimal and note the badness metric being used
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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~7/4 = 952.295
* [[CTE]]: ~2 = 1200.000, ~7/4 = 952.295
: [[error map]]: {{val| 0.000 +2.635 -16.531 }}
: [[error map]]: {{val| 0.000 +2.635 -16.531 }}
* [[POTE]]: ~2 = 1\1, ~7/4 = 949.615
* [[POTE]]: ~2 = 1200.000, ~7/4 = 949.615
: error map: {{val| 0.000 -2.724 -19.211 }}
: error map: {{val| 0.000 -2.724 -19.211 }}


Line 66: Line 66:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~7/4 = 948.796
* [[CTE]]: ~2 = 1200.000, ~7/4 = 948.796
: [[error map]]: {{val| 0.000 -4.363 +4.054 -20.030 }}
: [[error map]]: {{val| 0.000 -4.363 +4.054 -20.030 }}
* [[POTE]]: ~2 = 1\1, ~7/4 = 947.365
* [[POTE]]: ~2 = 1200.000, ~7/4 = 947.365
: error map: {{val| 0.000 -7.225 -7.394 -21.461 }}
: error map: {{val| 0.000 -7.225 -7.394 -21.461 }}


Line 77: Line 77:
{{Optimal ET sequence|legend=1| 5, 14c, 19 }}
{{Optimal ET sequence|legend=1| 5, 14c, 19 }}


[[Badness]]: 0.026747
[[Badness]] (Smith): 0.026747


=== 11-limit ===
=== 11-limit ===
Line 87: Line 87:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~7/4 = 947.456
* CTE: ~2 = 1200.000, ~7/4 = 947.456
* POTE: ~2 = 1\1, ~7/4 = 945.973
* POTE: ~2 = 1200.000, ~7/4 = 945.973


Tuning ranges:  
Tuning ranges:  
Line 96: Line 96:
Optimal ET sequence: {{Optimal ET sequence| 14c, 19, 33cd }}
Optimal ET sequence: {{Optimal ET sequence| 14c, 19, 33cd }}


Badness: 0.028947
Badness (Smith): 0.028947


==== 13-limit ====
==== 13-limit ====
Line 106: Line 106:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~7/4 = 947.888
* CTE: ~2 = 1200.000, ~7/4 = 947.888
* POTE: ~2 = 1\1, ~7/4 = 946.397
* POTE: ~2 = 1200.000, ~7/4 = 946.397


Tuning ranges:  
Tuning ranges:  
Line 115: Line 115:
Optimal ET sequence: {{Optimal ET sequence| 14cf, 19, 33cdff }}
Optimal ET sequence: {{Optimal ET sequence| 14cf, 19, 33cdff }}


Badness: 0.022503
Badness (Smith): 0.022503


=== Semafour ===
=== Semafour ===
Line 125: Line 125:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~7/4 = 948.209
* CTE: ~2 = 1200.000, ~7/4 = 948.209
* POTE: ~2 = 1\1, ~7/4 = 945.958
* POTE: ~2 = 1200.000, ~7/4 = 945.958


Optimal ET sequence: {{Optimal ET sequence| 14c, 19e, 33cdee, 52cdeee }}
Optimal ET sequence: {{Optimal ET sequence| 14c, 19e, 33cdee, 52cdeee }}


Badness: 0.028510
Badness (Smith): 0.028510


=== Varan ===
=== Varan ===
Line 140: Line 140:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~7/4 = 949.616
* CTE: ~2 = 1200.000, ~7/4 = 949.616
* POTE: ~2 = 1\1, ~7/4 = 948.921
* POTE: ~2 = 1200.000, ~7/4 = 948.921


Optimal ET sequence: {{Optimal ET sequence| 19e, 24, 43de }}
Optimal ET sequence: {{Optimal ET sequence| 19e, 24, 43de }}


Badness: 0.039647
Badness (Smith): 0.039647


==== 13-limit ====
==== 13-limit ====
Line 155: Line 155:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~7/4 = 949.525
* CTE: ~2 = 1200.000, ~7/4 = 949.525
* POTE: ~2 = 1\1, ~7/4 = 948.835
* POTE: ~2 = 1200.000, ~7/4 = 948.835


Optimal ET sequence: {{Optimal ET sequence| 19e, 24, 43de }}
Optimal ET sequence: {{Optimal ET sequence| 19e, 24, 43de }}


Badness: 0.025676
Badness (Smith): 0.025676


=== Baragon ===
=== Baragon ===
Line 170: Line 170:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~7/4 = 949.031
* CTE: ~2 = 1200.000, ~7/4 = 949.031
* POTE: ~2 = 1\1, ~7/4 = 948.827
* POTE: ~2 = 1200.000, ~7/4 = 948.827


Optimal ET sequence: {{Optimal ET sequence| 5, 19, 24 }}
Optimal ET sequence: {{Optimal ET sequence| 5, 19, 24 }}


Badness: 0.035673
Badness (Smith): 0.035673


==== 13-limit ====
==== 13-limit ====
Line 185: Line 185:


Optimal tunings:
Optimal tunings:
* CTE: ~2 = 1\1, ~7/4 = 949.067
* CTE: ~2 = 1200.000, ~7/4 = 949.067
* POTE: ~2 = 1\1, ~7/4 = 948.802
* POTE: ~2 = 1200.000, ~7/4 = 948.802


Optimal ET sequence: {{Optimal ET sequence| 5, 19, 24 }}
Optimal ET sequence: {{Optimal ET sequence| 5, 19, 24 }}


Badness: 0.026703
Badness (Smith): 0.026703


== Helayo ==
== Helayo ==
Line 204: Line 204:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~7/4 = 947.097
* [[CTE]]: ~2 = 1200.000, ~7/4 = 947.097
: [[error map]]: {{val| 0.000 -7.761 -4.380 -21.729 }}
: [[error map]]: {{val| 0.000 -7.761 -4.380 -21.729 }}
* [[CWE]]: ~2 = 1\1, ~7/4 = 947.505
* [[CWE]]: ~2 = 1200.000, ~7/4 = 947.505
: error map: {{val| 0.000 -6.946 -8.866 -21.321 }}
: error map: {{val| 0.000 -6.946 -8.866 -21.321 }}


{{Optimal ET sequence|legend=1| 5c, 14, 19 }}
{{Optimal ET sequence|legend=1| 5c, 14, 19 }}


[[Badness]]: 0.0791
[[Badness]] (Smith): 0.0791


; Music
; Music
Line 226: Line 226:


[[Optimal tuning]]s:
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~7/4 = 939.030
* [[CTE]]: ~2 = 1200.000, ~7/4 = 939.030
: [[error map]]: {{val| 0.000 -23.896 -20.492 -29.796 }}
: [[error map]]: {{val| 0.000 -23.896 -20.492 -29.796 }}
* [[POTE]]: ~2 = 1\1, ~7/4 = 940.048
* [[POTE]]: ~2 = 1200.000, ~7/4 = 940.048
: error map: {{val| 0.000 -21.859 -26.602 -28.778 }}
: error map: {{val| 0.000 -21.859 -26.602 -28.778 }}


{{Optimal ET sequence|legend=1| 9, 14c, 23d, 37bcd, 60bbccdd }}
{{Optimal ET sequence|legend=1| 9, 14c, 23d, 37bcd, 60bbccdd }}


[[Badness]]: 0.058216
[[Badness]] (Smith): 0.058216


=== 11-limit ===
=== 11-limit ===
Line 243: Line 243:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~7/4 = 938.467
* CTE: ~2 = 1200.000, ~7/4 = 938.467
* POTE: ~2 = 1\1, ~7/4 = 940.041
* POTE: ~2 = 1200.000, ~7/4 = 940.041


Optimal ET sequence: {{Optimal ET sequence|| 9, 14c, 23de, 37bcde }}
Optimal ET sequence: {{Optimal ET sequence|| 9, 14c, 23de, 37bcde }}


Badness: 0.028535
Badness (Smith): 0.028535


; Music
; Music
Line 261: Line 261:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~7/4 = 968.739
* [[CTE]]: ~2 = 1200.000, ~7/4 = 968.739
: [[error map]]: {{val| 0.000 +35.523 +76.208 -0.087 }}
: [[error map]]: {{val| 0.000 +35.523 +76.208 -0.087 }}
* [[POTE]]: ~2 = 1\1, ~7/4 = 973.296
* [[POTE]]: ~2 = 1200.000, ~7/4 = 973.296
: error map: {{val| 0.000 +42.644 +69.088 +3.473 }}
: error map: {{val| 0.000 +42.644 +69.088 +3.473 }}


Line 270: Line 270:
{{Optimal ET sequence|legend=1| 5, 11b, 16bc }}
{{Optimal ET sequence|legend=1| 5, 11b, 16bc }}


[[Badness]]: 0.044321
[[Badness]] (Smith): 0.044321


=== 11-limit ===
=== 11-limit ===
Line 280: Line 280:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~7/4 = 972.258
* CTE: ~2 = 1200.000, ~7/4 = 972.258
* POTE: ~2 = 1\1, ~7/4 = 978.164
* POTE: ~2 = 1200.000, ~7/4 = 978.164


Optimal ET sequence: {{Optimal ET sequence| 5, 11b }}
Optimal ET sequence: {{Optimal ET sequence| 5, 11b }}


Badness: 0.036538
Badness (Smith): 0.036538


== Negri ==
== Negri ==
Line 303: Line 303:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~16/15 = 125.396
* [[CTE]]: ~2 = 1200.000, ~16/15 = 125.396
: [[error map]]: {{val| 0.000 -3.539 -10.126 }}
: [[error map]]: {{val| 0.000 -3.539 -10.126 }}
* [[POTE]]: ~2 = 1\1, ~16/15 = 125.755
* [[POTE]]: ~2 = 1200.000, ~16/15 = 125.755
: error map: {{val| 0.000 -4.975 -9.049 }}
: error map: {{val| 0.000 -4.975 -9.049 }}


{{Optimal ET sequence|legend=1| 9, 10, 19, 67c, 86c, 105c }}
{{Optimal ET sequence|legend=1| 9, 10, 19, 67c, 86c, 105c }}


[[Badness]]: 0.086856
[[Badness]] (Smith): 0.086856


=== 7-limit ===
=== 7-limit ===
Line 322: Line 322:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~15/14 = 124.813
* [[CTE]]: ~2 = 1200.000, ~15/14 = 124.813
: [[error map]]: {{val| 0.000 -1.209 -11.874 -18.453 }}
: [[error map]]: {{val| 0.000 -1.209 -11.874 -18.453 }}
* [[POTE]]: ~2 = 1\1, ~15/14 = 125.608
* [[POTE]]: ~2 = 1200.000, ~15/14 = 125.608
: error map: {{val| 0.000 -4.387 -9.490 -20.042 }}
: error map: {{val| 0.000 -4.387 -9.490 -20.042 }}


{{Optimal ET sequence|legend=1| 9, 10, 19, 48d, 67cdd, 86cdd }}
{{Optimal ET sequence|legend=1| 9, 10, 19, 48d, 67cdd, 86cdd }}


[[Badness]]: 0.026483
[[Badness]] (Smith): 0.026483


==== 2.3.5.7.13 subgroup (negra) ====
==== 2.3.5.7.13 subgroup (negra) ====
Line 343: Line 343:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~14/13 = 124.457
* CTE: ~2 = 1200.000, ~14/13 = 124.457
* POTE: ~2 = 1\1, ~14/13 = 125.567
* POTE: ~2 = 1200.000, ~14/13 = 125.567


Optimal ET sequence: {{Optimal ET sequence| 9, 10, 19, 48df, 67cddf, 86cddff }}
Optimal ET sequence: {{Optimal ET sequence| 9, 10, 19, 48df, 67cddf, 86cddff }}
Line 358: Line 358:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~15/14 = 125.780
* CTE: ~2 = 1200.000, ~15/14 = 125.780
* POTE: ~2 = 1\1, ~15/14 = 126.474
* POTE: ~2 = 1200.000, ~15/14 = 126.474


Optimal ET sequence: {{Optimal ET sequence| 9, 10, 19 }}
Optimal ET sequence: {{Optimal ET sequence| 9, 10, 19 }}


Badness: 0.026190
Badness (Smith): 0.026190


==== 13-limit ====
==== 13-limit ====
Line 373: Line 373:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~14/13 = 125.434
* CTE: ~2 = 1200.000, ~14/13 = 125.434
* POTE: ~2 = 1\1, ~14/13 = 126.431
* POTE: ~2 = 1200.000, ~14/13 = 126.431


Optimal ET sequence: {{Optimal ET sequence| 9, 10, 19 }}
Optimal ET sequence: {{Optimal ET sequence| 9, 10, 19 }}


Badness: 0.017833
Badness (Smith): 0.017833


=== Negril ===
=== Negril ===
Line 390: Line 390:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~15/14 = 124.998
* CTE: ~2 = 1200.000, ~15/14 = 124.998
* POTE: ~2 = 1\1, ~15/14 = 124.767
* POTE: ~2 = 1200.000, ~15/14 = 124.767


Optimal ET sequence: {{Optimal ET sequence| 10e, 19, 29, 48d, 77cdd }}
Optimal ET sequence: {{Optimal ET sequence| 10e, 19, 29, 48d, 77cdd }}


Badness: 0.038679
Badness (Smith): 0.038679


==== 13-limit ====
==== 13-limit ====
Line 405: Line 405:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~14/13 = 124.865
* CTE: ~2 = 1200.000, ~14/13 = 124.865
* POTE: ~2 = 1\1, ~14/13 = 124.716
* POTE: ~2 = 1200.000, ~14/13 = 124.716


Optimal ET sequence: {{Optimal ET sequence| 10e, 19, 29, 48df, 77cddf }}
Optimal ET sequence: {{Optimal ET sequence| 10e, 19, 29, 48df, 77cddf }}


Badness: 0.024383
Badness (Smith): 0.024383


=== Negric ===
=== Negric ===
Line 422: Line 422:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~15/14 = 126.574
* CTE: ~2 = 1200.000, ~15/14 = 126.574
* POTE: ~2 = 1\1, ~15/14 = 127.039
* POTE: ~2 = 1200.000, ~15/14 = 127.039


Optimal ET sequence: {{Optimal ET sequence| 9, 19e }}
Optimal ET sequence: {{Optimal ET sequence| 9, 19e }}


Badness: 0.030617
Badness (Smith): 0.030617


==== 13-limit ====
==== 13-limit ====
Line 437: Line 437:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~14/13 = 126.153
* CTE: ~2 = 1200.000, ~14/13 = 126.153
* POTE: ~2 = 1\1, ~14/13 = 127.039
* POTE: ~2 = 1200.000, ~14/13 = 127.039


Optimal ET sequence: {{Optimal ET sequence| 9, 19e }}
Optimal ET sequence: {{Optimal ET sequence| 9, 19e }}


Badness: 0.020205
Badness (Smith): 0.020205


=== Negroni ===
=== Negroni ===
Line 452: Line 452:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~15/14 = 123.735
* CTE: ~2 = 1200.000, ~15/14 = 123.735
* POTE: ~2 = 1\1, ~15/14 = 124.539
* POTE: ~2 = 1200.000, ~15/14 = 124.539


Optimal ET sequence: {{Optimal ET sequence| 10, 19e, 29, 77cddee }}
Optimal ET sequence: {{Optimal ET sequence| 10, 19e, 29, 77cddee }}


Badness: 0.035296
Badness (Smith): 0.035296


==== 13-limit ====
==== 13-limit ====
Line 467: Line 467:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~15/14 = 123.644
* CTE: ~2 = 1200.000, ~15/14 = 123.644
* POTE: ~2 = 1\1, ~14/13 = 124.545
* POTE: ~2 = 1200.000, ~14/13 = 124.545


Optimal ET sequence: {{Optimal ET sequence| 10, 19e, 29, 77cddeef }}
Optimal ET sequence: {{Optimal ET sequence| 10, 19e, 29, 77cddeef }}


Badness: 0.021559
Badness (Smith): 0.021559


=== Wilsec ===
=== Wilsec ===
Line 484: Line 484:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~11/8 = 537.627
* CTE: ~2 = 1200.000, ~11/8 = 537.627
* POTE: ~2 = 1\1, ~11/8 = 537.186
* POTE: ~2 = 1200.000, ~11/8 = 537.186


Optimal ET sequence: {{Optimal ET sequence| 9, 20, 29, 38d, 67cdde, 105cdddee }}
Optimal ET sequence: {{Optimal ET sequence| 9, 20, 29, 38d, 67cdde, 105cdddee }}


Badness: 0.041886
Badness (Smith): 0.041886


==== 13-limit ====
==== 13-limit ====
Line 499: Line 499:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~11/8 = 537.802
* CTE: ~2 = 1200.000, ~11/8 = 537.802
* POTE: ~2 = 1\1, ~11/8 = 537.208
* POTE: ~2 = 1200.000, ~11/8 = 537.208


Optimal ET sequence: {{Optimal ET sequence| 9, 20, 29, 38df, 67cddef, 105cdddeefff }}
Optimal ET sequence: {{Optimal ET sequence| 9, 20, 29, 38df, 67cddef, 105cdddeefff }}


Badness: 0.025192
Badness (Smith): 0.025192


==== 17-limit ====
==== 17-limit ====
Line 514: Line 514:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~11/8 = 537.492
* CTE: ~2 = 1200.000, ~11/8 = 537.492
* POTE: ~2 = 1\1, ~11/8 = 537.230
* POTE: ~2 = 1200.000, ~11/8 = 537.230


Optimal ET sequence: {{Optimal ET sequence| 9, 20g, 29g, 38df, 67cddefg }}
Optimal ET sequence: {{Optimal ET sequence| 9, 20g, 29g, 38df, 67cddefg }}


Badness: 0.021778
Badness (Smith): 0.021778


==== 19-limit ====
==== 19-limit ====
Line 529: Line 529:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~11/8 = 537.553
* CTE: ~2 = 1200.000, ~11/8 = 537.553
* POTE: ~2 = 1\1, ~11/8 = 537.214
* POTE: ~2 = 1200.000, ~11/8 = 537.214


Optimal ET sequence: {{Optimal ET sequence| 9, 20g, 29g, 38df, 67cddefgh }}
Optimal ET sequence: {{Optimal ET sequence| 9, 20g, 29g, 38df, 67cddefgh }}


Badness: 0.016828
Badness (Smith): 0.016828


== Nuke ==
== Nuke ==
Line 544: Line 544:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1\1, ~16/15 = 81.345
* [[CTE]]: ~2 = 1200.000, ~16/15 = 81.345
: [[error map]]: {{val| 0.000 +9.975 +20.411 -12.861 }}
: [[error map]]: {{val| 0.000 +9.975 +20.411 -12.861 }}
* [[POTE]]: ~2 = 1\1, ~16/15 = 80.954
* [[POTE]]: ~2 = 1200.000, ~16/15 = 80.954
: error map: {{val| 0.000 +12.322 +18.456 -11.688 }}
: error map: {{val| 0.000 +12.322 +18.456 -11.688 }}


{{Optimal ET sequence|legend=1| 14, 15 }}
{{Optimal ET sequence|legend=1| 14, 15 }}


[[Badness]]: 0.129339
[[Badness]] (Smith): 0.129339


=== 11-limit ===
=== 11-limit ===
Line 561: Line 561:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~16/15 = 80.908
* CTE: ~2 = 1200.000, ~16/15 = 80.908
* POTE: ~2 = 1\1, ~16/15 = 80.817
* POTE: ~2 = 1200.000, ~16/15 = 80.817


Optimal ET sequence: {{Optimal ET sequence| 14e, 15 }}
Optimal ET sequence: {{Optimal ET sequence| 14e, 15 }}


Badness: 0.069398
Badness (Smith): 0.069398


=== 13-limit ===
=== 13-limit ===
Line 576: Line 576:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1\1, ~16/15 = 81.320
* CTE: ~2 = 1200.000, ~16/15 = 81.320
* POTE: ~2 = 1\1, ~16/15 = 81.024
* POTE: ~2 = 1200.000, ~16/15 = 81.024


Optimal ET sequence: {{Optimal ET sequence| 14e, 15 }}
Optimal ET sequence: {{Optimal ET sequence| 14e, 15 }}


Badness: 0.048553
Badness (Smith): 0.048553


[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Semaphoresmic clan| ]] <!-- main article -->
[[Category:Semaphoresmic clan| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Revision as of 13:44, 2 December 2024

The semaphoresmic clan (or semaphore family) tempers out the slendro diesis, or semaphoresma, 49/48, a triprime comma with factors of 2, 3 and 7.

Semaphore

Main article: Semaphore and godzilla

Subgroup: 2.3.7

Comma list: 49/48

Subgroup-val mapping: [⟨1 0 2], ⟨0 2 1]]

sval mapping generators: ~2, ~7/4

Gencom mapping: [⟨1 0 0 2], ⟨0 2 0 1]]

gencom: [2 7/4; 49/48]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/4 = 952.295
error map: ⟨0.000 +2.635 -16.531]
  • POTE: ~2 = 1200.000, ~7/4 = 949.615
error map: ⟨0.000 -2.724 -19.211]

Optimal ET sequence: 5, 14, 19, 24, 67dd, 91dd, 115ddd

Scales: semaphore5, semaphore9, semaphore14

Overview to extensions

The second comma of the comma list defines which 7-limit family member we are looking at.

Godzilla adds 81/80. Immunity adds 2240/2187. Helayo adds 3645/3584. Superpelog adds 135/128. Beep adds 21/20. Baba adds 16/15. These all use the same nominal generator as semaphore, though some of them are of very low accuracy.

Decimal adds 25/24. Anguirus adds 2048/2025. Those split the octave in two. Negri adds 225/224, splitting the hemifourth in two. Triforce adds 128/125, splitting the octave in three. Keemun adds 126/125, splitting the hemitwelfth in three. Nautilus adds 250/243, splitting the hemifourth in three. Nuke is like nautilus, but adds 3584/3375 instead. Hemidim adds 648/625 with a 1/4-octave period. Blacksmith adds 28/27, splitting the octave in five. Spell adds 3125/3072, splitting the hemitwelfth in five. Hemiripple adds 6561/6250, splitting the hemifourth in five. Finally, semabila adds 28672/28125 and splits an interval of two octaves plus a hemifourth in five.

Discussed elsewhere are

Considered below are godzilla, helayo, superpelog, baba, negri, and nuke.

Godzilla

Deutsch

Main article: Semaphore and godzilla

Godzilla tempers out 81/80, equating 9/8 and 10/9, so it finds the prime 5 at a stack of four fifths, as does any temperament in the meantone family. 19edo is close to being the optimal generator tuning; hence it can be more or less equated with taking 4\19 as a generator. Mos scales are of 5, 9, or 14 notes.

Subgroup: 2.3.5.7

Comma list: 49/48, 81/80

Mapping: [⟨1 0 -4 2], ⟨0 2 8 1]]

mapping generators: ~2, ~7/4

Wedgie: ⟨⟨ 2 8 1 8 -4 -20 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/4 = 948.796
error map: ⟨0.000 -4.363 +4.054 -20.030]
  • POTE: ~2 = 1200.000, ~7/4 = 947.365
error map: ⟨0.000 -7.225 -7.394 -21.461]

Tuning ranges:

Optimal ET sequence: 5, 14c, 19

Badness (Smith): 0.026747

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 49/48, 81/80

Mapping: [⟨1 0 -4 2 -6], ⟨0 2 8 1 12]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/4 = 947.456
  • POTE: ~2 = 1200.000, ~7/4 = 945.973

Tuning ranges:

  • 11-odd-limit diamond monotone: ~7/4 = [942.857, 947.368] (11\14 to 15\19)
  • 11-odd-limit diamond tradeoff: ~7/4 = [933.129, 968.826]

Optimal ET sequence: 14c, 19, 33cd

Badness (Smith): 0.028947

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 49/48, 78/77, 81/80

Mapping: [⟨1 0 -4 2 -6 -5], ⟨0 2 8 1 12 11]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/4 = 947.888
  • POTE: ~2 = 1200.000, ~7/4 = 946.397

Tuning ranges:

  • 13- and 15-odd-limit diamond monotone: ~7/4 = 947.368 (15\19)
  • 13- and 15-odd-limit diamond tradeoff: ~7/4 = [910.890, 968.826]

Optimal ET sequence: 14cf, 19, 33cdff

Badness (Smith): 0.022503

Semafour

Subgroup: 2.3.5.7.11

Comma list: 33/32, 49/48, 55/54

Mapping: [⟨1 0 -4 2 5], ⟨0 2 8 1 -2]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/4 = 948.209
  • POTE: ~2 = 1200.000, ~7/4 = 945.958

Optimal ET sequence: 14c, 19e, 33cdee, 52cdeee

Badness (Smith): 0.028510

Varan

Subgroup: 2.3.5.7.11

Comma list: 49/48, 77/75, 81/80

Mapping: [⟨1 0 -4 2 -10], ⟨0 2 8 1 17]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/4 = 949.616
  • POTE: ~2 = 1200.000, ~7/4 = 948.921

Optimal ET sequence: 19e, 24, 43de

Badness (Smith): 0.039647

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 66/65, 77/75, 81/80

Mapping: [⟨1 0 -4 2 -10 -5], ⟨0 2 8 1 17 11]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/4 = 949.525
  • POTE: ~2 = 1200.000, ~7/4 = 948.835

Optimal ET sequence: 19e, 24, 43de

Badness (Smith): 0.025676

Baragon

Subgroup: 2.3.5.7.11

Comma list: 49/48, 56/55, 81/80

Mapping: [⟨1 0 -4 2 9], ⟨0 2 8 1 -7]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/4 = 949.031
  • POTE: ~2 = 1200.000, ~7/4 = 948.827

Optimal ET sequence: 5, 19, 24

Badness (Smith): 0.035673

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 81/80, 91/90

Mapping: [⟨1 0 -4 2 9 -5], ⟨0 2 8 1 -7 11]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/4 = 949.067
  • POTE: ~2 = 1200.000, ~7/4 = 948.802

Optimal ET sequence: 5, 19, 24

Badness (Smith): 0.026703

Helayo

For the 5-limit version of this temperament see High badness temperaments #Hogzilla.

Subgroup: 2.3.5.7

Comma list: 49/48, 3645/3584

Mapping: [⟨1 0 11 2], ⟨0 2 -11 1]]

Wedgie: ⟨⟨ 2 -6 1 -14 -4 19 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/4 = 947.097
error map: ⟨0.000 -7.761 -4.380 -21.729]
  • CWE: ~2 = 1200.000, ~7/4 = 947.505
error map: ⟨0.000 -6.946 -8.866 -21.321]

Optimal ET sequence: 5c, 14, 19

Badness (Smith): 0.0791

Music

Superpelog

Subgroup: 2.3.5.7

Comma list: 49/48, 135/128

Mapping: [⟨1 0 7 2], ⟨0 2 -6 1]]

Wedgie: ⟨⟨ 2 -6 1 -14 -4 19 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/4 = 939.030
error map: ⟨0.000 -23.896 -20.492 -29.796]
  • POTE: ~2 = 1200.000, ~7/4 = 940.048
error map: ⟨0.000 -21.859 -26.602 -28.778]

Optimal ET sequence: 9, 14c, 23d, 37bcd, 60bbccdd

Badness (Smith): 0.058216

11-limit

Subgroup: 2.3.5.7.11

Comma list: 33/32, 45/44, 49/48

Mapping: [⟨1 0 7 2 5], ⟨0 2 -6 1 -2]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/4 = 938.467
  • POTE: ~2 = 1200.000, ~7/4 = 940.041

Optimal ET sequence:

Badness (Smith): 0.028535

Music
Mindaugas Rex Lithuaniae by Chris Vaisvil (blog) (superpelog[9] in 23edo tuning)

Baba

Subgroup: 2.3.5.7

Comma list: 16/15, 49/45

Mapping: [⟨1 0 4 2], ⟨0 2 -2 1]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/4 = 968.739
error map: ⟨0.000 +35.523 +76.208 -0.087]
  • POTE: ~2 = 1200.000, ~7/4 = 973.296
error map: ⟨0.000 +42.644 +69.088 +3.473]

Wedgie: ⟨⟨ 2 -2 1 -8 -4 8 ]]

Optimal ET sequence: 5, 11b, 16bc

Badness (Smith): 0.044321

11-limit

Subgroup: 2.3.5.7.11

Comma list: 16/15, 22/21, 49/45

Mapping: [⟨1 0 4 2 1], ⟨0 2 -2 1 3]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/4 = 972.258
  • POTE: ~2 = 1200.000, ~7/4 = 978.164

Optimal ET sequence: 5, 11b

Badness (Smith): 0.036538

Negri

Main article: Negri

Negri tempers out the negri comma in the 5-limit, 49/48 and 225/224 in the 7-limit. It can be extended naturally to the 2.3.5.7.13 subgroup by adding 91/90 to the comma list; this will be discussed below under the title of negra.

Subgroup: 2.3.5

Comma list: 16875/16384

Mapping: [⟨1 2 2], ⟨0 -4 3]]

mapping generators: ~2, ~16/15

Wedgie: ⟨⟨ 4 -3 -14 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~16/15 = 125.396
error map: ⟨0.000 -3.539 -10.126]
  • POTE: ~2 = 1200.000, ~16/15 = 125.755
error map: ⟨0.000 -4.975 -9.049]

Optimal ET sequence: 9, 10, 19, 67c, 86c, 105c

Badness (Smith): 0.086856

7-limit

Subgroup: 2.3.5.7

Comma list: 49/48, 225/224

Mapping: [⟨1 2 2 3], ⟨0 -4 3 -2]]

Wedgie: ⟨⟨ 4 -3 2 -14 -8 13 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~15/14 = 124.813
error map: ⟨0.000 -1.209 -11.874 -18.453]
  • POTE: ~2 = 1200.000, ~15/14 = 125.608
error map: ⟨0.000 -4.387 -9.490 -20.042]

Optimal ET sequence: 9, 10, 19, 48d, 67cdd, 86cdd

Badness (Smith): 0.026483

2.3.5.7.13 subgroup (negra)

Subgroup: 2.3.5.7.13

Comma list: 49/48, 65/64, 91/90

Sval mapping: [⟨1 2 2 3 4], ⟨0 -4 3 -2 -3]]

Gencom mapping: [⟨1 2 2 3 0 4], ⟨0 -4 3 -2 0 -3]]

gencom: [2 14/13; 49/48 65/64 91/90]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~14/13 = 124.457
  • POTE: ~2 = 1200.000, ~14/13 = 125.567

Optimal ET sequence: 9, 10, 19, 48df, 67cddf, 86cddff

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 49/48, 56/55

Mapping: [⟨1 2 2 3 4], ⟨0 -4 3 -2 -5]]

Wedgie: ⟨⟨ 4 -3 2 5 -14 -8 -6 13 22 7 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~15/14 = 125.780
  • POTE: ~2 = 1200.000, ~15/14 = 126.474

Optimal ET sequence: 9, 10, 19

Badness (Smith): 0.026190

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 49/48, 56/55, 78/77

Mapping: [⟨1 2 2 3 4 4], ⟨0 -4 3 -2 -5 -3]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~14/13 = 125.434
  • POTE: ~2 = 1200.000, ~14/13 = 126.431

Optimal ET sequence: 9, 10, 19

Badness (Smith): 0.017833

Negril

Subgroup: 2.3.5.7.11

Comma list: 49/48, 100/99, 225/224

Mapping: [⟨1 2 2 3 2], ⟨0 -4 3 -2 14]]

Wedgie: ⟨⟨ 4 -3 2 -14 -14 -8 -36 13 -22 -46 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~15/14 = 124.998
  • POTE: ~2 = 1200.000, ~15/14 = 124.767

Optimal ET sequence: 10e, 19, 29, 48d, 77cdd

Badness (Smith): 0.038679

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 65/64, 91/90, 875/858

Mapping: [⟨1 2 2 3 2 4], ⟨0 -4 3 -2 14 -3]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~14/13 = 124.865
  • POTE: ~2 = 1200.000, ~14/13 = 124.716

Optimal ET sequence: 10e, 19, 29, 48df, 77cddf

Badness (Smith): 0.024383

Negric

Subgroup: 2.3.5.7.11

Comma list: 33/32, 49/48, 77/75

Mapping: [⟨1 2 2 3 3], ⟨0 -4 3 -2 4]]

Wedgie: ⟨⟨ 4 -3 2 -4 -14 -8 -20 13 1 -18 ]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~15/14 = 126.574
  • POTE: ~2 = 1200.000, ~15/14 = 127.039

Optimal ET sequence: 9, 19e

Badness (Smith): 0.030617

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 33/32, 49/48, 65/64, 91/90

Mapping: [⟨1 2 2 3 3 4], ⟨0 -4 3 -2 4 -3]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~14/13 = 126.153
  • POTE: ~2 = 1200.000, ~14/13 = 127.039

Optimal ET sequence: 9, 19e

Badness (Smith): 0.020205

Negroni

Subgroup: 2.3.5.7.11

Comma list: 49/48, 55/54, 225/224

Mapping: [⟨1 2 2 3 5], ⟨0 -4 3 -2 -15]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~15/14 = 123.735
  • POTE: ~2 = 1200.000, ~15/14 = 124.539

Optimal ET sequence: 10, 19e, 29, 77cddee

Badness (Smith): 0.035296

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 55/54, 65/64, 91/90

Mapping: [⟨1 2 2 3 5 4], ⟨0 -4 3 -2 -15 -3]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~15/14 = 123.644
  • POTE: ~2 = 1200.000, ~14/13 = 124.545

Optimal ET sequence: 10, 19e, 29, 77cddeef

Badness (Smith): 0.021559

Wilsec

Subgroup: 2.3.5.7.11

Comma list: 49/48, 121/120, 225/224

Mapping: [⟨1 6 -1 5 4], ⟨0 -8 6 -4 -1]]

mapping generators: ~2, ~16/11

Optimal tunings:

  • CTE: ~2 = 1200.000, ~11/8 = 537.627
  • POTE: ~2 = 1200.000, ~11/8 = 537.186

Optimal ET sequence: 9, 20, 29, 38d, 67cdde, 105cdddee

Badness (Smith): 0.041886

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 65/64, 91/90, 121/120

Mapping: [⟨1 6 -1 5 4 7], ⟨0 -8 6 -4 -1 -6]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~11/8 = 537.802
  • POTE: ~2 = 1200.000, ~11/8 = 537.208

Optimal ET sequence: 9, 20, 29, 38df, 67cddef, 105cdddeefff

Badness (Smith): 0.025192

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 49/48, 65/64, 91/90, 121/120, 154/153

Mapping: [⟨1 6 -1 5 4 7 -2], ⟨0 -8 6 -4 -1 -6 11]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~11/8 = 537.492
  • POTE: ~2 = 1200.000, ~11/8 = 537.230

Optimal ET sequence: 9, 20g, 29g, 38df, 67cddefg

Badness (Smith): 0.021778

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 49/48, 65/64, 77/76, 91/90, 121/120, 154/153

Mapping: [⟨1 6 -1 5 4 7 -2 7], ⟨0 -8 6 -4 -1 -6 11 -5]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~11/8 = 537.553
  • POTE: ~2 = 1200.000, ~11/8 = 537.214

Optimal ET sequence: 9, 20g, 29g, 38df, 67cddefgh

Badness (Smith): 0.016828

Nuke

Subgroup: 2.3.5.7

Comma list: 49/48, 3584/3375

Mapping: [⟨1 2 2 3], ⟨0 -6 5 -3]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~16/15 = 81.345
error map: ⟨0.000 +9.975 +20.411 -12.861]
  • POTE: ~2 = 1200.000, ~16/15 = 80.954
error map: ⟨0.000 +12.322 +18.456 -11.688]

Optimal ET sequence: 14, 15

Badness (Smith): 0.129339

11-limit

Subgroup: 2.3.5.7.11

Comma list: 49/48, 77/75, 512/495

Mapping: [⟨1 2 2 3 3], ⟨0 -6 5 -3 7]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~16/15 = 80.908
  • POTE: ~2 = 1200.000, ~16/15 = 80.817

Optimal ET sequence: 14e, 15

Badness (Smith): 0.069398

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 66/65, 77/75, 448/429

Mapping: [⟨1 2 2 3 3 4], ⟨0 -6 5 -3 7 -4]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~16/15 = 81.320
  • POTE: ~2 = 1200.000, ~16/15 = 81.024

Optimal ET sequence: 14e, 15

Badness (Smith): 0.048553

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