Xenharmonic Wiki

Map of rank-2 temperaments: Difference between revisions

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This is intended to be a map of all interesting rank-2 temperaments that are compatible with octave equivalence. The only rank-2 temperaments not appearing here should be ones like [[Bohlen-Pierce|Bohlen-Pierce]] that completely lack octaves.
This is intended to be a map of all interesting rank-2 temperaments that are compatible with octave equivalence. The only rank-2 temperaments not appearing here should be ones like [[Bohlen–Pierce]] that completely lack octaves.


Please make sure each fraction of an octave is always the [[mediant|mediant]] of the ones directly above and below.
Please make sure each fraction of an octave is always the [[mediant]] of the ones directly above and below.


==One period per octave==
== One period per octave ==
Since this is the largest subset, it has its own page: [[Map_of_linear_temperaments|Map of linear temperaments]].
Since this is the largest subset, it has its own page: [[map of linear temperaments]].


==Two periods per octave==
== Two periods per octave ==
 
{| class="wikitable sortable mw-collapsible"
{| class="wikitable"
|-
|-
! colspan="12" | Generator
! colspan="9" | Generator
! | Cents
! Cents
! | Comments
! Comments
|-
|-
| | 0\2
| 0\2
| |
|  
| |
|  
| |
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 0.000
| |  
|  
| |  
| | 0.000
| |  
|-
|-
| |
|  
| |
|  
| |
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 1\26
| |  
| 46.154
| |  
|  
| | 1\26
| | 46.154
| |  
|-
|-
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |
| 1\24
| |
|  
| |
| 50.000
| | 1\24
| [[Shrutar]]
| |  
| | 50.000
| | [[Shrutar|Shrutar]]
|-
|-
| |
|  
| |
|  
| |
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 2\46
| |  
| 52.174
| |  
| Shrutar
| | 2\46
| | 52.174
| | Shrutar
|-
|-
| |
|  
| |
|  
| |
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 1\22
| |  
|  
| |  
|  
| | 1\22
| 54.545
| |  
| Shrutar
| |  
| | 54.545
| | Shrutar
|-
|-
| |
|  
| |
|  
| |
|  
| |  
|  
| |  
|  
| |  
| 1\20
| |  
|  
| |  
|  
| | 1\20
|  
| |  
| 60.000
| |  
|  
| |  
| | 60.000
| |  
|-
|-
| |
|  
| |
|  
| |
|  
| |  
|  
| |  
| 1\18
| |  
|  
| |  
|  
| | 1\18
|  
| |  
|  
| |  
| 66.667
| |  
|  
| |  
| | 66.667
| |  
|-
|-
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |
| 2\34
| |
|  
| |
|  
| | 2\34
|  
| |  
| 70.588
| |  
| [[Vishnu]]
| |  
| | 70.588
| | [[Vishnu|Vishnu]]
|-
|-
| |
|  
| |
|  
| |
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 7\118
| |  
| 71.186
| |  
| Vishnu
| | 7\118
| | 71.186
| | Vishnu
|-
|-
| |
|  
| |
|  
| |
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 5\84
| |  
|  
| |  
| 71.429
| | 5\84
| Vishnu
| |  
| | 71.429
| | Vishnu
|-
|-
| |
|  
| |
|  
| |
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 3\50
| |  
|  
| |  
|  
| | 3\50
| 72.000
| |  
|  
| |  
| | 72.000
| |  
|-
|-
| |
|  
| |
|  
| |
|  
| |  
| 1\16
| |  
|  
| |  
|  
| | 1\16
|  
| |  
|  
| |  
|  
| |  
| 75.000
| |  
|  
| |  
| | 75.000
| |  
|-
|-
| |
|  
| |
|  
| |
|  
| |  
|  
| |  
| 2\30
| |  
|  
| |  
|  
| | 2\30
|  
| |  
|  
| |  
| 80.000
| |  
|  
| |  
| | 80.000
| |  
|-
|-
| |
|  
| |
|  
| |
|  
| |  
|  
| |  
|  
| |  
| 3\44
| |  
|  
| |  
|  
| | 3\44
|  
| |  
| 81.818
| |  
|  
| |  
| | 81.818
| |  
|-
|-
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |
| 4\58
| |
|  
| |
|  
| | 4\58
| 82.759
| |  
| [[Harry]]
| |  
| | 82.759
| | [[Harry|Harry]]
|-
|-
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|
| 9\130
|
| 83.077
|
| Harry
|9\130
|83.077
|Harry
|-
|-
| |
|  
| |
|  
| |
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 5\72
| |  
|  
| |  
| 83.333
| | 5\72
| Harry
| |  
| | 83.333
| | Harry
|-
|-
| |
|  
| |
|  
| |
| 1\14
| |  
|  
| |  
|  
| | 1\14
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 85.714
| |  
|  
| |  
| | 85.714
| |  
|-
|-
| |
|  
| |  
|  
| |  
|  
| |  
| 2\26
| |
|  
| |
|  
| | 2\26
|  
| |  
|  
| |  
|  
| |  
| 92.308
| |  
| [[Injera]]
| |  
| | 92.308
| | [[Injera|Injera]]
|-
|-
| |
|  
| |
|  
| |
|  
| |  
|  
| |  
| 3\38
| |  
|  
| |  
|  
| | 3\38
|  
| |  
|  
| |  
| 94.737
| |  
| Injera
| |  
| | 94.737
| | Injera
|-
|-
| |
|  
| |
| 1\12
| |
|  
| |  
|  
| | 1\12
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 100.000
| |  
| Injera/[[Diaschismic]]
| |  
| | 100.000
| | Injera/[[diaschismic]]
|-
|-
|
|  
|
|  
|
|  
|
|  
|
|  
|
| 5\58
|
|  
|
|  
|5\58
|  
|
| 103.448
|
| Diaschismic
|
|103.448
|Diaschismic
|-
|-
| |
|  
| |
|  
| |
|  
| |  
|  
| |  
| 4\46
| |  
|  
| |  
|  
| | 4\46
|  
| |  
|  
| |  
| 104.348
| |  
| Diaschismic/[[Srutal]]
| |  
| | 104.348
| | Diaschismic/[[srutal]]
|-
|-
|
|  
|
|  
|
|  
|
|  
|
|  
|
| 7\80
|
|  
|
|  
|7\80
|  
|
| 105.000
|
| Srutal
|
|105.000
|Srutal
|-
|-
| |
|  
| |
|  
| |
|  
| |  
| 3\34
| |  
|  
| |  
|  
| | 3\34
|  
| |  
|  
| |  
|  
| |  
| 105.882
| |  
| Srutal/[[Pajara]]
| |  
| | 105.882
| | Srutal/[[pajara]]  
|-
|-
|
|  
|
|  
|
|  
|
|  
|
| 5\56
|
|  
|
|  
|5\56
|  
|
|  
|
| 107.143
|
| Pajara
|
|107.143
|Pajara
|-
|-
| |
|  
| |
|  
| |
| 2\22
| |  
|  
| |  
|  
| | 2\22
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 109.091
| |  
| Pajara
| |  
| | 109.091
| | Pajara
|-
|-
|
|  
|
|  
|
|  
|
| 3\32
|
|  
|
|  
|3\32
|  
|
|  
|
|  
|
| 112.500
|
|  
|
|112.500
|
|-
|-
| |
| 1\10
| |
|  
| |
|  
| | 1\10
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 120.000
| |  
| [[Octokaidecal]]
| |  
| | 120.000
| | [[Octokaidecal]]/[[Nimona]]
|-
|-
|
|  
|
|  
|
| 3\28
|
|  
|
|  
|3\28
|  
|
|  
|
|  
|
|  
|
| 128.571
|
| Octokaidecal
|
|128.571
|Octokaidecal/Nimona
|-
|-
| |
|  
| |
| 2\18
| |
|  
| |  
|  
| | 2\18
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 133.333
| |  
| Octokaidecal
| |  
| | 133.333
| | Octokaidecal/Nimona
|-
|-
| |
|  
| |
|  
| |
| 3\26
| |  
|  
| |  
|  
| | 3\26
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 138.462
| |  
|  
| |  
| | 138.462
| |  
|-
|-
|
|  
|
|  
|
|  
|
|  
|
| 6\60
|
|  
|
|  
|6\60
|  
|
|  
|
| 140.000
|
|  
|
|140.000
|
|-
|-
|
|  
|
|  
|
|  
|
|  
|
|  
|
| 11\94
|
|  
|
|  
|11\94
|  
|
| 140.126
|
| Fifive
|
|140.126
|Fifive
|-
|-
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|
| 15\128
|
|  
|
|  
|15\128
| 140.625
|
| Fifive
|
|140.625
|Fifive
|-
|-
| |
|  
| |  
|  
| |  
|  
| |  
| 4\34
| |
|  
| |
|  
| | 4\34
|  
| |  
|  
| |  
|  
| |  
| 141.176
| |  
| [[Fifive]]
| |  
| | 141.176
| | [[Fifive|Fifive]]
|-
|-
|
|  
|
|  
|
|  
|
|  
|
| 5\42
|
|  
|
|  
|5\42
|  
|
|  
|
| 142.857
|
|  
|
|142.857
|
|-
|-
| |
| 1\8
| |
|  
| | 1\8
|  
| |
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 150.000
| |  
|  
| |  
| | 150.000
| |  
|-
|-
| |
|  
| |
|  
| |
| 4\30
| |  
|  
| |  
|  
| | 4\30
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 160.000
| |  
|  
| |  
| | 160.000
| |  
|-
|-
| |
|  
| |
| 3\22
| |
|  
| |  
|  
| | 3\22
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 163.636
| |  
| [[Hedgehog]]/[[Echidna]]
| |  
| | 163.636
| | [[Hedgehog|Hedgehog]]/[[Echidna|echidna]]
|-
|-
|
|  
|
|  
|
|  
|
|  
|
| 11\80
|
|  
|
|  
|11\80
|  
|
|  
|
| 165.000
|
| Hedgehog/Echidna
|
|165.000
|Hedgehog/echidna
|-
|-
| |
|  
| |
|  
| |
|  
| |  
| 8\58
| |  
|  
| |  
|  
| | 8\58
|  
| |  
|  
| |  
|  
| |  
| 165.517
| |  
| Hedgehog/Echidna
| |  
| | 165.517
| | Hedgehog/echidna
|-
|-
| |
|  
| |
|  
| |
| 5\36
| |  
|  
| |  
|  
| | 5\36
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 166.667
| |  
|  
| |  
| | 166.667
| |  
|-
|-
| |
| 2\14
| |
|  
| |
|  
| | 2\14
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 171.429
| |  
|  
| |  
| | 171.429
| |  
|-
|-
| |
|  
| |
| 3\20
| |
|  
| |  
|  
| | 3\20
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 180.000
| |  
|  
| |  
| | 180.000
| |  
|-
|-
| |
|  
| |
|  
| |
|  
| |  
| 7\46
| |  
|  
| |  
|  
| | 7\46
|  
| |  
|  
| |  
|  
| |  
| 182.609
| |  
| [[Unidec]]/[[Hendec]]
| |  
| | 182.609
| | [[Unidec|Unidec]]/[[Hendec|hendec]]
|-
|-
| |
|  
| |
|  
| |
|  
| |  
|  
| |  
| 11\72
| |  
|  
| |  
|  
| | 11\72
|  
| |  
|  
| |  
| 183.333
| |  
| Unidec/Hendec
| |  
| | 183.333
| | Unidec/hendec
|-
|-
| |
|  
| |
|  
| |
| 4\26
| |  
|  
| |  
|  
| | 4\26
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 184.615
| |  
| Unidec/Hendec
| |  
| | 184.615
| | Unidec/hendec
|-
|-
| |
|  
| |
|  
| |
|  
| |  
| 5\32
| |  
|  
| |  
|  
| | 5\32
|  
| |  
|  
| |  
|  
| |  
| 187.500
| |  
|  
| |  
| | 187.500
| |  
|-
|-
| |
| 1\6
| | 1\6
|  
| |
|  
| |
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 200.000
| |  
|  
| |  
| | 200.000
| |  
|-
|-
| |
|  
| |
|  
| |
|  
| |  
| 6\34
| |  
|  
| |  
|  
| | 6\34
|  
| |  
|  
| |  
|  
| |  
| 211.765
| |  
|  
| |  
| | 211.765
| |  
|-
|-
| |
|  
| |
|  
| |
| 5\28
| |  
|  
| |  
|  
| | 5\28
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 214.286
| |  
| [[Antikythera]]
| |  
| | 214.286
| | [[Antikythera]]
|-
|-
| |
|  
| |
|  
| |
|  
| |  
| 9\50
| |  
|  
| |  
|  
| | 9\50
|  
| |  
|  
| |  
|  
| |  
| 216.000
| |  
| Antikythera/[[Wizard]]
| |  
| | 216.000
| | Antikythera/[[Wizard]]
|-
|-
| |
|  
| |
|  
| |
|  
| |  
|  
| |  
| 13\72
| |  
|  
| |  
|  
| | 13\72
|  
| |  
|  
| |  
| 216.667
| |  
| Antikythera/Wizard
| |  
| | 216.667
| | Antikythera/Wizard
|-
|-
| |
|  
| |
| 4\22
| |
|  
| |  
|  
| | 4\22
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 218.182
| |  
| Antikythera/Wizard/[[Astrology]]
| |  
| | 218.182
| | Antikythera/Wizard/[[Astrology]]
|-
|-
|
|  
|
|  
|
|  
|
|  
|
| 15\82
|
|  
|
|  
|15\82
|  
|
|  
|
| 219.512
|
| Astrology
|
|219.512
|Astrology
|-
|-
|
|  
|
|  
|
|  
|
| 11\60
|
|  
|
|  
|11\60
|  
|
|  
|
|  
|
| 220.000
|
| Astrology
|
|220.000
|Astrology
|-
|-
|
|  
|
|  
|
| 7\38
|
|  
|
|  
|7\38
|  
|
|  
|
|  
|
|  
|
| 221.053
|
|  
|
|221.053
|
|-
|-
| |
|  
| |
|  
| |
|  
| | 3\16
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 225.000
| |  
| [[Lemba]]
| |  
| | 225.000
| | [[Lemba]]
|-
|-
| |
|  
| |
| 5\26
| |
|  
| |  
|  
| | 5\26
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 230.769
| |  
| Lemba
| |  
| | 230.769
| | Lemba
|-
|-
|
|  
|
|  
|
|  
|
| 12\62
|
|  
|
|  
|12\62
|  
|
|  
|
|  
|
| 232.258
|
| Lemba
|
|232.258
|Lemba
|-
|-
| |
|  
| |
|  
| |
| 7\36
| |  
|  
| |  
|  
| | 7\36
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 233.333
| |  
|  
| |  
| | 233.333
| |  
|-
|-
| |
|  
| |  
|  
| |  
|  
| |  
| 9\46
| |
|  
| |
|  
| | 9\46
|  
| |  
|  
| |  
|  
| |  
| 234.783
| |  
| [[Echidnic]]
| |  
| | 234.783
| | [[Echidnic|Echidnic]]
|-
|-
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|
| 29\148
|
|  
|
|  
|29\148
| 235.135
|
| Echidnic
|
|235.135
|Echidnic
|-
|-
|
|  
|
|  
|
|  
|
|  
|
|  
|
| 20\102
|
|  
|
|  
|20\102
|  
|
| 235.294
|
| Echidnic
|
|235.294
|Echidnic
|-
|-
|
|  
|
|  
|
|  
|
|  
|
| 11\56
|
|  
|
|  
|11\56
|  
|
|  
|
| 235.714
|
|  
|
|235.714
|
|-
|-
| |
| 2\10
| |
|  
| | 2\10
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |
|  
| |  
| 240.000
| |  
| [[Decimal]]
| |  
| | 240.000
| | [[Decimal|Decimal]]
|-
|-
| |
|  
| |
| 5\24
| |
|  
| |  
|  
| | 5\24
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 250.000
| |  
| Decimal
| |  
| | 250.000
| | Decimal
|-
|-
|
|  
|
|  
|
| 8\38
|
|  
|
|  
|8\38
|  
|
|  
|
|  
|
|  
|
| 252.632
|
| Decimal
|
|252.632
|Decimal
|-
|-
| |
|  
| |
|  
| |
|  
| | 3\14
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 257.143
| |  
| Decimal
| |  
| | 257.143
| | Decimal
|-
|-
| |
|  
| |
| 4\18
| |
|  
| |  
|  
| | 4\18
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 266.667
| |  
|  
| |  
| | 266.667
| |  
|-
|-
| |
|  
| |  
|  
| |  
| 5\22
| |
|  
| |
|  
| | 5\22
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 272.727
| |  
| [[Doublewide]]
| |  
| | 272.727
| | [[Doublewide|Doublewide]]
|-
|-
|
|  
|
|  
|
|  
|
|  
|
|  
|
| 16\70
|
|  
|
|  
|16\70
|  
|
| 274.286
|
| Doublewide
|
|274.286
|Doublewide
|-
|-
| |
|  
| |
|  
| |
|  
| |  
|  
| |  
| 11\48
| |  
|  
| |  
|  
| | 11\48
|  
| |  
|  
| |  
| 275.000
| |  
| Doublewide
| |  
| | 275.000
| | Doublewide
|-
|-
| |
|  
| |
|  
| |
|  
| |  
| 6\26
| |  
|  
| |  
|  
| | 6\26
|  
| |  
|  
| |  
|  
| |  
| 276.923
| |  
|  
| |  
| | 276.923
| |  
|-
|-
| | 1\4
| 1\4
| |
|  
| |
|  
| |
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 300.000
| |  
|  
| |  
| | 300.000
| |  
|}
|}


==Three periods per octave==
== Three periods per octave ==
 
{{todo| complete table| comment=Add hemisemiaug (optimal ETs 18, 24, 42), add landscape microtemperaments}}
{| class="wikitable"
{| class="wikitable sortable mw-collapsible"
|-
|-
! colspan="9" | Generator
! colspan="10" | Generator
!
! Cents
! | Cents
! Comments
! | Comments
|-
|-
| | 0\3
| 0\3
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
|
|  
| | 0.000
| 0.000
| |  
|  
|-
|-
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |
| 1\30
| |  
|  
| | 1\30
|  
|
|  
| | 40.000
| 40.000
| |  
|  
|-
|-
| |
|  
| |
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 1\27
| |  
|  
| | 1\27
|  
| |  
|  
|
|  
| | 44.444
| 44.444
| | [[Semiaug|Semiaug]]
| [[Semiaug]]
|-
|-
| |  
|  
| |  
|
| |  
|
| |  
|
| |  
|
| |  
|
| |  
| 2\51
| |  
|
| | 2\51
|
|
|
| | 47.059
| 47.059
| | Semiaug
| Semiaug
|-
|  
|  
|  
|  
| 1\24
|  
|  
|  
|  
|  
| 50.000
| Semiaug/[[Augmented family#Hemiug| Hemiug]]
|-
|  
|  
|
|
|
| 2\45
|
|
|  
|  
| 53.333
| Hemiug
|-
|-
| |  
|  
| |  
|  
| |  
|  
| |
| 1\21
| |  
|  
| |  
|  
| | 1\24
|  
| |  
|  
| |  
|  
|
|  
| | 50.000
| 57.143
| | Semiaug
| Hemiug
|-
|-
| |  
|  
| |  
|  
| |
| 1\18
| |  
|  
| |  
|  
| | 1\21
|  
| |  
|  
| |  
|  
| |  
|  
|
|  
| | 57.143
| 66.667
| |  
|  
|-
|-
| |  
|  
| |
| 1\15
| |  
|  
| |  
|  
| | 1\18
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
|
|  
| | 66.667
| 80.000
| |  
|  
|-
|-
| |  
|  
| |  
|  
| |  
|  
| | 1\15
|  
| |  
|  
| |  
|  
| |
| 6\87
| |  
|  
| |  
|  
|
|  
| | 80.000
| 82.759
| |  
|  
|-
|-
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |
| 11\159
| | 6\87
|  
|
|  
| | 82.759
| 83.019
| |
| [[Tritikleismic]]
|-
|-
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|
| 16\231
|11\159
|  
|83.019
| 83.117
|[[Tritikleismic]]
| Tritikleismic
|-
|-
| |
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 5\72
| |
|  
| | 5\72
|  
| |  
|  
|
|  
| | 83.333
| 83.333
| | Tritikleismic
| Tritikleismic
|-
|-
| |
|  
| |  
|  
| |  
|  
| |  
|  
| |  
| 4\57
| |
|  
| | 4\57
|  
| |  
|  
| |  
|  
|
|  
| | 84.211
| 84.211
| | Tritikleismic
| Tritikleismic
|-
|-
| |
|  
| |  
|  
| |  
|  
| |  
| 3\42
| |
|  
| | 3\42
|  
| |  
|  
| |  
|  
| |  
|  
|
|  
| | 85.714
| 85.714
| |  
|  
|-
|-
| |
|  
| |
|  
| |  
| 2\27
| |  
|  
| | 2\27
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
|
|  
| | 88.889
| 88.889
| |[[Augmented]]/[[augene]]
| | [[Augmented]]/[[Augene]]
|-
|-
|
|  
|
|  
|
|  
|
| 3\39
|
|  
|3\39
|  
|
|  
|
|  
|
|  
|
|  
|92.308
| 92.308
|Augmented/augene
| Augmented/Augene/[[Fog]]
|-
|-
| |  
|  
| |  
|  
| | 1\12
|  
| |  
|  
| |
| 4\51
| |  
|  
| |  
|  
| |  
|  
| |  
|  
|
|  
| | 100.000
| 94.118
| | Augmented/augene/[[august]]
| Augmented/Augene/Fog
|-
|-
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| | 3\33
|  
| |  
| 5\63
| |  
|  
| |
|  
| |  
|  
|
|  
| | 109.091
| 95.238
| | August
| Augmented/Augene/Fog
|-
|-
| |  
|  
| |  
|  
| |  
|  
| | 2\21
|  
| |  
|  
| |  
|  
| |  
|  
| |
| 11\138
| |  
|  
|
|  
| | 114.286
| 95.652
| | August
| Augmented Augene/Fog
|-
|-
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| | 3\30
|  
| |  
|  
| |
| 6\75
| |  
|  
| |  
|  
|
|  
| | 120.000
| 96
| |
| Augmented/Augene/Fog
|-
|-
| |  
|  
| | 1\9
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |
| 7\87
| |  
|  
|
|  
| | 133.333
| 96.552
| |
| Augmented/Augene/[[Misty]]
|-
|-
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| | 4\33
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
|
| 15\186
| | 145.455
| 96.774
| |
| Augmented/Augene/Misty
|-
|-
| |  
|  
| |  
|  
| |  
|  
| | 3\24
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |
| 8\99
|
|  
| | 150.000
| 96.97
| |[[Triforce]]
| Augmented/Augene/Misty
|-
|-
|
| 1\12
|
|  
|
|  
|
|  
|5\39
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|153.846
| 100.000
|Triforce
| Augmented/Augene/[[August]]
|-
|-
| |  
|  
| |  
|  
| | 2\15
| 3\33
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
|
|  
| | 160.000
| 109.091
| | Triforce
| August
|-
|-
| |  
|  
| |
| 2\21
| |  
|  
| | 3/21
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
| |  
|  
|
|  
| | 171.429
| 114.286
| |
| August
|-
|-
| | 1\6
|  
| |  
|
| |  
| 3\30
| |  
|
| |  
|
| |  
|
| |  
|
| |  
|
| |  
|
|
|
| | 200.000
| 120.000
| |  
|
|-
| 1\9
|
|
|
|
|
|
|
|
|
| 133.333
|
|-
|
|
| 4\33
|
|
|
|
|
|
|
| 145.455
|
|-
|
| 3\24
|
|
|
|
|
|
|
|
| 150.000
| [[Triforce]]
|-
|
|
| 5\39
|
|
|
|
|
|
|
| 153.846
| Triforce
|-
| 2\15
|
|
|
|
|
|
|
|
|
| 160.000
| Triforce
|-
|
| 3\21
|
|
|
|
|
|
|
|
| 171.429
| [[Oodako]]
|-
|
|
|
| 7\48
|
|
|
|
|
|
| 175
| Oodako
|-
|
|
|
|
| 11\75
|
|
|
|
|
| 176
| Oodako
|-
|
|
|
|
|
| 15\102
|
|
|
|
| 176.471
| Oodako
|-
|
|
|
|
|
|
| 19\129
|
|
|
| 176.744
| Oodako
|-
|
|
| 4\27
|
|
|  
|  
|  
|  
|  
| 177.778
| Oodako
|-
| 1\6
|  
|  
|  
|  
|  
|  
|  
|  
|  
| 200.000
|  
|}
|}


==Four periods per octave==
== Four periods per octave ==
{| class="wikitable"
{| class="wikitable sortable mw-collapsible"
! colspan="9" | Generator !! Cents !! Comments
! colspan="6" | Generator
! Cents
! Comments
|-
|-
| 0\4 || || ||  ||  ||  ||  ||  ||  || 0 ||  
| 0\4
|  
|  
|  
|  
|  
| 0
|  
|-
|-
| || ||  ||  ||  ||  ||  ||  || 1\76 || 15.789 ||  
|  
|  
|  
| 1\76
|  
|
| 15.789
|  
|-
|-
| || ||  ||  ||  ||  ||  || 1\72 || || 16.667 || [[Quadritikleismic]]
|  
|  
|  
| 1\72
|  
|  
| 16.667
| [[Quadritikleismic]]
|-
|-
| || || ||  ||  ||  ||  ||  || 2\140 || 17.143 || Quadritikleismic
|  
|  
|  
|  
|  
| 3\212
| 16.981
| Quadritikleismic
|-
|-
| || || ||  ||  ||  ||  || 1\68 ||  || 17.647 || Quadritikleismic
|  
|  
|  
|  
| 2\140
|  
| 17.143
| Quadritikleismic
|-
|-
| || ||  ||  ||  ||  || 1\64 ||  || || 18.75 ||  
|  
|  
|  
| 1\68
|  
|  
| 17.647
| Quadritikleismic
|-
|-
| || ||  ||  ||  || 1\60 ||  ||  ||  || 20 ||  
|  
|  
|  
| 1\64
|  
|  
| 18.75
|  
|-
|-
| || ||  ||  ||  || 1\56 ||  ||  ||  || 21.429 ||  
|  
|  
|  
| 1\60
|  
|  
| 20
|  
|-
|-
| || || || ||  || 1\52 ||  ||  ||  || 23.068 ||  
|  
|  
|  
| ...
|  
|  
| ...
|  
|-
|-
| || ||  ||  ||  || 1\48 ||  ||  ||  || 25 ||  
|  
|  
|  
| 1\36
|  
|  
| 33.333
|  
|-
|-
| || ||  ||  ||  || 1\44 || ||  ||  || 27.273 ||  
|  
|  
|  
| 1\32
|  
|  
| 37.5
|  
|-
|-
| || ||  ||  ||  || 1\40 ||  ||  ||  || 30 ||  
|  
|  
|  
| 1\28
|  
|  
| 42.857
|  
|-
|-
| || || ||  ||  || 1\36 || ||  ||  || 33.333 ||  
|  
|  
|  
|  
| 2\52
|  
| 46.154
| [[Hemidim]]
|-
|-
| || || ||  ||  || 1\32 ||  ||  ||  || 37.5 ||  
|  
|  
|  
|  
|  
| 3\76
| 47.368
| Hemidim
|-
|-
| ||  ||  ||  ||  || 1\28 ||  ||  || || 42.857 ||  
|  
|  
| 1\24
|  
|  
|  
| 50
| Hemidim/[[4th-octave temperaments#Darian calendar| Darian calendar]]
|-
|-
| ||  ||  ||  || 1\24 ||  ||  ||  || || 50 || [[4th-octave temperaments#Darian calendar|Darian calendar]]
|  
| 1\20
|  
|  
|  
|  
| 60
|  
|-
|-
|  ||  ||  || 1\20 ||  ||  ||  || || || 60 ||  
| 1\16
|  
|  
|  
|  
|  
| 75
|  
|-
|-
| || || 1\16 || || || || || || || 75 ||  
|  
| 2\28
|
|
|
|
| 85.714
| [[Demolished]]
|-
|
|
| 3\40
|  
|
|
| 90
| Demolished
|-
| 1\12
|
|  
|  
|  
|  
| 100
| Demolished/[[Diminished]]
|-
|  
|  
|  
| 14\164
|  
|  
| 102.439
| Diminished
|-
|
|
|
|
| 27\316
|
| 102.532
| Diminished/[[Unlit]]
|-
|
|
|
|
|
| 40\468
| 102.564
| Diminished/Unlit
|-
|
|
|
| 13\152
|
|
| 102.632
| Diminished/Unlit/[[Undim]]
|-
|
|
|
|
|
| 38\444
| 102.703
| Diminished/Undim
|-
|
|
|
|
| 25\292
|
| 102.740
| Diminished/Undim
|-
|
|
|
| 12\140
|
|
| 102.857
| Diminished/Undim
|-
|
|
|
| 11\128
|
|
| 103.125
| Diminished
|-
|
|
|
| 10\116
|
|
| 103.448
| Diminished
|-
|
|
|
| 9\104
|
|
| 103.846
| Diminished
|-
|
|
|
| 8\92
|
|
| 104.348
| Diminished
|-
|
|
|
| 7\80
|
|
| 105
| Diminished/[[Bidia]]
|-
|
|
|
|
|
| 20\228
| 105.263
| Diminished/Bidia
|-
|-
| || 1\12 || ||  ||  ||  ||  ||  ||  || 100 || [[Diminished]]
|  
|  
|  
|  
| 13\148
|  
| 105.405
| Diminished/Bidia
|-
|-
| || ||  ||  ||  || 7\80 || ||  ||  || 105 || Diminished/[[Bidia]]
|  
|  
|  
| 6\68
|  
|  
| 105.882
| Diminished/Bidia
|-
|-
| || ||  ||  ||  || 6\68 || ||  ||  || 105.882 || Diminished/Bidia
|  
|  
|  
| 5\56
|  
|  
| 107.143
| Diminished/Bidia
|-
|-
| || ||  ||  ||  || 5\56 ||  || || || 107.143 || Diminished/Bidia
|  
|  
| 4\44
|  
|  
|  
| 109.091
| Diminished
|-
|-
| ||  ||  ||  || 4\44 || || ||  ||  || 109.091 || Diminished
|  
| 3\32
|  
|  
|  
|  
| 112.5
| Diminished
|-
|-
| ||  ||  || 3\32 ||  ||  ||  || || || 112.5 || Diminished
| 2\20
|  
|  
|  
|  
|  
| 120
| Diminished
|-
|-
| ||  || 2\20 ||  ||  ||  ||  ||  || || 120 || Diminished
|  
| 3\28
|  
|  
|  
|  
| 128.571
|  
|-
|-
| 1\8 || || ||  ||  ||  ||  ||  ||  || 150 || Diminished
| 1\8  
|  
|  
|  
|  
|  
| 150  
|  
|}
|}


==Five periods per octave==
== Five periods per octave ==
{| class="wikitable"
{{todo| complete table| comment=add trisedodge (optimal ETs 15, 50, 65, 80, 340, 405) and countdown (15, 35, 50, 65, 210, 275, 340)}}
! colspan="9" |Generator!!Cents !!Comments
{| class="wikitable sortable mw-collapsible"
! colspan="9" | Generator
! Cents
! Comments
|-
| 0\5
|
|
|
|
|
|
|
|
| 0
|
|-
|
|
|
|
|
| 1\140
|
|
|
| 8.571
|
|-
|
|
|
|
|
| 1\135
|
|
|
| 8.889
| [[5th-octave temperaments#Slendroschismic| Slendroschismic]]
|-
|
|
|
|
|
|
|
| 3\400
|
| 9
| Slendroschismic
|-
|
|
|
|
|
|
|
|
| 5\665
| 9.023
| Slendroschismic
|-
|
|
|
|
|
|
| 2\265
|
|
| 9.057
| Slendroschismic
|-
|
|
|
|
|
| 1\130
|
|
|
| 9.231
| Slendroschismic/[[Hemipental]]
|-
|
|
|
|
|
|
|
|
| 4\515
| 9.32
| Hemipental
|-
|
|
|
|
|
|
|
| 3\385
|
| 9.351
| Hemipental
|-
|
|
|
|
|
|
|
|
| 5\640
| 9.375
| Hemipental
|-
|-
|0\5 || || || || ||
|
|  
|  
| || 0||
|  
|  
|  
|  
|  
| 2\255
|
|
| 9.412
| Hemipental
|-
|-
|
|  
|
|  
|
|  
|
|  
|
|  
|1\35
| 1\125
|
|  
|
|  
|
|  
|34.286
| 9.6
|
| Hemipental
|-
|-
|
|
|
|  
|
|  
|
|  
|1\30
|  
|
| 1\120
|
|  
|
|  
|
|  
|40
| 10
|
|  
|-
|-
|
|  
|
|  
|
|  
|
|  
|
|  
|2\55
| 1\115
|
|  
|
|  
|
|  
|43.636
| 10.435
|
|  
|-
|-
|
|  
|
|  
|
|  
|1\25
|  
|
|  
|
| 1\70
|
|  
|
|  
|
|  
|48
| 17.143
|
|  
|-
|-
|
|  
|
|  
|
|  
|
|  
|
|  
|3\70
| 1\65
|
|  
|
|  
|
|  
|51.429
| 18.462
|
| [[Pental (temperament)| Pental]]
|-
|-
|
|  
|
|  
|
|  
|
|  
|2\45
|  
|
|  
|
|  
|
|  
|
| 4\255
|53.3
| 18.824
|
| Pental
|-
|-
|
|  
|
|  
|
|  
|
|  
|
|  
|3\65
|  
|
|  
|
| 3\190
|
|  
|55.385
| 18.947
|
| Pental
|-
|-
|
|  
|
|  
|1\20
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|
| 5\315
|60
| 19.048
|
| Pental
|-
|-
|
|  
|
|  
|
|  
|
|  
|
|  
|4\75
|  
|
| 2\125
|
|  
|
|  
|64
| 19.2
|
| Pental
|-
|-
|
|  
|
|  
|
|  
|
|  
|3\55
|  
|
|  
|
|  
|
| 3\185
|
|  
|65.455
| 19.459
|
| Pental
|-
|-
|
|  
|
|  
|
|  
|
|  
|
|  
|5\90
| 1\60
|
|  
|
|  
|
|  
|66.667
| 20
|
| Pental
|-
|-
|
|  
|
|  
|
|  
|2\35
|  
|
|  
|
| 1\55
|
|  
|
|  
|
|  
|68.571
| 21.818
|
|  
|-
|-
|
|  
|
|  
|
|  
|
|  
|
|  
|5\85
| 1\50
|
|  
|
|  
|
|  
|70.588
| 24
|
|  
|-
|-
|
|  
|
|  
|
|  
|
|  
|3\50
|  
|
| 1\45
|
|  
|
|  
|
|  
|72
| 26.67
|
|  
|-
|-
|
|  
|
|  
|
|  
|
|  
|
| 1\40
|4\65
|  
|
|  
|
|  
|
|  
|73.846
| 30
|
|  
|-
|-
|  || 1\15||  ||  ||  ||
|
|  
|  
| || 80||[[Blackwood]]/[[Blacksmith]]
|  
|  
| 1\35
|  
|
|
|
|
| 34.286
|  
|-
|-
|
|  
|
|  
|
| 1\30
|
|  
|
|  
|5\70
|  
|
|  
|
|  
|
|  
|85.714
| 40
|Blackwood/Blacksmith
|  
|-
|-
|
|  
|
| 1\25
|
|  
|
|  
|4\55
|  
|
|  
|
|  
|
|  
|
|  
|87.273
| 48
|Blackwood/Blacksmith
|  
|-
|-
|
| 1\20
|
|  
|
|  
|
|  
|
|  
|7\95
|  
|
|  
|
|  
|
|  
|88.421
| 60
|Blackwood/Blacksmith
|  
|-
|-
|
|  
|
| 2\35
|
|  
|3\40
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|90
| 68.571
|Blackwood/Blacksmith
| [[Coblack]]
|-
|-
|
|  
|
|  
|
|  
|
| 5\85
|
|  
|8\105
|  
|
|  
|
|  
|
|  
|91.429
| 70.588
|Blackwood/Blacksmith
| Coblack
|-
|-
|
|  
|
|  
|
| 3\50
|
|  
|5\65
|  
|
|  
|
|  
|
|  
|
|  
|92.308
| 72
|Blackwood/Blacksmith
| Coblack
|-
|-
|
|  
|
|  
|
|  
|
|  
|
| 7\115
|7\90
|  
|
|  
|
|  
|
|  
|93.333
| 73.043
|Blackwood/Blacksmith
| Coblack
|-
|-
|
|  
|
|  
|2\25
|  
|
| 4\65
|
|  
|
|  
|
|  
|
|  
|
|  
|96
| 73.846
|Blackwood/Blacksmith
| Coblack
|-
|-
|
| 1\15
|
|  
|
|  
|
|  
|5\60
|  
|
|  
|
|  
|
|  
|
|  
|100
| 80
|Blackwood
| [[Blackwood]]/[[Blacksmith]]
|-
|-
|
|  
|
|  
|
|
|
| 5\70
|
|
|8\95
|
|
|
|
|
|
|
|101.053
| 85.714
|Blackwood
| Blackwood/Blacksmith
|-
|
|
| 4\55
|
|
|
|
|
|
| 87.273
| Blackwood/Blacksmith
|-
|
|
|
| 7\95
|
|
|
|
|
| 88.421
| Blackwood/Blacksmith
|-
|
| 3\40
|
|
|
|
|
|
|
| 90
| Blackwood/Blacksmith
|-
|
|
|
| 8\105
|
|
|
|
|
| 91.429
| Blackwood/Blacksmith
|-
|
|
| 5\65
|
|
|
|
|
|
| 92.308
| Blackwood/Blacksmith
|-
|
|
|
| 7\90
|
|
|
|
|
| 93.333
| Blackwood/Blacksmith
|-
| 2\25
|
|
|
|
|
|
|
|
| 96
| Blackwood/Blacksmith
|-
|
|
| 5\60
|
|
|
|
|
|
| 100
| Blackwood
|-
|  
|  
|  
| 8\95
|
|
|  
|  
|  
| 101.053
| Blackwood
|-
|
| 3\35
|
|
|
|
|
|
|
| 102.857
| Blackwood
|-
|
|
|
| 7\80
|
|
|
|
|
| 105
| Blackwood
|-
|
|
| 4\45
|
|
|
|
|
|
| 106.667
| Blackwood
|-
|
|
|
| 5\55
|
|
|
|
|
| 109.091
| Blackwood
|-
|
|
|
|
| 6\65
|
|
|
|
| 110.769
| Blackwood/[[Quintosec]]/[[Sengasec]]
|-
|-
|
|  
|
|  
|
|  
|3\35
|  
|
|  
|
|  
|
|  
|
|  
|
| 25\270
|102.857
| 111.111
|Blackwood
| Blackwood/Quintosec/Sengasec/[[Qintosec]]
|-
|-
|
|  
|
|  
|
|  
|
|  
|
|  
|7\80
|  
|
|  
|
| 19\205
|
|  
|105
| 111.220
|Blackwood
| Blackwood/Quintosec/SengasecQintosec
|-
|-
|
|  
|
|  
|
|  
|
|  
|4\45
|  
|
|  
|
| 13\140
|
|  
|
|  
|106.667
| 111.429
|Blackwood
| Blackwood/Quintosec/Sengasec/Qintosec
|-
|-
|
|  
|
|  
|
|  
|
|  
|
|  
|5\55
|  
|
|  
|
| 20\215
|
|  
|109.091
| 111.628
|Blackwood
| Blackwood/Qintosec
|-
|-
|
|  
|
|  
|
|  
|
|  
|
|  
|
| 7\75
|6\65
|  
|
|  
|
|  
|110.769
| 112
|Blackwood
| Blackwood/[[Qinto]]
|-
|-
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|7\75
| 15\160
|
|  
|112
| 112.5
|Blackwood
| Blackwood/Qinto
|-
|-
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|
| 8\85
|
|  
|8\85
|  
|112.941
| 112.941
|Blackwood/[[Phi as a Generator|Elderthing]]
| Blackwood/[[Phi as a Generator| Elderthing]]
|-
|-
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|
| 9\95
|
|  
|9\95
|  
|113.684
| 113.684
|Blackwood
| Blackwood
|-
|-
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|
| 10\105
|
|  
|10\105
|  
|114.286
| 114.286
|Blackwood
| Blackwood
|-
|-
|1\10
| 1\10
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|120
| 120
|Blackwood
| Blackwood
|}
|}
==Six periods per octave ==
 
{| class="wikitable"
== Six periods per octave ==
! colspan="5" |Generator!!Cents !!Comments
{| class="wikitable sortable mw-collapsible"
! colspan="7" | Generator
! Cents
! Comments
|-
| 0\6
|
|
|
|
|
|
| 0
|
|-
|-
|0\6|| || || || ||0||
|  
|  
|  
| 1\288
|  
|  
|  
| 4.167
|  
|-
|-
|
|  
|
|  
|
|
|
| 1\282
|1\36
|  
|33.333
|  
|
|  
| 4.255
|  
|-
|-
|
|  
|
|  
|
|  
|1\30
| 1\276
|
| 2\552
|40
|  
|
|  
| 4.348
| [[Sextile]]
|-
|-
|
|  
|
|  
|
|  
|
|  
|2\54
|  
|44.444
| 3\822
|
|
| 4.380
| Sextile
|-
|-
|
|
|
|
|1\24
|
|
|
|
|
|50
|
| 4\1092
| 4.396
| Sextile
|-
|
|  
|  
| 1\270
|
|
|
| 4.444
| Sextile
|-
|
|
|
| 1\264
|
|  
|  
| 4.545
|  
|  
|-
|-
|
|  
|
|  
|
|  
|
| ...
|3\66
|  
|54.545
|  
|
|
| ...
|  
|-
|-
|
|
|
|
|
|
|2\42
| 1\90
|
|
| 57.143
|
|
|
| 13.333
|
|-
|
|
|
| 1\84
|
|
|
| 14.286
|
|-
|
|
|
| 1\78
|
|
|
| 15.385
|
|-
|
|
|
| 1\72
|
|
|
| 16.667
| [[Stearnscape]]
|-
|
|
|
|
|
|
| 4\282
| 17.021
| Stearnscape
|-
|
|
|
|
|
| 3\210
|
| 17.143
| Stearnscape
|-
|
|  
|  
|  
| 2\138
|
|
| 17.391
|
|-
|
|
|
| 1\66
|
|
|
| 18.182
|
|-
|
|
|
| 1\60
|
|
|
| 20
|
|-
|
|
|
| 1\54
|
|  
|  
| 22.222
|
|-
|
|
|
| 1\48
|
|
|
| 25
|  
|-
|-
|
|  
|
|  
|
|
|
| 1\42
|3\60
|  
|60
|  
|
|  
| 28.571
|  
|-
|-
|
|
|1\18
|
|
|  
|
| 1\36
|
|  
| 66.667
|  
|
|  
| 33.333
|  
|-
|-
|
|  
|
|
|
| 1\30
|
|  
|4\66
|  
|72.727
|  
|
|  
| 40
|  
|-
|-
|
|
|
| 1\24
|
|  
|3\48
|  
|
|  
| 75
|  
|  
| 50
|  
|  
|-
|-
|
| 1\18
|
|
|
|  
|
|  
|5\78
|  
|76.923
|  
|
|  
| 66.667
|  
|-
|-
|
|  
|
| 2\30
|2\30
|
|
|
|
|
|  
|  
| 80
| 80
|[[Hexe]]
| [[Hexe]]
|-
|-
|
|  
|
|  
|
|  
|
| 5\72
|5\72
|
|83.333
|
|Hexe
|
| 83.333
| Hexe
|-
|-
|
|  
|
|  
|
| 3\42
|3\42
|
|
|
|85.714
|
|  
| 85.714
| Hexe
| Hexe
|-
|-
|
|  
|
|  
|
|  
|
| 4\54
|4\54
|
|88.889
|
|Hexe
|
| 88.889
| Hexe
|-
|-
|1\12
| 1\12
|
|
|
|
|
|  
|
|  
|  
|  
| 100
| 100
|Hexe
| Hexe
|}
|}
==Seven periods per octave==
 
{| class="wikitable"
== Seven periods per octave ==
! colspan="9" |Generator!! Cents!!Comments
{{todo| complete table| comment=add akjaysmic (optimal ETs 140, 224, 301), nitrogen (140, 1407, 1547) and brahmagupta (217, 224, 441, 665)}}
{| class="wikitable sortable mw-collapsible"
! colspan="8" | Generator
! Cents
! Comments
|-
| 0\7
|
|
|
|
|
|
|
| 0
|
|-
|
|
|
|
| 1\91
|
|
|
| 13.187
|
|-
|-
| 0\7|| || || ||
|  
|
|  
|
|  
|
|  
| ||0||
| 1\84
|  
|  
|  
| 14.286
| [[Absurdity]]
|-
|-
|
|  
|
|  
|
|  
|
|  
|
|  
|1\91
|  
|
|  
|
| 4\329
|
| 14.59
|13.187
| Absurdity
|
|-
|-
|
|  
|
|  
|
|  
|
|  
|
|  
|1\84
|  
|
| 3\245
|
|  
|
| 14.69
|14.286
|  
|[[Absurdity]]
|-
|-
|
|  
|
|  
|
|  
|
|  
|
|  
|
| 2\161
|
|  
|
|  
|4\329
| 14.907
|14.59
|  
|Absurdity
|-
|-
|
|  
|
|  
|
|  
|
|  
|
| 1\77
|
|  
|
|  
|3\245
|  
|
| 15.584
|14.69
| [[Septant]]
|
|-
|-
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|2\161
|  
|
| 4\301
|
| 15.947
|14.907
| Septant
|
|-
|-
|
|  
|
|  
|
|  
|
|  
|
|  
|1\77
|  
|
| 3\224
|
|  
|
| 16.071
|15.584
| Septant
|
|-
|-
|
|  
|
|
|
|
|
|
|
|
|1\70
| 2\147
|
|
|
|
|
| 16.327
|17.143
| Septant
|
|-
|  
|  
|  
|  
| 1\70
|  
|  
|  
| 17.143
|  
|-
|-
|
|  
|
|  
|
|  
|
|  
|
| 1\63
|1\63
|  
|
|  
|
|  
|
| 19.048
|19.048
| [[Sevond]]
|[[Sevond]]
|-
|-
|
|  
|
|  
|
|  
|
|  
|
|  
|
| 2\119
|2\119
|  
|
|  
|
| 20.168
|20.168
| Sevond
|Sevond
|-
|-
|
|  
|
|  
|
|  
|
|  
|
| 1\56
|1\56
|  
|
|  
|
|  
|
| 21.429
|21.429
| Sevond
|Sevond
|-
|-
|
|  
|
|  
|
|  
|
|  
|
| 1\49
|1\48
|  
|
|  
|
|  
|
| 24.49
|25
|  
|
|-
|-
|
|  
|
|  
|
|  
|
| 1\42
|1\42
|  
|
|  
|
|  
|
|  
|
| 28.571
|28.571
| [[Mistismic temperaments#Neutron| Neutron]]
|
|-
|-
|
|  
|
|  
|
|  
|1\35
|  
|
|  
|
| 3\119
|
|  
|
|  
|
| 30.252
|34.286
| Neutron
|[[Whitewood]]
|-
|-
|
|  
|
|
|
|
|
|
|
|
| 5\196
|
| 30.612
| Neutron
|-
|
|
|
|
|
|
|
| 7\273
| 30.769
| Neutron
|-
|
|
|
|
| 2\77
|
|
|
| 31.169
| Neutron
|-
|
|
| 1\35
|
|
|
|
|
| 34.286
| [[Whitewood family#Redwood| Redwood]]
|-
|  
|  
|  
| 2\63
| 2\63
|
|  
|
|  
|
|  
|
|  
|38.095
| 38.095
|Whitewood
| Redwood
|-
|-
|
|  
|
| 1\28
| 1\28
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|42.857
| 42.857
|Whitewood
| Redwood/[[Whitewood]]
|-
|-
|
|  
|
|  
|
|  
|
| 3\77
| 3\77
|
|  
|
|  
|
|  
|
|  
| 46.753
| 46.753
|Whitewood
| Whitewood
|-
|-
|
|  
|
|  
|
| 2\49
|2\49
|  
|
|  
|
|  
|
|  
|
|  
|
| 48.980
|48.980
| Whitewood
|Whitewood
|-
|-
|
|  
|
|  
|
|  
|
| 3\70
|3\70
|  
|
|  
|
|  
|
|  
|
| 51.429
| 51.429
|Whitewood
| Whitewood
|-
|-
|
| 1\21
|1\21
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|
| 57.143
|57.143
| Whitewood/[[Jamesbond]]/[[7th-octave temperaments#Septimal| Septimal]]
|Whitewood/[[Jamesbond]]/[[7th-octave temperaments#Septimal|Septimal]]
|-
|-
|
|  
|
|  
|
|  
|
| 4\77
| 4\77
|
|  
|
|  
|
|  
|
|  
|62.338
| 62.338
|Jamesbond/Septimal
| Jamesbond/Septimal
|-
|-
|
|  
|
|  
|
| 3\56
|3\56
|  
|
|  
|
|  
|
|  
|
|  
|
| 64.286
|64.286
| Jamesbond/Septimal
| Jamesbond/Septimal
|-
|-
|
|  
|
|  
|
|  
|
| 5\91
|5\91
|  
|
|  
|
|  
|
|  
|
| 65.934
|65.934
| Jamesbond/Septimal
|Jamesbond/Septimal
|-
|-
|
|  
|
| 2\35
|2\35
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|
| 68.571
| 68.571
|Jamesbond/Septimal
| Jamesbond/Septimal/[[Greenwood]]
|-
|-
|
|  
|
|  
|
|  
|
| 5\84
|5\84
|  
|
|  
|
|  
|
|  
|
| 71.429
|71.429
| Jamesbond/Septimal/Greenwood
|Jamesbond/Septimal
|-
|-
|
|  
|
|  
|
| 3\49
|3\49
|  
|
|  
|
|  
|
|  
|
|  
|
| 73.469
| 73.469
|Jamesbond/Septimal
| Jamesbond/Septimal/Greenwood
|-
|-
|
|  
|
|  
|
|  
|
| 4\63
|4\63
|  
|
|  
|
|  
|
|  
|
| 76.190
|76.190
| Jamesbond/Septimal/Greenwood
|Jamesbond/Septimal
|-
|-
| 1\14
| 1\14
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|
|  
|
| 85.714
|85.714
| Jamesbond/Septimal/Greenwood
|Jamesbond/Septimal
|}
 
== Eight periods per octave ==
{{todo| complete table| comment=Add Twilight temperament (optimal ETs 152, 328, 480, 808) and Octium temperament (472, 1400, 1872)}}
{| class="wikitable sortable mw-collapsible"
! colspan="8" | Generator
! Cents
! Comments
|-
| 0\8
|
|
|
|
|
|
|
| 0
|
|-
|
|
|
|
| 1\480
|
|
|
| 2.5
|
|-
|
|
|
|
| 1\472
|
|
|
| 2.542
| [[Octium]]
|-
|
|
|
|
|
|
| 4\1872
|
| 2.564
| Octium
|-
|
|
|
|
|
|
|
| 7\3272
| 2.567
| Octium
|-
|
|
|
|
|
| 3\1400
|
|
| 2.571
| Octium
|-
|
|
|
|
| 1\464
|
|
|
| 2.586
|
|-
|
|
|
|
| …
|
|
|
|
|
|-
|
|
|
|
| 1\96
|
|
|
| 12.5
|
|-
|
|
|
|
| 1\88
|
|
|
| 13.635
|
|-
|
|
|
|
| 1\80
|
|
|
| 15
|
|-
|
|
|
|
|
| 2\152
|
|
| 15.789
| [[Octoid]]
|-
|
|
|
|
|
|
| 3\224
|
| 16.071
| Octoid
|-
|
|
|
|
| 1\72
|
|
|
| 16.667
| Octoid
|-
|
|
|
|
| 1\64
|
|
|
| 18.75
|
|-
|
|
|
| 1\56
|
|
|
|
| 21.429
|
|-
|
|
| 1\48
|
|
|
|
|
| 25
|
|-
|
| 1\40
|
|
|
|
|
|
| 30
|
|-
|
|
| 2\72
|
|
|
|
|
| 33.333
|
|-
| 1\32
|
|
|
|
|
|
|
| 37.5
| [[Semidim]]
|-
|
|
| 3\88
|
|
|
|
|
| 40.909
| Semidim
|-
|
| 2\56
|
|
|
|
|
|
| 42.857
| Semidim
|-
|
|
| 3\80
|
|
|
|
|
| 45
| Semidim
|-
|
|
|
| 4\104
|
|
|
|
| 46.154
| Semidim
|-
|
|
|
|
| 5\128
|
|
|
| 46.875
| Semidim
|-
|
|
|
|
| 6\152
|
|
|
| 47.368
| Semidim
|-
|
|
|
|
|
| 7\176
|
|
| 47.727
| Semidim
|-
|
|
|
|
|
| 8\200
|
|
| 48
| Semidim
|-
|
|
|
|
|
| 9\224
|
|
| 48.214
| Semidim/[[Octant]]
|-
|
|
|
|
|
|
|
| 28\696
| 48.276
| Semidim/Octant
|-
|
|
|
|
|
|
| 19\472
|
| 48.305
| Semidim/Octant
|-
|
|
|
|
|
| 10\248
|
|
| 48.387
| Semidim
|-
| 1\24
|
|
|
|
|
|
|
| 50
| Semidim
|-
| 2\40
|
|
|
|
|
|
|
| 60
|
|-
| 1\16
|
|
|
|
|
|
|
| 75
|
|}
 
== Nine periods per octave ==
{| class="wikitable sortable mw-collapsible"
! colspan="8" | Generator
! Cents
! Comments
|-
| 0\9
|
|
|
|
|
|
|
| 0
|
|-
|
|
|
|
| 1\72
|
|
|
| 16.667
| [[Enneaportent]]
|-
|
|
|
|
|
|
|
| 4\279
| 17.204
| Enneaportent
|-
|
|
|
|
|
|
| 3\207
|
| 17.391
| Enneaportent
|-
|
|
|
|
|
| 2\135
|
|
| 17.778
| Enneaportent
|-
|
|
|
| 1\63
|
|
|
|
| 19.048
| Enneaportent
|-
|
|
| 1\54
|
|
|
|
|
| 22.222
|
|-
|
| 1\45
|
|
|
|
|
|
| 26.667
|
|-
| 1\36
|
|
|
|
|
|
|
| 33.333
| [[Tritrizo clan#No-five tritrizo| No-five tritrizo]]
|-
|
|
|
| 4\135
|
|
|
|
| 35.556
| No-five tritrizo
|-
|
|
| 3\99
|
|
|
|
|
| 36.364
| No-five tritrizo
|-
|
| 2\63
|
|
|
|
|
|
| 38.095
| [[Niner]]
|-
|
|
| 3\90
|
|
|
|
|
| 40
| Niner
|-
| 1\27
|
|
|
|
|
|
|
| 44.444
| Niner
|-
|
|
| 4\99
|
|
|
|
|
| 48.485
| [[Ennealimmal]]
|-
|
|
|
| 7\171
|
|
|
|
| 49.123
| Ennealimmal
|-
|
| 3\72
|
|
|
|
|
|
| 50
| Ennealimmal
|-
|
|
|
|
|
|
|
| 20\477
| 50.314
| [[Novemkleismic]]
|-
|
|
|
|
|
|
| 17\405
|
| 50.370
| Novemkleismic
|-
|
|
|
|
|
| 14\333
|
|
| 50.451
| Novemkleismic
|-
|
|
|
|
| 11\261
|
|
|
| 50.575
| Novemkleismic
|-
|
|
|
| 8\189
|
|
|
|
| 50.794
| Novemkleismic
|-
|
|
| 5\117
|
|
|
|
|
| 51.282
|
|-
| 2\45
|
|
|
|
|
|
|
| 53.333
|
|-
| 1\18
|
|
|
|
|
|
|
| 66.667
|
|}
|}
==Eight periods per octave==
<ul><li>[[Octoid]] - 16-cent generator, sub-cent accuracy.</li></ul>
==Nine periods per octave==
<ul><li>[[Ennealimmal]] - The generator is 49.02 cents, and don't forget the ".02" because it really is that accurate.</li></ul>
==Twelve periods per octave==
See also: [[Pythagorean family]]


Temperaments in this family are interesting because they can be thought of as [[12edo]] with microtonal alterations.
== Ten periods per octave==
{| class="wikitable sortable mw-collapsible"
! colspan="7" | Generator
! Cents
! Comments
|-
| 0\10
|
|
|
|
|
|
| 0
|
|-
| 1\110
|
|
|
|
|
|
| 10.909
|
|-
| 1\100
|
|
|
|
|
|
| 12
|
|-
|
| 2\190
|
|
|
|
|
| 12.632
| [[Deca]]
|-
| 1\90
|
|
|
|
|
|
| 13.333
| Deca
|-
| 1\80
|
|
|
|
|
|
| 15
| Deca
|-
| 1\70
|
|
|
|
|
|
| 17.143
|
|-
| 1\60
|
|
|
|
|
|
| 20
|
|-
| 1\50
|
|
|
|
|
|
| 24
| [[Decimetra]]
|-
|
|
|
| 4\190
|
|
|
| 25.263
| Decimetra
|-
|
|
|
|
| 7\330
|
|
| 25.455
| Decimetra
|-
|
|
|
|
|
|
| 17\800
| 25.5
| Decimetra
|-
|
|
|
|
|
| 10\470
|
| 25.532
| Decimetra
|-
|
|
| 3\140
|
|
|
|
| 25.714
| Decimetra
|-
|
| 2\90
|
|
|
|
|
| 26.667
| Decimetra
|-
| 1\40
|
|
|
|
|
|
| 30
| [[Decic]]
|-
|
|
| 3\110
|
|
|
|
| 32.727
| Decic
|-
|
| 2\70
|
|
|
|
|
| 34.286
| Decic
|-
|
|
|
| 4\170
|
|
|
| 28.235
| Decic
|-
|
|
| 3\100
|
|
|
|
| 36
| Decic
|-
| 1\30
|
|
|
|
|
|
| 40
| Decic/[[Decal]]
|-
|
| 3\80
|
|
|
|
|
| 45
| Decal/[[Decistearn]]
|-
|
|
|
| 8\210
|
|
|
| 45.714
| Decal/Decistearn
|-
|
|
|
|
| 13\340
|
|
| 45.882
| Decal/Decistearn
|-
|
|
|
|
|
| 18\470
|
| 45.957
| Decal/Decistearn
|-
|
|
|
|
|
|
| 23\600
| 46
| Decal/Decistearn
|-
|
|
| 5\130
|
|
|
|
| 46.154
| Decal/Decistearn
|-
| 2\50
|
|
|
|
|
|
| 48
| Decistearn/[[Decavish]]
|-
|
|
|
|
|
| 11\270
|
| 48.889
| Decavish
|-
|
|
|
|
| 9\220
|
|
| 49.091
| Decavish
|-
|
|
|
| 7\170
|
|
|
| 49.412
| Decavish
|-
|
|
| 5\120
|
|
|
|
| 50
| Decavish
|-
|
| 3\70
|
|
|
|
|
| 51.429
| Decavish
|-
|
|
| 4\90
|
|
|
|
| 53.333
|
|-
|
|
|
| 5\110
|
|
|
| 54.545
|
|-
|
|
|
|
| 6\130
|
|
| 55.385
| [[Decoid]]
|-
| 1\20
|
|
|
|
|
|
| 60
|
|}


<ul><li>[[Compton]] - 3-limit as in 12edo; intervals of 5 are off by one generator. In the 7-limit (sometimes called [[waage]]), intervals of 7 are off by two generators. In the 11-limit, intervals of 11 are off by 3 generators. Thinking of [[72edo]] might make this more concrete.</li><li>[[Catler]] - 5-limit as in 12edo; intervals of 7 are off by one generator.</li><li>[[Atomic]] - Does not temper out the schisma, so 3/2 is one schisma sharp of its 12edo value. In atomic, since twelve fifths are sharp of seven octaves by twelve schismas, the Pythagorean comma is twelve schismas, and hence 81/80, the Didymus comma, is eleven schismas. In fact eleven schismas is sharp of 81/80, and twelve schismas of the Pythaorean comma, by the microscopic interval of the atom, which atomic tempers out. Extremely accurate.</li></ul>
== Eleven periods per octave==
{| class="wikitable sortable mw-collapsible"
! colspan="7" | Generator
! Cents
! Comments
|-
| 0\11
|
|
|
|
|
|
| 0
|
|-
|
| 1\55
|
|
|
|
|
| 21.818
|
|-
| 1\44
|
|
|
|
|
|
| 27.273
|
|-
|
| 2\77
|
|
|
|
|
| 31.169
|
|-
| 1\33
|
|
|
|
|
|
| 36.364
|
|-
|
| 3\88
|
|
|
|
|
| 40.909
|
|-
| 2\55
|
|
|
|
|
|
| 42.636
| [[Hendecatonic]]
|-
|
| 3\77
|
|
|
|
|
| 46.753
| Hendecatonic
|-
|
|
|
| 7\176
|
|
|
| 47.727
| Hendecatonic
|-
|
|
|
|
| 11\275
|
|
| 48
| Hendecatonic
|-
|
|
| 4\99
|
|
|
|
| 48.485
| Hendecatonic/[[Cohendecatonic]]
|-
|
|
|
|
| 9\220
|
|
| 49.091
| Cohendecatonic
|-
|
|
|
|
|
| 14\341
|
| 49.267
| Cohendecatonic
|-
|
|
|
| 5\121
|
|
|
| 49.587
| Cohendecatonic
|-
|
|
|
|
| 6\143
|
|
| 50.350
|
|-
|
|
|
|
|
| 7\165
|
| 50.909
|
|-
|
|
|
|
|
|
| 8\187
| 51.337
| [[Undeka]]
|-
|
|
|
|
|
|
| 9\209
| 51.675
| Undeka
|-
|
|
|
|
|
|
| 10\231
| 51.948
| Undeka
|-
|
|
|
|
|
|
| 11\253
| 52.174
| Undeka
|-
| 1\22
|
|
|
|
|
|
| 54.545
| Undeka
|}


==See also==
==Twelve periods per octave==
{{See also| Compton family}}
{| class="wikitable sortable mw-collapsible"
! colspan="8" | Generator
! Cents
! Comments
|-
| 0\12
|
|
|
|
|
|
|
| 0
|
|-
|
|
|
| 1\636
|
|
|
|
| 1.887
|
|-
|
|
|
| 1\624
|
|
|
|
| 1.923
|
|-
|
|
|
|
| 2\1236
|
|
|
| 1.942
| [[Atomic]]
|-
|
|
|
|
|
| 3\1848
|
|
| 1.948
| Atomic
|-
|
|
|
|
|
|
| 4\2460
|
| 1.951
| Atomic
|-
|
|
|
|
|
|
|
| 5\3072
| 1.953
| Atomic
|-
|
|
|
| 1\612
|
|
|
|
| 1.961
| Atomic
|-
|
|
|
| 1\600
|
|
|
|
| 2
|
|-
|
|
|
| 1\588
|
|
|
|
| 2.041
|
|-
|
|
|
| 1\576
|
|
|
|
| 2.083
|
|-
|
|
|
| ...
|
|
|
|
| ...
|
|-
|
|
|
| 1\108
|
|
|
|
| 11.111
|
|-
|
|
|
| 1\96
|
|
|
|
| 12.5
| [[Compton]]
|-
|
|
|
| 1\84
|
|
|
|
| 14.286
| Compton
|-
|
|
|
|
|
| 3\240
|
|
| 15
| Compton
|-
|
|
|
|
|
|
|
| 8\636
| 15.094
| Compton
|-
|
|
|
|
|
|
| 5\396
|
| 15.152
| Compton
|-
|
|
|
|
|
|
|
| 7\552
| 15.217
| Compton
|-
|
|
|
|
| 2\156
|
|
|
| 15.385
| Compton
|-
|
|
|
|
|
|
|
| 7\540
| 15.556
| Compton
|-
|
|
|
|
|
|
| 5\384
|
| 15.625
| Compton
|-
|
|
|
|
|
|
|
| 8\612
| 15.686
| Compton
|-
|
|
|
|
|
| 3\228
|
|
| 15.789
| Compton
|-
|
|
|
|
|
|
|
| 7\528
| 15.909
| Compton
|-
|
|
|
|
|
|
| 4\300
|
| 16
| Compton
|-
|
|
|
|
|
|
|
| 5\372
| 16.129
| Compton
|-
|
|
|
| 1\72
|
|
|
|
| 16.667
| Compton
|-
|
|
| 1\60
|
|
|
|
|
| 20
| Compton
|-
|
|
|
| 2\108
|
|
|
|
| 22.222
|
|-
|
| 1\48
|
|
|
|
|
|
| 25
| [[Catler]]
|-
|
|
|
| 3\132
|
|
|
|
| 27.273
| Catler
|-
|
|
| 2\84
|
|
|
|
|
| 28.571
| Catler
|-
|
|
|
| 3\120
|
|
|
|
| 30
| Catler
|-
| 1\36
|
|
|
|
|
|
|
| 33.333
| Catler/[[Catnip]]
|-
|
|
|
| 4\132
|
|
|
|
| 36.364
| Catnip
|-
|
|
| 3\96
|
|
|
|
|
| 37.5
| Catnip
|-
|
|
|
| 5\156
|
|
|
|
| 38.462
| Catnip
|-
|
| 2\60
|
|
|
|
|
|
| 40
| Catnip
|-
|
|
|
| 5\144
|
|
|
|
| 41.667
|
|-
|
|
| 3\84
|
|
|
|
|
| 42.857
| [[Magnesium]]
|-
|
|
|
|
|
|
|
| 16\444
| 43.243
| Magnesium
|-
|
|
|
|
|
|
| 13\360
|
| 43.333
| Magnesium
|-
|
|
|
|
|
| 10\276
|
|
| 43.378
|
|-
|
|
|
|
| 7\192
|
|
|
| 43.75
|
|-
|
|
|
| 4\108
|
|
|
|
| 44.444
|
|-
| 1\24
|
|
|
|
|
|
|
| 50
|
|}


*[[Fractional-octave temperaments]]: Lists dozens of temperaments that have not yet been listed here.
== See also ==
* [[Fractional-octave temperaments]] &ndash; lists dozens more temperaments beyond what can be listed here.


[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Lists of temperaments]]
[[Category:Lists of temperaments]]
Retrieved from "https://en.xen.wiki/w/Map_of_rank-2_temperaments"