5L 2s/Temperaments: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
5,013 edits
mNo edit summary
Phase out wedgies
 
(11 intermediate revisions by 4 users not shown)
Line 1: Line 1:
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 ==
== Meantone ==
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]]:  
* [[Diamond monotone]] range: [685.714, 720.000] (4\7 to 3\5)
* 5-odd-limit [[diamond monotone]]: ~3/2 = [685.714, 720.000] (4\7 to 3\5)
* [[Diamond tradeoff]] range: [694.786, 701.955]
* 5-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 701.955]
* Diamond monotone and tradeoff: [694.786, 701.955]
* 5-odd-limit diamond monotone and tradeoff: ~3/2 = [694.786, 701.955]


{{Vals|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b, 212bb, 293bb }}
{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b, 212bb, 293bb }}


[[Badness]]: 0.007381
[[Badness]]: 0.007381
Line 82: 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 140: Line 138:
| [[9/7]]
| [[9/7]]
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
 
<nowiki />** 2.3.5.7.13, odd limit &le; 27</div></div>


[[Comma list]]: 65/64, 81/80, 105/104
[[Comma list]]: 65/64, 81/80, 105/104
Line 150: Line 150:
[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]: ~3/2 = {{Monzo| 8/13 0 1/13 -1/13 }}
* [[7-odd-limit]]: ~3/2 = {{Monzo| 8/13 0 1/13 -1/13 }}
: [[Eigenmonzo]]s (unchanged intervals): 2, 7/5
: [[Eigenmonzo]]s (unchanged-intervals): 2, 7/5
* [[9-odd-limit]]: ~3/2 = {{monzo| 6/11 2/11 0 -1/11 }}
* [[9-odd-limit]]: ~3/2 = {{monzo| 6/11 2/11 0 -1/11 }}
: Eigenmonzos (unchanged intervals): 2, 9/7
: Eigenmonzos (unchanged-intervals): 2, 9/7


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* [[Diamond monotone]] range: [692.308, 694.737] (15\26 to 11\19)
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* 7-odd-limit [[diamond tradeoff]]: [692.353, 701.955]
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [692.353, 701.955]
* 9-odd-limit diamond tradeoff: [691.202, 701.955]
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]
* 7-odd-limit diamond monotone and tradeoff: [692.353, 694.737]
* 7-odd-limit diamond monotone and tradeoff: ~3/2 = [692.353, 694.737]
* 9-odd-limit diamond monotone and tradeoff: [692.308, 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, 45f }}
{{Optimal ET sequence|legend=1| 7, 19, 26, 45f }}


[[Tp_tuning#T2 tuning|RMS error]]: 1.742 cents
[[Tp_tuning#T2 tuning|RMS error]]: 1.742 cents
Line 178: 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 236: 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 245: Line 246:
[[Minimax tuning]]:  
[[Minimax tuning]]:  
* 7- and [[9-odd-limit]]: ~3/2 = {{Monzo| 0 0 1/4 }}
* 7- and [[9-odd-limit]]: ~3/2 = {{Monzo| 0 0 1/4 }}
: [[Eigenmonzo]]s (unchanged intervals): 2, 5
: [[Eigenmonzo]]s (unchanged-intervals): 2, 5


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* [[Diamond monotone]] range: [694.737, 700.000] (11\19 to 7\12)
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [694.737, 700.000] (11\19 to 7\12)
* 7-odd-limit [[diamond tradeoff]]: [694.786, 701.955]
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 701.955]
* 9-odd-limit diamond tradeoff: [691.202, 701.955]
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]
* 7-odd-limit diamond monotone and tradeoff: [694.786, 700.000]
* 7-odd-limit diamond monotone and tradeoff: ~3/2 = [694.786, 700.000]
* 9-odd-limit diamond monotone and tradeoff: [694.737, 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]]: 0.013707
[[Badness]]: 0.013707
Line 279: Line 280:
Minimax tuning:  
Minimax tuning:  
* [[11-odd-limit]]: ~3/2 = {{Monzo| 0 0 1/4 }}
* [[11-odd-limit]]: ~3/2 = {{Monzo| 0 0 1/4 }}
: [[Eigenmonzo]]s (unchanged intervals): 2, 5
: [[Eigenmonzo]]s (unchanged-intervals): 2, 5


Tuning ranges:  
Tuning ranges:  
* [[Diamond monotone]] range: [694.737, 696.774] (11\19 to 18\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: [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]]: 0.021543
[[Badness]]: 0.021543
Line 311: Line 312:
Minimax tuning:  
Minimax tuning:  
* [[11-odd-limit]]: ~3/2 = {{Monzo| 9/16 -1/8 0 0 1/16 }}
* [[11-odd-limit]]: ~3/2 = {{Monzo| 9/16 -1/8 0 0 1/16 }}
: [[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] (18\31 to 7\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: [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]]: 0.017027
[[Badness]]: 0.017027
Line 335: 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 461: Line 461:
| 13/10
| 13/10
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
 
<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
[[Comma list]]: 190/189, 209/208, 225/224, 275/273, 325/324


Line 468: Line 470:
[[Gencom]] [[mapping]]: [{{val|1 1 7 11 -10 -8 6 }}, {{val|0 1 -8 -14 23 20 -3 }}]
[[Gencom]] [[mapping]]: [{{val|1 1 7 11 -10 -8 6 }}, {{val|0 1 -8 -14 23 20 -3 }}]


{{Val list|legend=1| 41, 53, 94 }}
{{Optimal ET sequence|legend=1| 41, 53, 94 }}


[[Tp_tuning#T2 tuning|RMS error]]: 0.6486 cents
[[Tp_tuning#T2 tuning|RMS error]]: 0.6486 cents
Line 481: 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 559: Line 560:
| 21/16
| 21/16
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
 
<nowiki />** 2.3.7.11.13, odd limit &le; 27</div></div>
[[Comma list]]: 169/168, 352/351, 364/363
[[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]]: [{{val| 1 1 -6 -3 -1 }}, {{val| 0 1 15 11 8 }}]
[[Gencom]] [[mapping]]: [{{val| 1 1 -6 -3 -1 }}, {{val| 0 1 15 11 8 }}]


{{Val list|legend=1| 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 581: 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 619: Line 621:
|  
|  
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
 
<nowiki />** 2.3.7, odd limit &le; 27</div></div>
[[Comma]]: 64/63
[[Comma]]: 64/63


Line 626: Line 630:
[[Gencom mapping]]: [{{val| 1 1 4 }}, {{val| 0 1 -2 }}]
[[Gencom mapping]]: [{{val| 1 1 4 }}, {{val| 0 1 -2 }}]


{{Val list|legend=1| 5, 12, 17, 22, 27, 137bd }}
{{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
Line 640: 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 698: Line 701:
|  
|  
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
 
<nowiki />** 2.3.7.11, odd limit &le; 27</div></div>
Comma list: 64/63, 99/98
Comma list: 64/63, 99/98


Line 705: Line 710:
Gencom mapping: [{{val| 1 1 4 7 }}, {{val| 0 1 -2 -6 }}]
Gencom mapping: [{{val| 1 1 4 7 }}, {{val| 0 1 -2 -6 }}]


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


RMS error: 1.977 cents
RMS error: 1.977 cents
Line 716: 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 776: Line 780:
| 10/7
| 10/7
|}
|}
<references/></div></div>
<nowiki />* Octave-reduced
 
<nowiki />** 2.3.5.7, odd limit &le; 27</div></div>
[[Comma list]]: 64/63, 245/243
[[Comma list]]: 64/63, 245/243


[[Mapping]]: [{{val|1 0 -12 6}}, {{val|0 1 9 -2}}]
[[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
 
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 }}]


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


Badness: 0.032318
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}}

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

Retrieved from "https://en.xen.wiki/index.php?title=5L_2s/Temperaments&oldid=199376"