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	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

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 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 "supraminor third".