5L 2s/Temperaments: Difference between revisions

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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.  


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


Period: 1\1
Period: 1\1
Line 12: 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 51: 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 58: 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


=== Flattone (19&26, 2.3.5.7.13) ===
Period: 1\1
Period: 1\1


Line 80: 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 138: 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: 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


=== Septimal meantone (19&12, 2.3.5.7) ===
=== Septimal meantone ===
Subgroup: 2.3.5.7


Period: 1\1
Period: 1\1
Line 179: 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 237: 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 243: 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
 
==== Meanpop ====
Subgroup: 2.3.5.7.11


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


Line 278: 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 {{nowrap|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 3''x''<sup>3</sup> + 6''x'' - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.
{{Optimal ET sequence|legend=1| 12e, 19, 31, 81 }}


{{Vals|legend=1| 12e, 19, 31, 81 }}
[[Badness]]: 0.021543


[[Badness]]: 0.0215
==== Huygens ====
Subgroup: 2.3.5.7.11


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


Line 302: 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 309: 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 {{nowrap|''x''<sup>4</sup> + 2''x'' − 13}}, or 696.9529{{c}}.


[[Algebraic generator]]: Traverse, the positive real root of ''x''<sup>4</sup> + 2''x'' - 13, or 696.9529 cents.
{{Optimal ET sequence|legend=1| 12, 19e, 31, 105, 136b, 167be, 198be }}


{{Vals|legend=1| 12, 19e, 31, 105, 136b, 167be, 198be }}
[[Badness]]: 0.017027


Badness: 0.0170
== 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 333: 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 459: Line 461:
| 13/10
| 13/10
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
Comma list: 225/224, 275/273, 325/324, 385/384, 513/512


Mapping: [{{val| 1 0 15 25 -33 -28 9 }}, {{val| 0 1 -8 -14 23 20 -3 }}]
<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


Mapping generators: ~2, ~3
[[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 }}]
 
{{Optimal ET sequence|legend=1| 41, 53, 94 }}
 
[[Tp_tuning#T2 tuning|RMS error]]: 0.6486 cents
 
== Parapyth ==
Subgroup: 2.3.7.11.13


{{Val list|legend=1| 41, 53, 94}}
== Parapyth (29&17, 2.3.7.11.13) ==
Period: 1\1
Period: 1\1


Line 474: 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 552: Line 560:
| 21/16
| 21/16
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
[[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


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


Line 574: 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 612: Line 621:
|  
|  
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
[[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
=== Supra (17&22, 2.3.7.11) ===
 
=== Supra ===
Subgroup: 2.3.7.11
 
Period: 1\1
Period: 1\1


Line 632: 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 690: Line 701:
|  
|  
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
[[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 }}]
 
{{Optimal ET sequence|legend=1| 5, 12, 17, 39c, 56d }}


EDOs: 5, 12, 17, 39c, 56c
RMS error: 1.977 cents


[[Tp_tuning#T2 tuning|RMS error]]: 1.977 cents
=== 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 768: 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:Abstract MOS patterns]]
{{todo|rework}}
{{todo|rework}}

Latest revision as of 04:02, 12 June 2025

< 5L 2s

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

Main article: Meantone

Subgroup: 2.3.5

Period: 1\1

Optimal (POTE) generator: ~3/2 = 696.239

EDO generators: 7\12, 11\19, 18\31, 25\43, 29\50

Scales (Scala files): Meantone5, Meantone7, Meantone12

Interval table (7-note MOS, 2.3.5.7 POTE tuning)
# Cents* Approximate ratios**
0 0.00 1/1
1 696.2 3/2
2 192.5 9/8, 10/9
3 888.7 5/3
4 385.0 5/4
5 1081.2 15/8
6 577.4 25/18

* Octave-reduced

* 2.3.5, odd limit ≤ 27

Comma list: 81/80

Mapping: [⟨1 0 -4], ⟨0 1 4]]

Mapping generators: ~2, ~3

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]

Optimal ET sequence: 5, 7, 12, 19, 31, 50, 81, 131b, 212bb, 293bb

Badness: 0.007381

Flattone

Subgroup: 2.3.5.7 or 2.3.5.7.13

Period: 1\1

Optimal (POTE) generator: ~3/2 = 693.7498

EDO generators: 11\19, 15\26, 26\45, 37\64

Scales (Scala files): Flattone12

Interval table (12-note MOS, 2.3.5.7.13 POTE tuning)
#Gens up Cents* Approximate ratios**
0 0.00 1/1
1 693.7 3/2
2 187.5 9/8, 10/9
3 881.2 5/3
4 375.0 5/4, 16/13
5 1068.7 15/8, 24/13
6 562.5 18/13
7 56.2
8 750.0 20/13
9 243.7 8/7
10 937.5 12/7
11 431.2 9/7

* Octave-reduced

** 2.3.5.7.13, odd limit ≤ 27

Comma list: 65/64, 81/80, 105/104

Gencom: [2 3/2; 65/64 81/80 105/104]

Gencom mapping: [⟨1 1 0 8 6], ⟨0 1 4 -9 -4]]

Minimax tuning:

Eigenmonzos (unchanged-intervals): 2, 7/5
Eigenmonzos (unchanged-intervals): 2, 9/7

Tuning ranges:

  • 7- and 9-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
  • 7-odd-limit diamond tradeoff: ~3/2 = [692.353, 701.955]
  • 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 8x2 − 4x − 9, at 506.3239 ¢, equal to [math]\displaystyle{ \frac{1+\sqrt{19}}{4} }[/math].

Optimal ET sequence: 7, 19, 26, 45f

RMS error: 1.742 cents

Septimal meantone

Subgroup: 2.3.5.7

Period: 1\1

Optimal (POTE) generator: 696.495

EDO generators: 7\12, 11\19, 18\31, 25\43, 29\50

Scales (Scala files): Meantone5, Meantone7, Meantone12

Interval table (12-note MOS, 2.3.5.7 POTE tuning)
#Gens up Cents* Approximate ratios**
0 0.00 1/1
1 696.5 3/2
2 193.0 9/8, 10/9
3 889.5 5/3
4 386.0 5/4
5 1082.5 15/8, 28/15
6 579.0 7/5
7 75.5 21/20, 25/24, 28/27
8 772.0 14/9, 25/16
9 268.5 7/6
10 965.0 7/4
11 461.4 21/16

* Octave-reduced

** 2.3.5.7, odd limit ≤ 27

Comma list: 81/80, 126/125

Mapping: [⟨1 0 -4 -13], ⟨0 1 4 10]]

Mapping generators: ~2, ~3

Minimax tuning:

Eigenmonzos (unchanged-intervals): 2, 5

Tuning ranges:

  • 7- and 9-odd-limit diamond monotone: ~3/2 = [694.737, 700.000] (11\19 to 7\12)
  • 7-odd-limit diamond tradeoff: ~3/2 = [694.786, 701.955]
  • 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 15x3 − 10x2 − 18, which comes to 503.4257 ¢. The recurrence converges quickly.

Optimal ET sequence: 12, 19, 31, 81, 112b, 143b

Badness: 0.013707

Meanpop

Subgroup: 2.3.5.7.11

Period: 1\1

Optimal (POTE) generator: ~3/2 = 696.434

EDO generators: 29\50, 40\69, 47\81

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

Comma list: 81/80, 126/125, 385/384

Mapping: [⟨1 0 -4 -13 24], ⟨0 1 4 10 -13]]

Mapping generator: ~2, ~3

Minimax tuning:

Eigenmonzos (unchanged-intervals): 2, 5

Tuning ranges:

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

Algebraic generator: Cybozem; or else Radieubiz, the real root of 3x3 + 6x − 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.

Optimal ET sequence: 12e, 19, 31, 81

Badness: 0.021543

Huygens

Subgroup: 2.3.5.7.11

Period: 1\1

Optimal (POTE) generator: ~3/2 = 696.967

EDO generators: 25\43, 43\74

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

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

Mapping: [⟨1 0 -4 -13 -25], ⟨0 1 4 10 18]]

Mapping generators: ~2, ~3

Minimax tuning:

Eigenmonzos (unchanged-intervals): 2, 11/9

Tuning ranges:

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

Algebraic generator: Traverse, the positive real root of x4 + 2x − 13, or 696.9529 ¢.

Optimal ET sequence: 12, 19e, 31, 105, 136b, 167be, 198be

Badness: 0.017027

Schismic

Subgroup: 2.3.5.7.11.13.19

Period: 1\1

Optimal (POTE) generator: ~3/2 = 702.1044

EDO generators: 24\41, 31\53, 55\94

Scales: Garibaldi12, Garibaldi17

Interval table (29-note MOS, 2.3.5.7.11.13.19 POTE tuning)
#Gens up Cents* Approximate ratios**
0 0.00 1/1
1 702.10 3/2
2 204.21 9/8
3 906.31 27/16, 32/19
4 408.42
5 1110.52
6 612.63 10/7
7 114.73 15/14, 16/15
8 816.84 8/5
9 318.94 6/5
10 1021.04 9/5
11 523.15 27/20
12 25.25 81/80
13 727.36 32/21
14 229.462 8/7
15 931.57 12/7
16 433.67 9/7
17 1135.77 54/28
18 637.88 13/9
19 139.98 13/12
20 842.09 13/8
21 344.19 11/9, 39/32
22 1046.30 11/6
23 548.40 11/8, 26/19
24 50.51 33/32
25 752.61
26 254.714 22/19
27 956.82 26/15
28 458.92 13/10

* Octave-reduced

** In the 2.3.5.7.11.13.19 subgroup

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]

Gencom mapping: [⟨1 1 7 11 -10 -8 6], ⟨0 1 -8 -14 23 20 -3]]

Optimal ET sequence: 41, 53, 94

RMS error: 0.6486 cents

Parapyth

Subgroup: 2.3.7.11.13

Period: 1\1

Optimal (POTE) generator: ~3/2 = 704.745

EDO generators: 10\17, 17\29, 27\46

Interval table (17-note MOS, 2.3.7.11.13 POTE tuning)
#Gens up Cents* Approximate ratios**
0 0.00 1/1
1 704.7 3/2
2 209.5 9/8
3 914.2 22/13
4 419.0 14/11
5 1123.7
6 628.5 13/9, (23/16)
7 133.2 13/12, 14/13
8 838.0 13/8
9 342.7 11/9
10 1047.5 11/6
11 552.2 11/8
12 56.9 28/27
13 761.7 14/9
14 266.4 7/6
15 971.2 7/4
16 475.9 21/16

* Octave-reduced

** 2.3.7.11.13, odd limit ≤ 27

Comma list: 169/168, 352/351, 364/363

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

Gencom mapping: [⟨1 1 -6 -3 -1], ⟨0 1 15 11 8]]

Optimal ET sequence: 17, 46, 63

RMS error: 0.7541 cents

Archy

Subgroup: 2.3.7

Period: 1\1

Optimal (POTE) generator: ~3/2 = 709.321

EDO generators: 10\17, 13\22, 16\27

Scales: Archy5, Archy7, Archy12

Interval table (7-note MOS, 2.3.7 POTE tuning)
#Gens up Cents* Approximate ratios**
0 0.00 1/1
1 709.3 3/2
2 218.6 9/8, 8/7
3 927.8 12/7
4 437.3 9/7
5 1146.6 27/14
6 655.9

* Octave-reduced

** 2.3.7, odd limit ≤ 27

Comma: 64/63

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

Gencom mapping: [⟨1 1 4], ⟨0 1 -2]]

Optimal ET sequence: 5, 12, 17, 22, 27, 137bd

RMS error: 1.856 cents

Supra

Subgroup: 2.3.7.11

Period: 1\1

Optimal (POTE) generator: ~3/2 = 707.192

EDO generators: 10\17, 13\22, 23\39

Scales: Supra7, Supra12

Interval table (12-note MOS, 2.3.7.11 POTE tuning)
#Gens up Cents* Approximate ratios**
0 0.00 1/1
1 707.2 3/2
2 214.4 9/8, 8/7
3 921.6 12/7
4 428.8 9/7, 14/11
5 1136.0 27/14
6 643.2 16/11
7 150.3 12/11
8 857.5 18/11
9 364.7
10 1071.9
11 579.1

* Octave-reduced

** 2.3.7.11, odd limit ≤ 27

Comma list: 64/63, 99/98

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

Gencom mapping: [⟨1 1 4 7], ⟨0 1 -2 -6]]

Optimal ET sequence: 5, 12, 17, 39c, 56d

RMS error: 1.977 cents

Superpyth

Subgroup: 2.3.5.7

Period: 1\1

Optimal (POTE) generator: ~3/2 = 710.291

EDO generators: 13\22, 16\27, 29\49

Interval table (12-note MOS, 2.3.5.7 POTE tuning)
#Gens up Cents* Approximate ratios**
0 0.00 1/1
1 710.3 3/2
2 220.6 9/8, 8/7
3 930.9 12/7
4 441.2 9/7
5 1151.5
6 661.7 40/27
7 172.0 10/9
8 882.3 5/3
9 392.6 5/4
10 1102.9 15/8
11 613.2 10/7

* Octave-reduced

** 2.3.5.7, odd limit ≤ 27

Comma list: 64/63, 245/243

Mapping: [⟨1 1 -3 4], ⟨0 1 9 -2]]

Vals: 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: 19\32, 22\37

Interval table (22-note MOS, 2.3.5.7.13 POTE tuning)
#Gens up Cents* 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

* Octave-reduced

** 2.3.5.7.13, odd limit ≤ 15

Comma list: 64/63, 91/90, 4394/4375

Gencom: [2 3/2; 64/63 91/90 4394/4375]

Gencom mapping: [⟨1 1 -6 4 -7], ⟨0 1 14 -2 18]]

Optimal ET sequence: 5, 32, 37

RMS error: 2.318 cents

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