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'''Trackdye''', short for "[[Stretching and compression|tract-aberracted]] diatonic", is a term for seven [[Aberrisma|aberrismic]] ternary scales of the form 5L 2m 8s that may be constructed by compressing a diatonic scale and duplicating it offset by the amount it was compressed by. It is a superset of [[blackdye]] and [[pinedye]], which makes it useful as both a versatile tempered 5-limit scale and a [[straddle-3]] scale depending on the tuning.
'''Trackdye''', short for "[[Stretching and compression|tract-aberracted]] diatonic", is a term for seven [[Aberrisma|aberrismic]] ternary scales of the form 5L 2m 8s that may be constructed by compressing a diatonic scale and duplicating it offset by the amount it was compressed by. It is a superset of [[blackdye]] and [[pinedye]], which makes it useful as both a versatile tempered 5-limit scale and a [[straddle-3]] scale depending on the tuning.


Its steps may be expressed in generator-offset notation, with p as the period, g as the generator, and a as the aberrismic offset. As in most aberrismic scales, the offset is the smallest step in the scale. The equation for 5L 2m 8s trackdye is 5(-p +2g -2a) + 2(3p -5g +a) + 8a = p.
Its steps may be expressed in generator-offset notation, with p as the octave period, g as the perfect fifth generator, and a as the aberrismic offset. As in most aberrismic scales, the offset is the smallest step in the scale. The equation for 5L 2m 8s trackdye is 5(-p +2g -2a) + 2(3p -5g +a) + 8a = p.
{| class="wikitable"
{| class="wikitable"
|+Monotone-MOS Analysis
|+Monotone-MOS Analysis
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* It fills the space between 6/5 and 5/4 with as few distinct intervals as possible, allowing for smaller edos.
* It fills the space between 6/5 and 5/4 with as few distinct intervals as possible, allowing for smaller edos.
* As a generator, 2 of them makes the sharp fifth but -3 usually makes the m+s supraminor second.
* As a generator, 2 of them makes the sharp fifth but -3 usually makes the m+s supraminor second.
Suppose the supraminor second generated this way is equal to m+s: p -3g/2 = (3p -5g +a) + a -> -4p +7g -4a = 0.


=== Near-supraminor thirds ===
=== Near-supraminor thirds ===
Trackdye features two intervals that are usually tuned close to a supraminor third: LsL and sLsms. They can be expressed with equations for generator-offset notation:
Trackdye features two intervals that are usually tuned close to a supraminor third: LsL and sLsms. They can be expressed with equations for generator-offset notation:
* LsL: -2p +4g -3a
* LsL: 2(-p +2g -2a) + a = -2p +4g -3a
* sLsms: 2p -3g +2a
* sLsms: (-p +2g -2a) + (3p -5g +a) + 3a = 2p -3g +2a


Equating the supraminor thirds with units of cents gives -2p +4g -3a = 2p -3g +2a when p = 1200¢, simplifying to -4800¢ +7g -5a = 0, so the diatonic augmented unison is equal to 5 aberrismas. The line goes from 0¢, 4\7 to 48¢, 3\5.
Equating the supraminor thirds with units of cents gives -2p +4g -3a = 2p -3g +2a when p = 1200¢, simplifying to -4800¢ +7g -5a = 0, so the diatonic augmented unison is equal to 5 aberrismas. The line goes from 0¢, 4\7 to 48¢, 3\5.
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|1200
|1200
|}
|}
98bcc was selected because it's approximately the average of the tunings in the tuning range table, and does not equate the two types of "supraminor third".
98bcc was selected because it's approximately the average of the tunings in the tuning range table, and does not equate the two types of near-supraminor third.

Revision as of 05:03, 9 November 2026

Trackdye, short for "tract-aberracted diatonic", is a term for seven aberrismic ternary scales of the form 5L 2m 8s that may be constructed by compressing a diatonic scale and duplicating it offset by the amount it was compressed by. It is a superset of blackdye and pinedye, which makes it useful as both a versatile tempered 5-limit scale and a straddle-3 scale depending on the tuning.

Its steps may be expressed in generator-offset notation, with p as the octave period, g as the perfect fifth generator, and a as the aberrismic offset. As in most aberrismic scales, the offset is the smallest step in the scale. The equation for 5L 2m 8s trackdye is 5(-p +2g -2a) + 2(3p -5g +a) + 8a = p.

Monotone-MOS Analysis
Subcondition MOS Gen Range Name
L=m 7L 8s 2\15 - 1\7 Porcupine
m=s 5L 10s 0\5 - 1\15 Hard Blackwood
s=0 5L 2s 2\5 - 3\7 Diatonic

This shows that trackdye tunings tend to have a sharp diatonic fifth.

Step orders

The Seven Forms of Trackdye
(MOS if L=m and if s=0)
Diatonic Mode Tract-Aberrated
Lydian LLLsLLs sLsLsLsMsLsLsMs
Ionian LLsLLLs sLsLsMsLsLsLsMs
Mixolydian LLsLLsL sLsLsMsLsLsMsLs
Dorian LsLLLsL sLsMsLsLsLsMsLs
Aeolian LsLLsLL sLsMsLsLsMsLsLs
Phrygian sLLLsLL sMsLsLsLsMsLsLs
Locrian sLLsLLL sMsLsLsMsLsLsLs

Each of these forms has its own set of 15 modes. The modes given in this table are all the only one with s as the first and last step, which is the standard form for tract-aberrated scales. M is capitalized for visual distinction.

There are also four step orders that are MOS if m=s and if s=0: sLsmLssLssLsmLs, sLsmLssLssLmsLs, sLmsLssLssLsmLs, and sLmsLssLssLmsLs, shown with ssLss in the middle. They are not named nor discussed in this article. There is no step order that satisfies all three monotone-MOS subconditions.

Tuning range

g_
I split trackdye into two categories based on how it accesses LCJI intervals:
  • Pental
    • ~2.3.5.(7.11.13.17.19)
    • Pental intervals are one aberrisma from diatonic
    • A superset of pental blackdye/pinedye which functions predictably
  • Straddle-3
    • ~2.<<3.>3.5.7.11.13.(17.19)
    • Pental intervals are two aberrismas from diatonic
    • Approximates a roughly basic tuning of diatonic, but warped to access pental and septal intervals in either direction
Selected Tunings of Tract-Aberrated Dorian
Pental Compromise Straddle-3
2.3.5 JI Hemififth-Based
Ratio Cents 27P 27P+34P = 61P 34P 61P+37cc = 98bcc 39P 37cc 67bcccc
81/80 21.506 44.444 39.344 35.294 36.735 30.769 32.432 35.821
10/9 182.404 177.778 177.049 176.471 183.673 184.615 194.595 197.015
9/8 203.910 222.222 216.393 211.765 220.408 215.385 227.027 232.836
6/5 315.641 311.111 314.754 317.647 306.122 307.692 291.892 286.567
243/200 337.148 355.556 354.098 352.941 342.857 338.462 324.324 322.388
4/3 498.045 488.889 491.803 494.118 489.796 492.308 486.486 483.582
27/20 519.551 533.333 531.148 529.412 526.531 523.077 518.919 519.403
40/27 680.449 666.667 668.852 670.588 673.469 676.923 681.081 680.597
3/2 701.955 711.111 708.197 705.882 710.204 707.692 713.514 716.418
400/243 862.852 844.444 845.902 847.059 857.143 861.538 875.676 877.612
5/3 884.359 888.889 885.246 882.353 893.878 892.308 908.108 913.433
16/9 996.090 977.778 983.607 988.235 979.592 984.615 972.973 967.164
9/5 1017.596 1022.222 1022.951 1023.529 1016.327 1015.385 1005.405 1002.985
160/81 1178.494 1155.556 1160.656 1164.706 1163.265 1169.231 1167.568 1164.179
2/1 1200 1200 1200 1200 1200 1200 1200 1200

The sharp fifth is the generator in generator-offset notation. Edo tunings 27, 34, 37, and 67 all feature a significant number of intervals that split it in half, and all of these edos are 10n+7 except for 34 which is 10+7*2. These imply a very sharp and sharp diatonic fifth respectively. Splitting that into a sharp hemififth is useful for two reasons:

  • It fills the space between 6/5 and 5/4 with as few distinct intervals as possible, allowing for smaller edos.
  • As a generator, 2 of them makes the sharp fifth but -3 usually makes the m+s supraminor second.

Suppose the supraminor second generated this way is equal to m+s: p -3g/2 = (3p -5g +a) + a -> -4p +7g -4a = 0.

Near-supraminor thirds

Trackdye features two intervals that are usually tuned close to a supraminor third: LsL and sLsms. They can be expressed with equations for generator-offset notation:

  • LsL: 2(-p +2g -2a) + a = -2p +4g -3a
  • sLsms: (-p +2g -2a) + (3p -5g +a) + 3a = 2p -3g +2a

Equating the supraminor thirds with units of cents gives -2p +4g -3a = 2p -3g +2a when p = 1200¢, simplifying to -4800¢ +7g -5a = 0, so the diatonic augmented unison is equal to 5 aberrismas. The line goes from 0¢, 4\7 to 48¢, 3\5.

Intervals

Intervals of 98bcc Tract-Aberrated Dorian
On↓ #→ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 0 37 184 220 306 343 490 527 674 710 857 894 980 1016 1163 1200
37 0 147 184 269 306 453 490 637 674 820 857 943 980 1127 1163 1200
184 0 37 122 159 306 343 490 527 674 710 796 833 980 1016 1053 1200
220 0 86 122 269 306 453 490 637 674 759 796 943 980 1016 1163 1200
306 0 37 184 220 367 404 551 588 674 710 857 894 931 1078 1114 1200
343 0 147 184 331 367 514 551 637 674 820 857 894 1041 1078 1163 1200
490 0 37 184 220 367 404 490 527 674 710 747 894 931 1016 1053 1200
527 0 147 184 331 367 453 490 637 674 710 857 894 980 1016 1163 1200
674 0 37 184 220 306 343 490 527 563 710 747 833 869 1016 1053 1200
710 0 147 184 269 306 453 490 527 674 710 796 833 980 1016 1163 1200
857 0 37 122 159 306 343 380 527 563 649 686 833 869 1016 1053 1200
894 0 86 122 269 306 343 490 527 612 649 796 833 980 1016 1163 1200
980 0 37 184 220 257 404 441 527 563 710 747 894 931 1078 1114 1200
1016 0 147 184 220 367 404 490 527 674 710 857 894 1041 1078 1163 1200
1163 0 37 74 220 257 343 380 527 563 710 747 894 931 1016 1053 1200

98bcc was selected because it's approximately the average of the tunings in the tuning range table, and does not equate the two types of near-supraminor third.

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