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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


Equal Temperament Mappings
2 113 161 177 193
{| class="wikitable"
[ ⟨ 8 55 59 60 61 ]
!
⟨ 12 82 88 90 91 ] ⟩
!2
!113
!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 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
{| 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 277: Line 198:
*[-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===
===...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====
Melodic shape does not closely resemble any major temperament. Uses flatter substitutes for prime 3, 5, 7 and 11.
Uses flatter substitutes for prime 3, 5, 7 and 11.




Line 291: Line 791:


Equal Temperament Mappings
Equal Temperament Mappings
{| class="wikitable"
 
!
2 111 159 175 191
!2
[ ⟨ 9 61 66 67 68 ]
!111
⟨ 19 129 139 142 144 ] ⟩
!159
!175
!191
!
|-
|[ ⟨
|9
|61
|66
|67
|68
|]
|-
|⟨
|19
|129
|139
|142
|144
|] ⟩
|}




Reduced Mapping
Reduced Mapping
{| class="wikitable"
 
!
2 111 159 175 191
!2
[ ⟨ 1 7 7 8 8 ]
!111
⟨ 0 -2 3 -5 -4 ] ⟩
!159
!175
!191
!
|-
|[ ⟨
|1
|7
|7
|8
|8
|]
|-
|⟨
|0
| -2
|3
| -5
| -4
|] ⟩
|}




Line 368: 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]]
[[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]]
Retrieved from "https://en.xen.wiki/w/User:BudjarnLambeth/Substitute_harmonic"