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'''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 [[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. 

{| class="wikitable right-2 right-3 right-4 right-5"
|+ Interval sizes in blackdye
|-
! | Interval class
! Sizes
! [[5-limit]] JI
! [[32edo]] (L:m:s = 5:2:1)
! [[34edo]] (L:m:s = 5:3:1)
|-
!|[[TAMNAMS|1-steps]]
| s<br/>m<br/>L 
| 81/80, 21.51¢<br/>16/15, 111.73¢<br/>10/9, 182.40¢
| 1\32, 37.50¢<br/>2\32, 75.00¢<br/>5\32, 187.50¢
| 1\34, 35.29¢<br/>3\34, 105.88¢<br/>5\34, 176.47¢
|-
!|[[TAMNAMS|2-steps]]
| L + s<br/>L + m 
| 9/8, 203.91¢<br/>32/27, 294.14¢
| 6\32, 225.00¢<br/>7\32, 262.50¢
| 6\34, 211.77¢<br/>8\34, 282.35¢
|- 
!|[[TAMNAMS|3-steps]]
| 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¢
| 7\32, 262.50¢<br/>8\32, 300.00¢<br/>11\32, 412.50¢<br/>12\32, 450.00¢
| 7\34, 247.06¢<br/>9\34, 317.65¢<br/>11\34, 388.24¢<br/>13\34, 458.82¢
|- 
!|[[TAMNAMS|4-steps]]
| 2L + 2s<br/>2L + m + s
| 81/64, 407.82¢<br/>4/3, 498.04¢
| 12\32, 450.00¢<br/>13\32, 487.50¢
| 12\34, 423.53¢<br/>14\34, 494.12¢
|-
!|[[TAMNAMS|5-steps]]
| 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¢
| 14\32, 525.00¢<br/>15\32, 562.50¢<br/>17\32, 637.50¢<br/>18\32, 675.00¢
| 15\34, 529.41¢<br/>17\34, 600.00¢<br/>17\34, 600.00¢<br/>19\34, 670.59¢
|-
!|[[TAMNAMS|6-steps]]
| 3L + m + 2s<br/>3L + 2m + s
| 3/2, 701.96¢<br/>128/81, 792.18¢
| 19\32, 712.50¢<br/>20\32, 750.00¢
| 20\34, 705.88¢<br/>22\34, 776.47¢
|- 
!|[[TAMNAMS|7-steps]]
| 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¢
| 20\32, 750.00¢<br/>21\32, 787.50¢<br/>24\32, 900.00¢<br/>25\32, 937.50¢
| 21\34, 741.18¢<br/>23\34, 811.77¢<br/>25\34, 882.35¢<br/>27\34, 952.94¢
|- 
!|[[TAMNAMS|8-steps]]
| 4L + m + 3s<br/>4L + 2m + 2s
| 27/16, 905.87¢<br/>16/9, 996.09¢
| 25\32, 937.50¢<br/>26\32, 975.00¢
| 26\34, 917.65¢<br/>28\34, 988.24¢
|-
!|[[TAMNAMS|9-steps]]
| 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¢
| 27\32, 1012.50¢<br/>30\32, 1125.00¢<br/>31\32, 1162.50¢
| 29\34, 1023.53¢<br/>31\34, 1094.12¢<br/>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.)
{| class="wikitable"
|-
! style="text-align:center;" |Pattern
! style="text-align:center;" |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.<!--
#* (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 + 3s > 2m
#* equal to 700¢, if L + 3s = 2m
#* flatter than 700¢, if L + 3s < 2m

== Tunings ==
{| class="wikitable sortable right-2 right-3 right-4 right-5 right-6 right-7 right-8 right-9 right-10"
|+ Blackdye tunings
! rowspan="2" |Tuning
! rowspan="2" |L:m:s
! colspan="9" |Degrees of the mode sLmLsLmLsL
|-
!1!!2!!3!!4!!5!!6!!7!!8!!9
|-
| colspan=2 align=center | 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 ([[: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 29edo.mp3]] [[29edo]] (L:m:s = 4:3:1)

[[File:Blackdye Aeolian Example 32edo 5_2_1.mp3]] [[32edo]] (L:m:s = 5:2:1)

[[File:Blackdye Aeolian Example 34edo.mp3]] [[34edo]] (L:m:s = 5:3:1)

[[File:Blackdye Aeolian Example 36edo.mp3]] [[36edo]] (L:m:s = 5:4:1)

A fugue exposition:

[[File:34edo blackdye fugue expo.mp3]]

== 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 ==
* [[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:GO scales]]
[[Category:Rank-3 scales]]
[[Category:Step-nested scales]]
[[Category:Listen]]
[[Category:bipentatonic scales]]
[[Category:Flought scales]]
[[Category:Aberrismic theory]]