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== Temperament pages ==
Databoxes has been canceled, but the cleanup will continue
Note:
# Order: subgroup, [[comma list]], [[mapping]], mapping generators, gencom mapping, [[gencom]], map to lattice, lattice basis, [[wedgie]], [[minimax tuning]], [[tuning ranges]], [[algebraic generator]], vals, [[badness]], [[complexity spectrum]].
# Comma list shows the simplest commas sufficient to define the temperament, stated in [[Normal lists #Normal interval list]].
# Mapping generators should show all the ratios as used in the mapping, including the period.
# Minimax tuning are based on tonality diamond, so it should explicitly state the ''odd'' limit, or a diamond function of ratios.
# Use [[Template:Val list]].
# For subgroup temperaments, "mapping" becomes "sval mapping", add "gencom mapping" and "gencom". If TE is TE is TE (''sic''), simply show "POTE", otherwise show "POL2" or "POT2" instead of "POTE".
Get a family for:
* <s>Ripple (3 different 7-limit extensions)</s>
* <s>Smate (2 different 7-limit extensions)</s>
* Maybe [[parakleismic]] (2 different 7-limit extensions)
* Maybe [[superpyth]] (2 different 7-limit extensions)
Who's next?
* <s>Meantone family</s>
* <s>Archytas clan</s>
* <s>Father family</s>
* <s>Trienstonic clan</s>
* <s>Septisemi temperaments</s>
* <s>Archytas family</s>
* <s>Slendro clan</s>
* <s>Semiphore family</s>
* <s>Marvel temperaments</s>
* <s>Marvel family</s>
* <s>Mint temperaments</s>
* <s>Mint family</s>
* [[Gamelismic clan]]
* <s>Gamelismic family</s>
* <s>Jubilismic clan</s>
* <s>Jubilismic family</s>
* <s>Didymus rank three family</s>
* <s>Kleismic family</s>
* [[Kleismic rank three family]]
* <s>Shibboleth family</s>
* <s>Keemic temperaments</s>
* <s>Keemic family</s>
* <s>Schismatic family</s>
* <s>Hemimean clan</s>
* <s>Hemimean family</s>
* <s>Luna family</s>
* <s>Canousmic temperaments</s>
* <s>Canou family</s>
* [[Starling temperaments]]
* [[Starling family]]
* <s>Sensipent family</s>
* [[Sensamagic clan]]
* [[Sensamagic family]]
* [[Magic family]]
* [[Unicorn family]]
* [[Trisedodge family]]
== Septimal meantone ==
<span style="display: block; text-align: right;">[[:de:septimal-mitteltönig|Deutsch]]</span>
{{main| Meantone }}
{{see also| Wikipedia: Septimal meantone temperament }}
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.
Subgroup: 2.3.5.7
[[Comma list]]: 81/80, 126/125
[[Mapping]]: [{{val|1 0 -4 -13}}, {{val|0 1 4 10}}]
Mapping generators: ~2, ~3
{{Multival|legend=1| 1 4 10 4 13 12 }}
[[POTE generator]]: ~3/2 = 696.495
[[Minimax tuning]]:
* 7- and [[9-odd-limit]]
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| -3 0 5/2 0 }}]
: [[Eigenmonzo]]s: 2, 5
[[Tuning ranges]]:
* valid range: [694.737, 700.000] (11\19 to 7\12)
* nice range: [694.786, 701.955]
* strict range: [694.786, 700.000]
[[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.
{{Val list|legend=1| 12, 19, 31, 81, 112b, 143b }}
[[Badness]]: 0.0137
Scales: [[meantone5]], [[meantone7]], [[meantone12]]
== Archytas ==
{{main| Archytas }}
Subgroup: 2.3.5.7
[[Comma list]]: 64/63
[[Mapping]]: [{{val| 1 0 0 6 }}, {{val| 0 1 0 -2 }}, {{val| 0 0 1 0 }}]
Mapping generators: ~2, ~3, ~5
Map to lattice: [{{val| 0 1 0 -2 }}, {{val| 0 0 1 0 }}]
Lattice basis:
: 3/2 length = 1.0508, 5/4 length = 2.3219
: Angle (3/2, 5/4) = 90 degrees
[[POTE generator]]s: ~3/2 = 709.3213, ~5/4 = 393.3747
[[Minimax tuning]]:
* [[7-odd-limit]]
: [{{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 }}]
: [[Eigenmonzo]]s: 2, 6/5, 7/5
* [[9-odd-limit]]
: [{{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 }}]
: [[Eigenmonzo]]s: 2, 6/5, 9/7
{{Val list|legend=1| 5, 7, 10, 12, 15, 22, 27, 49 }}
Scales: [[archytas12]], [[archytas12synch]]
== Scale tree ==
== Scale tree ==
Version 1 (bounded by branch depth = 7)
(Bounded by branch depth = 7)


{| class="wikitable center-all"
{| class="wikitable center-all"
Line 96: Line 220:
| || || || || || 44\75 || || 704.000 || 13 || 5 || 2.600 ||
| || || || || || 44\75 || || 704.000 || 13 || 5 || 2.600 ||
|-
|-
| || || || || || || 71\121 || 704.132 || 21 || 8 || 2.625 || Golden parapyth
| || || || || || || 71\121 || 704.132 || 21 || 8 || 2.625 || Golden neogothic
|-
|-
| || || || || 27\46 || || || 704.348 || 8 || 3 || 2.667 ||
| || || || || 27\46 || || || 704.348 || 8 || 3 || 2.667 ||
Line 140: Line 264:
| 3\5 || || || || || || || 720.000 || 1 || 0 || → inf ||
| 3\5 || || || || || || || 720.000 || 1 || 0 || → inf ||
|}
|}
Version 2 (bounded by integer limit = 10)
{| class="wikitable center-all"
! colspan="10" | Generator
! Cents
! L
! s
! L/s
! Comments
|-
| 4\7 || || || || || || || || || || 685.714 || 1 || 1 || 1.000 ||
|-
| || || || || || || || || || 39\68 || 688.235 || 10 || 9 || 1.111 ||
|-
| || || || || || || || || 35\61 || || 688.525 || 9 || 8 || 1.125 ||
|-
| || || || || || || || 31\54 || || || 688.889 || 8 || 7 || 1.143 ||
|-
| || || || || || || 27\47 || || || || 689.362 || 7 || 6 || 1.167 ||
|-
| || || || || || 23\40 || || || || || 690.000 || 6 || 5 || 1.200 ||
|-
| || || || || 19\33 || || || || || || 690.909 || 5 || 4 || 1.250 ||
|-
| || || || || || 34\59 || || || || || 691.525 || 9 || 7 || 1.286 ||
|-
| || || || 15\26 || || || || || || || 692.308 || 4 || 3 || 1.333 ||
|-
| || || || || 26\45 || || || || || || 693.333 || 7 || 5 || 1.400 ||
|-
| || || || || || 37\64 || || || || || 693.750 || 10 || 7 || 1.429 ||
|-
| || || 11\19 || || || || || || || || 694.737 || 3 || 2 || 1.500 || L/s = 3/2
|-
| || || || || 29\50 || || || || || || 696.000 || 8 || 5 || 1.600 || Golden meantone
|-
| || || || 18\31 || || || || || || || 696.774 || 5 || 3 || 1.667 || Meantone is in this region
|-
| || || || || 25\43 || || || || || || 697.674 || 7 || 4 || 1.750 ||
|-
| || || || || ||32\55 || || || || || 698.182 || 9 || 5 || 1.800 ||
|-
| || 7\12 || || || || || || || || || 700.000 || 2 || 1 || 2.000 || Basic diatonic<br>(Generators smaller than this are proper)
|-
| || || || || || 31\53 || || || || || 701.887 || 9 || 4 || 2.250 ||
|-
| || || || || 24\41 || || || || || || 702.409 || 7 || 3 || 2.333 ||
|-
| || || || 17\29 || || || || || || || 703.448 || 5 || 2 || 2.500 ||
|-
| || || || || 27\46 || || || || || || 704.348 || 8 || 3 || 2.667 || Golden parapyth
|-
| || || 10\17 || || || || || || || || 705.882 || 3 || 1 || 3.000 || L/s = 3/1
|-
| || || || || || 33\56 || || || || || 707.143 || 10 || 3 || 3.333 ||
|-
| || || || || 23\39 || || || || || || 707.692 || 7 || 2 || 3.500 ||
|-
| || || || 13\22 || || || || || || || 709.091 || 4 || 1 || 4.000 || Archy is in this region
|-
| || || || || || 29\49 || || || || || 710.204 || 9 || 2 || 4.500 ||
|-
| || || || || 16\27 || || || || || || 711.111 || 5 || 1 || 5.000 ||
|-
| || || || || || 19\32 || || || || || 712.500 || 6 || 1 || 6.000 ||
|-
| || || || || || || 22\37 || || || || 713.514 || 7 || 1 || 7.000 ||
|-
| || || || || || || || 25\42 || || ||714.286 || 8 || 1 || 8.000 ||
|-
| || || || || || || || || 28\47 || ||714.894 || 9 || 1 || 9.000 ||
|-
| || || || || || || || || || 31\52 ||715.385 || 10 || 1 || 10.000 ||
|-
| 3\5 || || || || || || || || || || 720.000 || 1 || 0 || → inf ||
|}
== Temperament pages ==
Databoxes has been canceled, but the cleanup will continue
Note:
# Order: subgroup, [[comma list]], [[mapping]], mapping generators, gencom mapping, [[gencom]], map to lattice, lattice basis, [[wedgie]], [[minimax tuning]], [[tuning ranges]], [[algebraic generator]], vals, [[badness]], [[complexity spectrum]].
# Comma list shows the simplest commas sufficient to define the temperament, stated in [[Normal lists #Normal interval list]].
# Mapping generators should show all the ratios as used in the mapping, including the period.
# Minimax tuning are based on tonality diamond, so it should explicitly state the ''odd'' limit, or a diamond function of ratios.
# Use [[Template:Val list]].
# For subgroup temperaments, "mapping" becomes "sval mapping", add "gencom mapping" and "gencom". If TE is TE is TE (''sic''), simply show "POTE", otherwise show "POL2" or "POT2" instead of "POTE".
Get a family for:
* <s>Ripple (3 different 7-limit extensions)</s>
* <s>Smate (2 different 7-limit extensions)</s>
* Maybe [[parakleismic]] (2 different 7-limit extensions)
* Maybe [[superpyth]] (2 different 7-limit extensions)
Who's next?
* <s>Meantone family</s>
* <s>Archytas clan</s>
* <s>Father family</s>
* <s>Trienstonic clan</s>
* <s>Septisemi temperaments</s>
* <s>Archytas family</s>
* <s>Slendro clan</s>
* <s>Semiphore family</s>
* <s>Marvel temperaments</s>
* <s>Marvel family</s>
* <s>Mint temperaments</s>
* <s>Mint family</s>
* [[Gamelismic clan]]
* <s>Gamelismic family</s>
* <s>Jubilismic clan</s>
* <s>Jubilismic family</s>
* <s>Didymus rank three family</s>
* <s>Kleismic family</s>
* [[Kleismic rank three family]]
* <s>Shibboleth family</s>
* <s>Keemic temperaments</s>
* <s>Keemic family</s>
* <s>Schismatic family</s>
* <s>Hemimean clan</s>
* <s>Hemimean family</s>
* <s>Luna family</s>
* <s>Canousmic temperaments</s>
* <s>Canou family</s>
* [[Starling temperaments]]
* [[Starling family]]
* <s>Sensipent family</s>
* [[Sensamagic clan]]
* [[Sensamagic family]]
* [[Magic family]]
== Septimal meantone ==
<span style="display: block; text-align: right;">[[:de:septimal-mitteltönig|Deutsch]]</span>
{{main| Meantone }}
{{see also| Wikipedia: Septimal meantone temperament }}
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.
Subgroup: 2.3.5.7
[[Comma list]]: 81/80, 126/125
[[Mapping]]: [{{val|1 0 -4 -13}}, {{val|0 1 4 10}}]
Mapping generators: ~2, ~3
{{Multival|legend=1| 1 4 10 4 13 12 }}
[[POTE generator]]: ~3/2 = 696.495
[[Minimax tuning]]:
* 7- and [[9-odd-limit]]
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| -3 0 5/2 0 }}]
: [[Eigenmonzo]]s: 2, 5
[[Tuning ranges]]:
* valid range: [694.737, 700.000] (11\19 to 7\12)
* nice range: [694.786, 701.955]
* strict range: [694.786, 700.000]
[[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.
{{Val list|legend=1| 12, 19, 31, 81, 112b, 143b }}
[[Badness]]: 0.0137
Scales: [[meantone5]], [[meantone7]], [[meantone12]]
== Archytas ==
{{main| Archytas }}
Subgroup: 2.3.5.7
[[Comma list]]: 64/63
[[Mapping]]: [{{val| 1 0 0 6 }}, {{val| 0 1 0 -2 }}, {{val| 0 0 1 0 }}]
Mapping generators: ~2, ~3, ~5
Map to lattice: [{{val| 0 1 0 -2 }}, {{val| 0 0 1 0 }}]
Lattice basis:
: 3/2 length = 1.0508, 5/4 length = 2.3219
: Angle (3/2, 5/4) = 90 degrees
[[POTE generator]]s: ~3/2 = 709.3213, ~5/4 = 393.3747
[[Minimax tuning]]:
* [[7-odd-limit]]
: [{{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 }}]
: [[Eigenmonzo]]s: 2, 6/5, 7/5
* [[9-odd-limit]]
: [{{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 }}]
: [[Eigenmonzo]]s: 2, 6/5, 9/7
{{Val list|legend=1| 5, 7, 10, 12, 15, 22, 27, 49 }}
Scales: [[archytas12]], [[archytas12synch]]


== Commas ==
== Commas ==
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