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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 ==


===Daggerminished===
===...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.
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
Line 96: Line 104:


Equal Temperament Mappings
Equal Temperament Mappings
{| class="wikitable"
 
!
2 113 161 177 193
!2
[ ⟨ 8 55 59 60 61 ]
!113
⟨ 12 82 88 90 91 ] ⟩
!161
!177
!193
!
|-
|[ ⟨
|8
|55
|59
|60
|61
|]
|-
|⟨
|12
|82
|88
|90
|91
|] ⟩
|}




Reduced Mapping
Reduced Mapping
{| class="wikitable"
 
!
2 113 161 177 193
!2
[ ⟨ 4 28 30 30 31 ]
!113
⟨ 0 -1 -1 0 -1 ] ⟩
!161
!177
!193
!
|-
|[ ⟨
|4
|28
|30
|30
|31
|]
|-
|⟨
|0
| -1
| -1
|0
| -1
|] ⟩
|}




Line 173: Line 139:
Unison Vectors
Unison Vectors


* [7, -1, 1, -1, 0⟩ (20608:20001)
*[7, -1, 1, -1, 0⟩ (20608:20001)
* [-1, -2, 2, 0, 0⟩ (25921:25538)
*[-1, -2, 2, 0, 0⟩ (25921:25538)
* [8, 1, -1, -1, 0⟩ (28928:28497)
*[8, 1, -1, -1, 0⟩ (28928:28497)
* [15, 0, 0, -2, 0⟩ (32768:31329)
*[15, 0, 0, -2, 0⟩ (32768:31329)
* [-1, -1, -1, 0, 2⟩ (37249:36386)
*[-1, -1, -1, 0, 2⟩ (37249:36386)
* [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 190: Line 157:


Equal Temperament Mappings
Equal Temperament Mappings
{| class="wikitable"
 
!
2 113 161 177 193
!2
[ ⟨ 22 150 161 164 167 ]
!113
⟨ 12 82 88 90 91 ] ⟩
!161
!177
!193
!
|-
|[ ⟨
|22
|150
|161
|164
|167
|]
|-
|⟨
|12
|82
|88
|90
|91
|] ⟩
|}




Reduced Mapping
Reduced Mapping
{| class="wikitable"
 
!
2 113 161 177 193
!2
[ ⟨ 2 14 15 16 15 ]
!113
⟨ 0 -2 -2 -6 1 ] ⟩
!161
!177
!193
!
|-
|[ ⟨
|2
|14
|15
|16
|15
|]
|-
|⟨
|0
| -2
| -2
| -6
|1
|] ⟩
|}




Line 267: Line 192:
Unison Vectors
Unison Vectors


* [-1, -2, 2, 0, 0⟩ (25921:25538)
*[-1, -2, 2, 0, 0⟩ (25921:25538)
* [13, -3, 0, 1, 0⟩ (1449984:1442897)
*[13, -3, 0, 1, 0⟩ (1449984:1442897)
* [-14, 1, 2, -1, 0⟩ (2929073:2899968)
*[-14, 1, 2, -1, 0⟩ (2929073:2899968)
* [-22, 1, 0, 0, 2⟩ (4209137:4194304)
*[-22, 1, 0, 0, 2⟩ (4209137:4194304)
* [-9, -2, 0, 1, 2⟩ (6593073:6537728)
*[-9, -2, 0, 1, 2⟩ (6593073:6537728)
* [8, 0, 2, -1, -2⟩ (6635776:6593073)
*[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 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”.
 
 
===== 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:
* [[Ed255/128]] and [[Ed257/128]]
* [[Intercom scales]]


==See also==
Other related concepts:
*[[Dual-fifth temperaments]]
* [[Shadow]]
*[[List of octave-reduced harmonics]]
* [[Subgroup temperaments]]
*[[Naughty and nice harmonics]]
** [[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]]
[[Category:Octave-reduced harmonics]]
[[Category:Octave-reduced harmonics]]
[[Category:Regular temperament theory]]
[[Category:Combination product sets]]
[[Category:Subgroup temperaments]]
[[Category:Subgroup temperaments]]
[[Category:Temperament collections]]
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