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[[Mapping]]: [{{val| 1 0 15 }}, {{val| 0 1 -8 }}]
[[Mapping]]: [{{val| 1 0 15 }}, {{val| 0 1 -8 }}]


Mapping generators: ~2, ~3
: mapping generators: ~2, ~3


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.736
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.736
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{{Main| Garibaldi temperament }}
{{Main| Garibaldi temperament }}


''Garibaldi'' tempers out the [[garischisma]], equating the [[septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double diminished octave (C-Cbb). It necessitates a sharper fifth than pure.  
Garibaldi tempers out the [[garischisma]], equating the [[septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double diminished octave (C-Cbb). It necessitates a sharper fifth than pure.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Mapping]]: [{{val| 1 0 15 25 }}, {{val| 0 1 -8 -14 }}]
[[Mapping]]: [{{val| 1 0 15 25 }}, {{val| 0 1 -8 -14 }}]


Mapping generators: ~2, ~3
: mapping generators: ~2, ~3


{{Multival|legend=1| 1 -8 -14 -15 -25 -10 }}
{{Multival|legend=1| 1 -8 -14 -15 -25 -10 }}
Line 71: Line 71:
* [[7-odd-limit]]: ~3/2 = {{monzo| 2/3 1/15 0 -1/15 }}
* [[7-odd-limit]]: ~3/2 = {{monzo| 2/3 1/15 0 -1/15 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 5/3 1/15 0 -1/15 }}, {{monzo| 5/3 -8/15 0 8/15 }}, {{monzo| 5/3 -14/15 0 14/15 }}]
: [{{monzo| 1 0 0 0 }}, {{monzo| 5/3 1/15 0 -1/15 }}, {{monzo| 5/3 -8/15 0 8/15 }}, {{monzo| 5/3 -14/15 0 14/15 }}]
: [[Eigenmonzo basis]] ([[unchanged-interval basis]]): 2.7/3
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.7/3
* [[9-odd-limit]]: ~3/2 = {{monzo| 9/16 1/8 0 -1/16 }}
* [[9-odd-limit]]: ~3/2 = {{monzo| 9/16 1/8 0 -1/16 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 25/16 1/8 0 -1/16 }}, {{monzo| 5/2 -1 0 1/2 }}, {{monzo| 25/8 -7/4 0 7/8 }}]
: [{{monzo| 1 0 0 0 }}, {{monzo| 25/16 1/8 0 -1/16 }}, {{monzo| 5/2 -1 0 1/2 }}, {{monzo| 25/8 -7/4 0 7/8 }}]
: Eigenmonzo basis (unchanged-interval basis): 2.7/3
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.9/7


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 86: Line 86:


=== Cassandra ===
=== Cassandra ===
''Cassandra'' is one of the best extension of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup.  
Cassandra is one of the best extension of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup.  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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Mapping: [{{val| 1 0 15 25 -33 }}, {{val| 0 1 -8 -14 23 }}]
Mapping: [{{val| 1 0 15 25 -33 }}, {{val| 0 1 -8 -14 23 }}]
Mapping generators: ~2, ~3


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.157
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.157
Line 100: Line 98:
Minimax tuning:
Minimax tuning:
* 11-odd-limit: ~3/2 = {{monzo| 9/16 1/8 0 -1/16 }}
* 11-odd-limit: ~3/2 = {{monzo| 9/16 1/8 0 -1/16 }}
: Eigenmonzo basis (unchanged-interval basis): 2.9/7
: Eigenmonzo (unchanged-interval) basis: 2.9/7


Tuning ranges:  
Tuning ranges:  
Line 122: Line 120:
Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit: ~3/2 = {{monzo| 19/34 0 0 -1/34 0 1/34 }}
* 13- and 15-odd-limit: ~3/2 = {{monzo| 19/34 0 0 -1/34 0 1/34 }}
: Eigenmonzo basis (unchanged-interval basis): 2.13/7
: Eigenmonzo (unchanged-interval) basis: 2.13/7


Tuning ranges:  
Tuning ranges:  
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Mapping: [{{val| 1 0 15 25 32 }}, {{val| 0 1 -8 -14 -18 }}]
Mapping: [{{val| 1 0 15 25 32 }}, {{val| 0 1 -8 -14 -18 }}]
Mapping generators: ~2, ~3


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.321
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.321
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Minimax tuning:  
Minimax tuning:  
* 11-odd-limit: ~3/2 = {{monzo| 3/5 1/10 0 0 -1/20 }}
* 11-odd-limit: ~3/2 = {{monzo| 3/5 1/10 0 0 -1/20 }}
: Eigenmonzo basis (unchanged-interval basis): 2.11/9
: Eigenmonzo (unchanged-interval) basis: 2.11/9


Tuning ranges:  
Tuning ranges:  
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Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit: ~3/2 = {{monzo| 14/23 2/23 0 0 0 -1/23 }}
* 13- and 15-odd-limit: ~3/2 = {{monzo| 14/23 2/23 0 0 0 -1/23 }}
: Eigenmonzo basis (unchanged-interval basis): 2.13/9
: Eigenmonzo (unchanged-interval) basis: 2.13/9


Tuning ranges:  
Tuning ranges:  
Line 343: Line 339:


Mapping: [{{val| 1 0 15 25 51 }}, {{val| 0 1 -8 -14 -30 }}]
Mapping: [{{val| 1 0 15 25 51 }}, {{val| 0 1 -8 -14 -30 }}]
Mapping generators: ~2, ~3


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.725
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.725
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Minimax tuning:  
Minimax tuning:  
* 11-odd-limit: ~3/2 = {{monzo| 19/32 1/16 0 0 -1/32 }}
* 11-odd-limit: ~3/2 = {{monzo| 19/32 1/16 0 0 -1/32 }}
: Eigenmonzo basis (unchanged-interval basis): 2.11/9
: Eigenmonzo (unchanged-interval) basis: 2.11/9


Optimal GPV sequence: {{Val list| 12, 41e, 53, 118d, 171de }}
Optimal GPV sequence: {{Val list| 12, 41e, 53, 118d, 171de }}
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Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit: ~3/2 = {{monzo| 19/32 1/16 0 0 -1/32 }}
* 13- and 15-odd-limit: ~3/2 = {{monzo| 19/32 1/16 0 0 -1/32 }}
: Eigenmonzo basis (unchanged-interval basis): 2.11/9
: Eigenmonzo (unchanged-interval) basis: 2.11/9


Optimal GPV sequence: {{Val list| 12f, 41ef, 53, 118d, 171de }}
Optimal GPV sequence: {{Val list| 12f, 41ef, 53, 118d, 171de }}
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Mapping: [{{val| 1 0 15 25 9 }}, {{val| 0 2 -16 -28 -7 }}]
Mapping: [{{val| 1 0 15 25 9 }}, {{val| 0 2 -16 -28 -7 }}]


Mapping generators: ~2, ~110/63
: mapping generators: ~2, ~110/63


Optimal tuning (POTE): ~2 = 1\1, ~110/63 = 951.082 (~63/55 = 248.918)
Optimal tuning (POTE): ~2 = 1\1, ~110/63 = 951.082 (~63/55 = 248.918)
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Mapping: [{{val| 1 0 15 25 9 14 }}, {{val| 0 2 -16 -28 -7 -13 }}]
Mapping: [{{val| 1 0 15 25 9 14 }}, {{val| 0 2 -16 -28 -7 -13 }}]
Mapping generators: ~2, ~26/15


Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 951.082 (~15/13 = 248.918)
Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 951.082 (~15/13 = 248.918)
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Mapping: [{{val| 1 1 7 11 2 }}, {{val| 0 2 -16 -28 5 }}]
Mapping: [{{val| 1 1 7 11 2 }}, {{val| 0 2 -16 -28 5 }}]


Mapping generators: ~2, ~11/9
: mapping generators: ~2, ~11/9


Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.994
Optimal tuning (POTE): ~2 = 1\1, ~11/9 = 350.994
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Mapping: [{{val| 1 2 -1 -3 0 }}, {{val| 0 -3 24 42 25 }}]
Mapping: [{{val| 1 2 -1 -3 0 }}, {{val| 0 -3 24 42 25 }}]


Mapping generators: ~2, ~11/10
: mapping generators: ~2, ~11/10


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.974
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.974
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Mapping: [{{val| 1 2 -1 -3 0 -1 }}, {{val| 0 -3 24 42 25 34 }}]
Mapping: [{{val| 1 2 -1 -3 0 -1 }}, {{val| 0 -3 24 42 25 34 }}]
Mapping generators: ~2, ~11/10


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.963
Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 165.963
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{{See also| Archytas clan #Schism }}
{{See also| Archytas clan #Schism }}


''Schism'' is a low-accuracy extension as it tempers out the septimal comma. The 7/4 is found at -2 fifths, represented by the minor seventh.  
Schism is a low-accuracy extension as it tempers out the septimal comma. The 7/4 is found at -2 fifths, represented by the minor seventh.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Mapping]]: [{{val| 1 0 15 6 }}, {{val| 0 1 -8 -2 }}]
[[Mapping]]: [{{val| 1 0 15 6 }}, {{val| 0 1 -8 -2 }}]


Mapping generators: ~2, ~3
: mapping generators: ~2, ~3


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.556
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.556
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Mapping: [{{val| 1 0 15 6 13 }}, {{val| 0 1 -8 -2 -6 }}]
Mapping: [{{val| 1 0 15 6 13 }}, {{val| 0 1 -8 -2 -6 }}]
Mapping generators: ~2, ~3


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.136
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.136
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{{Main| Pontiac }}
{{Main| Pontiac }}


''Pontiac'' tempers out the [[nanisma]], rendering a very accurate 7-limit microtemperament. The 7/4 is found at +39 fifths, represented by the quintuple augmented third (C-Exx#).  
Pontiac tempers out the [[nanisma]], rendering a very accurate 7-limit microtemperament. The 7/4 is found at +39 fifths, represented by the quintuple augmented third (C-Exx#).  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[Mapping]]: [{{val| 1 0 15 -59 }}, {{val| 0 1 -8 39 }}]
[[Mapping]]: [{{val| 1 0 15 -59 }}, {{val| 0 1 -8 39 }}]


Mapping generators: ~2, ~3
: mapping generators: ~2, ~3


{{Multival|legend=1| 1 -8 39 -15 59 113 }}
{{Multival|legend=1| 1 -8 39 -15 59 113 }}
Line 545: Line 533:
* [[7-odd-limit]]: ~3/2 = {{monzo| 27/47 0 -1/47 1/47 }}
* [[7-odd-limit]]: ~3/2 = {{monzo| 27/47 0 -1/47 1/47 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 74/47 0 -1/47 1/47 }}, {{monzo| 113/47 0 8/47 -8/47 }}, {{monzo| 113/47 0 -39/47 39/47 }}]
: [{{monzo| 1 0 0 0 }}, {{monzo| 74/47 0 -1/47 1/47 }}, {{monzo| 113/47 0 8/47 -8/47 }}, {{monzo| 113/47 0 -39/47 39/47 }}]
: [[Eigenmonzo basis]] ([[unchanged-interval basis]]): 2.7/5
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.7/5
* [[9-odd-limit]]: ~3/2 = {{monzo| 1/2 1/5 -1/10 }}
* [[9-odd-limit]]: ~3/2 = {{monzo| 1/2 1/5 -1/10 }}
: [{{monzo| 1 0 0 0 }}, {{monzo| 3/2 1/5 -1/10 0 }}, {{monzo| 3 -8/5 4/5 0 }}, {{monzo| -1/2 39/5 -39/10 0 }}]
: [{{monzo| 1 0 0 0 }}, {{monzo| 3/2 1/5 -1/10 0 }}, {{monzo| 3 -8/5 4/5 0 }}, {{monzo| -1/2 39/5 -39/10 0 }}]
: Eigenmonzo basis (unchanged-interval basis): 2.9/5
: [[Eigenmonzo basis|Eigenmonzo (unchanged-interval) basis]]: 2.9/5


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 572: Line 560:
Minimax tuning:  
Minimax tuning:  
* 11-odd-limit: ~3/2 = {{monzo| 41/69 0 0 1/69 -1/69 }}
* 11-odd-limit: ~3/2 = {{monzo| 41/69 0 0 1/69 -1/69 }}
: Eigenmonzo basis (unchanged-interval basis): 2.11/7
: Eigenmonzo (unchanged-interval) basis: 2.11/7


Optimal GPV sequence: {{Val list| 53, 118, 289e, 407de }}
Optimal GPV sequence: {{Val list| 53, 118, 289e, 407de }}
Line 589: Line 577:
Minimax tuning:  
Minimax tuning:  
* 13- and 15-odd-limit: ~3/2 = {{monzo| 43/72 0 0 1/72 -1/72 }}
* 13- and 15-odd-limit: ~3/2 = {{monzo| 43/72 0 0 1/72 -1/72 }}
: Eigenmonzo basis (unchanged-interval basis): 2.13/7
: Eigenmonzo (unchanged-interval) basis: 2.13/7


Optimal GPV sequence: {{Val list| 53, 118, 171e }}
Optimal GPV sequence: {{Val list| 53, 118, 171e }}
Line 606: Line 594:
Minimax tuning:  
Minimax tuning:  
* 17-odd-limit: ~3/2 = {{monzo| 18/31 0 0 0 0 -1/93 1/93 }}
* 17-odd-limit: ~3/2 = {{monzo| 18/31 0 0 0 0 -1/93 1/93 }}
: Eigenmonzo basis (unchanged-interval basis): 2.17/13
: Eigenmonzo (unchanged-interval) basis: 2.17/13


Optimal GPV sequence: {{Val list| 53, 118, 171e, 289ef, 460eef }}
Optimal GPV sequence: {{Val list| 53, 118, 171e, 289ef, 460eef }}
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=== Ponta ===
=== Ponta ===
The ''ponta'' temperament (53 & 171) tempers out the [[540/539|swetisma]] and the ragisma.
The ponta temperament (53 & 171) tempers out the [[540/539|swetisma]] and the ragisma.


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 664: Line 652:
Minimax tuning:  
Minimax tuning:  
* 11-odd-limit: ~3/2 = {{monzo| 36/61 0 0 1/122 -1/122 }}
* 11-odd-limit: ~3/2 = {{monzo| 36/61 0 0 1/122 -1/122 }}
: Eigenmonzo basis (unchanged-interval basis): 2.11/7
: Eigenmonzo (unchanged-interval) basis: 2.11/7


Optimal GPV sequence: {{Val list| 53, 171, 224, 1291cde, 1515cde, 1739cddee, 1963cddee, 2187ccddee }}
Optimal GPV sequence: {{Val list| 53, 171, 224, 1291cde, 1515cde, 1739cddee, 1963cddee, 2187ccddee }}
Line 681: Line 669:
Minimax tuning:  
Minimax tuning:  
* 13 and 15-odd-limit: ~3/2 = {{monzo| 36/61 0 0 1/122 -1/122 }}
* 13 and 15-odd-limit: ~3/2 = {{monzo| 36/61 0 0 1/122 -1/122 }}
: Eigenmonzo basis (unchanged-interval basis): 2.11/7
: Eigenmonzo (unchanged-interval) basis: 2.11/7


Optimal GPV sequence: {{Val list| 53, 171, 224 }}
Optimal GPV sequence: {{Val list| 53, 171, 224 }}
Line 698: Line 686:
Minimax tuning:  
Minimax tuning:  
* 17-odd-limit: ~3/2 = {{monzo| 83/143 0 0 0 -1/143 0 1/143 }}
* 17-odd-limit: ~3/2 = {{monzo| 83/143 0 0 0 -1/143 0 1/143 }}
: Eigenmonzo basis (unchanged-interval basis): 2.17/11
: Eigenmonzo (unchanged-interval) basis: 2.17/11


Optimal GPV sequence: {{Val list| 53, 171, 224, 395e, 619eg }}
Optimal GPV sequence: {{Val list| 53, 171, 224, 395e, 619eg }}
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=== Pontic ===
=== Pontic ===
The ''pontic'' temperament (118 & 171) tempers out the [[441/440|werckisma]] and the ragisma.
The pontic temperament (118 & 171) tempers out the [[441/440|werckisma]] and the ragisma.


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 717: Line 705:
Minimax tuning:  
Minimax tuning:  
* 11-odd-limit: ~3/2 = {{monzo| 6/11 0 0 0 1/88 }}
* 11-odd-limit: ~3/2 = {{monzo| 6/11 0 0 0 1/88 }}
: Eigenmonzo basis (unchanged-interval basis): 2.11
: Eigenmonzo (unchanged-interval) basis: 2.11


Optimal GPV sequence: {{Val list| 53e, 118, 289, 407d, 696d }}
Optimal GPV sequence: {{Val list| 53e, 118, 289, 407d, 696d }}
Line 734: Line 722:
Minimax tuning:  
Minimax tuning:  
* 13 and 15-odd-limit: ~3/2 = {{monzo| 71/121 0 0 0 1/121 -1/121 }}
* 13 and 15-odd-limit: ~3/2 = {{monzo| 71/121 0 0 0 1/121 -1/121 }}
: Eigenmonzo basis (unchanged-interval basis): 2.13/11
: Eigenmonzo (unchanged-interval) basis: 2.13/11


Optimal GPV sequence: {{Val list| 53e, 118, 171, 289f, 460ef }}
Optimal GPV sequence: {{Val list| 53e, 118, 171, 289f, 460ef }}
Line 751: Line 739:
Minimax tuning:  
Minimax tuning:  
* 17-odd-limit: ~3/2 = {{monzo| 71/121 0 0 0 1/121 -1/121 }}
* 17-odd-limit: ~3/2 = {{monzo| 71/121 0 0 0 1/121 -1/121 }}
: Eigenmonzo basis (unchanged-interval basis): 2.13/11
: Eigenmonzo (unchanged-interval) basis: 2.13/11


Optimal GPV sequence: {{Val list| 53e, 118, 171, 289f, 460ef }}
Optimal GPV sequence: {{Val list| 53e, 118, 171, 289f, 460ef }}
Line 784: Line 772:


=== Bipont ===
=== Bipont ===
The ''bipont'' temperament (118 & 224) has a period of half octave and tempers out the [[3025/3024|lehmerisma]], 3025/3024 and the [[9801/9800|kalisma]], 9801/9800.
The bipont temperament (118 & 224) has a period of half octave and tempers out the [[3025/3024|lehmerisma]], 3025/3024 and the [[9801/9800|kalisma]], 9801/9800.


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 874: Line 862:
Mapping: [{{val| 4 0 60 -236 -170 -131 }}, {{val| 0 1 -8 39 29 23 }}]
Mapping: [{{val| 4 0 60 -236 -170 -131 }}, {{val| 0 1 -8 39 29 23 }}]


Mapping generators: ~208/175, ~3
: mapping generators: ~208/175, ~3


Optimal tuning (POTE): ~208/175 = 1\4, ~3/2 = 701.756
Optimal tuning (POTE): ~208/175 = 1\4, ~3/2 = 701.756
Line 883: Line 871:


== Grackle ==
== Grackle ==
''Grackle'' tempers out {{monzo| -44 26 0 1 }}. The 7/4 is found at -26 fifths, represented by the triple diminished ninth (C-Dbbbb).  
Grackle tempers out {{monzo| -44 26 0 1 }}. The 7/4 is found at -26 fifths, represented by the triple diminished ninth (C-Dbbbb).  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 891: Line 879:
[[Mapping]]: [{{val| 1 0 15 44 }}, {{val| 0 1 -8 -26 }}]
[[Mapping]]: [{{val| 1 0 15 44 }}, {{val| 0 1 -8 -26 }}]


Mapping generators: ~2, ~3
: mapping generators: ~2, ~3


{{Multival|legend=1| 1 -8 -26 -15 -44 -38 }}
{{Multival|legend=1| 1 -8 -26 -15 -44 -38 }}
Line 1,016: Line 1,004:
[[Mapping]]: [{{val| 2 0 30 69 }}, {{val| 0 1 -8 -20 }}]
[[Mapping]]: [{{val| 2 0 30 69 }}, {{val| 0 1 -8 -20 }}]


Mapping generators: ~567/400, ~3
: mapping generators: ~567/400, ~3


{{Multival|legend=1| 2 -16 -40 -30 -69 -48 }}
{{Multival|legend=1| 2 -16 -40 -30 -69 -48 }}
Line 1,102: Line 1,090:
[[Mapping]]: [{{val| 2 1 22 -15 }}, {{val| 0 2 -16 19 }}]
[[Mapping]]: [{{val| 2 1 22 -15 }}, {{val| 0 2 -16 19 }}]


Mapping generators: ~1225/864, ~35/24
: mapping generators: ~1225/864, ~35/24


{{Multival|legend=1| 4 -32 38 -60 49 178 }}
{{Multival|legend=1| 4 -32 38 -60 49 178 }}


[[Optimal tuning]] ([[POTE]]): ~1225/864 = 1\2, ~36/35 = 50.920
[[Optimal tuning]] ([[POTE]]): ~1225/864 = 1\2, ~35/24 = 650.920 (~36/35 = 50.920)


{{Val list|legend=1| 24, 70c, 94, 118, 212, 330, 542d, 872cd }}
{{Val list|legend=1| 24, 70c, 94, 118, 212, 330, 542d, 872cd }}
Line 1,119: Line 1,107:
Mapping: [{{val| 2 1 22 -15 8 }}, {{val| 0 2 -16 19 -1 }}]
Mapping: [{{val| 2 1 22 -15 8 }}, {{val| 0 2 -16 19 -1 }}]


Optimal tuning (POTE): ~99/70 = 1\2, ~36/35 = 50.918
Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.918 (~36/35 = 50.918)


Optimal GPV sequence: {{Val list| 24, 70c, 94, 118, 212, 330e, 542de }}
Optimal GPV sequence: {{Val list| 24, 70c, 94, 118, 212, 330e, 542de }}
Line 1,132: Line 1,120:
Mapping: [{{val| 2 1 22 -15 8 15 }}, {{val| 0 2 -16 19 -1 -7 }}]
Mapping: [{{val| 2 1 22 -15 8 15 }}, {{val| 0 2 -16 19 -1 -7 }}]


Optimal tuning (POTE): ~99/70 = 1\2, ~36/35 = 50.938
Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.938 (~36/35 = 50.938)


Optimal GPV sequence: {{Val list| 24, 70c, 94, 118, 212f }}
Optimal GPV sequence: {{Val list| 24, 70c, 94, 118, 212f }}
Line 1,145: Line 1,133:
Mapping: [{{val| 2 1 22 -15 8 15 6 }}, {{val| 0 2 -16 19 -1 -7 2 }}]
Mapping: [{{val| 2 1 22 -15 8 15 6 }}, {{val| 0 2 -16 19 -1 -7 2 }}]


Optimal tuning (POTE): ~99/70 = 1\2, ~36/35 = 50.942
Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.942 (~36/35 = 50.942)


Optimal GPV sequence: {{Val list| 24, 70c, 94, 118, 212fg }}
Optimal GPV sequence: {{Val list| 24, 70c, 94, 118, 212fg }}
Line 1,158: Line 1,146:
Mapping: [{{val| 2 1 22 -15 8 -36 }}, {{val| 0 2 -16 19 -1 40 }}]
Mapping: [{{val| 2 1 22 -15 8 -36 }}, {{val| 0 2 -16 19 -1 40 }}]


Optimal tuning (POTE): ~99/70 = 1\2, ~36/35 = 50.951
Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.951 (~36/35 = 50.951)


Optimal GPV sequence: {{Val list| 24f, 70cf, 94, 118f, 212 }}
Optimal GPV sequence: {{Val list| 24f, 70cf, 94, 118f, 212 }}
Line 1,171: Line 1,159:
Mapping: [{{val| 2 1 22 -15 8 -36 6 }}, {{val| 0 2 -16 19 -1 40 2 }}]
Mapping: [{{val| 2 1 22 -15 8 -36 6 }}, {{val| 0 2 -16 19 -1 40 2 }}]


Optimal tuning (POTE): ~99/70 = 1\2, ~36/35 = 50.948
Optimal tuning (POTE): ~99/70 = 1\2, ~35/24 = 650.948 (~36/35 = 50.948)


Optimal GPV sequence: {{Val list| 24f, 70cf, 94, 118f, 212g }}
Optimal GPV sequence: {{Val list| 24f, 70cf, 94, 118f, 212g }}
Line 1,183: Line 1,171:


[[Mapping]]: [{{val| 1 0 15 -17 }}, {{val| 0 2 -16 25 }}]
[[Mapping]]: [{{val| 1 0 15 -17 }}, {{val| 0 2 -16 25 }}]
: mapping generators: ~2, ~140/81


{{Multival|legend=1| 2 -16 25 -30 34 103 }}
{{Multival|legend=1| 2 -16 25 -30 34 103 }}


[[POTE generator]]: ~81/70 = 249.203
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~140/81 = 950.797


{{Val list|legend=1| 24, 53, 130, 183, 313 }}
{{Val list|legend=1| 24, 53, 130, 183, 313 }}
Line 1,199: Line 1,189:
Mapping: [{{val| 1 0 15 -17 51 }}, {{val| 0 2 -16 25 -60 }}]
Mapping: [{{val| 1 0 15 -17 51 }}, {{val| 0 2 -16 25 -60 }}]


POTE generator: ~81/70 = 249.199
Optimal tuning (POTE): ~2 = 1\1, ~140/81 = 950.801


Optimal GPV sequence: {{Val list| 24e, 53, 130, 183, 313 }}
Optimal GPV sequence: {{Val list| 24e, 53, 130, 183, 313 }}
Line 1,212: Line 1,202:
Mapping: [{{val| 1 0 15 -17 51 14 }}, {{val| 0 2 -16 25 -60 -13 }}]
Mapping: [{{val| 1 0 15 -17 51 14 }}, {{val| 0 2 -16 25 -60 -13 }}]


POTE generator: ~15/13 = 249.199
Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 950.801


Optimal GPV sequence: {{Val list| 24e, 53, 130, 183, 313 }}
Optimal GPV sequence: {{Val list| 24e, 53, 130, 183, 313 }}
Line 1,225: Line 1,215:
Mapping: [{{val| 1 0 15 -17 51 14 -49 }}, {{val| 0 2 -16 25 -60 -13 67 }}]
Mapping: [{{val| 1 0 15 -17 51 14 -49 }}, {{val| 0 2 -16 25 -60 -13 67 }}]


POTE generator: ~15/13 = 249.190
Optimal tuning (POTE): ~2 = 1\1, ~26/15 = 950.810


Optimal GPV sequence: {{Val list| 53, 130, 183, 679df }}
Optimal GPV sequence: {{Val list| 53, 130, 183, 679df }}
Line 1,275: Line 1,265:


== Tertiaschis ==
== Tertiaschis ==
The ''tertiaschis'' temperament (94&159) has a ~11/10 generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel]], but tempers out 1071785/1062882 for prime 7.  
The tertiaschis temperament (94 & 159) has a ~11/10 generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel]], but tempers out 1071785/1062882 for prime 7.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,331: Line 1,321:


== Countertertiaschis ==
== Countertertiaschis ==
The ''countertertiaschis'' temperament (159 & 224) has a ~11/10 generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel]], but tempers out 244140625/243045684 for prime 7.  
The countertertiaschis temperament (159 & 224) has a ~11/10 generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel]], but tempers out 244140625/243045684 for prime 7.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,425: Line 1,415:
[[Mapping]]: [{{val| 3 0 45 94 }}, {{val| 0 1 -8 -18 }}]
[[Mapping]]: [{{val| 3 0 45 94 }}, {{val| 0 1 -8 -18 }}]


Mapping generators: ~63/50, ~3
: mapping generators: ~63/50, ~3


{{Multival|legend=1| 3 -24 -54 -45 -94 -58 }}
{{Multival|legend=1| 3 -24 -54 -45 -94 -58 }}
Line 1,432: Line 1,422:


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]] [[Eigenmonzo basis]] ([[unchanged-interval basis]]): 2.5/3
* [[7-odd-limit]] [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis)]]: 2.5/3
* [[9-odd-limit]] Eigenmonzo basis (unchanged-interval basis): 2.9/7
* [[9-odd-limit]] eigenmonzo (unchanged-interval) basis: 2.9/7


{{Val list|legend=1| 12, 147d, 159, 171, 867, 1038, 1209, 1380, 1551, 1722 }}
{{Val list|legend=1| 12, 147d, 159, 171, 867, 1038, 1209, 1380, 1551, 1722 }}
Line 1,440: Line 1,430:


=== Terminal ===
=== Terminal ===
The ''terminal'' temperament (12 & 159) tempers out 441/440 and 4375/4356. In this temperament, 44/35 and 63/50 are represented as one period of 1/3 octave.  
The terminal temperament (12 & 159) tempers out 441/440 and 4375/4356. In this temperament, 44/35 and 63/50 are represented as one period of 1/3 octave.  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 1,520: Line 1,510:


=== Semiterm ===
=== Semiterm ===
The ''semiterm'' temperament (12 & 342) has a period of 1/6 octave and tempers out [[9801/9800]] (kalisma) and 151263/151250 (odiheim comma).
The semiterm temperament (12 & 342) has a period of 1/6 octave and tempers out [[9801/9800]] (kalisma) and 151263/151250 (odiheim comma).


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
Line 1,558: Line 1,548:
Mapping: [{{val| 3 0 45 94 8 }}, {{val| 0 2 -16 -36 1 }}]
Mapping: [{{val| 3 0 45 94 8 }}, {{val| 0 2 -16 -36 1 }}]


Mapping generators: ~63/50, ~693/400
: mapping generators: ~63/50, ~693/400


Optimal tuning (POTE): ~63/50 = 1\3, ~693/400 = 950.872 (~12/11 = 150.872)
Optimal tuning (POTE): ~63/50 = 1\3, ~693/400 = 950.872 (~12/11 = 150.872)
Line 1,599: Line 1,589:
[[Mapping]]: [{{val| 3 0 45 -32 }}, {{val| 0 2 -16 17 }}]
[[Mapping]]: [{{val| 3 0 45 -32 }}, {{val| 0 2 -16 17 }}]


Mapping generators: ~1536/1225, ~34300/19683
: mapping generators: ~1536/1225, ~34300/19683


[[Optimal tuning]] ([[CTE]]): ~1536/1225 = 1\3, ~34300/19683 = 950.9654
[[Optimal tuning]] ([[CTE]]): ~1536/1225 = 1\3, ~34300/19683 = 950.9654
Line 1,627: Line 1,617:
[[Mapping]]: [{{val| 1 1 7 5 }}, {{val| 0 4 -32 -15 }}]
[[Mapping]]: [{{val| 1 1 7 5 }}, {{val| 0 4 -32 -15 }}]


Mapping generators: ~2, ~448/405
: mapping generators: ~2, ~448/405


{{Multival|legend=1| 4 -32 -15 -60 -35 55 }}
{{Multival|legend=1| 4 -32 -15 -60 -35 55 }}
Line 1,634: Line 1,624:


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]] eigenmonzos (unchanged-intervals): 2, 7/6
* [[7-odd-limit]] [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/3
* [[9-odd-limit]] eigenmonzos (unchanged-intervals): 2, 9/7
* [[9-odd-limit]] eigenmonzo (unchanged-interval) basis: 2.9/7


{{Val list|legend=1| 41, 89, 130, 171, 814, 985, 1156, 1327, 1498, 2825bd }}
{{Val list|legend=1| 41, 89, 130, 171, 814, 985, 1156, 1327, 1498, 2825bd }}
Line 1,647: Line 1,637:


Mapping: [{{val| 1 1 7 5 2 }}, {{val| 0 4 -32 -15 10 }}]
Mapping: [{{val| 1 1 7 5 2 }}, {{val| 0 4 -32 -15 10 }}]
Mapping generators: ~2, ~256/231


Optimal tuning (POTE): ~2 = 1\1, ~256/231 = 175.406
Optimal tuning (POTE): ~2 = 1\1, ~256/231 = 175.406
Line 1,675: Line 1,663:


Mapping: [{{val| 2 2 14 10 23 }}, {{val| 0 4 -32 -15 -55 }}]
Mapping: [{{val| 2 2 14 10 23 }}, {{val| 0 4 -32 -15 -55 }}]
: mapping generators: ~99/70, ~448/405


Optimal tuning (POTE): ~99/70 = 1\2, ~448/405 = 175.435
Optimal tuning (POTE): ~99/70 = 1\2, ~448/405 = 175.435
Line 1,683: Line 1,673:


== Quintilipyth ==
== Quintilipyth ==
The ''quintilipyth'' temperament (12 & 253, formerly ''quintilischis'') slices the pythagorean fourth ([[4/3]]) into five semitones and tempers out the compass comma (9765625/9680832) in the 7-limit.
The quintilipyth temperament (12 & 253, formerly ''quintilischis'') slices the pythagorean fourth ([[4/3]]) into five semitones and tempers out the compass comma (9765625/9680832) in the 7-limit.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,752: Line 1,742:


== Quintaschis ==
== Quintaschis ==
The ''quintaschis'' temperament (12 & 289) slices the fourth (4/3) into five semitones and tempers out 49009212/48828125 in the 7-limit.
The quintaschis temperament (12 & 289) slices the fourth (4/3) into five semitones and tempers out 49009212/48828125 in the 7-limit.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 1,920: Line 1,910:
[[Mapping]]: [{{val| 1 2 -1 -1 }}, {{val| 0 -6 48 55 }}]
[[Mapping]]: [{{val| 1 2 -1 -1 }}, {{val| 0 -6 48 55 }}]


Mapping generators: ~2, ~21/20
: mapping generators: ~2, ~21/20


{{Multival|legend=1| 6 -48 -55 -90 -104 7 }}
{{Multival|legend=1| 6 -48 -55 -90 -104 7 }}
Line 1,957: Line 1,947:


== Septiquarschis ==
== Septiquarschis ==
The ''septiquarschis'' temperament (89 & 94) splits septimal minor seventh ([[7/4]]) into four generators and tempers out 829440/823543 (mynaslender comma) and 67108864/66706983 (septiness comma).
The septiquarschis temperament (89 & 94) splits septimal minor seventh ([[7/4]]) into four generators and tempers out 829440/823543 (mynaslender comma) and 67108864/66706983 (septiness comma).


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 2,008: Line 1,998:
[[Mapping]]: [{{val| 1 1 7 0 }}, {{val| 0 5 -40 24 }}]
[[Mapping]]: [{{val| 1 1 7 0 }}, {{val| 0 5 -40 24 }}]


Mapping generators: ~2, ~243/224
: mapping generators: ~2, ~243/224


{{Multival|legend=1| 5 -40 24 -75 24 168 }}
{{Multival|legend=1| 5 -40 24 -75 24 168 }}
Line 2,051: Line 2,041:
[[Mapping]]: [{{val| 1 0 15 12 }}, {{val| 0 5 -40 -29 }}]
[[Mapping]]: [{{val| 1 0 15 12 }}, {{val| 0 5 -40 -29 }}]


Mapping generators: ~2, ~56/45
: mapping generators: ~2, ~56/45


{{Multival|legend=1| 5 -40 -29 -75 -60 45 }}
{{Multival|legend=1| 5 -40 -29 -75 -60 45 }}
Line 2,067: Line 2,057:


Mapping: [{{val| 1 0 15 12 -7 }}, {{val| 0 5 -40 -29 33 }}]
Mapping: [{{val| 1 0 15 12 -7 }}, {{val| 0 5 -40 -29 33 }}]
Mapping generators: ~2, ~56/45


Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 380.352
Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 380.352
Line 2,082: Line 2,070:


Mapping: [{{val| 1 0 15 12 -7 -15 }}, {{val| 0 5 -40 -29 33 59 }}]
Mapping: [{{val| 1 0 15 12 -7 -15 }}, {{val| 0 5 -40 -29 33 59 }}]
Mapping generators: ~2, ~56/45


Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 380.351
Optimal tuning (POTE): ~2 = 1\1, ~56/45 = 380.351
Line 2,100: Line 2,086:
[[Mapping]]: [{{val| 4 0 60 119 }}, {{val| 0 1 -8 -17 }}]
[[Mapping]]: [{{val| 4 0 60 119 }}, {{val| 0 1 -8 -17 }}]


Mapping generators: ~25/21, ~3
: mapping generators: ~25/21, ~3


{{Multival|legend=1| 4 -32 -68 -60 -119 -68 }}
{{Multival|legend=1| 4 -32 -68 -60 -119 -68 }}
Line 2,145: Line 2,131:
[[Mapping]]: [{{val| 7 0 105 -56 }}, {{val| 0 1 -8 7 }}]
[[Mapping]]: [{{val| 7 0 105 -56 }}, {{val| 0 1 -8 7 }}]


Mapping generators: ~8575/7776, ~3
: mapping generators: ~8575/7776, ~3


{{Multival|legend=1| 7 -56 49 -105 58 271 }}
{{Multival|legend=1| 7 -56 49 -105 58 271 }}
Line 2,190: Line 2,176:
[[Mapping]]: [{{val| 8 0 120 -117 }}, {{val| 0 1 -8 11 }}]
[[Mapping]]: [{{val| 8 0 120 -117 }}, {{val| 0 1 -8 11 }}]


Mapping generators: ~42875/39366, ~3
: mapping generators: ~42875/39366, ~3


{{Multival|legend=1| 8 -64 88 -120 117 384 }}
{{Multival|legend=1| 8 -64 88 -120 117 384 }}
Line 2,227: Line 2,213:


== Tridecafifths ==
== Tridecafifths ==
''Tridecafifths'' divides the perfect 3/2 into 13 quartertones.  
Tridecafifths divides the perfect 3/2 into 13 quartertones.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 2,235: Line 2,221:
[[Mapping]]: [{{val| 1 1 7 6 }}, {{val| 0 13 -104 -71 }}]
[[Mapping]]: [{{val| 1 1 7 6 }}, {{val| 0 13 -104 -71 }}]


Mapping generators: ~2, ~1323/1280
: mapping generators: ~2, ~1323/1280


[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~1323/1280 = 53.9741
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~1323/1280 = 53.9741
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