Trackdye

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 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.

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

Selected Tunings of Tract-Aberrated Dorian

2.3.5 JI		~2.3.5.(7).13 (Hemififth)			Compromise	~2.<<3.>3.5.7.11.13 (Straddle)
Ratio	Cents	27P	61P	34P	39P	37cc	67bcccc
81/80	21.506	44.444	39.344	35.294	30.769	32.432	35.821
10/9	182.404	177.778	177.049	176.471	184.615	194.595	197.015
9/8	203.910	222.222	216.393	211.765	215.385	227.027	232.836
6/5	315.641	311.111	314.754	317.647	307.692	291.892	286.567
243/200	337.148	355.556	354.098	352.941	338.462	324.324	322.388
4/3	498.045	488.889	491.803	494.118	492.308	486.486	483.582
27/20	519.551	533.333	531.148	529.412	523.077	518.919	519.403
40/27	680.449	666.667	668.852	670.588	676.923	681.081	680.597
3/2	701.955	711.111	708.197	705.882	707.692	713.514	716.418
400/243	862.852	844.444	845.902	847.059	861.538	875.676	877.612
5/3	884.359	888.889	885.246	882.353	892.308	908.108	913.433
16/9	996.090	977.778	983.607	988.235	984.615	972.973	967.164
9/5	1017.596	1022.222	1022.951	1023.529	1015.385	1005.405	1002.985
160/81	1178.494	1155.556	1160.656	1164.706	1169.231	1167.568	1164.179
2/1	1200	1200	1200	1200.000	1200.000	1200.000	1200.000

Supraminor thirds

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

• LsL: -2p +4g -3a
• sLsms: 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 37cc Tract-Aberrated Dorian

	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	-15
0	0	32	195	227	292	324	487	519	681	714	876	908	973	1005	1168	1200
32	0	162	195	260	292	454	487	649	681	843	876	941	973	1135	1168	1200
195	0	32	97	130	292	324	487	519	681	714	778	811	973	1005	1038	1200
227	0	65	97	260	292	454	487	649	681	746	778	941	973	1005	1168	1200
292	0	32	195	227	389	422	584	616	681	714	876	908	941	1103	1135	1200
324	0	162	195	357	389	551	584	649	681	843	876	908	1070	1103	1168	1200
487	0	32	195	227	389	422	487	519	681	714	746	908	941	1005	1038	1200
519	0	162	195	357	389	454	487	649	681	714	876	908	973	1005	1168	1200
681	0	32	195	227	292	324	487	519	551	714	746	811	843	1005	1038	1200
714	0	162	195	260	292	454	487	519	681	714	778	811	973	1005	1168	1200
876	0	32	97	130	292	324	357	519	551	616	649	811	843	1005	1038	1200
908	0	65	97	260	292	324	487	519	584	616	778	811	973	1005	1168	1200
973	0	32	195	227	260	422	454	519	551	714	746	908	941	1103	1135	1200
1005	0	162	195	227	389	422	487	519	681	714	876	908	1070	1103	1168	1200
1168	0	32	65	227	260	324	357	519	551	714	746	908	941	1005	1038	1200

37edo was selected because it distinguishes the sLsms 324¢ and LsL 357¢ intervals, which are equal in many tunings.