Semaphoresmic 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 '''semaphoresmic family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] [[49/48]] in the full [[7-limit]], and thereby identifies the septimal minor third, [[7/6]], and the septimal whole tone, [[8/7]]. It also splits the fourth into two of these intervals; hence the name, which sounds like "semi-fourth". Related to this is the 2.3.7-subgroup {49/48} temperament [[Semaphoresmic clan #Semaphore|semaphore]], and the 7-limit {49/48, 81/80} temperament [[Semaphoresmic clan #Godzilla|godzilla]].
: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2013-08-03 11:40:08 UTC</tt>.<br>
: The original revision id was <tt>443283010</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|flat]]
Semiphore temperament tempers out 49/48 and thereby identifies the septimal minor third, 7/6, and the septimal whole tone, 8/7. It also splits the fourth into two of these intervals; hence the name, which sounds like "semi-fourth". Related to this is the 2.3.7-limit 49/48 temperament called "[[xenharmonic/Chromatic pairs#Semaphore|semaphore]]," and the 2.3.5.7 49/48 and 81/80 temperament called [[meantone family|"godzilla"]].


Comma: 49/48
== Semaphoresmic a.k.a. semiphore ==
[[Subgroup]]: 2.3.5.7


Lattice basis: 7/6 0.7627 5/4 2.322
[[Comma list]]: 49/48
Angle(7/6, 5/4) = 90 degrees


Map: [&lt;1 0 2 2|, &lt;0 2 0 1|, &lt;0 0 1 0|]
{{Mapping|legend=1| 1 0 2 2 | 0 2 0 1 | 0 0 1 0 }}
Generators: 2, 7/4, 5
EDOs: 4, 5, 9, 10, 15, 19
Badness: 0.000116


=Selenium=
: mapping generators: ~2, ~7/4, ~5
Commas: 49/48, 56/55


Map: [&lt;1 0 2 2 5|, &lt;0 2 0 1 1|, &lt;0 0 1 0 -1|]
Lattice basis:  
EDOs: 4, 5, 9, 10, 15, 19, 24, 34
: 7/6 length = 0.7627, 5/4 length = 2.322
Badness: 0.000665
: Angle (7/6, 5/4) = 90 degrees


==13-limit==
[[Optimal tuning]]s:
Commas: 49/48, 56/55, 91/90
* [[WE]]: ~2 = 1202.8324{{c}}, ~7/4 = 951.8567{{c}}, ~5/4 = 380.5997{{c}}
: [[error map]]: {{val| +2.832 +1.758 -0.049 -11.304 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 950.6890{{c}}, ~5/4 = 382.3522{{c}}
: error map: {{val| 0.000 -0.577 -3.962 -18.137 }}


Map: [&lt;1 0 2 2 5 -1|, &lt;0 2 0 1 1 3|, &lt;0 0 1 0 -1 1|]
{{Optimal ET sequence|legend=1| 4, 5, 9, 10, 14c, 15, 19, 48d, 53d, 67cdd, 86cdd }}
EDOs: 5, 9, 10, 15, 19, 24, 34
Badness: 0.000787


=Negric=
[[Badness]] (Sintel): 0.510
Commas: 49/48, 225/224


Map: [&lt;1 2 2 3 0|, &lt;0 -4 3 -2 0|, &lt;0 0 0 0 1|]
=== 2.3.5.7.13 subgroup ===
EDOs: 9, 10, 19, 29, 38d, 67cde
Subgroup: 2.3.5.7.13
Badness: 0.001087


==13-limit==
Comma list: 49/48, 91/90
Commas: 49/48, 65/64, 91/90


Map: [&lt;1 2 2 3 0 4|, &lt;0 -4 3 -2 0 -3|, &lt;0 0 0 0 1 0|]
Sval mapping: {{mapping| 1 0 0 2 -1 | 0 2 0 1 3 | 0 0 1 0 1 }}
EDOs: 9, 10, 19, 29, 38df, 67cdef
 
Badness: 80.756</pre></div>
Optimal tunings:  
<h4>Original HTML content:</h4>
* WE: ~2 = 1202.7771{{c}}, ~7/4 = 951.9629{{c}}, ~5/4 = 381.0131{{c}}
<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;Semiphore family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:8:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:8 --&gt;&lt;!-- ws:start:WikiTextTocRule:9: --&gt;&lt;a href="#Selenium"&gt;Selenium&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:9 --&gt;&lt;!-- ws:start:WikiTextTocRule:10: --&gt;&lt;!-- ws:end:WikiTextTocRule:10 --&gt;&lt;!-- ws:start:WikiTextTocRule:11: --&gt; | &lt;a href="#Negric"&gt;Negric&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:11 --&gt;&lt;!-- ws:start:WikiTextTocRule:12: --&gt;&lt;!-- ws:end:WikiTextTocRule:12 --&gt;&lt;!-- ws:start:WikiTextTocRule:13: --&gt;
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 950.6703{{c}}, ~5/4 = 382.2972{{c}}
&lt;!-- ws:end:WikiTextTocRule:13 --&gt;Semiphore temperament tempers out 49/48 and thereby identifies the septimal minor third, 7/6, and the septimal whole tone, 8/7. It also splits the fourth into two of these intervals; hence the name, which sounds like &amp;quot;semi-fourth&amp;quot;. Related to this is the 2.3.7-limit 49/48 temperament called &amp;quot;&lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Chromatic%20pairs#Semaphore"&gt;semaphore&lt;/a&gt;,&amp;quot; and the 2.3.5.7 49/48 and 81/80 temperament called &lt;a class="wiki_link" href="/meantone%20family"&gt;&amp;quot;godzilla&amp;quot;&lt;/a&gt;.&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=0| 5, 9, 10, 14cf, 15, 19, 48df, 53d, 67cddf, 72dd }}
Comma: 49/48&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 0.380
Lattice basis: 7/6 0.7627 5/4 2.322&lt;br /&gt;
 
Angle(7/6, 5/4) = 90 degrees&lt;br /&gt;
== Selenium ==
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7.11
Map: [&amp;lt;1 0 2 2|, &amp;lt;0 2 0 1|, &amp;lt;0 0 1 0|]&lt;br /&gt;
 
Generators: 2, 7/4, 5&lt;br /&gt;
[[Comma list]]: 49/48, 56/55
EDOs: 4, 5, 9, 10, 15, 19&lt;br /&gt;
 
Badness: 0.000116&lt;br /&gt;
{{Mapping|legend=1| 1 0 2 2 5 | 0 2 0 1 1 | 0 0 1 0 -1 }}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Selenium"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Selenium&lt;/h1&gt;
[[Optimal tuning]]s:
Commas: 49/48, 56/55&lt;br /&gt;
* [[WE]]: ~2 = 1200.0412{{c}}, ~7/4 = 951.7616{{c}}, ~5/4 = 390.6621{{c}}
&lt;br /&gt;
: [[error map]]: {{val| +0.041 +1.568 +4.431 -16.982 +9.905 }}
Map: [&amp;lt;1 0 2 2 5|, &amp;lt;0 2 0 1 1|, &amp;lt;0 0 1 0 -1|]&lt;br /&gt;
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 951.7409{{c}}, ~5/4 = 390.6582{{c}}
EDOs: 4, 5, 9, 10, 15, 19, 24, 34&lt;br /&gt;
: error map: {{val| 0.000 +1.527 +4.344 -17.085 +9.765 }}
Badness: 0.000665&lt;br /&gt;
 
&lt;br /&gt;
{{Optimal ET sequence|legend=1| 4, 5, 9, 10, 15, 19, 24, 34 }}
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Selenium-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;13-limit&lt;/h2&gt;
 
Commas: 49/48, 56/55, 91/90&lt;br /&gt;
[[Badness]] (Sintel): 0.800
&lt;br /&gt;
 
Map: [&amp;lt;1 0 2 2 5 -1|, &amp;lt;0 2 0 1 1 3|, &amp;lt;0 0 1 0 -1 1|]&lt;br /&gt;
=== 13-limit ===
EDOs: 5, 9, 10, 15, 19, 24, 34&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
Badness: 0.000787&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 49/48, 56/55, 91/90
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Negric"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Negric&lt;/h1&gt;
 
Commas: 49/48, 225/224&lt;br /&gt;
Mapping: {{mapping| 1 0 2 2 5 -1 | 0 2 0 1 1 3 | 0 0 1 0 -1 1 }}
&lt;br /&gt;
 
Map: [&amp;lt;1 2 2 3 0|, &amp;lt;0 -4 3 -2 0|, &amp;lt;0 0 0 0 1|]&lt;br /&gt;
Optimal tunings:
EDOs: 9, 10, 19, 29, 38d, 67cde&lt;br /&gt;
* WE: ~2 = 1200.0790{{c}}, ~7/4 = 951.3067{{c}}, ~5/4 = 389.6213{{c}}
Badness: 0.001087&lt;br /&gt;
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 951.2662{{c}}, ~5/4 = 389.6120{{c}}
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Negric-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;13-limit&lt;/h2&gt;
{{Optimal ET sequence|legend=0| 5, 9, 10, 15, 19, 24, 34 }}
Commas: 49/48, 65/64, 91/90&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 0.736
Map: [&amp;lt;1 2 2 3 0 4|, &amp;lt;0 -4 3 -2 0 -3|, &amp;lt;0 0 0 0 1 0|]&lt;br /&gt;
 
EDOs: 9, 10, 19, 29, 38df, 67cdef&lt;br /&gt;
== Negric ==
Badness: 80.756&lt;/body&gt;&lt;/html&gt;</pre></div>
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 49/48, 225/224
 
{{Mapping|legend=1| 1 2 2 3 0 | 0 -4 3 -2 0 | 0 0 0 0 1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.4810{{c}}, ~15/14 = 125.9724{{c}}, ~11/8 = 540.7985{{c}}
: [[error map]]: {{val| +3.481 +1.118 -1.435 -10.328 -0.076 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/14 = 125.4347{{c}}, ~11/8 = 541.8688{{c}}
: error map: {{val| 0.000 -3.694 -10.009 -19.695 -9.449 }}
 
{{Optimal ET sequence|legend=1| 9, 10, 19, 29, 38d, 67cdde, 105cdddee }}
 
[[Badness]] (Sintel): 1.31
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 49/48, 65/64, 91/90
 
Mapping: {{mapping| 1 2 2 3 0 4 | 0 -4 3 -2 0 -3 | 0 0 0 0 1 0 }}
 
Optimal tunings:
* WE: ~2 = 1203.6981{{c}}, ~14/13 = 125.9545{{c}}, ~11/8 = 540.1440{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 125.3543{{c}}, ~11/8 = 540.9490{{c}}
 
{{Optimal ET sequence|legend=0| 9, 10, 19, 29, 38df, 67cddef, 105cdddeefff }}
 
Badness (Sintel): 0.755
 
[[Category:Semaphoresmic family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-3 temperaments]]

Latest revision as of 11:48, 16 August 2026

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 semaphoresmic family of rank-3 temperaments tempers out 49/48 in the full 7-limit, and thereby identifies the septimal minor third, 7/6, and the septimal whole tone, 8/7. It also splits the fourth into two of these intervals; hence the name, which sounds like "semi-fourth". Related to this is the 2.3.7-subgroup {49/48} temperament semaphore, and the 7-limit {49/48, 81/80} temperament godzilla.

Semaphoresmic a.k.a. semiphore

Subgroup: 2.3.5.7

Comma list: 49/48

Mapping: [⟨1 0 2 2], ⟨0 2 0 1], ⟨0 0 1 0]]

mapping generators: ~2, ~7/4, ~5

Lattice basis:

7/6 length = 0.7627, 5/4 length = 2.322
Angle (7/6, 5/4) = 90 degrees

Optimal tunings:

  • WE: ~2 = 1202.8324 ¢, ~7/4 = 951.8567 ¢, ~5/4 = 380.5997 ¢
error map: ⟨+2.832 +1.758 -0.049 -11.304]
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 950.6890 ¢, ~5/4 = 382.3522 ¢
error map: ⟨0.000 -0.577 -3.962 -18.137]

Optimal ET sequence: 4, 5, 9, 10, 14c, 15, 19, 48d, 53d, 67cdd, 86cdd

Badness (Sintel): 0.510

2.3.5.7.13 subgroup

Subgroup: 2.3.5.7.13

Comma list: 49/48, 91/90

Sval mapping: [⟨1 0 0 2 -1], ⟨0 2 0 1 3], ⟨0 0 1 0 1]]

Optimal tunings:

  • WE: ~2 = 1202.7771 ¢, ~7/4 = 951.9629 ¢, ~5/4 = 381.0131 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 950.6703 ¢, ~5/4 = 382.2972 ¢

Optimal ET sequence: 5, 9, 10, 14cf, 15, 19, 48df, 53d, 67cddf, 72dd

Badness (Sintel): 0.380

Selenium

Subgroup: 2.3.5.7.11

Comma list: 49/48, 56/55

Mapping: [⟨1 0 2 2 5], ⟨0 2 0 1 1], ⟨0 0 1 0 -1]]

Optimal tunings:

  • WE: ~2 = 1200.0412 ¢, ~7/4 = 951.7616 ¢, ~5/4 = 390.6621 ¢
error map: ⟨+0.041 +1.568 +4.431 -16.982 +9.905]
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 951.7409 ¢, ~5/4 = 390.6582 ¢
error map: ⟨0.000 +1.527 +4.344 -17.085 +9.765]

Optimal ET sequence: 4, 5, 9, 10, 15, 19, 24, 34

Badness (Sintel): 0.800

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 91/90

Mapping: [⟨1 0 2 2 5 -1], ⟨0 2 0 1 1 3], ⟨0 0 1 0 -1 1]]

Optimal tunings:

  • WE: ~2 = 1200.0790 ¢, ~7/4 = 951.3067 ¢, ~5/4 = 389.6213 ¢
  • CWE: ~2 = 1200.0000 ¢, ~7/4 = 951.2662 ¢, ~5/4 = 389.6120 ¢

Optimal ET sequence: 5, 9, 10, 15, 19, 24, 34

Badness (Sintel): 0.736

Negric

Subgroup: 2.3.5.7.11

Comma list: 49/48, 225/224

Mapping: [⟨1 2 2 3 0], ⟨0 -4 3 -2 0], ⟨0 0 0 0 1]]

Optimal tunings:

  • WE: ~2 = 1203.4810 ¢, ~15/14 = 125.9724 ¢, ~11/8 = 540.7985 ¢
error map: ⟨+3.481 +1.118 -1.435 -10.328 -0.076]
  • CWE: ~2 = 1200.0000 ¢, ~15/14 = 125.4347 ¢, ~11/8 = 541.8688 ¢
error map: ⟨0.000 -3.694 -10.009 -19.695 -9.449]

Optimal ET sequence: 9, 10, 19, 29, 38d, 67cdde, 105cdddee

Badness (Sintel): 1.31

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 65/64, 91/90

Mapping: [⟨1 2 2 3 0 4], ⟨0 -4 3 -2 0 -3], ⟨0 0 0 0 1 0]]

Optimal tunings:

  • WE: ~2 = 1203.6981 ¢, ~14/13 = 125.9545 ¢, ~11/8 = 540.1440 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/13 = 125.3543 ¢, ~11/8 = 540.9490 ¢

Optimal ET sequence: 9, 10, 19, 29, 38df, 67cddef, 105cdddeefff

Badness (Sintel): 0.755

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