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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 278: Line 199:
*[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]] 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.
==== 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
Subgroup: 2.3.43


Recommended ETs: [[144edo]], [[156edo]]
Recommended ETs: '''[[144edo]]''', [[156edo]], [[168edo]]




Line 333: Line 256:


[-19, 12, 0⟩ (531441:524288)
[-19, 12, 0⟩ (531441:524288)


===== Sitcom =====
===== Sitcom =====
Subgroup: 2.3.85
Subgroup: 2.3.85


Recommended ETs: [[96edo]]
Recommended ETs: '''[[96edo]]'''




Line 384: Line 308:


[-19, 12, 0⟩ (531441:524288)
[-19, 12, 0⟩ (531441:524288)


===== Romcom =====
===== Romcom =====
Subgroup: 2.3.91
Subgroup: 2.3.91


Recommended ETs: [[228edo]]
Recommended ETs: '''[[228edo]]''', [[216edo]], [[240edo]]




Line 435: Line 360:


[-19, 12, 0⟩ (531441:524288)
[-19, 12, 0⟩ (531441:524288)


===== Income =====
===== Income =====
Subgroup: 2.3.135
Subgroup: 2.3.135


Recommended ETs: [[96edo]]
Recommended ETs: '''[[96edo]]'''




Line 486: Line 412:


[-19, 12, 0⟩ (531441:524288)
[-19, 12, 0⟩ (531441:524288)


===== Outcome =====
===== Outcome =====
Subgroup: 2.3.143
Subgroup: 2.3.143


Recommended ETs: [[96edo]]
Recommended ETs: '''[[96edo]]'''




Line 537: Line 464:


[-19, 12, 0⟩ (531441:524288)
[-19, 12, 0⟩ (531441:524288)


===== Telecom =====
===== Telecom =====
Subgroup: 2.3.191
Subgroup: 2.3.191


Recommended ETs: [[108edo]]
Recommended ETs: '''[[108edo]]'''




Line 588: Line 516:


[-19, 12, 0⟩ (531441:524288)
[-19, 12, 0⟩ (531441:524288)


===== Intercom =====
===== Intercom =====
Subgroup: 2.3.193
Subgroup: 2.3.193


Recommended ETs: [[192edo]], [[180edo]], [[204edo]]
Recommended ETs: '''[[192edo]]''', [[180edo]], [[204edo]]




Line 639: Line 568:


[-19, 12, 0⟩ (531441:524288)
[-19, 12, 0⟩ (531441:524288)


===== Newcome =====
===== Newcome =====
Subgroup: 2.3.217
Subgroup: 2.3.217


Recommended ETs: [[132edo]]
Recommended ETs: '''[[132edo]]''', [[120edo]]




Line 690: Line 620:


[-19, 12, 0⟩ (531441:524288)
[-19, 12, 0⟩ (531441:524288)


===== Satcom =====
===== Satcom =====
Subgroup: 2.3.227
Subgroup: 2.3.227


Recommended ETs: [[96edo]]
Recommended ETs: '''[[96edo]]'''




Line 741: Line 672:


[-19, 12, 0⟩ (531441:524288)
[-19, 12, 0⟩ (531441:524288)


===== Minicom =====
===== Minicom =====
Subgroup: 2.3.245
Subgroup: 2.3.245


Recommended ETs: [[252edo]], [[264edo]]
Recommended ETs: '''[[252edo]]''', [[264edo]], [[276edo]], [[288edo]], [[300edo]]




Line 792: Line 724:


[-19, 12, 0⟩ (531441:524288)
[-19, 12, 0⟩ (531441:524288)


===== Glycome =====
===== Glycome =====
Subgroup: 2.3.255
Subgroup: 2.3.255


Recommended ETs: [[108edo]]
Recommended ETs: '''[[108edo]]'''




Line 846: 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====
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 857: 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 934: 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]]
*[[Subgroup temperaments]]
 
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]]
Retrieved from "https://en.xen.wiki/w/User:BudjarnLambeth/Substitute_harmonic"