Xenharmonic Wiki

Gammic family: Difference between revisions

VisualWikitext
Wikispaces>genewardsmith
**Imported revision 152023425 - Original comment: **
 
m Text replacement - "Subgroup-val mapping: {{mapping| " to "{{Mapping|legend=2| "
Tags: Mobile edit Mobile web edit
 
(48 intermediate revisions by 13 users not shown)
Line 1: Line 1:
<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 '''gammic family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[gammic comma]] ({{monzo|legend=1| -29 -11 20 }}), a [[5-limit]] comma of about 4.77 cents in size.  
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-07-09 00:13:10 UTC</tt>.<br>
: The original revision id was <tt>152023425</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 [[Carlos Gamma]] rank one temperament divides 3/2 into 20 equal parts, 11 of which give a 5/4. This is closely related to the rank two microtemperament tempering out |-29 -11 20&gt;. This temperament, gammic, takes five generator steps to reach 5/4, and 20 to reach 3/2.The generator in question is 1990656/1953125 = |13 5 -9&gt;, which when suitably tempered is very close to 5/171 octaves, which makes for an ideal gammic tuning. As a 5-limit temperament supported by 171-et, [[Schismatic family|schismatic]] temperament makes for a natural comparison. Schismatic, with a wedgie of &lt;&lt;1 -8 -15|| is plainly much less complex than gammic with wedgie &lt;&lt;20 11 -29||, but people seeking the exotic might prefer gammic even so. The 34-note MOS is interesting, being a 1L33s refinement of the [[34edo]] tuning. Of course gammic can be tuned to 34, which makes the two equivalent, and would rather remove the point of Carlos Gamma if used for it.


Because 171 is such a strong 7-limit system, it is natural to extend gammic to the 7-limit. This we may do by adding 4375/4374 to the comma list, giving a wedgie of &lt;&lt;20 11 96 -29 96 192||. 96 gammic generators finally reach 7, which is a long way to go compared to the 39 generator steps of pontiac. If someone wants to make the trip, a 103-note MOS is possible.
== Gammic ==
The [[Carlos Gamma]] rank-1 temperament divides a [[~]][[3/2]] into 20 equal parts, 11 of which give a ~[[5/4]]. This is closely related to the rank-2 [[microtemperament]] tempering out {{monzo| -29 -11 20 }}, the gammic comma. This temperament, gammic, takes 11 [[generator]] steps to reach 5/4, and 20 to reach 3/2. The generator in question is ~[[1990656/1953125]] ({{monzo| 13 5 -9 }}), which when suitably tempered is very close to [[171edo|5\171]], which makes for an ideal gammic tuning.  


===Neptune===
As a 5-limit temperament supported by 171edo, the [[schismic]] temperament makes for a natural comparison. Schismic, tempering out the [[schisma]] ({{monzo| -15 8 1 }}), is plainly much less complex than gammic, but people seeking the exotic might prefer gammic even so.  
A more interesting extension is to Neptune, which divides an octave plus a gammic generator in half, to get a 10/7 generator. Neptune adds 2401/2400 to the gammic comma, and may be described as the 68&amp;171 temperament, with wedgie &lt;&lt;40 22 21 -58 -79 -13||. The generator chain goes merrily on, stacking one 10/7 over another. until after eighteen generator steps 6/5 (up nine octaves) is reached. Then in succession we get 12/7, the neutral third, 7/4 and 5/4. Two neutral thirds then gives a fifth, and these intervals with their inverses are the full set of septimal consonances. [[171edo]] makes a good tuning, and we can also choose to make any of the consonances besides 7/5 and 10/7 just, including the fifth, which gives a tuning extending Carlos Gamma.  


Adding 385/384 or 1375/1372 to the list of commas allows for an extension to the 11-limit, where (7/5)^3 equates to 11/4. This may be described as &lt;&lt;40 22 21 -3 ...|| or 68&amp;103, and 171 can still be used as a tuning, with val &lt;171 271 397 480  591|.
The 34-note [[mos]] is interesting, being a 1L 33s refinement of the [[34edo]] tuning. Of course gammic can be tuned to 34edo, which makes the two equivalent, and would rather remove the point of Carlos Gamma if used for it.


An article on Neptune as an analog of miracle can be found [[http://tech.groups.yahoo.com/group/tuning-math/message/6001|here]].
[[Subgroup]]: 2.3.5
</pre></div>
 
<h4>Original HTML content:</h4>
[[Comma list]]: {{monzo| -29 -11 20 }}
<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;Gammic family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;a class="wiki_link" href="/Carlos%20Gamma"&gt;Carlos Gamma&lt;/a&gt; rank one temperament divides 3/2 into 20 equal parts, 11 of which give a 5/4. This is closely related to the rank two microtemperament tempering out |-29 -11 20&amp;gt;. This temperament, gammic, takes five generator steps to reach 5/4, and 20 to reach 3/2.The generator in question is 1990656/1953125 = |13 5 -9&amp;gt;, which when suitably tempered is very close to 5/171 octaves, which makes for an ideal gammic tuning. As a 5-limit temperament supported by 171-et, &lt;a class="wiki_link" href="/Schismatic%20family"&gt;schismatic&lt;/a&gt; temperament makes for a natural comparison. Schismatic, with a wedgie of &amp;lt;&amp;lt;1 -8 -15|| is plainly much less complex than gammic with wedgie &amp;lt;&amp;lt;20 11 -29||, but people seeking the exotic might prefer gammic even so. The 34-note MOS is interesting, being a 1L33s refinement of the &lt;a class="wiki_link" href="/34edo"&gt;34edo&lt;/a&gt; tuning. Of course gammic can be tuned to 34, which makes the two equivalent, and would rather remove the point of Carlos Gamma if used for it.&lt;br /&gt;
 
&lt;br /&gt;
{{Mapping|legend=1| 1 1 2 | 0 20 11 }}
Because 171 is such a strong 7-limit system, it is natural to extend gammic to the 7-limit. This we may do by adding 4375/4374 to the comma list, giving a wedgie of &amp;lt;&amp;lt;20 11 96 -29 96 192||. 96 gammic generators finally reach 7, which is a long way to go compared to the 39 generator steps of pontiac. If someone wants to make the trip, a 103-note MOS is possible.&lt;br /&gt;
: mapping generators: ~2, ~1990656/1953125
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc0"&gt;&lt;a name="x--Neptune"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Neptune&lt;/h3&gt;
[[Optimal tuning]]s:  
A more interesting extension is to Neptune, which divides an octave plus a gammic generator in half, to get a 10/7 generator. Neptune adds 2401/2400 to the gammic comma, and may be described as the 68&amp;amp;171 temperament, with wedgie &amp;lt;&amp;lt;40 22 21 -58 -79 -13||. The generator chain goes merrily on, stacking one 10/7 over another. until after eighteen generator steps 6/5 (up nine octaves) is reached. Then in succession we get 12/7, the neutral third, 7/4 and 5/4. Two neutral thirds then gives a fifth, and these intervals with their inverses are the full set of septimal consonances. &lt;a class="wiki_link" href="/171edo"&gt;171edo&lt;/a&gt; makes a good tuning, and we can also choose to make any of the consonances besides 7/5 and 10/7 just, including the fifth, which gives a tuning extending Carlos Gamma. &lt;br /&gt;
* [[WE]]: ~2 = 1200.0419{{c}}, ~1990656/1953125 = 35.0977{{c}}
&lt;br /&gt;
: [[error map]]: {{val| +0.042 +0.399 -0.156 }}
Adding 385/384 or 1375/1372 to the list of commas allows for an extension to the 11-limit, where (7/5)^3 equates to 11/4. This may be described as &amp;lt;&amp;lt;40 22 21 -3 ...|| or 68&amp;amp;103, and 171 can still be used as a tuning, with val &amp;lt;171 271 397 480  591|.&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1990656/1953125 = 35.0981{{c}}
&lt;br /&gt;
: error map: {{val| 0.000 +0.008 -0.234 }}
An article on Neptune as an analog of miracle can be found &lt;a class="wiki_link_ext" href="http://tech.groups.yahoo.com/group/tuning-math/message/6001" rel="nofollow"&gt;here&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
{{Optimal ET sequence|legend=1| 34, 103, 137, 171, 547, 718, 889, 1607 }}
 
[[Badness]] (Sintel): 2.06
 
=== Overview to extensions ===
==== 7-limit extensions ====
Because 171 is such a strong [[7-limit]] system, it is well motivated to extend gammic to the 7-limit. This we may do by adding [[4375/4374]] to the comma list. 96 gammic generators finally reach 7, which is a long way to go compared to the 39 generator steps of [[pontiac]]. If someone wants to make the trip, a 103-note generator chain is possible.
 
==== Subgroup extensions ====
Gammic also naturally extends with the [[17/1|17th harmonic]], as is given in [[#Subgroup extensions_2|#Subgroup extensions]].
 
== Septimal gammic ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 4375/4374, 6591796875/6576668672
 
{{Mapping|legend=1| 1 1 2 0 | 0 20 11 96 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0712{{c}}, ~234375/229376 = 35.0924{{c}}
: [[error map]]: {{val| +0.071 -0.035 -0.154 +0.049 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~234375/229376 = 35.0913{{c}}
: error map: {{val| 0.000 -0.130 -0.310 -0.065 }}
 
{{Optimal ET sequence|legend=1| 34d, …, 137d, 171, 1402, 1573, 1744, 1915, 2086c, …, 2599c, 5369bccd }}
 
[[Badness]] (Sintel): 1.20
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 4375/4356, 100352/99825
 
{{Mapping|legend=0| 1 1 2 0 2 | 0 20 11 96 50 }}
 
Optimal tunings:
* WE: ~2 = 1199.8949{{c}}, ~45/44 = 35.0855{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 35.0872{{c}}
 
{{Optimal ET sequence|legend=0| 34d, …, 137d, 171 }}
 
Badness (Sintel): 3.21
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 243/242, 364/363, 625/624, 2200/2197
 
{{Mapping|legend=0| 1 1 2 0 2 3 | 0 20 11 96 50 24 }}
 
Optimal tunings:
* WE: ~2 = 1199.8098{{c}}, ~45/44 = 35.0855{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 35.0888{{c}}
 
{{Optimal ET sequence|legend=0| 34d, 137d, 171 }}
 
Badness (Sintel): 1.98
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 243/242, 364/363, 375/374, 595/594, 2200/2197
 
{{Mapping|legend=0| 1 1 2 0 2 3 4 | 0 20 11 96 50 24 3 }}
 
Optimal tunings:
* WE: ~2 = 1199.8393{{c}}, ~45/44 = 35.0851{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 35.0882{{c}}
 
{{Optimal ET sequence|legend=0| 34d, 137d, 171 }}
 
Badness (Sintel): 1.60
 
== Gammy ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 225/224, 94143178827/91913281250
 
{{Mapping|legend=1| 1 1 2 1 | 0 20 11 62 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.5055{{c}}, ~1990656/1953125 = 34.9984{{c}}
: [[error map]]: {{val| +0.506 -1.482 -0.321 +1.577 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1990656/1953125 = 34.9947{{c}}
: error map: {{val| 0.000 -2.060 -1.372 +0.848 }}
 
{{Optimal ET sequence|legend=1| 34d, 69d, 103, 240, 343b }}
 
[[Badness]] (Sintel): 5.84
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 225/224, 243/242, 215622/214375
 
{{Mapping|legend=0| 1 1 2 1 2 | 0 20 11 62 50 }}
 
Optimal tunings:  
* WE: ~2 = 1200.5129{{c}}, ~45/44 = 34.9999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 34.9967{{c}}
 
{{Optimal ET sequence|legend=0| 34d, 69de, 103, 240, 343be }}
 
Badness (Sintel): 2.16
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 225/224, 243/242, 351/350, 1188/1183
 
{{Mapping|legend=0| 1 1 2 1 2 3 | 0 20 11 62 50 24 }}
 
Optimal tunings:
* WE: ~2 = 1200.4356{{c}}, ~45/44 = 35.0008{{c}}
* CWE: ~2 = 1200.000{{c}}, ~45/44 = 34.9975{{c}}
 
{{Optimal ET sequence|legend=0| 34d, 69de, 103, 240, 343be }}
 
Badness (Sintel): 1.38
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 225/224, 243/242, 351/350, 375/374, 1188/1183
 
{{Mapping|legend=0| 1 1 2 1 2 3 4 | 0 20 11 62 50 24 3 }}
 
Optimal tunings:
* WE: ~2 = 1200.2936{{c}}, ~45/44 = 35.0057{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 35.0021{{c}}
 
{{Optimal ET sequence|legend=0| 34d, 69de, 103, 137, 240 }}
 
Badness (Sintel): 1.28
 
== Neptune ==
A more interesting extension is to neptune, which divides an octave plus a gammic generator in half, to get a 10/7 generator. Neptune adds [[2401/2400]] to the gammic comma, and may be described as the {{nowrap| 68 & 171 }} temperament. The generator chain goes merrily on, stacking one 10/7 over another, until after eighteen generator steps [[6/5]] (up nine octaves) is reached. Then in succession we get [[12/7]], the neutral third, [[7/4]] and [[5/4]]. Two neutral thirds then gives a fifth, and these intervals with their inverses are the full set of [[7-odd-limit]] consonances. [[171edo]] makes a good tuning, and we can also choose to make any of the consonances besides 7/5 and 10/7 just, including the fifth, which gives a tuning extending [[Carlos Gamma]].
 
Adding 385/384 or 1375/1372 to the list of commas allows for an extension to the [[11-limit]], where (7/5)<sup>3</sup> equates to 11/4.
 
[[Gene Ward Smith]] once described [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_6001.html neptune as an analog of miracle].
 
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 48828125/48771072
 
{{Mapping|legend=1| 1 -19 -9 -8 | 0 40 22 21 }}
: mapping generators: 2, ~10/7
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0660{{c}}, ~10/7 = 617.5815{{c}}
: [[error map]]: {{val| +0.066 +0.053 -0.114 -0.141 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 617.5489{{c}}
: error map: {{val| 0.000 +0.000 -0.238 -0.299 }}
 
{{Optimal ET sequence|legend=1| 35, 68, 103, 171, 1094, 1265, 1436, 1607, 1778, 1949d, 3727cdd, 5676ccddd }}
 
[[Badness]] (Sintel): 0.593
 
==== 2.3.5.7.17 subgroup ====
Extending 2.3.5.17 gammic via neptune, we find that both 2401/2400 ({{S|49}}) and 2500/2499 (S50) are tempered out; their product, 1225/1224 (S35) is therefore also tempered out.
 
Subgroup: 2.3.5.7.17
 
Comma list: 1225/1224, 2401/2400, 24576/24565
 
{{Mapping|legend=2| 1 -19 -9 -8 1 | 0 40 22 21 6 }}
 
Optimal tunings:
* WE: ~2 = 1200.0136{{c}}, ~10/7 = 617.5572{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 617.5504{{c}}
 
{{Optimal ET sequence|legend=0| 35, 68, 103, 171, 581, 752, 923, 1094 }}
 
Badness (Sintel): 0.404
 
==== 2.3.5.7.17.31 subgroup ====
Since neptune splits the interval of [[5/3]] into two, we can accurately map each part to [[40/31]]~[[31/24]] by tempering out [[961/960]] (S31). This is especially natural, as combined with tempering out 1225/1224 (S35) and 24576/24565 (S16/S17), we can map (17/16)<sup>2</sup> (6 gammic generators) to [[35/31]]. This also gives us its complement with respect to [[5/4]], the interval of 5 gammic generators representing a quarter of a perfect fifth, as [[31/28]].
 
Subgroup: 2.3.5.7.17.31
 
Comma list: 868/867, 961/960, 1225/1224, 2401/2400
 
{{Mapping|legend=2| 1 -19 -9 -8 1 -11 | 0 40 22 21 6 31 }}
 
Optimal tunings:
* WE: ~2 = 1200.0519{{c}}, ~10/7 = 617.5760{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 617.5501{{c}}
 
{{Optimal ET sequence|legend=0| 35, 68, 103, 171, 752k, 923k }}
 
Badness (Sintel): 0.393
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 1375/1372, 78408/78125
 
{{Mapping|legend=0| 1 -19 -9 -8 5 | 0 40 22 21 -3 }}
 
Optimal tunings:
* WE: ~2 = 1200.4655{{c}}, ~10/7 = 617.7648{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 617.5317{{c}}
 
{{Optimal ET sequence|legend=0| 35, 68, 103, 171e, 274e, 445ee }}
 
Badness (Sintel): 2.10
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 385/384, 625/624, 1188/1183, 1375/1372
 
{{Mapping|legend=0| 1 -19 -9 -8 5 -21 | 0 40 22 21 -3 48 }}
 
Optimal tunings:
* WE: ~2 = 1200.4067{{c}}, ~10/7 = 617.7290{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 617.5257{{c}}
 
{{Optimal ET sequence|legend=0| 35f, 68, 103, 171e, 274e }}
 
Badness (Sintel): 1.54
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 385/384, 561/560, 625/624, 715/714, 1188/1183
 
{{Mapping|legend=0| 1 -19 -9 -8 5 -21 1 | 0 40 22 21 -3 48 6 }}
 
Optimal tunings:
* WE: ~2 = 1200.2971{{c}}, ~10/7 = 617.6784{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 617.5291{{c}}
 
{{Optimal ET sequence|legend=0| 35f, 68, 103, 171e, 274e }}
 
Badness (Sintel): 1.32
 
=== Salacia ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 9765625/9732096
 
{{Mapping|legend=0| 1 -19 -9 -8 -48 | 0 40 22 21 100 }}
 
Optimal tunings:
* WE: ~2 = 1200.2180{{c}}, ~10/7 = 617.6341{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 617.5253{{c}}
 
{{Optimal ET sequence|legend=0| 68e, 103, 171, 274 }}
 
Badness (Sintel): 2.30
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 243/242, 441/440, 625/624, 2200/2197
 
{{Mapping|legend=0| 1 -19 -9 -8 -48 -21 | 0 40 22 21 100 48 }}
 
Optimal tunings:
* WE: ~2 = 1200.1492{{c}}, ~10/7 = 617.5993{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 617.5249{{c}}
 
{{Optimal ET sequence|legend=0| 68e, 103, 171, 274 }}
 
Badness (Sintel): 1.45
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 243/242, 375/374, 441/440, 625/624, 2200/2197
 
{{Mapping|legend=0| 1 -19 -9 -8 -48 -21 1 | 0 40 22 21 100 48 6 }}
 
Optimal tunings:
* WE: ~2 = 1200.0872{{c}}, ~10/7 = 617.5702{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 617.5264{{c}}
 
{{Optimal ET sequence|legend=0| 68e, 103, 171, 274, 445e }}
 
Badness (Sintel): 1.25
 
=== Poseidon ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 9801/9800, 9453125/9437184
 
{{Mapping|legend=0| 2 2 4 5 8 | 0 40 22 21 -37 }}
: mapping generators: ~99/70, ~99/98
 
Optimal tunings:
* WE: ~99/70 = 600.0509{{c}}, ~99/98 = 17.5466{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~99/98 = 17.5458{{c}}
 
{{Optimal ET sequence|legend=0| 68, 206b, 274, 342, 2804cdee, 3146cdee, …, 5198bccdddeeee }}
 
Badness (Sintel): 1.38
 
== Subgroup extensions ==
=== Gammic (2.3.5.17) ===
The interval of 3 generators represents 1/3 of [[6/5]], which is very close to [[17/16]], with the comma between 6/5 and (17/16)<sup>3</sup> being [[24576/24565]] ([[S-expression|S16/S17]]). This then naturally interprets the generator as [[51/50]] with two generators representing [[25/24]], tempering out [[15625/15606]] (S49⋅S50<sup>2</sup>).
 
Subgroup: 2.3.5.17
 
Comma list: 15625/15606, 24576/24565
 
{{Mapping|legend=2| 1 1 2 4 | 0 20 11 3 }}
: mapping generators: ~2, ~51/50
 
Optimal tunings:
* WE: ~2 = 1199.9899{{c}}, ~51/50 = 35.1008{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~51/50 = 35.1008{{c}}
 
{{Optimal ET sequence|legend=0| 34, 103, 137, 171, 376, 547 }}
 
Badness (Sintel): 0.320
 
[[Category:Gammic family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Gammic_family"