User:BudjarnLambeth/Substitute harmonic: Difference between revisions
→...with mostly sharp substitutes: Added Compton substitute templates |
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{{Technical data page}} | |||
{{Editable user page}} | |||
A '''substitute harmonic'''{{idiosyncratic}} is a more complex [[harmonic]] which is used to substitute for a simpler one. | A '''substitute harmonic'''{{idiosyncratic}} is a more complex [[harmonic]] which is used to substitute for a simpler one. | ||
| Line 4: | Line 6: | ||
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}}. For example, you could convert a [[Dual-fifth temperaments|2.3-.3+.5 subgroup temperament]] into a 2.5.767.769 subgroup temperament. | 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}}. For example, you could convert a [[Dual-fifth temperaments|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 == | == List of substitute harmonics == | ||
| Line 25: | Line 29: | ||
=== Substitutes for the 3rd harmonic (~702) === | === Substitutes for the 3rd harmonic (~702) === | ||
*the 381st harmonic (~688) | * the 381st harmonic (~688) | ||
*the 763rd harmonic (~691) | * the 763rd harmonic (~691) | ||
*the 191st harmonic (~693) | * the 191st harmonic (~693) | ||
*the 765th harmonic (~695) | * the 765th harmonic (~695) | ||
*the 383rd harmonic (~697) | * the 383rd harmonic (~697) | ||
*the 767th harmonic (~700) | * the 767th harmonic (~700) | ||
*the 769th harmonic (~704) | * the 769th harmonic (~704) | ||
*the 385th harmonic (~706) | * the 385th harmonic (~706) | ||
*the 771st harmonic (~709) | * the 771st harmonic (~709) | ||
*the 193rd harmonic (~711) | * the 193rd harmonic (~711) | ||
*the 773rd harmonic (~713) | * the 773rd harmonic (~713) | ||
*the 387th harmonic (~715) | * the 387th harmonic (~715) | ||
=== Substitutes for the 5th harmonic (~386) === | === Substitutes for the 5th harmonic (~386) === | ||
*the 317th harmonic (~370) | * the 317th harmonic (~370) | ||
*the 635th harmonic (~373) | * the 635th harmonic (~373) | ||
*the 159th harmonic (~375) | * the 159th harmonic (~375) | ||
*the 637th harmonic (~378) | * the 637th harmonic (~378) | ||
*the 319th harmonic (~381) | * the 319th harmonic (~381) | ||
*the 639th harmonic (~384) | * the 639th harmonic (~384) | ||
*the 641st harmonic (~389) | * the 641st harmonic (~389) | ||
*the 321st harmonic (~392) | * the 321st harmonic (~392) | ||
*the 643rd harmonic (~394) | * the 643rd harmonic (~394) | ||
*the 161st harmonic (~397) | * the 161st harmonic (~397) | ||
*the 645th harmonic (~400) | * the 645th harmonic (~400) | ||
*the 323rd harmonic (~402) | * the 323rd harmonic (~402) | ||
=== Substitutes for the 7th harmonic (~969) === | === Substitutes for the 7th harmonic (~969) === | ||
*the 111th harmonic (~953) | * the 111th harmonic (~953) | ||
*the 889th harmonic (~955) | * the 889th harmonic (~955) | ||
*the 445th harmonic (~957) | * the 445th harmonic (~957) | ||
*the 891st harmonic (~959) | * the 891st harmonic (~959) | ||
*the 223rd harmonic (~961) | * the 223rd harmonic (~961) | ||
*the 893rd harmonic (~963) | * the 893rd harmonic (~963) | ||
*the 447th harmonic (~965) | * the 447th harmonic (~965) | ||
*the 895th harmonic (~967) | * the 895th harmonic (~967) | ||
*the 897th harmonic (~971) | * the 897th harmonic (~971) | ||
*the 449th harmonic (~973) | * the 449th harmonic (~973) | ||
*the 899th harmonic (~975) | * the 899th harmonic (~975) | ||
*the 225th harmonic (~977) | * the 225th harmonic (~977) | ||
*the 901st harmonic (~978) | * the 901st harmonic (~978) | ||
*the 451st harmonic (~980) | * the 451st harmonic (~980) | ||
*the 903rd harmonic (~982) | * the 903rd harmonic (~982) | ||
*the 113th harmonic (~984) | * the 113th harmonic (~984) | ||
=== Substitutes for the 11th harmonic (~551) === | === Substitutes for the 11th harmonic (~551) === | ||
*the 349th harmonic (~537) | * the 349th harmonic (~537) | ||
*the 699th harmonic (~539) | * the 699th harmonic (~539) | ||
*the 175th harmonic (~541) | * the 175th harmonic (~541) | ||
*the 701st harmonic (~544) | * the 701st harmonic (~544) | ||
*the 351st harmonic (~546) | * the 351st harmonic (~546) | ||
*the 703rd harmonic (~549) | * the 703rd harmonic (~549) | ||
*the 705th harmonic (~554) | * the 705th harmonic (~554) | ||
*the 353rd harmonic (~556) | * the 353rd harmonic (~556) | ||
*the 707th harmonic (~559) | * the 707th harmonic (~559) | ||
*the 177th harmonic (~561) | * the 177th harmonic (~561) | ||
*the 709th harmonic (~564) | * the 709th harmonic (~564) | ||
*the 355th harmonic (~566) | * the 355th harmonic (~566) | ||
== Temperaments using substitute harmonics == | == Temperaments using substitute harmonics == | ||
| Line 89: | Line 93: | ||
Name these after sharp weapons. If they closely resemble another temperament, reference that temperament in the name. | Name these after sharp weapons. If they closely resemble another temperament, reference that temperament in the name. | ||
====Daggerminished==== | ==== Daggerminished ==== | ||
Same melodic shape as [[diminished]]. Uses sharper substitutes for prime 3, 5, 7 and 11. | Same melodic shape as [[diminished]]. Uses sharper substitutes for prime 3, 5, 7 and 11. | ||
Subgroup | Subgroup | ||
2.113.161.177.193 | 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 | Reduced Mapping | ||
2 113 161 177 193 | |||
[ ⟨ 4 28 30 30 31 ] | |||
⟨ 0 -1 -1 0 -1 ] ⟩ | |||
| Line 184: | Line 146: | ||
*[6, -2, 0, -1, 2⟩ (2383936:2260113) | *[6, -2, 0, -1, 2⟩ (2383936:2260113) | ||
====Pajaraxe ==== | |||
==== Pajaraxe ==== | |||
Same melodic shape as [[pajara]]. Uses sharper substitutes for prime 3, 5, 7 and 11. | Same melodic shape as [[pajara]]. Uses sharper substitutes for prime 3, 5, 7 and 11. | ||
| Line 194: | Line 157: | ||
Equal Temperament Mappings | Equal Temperament Mappings | ||
2 113 161 177 193 | |||
[ ⟨ 22 150 161 164 167 ] | |||
⟨ 12 82 88 90 91 ] ⟩ | |||
Reduced Mapping | Reduced Mapping | ||
2 113 161 177 193 | |||
[ ⟨ 2 14 15 16 15 ] | |||
⟨ 0 -2 -2 -6 1 ] ⟩ | |||
| Line 278: | Line 199: | ||
*[8, 0, 2, -1, -2⟩ (6635776:6593073) | *[8, 0, 2, -1, -2⟩ (6635776:6593073) | ||
==== Narrowed | |||
These temperaments are like [[compton]] but with a smaller generator. They reduce the incidence of [[wolf]] fifths, and allow the melodic shape of [[compton]] to be used in tunings (especially [[edo]]s that might not otherwise support it. | ==== Narrowed compton ==== | ||
These temperaments are like [[compton]] but with a smaller generator. They reduce the incidence of [[Wolf interval|wolf]] fifths, especially in the smaller 24- and 36-tone [[MOS scale]]s, and allow the melodic shape of [[compton]] to be used in tunings (especially [[edo]]s) 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”. | 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 ===== | ===== Dotcom ===== | ||
Subgroup: 2.3.43 | |||
Recommended ETs: '''[[144edo]]''', [[156edo]], [[168edo]] | |||
Equal Temperament Mappings | Equal Temperament Mappings | ||
| Line 290: | Line 218: | ||
⟨ 48 76 261 ] ⟩ | ⟨ 48 76 261 ] ⟩ | ||
Reduced Mapping | Reduced Mapping | ||
| Line 296: | Line 225: | ||
⟨ 0 0 1 ] ⟩ | ⟨ 0 0 1 ] ⟩ | ||
POTE Generator Tunings (cents) | POTE Generator Tunings (cents) | ||
⟨100.0000, 8.1729] | ⟨100.0000, 8.1729] | ||
POTE Step Tunings (cents) | POTE Step Tunings (cents) | ||
⟨67.30854, 8.17286] | ⟨67.30854, 8.17286] | ||
POTE Tuning Map (cents) | POTE Tuning Map (cents) | ||
⟨1200.000, 1900.000, 6508.173] | ⟨1200.000, 1900.000, 6508.173] | ||
POTE Mistunings (cents) | POTE Mistunings (cents) | ||
⟨0.000, -1.955, -3.345] | ⟨0.000, -1.955, -3.345] | ||
Complexity 1.041959 | Complexity 1.041959 | ||
| Line 318: | Line 252: | ||
TE Error 0.503820 cents/octave | TE Error 0.503820 cents/octave | ||
Unison Vector | Unison Vector | ||
[-19, 12, 0⟩ (531441:524288) | [-19, 12, 0⟩ (531441:524288) | ||
===== Sitcom ===== | ===== Sitcom ===== | ||
Subgroup: 2.3.85 | |||
Recommended ETs: '''[[96edo]]''' | |||
Equal Temperament Mappings | Equal Temperament Mappings | ||
| Line 330: | Line 270: | ||
⟨ 48 76 307 ] ⟩ | ⟨ 48 76 307 ] ⟩ | ||
Reduced Mapping | Reduced Mapping | ||
| Line 336: | Line 277: | ||
⟨ 0 0 -1 ] ⟩ | ⟨ 0 0 -1 ] ⟩ | ||
POTE Generator Tunings (cents) | POTE Generator Tunings (cents) | ||
⟨100.0000, 12.6817] | ⟨100.0000, 12.6817] | ||
POTE Step Tunings (cents) | POTE Step Tunings (cents) | ||
⟨49.27306, 12.68173] | ⟨49.27306, 12.68173] | ||
POTE Tuning Map (cents) | POTE Tuning Map (cents) | ||
⟨1200.000, 1900.000, 7687.318] | ⟨1200.000, 1900.000, 7687.318] | ||
POTE Mistunings (cents) | POTE Mistunings (cents) | ||
⟨0.000, -1.955, -3.951] | ⟨0.000, -1.955, -3.951] | ||
Complexity 0.882135 | Complexity 0.882135 | ||
| Line 358: | Line 304: | ||
TE Error 0.503820 cents/octave | TE Error 0.503820 cents/octave | ||
Unison Vector | Unison Vector | ||
[-19, 12, 0⟩ (531441:524288) | [-19, 12, 0⟩ (531441:524288) | ||
===== Romcom ===== | ===== Romcom ===== | ||
Subgroup: 2.3.91 | |||
Recommended ETs: '''[[228edo]]''', [[216edo]], [[240edo]] | |||
Equal Temperament Mappings | Equal Temperament Mappings | ||
| Line 370: | Line 322: | ||
⟨ 36 57 235 ] ⟩ | ⟨ 36 57 235 ] ⟩ | ||
Reduced Mapping | Reduced Mapping | ||
| Line 376: | Line 329: | ||
⟨ 0 0 1 ] ⟩ | ⟨ 0 0 1 ] ⟩ | ||
POTE Generator Tunings (cents) | POTE Generator Tunings (cents) | ||
⟨100.0000, 5.3421] | ⟨100.0000, 5.3421] | ||
POTE Step Tunings (cents) | POTE Step Tunings (cents) | ||
⟨83.97384, 5.34205] | ⟨83.97384, 5.34205] | ||
POTE Tuning Map (cents) | POTE Tuning Map (cents) | ||
⟨1200.000, 1900.000, 7805.342] | ⟨1200.000, 1900.000, 7805.342] | ||
POTE Mistunings (cents) | POTE Mistunings (cents) | ||
⟨0.000, -1.955, -4.012] | ⟨0.000, -1.955, -4.012] | ||
Complexity 0.868796 | Complexity 0.868796 | ||
| Line 398: | Line 356: | ||
TE Error 0.503820 cents/octave | TE Error 0.503820 cents/octave | ||
Unison Vector | Unison Vector | ||
[-19, 12, 0⟩ (531441:524288) | [-19, 12, 0⟩ (531441:524288) | ||
===== Income ===== | ===== Income ===== | ||
Subgroup: 2.3.135 | |||
Recommended ETs: '''[[96edo]]''' | |||
Equal Temperament Mappings | Equal Temperament Mappings | ||
| Line 410: | Line 374: | ||
⟨ 48 76 339 ] ⟩ | ⟨ 48 76 339 ] ⟩ | ||
Reduced Mapping | Reduced Mapping | ||
| Line 416: | Line 381: | ||
⟨ 0 0 -1 ] ⟩ | ⟨ 0 0 -1 ] ⟩ | ||
POTE Generator Tunings (cents) | POTE Generator Tunings (cents) | ||
⟨100.0000, 12.1836] | ⟨100.0000, 12.1836] | ||
POTE Step Tunings (cents) | POTE Step Tunings (cents) | ||
⟨51.26579, 12.18355] | ⟨51.26579, 12.18355] | ||
POTE Tuning Map (cents) | POTE Tuning Map (cents) | ||
⟨1200.000, 1900.000, 8487.816] | ⟨1200.000, 1900.000, 8487.816] | ||
POTE Mistunings (cents) | POTE Mistunings (cents) | ||
⟨0.000, -1.955, -4.362] | ⟨0.000, -1.955, -4.362] | ||
Complexity 0.798940 | Complexity 0.798940 | ||
| Line 438: | Line 408: | ||
TE Error 0.503820 cents/octave | TE Error 0.503820 cents/octave | ||
Unison Vector | Unison Vector | ||
[-19, 12, 0⟩ (531441:524288) | [-19, 12, 0⟩ (531441:524288) | ||
===== Outcome ===== | ===== Outcome ===== | ||
Subgroup: 2.3.143 | |||
Recommended ETs: '''[[96edo]]''' | |||
Equal Temperament Mappings | Equal Temperament Mappings | ||
| Line 450: | Line 426: | ||
⟨ 48 76 343 ] ⟩ | ⟨ 48 76 343 ] ⟩ | ||
Reduced Mapping | Reduced Mapping | ||
| Line 456: | Line 433: | ||
⟨ 0 0 -1 ] ⟩ | ⟨ 0 0 -1 ] ⟩ | ||
POTE Generator Tunings (cents) | POTE Generator Tunings (cents) | ||
⟨100.0000, 12.5679] | ⟨100.0000, 12.5679] | ||
POTE Step Tunings (cents) | POTE Step Tunings (cents) | ||
⟨49.72855, 12.56786] | ⟨49.72855, 12.56786] | ||
POTE Tuning Map (cents) | POTE Tuning Map (cents) | ||
⟨1200.000, 1900.000, 8587.432] | ⟨1200.000, 1900.000, 8587.432] | ||
POTE Mistunings (cents) | POTE Mistunings (cents) | ||
⟨-0.000, -1.955, -4.413] | ⟨-0.000, -1.955, -4.413] | ||
Complexity 0.789672 | Complexity 0.789672 | ||
| Line 478: | Line 460: | ||
TE Error 0.503820 cents/octave | TE Error 0.503820 cents/octave | ||
Unison Vector | Unison Vector | ||
[-19, 12, 0⟩ (531441:524288) | [-19, 12, 0⟩ (531441:524288) | ||
===== Telecom ===== | ===== Telecom ===== | ||
Subgroup: 2.3.191 | |||
Recommended ETs: '''[[108edo]]''' | |||
Equal Temperament Mappings | Equal Temperament Mappings | ||
| Line 490: | Line 478: | ||
⟨ 48 76 363 ] ⟩ | ⟨ 48 76 363 ] ⟩ | ||
Reduced Mapping | Reduced Mapping | ||
| Line 496: | Line 485: | ||
⟨ 0 0 -1 ] ⟩ | ⟨ 0 0 -1 ] ⟩ | ||
POTE Generator Tunings (cents) | POTE Generator Tunings (cents) | ||
⟨100.0000, 11.7563] | ⟨100.0000, 11.7563] | ||
POTE Step Tunings (cents) | POTE Step Tunings (cents) | ||
⟨52.97495, 11.75626] | ⟨52.97495, 11.75626] | ||
POTE Tuning Map (cents) | POTE Tuning Map (cents) | ||
⟨1200.000, 1900.000, 9088.244] | ⟨1200.000, 1900.000, 9088.244] | ||
POTE Mistunings (cents) | POTE Mistunings (cents) | ||
⟨0.000, -1.955, -4.671] | ⟨0.000, -1.955, -4.671] | ||
Complexity 0.746156 | Complexity 0.746156 | ||
| Line 518: | Line 512: | ||
TE Error 0.503820 cents/octave | TE Error 0.503820 cents/octave | ||
Unison Vector | Unison Vector | ||
[-19, 12, 0⟩ (531441:524288) | [-19, 12, 0⟩ (531441:524288) | ||
===== Intercom ===== | ===== Intercom ===== | ||
Subgroup: 2.3.193 | |||
Recommended ETs: '''[[192edo]]''', [[180edo]], [[204edo]] | |||
Equal Temperament Mappings | Equal Temperament Mappings | ||
| Line 530: | Line 530: | ||
⟨ 36 57 274 ] ⟩ | ⟨ 36 57 274 ] ⟩ | ||
Reduced Mapping | Reduced Mapping | ||
| Line 536: | Line 537: | ||
⟨ 0 0 1 ] ⟩ | ⟨ 0 0 1 ] ⟩ | ||
POTE Generator Tunings (cents) | POTE Generator Tunings (cents) | ||
⟨100.0000, 6.2683] | ⟨100.0000, 6.2683] | ||
POTE Step Tunings (cents) | POTE Step Tunings (cents) | ||
⟨81.19502, 6.26833] | ⟨81.19502, 6.26833] | ||
POTE Tuning Map (cents) | POTE Tuning Map (cents) | ||
⟨1200.000, 1900.000, 9106.268] | ⟨1200.000, 1900.000, 9106.268] | ||
POTE Mistunings (cents) | POTE Mistunings (cents) | ||
⟨-0.000, -1.955, -4.680] | ⟨-0.000, -1.955, -4.680] | ||
Complexity 0.744680 | Complexity 0.744680 | ||
| Line 558: | Line 564: | ||
TE Error 0.503820 cents/octave | TE Error 0.503820 cents/octave | ||
Unison Vector | Unison Vector | ||
[-19, 12, 0⟩ (531441:524288) | [-19, 12, 0⟩ (531441:524288) | ||
===== Newcome ===== | ===== Newcome ===== | ||
Subgroup: 2.3.217 | |||
Recommended ETs: '''[[132edo]]''', [[120edo]] | |||
Equal Temperament Mappings | Equal Temperament Mappings | ||
| Line 570: | Line 582: | ||
⟨ 36 57 280 ] ⟩ | ⟨ 36 57 280 ] ⟩ | ||
Reduced Mapping | Reduced Mapping | ||
| Line 576: | Line 589: | ||
⟨ 0 0 1 ] ⟩ | ⟨ 0 0 1 ] ⟩ | ||
POTE Generator Tunings (cents) | POTE Generator Tunings (cents) | ||
⟨100.0000, 9.0771] | ⟨100.0000, 9.0771] | ||
POTE Step Tunings (cents) | POTE Step Tunings (cents) | ||
⟨72.76862, 9.07713] | ⟨72.76862, 9.07713] | ||
POTE Tuning Map (cents) | POTE Tuning Map (cents) | ||
⟨1200.000, 1900.000, 9309.077] | ⟨1200.000, 1900.000, 9309.077] | ||
POTE Mistunings (cents) | POTE Mistunings (cents) | ||
⟨0.000, -1.955, -4.784] | ⟨0.000, -1.955, -4.784] | ||
Complexity 0.728456 | Complexity 0.728456 | ||
| Line 598: | Line 616: | ||
TE Error 0.503820 cents/octave | TE Error 0.503820 cents/octave | ||
Unison Vector | Unison Vector | ||
[-19, 12, 0⟩ (531441:524288) | [-19, 12, 0⟩ (531441:524288) | ||
===== Satcom ===== | ===== Satcom ===== | ||
Subgroup: 2.3.227 | |||
Recommended ETs: '''[[96edo]]''' | |||
Equal Temperament Mappings | Equal Temperament Mappings | ||
| Line 610: | Line 634: | ||
⟨ 48 76 375 ] ⟩ | ⟨ 48 76 375 ] ⟩ | ||
Reduced Mapping | Reduced Mapping | ||
| Line 616: | Line 641: | ||
⟨ 0 0 -1 ] ⟩ | ⟨ 0 0 -1 ] ⟩ | ||
POTE Generator Tunings (cents) | POTE Generator Tunings (cents) | ||
⟨100.0000, 12.9662] | ⟨100.0000, 12.9662] | ||
POTE Step Tunings (cents) | POTE Step Tunings (cents) | ||
⟨48.13507, 12.96623] | ⟨48.13507, 12.96623] | ||
POTE Tuning Map (cents) | POTE Tuning Map (cents) | ||
⟨1200.000, 1900.000, 9387.034] | ⟨1200.000, 1900.000, 9387.034] | ||
POTE Mistunings (cents) | POTE Mistunings (cents) | ||
⟨0.000, -1.955, -4.824] | ⟨0.000, -1.955, -4.824] | ||
Complexity 0.722406 | Complexity 0.722406 | ||
| Line 638: | Line 668: | ||
TE Error 0.503820 cents/octave | TE Error 0.503820 cents/octave | ||
Unison Vector | Unison Vector | ||
[-19, 12, 0⟩ (531441:524288) | [-19, 12, 0⟩ (531441:524288) | ||
===== Minicom ===== | ===== Minicom ===== | ||
Subgroup: 2.3.245 | |||
Recommended ETs: '''[[252edo]]''', [[264edo]], [[276edo]], [[288edo]], [[300edo]] | |||
Equal Temperament Mappings | Equal Temperament Mappings | ||
| Line 650: | Line 686: | ||
⟨ 24 38 191 ] ⟩ | ⟨ 24 38 191 ] ⟩ | ||
Reduced Mapping | Reduced Mapping | ||
| Line 656: | Line 693: | ||
⟨ 0 0 1 ] ⟩ | ⟨ 0 0 1 ] ⟩ | ||
POTE Generator Tunings (cents) | POTE Generator Tunings (cents) | ||
⟨100.0000, 4.8900] | ⟨100.0000, 4.8900] | ||
POTE Step Tunings (cents) | POTE Step Tunings (cents) | ||
⟨90.21997, 4.89002] | ⟨90.21997, 4.89002] | ||
POTE Tuning Map (cents) | POTE Tuning Map (cents) | ||
⟨1200.000, 1900.000, 9504.890] | ⟨1200.000, 1900.000, 9504.890] | ||
POTE Mistunings (cents) | POTE Mistunings (cents) | ||
⟨0.000, -1.955, -4.885] | ⟨0.000, -1.955, -4.885] | ||
Complexity 0.713449 | Complexity 0.713449 | ||
| Line 678: | Line 720: | ||
TE Error 0.503820 cents/octave | TE Error 0.503820 cents/octave | ||
Unison Vector | Unison Vector | ||
[-19, 12, 0⟩ (531441:524288) | [-19, 12, 0⟩ (531441:524288) | ||
===== Glycome ===== | ===== Glycome ===== | ||
Subgroup: 2.3.255 | |||
Recommended ETs: '''[[108edo]]''' | |||
Equal Temperament Mappings | Equal Temperament Mappings | ||
| Line 690: | Line 738: | ||
⟨ 48 76 383 ] ⟩ | ⟨ 48 76 383 ] ⟩ | ||
Reduced Mapping | Reduced Mapping | ||
| Line 696: | Line 745: | ||
⟨ 0 0 -1 ] ⟩ | ⟨ 0 0 -1 ] ⟩ | ||
POTE Generator Tunings (cents) | POTE Generator Tunings (cents) | ||
⟨100.0000, 11.7037] | ⟨100.0000, 11.7037] | ||
POTE Step Tunings (cents) | POTE Step Tunings (cents) | ||
⟨53.18508, 11.70373] | ⟨53.18508, 11.70373] | ||
POTE Tuning Map (cents) | POTE Tuning Map (cents) | ||
⟨1200.000, 1900.000, 9588.296] | ⟨1200.000, 1900.000, 9588.296] | ||
POTE Mistunings (cents) | POTE Mistunings (cents) | ||
⟨0.000, -1.955, -4.928] | ⟨0.000, -1.955, -4.928] | ||
Complexity 0.707243 | Complexity 0.707243 | ||
| Line 718: | Line 772: | ||
TE Error 0.503820 cents/octave | TE Error 0.503820 cents/octave | ||
Unison Vector | Unison Vector | ||
| Line 724: | Line 779: | ||
===...with mostly flat substitutes=== | ===...with mostly flat substitutes=== | ||
Name these after flat regions like deserts. If they closely resemble another temperament, reference that temperament in the name. | Name these after flat regions like deserts. If they closely resemble another temperament, reference that temperament in the name. | ||
====Sahara==== | ====Sahara==== | ||
Uses flatter substitutes for prime 3, 5, 7 and 11. | |||
| Line 735: | Line 791: | ||
Equal Temperament Mappings | Equal Temperament Mappings | ||
2 111 159 175 191 | |||
[ ⟨ 9 61 66 67 68 ] | |||
⟨ 19 129 139 142 144 ] ⟩ | |||
Reduced Mapping | Reduced Mapping | ||
2 111 159 175 191 | |||
[ ⟨ 1 7 7 8 8 ] | |||
⟨ 0 -2 3 -5 -4 ] ⟩ | |||
| Line 812: | Line 826: | ||
Unison Vectors | Unison Vectors | ||
* [-6, 2, 0, 0, -1⟩ (12321:12224) | *[-6, 2, 0, 0, -1⟩ (12321:12224) | ||
* [8, 1, -1, -1, 0⟩ (9472:9275) | *[8, 1, -1, -1, 0⟩ (9472:9275) | ||
* [14, -1, -1, -1, 1⟩ (3129344:3088575) | *[14, -1, -1, -1, 1⟩ (3129344:3088575) | ||
* [-7, 1, 0, -2, 2⟩ (4049391:3920000) | *[-7, 1, 0, -2, 2⟩ (4049391:3920000) | ||
* [2, 3, -1, -1, -1⟩ (1823508:1771525) | *[2, 3, -1, -1, -1⟩ (1823508:1771525) | ||
* [-15, 0, 1, -1, 2⟩ (5800479:5734400) | *[-15, 0, 1, -1, 2⟩ (5800479:5734400) | ||
'''Sahara Septatonic scale''': A nice subset of Sahara[9]. Try noodling with it in Scale Workshop: | '''Sahara Septatonic scale''': A nice subset of Sahara[9]. Try noodling with it in Scale Workshop: | ||
* 256.237 | |||
* 384.356 | *256.237 | ||
* 512.475 | *384.356 | ||
* 687.525 | *512.475 | ||
* 943.762 | *687.525 | ||
* 1071.881 | *943.762 | ||
* 1200. | *1071.881 | ||
*1200. | |||
===...with an even mix of both=== | ===...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. | 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== | == See also == | ||
*[[Dual-fifth temperaments]] | Scales that make use of substitute harmonics: | ||
*[[List of octave-reduced harmonics]] | * [[Ed255/128]] and [[Ed257/128]] | ||
*[[Naughty and nice harmonics | * [[Intercom scales]] | ||
Other related concepts: | |||
* [[Shadow]] | |||
* [[Subgroup temperaments]] | |||
** [[Equalizer subgroup]]s | |||
** [[Dual-fifth temperaments]] | |||
** [[Half-prime subgroup]]s | |||
* [[List of octave-reduced harmonics]] | |||
* [[User:MasonGreen1/Naughty and nice harmonics]] | |||
[[Category:Harmonic series]] | [[Category:Harmonic series]] | ||