Blackdye: Difference between revisions
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'''Blackdye''' is a 10-note, [[ | '''Blackdye''' (also denoted 3s in [[groundfault]]'s [[aberrismic theory]]) is a 10-note, [[bipentatonic scale|bipentatonic]], [[generator-offset]] scale with [[step signature]] 5L 2m 3s ([[step pattern]] sLmLsLmLsL), equivalent to the [[5L 5s]] mos with two of the small steps made larger and the other three made smaller. Unlike [[diasem]], another diatonic-based scale with comma steps, blackdye is not [[chirality|chiral]]. It can be tuned as a [[5-limit]] JI scale or a tempered version thereof, where L represents [[10/9]], m represents [[16/15]], and s represents [[81/80]]. Under this interpretation, the scale has two chains of [[3/2]] generators with an offset of [[10/9]]. Blackdye can, of course, be used in 5-limit [[equal temperament]]s that do not temper out [[81/80]], such as [[22edo]] and [[34edo]]; however, there are more exotic tunings of the scale, such as [[32edo]]'s 5:2:1 and [[37edo]]'s 6:2:1, which have [[neogothic]] thirds and [[naiadic]]s instead of 5-limit thirds and Pythagorean thirds, and [[29edo]]'s 4:3:1 and [[36edo]]'s 5:4:1, which have submajor and supraminor thirds instead of 5-limit thirds. One can even interpret diasem as a [[negative-s blackdye|9-note subset of blackdye with a negative s step]]. | ||
Blackdye is a 10-note extension of the | Blackdye is a 10-note extension of the [[nicetone]] scale 3L 2M 2s, LMsLMLs. | ||
''Blackdye'', a name given by groundfault, is a pun on "Blackwood" and "diatonic" since its step sizes are intermediate between that of [[diatonic]] (5L 2s) and [[Blackwood]][10] ([[5L 5s]]). | |||
== Intervals == | == Intervals == | ||
Line 25: | Line 23: | ||
! [[34edo]] (L:m:s = 5:3:1) | ! [[34edo]] (L:m:s = 5:3:1) | ||
|- | |- | ||
!|[[TAMNAMS|1- | !|[[TAMNAMS|1-steps]] | ||
| s<br/>m<br/>L | | s<br/>m<br/>L | ||
| 81/80, 21.51¢<br/>16/15, 111.73¢<br/>10/9, 182.40¢ | | 81/80, 21.51¢<br/>16/15, 111.73¢<br/>10/9, 182.40¢ | ||
Line 31: | Line 29: | ||
| 1\34, 35.29¢<br/>3\34, 105.88¢<br/>5\34, 176.47¢ | | 1\34, 35.29¢<br/>3\34, 105.88¢<br/>5\34, 176.47¢ | ||
|- | |- | ||
!|[[TAMNAMS|2- | !|[[TAMNAMS|2-steps]] | ||
| L + s<br/>L + m | | L + s<br/>L + m | ||
| 9/8, 203.91¢<br/>32/27, 294.14¢ | | 9/8, 203.91¢<br/>32/27, 294.14¢ | ||
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| 6\34, 211.77¢<br/>8\34, 282.35¢ | | 6\34, 211.77¢<br/>8\34, 282.35¢ | ||
|- | |- | ||
!|[[TAMNAMS|3- | !|[[TAMNAMS|3-steps]] | ||
| L + 2s<br/>L + m + s<br/>2L + s<br/>2L + m | | L + 2s<br/>L + m + s<br/>2L + s<br/>2L + m | ||
| 729/640‚ 225.42¢<br/>6/5, 315.64¢<br/>5/4, 386.31¢<br/>320/243, 476.54¢ | | 729/640‚ 225.42¢<br/>6/5, 315.64¢<br/>5/4, 386.31¢<br/>320/243, 476.54¢ | ||
Line 43: | Line 41: | ||
| 7\34, 247.06¢<br/>9\34, 317.65¢<br/>11\34, 388.24¢<br/>13\34, 458.82¢ | | 7\34, 247.06¢<br/>9\34, 317.65¢<br/>11\34, 388.24¢<br/>13\34, 458.82¢ | ||
|- | |- | ||
!|[[TAMNAMS|4- | !|[[TAMNAMS|4-steps]] | ||
| 2L + 2s<br/>2L + m + s | | 2L + 2s<br/>2L + m + s | ||
| 81/64, 407.82¢<br/>4/3, 498.04¢ | | 81/64, 407.82¢<br/>4/3, 498.04¢ | ||
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| 12\34, 423.53¢<br/>14\34, 494.12¢ | | 12\34, 423.53¢<br/>14\34, 494.12¢ | ||
|- | |- | ||
!|[[TAMNAMS|5- | !|[[TAMNAMS|5-steps]] | ||
| 2L + m + 2s<br/>2L + 2m + s<br/>3L + 2s<br/>3L + m + s | | 2L + m + 2s<br/>2L + 2m + s<br/>3L + 2s<br/>3L + m + s | ||
| 27/20, 519.55¢<br/>64/45, 609.78¢<br/>45/32, 590.22¢<br/>40/27, 680.45¢ | | 27/20, 519.55¢<br/>64/45, 609.78¢<br/>45/32, 590.22¢<br/>40/27, 680.45¢ | ||
Line 55: | Line 53: | ||
| 15\34, 529.41¢<br/>17\34, 600.00¢<br/>17\34, 600.00¢<br/>19\34, 670.59¢ | | 15\34, 529.41¢<br/>17\34, 600.00¢<br/>17\34, 600.00¢<br/>19\34, 670.59¢ | ||
|- | |- | ||
!|[[TAMNAMS|6- | !|[[TAMNAMS|6-steps]] | ||
| 3L + m + 2s<br/>3L + 2m + s | | 3L + m + 2s<br/>3L + 2m + s | ||
| 3/2, 701.96¢<br/>128/81, 792.18¢ | | 3/2, 701.96¢<br/>128/81, 792.18¢ | ||
Line 61: | Line 59: | ||
| 20\34, 705.88¢<br/>22\34, 776.47¢ | | 20\34, 705.88¢<br/>22\34, 776.47¢ | ||
|- | |- | ||
!|[[TAMNAMS|7- | !|[[TAMNAMS|7-steps]] | ||
| 3L + m + 3s<br/>3L + 2m + 2s<br/>4L + m + 2s<br/>4L + 2m + s | | 3L + m + 3s<br/>3L + 2m + 2s<br/>4L + m + 2s<br/>4L + 2m + s | ||
| 243/160, 723.46¢<br/>8/5, 813.69¢<br/>5/3, 884.36¢<br/>1280/729, 974.58¢ | | 243/160, 723.46¢<br/>8/5, 813.69¢<br/>5/3, 884.36¢<br/>1280/729, 974.58¢ | ||
Line 67: | Line 65: | ||
| 21\34, 741.18¢<br/>23\34, 811.77¢<br/>25\34, 882.35¢<br/>27\34, 952.94¢ | | 21\34, 741.18¢<br/>23\34, 811.77¢<br/>25\34, 882.35¢<br/>27\34, 952.94¢ | ||
|- | |- | ||
!|[[TAMNAMS|8- | !|[[TAMNAMS|8-steps]] | ||
| 4L + m + 3s<br/>4L + 2m + 2s | | 4L + m + 3s<br/>4L + 2m + 2s | ||
| 27/16, 905.87¢<br/>16/9, 996.09¢ | | 27/16, 905.87¢<br/>16/9, 996.09¢ | ||
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| 26\34, 917.65¢<br/>28\34, 988.24¢ | | 26\34, 917.65¢<br/>28\34, 988.24¢ | ||
|- | |- | ||
!|[[TAMNAMS|9- | !|[[TAMNAMS|9-steps]] | ||
| 5L + 2m + s<br/>5L + m + 2s<br/>4L + 2m + 2s | | 5L + 2m + s<br/>5L + m + 2s<br/>4L + 2m + 2s | ||
| 9/5, 1017.60¢<br/>15/8, 1088.27¢<br/>160/81, 1178.49¢ | | 9/5, 1017.60¢<br/>15/8, 1088.27¢<br/>160/81, 1178.49¢ | ||
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== Modes == | == Modes == | ||
=== Cyclic order === | |||
The modes arranged in cyclic order: (Note: The mode names are based on the 5-limit JI interpretation; modes in a less JI-like tuning may differ greatly from what these names suggest.) | The modes arranged in cyclic order: (Note: The mode names are based on the 5-limit JI interpretation; modes in a less JI-like tuning may differ greatly from what these names suggest.) | ||
{| class="wikitable" | {| class="wikitable" | ||
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! style="text-align:center;" |Name | ! style="text-align:center;" |Name | ||
|- | |- | ||
| | | | | sLmLsLmLsL | ||
| | Blackdye | | | Blackdye Acute Aeolian | ||
|- | |- | ||
| | | | | LmLsLmLsLs | ||
| | Blackdye | | | Blackdye Grave Aeolian | ||
|- | |- | ||
| | | | | mLsLmLsLsL | ||
| | Blackdye Locrian | | | Blackdye Locrian | ||
|- | |- | ||
| | | | | LsLmLsLsLm | ||
| | Blackdye Ionian | | | Blackdye Ionian | ||
|- | |- | ||
| | | | | sLmLsLsLmL | ||
| | Blackdye | | | Blackdye Acute Dorian | ||
|- | |- | ||
| | | | | LmLsLsLmLs | ||
| | Blackdye | | | Blackdye Grave Dorian | ||
|- | |- | ||
| | | | | mLsLsLmLsL | ||
| | Blackdye Phrygian | | | Blackdye Phrygian | ||
|- | |- | ||
| | | | | LsLsLmLsLm | ||
| | Blackdye Lydian | | | Blackdye Lydian | ||
|- | |- | ||
| | | | | sLsLmLsLmL | ||
| | Blackdye | | | Blackdye Acute Mixo | ||
|- | |- | ||
| | | | | LsLmLsLmLs | ||
| | Blackdye | | | Blackdye Grave Mixo | ||
|} | |} | ||
=== By generator chain === | |||
Because blackdye has two chains of 5 fifth generators, it has two chains of 5 modes each separated by one generator: | |||
* Lydian > Ionian > Grave Mixo > Grave Dorian > Grave Aeolian | |||
* Acute Mixo > Acute Dorian > Acute Aeolian > Phrygian > Locrian | |||
== Properties == | == Properties == | ||
# | # Assuming positive step sizes, the fifth (3L + m + 2s) of blackdye must be a diatonic fifth.<!-- | ||
#* (To see this: Consider the chain of fifths generated by the fifth. The diatonic chroma = 7 fifths - 4 octaves = 21L + | #* (To see this: Consider the chain of fifths generated by the fifth. The diatonic chroma = 7 fifths - 4 octaves = 21L + 7m + 14s - (20L + 8m + 12s) = L - m + 2s. Since L - m >= 0, this must always be nonnegative, so the fifth must be >= 4\7. The fifth cannot be sharper than 3\5, since it generates the [[2L 3s|2L' 3s']] subset of blackdye with L' = mL and s' = sL.)--> | ||
# | # Assuming positive step sizes, the fifth is: | ||
#* sharper than 700¢, if L + | #* sharper than 700¢, if L + 3s > 2m | ||
#* equal to 700¢, if L + | #* equal to 700¢, if L + 3s = 2m | ||
#* flatter than 700¢, if L + | #* flatter than 700¢, if L + 3s < 2m | ||
== Tunings == | == Tunings == | ||
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|+ Blackdye tunings | |+ Blackdye tunings | ||
! rowspan="2" |Tuning | ! rowspan="2" |Tuning | ||
! rowspan="2" |L: | ! rowspan="2" |L:m:s | ||
! colspan="9" |Degrees of the mode | ! colspan="9" |Degrees of the mode sLmLsLmLsL | ||
|- | |- | ||
!1!!2!!3!!4!!5!!6!!7!!8!!9 | !1!!2!!3!!4!!5!!6!!7!!8!!9 | ||
Line 204: | Line 207: | ||
== Example == | == Example == | ||
An example in Blackdye | An example in Blackdye Acute Aeolian, sLmLsLmLsL ([[:File:Blackdye Aeolian Example Score.pdf|score]]) | ||
[[File:Blackdye Aeolian Example 22edo.mp3]] [[22edo]] (L:m:s = 3:2:1) | |||
[[File:Blackdye Aeolian Example 27edo.mp3]] [[27edo]] (L:m:s = 4:2:1) | |||
[[File:Blackdye Aeolian Example | [[File:Blackdye Aeolian Example 29edo.mp3]] [[29edo]] (L:m:s = 4:3:1) | ||
[[File:Blackdye Aeolian Example | [[File:Blackdye Aeolian Example 32edo 5_2_1.mp3]] [[32edo]] (L:m:s = 5:2:1) | ||
[[File:Blackdye Aeolian Example | [[File:Blackdye Aeolian Example 34edo.mp3]] [[34edo]] (L:m:s = 5:3:1) | ||
[[File:Blackdye Aeolian Example | [[File:Blackdye Aeolian Example 36edo.mp3]] [[36edo]] (L:m:s = 5:4:1) | ||
[[ | == Scala files == | ||
* [[SNS ((2/1, 3/2)-5, 10/9)-10]] | |||
[ | == Music == | ||
* [https://www.youtube.com/watch?v=HGaiFU_S2WI ''squid''] (2024) by [[Abnormality]] | |||
== See also == | == See also == | ||
* [[Diasem]], a similar diatonic-like mos detempering but for 2.3.7 | * [[Diasem]], a similar diatonic-like mos detempering but for 2.3.7 | ||
* [[Nicetone]], a subset of blackdye | |||
== External links == | |||
* [https://sw3.lumipakkanen.com/scale/9kkiMRL__ 5-limit JI blackdye in Scale Workshop] | |||
* [https://sw3.lumipakkanen.com/scale/9kkCrGrsY 27edo blackdye in Scale Workshop] | |||
* [https://sw3.lumipakkanen.com/scale/9kkWpQX9k 32edo blackdye in Scale Workshop] | |||
* [https://sw3.lumipakkanen.com/scale/9kkLl2oWP 34edo blackdye in Scale Workshop] | |||
[[Category: | [[Category:GO scales]] | ||
[[Category:Rank-3 scales]] | [[Category:Rank-3 scales]] | ||
[[Category: | [[Category:Step-nested scales]] | ||
[[Category:Listen]] | |||
[[Category:bipentatonic scales]] | |||
[[Category:Flought scales]] | |||
[[Category:Aberrismic theory]] |
Latest revision as of 01:12, 29 September 2024
Blackdye (also denoted 3s in groundfault's aberrismic theory) is a 10-note, bipentatonic, generator-offset scale with step signature 5L 2m 3s (step pattern sLmLsLmLsL), equivalent to the 5L 5s mos with two of the small steps made larger and the other three made smaller. Unlike diasem, another diatonic-based scale with comma steps, blackdye is not chiral. It can be tuned as a 5-limit JI scale or a tempered version thereof, where L represents 10/9, m represents 16/15, and s represents 81/80. Under this interpretation, the scale has two chains of 3/2 generators with an offset of 10/9. Blackdye can, of course, be used in 5-limit equal temperaments that do not temper out 81/80, such as 22edo and 34edo; however, there are more exotic tunings of the scale, such as 32edo's 5:2:1 and 37edo's 6:2:1, which have neogothic thirds and naiadics instead of 5-limit thirds and Pythagorean thirds, and 29edo's 4:3:1 and 36edo's 5:4:1, which have submajor and supraminor thirds instead of 5-limit thirds. One can even interpret diasem as a 9-note subset of blackdye with a negative s step.
Blackdye is a 10-note extension of the nicetone scale 3L 2M 2s, LMsLMLs.
Blackdye, a name given by groundfault, is a pun on "Blackwood" and "diatonic" since its step sizes are intermediate between that of diatonic (5L 2s) and Blackwood[10] (5L 5s).
Intervals
The following is a table of blackdye intervals and their abstract sizes in terms of L, m and s. Given concrete sizes of L, m and s in edo steps or cents, you can compute the concrete size of any interval in blackdye using the following expressions.
Interval class | Sizes | 5-limit JI | 32edo (L:m:s = 5:2:1) | 34edo (L:m:s = 5:3:1) |
---|---|---|---|---|
1-steps | s m L |
81/80, 21.51¢ 16/15, 111.73¢ 10/9, 182.40¢ |
1\32, 37.50¢ 2\32, 75.00¢ 5\32, 187.50¢ |
1\34, 35.29¢ 3\34, 105.88¢ 5\34, 176.47¢ |
2-steps | L + s L + m |
9/8, 203.91¢ 32/27, 294.14¢ |
6\32, 225.00¢ 7\32, 262.50¢ |
6\34, 211.77¢ 8\34, 282.35¢ |
3-steps | L + 2s L + m + s 2L + s 2L + m |
729/640‚ 225.42¢ 6/5, 315.64¢ 5/4, 386.31¢ 320/243, 476.54¢ |
7\32, 262.50¢ 8\32, 300.00¢ 11\32, 412.50¢ 12\32, 450.00¢ |
7\34, 247.06¢ 9\34, 317.65¢ 11\34, 388.24¢ 13\34, 458.82¢ |
4-steps | 2L + 2s 2L + m + s |
81/64, 407.82¢ 4/3, 498.04¢ |
12\32, 450.00¢ 13\32, 487.50¢ |
12\34, 423.53¢ 14\34, 494.12¢ |
5-steps | 2L + m + 2s 2L + 2m + s 3L + 2s 3L + m + s |
27/20, 519.55¢ 64/45, 609.78¢ 45/32, 590.22¢ 40/27, 680.45¢ |
14\32, 525.00¢ 15\32, 562.50¢ 17\32, 637.50¢ 18\32, 675.00¢ |
15\34, 529.41¢ 17\34, 600.00¢ 17\34, 600.00¢ 19\34, 670.59¢ |
6-steps | 3L + m + 2s 3L + 2m + s |
3/2, 701.96¢ 128/81, 792.18¢ |
19\32, 712.50¢ 20\32, 750.00¢ |
20\34, 705.88¢ 22\34, 776.47¢ |
7-steps | 3L + m + 3s 3L + 2m + 2s 4L + m + 2s 4L + 2m + s |
243/160, 723.46¢ 8/5, 813.69¢ 5/3, 884.36¢ 1280/729, 974.58¢ |
20\32, 750.00¢ 21\32, 787.50¢ 24\32, 900.00¢ 25\32, 937.50¢ |
21\34, 741.18¢ 23\34, 811.77¢ 25\34, 882.35¢ 27\34, 952.94¢ |
8-steps | 4L + m + 3s 4L + 2m + 2s |
27/16, 905.87¢ 16/9, 996.09¢ |
25\32, 937.50¢ 26\32, 975.00¢ |
26\34, 917.65¢ 28\34, 988.24¢ |
9-steps | 5L + 2m + s 5L + m + 2s 4L + 2m + 2s |
9/5, 1017.60¢ 15/8, 1088.27¢ 160/81, 1178.49¢ |
27\32, 1012.50¢ 30\32, 1125.00¢ 31\32, 1162.50¢ |
29\34, 1023.53¢ 31\34, 1094.12¢ 33\34, 1164.71¢ |
The octave in blackdye can be called the "perfect 10-step" in TAMNAMS.
Modes
Cyclic order
The modes arranged in cyclic order: (Note: The mode names are based on the 5-limit JI interpretation; modes in a less JI-like tuning may differ greatly from what these names suggest.)
Pattern | Name |
---|---|
sLmLsLmLsL | Blackdye Acute Aeolian |
LmLsLmLsLs | Blackdye Grave Aeolian |
mLsLmLsLsL | Blackdye Locrian |
LsLmLsLsLm | Blackdye Ionian |
sLmLsLsLmL | Blackdye Acute Dorian |
LmLsLsLmLs | Blackdye Grave Dorian |
mLsLsLmLsL | Blackdye Phrygian |
LsLsLmLsLm | Blackdye Lydian |
sLsLmLsLmL | Blackdye Acute Mixo |
LsLmLsLmLs | Blackdye Grave Mixo |
By generator chain
Because blackdye has two chains of 5 fifth generators, it has two chains of 5 modes each separated by one generator:
- Lydian > Ionian > Grave Mixo > Grave Dorian > Grave Aeolian
- Acute Mixo > Acute Dorian > Acute Aeolian > Phrygian > Locrian
Properties
- Assuming positive step sizes, the fifth (3L + m + 2s) of blackdye must be a diatonic fifth.
- Assuming positive step sizes, the fifth is:
- sharper than 700¢, if L + 3s > 2m
- equal to 700¢, if L + 3s = 2m
- flatter than 700¢, if L + 3s < 2m
Tunings
Tuning | L:m:s | Degrees of the mode sLmLsLmLsL | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||
5-Limit Interpretation | 81/80 | 9/8 | 6/5 | 4/3 | 27/20 | 3/2 | 8/5 | 16/9 | 9/5 | |
JI | 8.481:5.195:1 | 21.506 | 203.910 | 315.641 | 498.045 | 519.551 | 701.955 | 813.686 | 996.090 | 1017.596 |
22edo | 3:2:1 | 54.545 | 218.182 | 327.273 | 490.909 | 545.455 | 709.090 | 818.182 | 981.818 | 1036.364 |
27edo | 4:2:1 | 44.444 | 222.222 | 311.111 | 488.889 | 533.333 | 711.111 | 800.000 | 977.778 | 1022.222 |
29edo | 4:3:1 | 41.379 | 206.897 | 331.034 | 496.551 | 537.931 | 703.448 | 827.586 | 993.103 | 1034.483 |
32edo | 4:3:2 | 75.000 | 225.000 | 337.500 | 487.500 | 562.500 | 712.500 | 825.000 | 975.000 | 1050.000 |
32edo | 5:2:1 | 37.500 | 225.000 | 300.000 | 487.500 | 525.000 | 712.500 | 787.500 | 975.000 | 1012.500 |
34edo | 5:3:1 | 35.294 | 211.765 | 317.647 | 494.117 | 529.412 | 705.882 | 811.764 | 988.235 | 1023.529 |
37edo | 5:3:2 | 64.865 | 227.027 | 324.324 | 486.486 | 551.351 | 713.514 | 810.810 | 972.973 | 1037.838 |
36edo | 5:4:1 | 33.333 | 200.000 | 333.333 | 500.000 | 533.333 | 700.000 | 833.333 | 1000.000 | 1033.333 |
39edo | 5:4:2 | 61.538 | 215.385 | 338.462 | 492.308 | 553.846 | 707.692 | 830.769 | 984.615 | 1046.154 |
42edo | 5:4:3 | 85.714 | 228.571 | 342.857 | 485.714 | 571.429 | 714.286 | 828.571 | 971.429 | 1057.143 |
39edo | 5:4:2 | 61.538 | 215.385 | 338.462 | 492.308 | 553.846 | 707.692 | 830.769 | 984.615 | 1046.154 |
37edo | 6:2:1 | 32.432 | 227.027 | 291.892 | 486.486 | 518.919 | 713.514 | 778.378 | 972.973 | 1005.405 |
39edo | 6:3:1 | 30.769 | 215.385 | 307.692 | 492.308 | 523.077 | 707.692 | 800.000 | 984.615 | 1015.384 |
42edo | 6:3:2 | 57.143 | 228.571 | 314.286 | 485.714 | 542.857 | 714.286 | 800.000 | 971.429 | 1028.571 |
41edo | 6:4:1 | 29.268 | 204.878 | 321.951 | 497.561 | 526.829 | 702.439 | 819.512 | 995.122 | 1024.390 |
47edo | 6:4:3 | 76.596 | 229.787 | 331.915 | 485.106 | 561.702 | 702.439 | 714.894 | 970.213 | 1046.809 |
42edo | 7:2:1 | 28.571 | 228.571 | 285.714 | 485.714 | 514.286 | 714.286 | 771.429 | 971.429 | 1000.000 |
44edo | 7:3:1 | 27.273 | 218.182 | 300.000 | 490.909 | 518.182 | 709.091 | 790.901 | 981.818 | 1009.091 |
46edo | 7:4:1 | 26.087 | 208.696 | 313.043 | 495.652 | 521.739 | 704.348 | 808.696 | 991.304 | 1017.391 |
53edo | 8:5:1 | 22.642 | 203.773 | 316.981 | 498.113 | 520.755 | 701.887 | 815.094 | 996.226 | 1018.868 |
Example
An example in Blackdye Acute Aeolian, sLmLsLmLsL (score)
22edo (L:m:s = 3:2:1)
27edo (L:m:s = 4:2:1)
29edo (L:m:s = 4:3:1)
32edo (L:m:s = 5:2:1)
34edo (L:m:s = 5:3:1)
36edo (L:m:s = 5:4:1)
Scala files
Music
- squid (2024) by Abnormality