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Below are some important [[rank]]-2 [[temperaments]] with optimal [[generator]] size in the 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.  
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.  


=== Meantone (12&19, 2.3.5) ===
== Meantone ==
{{main| Meantone }}
{{main| Meantone }}
Subgroup: 2.3.5


Period: 1\1
Period: 1\1
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


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<nowiki />* 2.3.5, odd limit &le; 27</div></div>
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


Comma list: 81/80
Comma list: 81/80
Line 63: 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]


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


{{Vals|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b, 212bb, 293bb }}
[[Badness]]: 0.007381


[[Badness]]: 0.00736
=== Flattone ===
Subgroup: 2.3.5.7 or 2.3.5.7.13


</div></div>
==== Flattone (19&26, 2.3.5.7.13) ====
Period: 1\1
Period: 1\1


Line 87: 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 145: Line 138:
| [[9/7]]
| [[9/7]]
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


[[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: 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: 2, 9/7


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* valid 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)
* nice range: [692.353, 701.955]
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [692.353, 701.955]
* strict 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


</div></div>
=== Septimal meantone ===
 
Subgroup: 2.3.5.7
==== Septimal meantone (19&12, 2.3.5.7) ====


Period: 1\1
Period: 1\1
Line 191: 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 249: Line 235:
| 21/16
| 21/16
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">


<nowiki />** 2.3.5.7, odd limit &le; 27</div></div>
[[Comma]] list: 81/80, 126/125
[[Comma]] list: 81/80, 126/125


Line 259: 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: 2, 5


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* valid 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)
* nice range: [694.786, 701.955]
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 701.955]
* strict 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


</div></div>
==== Meanpop ====
Subgroup: 2.3.5.7.11


===== Meanpop (31&50, 2.3.5.7.11) =====
Period: 1\1
Period: 1\1


Line 289: Line 272:
Mapping: Same as septimal meantone, plus -13 gens = 11/8
Mapping: Same as septimal meantone, plus -13 gens = 11/8


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 81/80, 126/125, 385/384
Comma list: 81/80, 126/125, 385/384


Line 299: 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: 2, 5


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


</div></div>
==== Huygens ====
Subgroup: 2.3.5.7.11


===== Huygens (31&43, 2.3.5.7.11) =====
[[Period]]: 1\1
[[Period]]: 1\1


Line 325: Line 304:
Mapping: Same as septimal meantone, plus 18 gens = 11/8
Mapping: Same as septimal meantone, plus 18 gens = 11/8


<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
Comma list: 81/80, 99/98, 126/125
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
Comma list: 81/80, 126/125, 99/98


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 335: 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: 2, 11/9


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


</div></div>
== Schismic ==
Subgroup: 2.3.5.7.11.13.19


=== Schismic (41&53, 2.3.5.7.11.13.19) ===
Period: 1\1
Period: 1\1


Line 361: 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 487: Line 461:
| 13/10
| 13/10
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
 
<div style="line-height:1.6;">Technical data</div>
<nowiki />** In the 2.3.5.7.11.13.19 subgroup</div></div>
<div class="mw-collapsible-content">
[[Comma list]]: 190/189, 209/208, 225/224, 275/273, 325/324
Comma list: 225/224, 275/273, 325/324, 385/384, 513/512
 
[[Gencom]]: [2 3/2; 190/189 209/208 225/224 275/273 325/324]
 
[[Gencom]] [[mapping]]: [{{val|1 1 7 11 -10 -8 6 }}, {{val|0 1 -8 -14 23 20 -3 }}]


Mapping: [{{val| 1 0 15 25 -33 -28 9 }}, {{val| 0 1 -8 -14 23 20 -3 }}]
{{Optimal ET sequence|legend=1| 41, 53, 94 }}


Mapping generators: ~2, ~3
[[Tp_tuning#T2 tuning|RMS error]]: 0.6486 cents


{{Val list|legend=1| 41, 53, 94}}
== Parapyth ==
</div></div>
Subgroup: 2.3.7.11.13


=== Parapyth (29&17, 2.3.7.11.13) ===
Period: 1\1
Period: 1\1


Line 507: 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 585: Line 560:
| 21/16
| 21/16
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
[[Mapping|Period-generator mapping]]: [&lt;1 0 -21 -14 -9|, &lt;0 1 15 11 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;1 1 0 -6 -3 -1|, &lt;0 1 0 15 11 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
</div></div>


=== Archy (17&5, 2.3.7) ===
== Archy ==
Subgroup: 2.3.7
 
Period: 1\1
Period: 1\1


Line 611: 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 649: Line 621:
|  
|  
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
[[Mapping|Period-generator mapping]]: [&lt;1 2 2|, &lt;0 -1 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;1 1 0 4|, &lt;0 1 0 -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
</div></div>
 
==== Supra (17&22, 2.3.7.11) ====
=== Supra ===
Subgroup: 2.3.7.11
 
Period: 1\1
Period: 1\1


Line 673: 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 731: Line 701:
|  
|  
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div style="line-height:1.6;">Technical data</div>
<div class="mw-collapsible-content">
[[Mapping|Period-generator mapping]]: [&lt;1 0 6 13|, &lt;0 1 -2 -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;1 1 0 4 7|, &lt;0 1 0 -2 -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
</div></div>
 
=== Superpyth ===
Subgroup: 2.3.5.7


==== Superpyth (22&27, 2.3.5.7) ====
Period: 1\1
Period: 1\1


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 813: Line 780:
| 10/7
| 10/7
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
 
<div style="line-height:1.6;">Technical data</div>
<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">
<div class="mw-collapsible-content">
[[Mapping|Period-generator mapping]]: [&lt;1 0 -12 6|, &lt;0 1 9 -2|]
{| 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
</div></div>


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