Semicomma family: Difference between revisions

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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 [[5-limit]] parent [[comma]] for the '''semicomma family''' of [[regular temperament|temperaments]] is the [[semicomma]] ({{monzo|legend=1| -21 3 7 }}, [[ratio]]: 2109375/2097152). This is the amount by which three pure 3/1 twelfths exceed seven pure 8/5 minor sixths.
: This revision was by author [[User:hearneg|hearneg]] and made on <tt>2014-10-12 11:26:53 UTC</tt>.<br>
: The original revision id was <tt>526123066</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">[[toc]]
=Orson=
The 5-limit parent comma for the **semicomma family** is the semicomma, 2109375/2097152 = |-21 3 7&gt;. This is the amount by which three pure 3/1 twelfths exceed seven pure 8/5 minor sixths. **Orson**, the [[5-limit]] temperament tempering it out, has a [[generator]] of 75/64, which is sharper than 7/6 by 225/224 when untempered, and less sharp than that in any good orson tempering, for example [[53edo]] or [[84edo]]. These give tunings to the generator which are sharp of 7/6 by less than five [[cent]]s, making it hard to treat orson as anything other than an (at least) 7-limit system, leading to orwell.


Comma: 2109375/2097152
== Orson ==
'''Orson''', first discovered by [[Erv Wilson]]{{citation needed}}, is the [[5-limit]] temperament [[tempering out]] the semicomma. It has a [[generator]] of [[~]][[75/64]], seven of which give the [[3/1|perfect twelfth]]; its [[ploidacot]] is alpha-heptacot. The generator is sharper than [[7/6]] by [[225/224]] when untempered, and less sharp than that in any good orson tempering, for example [[53edo]] or [[84edo]]. These give tunings to the generator which are sharp of 7/6 by less than five [[cent]]s, making it hard to treat orson as anything other than an (at least) 7-limit system, leading to orwell.


[[Tuning Ranges of Regular Temperaments|valid range]]: [257.143, 276.923] (14b to 13)
[[Subgroup]]: 2.3.5
nice range: [271.229, 271.708]
strict range: [271.229, 271.708]


[[POTE tuning|POTE generator]]: ~75/64 = 271.627
[[Comma list]]: 2109375/2097152


Map: [&lt;1 0 3|, &lt;0 7 -3|]
{{Mapping|legend=1| 1 0 3 | 0 7 -3 }}
EDOs: 22, 31, 53, 190, 243, 296, 645c
Badness: 0.0408


==Seven limit children==
: mapping generators: ~2, ~75/64
The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Adding 65536/64625 leads to orwell, but we could also add 1029/1024, leading to the 31&amp;159 temperament with wedgie &lt;&lt;21 -9 -7 -63 -70 9||, or 67528125/67108864, giving the 31&amp;243 temperament with wedgie &lt;&lt;28 -12 1 -84 -77 36||, or 4375/4374, giving the 53&amp;243 temperament with wedgie &lt;&lt;7 -3 61 -21 77 150||.


=Orwell=
[[Optimal tuning]]s:
Main article: [[Orwell]]
* [[CTE]]: ~2 = 1200.000, ~75/64 = 271.670
So called because 19\84 (as a [[fraction of the octave]]) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It's compatible with [[22edo|22]], [[31edo|31]], [[53edo|53]] and [[84edo|84]] equal, and may be described as the 22&amp;31 temperament, or &lt;&lt;7 -3 8 -21 -7 27||. It's a good system in the [[7-limit]] and naturally extends into the [[11-limit]]. [[84edo]], with the 19\84 generator, provides a good tuning for the 5, 7 and 11 limits, but it does use its second-best 11. However, the 19\84 generator is remarkably close to the 11-limit [[POTE tuning]], as the generator is only 0.0024 cents sharper, and it is a good approximation to the 7-limit POTE generator also; hence 84 may be considered the most recommendable tuning in the 7-limit. [[53edo]] might be preferred in the 5-limit because of its nearly pure fifth and in the 11-limit because of it slightly better 11, though most of its 11-limit harmony is actually worse. Aside from the semicomma and 65625/65536, 7-limit orwell tempers out 2430/2401, the nuwell comma, 1728/1715, the orwellisma, 225/224, the septimal kleisma, and 6144/6125, the porwell comma.
: [[error map]]: {{val| 0.000 -0.264 -1.324 }}
* [[POTE]]: ~2 = 1200.000, ~75/64 = 271.627
: error map: {{val| 0.000 -0.564 -1.195 }}


The 11-limit version of orwell tempers out 99/98, which means that two of its sharpened 7/6 generators give a flattened 11/8, as well as 121/120, 176/175, 385/384 and 540/539. Despite lowered tuning accuracy, orwell comes into its own in the 11-limit, giving acceptable accuracy and relatively low complexity. Tempering out the orwellisma, 1728/1715, means that orwell interprets three stacked 7/6 generators as an 8/5, and the tempered 1-7/6-11/8-8/5 chord is natural to orwell.
[[Tuning ranges]]:
* 5-odd-limit [[diamond monotone]]: ~75/64 = [257.143, 276.923] (3\14 to 3\13)
* 5-odd-limit [[diamond tradeoff]]: ~75/64 = [271.229, 271.708] (1/3-comma to 2/7-comma)


Orwell has MOS of size 9, 13, 22 and 31. The 9-note MOS is small enough to be retained in the mind as a genuine scale, is pleasing melodically, and has [[Retuning 12edo to Orwell9|considerable harmonic resources]] despite its absence of 5-limit triads. The 13 note MOS has those, and of course the 22 and 31 note MOS are very well supplied with everything.
{{Optimal ET sequence|legend=1| 22, 31, 53, 190, 243, 296, 645c }}


[[Comma|Commas]]: 225/224, 1728/1715
[[Badness]] (Smith): 0.040807


7-limit
=== Overview to extensions ===
[|1 0 0 0&gt;, |14/11 0 -7/11 7/11&gt;,
The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at. Adding 65625/65536 (or 225/224) leads to orwell, but we could also add
|27/11 0 3/11 -3/11&gt;, |27/11 0 -8/11 8/11&gt;]
* 1029/1024, leading to the {{nowrap| 31 & 159 }} temperament (triwell), or
[[Fractional monzos|Eigenmonzos]]: 2, 7/5
* 2401/2400, giving the {{nowrap| 31 & 243 }} temperament (quadrawell), or
* 4375/4374, giving the {{nowrap| 53 & 243 }} temperament (sabric).


9-limit
== Orwell ==
[|1 0 0 0&gt;, |21/17 14/17 -7/17 0&gt;,
{{Main| Orwell }}
|42/17 -6/17 3/17 0&gt;, |41/17 16/17 -8/17 0&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 10/9


valid range: [266.667, 272.727] (9 to 22)
So called because 19\84 (as a fraction of the octave) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It is compatible with [[22edo|22]], [[31edo|31]], [[53edo|53]] and [[84edo|84]] equal, and may be described as the {{nowrap| 22 & 31 }} temperament. It is a good system in the [[7-limit]] and naturally extends into the [[11-limit]]. [[84edo]], with the 19\84 generator, provides a good tuning for the 5-, 7- and 11-limit, but it does use its second-closest approximation to 11. However, the 19\84 generator is remarkably close to the 11-limit [[POTE tuning]], as the generator is only 0.0024 cents sharper, and it is a good approximation to the 7-limit POTE generator also; hence 84 may be considered the most recommendable tuning in the 7-limit. [[53edo]] might be preferred in the 5-limit because of its nearly pure fifth and in the 11-limit because of it slightly better 11, though most of its 11-limit harmony is actually worse. Aside from the semicomma and 65625/65536, 7-limit orwell tempers out [[2430/2401]], the nuwell comma, [[1728/1715]], the orwellisma, [[225/224]], the septimal kleisma, and [[6144/6125]], the porwell comma.
nice range: [266.871, 271.708]
strict range: [266.871, 271.708]


[[POTE tuning|POTE generator]]: ~7/6 = 271.509
The 11-limit version of orwell tempers out [[99/98]], which means that two of its sharpened 7/6 generators give a flattened 11/8, as well as 121/120, 176/175, 385/384 and 540/539. Despite lowered tuning accuracy, orwell comes into its own in the 11-limit, giving acceptable accuracy and relatively low complexity. Tempering out the orwellisma, 1728/1715, means that orwell interprets three stacked 7/6 generators as an 8/5, and the tempered 1–7/6–11/8–8/5 chord is natural to orwell.
Algebraic generators: Sabra3, the real root of 12x^3-7x-48.


Map: [&lt;1 0 3 1|, &lt;0 7 -3 8|]
Orwell has [[mos scale]]s of size 9, 13, 22 and 31. The 9-note mos is small enough to be retained in the mind as a genuine scale, is pleasing melodically, and has [[Retuning 12edo to Orwell9|considerable harmonic resources]] despite its absence of 5-limit triads. The 13-note mos has those, and of course the 22- and 31-note mos are very well supplied with everything.
Wedgie: &lt;&lt;7 -3 8 -21 -7 27||
EDOs: 22, 31, 53, 84, 137, 221d, 358d
Badness: 0.0207


==11-limit==
[[Subgroup]]: 2.3.5.7
[[Comma|Commas]]: 99/98, 121/120, 176/175


[[Minimax tuning]]
[[Comma list]]: 225/224, 1728/1715
[|1 0 0 0 0&gt;, |14/11 0 -7/11 7/11 0&gt;, |27/11 0 3/11 -3/11 0&gt;,
|27/11 0 -8/11 8/11 0&gt;, |37/11 0 -2/11 2/11 0&gt;]
[[Eigenmonzo|Eigenmonzos]]: 2, 7/5


valid range: [270.968, 272.727] (31 to 22)
{{Mapping|legend=1| 1 0 3 1 | 0 7 -3 8 }}
nice range: [266.871, 275.659]
strict range: [270.968, 272.727]


[[POTE tuning|POTE generator]]: ~7/6 = 271.426
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~7/6 = 271.513
: [[error map]]: {{val| 0.000 -1.364 -0.853 +3.278 }}
* [[POTE]]: ~2 = 1200.000, ~7/6 = 271.509
: error map: {{val| 0.000 -1.394 -0.840 +3.243 }}


Map: [&lt;1 0 3 1 3|, &lt;0 7 -3 8 2|]
[[Minimax tuning]]:
[[edo|Edos]]: [[22edo|22]], [[31edo|31]], [[53edo|53]], [[84edo|84e]]
* [[7-odd-limit]]: ~7/6 = {{monzo| 2/11 0 -1/11 1/11 }}
Badness: 0.0152
: {{monzo list| 1 0 0 0 | 14/11 0 -7/11 7/11 | 27/11 0 3/11 -3/11 | 27/11 0 -8/11 8/11 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
* [[9-odd-limit]]: ~7/6 = {{monzo| 3/17 2/17 -1/17 }}
: {{monzo list| 1 0 0 0 | 21/17 14/17 -7/17 0 | 42/17 -6/17 3/17 0 | 41/17 16/17 -8/17 0 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5


==13-limit==
[[Tuning ranges]]:
Commas: 99/98, 121/120, 176/175, 275/273
* 7-odd-limit [[diamond monotone]]: ~7/6 = [266.667, 272.727] (2\9 to 5\22)
* 9-odd-limit diamond monotone: ~7/6 = [270.968, 272.727] (7\31 to 5\22)
* 7-odd-limit [[diamond tradeoff]]: ~7/6 = [266.871, 271.708]
* 9-odd-limit diamond tradeoff: ~7/6 = [266.871, 272.514]


valid range: [270.968, 271.698] (31 to 53)
[[Algebraic generator]]: Sabra3, the real root of 12''x<sup>3</sup> - 7''x'' - 48.
nice range: [266.871, 275.659]
strict range: [270.968, 271.698]


[[POTE tuning|POTE generator]]: ~7/6 = 271.546
{{Optimal ET sequence|legend=1| 9, 22, 31, 53, 84, 137, 221d, 358d }}


Map: [&lt;1 0 3 1 3 8|, &lt;0 7 -3 8 2 -19|]
[[Badness]] (Smith): 0.020735
EDOs: 22, 31, 53, 84e, 137e
Badness: 0.0197


[[Orwell#Music|Music in Orwell]]
=== 11-limit ===
Subgroup: 2.3.5.7.11


==Blair==
Comma list: 99/98, 121/120, 176/175
Commas: 65/64, 78/77, 91/90, 99/98


valid range: []
Mapping: {{mapping| 1 0 3 1 3 | 0 7 -3 8 2 }}
nice range: [265.357, 289.210]
strict range: []


POTE generator: ~7/6 = 271.301
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/6 = 271.560
* POTE: ~2 = 1200.000, ~7/6 = 271.426


Map: [&lt;1 0 3 1 3 3|, &lt;0 7 -3 8 2 3|]
Minimax tuning:
EDOs: 9, 22, 31f
* 11-odd-limit: ~7/6 = {{monzo| 2/11 0 -1/11 1/11 }}
Badness: 0.0231
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 14/11 0 -7/11 7/11 0 }}, {{monzo| 27/11 0 3/11 -3/11 0 }}, {{monzo| 27/11 0 -8/11 8/11 0 }}, {{monzo| 37/11 0 -2/11 2/11 0 }}]
: Unchanged-interval (eigenmonzo) basis: 2.7/5


==Newspeak==
Tuning ranges:
Commas: 225/224, 441/440, 1728/1715
* 11-odd-limit diamond monotone: ~7/6 = [270.968, 272.727] (7\31 to 5\22)
* 11-odd-limit diamond tradeoff: ~7/6 = [266.871, 275.659]


valid range: [270.968, 271.698] (31 to 53)
{{Optimal ET sequence|legend=0| 9, 22, 31, 53, 84e }}
nice range: [266.871, 272.514]
strict range: [270.968, 271.698]


POTE tuning: ~7/6 = 271.288
Badness (Smith): 0.015231


Map: [&lt;1 0 3 1 -4|, &lt;0 7 -3 8 33|]
==== 13-limit ====
EDOs: 31, 84, 115, 376b, 491bd, 606bde
Subgroup: 2.3.5.7.11.13
Badness: 0.0314


==Winston==
Comma list: 99/98, 121/120, 176/175, 275/273
Commas: 66/65, 99/98, 105/104, 121/120


valid range: [270.968, 272.727] (31 to 22f)
Mapping: {{mapping| 1 0 3 1 3 8 | 0 7 -3 8 2 -19 }}
nice range: [266.871, 281.691]
strict range: [270.968, 272.727]


[[POTE tuning|POTE generator]]: ~7/6 = 271.088
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/6 = 271.556
* POTE: ~2 = 1200.000, ~7/6 = 271.546


Map: [&lt;1 0 3 1 3 1|, &lt;0 7 -3 8 2 12|]
Tuning ranges:
EDOs: 22f, 31
* 13- and 15-odd-limit diamond monotone: ~7/6 = [270.968, 271.698] (7\31 to 12\53)
Badness: 0.0199
* 13- and 15-odd-limit diamond tradeoff: ~7/6 = [266.871, 275.659]


=Doublethink=
{{Optimal ET sequence|legend=0| 22, 31, 53, 84e }}
Commas: 99/98, 121/120, 169/168, 176/175


valid range: [135.484, 136.364] (62 to 44)
Badness (Smith): 0.019718
nice range: [128.298, 138.573]
strict range: [135.484, 136.364]


POTE tuning: ~13/12 = 135.723
==== Blair ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 0 3 1 3 2|, &lt;0 14 -6 16 4 15|]
Comma list: 65/64, 78/77, 91/90, 99/98
EDOs: 9, 35, 44, 53, 62, 115ef, 168ef
Badness: 0.0271


=Borwell=
Mapping: {{mapping| 1 0 3 1 3 3 | 0 7 -3 8 2 3 }}
Commas: 225/224, 243/242, 1728/1715


POTE generator: ~55/36 = 735.752
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/6 = 271.747
* POTE: ~2 = 1200.000, ~7/6 = 271.301


Map: [&lt;1 7 0 9 17|, &lt;0 -14 6 -16 -35|]
{{Optimal ET sequence|legend=0| 9, 22, 31f }}
EDOs: 31, 106, 137, 442bd
Badness: 0.0384


=Triwell=
Badness (Smith): 0.023086
Commas: 1029/1024, 235298/234375


POTE generator: ~448/375 = 309.472
==== Winston ====
Subgroup: 2.3.5.7.11.13


Map: [&lt;1 7 0 1|, &lt;0 -21 9 7]]
Comma list: 66/65, 99/98, 105/104, 121/120
Wedgie: &lt;&lt;21 -9 -7 -63 -70 9||
EDOs: 31, 97, 128, 159, 190
Badness: 0.0806


==11-limit==
Mapping: {{mapping| 1 0 3 1 3 1 | 0 7 -3 8 2 12 }}
Commas: 385/384, 441/440, 456533/455625


POTE generator: ~448/375 = 309.471
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/6 = 271.163
* POTE: ~2 = 1200.000, ~7/6 = 271.088


Map: [&lt;1 7 0 1 13|, &lt;0 -21 9 7 -37]]
Tuning ranges:
EDOs: 31, 97, 128, 159, 190
* 13- and 15-odd-limit diamond monotone: ~7/6 = [270.968, 272.727] (7\31 to 5\22)
Badness: 0.0298</pre></div>
* 13- and 15-odd-limit diamond tradeoff: ~7/6 = [266.871, 281.691]
<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;title&gt;Semicomma family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:24:&amp;lt;img id=&amp;quot;wikitext@@toc@@normal&amp;quot; class=&amp;quot;WikiMedia WikiMediaToc&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/normal?w=225&amp;amp;h=100&amp;quot;/&amp;gt; --&gt;&lt;div id="toc"&gt;&lt;h1 class="nopad"&gt;Table of Contents&lt;/h1&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Orson"&gt;Orson&lt;/a&gt;&lt;/div&gt;
{{Optimal ET sequence|legend=0| 9, 22f, 31 }}
&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Orson-Seven limit children"&gt;Seven limit children&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Orwell"&gt;Orwell&lt;/a&gt;&lt;/div&gt;
Badness (Smith): 0.019931
&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Orwell-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Orwell-13-limit"&gt;13-limit&lt;/a&gt;&lt;/div&gt;
==== Doublethink ====
&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Orwell-Blair"&gt;Blair&lt;/a&gt;&lt;/div&gt;
Doublethink is a weak extension of orwell to the 13-limit. It splits the generator of ~7/6 into two [[13/12]]~[[14/13]]'s by tempering out their difference, [[169/168]]. Its ploidacot is alpha-tetradecacot.
&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Orwell-Newspeak"&gt;Newspeak&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:31 --&gt;&lt;!-- ws:start:WikiTextTocRule:32: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Orwell-Winston"&gt;Winston&lt;/a&gt;&lt;/div&gt;
Subgroup: 2.3.5.7.11.13
&lt;!-- ws:end:WikiTextTocRule:32 --&gt;&lt;!-- ws:start:WikiTextTocRule:33: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Doublethink"&gt;Doublethink&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:33 --&gt;&lt;!-- ws:start:WikiTextTocRule:34: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Borwell"&gt;Borwell&lt;/a&gt;&lt;/div&gt;
Comma list: 99/98, 121/120, 169/168, 176/175
&lt;!-- ws:end:WikiTextTocRule:34 --&gt;&lt;!-- ws:start:WikiTextTocRule:35: --&gt;&lt;div style="margin-left: 1em;"&gt;&lt;a href="#Triwell"&gt;Triwell&lt;/a&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:35 --&gt;&lt;!-- ws:start:WikiTextTocRule:36: --&gt;&lt;div style="margin-left: 2em;"&gt;&lt;a href="#Triwell-11-limit"&gt;11-limit&lt;/a&gt;&lt;/div&gt;
Mapping: {{mapping| 1 0 3 1 3 2 | 0 14 -6 16 4 15 }}
&lt;!-- ws:end:WikiTextTocRule:36 --&gt;&lt;!-- ws:start:WikiTextTocRule:37: --&gt;&lt;/div&gt;
 
&lt;!-- ws:end:WikiTextTocRule:37 --&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Orson"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Orson&lt;/h1&gt;
Optimal tunings:
The 5-limit parent comma for the &lt;strong&gt;semicomma family&lt;/strong&gt; is the semicomma, 2109375/2097152 = |-21 3 7&amp;gt;. This is the amount by which three pure 3/1 twelfths exceed seven pure 8/5 minor sixths. &lt;strong&gt;Orson&lt;/strong&gt;, the &lt;a class="wiki_link" href="/5-limit"&gt;5-limit&lt;/a&gt; temperament tempering it out, has a &lt;a class="wiki_link" href="/generator"&gt;generator&lt;/a&gt; of 75/64, which is sharper than 7/6 by 225/224 when untempered, and less sharp than that in any good orson tempering, for example &lt;a class="wiki_link" href="/53edo"&gt;53edo&lt;/a&gt; or &lt;a class="wiki_link" href="/84edo"&gt;84edo&lt;/a&gt;. These give tunings to the generator which are sharp of 7/6 by less than five &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s, making it hard to treat orson as anything other than an (at least) 7-limit system, leading to orwell.&lt;br /&gt;
* CTE: ~2 = 1200.000, ~13/12 = 135.811
&lt;br /&gt;
* POTE: ~2 = 1200.000, ~13/12 = 135.723
Comma: 2109375/2097152&lt;br /&gt;
 
&lt;br /&gt;
Tuning ranges:
&lt;a class="wiki_link" href="/Tuning%20Ranges%20of%20Regular%20Temperaments"&gt;valid range&lt;/a&gt;: [257.143, 276.923] (14b to 13)&lt;br /&gt;
* 13- and 15-odd-limit diamond monotone: ~13/12 = [135.484, 136.364] (7\62 to 5\44)
nice range: [271.229, 271.708]&lt;br /&gt;
* 13- and 15-odd-limit diamond tradeoff: ~13/12 = [128.298, 138.573]
strict range: [271.229, 271.708]&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 9, 35bd, 44, 53, 62, 115ef }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~75/64 = 271.627&lt;br /&gt;
 
&lt;br /&gt;
Badness (Smith): 0.027120
Map: [&amp;lt;1 0 3|, &amp;lt;0 7 -3|]&lt;br /&gt;
 
EDOs: 22, 31, 53, 190, 243, 296, 645c&lt;br /&gt;
=== Newspeak ===
Badness: 0.0408&lt;br /&gt;
Subgroup: 2.3.5.7.11
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Orson-Seven limit children"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Seven limit children&lt;/h2&gt;
Comma list: 225/224, 441/440, 1728/1715
The second comma of the &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal comma list&lt;/a&gt; defines which 7-limit family member we are looking at. Adding 65536/64625 leads to orwell, but we could also add 1029/1024, leading to the 31&amp;amp;159 temperament with wedgie &amp;lt;&amp;lt;21 -9 -7 -63 -70 9||, or 67528125/67108864, giving the 31&amp;amp;243 temperament with wedgie &amp;lt;&amp;lt;28 -12 1 -84 -77 36||, or 4375/4374, giving the 53&amp;amp;243 temperament with wedgie &amp;lt;&amp;lt;7 -3 61 -21 77 150||.&lt;br /&gt;
 
&lt;br /&gt;
Mapping: {{mapping| 1 0 3 1 -4 | 0 7 -3 8 33 }}
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Orwell"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Orwell&lt;/h1&gt;
 
Main article: &lt;a class="wiki_link" href="/Orwell"&gt;Orwell&lt;/a&gt;&lt;br /&gt;
Optimal tunings:  
So called because 19\84 (as a &lt;a class="wiki_link" href="/fraction%20of%20the%20octave"&gt;fraction of the octave&lt;/a&gt;) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It's compatible with &lt;a class="wiki_link" href="/22edo"&gt;22&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53&lt;/a&gt; and &lt;a class="wiki_link" href="/84edo"&gt;84&lt;/a&gt; equal, and may be described as the 22&amp;amp;31 temperament, or &amp;lt;&amp;lt;7 -3 8 -21 -7 27||. It's a good system in the &lt;a class="wiki_link" href="/7-limit"&gt;7-limit&lt;/a&gt; and naturally extends into the &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt;. &lt;a class="wiki_link" href="/84edo"&gt;84edo&lt;/a&gt;, with the 19\84 generator, provides a good tuning for the 5, 7 and 11 limits, but it does use its second-best 11. However, the 19\84 generator is remarkably close to the 11-limit &lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE tuning&lt;/a&gt;, as the generator is only 0.0024 cents sharper, and it is a good approximation to the 7-limit POTE generator also; hence 84 may be considered the most recommendable tuning in the 7-limit. &lt;a class="wiki_link" href="/53edo"&gt;53edo&lt;/a&gt; might be preferred in the 5-limit because of its nearly pure fifth and in the 11-limit because of it slightly better 11, though most of its 11-limit harmony is actually worse. Aside from the semicomma and 65625/65536, 7-limit orwell tempers out 2430/2401, the nuwell comma, 1728/1715, the orwellisma, 225/224, the septimal kleisma, and 6144/6125, the porwell comma.&lt;br /&gt;
* CTE: ~2 = 1200.000, ~7/6 = 271.316
&lt;br /&gt;
* POTE: ~2 = 1200.000, ~7/6 = 271.288
The 11-limit version of orwell tempers out 99/98, which means that two of its sharpened 7/6 generators give a flattened 11/8, as well as 121/120, 176/175, 385/384 and 540/539. Despite lowered tuning accuracy, orwell comes into its own in the 11-limit, giving acceptable accuracy and relatively low complexity. Tempering out the orwellisma, 1728/1715, means that orwell interprets three stacked 7/6 generators as an 8/5, and the tempered 1-7/6-11/8-8/5 chord is natural to orwell.&lt;br /&gt;
 
&lt;br /&gt;
Tuning ranges:
Orwell has MOS of size 9, 13, 22 and 31. The 9-note MOS is small enough to be retained in the mind as a genuine scale, is pleasing melodically, and has &lt;a class="wiki_link" href="/Retuning%2012edo%20to%20Orwell9"&gt;considerable harmonic resources&lt;/a&gt; despite its absence of 5-limit triads. The 13 note MOS has those, and of course the 22 and 31 note MOS are very well supplied with everything.&lt;br /&gt;
* 11-odd-limit diamond monotone: ~7/6 = [270.968, 271.698] (7\31 to 12\53)
&lt;br /&gt;
* 11-odd-limit diamond tradeoff: ~7/6 = [266.871, 272.514]
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 225/224, 1728/1715&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 22e, 31, 84, 115 }}
7-limit&lt;br /&gt;
 
[|1 0 0 0&amp;gt;, |14/11 0 -7/11 7/11&amp;gt;,&lt;br /&gt;
Badness (Smith): 0.031438
|27/11 0 3/11 -3/11&amp;gt;, |27/11 0 -8/11 8/11&amp;gt;]&lt;br /&gt;
 
&lt;a class="wiki_link" href="/Fractional%20monzos"&gt;Eigenmonzos&lt;/a&gt;: 2, 7/5&lt;br /&gt;
=== Borwell ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
9-limit&lt;br /&gt;
 
[|1 0 0 0&amp;gt;, |21/17 14/17 -7/17 0&amp;gt;,&lt;br /&gt;
Comma list: 225/224, 243/242, 1728/1715
|42/17 -6/17 3/17 0&amp;gt;, |41/17 16/17 -8/17 0&amp;gt;]&lt;br /&gt;
 
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 10/9&lt;br /&gt;
Mapping: {{mapping| 1 7 0 9 17 | 0 -14 6 -16 -35 }}
&lt;br /&gt;
 
valid range: [266.667, 272.727] (9 to 22)&lt;br /&gt;
: mapping generators: ~2, ~72/55
nice range: [266.871, 271.708]&lt;br /&gt;
 
strict range: [266.871, 271.708]&lt;br /&gt;
Optimal tunings:
&lt;br /&gt;
* CTE: ~2 = 1200.000, ~55/36 = 735.754
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~7/6 = 271.509&lt;br /&gt;
* POTE: ~2 = 1200.000, ~55/36 = 735.752
Algebraic generators: Sabra3, the real root of 12x^3-7x-48.&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 31, 75e, 106, 137 }}
Map: [&amp;lt;1 0 3 1|, &amp;lt;0 7 -3 8|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;7 -3 8 -21 -7 27||&lt;br /&gt;
Badness (Smith): 0.038377
EDOs: 22, 31, 53, 84, 137, 221d, 358d&lt;br /&gt;
 
Badness: 0.0207&lt;br /&gt;
== Sabric ==
&lt;br /&gt;
The sabric temperament ({{nowrap| 53 & 190 }}) tempers out the [[4375/4374|ragisma (4375/4374)]]. It is so named because it is closely related to the ''Sabra2 tuning'' (generator: 271.607278 cents).
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Orwell-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;11-limit&lt;/h2&gt;
 
&lt;a class="wiki_link" href="/Comma"&gt;Commas&lt;/a&gt;: 99/98, 121/120, 176/175&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
&lt;a class="wiki_link" href="/Minimax%20tuning"&gt;Minimax tuning&lt;/a&gt;&lt;br /&gt;
[[Comma list]]: 4375/4374, 2109375/2097152
[|1 0 0 0 0&amp;gt;, |14/11 0 -7/11 7/11 0&amp;gt;, |27/11 0 3/11 -3/11 0&amp;gt;,&lt;br /&gt;
 
|27/11 0 -8/11 8/11 0&amp;gt;, |37/11 0 -2/11 2/11 0&amp;gt;]&lt;br /&gt;
{{Mapping|legend=1| 1 0 3 -11 | 0 7 -3 61 }}
&lt;a class="wiki_link" href="/Eigenmonzo"&gt;Eigenmonzos&lt;/a&gt;: 2, 7/5&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:  
valid range: [270.968, 272.727] (31 to 22)&lt;br /&gt;
* [[CTE]]: ~2 = 1200.000, ~75/64 = 271.622
nice range: [266.871, 275.659]&lt;br /&gt;
: [[error map]]: {{val| 0.000 -0.599 -1.180 +0.131 }}
strict range: [270.968, 272.727]&lt;br /&gt;
* [[POTE]]: ~2 = 1200.000, ~75/64 = 271.607
&lt;br /&gt;
: error map: {{val| 0.000 -0.707 -1.134 -0.808 }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~7/6 = 271.426&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 53, 137d, 190, 243, 1511bccd }}
Map: [&amp;lt;1 0 3 1 3|, &amp;lt;0 7 -3 8 2|]&lt;br /&gt;
 
&lt;a class="wiki_link" href="/edo"&gt;Edos&lt;/a&gt;: &lt;a class="wiki_link" href="/22edo"&gt;22&lt;/a&gt;, &lt;a class="wiki_link" href="/31edo"&gt;31&lt;/a&gt;, &lt;a class="wiki_link" href="/53edo"&gt;53&lt;/a&gt;, &lt;a class="wiki_link" href="/84edo"&gt;84e&lt;/a&gt;&lt;br /&gt;
[[Badness]] (Smith): 0.088355
Badness: 0.0152&lt;br /&gt;
 
&lt;br /&gt;
== Triwell ==
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc4"&gt;&lt;a name="Orwell-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;13-limit&lt;/h2&gt;
The triwell temperament ({{nowrap| 31 & 159 }}) slices orwell major sixth ~128/75 into three generators, nine of which give the 5th harmonic.
Commas: 99/98, 121/120, 176/175, 275/273&lt;br /&gt;
 
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
valid range: [270.968, 271.698] (31 to 53)&lt;br /&gt;
 
nice range: [266.871, 275.659]&lt;br /&gt;
[[Comma list]]: 1029/1024, 235298/234375
strict range: [270.968, 271.698]&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 1 7 0 1 | 0 -21 9 7 }}
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~7/6 = 271.546&lt;br /&gt;
 
&lt;br /&gt;
: mapping generators: ~2, ~448/375
Map: [&amp;lt;1 0 3 1 3 8|, &amp;lt;0 7 -3 8 2 -19|]&lt;br /&gt;
 
EDOs: 22, 31, 53, 84e, 137e&lt;br /&gt;
[[Optimal tuning]]s:  
Badness: 0.0197&lt;br /&gt;
* [[CTE]]: ~2 = 1200.000, ~448/375 = 309.456
&lt;br /&gt;
: [[error map]]: {{val| 0.000 -0.522 -1.213 -2.637 }}
&lt;a class="wiki_link" href="/Orwell#Music"&gt;Music in Orwell&lt;/a&gt;&lt;br /&gt;
* [[POTE]]: ~2 = 1200.000, ~448/375 = 309.472
&lt;br /&gt;
: error map: {{val| 0.000 -0.872 -1.063 -2.520 }}
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="Orwell-Blair"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Blair&lt;/h2&gt;
 
Commas: 65/64, 78/77, 91/90, 99/98&lt;br /&gt;
{{Optimal ET sequence|legend=1| 31, 97, 128, 159, 190 }}
&lt;br /&gt;
 
valid range: []&lt;br /&gt;
[[Badness]] (Smith): 0.080575
nice range: [265.357, 289.210]&lt;br /&gt;
 
strict range: []&lt;br /&gt;
=== 11-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11
POTE generator: ~7/6 = 271.301&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 385/384, 441/440, 456533/455625
Map: [&amp;lt;1 0 3 1 3 3|, &amp;lt;0 7 -3 8 2 3|]&lt;br /&gt;
 
EDOs: 9, 22, 31f&lt;br /&gt;
Mapping: {{mapping| 1 7 0 1 13 | 0 -21 9 7 -37 }}
Badness: 0.0231&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc6"&gt;&lt;a name="Orwell-Newspeak"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;Newspeak&lt;/h2&gt;
* CTE: ~2 = 1200.000, ~448/375 = 309.444
Commas: 225/224, 441/440, 1728/1715&lt;br /&gt;
* POTE: ~2 = 1200.000, ~448/375 = 309.471
&lt;br /&gt;
 
valid range: [270.968, 271.698] (31 to 53)&lt;br /&gt;
{{Optimal ET sequence|legend=0| 31, 97, 128, 159, 190 }}
nice range: [266.871, 272.514]&lt;br /&gt;
 
strict range: [270.968, 271.698]&lt;br /&gt;
Badness (Smith): 0.029807
&lt;br /&gt;
 
POTE tuning: ~7/6 = 271.288&lt;br /&gt;
== Quadrawell ==
&lt;br /&gt;
The ''quadrawell'' temperament ({{nowrap| 31 & 212 }}) has an [[8/7]] generator of about 232 cents, twelve of which give the 5th harmonic.
Map: [&amp;lt;1 0 3 1 -4|, &amp;lt;0 7 -3 8 33|]&lt;br /&gt;
 
EDOs: 31, 84, 115, 376b, 491bd, 606bde&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
Badness: 0.0314&lt;br /&gt;
 
&lt;br /&gt;
[[Comma list]]: 2401/2400, 2109375/2097152
&lt;!-- ws:start:WikiTextHeadingRule:14:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc7"&gt;&lt;a name="Orwell-Winston"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:14 --&gt;Winston&lt;/h2&gt;
 
Commas: 66/65, 99/98, 105/104, 121/120&lt;br /&gt;
{{Mapping|legend=1| 1 7 0 3 | 0 -28 12 -1 }}
&lt;br /&gt;
 
valid range: [270.968, 272.727] (31 to 22f)&lt;br /&gt;
: mapping generators: ~2, ~8/7
nice range: [266.871, 281.691]&lt;br /&gt;
 
strict range: [270.968, 272.727]&lt;br /&gt;
[[Optimal tuning]]s:
&lt;br /&gt;
* [[CTE]]: ~2 = 1200.000, ~8/7 = 232.082
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~7/6 = 271.088&lt;br /&gt;
: [[error map]]: {{val| 0.000 -0.255 -1.328 -0.908 }}
&lt;br /&gt;
* [[POTE]]: ~2 = 1200.000, ~8/7 = 232.094
Map: [&amp;lt;1 0 3 1 3 1|, &amp;lt;0 7 -3 8 2 12|]&lt;br /&gt;
: error map: {{val| 0.000 -0.574 -1.191 -0.919 }}
EDOs: 22f, 31&lt;br /&gt;
 
Badness: 0.0199&lt;br /&gt;
{{Optimal ET sequence|legend=1| 31, 119, 150, 181, 212, 243, 698cd, 941cd }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:16:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc8"&gt;&lt;a name="Doublethink"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:16 --&gt;Doublethink&lt;/h1&gt;
[[Badness]] (Smith): 0.075754
Commas: 99/98, 121/120, 169/168, 176/175&lt;br /&gt;
 
&lt;br /&gt;
=== 11-limit ===
valid range: [135.484, 136.364] (62 to 44)&lt;br /&gt;
Subgroup: 2.3.5.7.11
nice range: [128.298, 138.573]&lt;br /&gt;
 
strict range: [135.484, 136.364]&lt;br /&gt;
Comma list: 385/384, 1375/1372, 14641/14580
&lt;br /&gt;
 
POTE tuning: ~13/12 = 135.723&lt;br /&gt;
Mapping: {{mapping| 1 7 0 3 11 | 0 -28 12 -1 -39 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 0 3 1 3 2|, &amp;lt;0 14 -6 16 4 15|]&lt;br /&gt;
Optimal tunings:
EDOs: 9, 35, 44, 53, 62, 115ef, 168ef&lt;br /&gt;
* CTE: ~2 = 1200.000, ~8/7 = 232.065
Badness: 0.0271&lt;br /&gt;
* POTE: ~2 = 1200.000, ~8/7 = 232.083
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc9"&gt;&lt;a name="Borwell"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;Borwell&lt;/h1&gt;
{{Optimal ET sequence|legend=0| 31, 119, 150, 181, 212, 455ee, 667cdee }}
Commas: 225/224, 243/242, 1728/1715&lt;br /&gt;
 
&lt;br /&gt;
Badness (Smith): 0.036493
POTE generator: ~55/36 = 735.752&lt;br /&gt;
 
&lt;br /&gt;
== Rainwell ==
Map: [&amp;lt;1 7 0 9 17|, &amp;lt;0 -14 6 -16 -35|]&lt;br /&gt;
The ''rainwell'' temperament ({{nowrap| 31 & 265 }}) tempers out the mirkwai comma, 16875/16807 and the [[rainy comma]], 2100875/2097152.
EDOs: 31, 106, 137, 442bd&lt;br /&gt;
 
Badness: 0.0384&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:20:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc10"&gt;&lt;a name="Triwell"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:20 --&gt;Triwell&lt;/h1&gt;
[[Comma list]]: 16875/16807, 2100875/2097152
Commas: 1029/1024, 235298/234375&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 1 14 -3 6 | 0 -35 15 -9 }}
POTE generator: ~448/375 = 309.472&lt;br /&gt;
 
&lt;br /&gt;
: mapping generators: ~2, ~2625/2048
Map: [&amp;lt;1 7 0 1|, &amp;lt;0 -21 9 7]]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;21 -9 -7 -63 -70 9||&lt;br /&gt;
[[Optimal tuning]]s:
EDOs: 31, 97, 128, 159, 190&lt;br /&gt;
* [[CTE]]: ~2 = 1200.000, ~2625/2048 = 425.666
Badness: 0.0806&lt;br /&gt;
: [[error map]]: {{val| 0.000 -0.278 -1.318 0.177 }}
&lt;br /&gt;
* [[POTE]]: ~2 = 1200.000, ~2625/2048 = 425.673
&lt;!-- ws:start:WikiTextHeadingRule:22:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc11"&gt;&lt;a name="Triwell-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:22 --&gt;11-limit&lt;/h2&gt;
: error map: {{val| 0.000 -0.526 -1.212 0.113 }}
Commas: 385/384, 441/440, 456533/455625&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 31, 172, 203, 234, 265, 296 }}
POTE generator: ~448/375 = 309.471&lt;br /&gt;
 
&lt;br /&gt;
[[Badness]] (Smith): 0.143488
Map: [&amp;lt;1 7 0 1 13|, &amp;lt;0 -21 9 7 -37]]&lt;br /&gt;
 
EDOs: 31, 97, 128, 159, 190&lt;br /&gt;
=== 11-limit ===
Badness: 0.0298&lt;/body&gt;&lt;/html&gt;</pre></div>
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 2100875/2097152
 
Mapping: {{mapping| 1 14 -3 6 29 | 0 -35 15 -9 -72 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~2625/2048 = 425.671
* POTE: ~2 = 1200.000, ~2625/2048 = 425.679
 
{{Optimal ET sequence|legend=0| 31, 234, 265, 296, 919bc }}
 
Badness (Smith): 0.052774
 
== Quinwell ==
The quinwell temperament ({{nowrap| 22 & 243 }}) slices orwell minor third into five generators and tempers out the wizma, 420175/419904.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 420175/419904, 2109375/2097152
 
{{Mapping|legend=1| 1 0 3 0 | 0 35 -15 62 }}
 
: mapping generators: ~2, ~405/392
 
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~405/392 = 54.335
: [[error map]]: {{val| 0.000 -0.233 -1.338 -0.061 }}
* [[POTE]]: ~2 = 1200.000, ~405/392 = 54.324
: error map: {{val| 0.000 -0.604 -1.178 -0.718 }}
 
{{Optimal ET sequence|legend=1| 22, …, 199d, 221, 243, 751c, 994cd, 1237bccd, 1480bccd }}
 
[[Badness]] (Smith): 0.168897
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 4375/4356, 2109375/2097152
 
Mapping: {{mapping| 1 0 3 0 5 | 0 35 -15 62 -34 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~33/32 = 54.338
* POTE: ~2 = 1200.000, ~33/32 = 54.334
 
{{Optimal ET sequence|legend=0| 22, 221, 243, 265 }}
 
Badness (Smith): 0.097202
 
=== Quinbetter ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 24057/24010, 43923/43750
 
Mapping: {{mapping| 1 0 3 0 4 | 0 35 -15 62 -12 }}
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~405/392 = 54.332
* POTE: ~2 = 1200.000, ~405/392 = 54.316
 
{{Optimal ET sequence|legend=0| 22, …, 199d, 221e, 243e, 707bcdeee }}
 
Badness (Smith): 0.078657
 
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Semicomma family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Orson]]
[[Category:Orwell]]

Latest revision as of 00:25, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The 5-limit parent comma for the semicomma family of temperaments is the semicomma (monzo: [-21 3 7⟩, ratio: 2109375/2097152). This is the amount by which three pure 3/1 twelfths exceed seven pure 8/5 minor sixths.

Orson

Orson, first discovered by Erv Wilson[citation needed], is the 5-limit temperament tempering out the semicomma. It has a generator of ~75/64, seven of which give the perfect twelfth; its ploidacot is alpha-heptacot. The generator is sharper than 7/6 by 225/224 when untempered, and less sharp than that in any good orson tempering, for example 53edo or 84edo. These give tunings to the generator which are sharp of 7/6 by less than five cents, making it hard to treat orson as anything other than an (at least) 7-limit system, leading to orwell.

Subgroup: 2.3.5

Comma list: 2109375/2097152

Mapping: [⟨1 0 3], ⟨0 7 -3]]

mapping generators: ~2, ~75/64

Optimal tunings:

  • CTE: ~2 = 1200.000, ~75/64 = 271.670
error map: ⟨0.000 -0.264 -1.324]
  • POTE: ~2 = 1200.000, ~75/64 = 271.627
error map: ⟨0.000 -0.564 -1.195]

Tuning ranges:

Optimal ET sequence: 22, 31, 53, 190, 243, 296, 645c

Badness (Smith): 0.040807

Overview to extensions

The second comma of the normal comma list defines which 7-limit family member we are looking at. Adding 65625/65536 (or 225/224) leads to orwell, but we could also add

  • 1029/1024, leading to the 31 & 159 temperament (triwell), or
  • 2401/2400, giving the 31 & 243 temperament (quadrawell), or
  • 4375/4374, giving the 53 & 243 temperament (sabric).

Orwell

Main article: Orwell

So called because 19\84 (as a fraction of the octave) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It is compatible with 22, 31, 53 and 84 equal, and may be described as the 22 & 31 temperament. It is a good system in the 7-limit and naturally extends into the 11-limit. 84edo, with the 19\84 generator, provides a good tuning for the 5-, 7- and 11-limit, but it does use its second-closest approximation to 11. However, the 19\84 generator is remarkably close to the 11-limit POTE tuning, as the generator is only 0.0024 cents sharper, and it is a good approximation to the 7-limit POTE generator also; hence 84 may be considered the most recommendable tuning in the 7-limit. 53edo might be preferred in the 5-limit because of its nearly pure fifth and in the 11-limit because of it slightly better 11, though most of its 11-limit harmony is actually worse. Aside from the semicomma and 65625/65536, 7-limit orwell tempers out 2430/2401, the nuwell comma, 1728/1715, the orwellisma, 225/224, the septimal kleisma, and 6144/6125, the porwell comma.

The 11-limit version of orwell tempers out 99/98, which means that two of its sharpened 7/6 generators give a flattened 11/8, as well as 121/120, 176/175, 385/384 and 540/539. Despite lowered tuning accuracy, orwell comes into its own in the 11-limit, giving acceptable accuracy and relatively low complexity. Tempering out the orwellisma, 1728/1715, means that orwell interprets three stacked 7/6 generators as an 8/5, and the tempered 1–7/6–11/8–8/5 chord is natural to orwell.

Orwell has mos scales of size 9, 13, 22 and 31. The 9-note mos is small enough to be retained in the mind as a genuine scale, is pleasing melodically, and has considerable harmonic resources despite its absence of 5-limit triads. The 13-note mos has those, and of course the 22- and 31-note mos are very well supplied with everything.

Subgroup: 2.3.5.7

Comma list: 225/224, 1728/1715

Mapping: [⟨1 0 3 1], ⟨0 7 -3 8]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/6 = 271.513
error map: ⟨0.000 -1.364 -0.853 +3.278]
  • POTE: ~2 = 1200.000, ~7/6 = 271.509
error map: ⟨0.000 -1.394 -0.840 +3.243]

Minimax tuning:

[[1 0 0 0⟩, [14/11 0 -7/11 7/11⟩, [27/11 0 3/11 -3/11⟩, [27/11 0 -8/11 8/11⟩]
unchanged-interval (eigenmonzo) basis: 2.7/5
[[1 0 0 0⟩, [21/17 14/17 -7/17 0⟩, [42/17 -6/17 3/17 0⟩, [41/17 16/17 -8/17 0⟩]
unchanged-interval (eigenmonzo) basis: 2.9/5

Tuning ranges:

  • 7-odd-limit diamond monotone: ~7/6 = [266.667, 272.727] (2\9 to 5\22)
  • 9-odd-limit diamond monotone: ~7/6 = [270.968, 272.727] (7\31 to 5\22)
  • 7-odd-limit diamond tradeoff: ~7/6 = [266.871, 271.708]
  • 9-odd-limit diamond tradeoff: ~7/6 = [266.871, 272.514]

Algebraic generator: Sabra3, the real root of 12x3 - 7x - 48.

Optimal ET sequence: 9, 22, 31, 53, 84, 137, 221d, 358d

Badness (Smith): 0.020735

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 121/120, 176/175

Mapping: [⟨1 0 3 1 3], ⟨0 7 -3 8 2]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/6 = 271.560
  • POTE: ~2 = 1200.000, ~7/6 = 271.426

Minimax tuning:

  • 11-odd-limit: ~7/6 = [2/11 0 -1/11 1/11⟩
[[1 0 0 0 0⟩, [14/11 0 -7/11 7/11 0⟩, [27/11 0 3/11 -3/11 0⟩, [27/11 0 -8/11 8/11 0⟩, [37/11 0 -2/11 2/11 0⟩]
Unchanged-interval (eigenmonzo) basis: 2.7/5

Tuning ranges:

  • 11-odd-limit diamond monotone: ~7/6 = [270.968, 272.727] (7\31 to 5\22)
  • 11-odd-limit diamond tradeoff: ~7/6 = [266.871, 275.659]

Optimal ET sequence: 9, 22, 31, 53, 84e

Badness (Smith): 0.015231

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 121/120, 176/175, 275/273

Mapping: [⟨1 0 3 1 3 8], ⟨0 7 -3 8 2 -19]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/6 = 271.556
  • POTE: ~2 = 1200.000, ~7/6 = 271.546

Tuning ranges:

  • 13- and 15-odd-limit diamond monotone: ~7/6 = [270.968, 271.698] (7\31 to 12\53)
  • 13- and 15-odd-limit diamond tradeoff: ~7/6 = [266.871, 275.659]

Optimal ET sequence: 22, 31, 53, 84e

Badness (Smith): 0.019718

Blair

Subgroup: 2.3.5.7.11.13

Comma list: 65/64, 78/77, 91/90, 99/98

Mapping: [⟨1 0 3 1 3 3], ⟨0 7 -3 8 2 3]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/6 = 271.747
  • POTE: ~2 = 1200.000, ~7/6 = 271.301

Optimal ET sequence: 9, 22, 31f

Badness (Smith): 0.023086

Winston

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 99/98, 105/104, 121/120

Mapping: [⟨1 0 3 1 3 1], ⟨0 7 -3 8 2 12]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/6 = 271.163
  • POTE: ~2 = 1200.000, ~7/6 = 271.088

Tuning ranges:

  • 13- and 15-odd-limit diamond monotone: ~7/6 = [270.968, 272.727] (7\31 to 5\22)
  • 13- and 15-odd-limit diamond tradeoff: ~7/6 = [266.871, 281.691]

Optimal ET sequence: 9, 22f, 31

Badness (Smith): 0.019931

Doublethink

Doublethink is a weak extension of orwell to the 13-limit. It splits the generator of ~7/6 into two 13/12~14/13's by tempering out their difference, 169/168. Its ploidacot is alpha-tetradecacot.

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 121/120, 169/168, 176/175

Mapping: [⟨1 0 3 1 3 2], ⟨0 14 -6 16 4 15]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~13/12 = 135.811
  • POTE: ~2 = 1200.000, ~13/12 = 135.723

Tuning ranges:

  • 13- and 15-odd-limit diamond monotone: ~13/12 = [135.484, 136.364] (7\62 to 5\44)
  • 13- and 15-odd-limit diamond tradeoff: ~13/12 = [128.298, 138.573]

Optimal ET sequence: 9, 35bd, 44, 53, 62, 115ef

Badness (Smith): 0.027120

Newspeak

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 1728/1715

Mapping: [⟨1 0 3 1 -4], ⟨0 7 -3 8 33]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~7/6 = 271.316
  • POTE: ~2 = 1200.000, ~7/6 = 271.288

Tuning ranges:

  • 11-odd-limit diamond monotone: ~7/6 = [270.968, 271.698] (7\31 to 12\53)
  • 11-odd-limit diamond tradeoff: ~7/6 = [266.871, 272.514]

Optimal ET sequence: 22e, 31, 84, 115

Badness (Smith): 0.031438

Borwell

Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242, 1728/1715

Mapping: [⟨1 7 0 9 17], ⟨0 -14 6 -16 -35]]

mapping generators: ~2, ~72/55

Optimal tunings:

  • CTE: ~2 = 1200.000, ~55/36 = 735.754
  • POTE: ~2 = 1200.000, ~55/36 = 735.752

Optimal ET sequence: 31, 75e, 106, 137

Badness (Smith): 0.038377

Sabric

The sabric temperament (53 & 190) tempers out the ragisma (4375/4374). It is so named because it is closely related to the Sabra2 tuning (generator: 271.607278 cents).

Subgroup: 2.3.5.7

Comma list: 4375/4374, 2109375/2097152

Mapping: [⟨1 0 3 -11], ⟨0 7 -3 61]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~75/64 = 271.622
error map: ⟨0.000 -0.599 -1.180 +0.131]
  • POTE: ~2 = 1200.000, ~75/64 = 271.607
error map: ⟨0.000 -0.707 -1.134 -0.808]

Optimal ET sequence: 53, 137d, 190, 243, 1511bccd

Badness (Smith): 0.088355

Triwell

The triwell temperament (31 & 159) slices orwell major sixth ~128/75 into three generators, nine of which give the 5th harmonic.

Subgroup: 2.3.5.7

Comma list: 1029/1024, 235298/234375

Mapping: [⟨1 7 0 1], ⟨0 -21 9 7]]

mapping generators: ~2, ~448/375

Optimal tunings:

  • CTE: ~2 = 1200.000, ~448/375 = 309.456
error map: ⟨0.000 -0.522 -1.213 -2.637]
  • POTE: ~2 = 1200.000, ~448/375 = 309.472
error map: ⟨0.000 -0.872 -1.063 -2.520]

Optimal ET sequence: 31, 97, 128, 159, 190

Badness (Smith): 0.080575

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 456533/455625

Mapping: [⟨1 7 0 1 13], ⟨0 -21 9 7 -37]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~448/375 = 309.444
  • POTE: ~2 = 1200.000, ~448/375 = 309.471

Optimal ET sequence: 31, 97, 128, 159, 190

Badness (Smith): 0.029807

Quadrawell

The quadrawell temperament (31 & 212) has an 8/7 generator of about 232 cents, twelve of which give the 5th harmonic.

Subgroup: 2.3.5.7

Comma list: 2401/2400, 2109375/2097152

Mapping: [⟨1 7 0 3], ⟨0 -28 12 -1]]

mapping generators: ~2, ~8/7

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 232.082
error map: ⟨0.000 -0.255 -1.328 -0.908]
  • POTE: ~2 = 1200.000, ~8/7 = 232.094
error map: ⟨0.000 -0.574 -1.191 -0.919]

Optimal ET sequence: 31, 119, 150, 181, 212, 243, 698cd, 941cd

Badness (Smith): 0.075754

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1375/1372, 14641/14580

Mapping: [⟨1 7 0 3 11], ⟨0 -28 12 -1 -39]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~8/7 = 232.065
  • POTE: ~2 = 1200.000, ~8/7 = 232.083

Optimal ET sequence: 31, 119, 150, 181, 212, 455ee, 667cdee

Badness (Smith): 0.036493

Rainwell

The rainwell temperament (31 & 265) tempers out the mirkwai comma, 16875/16807 and the rainy comma, 2100875/2097152.

Subgroup: 2.3.5.7

Comma list: 16875/16807, 2100875/2097152

Mapping: [⟨1 14 -3 6], ⟨0 -35 15 -9]]

mapping generators: ~2, ~2625/2048

Optimal tunings:

  • CTE: ~2 = 1200.000, ~2625/2048 = 425.666
error map: ⟨0.000 -0.278 -1.318 0.177]
  • POTE: ~2 = 1200.000, ~2625/2048 = 425.673
error map: ⟨0.000 -0.526 -1.212 0.113]

Optimal ET sequence: 31, 172, 203, 234, 265, 296

Badness (Smith): 0.143488

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 2100875/2097152

Mapping: [⟨1 14 -3 6 29], ⟨0 -35 15 -9 -72]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~2625/2048 = 425.671
  • POTE: ~2 = 1200.000, ~2625/2048 = 425.679

Optimal ET sequence: 31, 234, 265, 296, 919bc

Badness (Smith): 0.052774

Quinwell

The quinwell temperament (22 & 243) slices orwell minor third into five generators and tempers out the wizma, 420175/419904.

Subgroup: 2.3.5.7

Comma list: 420175/419904, 2109375/2097152

Mapping: [⟨1 0 3 0], ⟨0 35 -15 62]]

mapping generators: ~2, ~405/392

Optimal tunings:

  • CTE: ~2 = 1200.000, ~405/392 = 54.335
error map: ⟨0.000 -0.233 -1.338 -0.061]
  • POTE: ~2 = 1200.000, ~405/392 = 54.324
error map: ⟨0.000 -0.604 -1.178 -0.718]

Optimal ET sequence: 22, …, 199d, 221, 243, 751c, 994cd, 1237bccd, 1480bccd

Badness (Smith): 0.168897

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4375/4356, 2109375/2097152

Mapping: [⟨1 0 3 0 5], ⟨0 35 -15 62 -34]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~33/32 = 54.338
  • POTE: ~2 = 1200.000, ~33/32 = 54.334

Optimal ET sequence: 22, 221, 243, 265

Badness (Smith): 0.097202

Quinbetter

Subgroup: 2.3.5.7.11

Comma list: 385/384, 24057/24010, 43923/43750

Mapping: [⟨1 0 3 0 4], ⟨0 35 -15 62 -12]]

Optimal tunings:

  • CTE: ~2 = 1200.000, ~405/392 = 54.332
  • POTE: ~2 = 1200.000, ~405/392 = 54.316

Optimal ET sequence: 22, …, 199d, 221e, 243e, 707bcdeee

Badness (Smith): 0.078657

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