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| == Regular temperament properties == | | == Regular temperament properties == |
| {| class="wikitable center-4 center-5 center-6" | | {| class="wikitable center-4 center-5 center-6" |
| ! Subgroup | | ! rowspan="2" | Subgroup |
| ! Comma List | | ! rowspan="2" | [[Comma list]] |
| ! Mapping | | ! rowspan="2" | [[Mapping]] |
| ! TE Octave<br>Stretch (¢) | | ! rowspan="2" | Optimal<br>8ve stretch (¢) |
| ! TE Absolute<br>Error (¢) | | ! colspan="2" | Tuning error |
| ! TE Relative<br>Error (%) | | |- |
| | ! [[TE error|Absolute]] (¢) |
| | ! [[TE simple badness|Relative]] (%) |
| |- | | |- |
| | 2.3 | | | 2.3 |
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| | 3.84 | | | 3.84 |
| |} | | |} |
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| == Septimal meantone ==
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| <span style="display: block; text-align: right;">[[:de:septimal-mitteltönig|Deutsch]]</span>
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| {{main| Meantone }}
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| {{see also| Wikipedia: Septimal meantone temperament }}
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| The [[7/4]] of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are [[7/6]], C-D#, the augmented second, [[7/5]], C-F#, the tritone, and [[21/16]], C-E#, the augmented third. Septimal meantone tempers out the common 7-limit commas [[126/125]] and [[225/224]] and in fact can be defined as the 7-limit temperament that tempers out any two of 81/80, 126/125 and 225/224.
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| Subgroup: 2.3.5.7
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| [[Comma list]]: 81/80, 126/125
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| [[Mapping]]: [{{val|1 0 -4 -13}}, {{val|0 1 4 10}}]
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| Mapping generators: ~2, ~3
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| {{Multival|legend=1| 1 4 10 4 13 12 }}
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| [[POTE generator]]: ~3/2 = 696.495
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| [[Minimax tuning]]:
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| * 7- and [[9-odd-limit]]
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| : [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| -3 0 5/2 0 }}]
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| : [[Eigenmonzo]]s: 2, 5
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| [[Tuning ranges]]:
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| * valid range: [694.737, 700.000] (11\19 to 7\12)
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| * nice range: [694.786, 701.955]
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| * strict range: [694.786, 700.000]
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| [[Algebraic generator]]: Cybozem, the real root of 15''x''<sup>3</sup> - 10''x''<sup>2</sup> - 18, 503.4257 cents. The recurrence converges quickly.
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| {{Val list|legend=1| 12, 19, 31, 81, 112b, 143b }}
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| [[Badness]]: 0.0137
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| Scales: [[meantone5]], [[meantone7]], [[meantone12]]
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| == Archytas ==
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| {{main| Archytas }}
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| Subgroup: 2.3.5.7
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| [[Comma list]]: 64/63
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| [[Mapping]]: [{{val| 1 0 0 6 }}, {{val| 0 1 0 -2 }}, {{val| 0 0 1 0 }}]
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| Mapping generators: ~2, ~3, ~5
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| Map to lattice: [{{val| 0 1 0 -2 }}, {{val| 0 0 1 0 }}]
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| Lattice basis:
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| : 3/2 length = 1.0508, 5/4 length = 2.3219
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| : Angle (3/2, 5/4) = 90 degrees
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| [[POTE generator]]s: ~3/2 = 709.3213, ~5/4 = 393.3747
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| [[Minimax tuning]]:
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| * [[7-odd-limit]]
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| : [{{monzo| 1 0 0 0 }}, {{monzo| 2 1/3 0 -1/3 }}, {{monzo| 2 -2/3 1 -1/3 }}, {{monzo| 2 -2/3 0 2/3 }}]
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| : [[Eigenmonzo]]s: 2, 6/5, 7/5
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| * [[9-odd-limit]]
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| : [{{monzo| 1 0 0 0 }}, {{monzo| 3/2 1/2 0 -1/4 }}, {{monzo| 3/2 -1/2 1 -1/4 }}, {{monzo| 3 -1 0 1/2 }}]
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| : [[Eigenmonzo]]s: 2, 6/5, 9/7
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| {{Val list|legend=1| 5, 7, 10, 12, 15, 22, 27, 49 }}
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| Scales: [[archytas12]], [[archytas12synch]]
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| == Scale tree == | | == Scale tree == |