Substitute harmonic

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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.

The use of substitute harmonics is one kind of fudging.

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

...with mostly sharp substitutes

Name these after sharp weapons. If they closely resemble another temperament, reference that temperament in the name.


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)


Narrowed compton

These temperaments are like compton but with a smaller generator. They reduce the incidence of wolf fifths, especially in the smaller 24- and 36-tone MOS scales, and allow the melodic shape of compton to be used in tunings (especially edos) that might not otherwise support it.

These temperaments work by replacing the 5th harmonic with a slightly sharper substitute harmonic. These temperaments do not follow the naming conventions of other sharp-substitute temperaments. Instead these should be named using words that end with “com” or “come”.


Dotcom

Subgroup: 2.3.43

Recommended ETs: 144edo, 156edo, 168edo


Equal Temperament Mappings

2 3 43 [ ⟨ 12 19 65 ] ⟨ 48 76 261 ] ⟩


Reduced Mapping

2 3 43 [ ⟨ 12 19 65 ] ⟨ 0 0 1 ] ⟩


POTE Generator Tunings (cents)

⟨100.0000, 8.1729]


POTE Step Tunings (cents)

⟨67.30854, 8.17286]


POTE Tuning Map (cents)

⟨1200.000, 1900.000, 6508.173]


POTE Mistunings (cents)

⟨0.000, -1.955, -3.345]


Complexity 1.041959

Adjusted Error 2.733862 cents

TE Error 0.503820 cents/octave


Unison Vector

[-19, 12, 0⟩ (531441:524288)


Sitcom

Subgroup: 2.3.85

Recommended ETs: 96edo


Equal Temperament Mappings

2 3 85 [ ⟨ 12 19 77 ] ⟨ 48 76 307 ] ⟩


Reduced Mapping

2 3 85 [ ⟨ 12 19 77 ] ⟨ 0 0 -1 ] ⟩


POTE Generator Tunings (cents)

⟨100.0000, 12.6817]


POTE Step Tunings (cents)

⟨49.27306, 12.68173]


POTE Tuning Map (cents)

⟨1200.000, 1900.000, 7687.318]


POTE Mistunings (cents)

⟨0.000, -1.955, -3.951]


Complexity 0.882135

Adjusted Error 3.229180 cents

TE Error 0.503820 cents/octave


Unison Vector

[-19, 12, 0⟩ (531441:524288)


Romcom

Subgroup: 2.3.91

Recommended ETs: 228edo, 216edo, 240edo


Equal Temperament Mappings

2 3 91 [ ⟨ 12 19 78 ] ⟨ 36 57 235 ] ⟩


Reduced Mapping

2 3 91 [ ⟨ 12 19 78 ] ⟨ 0 0 1 ] ⟩


POTE Generator Tunings (cents)

⟨100.0000, 5.3421]


POTE Step Tunings (cents)

⟨83.97384, 5.34205]


POTE Tuning Map (cents)

⟨1200.000, 1900.000, 7805.342]


POTE Mistunings (cents)

⟨0.000, -1.955, -4.012]


Complexity 0.868796

Adjusted Error 3.278758 cents

TE Error 0.503820 cents/octave


Unison Vector

[-19, 12, 0⟩ (531441:524288)


Income

Subgroup: 2.3.135

Recommended ETs: 96edo


Equal Temperament Mappings

2 3 135 [ ⟨ 12 19 85 ] ⟨ 48 76 339 ] ⟩


Reduced Mapping

2 3 135 [ ⟨ 12 19 85 ] ⟨ 0 0 -1 ] ⟩


POTE Generator Tunings (cents)

⟨100.0000, 12.1836]


POTE Step Tunings (cents)

⟨51.26579, 12.18355]


POTE Tuning Map (cents)

⟨1200.000, 1900.000, 8487.816]


POTE Mistunings (cents)

⟨0.000, -1.955, -4.362]


Complexity 0.798940

Adjusted Error 3.565442 cents

TE Error 0.503820 cents/octave


Unison Vector

[-19, 12, 0⟩ (531441:524288)


Outcome

Subgroup: 2.3.143

Recommended ETs: 96edo


Equal Temperament Mappings

2 3 143 [ ⟨ 12 19 86 ] ⟨ 48 76 343 ] ⟩


Reduced Mapping

2 3 143 [ ⟨ 12 19 86 ] ⟨ 0 0 -1 ] ⟩


POTE Generator Tunings (cents)

⟨100.0000, 12.5679]


POTE Step Tunings (cents)

⟨49.72855, 12.56786]


POTE Tuning Map (cents)

⟨1200.000, 1900.000, 8587.432]


POTE Mistunings (cents)

⟨-0.000, -1.955, -4.413]


Complexity 0.789672

Adjusted Error 3.607288 cents

TE Error 0.503820 cents/octave


Unison Vector

[-19, 12, 0⟩ (531441:524288)


Telecom

Subgroup: 2.3.191

Recommended ETs: 108edo


Equal Temperament Mappings

2 3 191 [ ⟨ 12 19 91 ] ⟨ 48 76 363 ] ⟩


Reduced Mapping

2 3 191 [ ⟨ 12 19 91 ] ⟨ 0 0 -1 ] ⟩


POTE Generator Tunings (cents)

⟨100.0000, 11.7563]


POTE Step Tunings (cents)

⟨52.97495, 11.75626]


POTE Tuning Map (cents)

⟨1200.000, 1900.000, 9088.244]


POTE Mistunings (cents)

⟨0.000, -1.955, -4.671]


Complexity 0.746156

Adjusted Error 3.817661 cents

TE Error 0.503820 cents/octave


Unison Vector

[-19, 12, 0⟩ (531441:524288)


Intercom

Subgroup: 2.3.193

Recommended ETs: 192edo, 180edo, 204edo


Equal Temperament Mappings

2 3 193 [ ⟨ 12 19 91 ] ⟨ 36 57 274 ] ⟩


Reduced Mapping

2 3 193 [ ⟨ 12 19 91 ] ⟨ 0 0 1 ] ⟩


POTE Generator Tunings (cents)

⟨100.0000, 6.2683]


POTE Step Tunings (cents)

⟨81.19502, 6.26833]


POTE Tuning Map (cents)

⟨1200.000, 1900.000, 9106.268]


POTE Mistunings (cents)

⟨-0.000, -1.955, -4.680]


Complexity 0.744680

Adjusted Error 3.825233 cents

TE Error 0.503820 cents/octave


Unison Vector

[-19, 12, 0⟩ (531441:524288)


Newcome

Subgroup: 2.3.217

Recommended ETs: 132edo, 120edo


Equal Temperament Mappings

2 3 217 [ ⟨ 12 19 93 ] ⟨ 36 57 280 ] ⟩


Reduced Mapping

2 3 217 [ ⟨ 12 19 93 ] ⟨ 0 0 1 ] ⟩


POTE Generator Tunings (cents)

⟨100.0000, 9.0771]


POTE Step Tunings (cents)

⟨72.76862, 9.07713]


POTE Tuning Map (cents)

⟨1200.000, 1900.000, 9309.077]


POTE Mistunings (cents)

⟨0.000, -1.955, -4.784]


Complexity 0.728456

Adjusted Error 3.910426 cents

TE Error 0.503820 cents/octave


Unison Vector

[-19, 12, 0⟩ (531441:524288)


Satcom

Subgroup: 2.3.227

Recommended ETs: 96edo


Equal Temperament Mappings

2 3 227 [ ⟨ 12 19 94 ] ⟨ 48 76 375 ] ⟩


Reduced Mapping

2 3 227 [ ⟨ 12 19 94 ] ⟨ 0 0 -1 ] ⟩


POTE Generator Tunings (cents)

⟨100.0000, 12.9662]


POTE Step Tunings (cents)

⟨48.13507, 12.96623]


POTE Tuning Map (cents)

⟨1200.000, 1900.000, 9387.034]


POTE Mistunings (cents)

⟨0.000, -1.955, -4.824]


Complexity 0.722406

Adjusted Error 3.943173 cents

TE Error 0.503820 cents/octave


Unison Vector

[-19, 12, 0⟩ (531441:524288)


Minicom

Subgroup: 2.3.245

Recommended ETs: 252edo, 264edo, 276edo, 288edo, 300edo


Equal Temperament Mappings

2 3 243 [ ⟨ 12 19 95 ] ⟨ 24 38 191 ] ⟩


Reduced Mapping

2 3 243 [ ⟨ 12 19 95 ] ⟨ 0 0 1 ] ⟩


POTE Generator Tunings (cents)

⟨100.0000, 4.8900]


POTE Step Tunings (cents)

⟨90.21997, 4.89002]


POTE Tuning Map (cents)

⟨1200.000, 1900.000, 9504.890]


POTE Mistunings (cents)

⟨0.000, -1.955, -4.885]


Complexity 0.713449

Adjusted Error 3.992680 cents

TE Error 0.503820 cents/octave


Unison Vector

[-19, 12, 0⟩ (531441:524288)


Glycome

Subgroup: 2.3.255

Recommended ETs: 108edo


Equal Temperament Mappings

2 3 255 [ ⟨ 12 19 96 ] ⟨ 48 76 383 ] ⟩


Reduced Mapping

2 3 255 [ ⟨ 12 19 96 ] ⟨ 0 0 -1 ] ⟩


POTE Generator Tunings (cents)

⟨100.0000, 11.7037]


POTE Step Tunings (cents)

⟨53.18508, 11.70373]


POTE Tuning Map (cents)

⟨1200.000, 1900.000, 9588.296]


POTE Mistunings (cents)

⟨0.000, -1.955, -4.928]


Complexity 0.707243

Adjusted Error 4.027716 cents

TE Error 0.503820 cents/octave


Unison Vector

[-19, 12, 0⟩ (531441:524288)

...with mostly flat substitutes

Name these after flat regions like deserts. If they closely resemble another temperament, reference that temperament in the name.


Sahara

Uses flatter substitutes for prime 3, 5, 7 and 11.


Subgroup

2.111.159.175.191


Equal Temperament Mappings

2 111 159 175 191 [ ⟨ 9 61 66 67 68 ] ⟨ 19 129 139 142 144 ] ⟩


Reduced Mapping

2 111 159 175 191 [ ⟨ 1 7 7 8 8 ] ⟨ 0 -2 3 -5 -4 ] ⟩


POTE Generator Tunings (cents)

⟨1200.0000, 128.1188]


POTE Step Tunings (cents)

⟨34.25808, 46.93038]


POTE Tuning Map (cents)

⟨1200.000, 8143.762, 8784.357, 8959.406, 9087.525]


POTE Mistunings (cents)

⟨0.000, -9.537, 8.897, 17.952, -5.390]


Unison Vectors

  • [-6, 2, 0, 0, -1⟩ (12321:12224)
  • [8, 1, -1, -1, 0⟩ (9472:9275)
  • [14, -1, -1, -1, 1⟩ (3129344:3088575)
  • [-7, 1, 0, -2, 2⟩ (4049391:3920000)
  • [2, 3, -1, -1, -1⟩ (1823508:1771525)
  • [-15, 0, 1, -1, 2⟩ (5800479:5734400)


Sahara Septatonic scale: A nice subset of Sahara[9]. Try noodling with it in Scale Workshop:

  • 256.237
  • 384.356
  • 512.475
  • 687.525
  • 943.762
  • 1071.881
  • 1200.


...with an even mix of both

Name these after dishes which involve mixing things (e.g. stirfry, salad). If they closely resemble another temperament, reference that temperament in the name.


See also

Scales that make use of substitute harmonics:

Other related concepts:

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