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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
The '''diaschismic family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the diaschisma, [[2048/2025]].  
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-12-10 18:46:39 UTC</tt>.<br>
: The original revision id was <tt>187253471</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The 5-limit parent comma for the diaschismic family is 2048/2025, the diaschisma. Its monzo is |11 -4 -2&gt;, and flipping that yields &lt;&lt;2 -4 -11|| for the wedgie. This tells us the period is half an octave, the GCD of 2 and -4, and that the generator is a fifth. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34edo]] is a good tuning choice, with [[46edo]], [[56edo]], [[58edo]] or [[80edo]] being other possibilities. Both [[12edo]] and [[22edo]] support it, and retuning them to a MOS of diaschismic gives two scale possibilities.


[[POTE tuning|POTE generator]]: 704.898
== Diaschismic ==
{{Main| Diaschismic }}


Map: [&lt;2 0 11|, &lt;0 1 -2|]
The [[period]] of diaschismic is half an [[octave]], and the [[generator]] is a fifth; the [[ploidacot]] is diploid monocot. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34edo]] is a good tuning choice, with [[46edo]], [[56edo]], [[58edo]], or [[80edo]] being other possibilities. Both [[12edo]] and [[22edo]] support it, and retuning them to a [[mos]] of diaschismic gives two scale possibilities.


EDOs: 34, 46, 80, 286
This temperament is also known as '''srutal''' in the 5-limit, but that name more strictly speaking refers to the [[#Srutal|34d & 46 extension]] to the [[7-limit]] that adds [[4375/4374]] to the comma list.


==Seven limit children==
[[Subgroup]]: 2.3.5
The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Pajara derives from 64/63 and is a popular and well-known choice. Diaschismic adds 2097152/2066715 to obtain 7-limit harmony by more complex methods, but with greater accuracy. Keen adds 2240/2187, echidna 1728/1715 and shrutar 245/243, the sensamagic comma. The pajara, diaschismic and keen keep the same 1/2 octave period and fifth generator, but shrutar has a generator of a quarter-tone (which can be taken as [[36_35|36/35]], the septimal quarter-tone) and echidna has a generator of 9/7.


===Pajara===
[[Comma list]]: 2048/2025
Pajara, with wedgie &lt;&lt;2 -4 -4 -11 -12 2|| is closely associated with 22et (not to mention [[Paul Erlich]]) but other tunings are possible. The 1/2 octave period serves as both a [[10_7|10/7]] and a [[7_5|7/5]]. Aside from 22et, 34 with the val &lt;34 54 79 96| and 56 with the val &lt;56 89 130 158| are are interesting alternatives, with more accpetable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12et and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.


Pajara extends nicely to an 11-limit version, for which the 56 tuning can be used, but a good alternative is to make the major thirds pure by setting the fifth to be 706.843 cents. Now 99/98, 100/99, 176/175 and 896/891 are being tempered out.
{{Mapping|legend=1| 2 0 11 | 0 1 -2 }}
: mapping generators: ~45/32, ~3


Commas: 50/49, 64/63
[[Optimal tuning]]s:  
* [[WE]]: ~45/32 = 599.4107{{c}}, ~3/2 = 704.2059{{c}}
: [[error map]]: {{val| -1.179 +1.072 +1.150 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 704.9585{{c}}
: error map: {{val| 0.000 +3.003 +3.769 }}


[[POTE tuning|POTE generator]]: 707.048
[[Tuning ranges]]:  
* [[5-odd-limit]] [[diamond monotone]]: ~3/2 = [600.000 to 720.000] (1\2 to 6\10)
* 5-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 706.843]


Map: [&lt;2 0 11 12|, &lt;0 1 -2 -2|]
{{Optimal ET sequence|legend=1| 10, 12, 22, 34, 46, 80, 206c, 286bc }}


EDOs: 22, 34, 56
[[Badness]] (Sintel): 0.467


====11-limit====
=== Overview to extensions ===
Commas: 50/49, 64/63, 99/98
==== 7-limit extensions ====
To get the 7-limit extensions, we add another comma:
* Septimal diaschismic adds [[126/125]], the starling comma, to obtain 7-limit harmony by more complex methods than pajara, but with greater accuracy.
* Pajara adds [[50/49]] or [[64/63]] and is a popular and well-known choice.
* Srutal adds [[4375/4374]], the ragisma, which is about as accurate as septimal diaschismic but has a much more complex mapping of 7.
* Keen adds [[875/864]].


[[POTE tuning|POTE generator]]: 706.885
Those all keep the same half-octave period and fifth generator.  


Map: [&lt;2 0 11 12 26|, &lt;0 1 -2 -2 -6|]
Bidia adds [[3136/3125]], the hemimean comma, with a 1/4-octave period. Shrutar adds [[245/243]] and shru adds [[392/375]], with a quartertone generator. Sruti adds [[19683/19600]] and anguirus adds [[49/48]], with a neutral third or hemitwelfth generator. Those split the original generator in two. Echidna adds [[1728/1715]], the orwellisma, with a ~9/7 generator. Echidnic adds [[686/675]], the senga, with a ~8/7 generator. Those split the original generator in three. Finally, quadrasruta adds [[2401/2400]] and splits the original  generator in four.


EDOs: 22, 34, 56, 146
==== Subgroup extensions ====
Since the diaschisma factors into ([[256/255]])<sup>2</sup>([[289/288]]) in the 17-limit, it extends naturally to the 2.3.5.17 subgroup as ''srutal archagall'', considered in [[#Subgroup extensions_2|#Subgroup extensions]]. The [[S-expression]]-based comma list of this temperament is {[[256/255|S16]], [[289/288|S17]]}.


===Diaschismic===
== Septimal diaschismic ==
A simpler characterization than the one given by the normal comma list is that diaschismic adds 126/125 or 5120/5103 to the set of commas, and it can also be called 46&amp;58. However described, diaschismic has wedgie &lt;&lt;2 -4 -16 -11 -31 -26||, with a 1/2 period and a sharp fifth generator like pajara, but not so sharp, giving a more accurate but more complex temperament. [[58edo|58et]] provides an excellent tuning, but an alternative is to make [[7_4|7/4]] just by making the fifth 703.897 cents, as opposed to 703.448 cents for 58et.
{{Main| Diaschismic }}
{{See also| Srutal vs diaschismic }}


Diaschismic extends naturally to the 17-limit, for which the same tunings may be used, making it one of the most important of the higher limit rank two temperaments. Adding the 11-limit adds the commas 176/175, 896/891 and 441/440. The 13-limit yields 196/195, 351/350, and 364/363. The 17-limit adds 136/135, 221/220, and 442/441. If you want to explore higher limit harmonies, diaschismic is certainly one excellent way to do it; MOS of 34 notes and even more the 46 note MOS will encompass very great deal of it. Of course 46 or 58 equal provide alternatives which in many ways are similar, particularly in the case of 58.
A simpler characterization than the one given by the normal comma list is that septimal diaschismic adds [[126/125]] or [[5120/5103]] to the set of commas, and it can also be called {{nowrap| 46 & 58 }}. However described, septimal diaschismic has a 1/2-octave period and a sharp fifth generator like the 5-limit version, but not so sharp, giving a more accurate but more complex temperament. [[104edo]] with the 104c [[val]] provides an excellent tuning, which is close to tuning [[7/4]] just by making the fifth 703.897 cents.  


Commas: 126/125, 2048/2025
Diaschismic extends naturally to the 17-limit, for which the same tunings may be used, making it one of the most important of the higher-limit rank-2 temperaments. Adding the 11-limit adds the commas 176/175, 896/891 and 441/440. The 13-limit yields 196/195, 351/350, and 364/363; the 17-limit adds 136/135, 221/220, and 442/441. This mapping can also be rationalized by [[parapyth]], which makes sense due to the sharp fifth, and prime 17 is found as in srutal archagall. If you want to explore higher-limit harmonies, diaschismic is certainly one excellent way to do it; [[mos]] scales of 34 notes and even more the 46-note mos will encompass very great deal of it. Of course 46 or 58 equal provide alternatives which in many ways are similar, particularly in the case of 58.


[[POTE tuning|POTE generator]]: 703.681
[[Subgroup]]: 2.3.5.7


Map: [&lt;2 0 11 31|, &lt;0 1 -2 -8|]
[[Comma list]]: 126/125, 2048/2025


EDOs: 46, 58, 162
{{Mapping|legend=1| 2 0 11 31 | 0 1 -2 -8 }}


====11-limit====
[[Optimal tuning]]s:
Commas: 126/125, 176/175, 896/891
* [[WE]]: ~45/32 = 599.4449{{c}}, ~3/2 = 703.0299{{c}}
: [[error map]]: {{val| -1.110 -0.035 +3.740 -1.391 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7739{{c}}
: error map: {{val| 0.000 +1.819 +6.138 +0.983 }}


[[POTE tuning|POTE generator]]: 703.714
[[Tuning ranges]]:  
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [700.000, 705.882] (7\12 to 20\34)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 706.843]


Map: [&lt;2 0 11 31 45|, &lt;0 1 -2 -8 -12|]
{{Optimal ET sequence|legend=1| 12, 34, 46, 58, 104c, 162c }}


EDOs: 46, 58, 104, 162
[[Badness]] (Sintel): 0.959


====13-limit====
=== 11-limit ===
Commas: 126/125, 196/195, 364/363, 2048/2025
Subgroup: 2.3.5.7.11


[[POTE tuning|POTE generator]]: 703.704
Comma list: 126/125, 176/175, 896/891


Map: [&lt;2 0 11 31 45 55|, &lt;0 1 -2 -8 -12 -15|]
Mapping: {{mapping| 2 0 11 31 45 | 0 1 -2 -8 -12 }}


EDOs: [[46edo]], [[58edo]], [[162edo]]
Optimal tunings:  
* WE: ~45/32 = 599.4471{{c}}, ~3/2 = 703.0657{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7996{{c}}


====17-limit====
Tuning ranges:
Commas: 126/125, 136/135, 176/175, 196/195, 256/255
* 11-odd-limit diamond monotone: ~3/2 = [700.000, 704.348] (7\12 to 27\46)
* 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]


[[POTE tuning|POTE generator]]: 703.812
{{Optimal ET sequence|legend=0| 12, 34e, 46, 58, 104c, 162ce }}


Map: [&lt;2 0 11 31 45 55 5|, &lt;0 1 -2 -8 -12 -15 1|]
Badness (Sintel): 0.828


EDOs: 46, 58, 104
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


===Keen===
Comma list: 126/125, 176/175, 196/195, 364/363
Keen adds 875/864 as well as 2240/2187 to the set of commas, and has wedgie &lt;&lt;2 -4 18 -11 23 53||. It may also be described as the 22&amp;56 temperament. [[78edo|78et]] is a good tuning choice, and remains a good one in the 11-limit, where keen, &lt;&lt;2 -4 18 -12 ...||, is really more interesting, adding 100/99 and 385/384 to the commas.


Commas: 2048/2025, 875/864
Mapping: {{mapping| 2 0 11 31 45 55 | 0 1 -2 -8 -12 -15 }}


[[POTE tuning|POTE generator]]: 707.571
Optimal tunings:  
* WE: ~45/32 = 599.4451{{c}}, ~3/2 = 703.0528{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7813{{c}}


Map: [&lt;2 0 11 -23|, &lt;0 1 -2 9|]
Tuning ranges:  
* 13- and 15-odd-limit diamond monotone: ~3/2 = [703.448, 704.348] (34\58 to 27\46)
* 13-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
* 15-odd-limit diamond tradeoff: ~3/2 = [701.955, 711.731]


EDOs: 22, 56, 78
{{Optimal ET sequence|legend=0| 12f, 34ef, 46, 58, 104c, 162cef }}


====11-limit====
Badness (Sintel): 0.782
Commas: 100/99, 385/384, 1232/1215


[[POTE tuning|POTE generator]]: 707.609
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17


Map: [&lt;2 0 11 -23 26|, &lt;0 1 -2 9 -6|]
Comma list: 126/125, 136/135, 176/175, 196/195, 256/255


EDOs: 22, 56, 78
Mapping: {{mapping| 2 0 11 31 45 55 5 | 0 1 -2 -8 -12 -15 1 }}


===Echidna===
Optimal tunings:
Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It has a wedgie &lt;&lt;6 -12 10 -33 -1 57|| and may be called the 22&amp;58 temperament. [[58edo|58et]] or [[80edo|80et]] make for good tunings, or their vals can be add to &lt;138 219 321 388|.
* WE: ~17/12 = 599.6253{{c}}, ~3/2 = 703.3726{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 703.8520{{c}}


Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 896/891 or 540/539 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-limit diamond to within about six cents of error, within a compass of 24 notes. The 28 note 2MOS gives scope for this, and the 36 note MOS much more.
Tuning ranges:
* 17-odd-limit diamond monotone: ~3/2 = [703.448, 704.348] (34\58 to 27\46)
* 17-odd-limit diamond tradeoff: ~3/2 = [698.955, 711.731]


Commas: 2048/2025, 1728/1715
{{Optimal ET sequence|legend=0| 12f, 34ef, 46, 58, 104c }}


[[POTE tuning|POTE generator]]: 434.856
Badness (Sintel): 0.837


Map: [&lt;2 1 9 2|, &lt;0 3 -6 5|]
=== 2.3.5.7.11.13.17.23 subgroup (Na"Naa') ===
<b>Na"Naa'</b> is a remarkable subgroup temperament of {{nowrap| 46 & 58 }} with a prime harmonic of 23. It is yet to be found why it got this strange name.


EDOs: 22, 58, 80, 138, 218
Subgroup: 2.3.5.7.11.13.17.23


====11-limit====
Comma list: 126/125, 136/135, 176/175, 196/195, 231/230, 256/255
Commas: 176/175, 896/891, 540/539


[[POTE tuning|POTE generator]]: 434.852
Subgroup-val mapping: {{mapping| 2 0 11 31 45 55 5 63 | 0 1 -2 -8 -12 -15 1 -17 }}


Map: [&lt;2 1 9 2 12|, &lt;0 3 -6 5 -7|]
Optimal tunings:  
* WE: ~17/12 = 599.6272{{c}}, ~3/2 = 703.4326{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 703.9093{{c}}


EDOs: 22, 58, 80, 138, 218
{{Optimal ET sequence|legend=0| 12i, 34efi, 46, 58i, 104ci }}


===Shrutar===
Badness (Sintel): 0.882
Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. With wedgie &lt;&lt;4 -8 14 -22 11 55||, it can also be described as 22&amp;46. Its generator can be taken as either 36/35 or 35/24; the latter is interesting since along with 15/14 and 21/20, it connects opposite sides of a hexany. [[68edo]] makes for a good tuning, but another and excellent choice is a generator of 14^(1/7), making 7s just.


By adding 121/120 or 176/175 to the commas, shrutar can be extended to the 11-limit, which loses a bit of accuracy, but picks up low-complexity 11-limit harmony, making shrutar quite an interesting 11-limit system. 68, 114 or a 14^(1/7) generator can again be used as tunings.
== Pajara ==
{{Main| Pajara }}


Commas: 2048/2025, 245/243
Pajara is closely associated with 22edo (not to mention [[Paul Erlich]]) but other tunings are possible. The 1/2-octave period serves as both a [[10/7]] and a [[7/5]]. Aside from 22edo, 34 with the val {{val| 34 54 79 96 }} (34d) and 56 with the val {{val| 56 89 130 158 }} (56d) are interesting alternatives, with more acceptable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12edo and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.


[[POTE tuning|POTE generator]]: 52.811
Pajara extends nicely to an 11-limit version, for which the 56edo tuning can be used, but a good alternative is to make the major thirds pure by setting the fifth to be 706.843 cents. Now 99/98, 100/99, 176/175 and 896/891 are being tempered out.


Map: [&lt;2 1 9 -2|, &lt;0 2 -4 7|]
[[Subgroup]]: 2.3.5.7


EDOs: 22, 46, 68, 250
[[Comma list]]: 50/49, 64/63


====11-limit====
{{Mapping|legend=1| 2 0 11 12 | 0 1 -2 -2 }}
Commas: 2048/2025, 245/243, 121/120


[[POTE tuning|POTE generator]]: 52.680
[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 598.8483{{c}}, ~3/2 = 705.6906{{c}}
: [[error map]]: {{val| -2.303 +1.432 -5.756 +10.580 }}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~3/2 = 707.3438{{c}}
: error map: {{val| 0.000 +5.389 -1.001 +16.487 }}


Map: [&lt;2 1 9 -2 8|, &lt;0 2 -4 7 -1|]
[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [700.000, 720.000] (7\12 to 6\10)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 715.587]


EDOs: 22, 46, 68, 114, 410</pre></div>
{{Optimal ET sequence|legend=1| 10, 12, 22, 34d, 56d }}
<h4>Original HTML content:</h4>
 
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;
[[Badness]] (Sintel): 0.507
 
=== 2.3.5.7.17 subgroup ===
Subgroup: 2.3.5.7.17
 
Comma list: 50/49, 64/63, 85/84
 
Mapping: {{mapping| 2 0 11 12 5 | 0 1 -2 -2 1 }}
 
Optimal tunings:
* WE: ~7/5 = 599.053{{c}}, ~3/2 = 706.355{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 707.607{{c}}
 
{{Optimal ET sequence|legend=0| 10, 12, 22, 56d }}
 
Badness (Sintel): 0.438
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 50/49, 64/63, 99/98
 
Mapping: {{mapping| 2 0 11 12 26 | 0 1 -2 -2 -6 }}
 
Optimal tunings:
* WE: ~7/5 = 598.8485{{c}}, ~3/2 = 705.5285{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 707.1826{{c}}
 
Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [700.000, 709.091] (7\12 to 13\22)
* 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 715.587]
 
{{Optimal ET sequence|legend=0| 10e, 12, 22, 34d, 56d }}
 
Badness (Sintel): 0.673
 
==== 2.3.5.7.11.17 subgroup ====
Subgroup: 2.3.5.7.11.17
 
Comma list: 50/49, 64/63, 85/84, 99/98
 
Mapping: {{mapping| 2 0 11 12 26 5 | 0 1 -2 -2 -6 1 }}
 
Optimal tunings:
* WE: ~7/5 = 599.062{{c}}, ~3/2 = 706.095{{c}}
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 707.370{{c}}
 
{{Optimal ET sequence|legend=0| 10e, 12, 22, 34d, 56d }}
 
Badness (Sintel): 0.645
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 50/49, 64/63, 65/63, 99/98
 
Mapping: {{mapping| 2 0 11 12 26 1 | 0 1 -2 -2 -6 2 }}
 
Optimal tunings:
* WE: ~7/5 = 599.9732{{c}}, ~3/2 = 708.8873{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.9227{{c}}
 
{{Optimal ET sequence|legend=0| 10e, 12, 22 }}
 
Badness (Sintel): 1.14
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 50/49, 52/51, 64/63, 65/63, 99/98
 
Mapping: {{mapping| 2 0 11 12 26 1 5 | 0 1 -2 -2 -6 2 1 }}
 
Optimal tunings:
* WE: ~7/5 = 599.8871{{c}}, ~3/2 = 708.6725{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.8176{{c}}
 
{{Optimal ET sequence|legend=0| 10e, 12, 22 }}
 
Badness (Sintel): 1.06
 
==== Pajarina ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 50/49, 64/63, 78/77, 99/98
 
Mapping: {{mapping| 2 0 11 12 26 36 | 0 1 -2 -2 -6 -9 }}
 
Optimal tunings:
* WE: ~7/5 = 598.7732{{c}}, ~3/2 = 704.6889{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.3950{{c}}
 
{{Optimal ET sequence|legend=0| 12f, 22, 34d }}
 
Badness (Sintel): 0.923
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 50/49, 64/63, 78/77, 85/84, 99/98
 
Mapping: {{mapping| 2 0 11 12 26 36 5 | 0 1 -2 -2 -6 -9 1 }}
 
Optimal tunings:
* WE: ~7/5 = 599.0204{{c}}, ~3/2 = 705.
Retrieved from "https://en.xen.wiki/w/Diaschismic_family"