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¢ | ||
| Line 37: | Line 35: | ||
| 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¢ | ||
| Line 49: | Line 47: | ||
| 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¢ | ||
| Line 73: | Line 71: | ||
| 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¢ | ||
| Line 82: | Line 80: | ||
== 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" | ||
| Line 88: | Line 87: | ||
! 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 == | ||
| Line 131: | Line 134: | ||
|+ 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 140: | Line 143: | ||
|- | |- | ||
| JI || 8.481:5.195:1 | | JI || 8.481:5.195:1 | ||
|| 21. | || 21.506 || 203.910 || 315.641 || 498.045 || 519.551 || 701.955 || 813.686 || 996.090 || 1017.596 | ||
|- | |- | ||
| 22edo || 3:2:1 | | 22edo || 3:2:1 | ||
|| 54. | || 54.545 || 218.182 || 327.273 || 490.909 || 545.455 || 709.090 || 818.182 || 981.818 || 1036.364 | ||
|- | |- | ||
| 27edo || 4:2:1 | | 27edo || 4:2:1 | ||
|| 44. | || 44.444 || 222.222 || 311.111 || 488.889 || 533.333 || 711.111 || 800.000 || 977.778 || 1022.222 | ||
|- | |- | ||
| 29edo || 4:3:1 | | 29edo || 4:3:1 | ||
|| 41. | || 41.379 || 206.897 || 331.034 || 496.551 || 537.931 || 703.448 || 827.586 || 993.103 || 1034.483 | ||
|- | |- | ||
| 32edo || 4:3:2 | | 32edo || 4:3:2 | ||
| Line 158: | Line 161: | ||
|- | |- | ||
| 34edo || 5:3:1 | | 34edo || 5:3:1 | ||
|| 35. | || 35.294 || 211.765 || 317.647 || 494.117 || 529.412 || 705.882 || 811.764 || 988.235 || 1023.529 | ||
|- | |- | ||
| 37edo || 5:3:2 | | 37edo || 5:3:2 | ||
|| 64. | || 64.865 || 227.027 || 324.324 || 486.486 || 551.351 || 713.514 || 810.810 || 972.973 || 1037.838 | ||
|- | |- | ||
| 36edo || 5:4:1 | | 36edo || 5:4:1 | ||
|| 33. | || 33.333 || 200.000 || 333.333 || 500.000 || 533.333 || 700.000 || 833.333 || 1000.000 || 1033.333 | ||
|- | |- | ||
| 39edo || 5:4:2 | | 39edo || 5:4:2 | ||
|| 61. | || 61.538 || 215.385 || 338.462 || 492.308 || 553.846 || 707.692 || 830.769 || 984.615 || 1046.154 | ||
|- | |- | ||
| 42edo || 5:4:3 | | 42edo || 5:4:3 | ||
|| 85. | || 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 | | 37edo || 6:2:1 | ||
|| 32. | || 32.432 || 227.027 || 291.892 || 486.486 || 518.919 || 713.514 || 778.378 || 972.973 || 1005.405 | ||
|- | |- | ||
| 39edo || 6:3:1 | | 39edo || 6:3:1 | ||
|| 30. | || 30.769 || 215.385 || 307.692 || 492.308 || 523.077 || 707.692 || 800.000 || 984.615 || 1015.384 | ||
|- | |- | ||
| 42edo || 6:3:2 | | 42edo || 6:3:2 | ||
|| 57. | || 57.143 || 228.571 || 314.286 || 485.714 || 542.857 || 714.286 || 800.000 || 971.429 || 1028.571 | ||
|- | |- | ||
| 41edo || 6:4:1 | | 41edo || 6:4:1 | ||
|| 29. | || 29.268 || 204.878 || 321.951 || 497.561 || 526.829 || 702.439 || 819.512 || 995.122 || 1024.390 | ||
|- | |- | ||
| 47edo || 6:4:3 | | 47edo || 6:4:3 | ||
|| 76. | || 76.596 || 229.787 || 331.915 || 485.106 || 561.702 || 702.439 || 714.894 || 970.213 || 1046.809 | ||
|- | |- | ||
| 42edo || 7:2:1 | | 42edo || 7:2:1 | ||
|| 28. | || 28.571 || 228.571 || 285.714 || 485.714 || 514.286 || 714.286 || 771.429 || 971.429 || 1000.000 | ||
|- | |- | ||
| 44edo || 7:3:1 | | 44edo || 7:3:1 | ||
|| 27. | || 27.273 || 218.182 || 300.000 || 490.909 || 518.182 || 709.091 || 790.901 || 981.818 || 1009.091 | ||
|- | |- | ||
| 46edo || 7:4:1 | | 46edo || 7:4:1 | ||
|| 26. | || 26.087 || 208.696 || 313.043 || 495.652 || 521.739 || 704.348 || 808.696 || 991.304 || 1017.391 | ||
|- | |- | ||
| 53edo || 8:5:1 | | 53edo || 8:5:1 | ||
| Line 285: | 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 | [[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]] | |||