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Below are some important [[rank]]-2 [[temperaments]] with optimal [[generator]] size in the diatonic ([[5L 2s]]) range (the [[period]] is always 1\1 for temperaments with this MOS structure). The temperaments are listed following the 5L 2s scale tree, in order of increasing generator size. The top-level temperaments are the most important and obvious divisions in diatonic tunings. Child temperaments are higher-complexity extensions of low-complexity parent temperaments, with new JI readings for intervals further out in the generator chain. These are finer adjustments of the major, parent temperaments, thus are less useful when the composer chooses not to use a long generator chain in the music.  
{{breadcrumb}}
Below are some important [[rank]]-2 [[temperaments]] with optimal [[generator]] size in the diatonic ([[5L 2s]]) range (the [[period]] is always 1\1 for temperaments with this MOS structure). The temperaments are listed following the 5L 2s scale tree, in order of increasing generator size. The top-level temperaments are the most important and obvious divisions in diatonic tunings. Child temperaments are higher-complexity extensions of low-complexity parent temperaments, with new JI readings for intervals further out in the generator chain. These are finer adjustments of the major, parent temperaments, thus are less useful when the composer chooses not to use a long generator chain in the music.  


== 5-limit meantone ==
== Meantone ==
{{main| Meantone }}
{{main| Meantone }}
Subgroup: 2.3.5
Subgroup: 2.3.5
Line 13: Line 14:
Scales (Scala files): [[Meantone5]], [[Meantone7]], [[Meantone12]]
Scales (Scala files): [[Meantone5]], [[Meantone7]], [[Meantone12]]


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div class="toccolours mw-collapsible mw-collapsed" style="width: 600px; overflow: auto;">
<div style="line-height:1.6;">Interval table (7-note MOS, 2.3.5.7 POTE tuning)</div>
<div style="line-height: 1.6;">Interval table (7-note MOS, 2.3.5.7 POTE tuning)</div>
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
{| class="wikitable right-1 right-2 sortable"
|+
|-
|-
! #
! &#35;
! Cents<ref>octave-reduced</ref>
! Cents*
! class="unsortable"| Approximate ratios<ref>2.3.5, odd limit ≤ 27</ref>
! class="unsortable" | Approximate ratios**
|-
|-
| 0
| 0
Line 52: Line 51:
| 25/18
| 25/18
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
 
<nowiki />* 2.3.5, odd limit &le; 27</div></div>


Comma list: 81/80
Comma list: 81/80
Line 59: Line 60:


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
[[Wedgie]]: {{wedgie| 1 4 4 }}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* 5-odd-limit [[diamond monotone]]: ~3/2 = [685.714, 720.000] (4\7 to 3\5)
* 5-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 701.955]
* 5-odd-limit diamond monotone and tradeoff: ~3/2 = [694.786, 701.955]


* [[diamond monotone]] range: [685.714, 720.000] (7 to 5)
{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b, 212bb, 293bb }}
* [[diamond tradeoff]] range: [694.786, 701.955] (1/3 comma to Pythagorean)
* diamond monotone and tradeoff range: [694.786, 701.955]
 
{{Vals|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b, 212bb, 293bb }}


[[Badness]]: 0.00736
[[Badness]]: 0.007381


=== Flattone ===
=== Flattone ===
Subgroup: 2.3.5.7.13
Subgroup: 2.3.5.7 or 2.3.5.7.13


Period: 1\1
Period: 1\1
Line 83: Line 81:
Scales (Scala files): [[Flattone12]]
Scales (Scala files): [[Flattone12]]


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div class="toccolours mw-collapsible mw-collapsed" style="width: 600px; overflow: auto;">
<div style="line-height:1.6;">Interval table (12-note MOS, 2.3.5.7.13 POTE tuning)</div>
<div style="line-height: 1.6;">Interval table (12-note MOS, 2.3.5.7.13 POTE tuning)</div>
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
{| class="wikitable right-1 right-2 sortable"
|+
|-
|-
! #Gens up
! #Gens up
! Cents <ref>octave-reduced</ref>
! Cents*
! class="unsortable"| Approximate ratios<ref>2.3.5.7.13, odd limit ≤ 27</ref>
! class="unsortable" | Approximate ratios**
|-
|-
| 0
| 0
Line 141: Line 138:
| [[9/7]]
| [[9/7]]
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced


[[Comma]] list: 81/80, 525/512
<nowiki />** 2.3.5.7.13, odd limit &le; 27</div></div>


[[Mapping]]: [{{val|1 0 -4 17}}, {{val|0 1 4 -9}}]
[[Comma list]]: 65/64, 81/80, 105/104


Mapping generators: ~2, ~3
[[Gencom]]: [2 3/2; 65/64 81/80 105/104]


[[Wedgie]]: {{wedgie|1 4 -9 4 -17 -32}}
[[Gencom]] [[mapping]]: [{{val|1 1 0 8 6}}, {{val|0 1 4 -9 -4}}]


[[Minimax tuning]]:  
[[Minimax tuning]]:  
 
* [[7-odd-limit]]: ~3/2 = {{Monzo| 8/13 0 1/13 -1/13 }}
* [[7-odd-limit]]
: [[Eigenmonzo]]s (unchanged-intervals): 2, 7/5
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 21/13 0 1/13 -1/13 }}, {{Monzo| 32/13 0 4/13 -4/13 }}, {{Monzo| 32/13 0 -9/13 9/13 }}]
* [[9-odd-limit]]: ~3/2 = {{monzo| 6/11 2/11 0 -1/11 }}
: [[Eigenmonzo]]s (unchanged intervals): 2, 7/5
: Eigenmonzos (unchanged-intervals): 2, 9/7
 
* [[9-odd-limit]]
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 17/11 2/11 0 -1/11 }}, {{Monzo| 24/11 8/11 0 -4/11 }}, {{Monzo| 34/11 -18/11 0 9/11 }}]
: Eigenmonzos (unchanged intervals): 2, 9/7


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* [[diamond monotone]] range: [692.308, 694.737] (26 to 19)
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* [[diamond tradeoff]] range: [692.353, 701.955]
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [692.353, 701.955]
* diamond monotone and tradeoff range: [692.353, 694.737]
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]
* 7-odd-limit diamond monotone and tradeoff: ~3/2 = [692.353, 694.737]
* 9-odd-limit diamond monotone and tradeoff: ~3/2 = [692.308, 694.737]


Algebraic generator: Squarto, the positive root of 8''x''<sup>2</sup> - 4''x'' - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.
Algebraic generator: Squarto, the positive root of {{nowrap|8''x''<sup>2</sup> − 4''x'' − 9}}, at 506.3239{{c}}, equal to <math>\frac{1+\sqrt{19}}{4}</math>.


{{Vals|legend=1| 7, 19, 26, 45 }}
{{Optimal ET sequence|legend=1| 7, 19, 26, 45f }}


[[Badness]]: 0.0386
[[Tp_tuning#T2 tuning|RMS error]]: 1.742 cents


=== Septimal meantone ===
=== Septimal meantone ===
Line 183: Line 178:
Scales (Scala files): [[Meantone5]], [[Meantone7]], [[Meantone12]]
Scales (Scala files): [[Meantone5]], [[Meantone7]], [[Meantone12]]


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div class="toccolours mw-collapsible mw-collapsed" style="width: 600px; overflow: auto;">
<div style="line-height:1.6;">Interval table (12-note MOS, 2.3.5.7 POTE tuning)</div>
<div style="line-height: 1.6;">Interval table (12-note MOS, 2.3.5.7 POTE tuning)</div>
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
{| class="wikitable right-1 right-2 sortable"
|+
|-
|-
! #Gens up
! #Gens up
! Cents <ref>octave-reduced</ref>
! Cents*
! class="unsortable"| Approximate ratios<ref>2.3.5.7, odd limit ≤ 27</ref>
! class="unsortable" | Approximate ratios**
|-
|-
| 0
| 0
Line 241: Line 235:
| 21/16
| 21/16
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
 
<nowiki />** 2.3.5.7, odd limit &le; 27</div></div>
[[Comma]] list: 81/80, 126/125
[[Comma]] list: 81/80, 126/125


Line 247: Line 243:


Mapping generators: ~2, ~3
Mapping generators: ~2, ~3
[[Wedgie]]: {{wedgie| 1 4 10 4 13 12 }}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* 7- and [[9-odd-limit]]
* 7- and [[9-odd-limit]]: ~3/2 = {{Monzo| 0 0 1/4 }}
: [{{Monzo| 1 0 0 0 }}, {{Monzo| 1 0 1/4 0 }}, {{Monzo| 0 0 1 0 }}, {{Monzo| -3 0 5/2 0 }}]
: [[Eigenmonzo]]s (unchanged-intervals): 2, 5
: [[Eigenmonzo]]s (unchanged intervals): 2, 5


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* [[diamond monotone]] range: [694.737, 700.000] (19 to 12)
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [694.737, 700.000] (11\19 to 7\12)
* [[diamond tradeoff]] range: [694.786, 701.955]
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 701.955]
* diamond monotone and tradeoff range: [694.786, 700.000]
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]
* 7-odd-limit diamond monotone and tradeoff: ~3/2 = [694.786, 700.000]
* 9-odd-limit diamond monotone and tradeoff: ~3/2 = [694.737, 700.000]


Algebraic generator: Cybozem, the real root of 15''x''<sup>3</sup> - 10''x''<sup>2</sup> - 18, which comes to 503.4257 cents. The recurrence converges quickly.
Algebraic generator: Cybozem, the real root of {{nowrap|15''x''<sup>3</sup> − 10''x''<sup>2</sup> − 18}}, which comes to 503.4257{{c}}. The recurrence converges quickly.


{{Vals|legend=1| 12, 19, 31, 81, 112b, 143b }}
{{Optimal ET sequence|legend=1| 12, 19, 31, 81, 112b, 143b }}


[[Badness]]: 0.0137
[[Badness]]: 0.013707


==== Meanpop ====
==== Meanpop ====
Line 284: Line 279:


Minimax tuning:  
Minimax tuning:  
* [[11-odd-limit]]: 1/4 comma
* [[11-odd-limit]]: ~3/2 = {{Monzo| 0 0 1/4 }}
: [{{Monzo| 1 0 0 0 0 }}, {{Monzo| 1 0 1/4 0 0 }}, {{Monzo| 0 0 1 0 0 }}, {{Monzo| -3 0 5/2 0 0 }}, {{Monzo| 11 0 -13/4 0 0 }}]
: [[Eigenmonzo]]s (unchanged-intervals): 2, 5
: [[Eigenmonzo]]s (unchanged intervals): 2, 5


Tuning ranges:  
Tuning ranges:  
* [[diamond monotone]] range: [694.737, 696.774] (19 to 31)
* 11-odd-limit [[diamond monotone]]: ~3/2 = [694.737, 696.774] (11\19 to 18\31)
* [[diamond tradeoff]] range: [691.202, 701.955]
* 11-odd-limit [[diamond tradeoff]]: ~3/2 = [691.202, 701.955]
* diamond monotone and tradeoff range: [694.737, 696.774]
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [694.737, 696.774]


Algebraic generator: Cybozem; or else Radieubiz, the real root of 3''x''<sup>3</sup> + 6''x'' - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.
Algebraic generator: Cybozem; or else Radieubiz, the real root of {{nowrap|3''x''<sup>3</sup> + 6''x'' − 19}}. Unlike Cybozem, the recurrence for Radieubiz does not converge.


{{Vals|legend=1| 12e, 19, 31, 81 }}
{{Optimal ET sequence|legend=1| 12e, 19, 31, 81 }}


[[Badness]]: 0.0215
[[Badness]]: 0.021543


==== Huygens ====
==== Huygens ====
Line 310: Line 304:
Mapping: Same as septimal meantone, plus 18 gens = 11/8
Mapping: Same as septimal meantone, plus 18 gens = 11/8


Comma list: 81/80, 126/125, 99/98
Comma list: 81/80, 99/98, 126/125


Mapping: [{{val| 1 0 -4 -13 -25 }}, {{val| 0 1 4 10 18 }}]
Mapping: [{{val| 1 0 -4 -13 -25 }}, {{val| 0 1 4 10 18 }}]
Line 317: Line 311:


Minimax tuning:  
Minimax tuning:  
* [[11-odd-limit]]
* [[11-odd-limit]]: ~3/2 = {{Monzo| 9/16 -1/8 0 0 1/16 }}
: [{{Monzo| 1 0 0 0 0 }}, {{Monzo| 25/16 -1/8 0 0 1/16 }}, {{Monzo| 9/4 -1/2 0 0 1/4 }}, {{Monzo| 21/8 -5/4 0 0 5/8 }}, {{Monzo| 25/8 -9/4 0 0 9/8 }}]
: [[Eigenmonzo]]s (unchanged-intervals): 2, 11/9
: [[Eigenmonzo]]s (unchanged intervals): 2, 11/9


Tuning ranges:  
Tuning ranges:  
* [[diamond monotone]] range: [696.774, 700.000] (31 to 12)
* 11-odd-limit [[diamond monotone]]: ~3/2 = [696.774, 700.000] (18\31 to 7\12)
* [[diamond tradeoff]] range: [691.202, 701.955]
* 11-odd-limit [[diamond tradeoff]]: ~3/2 = [691.202, 701.955]
* diamond monotone and tradeoff range: [696.774, 700.000]
* 11-odd-limit diamond monotone and tradeoff: ~3/2 = [696.774, 700.000]


[[Algebraic generator]]: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.
[[Algebraic generator]]: Traverse, the positive real root of {{nowrap|''x''<sup>4</sup> + 2''x'' − 13}}, or 696.9529{{c}}.


{{Vals|legend=1| 12, 19e, 31, 105, 136b, 167be, 198be }}
{{Optimal ET sequence|legend=1| 12, 19e, 31, 105, 136b, 167be, 198be }}


Badness: 0.0170
[[Badness]]: 0.017027


== Schismic ==
== Schismic ==
Subgroup: 2.3.5.7.9.11.13.19
Subgroup: 2.3.5.7.11.13.19


Period: 1\1
Period: 1\1
Line 343: Line 336:
Scales: [[Garibaldi12]], [[Garibaldi17]]
Scales: [[Garibaldi12]], [[Garibaldi17]]


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div class="toccolours mw-collapsible mw-collapsed" style="width: 600px; overflow: auto;">
<div style="line-height:1.6;">Interval table (29-note MOS, 2.3.5.7.11.13.19 POTE tuning)</div>
<div style="line-height: 1.6;">Interval table (29-note MOS, 2.3.5.7.11.13.19 POTE tuning)</div>
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
{| class="wikitable right-1 right-2 sortable"
|+
|-
|-
! #Gens up
! #Gens up
! Cents <ref>octave-reduced</ref>
! Cents*
! class="unsortable"| Approximate ratios<ref>2.3.5.7.11.13.19</ref>
! class="unsortable" | Approximate ratios**
|-
|-
| 0
| 0
Line 469: Line 461:
| 13/10
| 13/10
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
Comma list: 225/224, 275/273, 325/324, 385/384, 513/512
 
<nowiki />** In the 2.3.5.7.11.13.19 subgroup</div></div>
[[Comma list]]: 190/189, 209/208, 225/224, 275/273, 325/324
 
[[Gencom]]: [2 3/2; 190/189 209/208 225/224 275/273 325/324]


Mapping: [{{val|2: 1, 3: 0, 5: 15, 7: 25, 11: -33, 13: -28, 19: 9 }}, {{val|2: 0, 3: 1, 5: -8, 7: -14, 11: 23, 13: 20, 19: -3 }}]
[[Gencom]] [[mapping]]: [{{val|1 1 7 11 -10 -8 6 }}, {{val|0 1 -8 -14 23 20 -3 }}]


Mapping generators: ~2, ~3
{{Optimal ET sequence|legend=1| 41, 53, 94 }}


{{Val list|legend=1| 41, 53, 94}}
[[Tp_tuning#T2 tuning|RMS error]]: 0.6486 cents


== Parapyth ==
== Parapyth ==
Line 487: Line 483:
EDO generators: [[17edo|10\17]], [[29edo|17\29]], [[46edo|27\46]]
EDO generators: [[17edo|10\17]], [[29edo|17\29]], [[46edo|27\46]]


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div class="toccolours mw-collapsible mw-collapsed" style="width: 600px; overflow: auto;">
<div style="line-height:1.6;">Interval table (17-note MOS, 2.3.7.11.13 POTE tuning)</div>
<div style="line-height: 1.6;">Interval table (17-note MOS, 2.3.7.11.13 POTE tuning)</div>
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
{| class="wikitable right-1 right-2 sortable"
|+
|-
|-
! #Gens up
! #Gens up
! Cents <ref>octave-reduced</ref>
! Cents*
! class="unsortable"| Approximate ratios<ref>2.3.7.11.13, odd limit ≤ 27</ref>
! class="unsortable" | Approximate ratios**
|-
|-
| 0
| 0
Line 565: Line 560:
| 21/16
| 21/16
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
[[Mapping|Period-generator mapping]]: [&lt;2: 1, 3: 0, 7: -21, 11: -14, 13: -9|, &lt;2: 0, 3: 1, 7: 15, 11: 11, 13: 8|]


Commas: 169/168, 352/351, 364/363
<nowiki />** 2.3.7.11.13, odd limit &le; 27</div></div>
[[Comma list]]: 169/168, 352/351, 364/363


Gencom: [2 3/2; 169/169 352/351 364/363]
[[Gencom]]: [2 3/2; 169/168 352/351 364/363]


Gencom mapping: [&lt;2: 1, 3: 1, 7: -6, 11: -3, 13: -1|, &lt;2: 0, 3: 1, 7: 15, 11: 11, 13: 8|]
[[Gencom]] [[mapping]]: [{{val| 1 1 -6 -3 -1 }}, {{val| 0 1 15 11 8 }}]


EDOs: 17, 46, 63
{{Optimal ET sequence|legend=1| 17, 46, 63 }}


[[Tp_tuning#T2 tuning|RMS error]]: 0.7541 cents
[[Tp_tuning#T2 tuning|RMS error]]: 0.7541 cents
Line 589: Line 584:
Scales: [[Archy5]], [[Archy7]], [[Archy12]]
Scales: [[Archy5]], [[Archy7]], [[Archy12]]


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div class="toccolours mw-collapsible mw-collapsed" style="width: 600px; overflow: auto;">
<div style="line-height:1.6;">Interval table (7-note MOS, 2.3.7 POTE tuning)</div>
<div style="line-height: 1.6;">Interval table (7-note MOS, 2.3.7 POTE tuning)</div>
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
{| class="wikitable right-1 right-2 sortable"
|+
|-
|-
! #Gens up
! #Gens up
! Cents <ref>octave-reduced</ref>
! Cents*
! class="unsortable"| Approximate ratios<ref>2.3.7, odd limit ≤ 27</ref>
! class="unsortable" | Approximate ratios**
|-
|-
| 0
| 0
Line 627: Line 621:
|  
|  
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
[[Mapping|Period-generator mapping]]: [&lt;2: 1 3: 2, 7: 2|, &lt;2: 0, 3: -1, 7: 2|]


Comma: 64/63
<nowiki />** 2.3.7, odd limit &le; 27</div></div>
[[Comma]]: 64/63


Gencom: [2 3/2; 64/63]
[[Gencom]]: [2 3/2; 64/63]


Gencom mapping: [&lt;2: 1, 3: 1, 7: 4|, &lt;2: 0, 3: 1, 7: -2|]
[[Gencom mapping]]: [{{val| 1 1 4 }}, {{val| 0 1 -2 }}]


EDOs: 5, 12, 17, 22, 27, 137bc
{{Optimal ET sequence|legend=1| 5, 12, 17, 22, 27, 137bd }}


[[Tp_tuning#T2 tuning|RMS error]]: 1.856 cents
[[Tp_tuning#T2 tuning|RMS error]]: 1.856 cents
=== Supra ===
=== Supra ===
Subgroup: 2.3.7.11
Subgroup: 2.3.7.11
Line 649: Line 644:


Scales: [[Supra7]], [[Supra12]]
Scales: [[Supra7]], [[Supra12]]
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div class="toccolours mw-collapsible mw-collapsed" style="width: 600px; overflow: auto;">
<div style="line-height:1.6;">Interval table (12-note MOS, 2.3.7.11 POTE tuning)</div>
<div style="line-height: 1.6;">Interval table (12-note MOS, 2.3.7.11 POTE tuning)</div>
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
{| class="wikitable right-1 right-2 sortable"
|+
|-
|-
! #Gens up
! #Gens up
! Cents <ref>octave-reduced</ref>
! Cents*
! class="unsortable"| Approximate ratios<ref>2.3.7.11, odd limit ≤ 27</ref>
! class="unsortable" | Approximate ratios**
|-
|-
| 0
| 0
Line 707: Line 701:
|  
|  
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
[[Mapping|Period-generator mapping]]: [&lt;2: 1, 3: 0, 7: 6, 11: 13|, &lt;2: 0, 3: 1, 7: -2, 11: -6|]


Commas: 64/63, 99/98
<nowiki />** 2.3.7.11, odd limit &le; 27</div></div>
Comma list: 64/63, 99/98


Gencom: [2 3/2; 64/63 99/98]
Gencom: [2 3/2; 64/63 99/98]


Gencom mapping: [&lt;2: 1, 3: 1, 7: 4, 11: 7|, &lt;2: 0, 3: 1, 7: -2, 11: -6|]
Gencom mapping: [{{val| 1 1 4 7 }}, {{val| 0 1 -2 -6 }}]


EDOs: 5, 12, 17, 39c, 56c
{{Optimal ET sequence|legend=1| 5, 12, 17, 39c, 56d }}


[[Tp_tuning#T2 tuning|RMS error]]: 1.977 cents
RMS error: 1.977 cents


=== Superpyth ===
=== Superpyth ===
Line 727: Line 721:
Optimal ([[POTE]]) generator: ~3/2 = 710.291
Optimal ([[POTE]]) generator: ~3/2 = 710.291


EDO generators: [[22edo|13\22]], [[27edo|18\27]], [[49edo|31\49]]
EDO generators: [[22edo|13\22]], [[27edo|16\27]], [[49edo|29\49]]


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div class="toccolours mw-collapsible mw-collapsed" style="width: 600px; overflow :auto;">
<div style="line-height:1.6;">Interval table (12-note MOS, 2.3.5.7 POTE tuning)</div>
<div style="line-height: 1.6;">Interval table (12-note MOS, 2.3.5.7 POTE tuning)</div>
<div class="mw-collapsible-content">
<div class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
{| class="wikitable right-1 right-2 sortable"
|+
|-
|-
! #Gens up
! #Gens up
! Cents <ref>octave-reduced</ref>
! Cents*
! class="unsortable"| Approximate ratios<ref>2.3.5.7, odd limit ≤ 27</ref>
! class="unsortable" | Approximate ratios**
|-
|-
| 0
| 0
Line 787: Line 780:
| 10/7
| 10/7
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
[[Mapping|Period-generator mapping]]: [&lt;1 0 -12 6|, &lt;0 1 9 -2|]
 
<nowiki />** 2.3.5.7, odd limit &le; 27</div></div>
[[Comma list]]: 64/63, 245/243
 
[[Mapping]]: [{{val|1 1 -3 4}}, {{val|0 1 9 -2}}]
 
[[Val]]s: {{Optimal ET sequence| 5, 17, 22, 27, 49 }}
 
[[Badness]]: 0.032318
 
=== Ultrapyth ===
Subgroup: 2.3.5.7.13
 
Period: 1\1
 
Optimal ([[POTE]]) generator: ~3/2 = 713.745
 
EDO generators: [[32edo|19\32]], [[37edo|22\37]]
 
<div class="toccolours mw-collapsible mw-collapsed" style="width: 600px; overflow: auto;">
<div style="line-height: 1.6;">Interval table (22-note MOS, 2.3.5.7.13 POTE tuning)</div>
<div class="mw-collapsible-content">
{| class="wikitable right-1 right-2 sortable"
|-
! #Gens up
! Cents*
! class="unsortable" | Approximate ratios**
|-
| 0
| 0.00
| 1/1
|-
| 1
| 713.7
| 3/2
|-
| 2
| 227.5
| 9/8, 8/7
|-
| 3
| 941.2
| 12/7, 26/15
|-
| 4
| 455.0
| 9/7, 13/10
|-
| 5
| 1168.7
|
|-
| 6
| 682.5
|
|-
| 7
| 196.2
|
|-
| 8
| 910.0
|
|-
| 9
| 423.7
|
|-
| 10
| 1137.5
|
|-
| 11
| 651.2
|
|-
| 12
| 164.9
| 10/9
|-
| 13
| 878.7
| 5/3
|-
| 14
| 392.4
| 5/4
|-
| 15
| 1106.2
| 15/8
|-
| 16
| 619.9
| 10/7, 13/9
|-
| 17
| 133.7
| 15/14, 13/12
|-
| 18
| 847.4
| 13/8
|-
| 19
| 361.2
|  
|-
| 20
| 1074.9
| 13/7
|-
| 21
| 588.7
|
|}
<nowiki />* Octave-reduced
 
<nowiki />** 2.3.5.7.13, odd limit &le; 15</div></div>
Comma list: 64/63, 91/90, 4394/4375


[[Comma]]s: 64/63, 245/243
Gencom: [2 3/2; 64/63 91/90 4394/4375]


[[Wedgie]]: {{wedgie|1 9 -2 12 -6 -30}}
Gencom mapping: [{{val| 1 1 -6 4 -7 }}, {{val| 0 1 14 -2 18 }}]


EDOs: 5, 17, 22, 27, 49
{{Optimal ET sequence|legend=1| 5, 32, 37 }}


Badness: 0.0323
RMS error: 2.318 cents


[[Category:Scales]]
[[Category:Lists of temperaments]]
[[Category:MOS Scales]]
[[Category:Diatonic]]
[[Category:Diatonic]]
[[Category:Regular temperament theory]]
{{todo|rework}}
{{todo|rework}}
Retrieved from "https://en.xen.wiki/w/5L_2s/Temperaments"