User:BudjarnLambeth/Substitute harmonic

A substitute harmonic^{[idiosyncratic term]} is a more complex harmonic which is used to substitute for a simpler one.

For example, you could substitute the 3rd harmonic out for the 769th harmonic, whose octave reduction is very close to the 3rd harmonic's. By doing this, you could convert a 2.3.5 subgroup temperament into a 2.5.769 subgroup temperament. Or, you could convert a 3.5.7 combination product set into a 5.7.769 combination product set.

You could also substitute a simpler harmonic n in a dual-n temperament for two more complex harmonics, to make a dual-substitute-n temperament^{[idiosyncratic term]}. For example, you could convert a 2.3-.3+.5 subgroup temperament into a 2.5.767.769 subgroup temperament.

List of substitute harmonics

Each harmonic is given in octave-reduced cents. This list is not exhaustive.

Substitutes for the 2nd harmonic (1200)

• the 1017th harmonic (~1188)
• the 509th harmonic (~1190)
• the 1019th harmonic (~1192)
• the 255th harmonic (~1193)
• the 1021st harmonic (~1195)
• the 511th harmonic (~1197)
• the 1023rd harmonic (~1198)
• the 1025th harmonic (~2)
• the 513th harmonic (~3)
• the 1027th harmonic (~5)
• the 257th harmonic (~7)
• the 1029th harmonic (~8)
• the 515th harmonic (~10)
• the 1031st harmonic (~12)

Substitutes for the 3rd harmonic (~702)

• the 381st harmonic (~688)
• the 763rd harmonic (~691)
• the 191st harmonic (~693)
• the 765th harmonic (~695)
• the 383rd harmonic (~697)
• the 767th harmonic (~700)
• the 769th harmonic (~704)
• the 385th harmonic (~706)
• the 771st harmonic (~709)
• the 193rd harmonic (~711)
• the 773rd harmonic (~713)
• the 387th harmonic (~715)

Substitutes for the 5th harmonic (~386)

• the 317th harmonic (~370)
• the 635th harmonic (~373)
• the 159th harmonic (~375)
• the 637th harmonic (~378)
• the 319th harmonic (~381)
• the 639th harmonic (~384)
• the 641st harmonic (~389)
• the 321st harmonic (~392)
• the 643rd harmonic (~394)
• the 161st harmonic (~397)
• the 645th harmonic (~400)
• the 323rd harmonic (~402)

Substitutes for the 7th harmonic (~969)

• the 111th harmonic (~953)
• the 889th harmonic (~955)
• the 445th harmonic (~957)
• the 891st harmonic (~959)
• the 223rd harmonic (~961)
• the 893rd harmonic (~963)
• the 447th harmonic (~965)
• the 895th harmonic (~967)
• the 897th harmonic (~971)
• the 449th harmonic (~973)
• the 899th harmonic (~975)
• the 225th harmonic (~977)
• the 901st harmonic (~978)
• the 451st harmonic (~980)
• the 903rd harmonic (~982)
• the 113th harmonic (~984)

Substitutes for the 11th harmonic (~551)

• the 349th harmonic (~537)
• the 699th harmonic (~539)
• the 175th harmonic (~541)
• the 701st harmonic (~544)
• the 351st harmonic (~546)
• the 703rd harmonic (~549)
• the 705th harmonic (~554)
• the 353rd harmonic (~556)
• the 707th harmonic (~559)
• the 177th harmonic (~561)
• the 709th harmonic (~564)
• the 355th harmonic (~566)

Temperaments using substitute harmonics

Daggerminished

Same melodic shape as diminished. Uses sharper substitutes for prime 3, 5, 7 and 11.

Subgroup

2.113.161.177.193

Equal Temperament Mappings

	2	113	161	177	193
[ ⟨	8	55	59	60	61	]
⟨	12	82	88	90	91	] ⟩

Reduced Mapping

	2	113	161	177	193
[ ⟨	4	28	30	30	31	]
⟨	0	-1	-1	0	-1	] ⟩

POTE Generator Tunings (cents)

⟨300.0000, 185.4640]

POTE Step Tunings (cents)

⟨43.60795, 70.92803]

POTE Tuning Map (cents)

⟨1200.000, 8214.536, 8814.536, 9000.000, 9114.536]

POTE Mistunings (cents)

⟨0.000, 30.321, 17.436, 38.873, 3.588]

Unison Vectors

• [7, -1, 1, -1, 0⟩ (20608:20001)
• [-1, -2, 2, 0, 0⟩ (25921:25538)
• [8, 1, -1, -1, 0⟩ (28928:28497)
• [15, 0, 0, -2, 0⟩ (32768:31329)
• [-1, -1, -1, 0, 2⟩ (37249:36386)
• [6, -2, 0, -1, 2⟩ (2383936:2260113)

Pajaraxe

Same melodic shape as pajara. Uses sharper substitutes for prime 3, 5, 7 and 11.

Subgroup

2.113.161.177.193

Equal Temperament Mappings

	2	113	161	177	193
[ ⟨	22	150	161	164	167	]
⟨	12	82	88	90	91	] ⟩

Reduced Mapping

	2	113	161	177	193
[ ⟨	2	14	15	16	15	]
⟨	0	-2	-2	-6	1	] ⟩

POTE Generator Tunings (cents)

⟨600.0000, 106.8797]

POTE Step Tunings (cents)

⟨41.27832, 24.32309]

POTE Tuning Map (cents)

⟨1200.000, 8186.241, 8786.241, 8958.722, 9106.880]

POTE Mistunings (cents)

⟨0.000, 2.026, -10.860, -2.405, -4.069]

Unison Vectors

• [-1, -2, 2, 0, 0⟩ (25921:25538)
• [13, -3, 0, 1, 0⟩ (1449984:1442897)
• [-14, 1, 2, -1, 0⟩ (2929073:2899968)
• [-22, 1, 0, 0, 2⟩ (4209137:4194304)
• [-9, -2, 0, 1, 2⟩ (6593073:6537728)
• [8, 0, 2, -1, -2⟩ (6635776:6593073)

See also

• Dual-fifth temperaments
• List of octave-reduced harmonics
• Naughty and nice harmonics