Trackdye

Trackdye, short for "tract-aberracted diatonic", is a term for the seven aberrismic ternary scales of MOS substitution type 8s(5L2m), which 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. It may be abbreviated to dia8s or just 8s.

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

Tract-aberrated diatonic generator splotches

The Seven Forms of Trackdye
(MOS if L=m and if s=0)

Diatonic Mode		Tract-Aberrated	2.3.5 JI
Lydian	LLLsLLs	sLsLsLsMsLsLsMs	81/80 10/9 9/8 100/81 5/4 1000/729 25/18 40/27 3/2 400/243 5/3 4000/2187 50/27 160/81 2/1
Ionian	LLsLLLs	sLsLsMsLsLsLsMs	81/80 10/9 9/8 100/81 5/4 4/3 27/20 40/27 3/2 400/243 5/3 4000/2187 50/27 160/81 2/1
Mixolydian (8s R)	LLsLLsL	sLsLsMsLsLsMsLs	81/80 10/9 9/8 100/81 5/4 4/3 27/20 40/27 3/2 400/243 5/3 16/9 9/5 160/81 2/1
Dorian	LsLLLsL	sLsMsLsLsLsMsLs	81/80 10/9 9/8 6/5 243/200 4/3 27/20 40/27 3/2 400/243 5/3 16/9 9/5 160/81 2/1
Aeolian (8s L)	LsLLsLL	sLsMsLsLsMsLsLs	81/80 10/9 9/8 6/5 243/200 4/3 27/20 40/27 3/2 8/5 81/50 16/9 9/5 160/81 2/1
Phrygian	sLLLsLL	sMsLsLsLsMsLsLs	81/80 27/25 2187/2000 6/5 243/200 4/3 27/20 40/27 3/2 8/5 81/50 16/9 9/5 160/81 2/1
Locrian	sLLsLLL	sMsLsLsMsLsLsLs	81/80 27/25 2187/2000 6/5 243/200 4/3 27/20 36/25 729/500 8/5 81/50 16/9 9/5 160/81 2/1

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.

Tract-aberrated Aeolian and Mixolydian are special because they are the only forms to contain blackdye as a subset, and to have four chains of perfect fifth generators rather than 5. They are assigned to the L and R aberrismic chirality abbreviations.

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 Aeolian

Pental					Compromise		Straddle-3
2.3.5 JI		Sharp Hemififth-Based
Ratio	Cents	27P (3:2:1)	27P+34P = 61P (7:5:2)	34P (4:3:1)	61P+37cc = 98bcc (12:7:3)	39P (5:3:1)	37cc (5:2:1)	67bcccc (9:3:2)	61cc (9:4:1)
81/80	21.5	44.4	39.3	35.3	36.7	30.8	32.4	35.8	19.7
10/9	182.4	177.8	177.0	176.5	183.7	184.6	194.6	197.0	196.7
9/8	203.9	222.2	216.4	211.8	220.4	215.4	227.0	232.8	216.4
6/5	315.6	311.1	314.8	317.6	306.1	307.7	291.9	286.6	295.1
243/200	337.1	355.6	354.1	352.9	342.9	338.5	324.3	322.4	314.8
4/3	498.0	488.9	491.8	494.1	489.8	492.3	486.5	483.6	491.8
27/20	519.6	533.3	531.1	529.4	526.5	523.1	518.9	519.4	511.5
40/27	680.4	666.7	668.9	670.6	673.5	676.9	681.1	680.6	688.5
3/2	702.0	711.1	708.2	705.9	710.2	707.7	713.5	716.4	708.2
8/5	813.7	800.0	806.6	811.8	795.9	800.0	778.4	770.1	786.9
81/50	835.2	844.4	845.9	847.1	832.7	830.8	810.8	806.0	806.6
16/9	996.1	977.8	983.6	988.2	979.6	984.6	973.0	967.2	983.6
9/5	1017.6	1022.2	1023.0	1023.5	1016.3	1015.4	1005.4	1003.0	1003.3
160/81	1178.5	1155.6	1160.7	1164.7	1163.3	1169.2	1167.6	1164.2	1180.3
2/1	1200.0	1200.0	1200.0	1200.0	1200.0	1200.0	1200.0	1200.0	1200.0

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 supraminor second.

Which supraminor second is generated this way varies with tuning category:

• ms: p -3g/2 = (3p -5g +a) + a ⇒ -4p +7g -4a = 0
• sms: p -3g/2 = (3p -5g +a) + 2a ⇒ -4p +7g -6a = 0

Range definitions

The simplest way to define the two ranges is:

• Pental (including JI): sLsLs ≈ 5/4
• Straddle-3: sLsL ≈ 5/4

It may be useful to compare the two side-by-side in one tuning. Equating the representations of 5/4 with a as the pental aberrisma and b as the straddle-3 aberrisma:

2(-p +2g -2a) + 3a = 2(-p +2g -2b) + 2b ⇒ a = 2b

The first edo to do this while keeping L≠m≠s is 42 with 4:3:2 and 6:2:1, which are similar enough to 27edo (3:2:1 Augmented) and 37edo (5:2:1 Ultrapyth), although not ideal. 61edo offers 7:5:2 and 9:4:1, which is better.

Equivalent intervals

The simplest common equivalences that lead to thirds being equated are:

• m = ss (3:2:1, 5:2:1)
  • Subminor thirds Lsm = sLss
• L = ms (3:2:1, 4:3:1)
  • Minor thirds LsL = sLsm
• L = sms (5:3:1)
  • Supraminor thirds LsL = sLsms
• L = smss (5:2:1)
  • This one occurs less often directly because there has to be an m step adjacent to the only ss interval.

For example, trackdye features two intervals that are usually tuned close to a supraminor third: LsL and either slsm or 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 above 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.

Doing the same with LsL and sLsm gives -4p +7g -4a = 0, the same equation as the tunings with a sharp hemififth.

Intervals

Intervals of 98bcc Tract-Aberrated Dorian

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