Sensi extensions: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Breadcrumb|Sensi}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-11-06 09:41:21 UTC</tt>.<br>
: The original revision id was <tt>379671118</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">[[Sensi]] temperament has various extensions to the 13 limit. These are all members of the [[sensipent family]], but in addition they are linear temperaments.


=Tuning Spectra=
[[Sensi]] has multiple competing [[extension]]s to the [[11-limit]]. The simplest [[7-limit]] [[comma]]s of sensi are [[126/125|starling (126/125)]] and [[245/243|sensamagic (245/243)]], and it can be viewed as the merge of the two corresponding [[rank-3 temperament]]s. These rank-3 temperaments are associated with distinct paths to the 11-limit. On one hand, [[starling]] strongly suggests tempering out [[176/175]], leading to thrush ({{nowrap|{126/125, 176/175}}}), since {{nowrap|126/125 {{=}} (176/175)(441/440)}}. On the other, [[sensamagic]] strongly suggests tempering out [[385/384]], leading to undecimal sensamagic ({{nowrap|{245/243, 385/384}}}), since {{nowrap|245/243 {{=}} (385/384)(896/891)}}. Taking either path for sensi leads us to one of the following entries:
==Spectrum of Sensor Tunings by Eigenmonzos==
* '''Sensor''' ({{nowrap| 19 & 27 }}) – tempering out 126/125, 245/243, and 385/384
* '''Sensus''' ({{nowrap| 19e & 27e }}) – tempering out 126/125, 176/175, and 245/243


Gencom: [2 9/7; 91/90 126/125 169/168 385/384]
The two unite in [[46edo|46et]], where both 176/175 and 385/384, as well as their sum, [[121/120]], are tempered out. They can be extended to the 13- and 17-limit naturally by adding [[91/90]] and [[154/153]] to the comma list in this order. Then the generator represents [[9/7]], [[13/10]], and [[22/17]].
Gencom map: [&lt;1 -1 -1 -2 9 0|, &lt;0 7 9 13 -15 10|]
||~ Eigenmonzo ||~ Supermajor Third ||
|| 9/7 || 435.084 ||
|| 15/14 || 439.814 ||
|| 18/13 || 440.846 ||
|| 15/13 || 441.290 ||
|| 6/5 || 442.179 ||
|| 14/13 || 442.766 ||
|| 5/4 || 442.924 ||
|| 16/15 || 443.017 ||
|| 11/10 || 443.125 ||
|| 15/11 || 443.127 ||
|| 4/3 || 443.136 ||
|| 11/9 || 443.193 ||
|| 12/11 || 443.211 ||
|| 11/8 || 443.245 ||
|| 14/11 || 443.482 ||
|| 10/9 || 443.519 ||
|| 13/11 || 443.568 ||
|| 8/7 || 443.756 ||
|| 16/13 || 444.053 ||
|| 7/6 || 444.478 ||
|| 7/5 || 445.628 ||
|| 13/12 || 446.191 ||
|| 13/10 || 454.214 ||


==Spectrum of Sensis Tunings by Eigenmonzos==
In addition, there are some low-complexity low-accuracy entries:
* '''Sensis''' ({{nowrap| 19 & 27e }}) – tempering out 56/55, 100/99, and 245/243
* '''Sensa''' ({{nowrap| 19e & 27 }}) – tempering out 55/54, 77/75, and 99/98


Gencom: [2 9/7; 56/55 78/77 91/90 100/99]
Another possible path which relates a sense of compromise is to temper out [[121/120]], leading to bisensi. This has the effect of slicing the octave in two, and is supported by [[38edo|38df]], 46, and [[54edo|54c]].  
Gencom map: [&lt;1 -1 -1 -2 2 0|, &lt;0 7 9 13 4 10|]
||~ Eigenmonzo ||~ Supermajor Third ||
|| 9/7 || 435.084 ||
|| 11/8 || 437.829 ||
|| 15/14 || 439.814 ||
|| 18/13 || 440.846 ||
|| 15/13 || 441.290 ||
|| 6/5 || 442.179 ||
|| 14/13 || 442.766 ||
|| 5/4 || 442.924 ||
|| 16/15 || 443.017 ||
|| 4/3 || 443.136 ||
|| 10/9 || 443.519 ||
|| 8/7 || 443.756 ||
|| 16/13 || 444.053 ||
|| 7/6 || 444.478 ||
|| 15/11 || 444.746 ||
|| 11/9 || 445.259 ||
|| 7/5 || 445.628 ||
|| 13/12 || 446.191 ||
|| 14/11 || 446.390 ||
|| 11/10 || 446.999 ||
|| 13/11 || 448.202 ||
|| 12/11 || 450.212 ||
|| 13/10 || 454.214 ||


==Spectrum of Sensus Tunings by Eigenmonzos==
See [[Sensipent family #Sensor]], [[Sensipent family #Sensus|#Sensus]], [[Sensipent family #Sensis|#Sensis]], and [[Sensipent family #Sensa|#Sensa]] for technical data.


Gencom: [2 9/7; 91/90 126/125 169/168 352/351]
== Interval chain ==
Gencom map: [&lt;1 -1 -1 -2 -8 0|, &lt;0 7 9 13 31 10|]
In the following table, odd harmonics and subharmonics 1–21 are in '''bold'''.  
||~ Eigenmonzo ||~ Supermajor Third ||
|| 9/7 || 435.084 ||
|| 15/14 || 439.814 ||
|| 18/13 || 440.846 ||
|| 15/13 || 441.290 ||
|| 6/5 || 442.179 ||
|| 14/13 || 442.766 ||
|| 5/4 || 442.924 ||
|| 16/15 || 443.017 ||
|| 4/3 || 443.136 ||
|| 13/11 || 443.371 ||
|| 14/11 || 443.472 ||
|| 10/9 || 443.519 ||
|| 11/8 || 443.591 ||
|| 12/11 || 443.723 ||
|| 8/7 || 443.756 ||
|| 11/10 || 443.864 ||
|| 11/9 || 443.965 ||
|| 16/13 || 444.053 ||
|| 15/11 || 444.203 ||
|| 7/6 || 444.478 ||
|| 7/5 || 445.628 ||
|| 13/12 || 446.191 ||
|| 13/10 || 454.214 ||


{| class="wikitable right-1 right-2"
|-
! rowspan="2" | #
! rowspan="2" | Cents*
! colspan="5" | Approximate ratios
|-
! Sensi
! Sensor
! Sensis
! Sensus
! Sensa
|-
| 0
| 0.0
| '''1/1'''
|
|
|
|
|-
| 1
| 443.4
| 9/7, 13/10, 22/17
|
|
|
| 14/11, 17/13
|-
| 2
| 886.8
| 5/3
|
|
|
| 17/10, 18/11, 22/13, 28/17
|-
| 3
| 130.2
| 13/12, 14/13, 15/14
|
| 12/11, '''17/16'''
|
| 11/10, 18/17
|-
| 4
| 573.6
| 7/5, 18/13
|
| '''11/8''', 24/17
|
| 15/11, 17/12
|-
| 5
| 1017.0
| 9/5
|
| 20/11
|
| 11/6, 30/17
|-
| 6
| 260.4
| 7/6, 15/13
|
| 13/11, 20/17
|
|
|-
| 7
| 703.8
| '''3/2'''
|
| 26/17
|
|
|-
| 8
| 1147.2
| 27/14, 35/18
|
|
|
|
|-
| 9
| 390.6
| '''5/4'''
|
| 14/11
|
|
|-
| 10
| 834.0
| '''13/8''', 21/13
|
| 18/11, 28/17
|
|
|-
| 11
| 77.4
| 21/20, 25/24
|
| 18/17
|
| '''17/16'''
|-
| 12
| 520.8
| 27/20
|
| 15/11
|
| '''11/8'''
|-
| 13
| 964.2
| '''7/4'''
|
| 30/17
|
|
|-
| 14
| 207.6
| '''9/8'''
|
|
| 17/15
|
|-
| 15
| 651.0
| 35/24
| '''16/11'''
|
| 22/15
|
|-
| 16
| 1094.5
| '''15/8'''
| '''32/17'''
|
| 17/9
|
|-
| 17
| 337.9
| 39/32
|
|
| 11/9, 17/14
|
|-
| 18
| 781.3
| 25/16
|
|
| 11/7
|
|-
| 19
| 24.7
| 49/48, 65/64, 81/80
|
|
|
|
|-
| 20
| 468.1
| '''21/16'''
|
|
| 17/13
|
|-
| 21
| 911.5
| 27/16
|
|
| 17/10, 22/13
|
|-
| 22
| 154.9
| 35/32
| 12/11
|
| 11/10
|
|-
| 23
| 598.3
| 45/32
| 24/17
|
| 17/12
|
|-
| 24
| 1041.7
| 117/64
| 20/11
|
| 11/6
|
|-
| 25
| 285.1
| 75/64
| 13/11, 20/17
|
|
|
|-
| 26
| 728.5
| 49/32
| 26/17
|
|
|
|-
| 27
| 1171.9
| 63/32
|
|
|
|
|-
| 28
| 415.3
| 81/64
| 14/11
|
|
|
|-
| 29
| 858.7
| 105/64
| 18/11, 28/17
|
|
|
|-
| 30
| 102.1
| 135/128
| 18/17
|
| '''17/16'''
|
|-
| 31
| 545.5
| 175/128
| 15/11
|
| '''11/8'''
|
|-
| 32
| 988.9
| 225/128
| 30/17
|
|
|
|}
<nowiki/>* In 2.3.5.7.13.17/11 subgroup CTE tuning


== Tuning spectra ==
=== Sensor ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generators
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Comments
|-
|
| 9/7
| 435.084
|
|-
| 4\11
|
| 436.364
| 11cdefgg val
|-
|
| 15/14
| 439.814
|
|-
|
| 13/9
| 440.846
|
|-
|
| 15/13
| 441.290
|
|-
| 7\19
|
| 442.105
|
|-
|
| 5/3
| 442.179
|
|-
|
| 13/7
| 442.766
|
|-
|
| 5/4
| 442.924
| 5-odd-limit minimax
|-
|
| 15/8
| 443.017
|
|-
|
| 11/10
| 443.125
|
|-
|
| 15/11
| 443.127
|
|-
|
| 3/2
| 443.136
| 15-odd-limit minimax
|-
|
| 11/9
| 443.193
|
|-
|
| 11/6
| 443.211
|
|-
|
| 11/8
| 443.245
|
|-
| 17\46
|
| 443.478
|
|-
|
| 11/7
| 443.482
| 11-odd-limit minimax
|-
|
| 9/5
| 443.519
| 9- and 13-odd-limit minimax
|-
|
| 13/11
| 443.568
|
|-
|
| 7/4
| 443.756
| 7-odd-limit minimax
|-
|
| 13/8
| 444.053
|
|-
| 10\27
|
| 444.444
|
|-
|
| 7/6
| 444.478
|
|-
|
| 117/70
| 444.649
|
|-
|
| 7/5
| 445.628
|
|-
|
| 13/12
| 446.191
|
|-
| 3\8
|
| 450.000
| 8deg val
|-
|
| 13/10
| 454.214
|
|}
<nowiki/>* Besides the octave


=== Sensis ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generators
! Unchanged interval<br>(eigenmonzo)*
! Generator (¢)
! Comments
|-
|
| 9/7
| 435.084
|
|-
| 4\11
|
| 436.364
| 11cdf val
|-
|
| 11/8
| 437.829
|
|-
|
| 15/14
| 439.814
|
|-
|
| 13/9
| 440.846
|
|-
|
| 15/13
| 441.290
|
|-
| 7\19
|
| 442.105
|
|-
|
| 5/3
| 442.179
|
|-
|
| 13/7
| 442.766
|
|-
|
| 5/4
| 442.924
| 5-odd-limit minimax
|-
|
| 15/8
| 443.017
|
|-
|
| 3/2
| 443.136
|
|-
| 17\46
|
| 443.478
| 46e val
|-
|
| 9/5
| 443.519
| 9-odd-limit minimax
|-
|
| 7/4
| 443.756
| 7- and 11-odd-limit minimax
|-
|
| 13/8
| 444.053
| 13- and 15-odd-limit minimax
|-
| 10\27
|
| 444.444
| 27e val
|-
|
| 7/6
| 444.478
|
|-
|
| 117/70
| 444.649
|
|-
|
| 15/11
| 444.746
|
|-
|
| 11/9
| 445.259
|
|-
|
| 7/5
| 445.628
|
|-
|
| 13/12
| 446.191
|
|-
|
| 11/7
| 446.390
|
|-
|
| 11/10
| 446.999
|
|-
|
| 13/11
| 448.202
|
|-
| 3\8
|
| 450.000
| 8d val
|-
|
| 11/6
| 450.212
|
|-
|
| 13/10
| 454.214
|
|}
<nowiki/>* Besides the octave


</pre></div>
=== Sensus ===
<h4>Original HTML content:</h4>
{| class="wikitable center-all left-4"
<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;Sensi extensions&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;a class="wiki_link" href="/Sensi"&gt;Sensi&lt;/a&gt; temperament has various extensions to the 13 limit. These are all members of the &lt;a class="wiki_link" href="/sensipent%20family"&gt;sensipent family&lt;/a&gt;, but in addition they are linear temperaments.&lt;br /&gt;
|-
&lt;br /&gt;
! Edo<br>generators
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Tuning Spectra"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Tuning Spectra&lt;/h1&gt;
! Unchanged interval<br>(eigenmonzo)*
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="Tuning Spectra-Spectrum of Sensor Tunings by Eigenmonzos"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Spectrum of Sensor Tunings by Eigenmonzos&lt;/h2&gt;
! Generator (¢)
&lt;br /&gt;
! Comments
Gencom: [2 9/7; 91/90 126/125 169/168 385/384]&lt;br /&gt;
|-
Gencom map: [&amp;lt;1 -1 -1 -2 9 0|, &amp;lt;0 7 9 13 -15 10|]&lt;br /&gt;
|
| 9/7
| 435.084
|
|-
| 4\11
|
| 436.364
| 11cdeeeefggg val
|-
|
| 15/14
| 439.814
|
|-
|
| 13/9
| 440.846
|
|-
|
| 15/13
| 441.290
|
|-
| 7\19
|
| 442.105
| 19eg val
|-
|
| 5/3
| 442.179
|
|-
|
| 13/7
| 442.766
|
|-
|
| 5/4
| 442.924
| 5-odd-limit minimax
|-
|
| 15/8
| 443.017
|
|-
|
| 3/2
| 443.136
|
|-
|
| 13/11
| 443.371
|
|-
|
| 11/7
| 443.472
|
|-
| 17\46
|
| 443.478
|
|-
|
| 9/5
| 443.519
| 9-odd-limit minimax
|-
|
| 11/8
| 443.591
|
|-
|
| 11/6
| 443.723
|
|-
|
| 7/4
| 443.756
| 7- and 11-odd-limit minimax
|-
|
| 11/10
| 443.864
|
|-
|
| 11/9
| 443.965
|
|-
|
| 13/8
| 444.053
| 13- and 15-odd-limit minimax
|-
|
| 15/11
| 444.203
|
|-
| 10\27
|
| 444.444
| 27eg val
|-
|
| 7/6
| 444.478
|
|-
|
| 117/70
| 444.649
|
|-
|
| 7/5
| 445.628
|
|-
|
| 13/12
| 446.191
|
|-
| 3\8
|
| 450.000
| 8deegg val
|-
|
| 13/10
| 454.214
|
|}
<nowiki/>* Besides the octave


=== Sensa ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generators
! Unchanged interval<br>(eigenmonzo)*
! Generator (¢)
! Comments
|-
|
| 11/7
| 417.508
|
|-
|
| 11/9
| 426.296
|
|-
|
| 15/11
| 434.238
|
|-
|
| 9/7
| 435.084
|
|-
| 4\11
|
| 436.364
| 11cdeef val
|-
|
| 15/14
| 439.814
|
|-
|
| 13/9
| 440.846
|
|-
|
| 15/13
| 441.290
|
|-
| 7\19
|
| 442.105
| 19e val
|-
|
| 5/3
| 442.179
|
|-
|
| 13/7
| 442.766
|
|-
|
| 5/4
| 442.924
| 5-odd-limit minimax
|-
|
| 15/8
| 443.017
|
|-
|
| 3/2
| 443.136
|
|-
| 17\46
|
| 443.478
| 46ee val
|-
|
| 9/5
| 443.519
| 9-odd-limit minimax
|-
|
| 7/4
| 443.756
| 7- and 11-odd-limit minimax
|-
|
| 13/8
| 444.053
| 13- and 15-odd-limit minimax
|-
| 10\27
|
| 444.444
|
|-
|
| 7/6
| 444.478
|
|-
|
| 117/70
| 444.649
|
|-
|
| 7/5
| 445.628
|
|-
|
| 11/8
| 445.943
|
|-
|
| 13/12
| 446.191
|
|-
|
| 11/6
| 449.873
|
|-
| 3\8
|
| 450.000
| 8d val
|-
|
| 13/10
| 454.214
|
|-
|
| 11/10
| 455.001
|
|-
|
| 13/11
| 455.395
|
|}
<nowiki/>* Besides the octave


&lt;table class="wiki_table"&gt;
[[Category:Sensi]]
    &lt;tr&gt;
[[Category:Temperament extensions]]
        &lt;th&gt;Eigenmonzo&lt;br /&gt;
[[Category:Rank-2 temperaments]]
&lt;/th&gt;
        &lt;th&gt;Supermajor Third&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;9/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;435.084&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;15/14&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;439.814&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;18/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;440.846&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;15/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;441.290&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;6/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;442.179&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;14/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;442.766&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;442.924&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;16/15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;443.017&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11/10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;443.125&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;15/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;443.127&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;443.136&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;443.193&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;12/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;443.211&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;443.245&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;14/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;443.482&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;10/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;443.519&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;443.568&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;8/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;443.756&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;16/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;444.053&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;444.478&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;445.628&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13/12&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;446.191&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13/10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;454.214&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Tuning Spectra-Spectrum of Sensis Tunings by Eigenmonzos"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Spectrum of Sensis Tunings by Eigenmonzos&lt;/h2&gt;
&lt;br /&gt;
Gencom: [2 9/7; 56/55 78/77 91/90 100/99]&lt;br /&gt;
Gencom map: [&amp;lt;1 -1 -1 -2 2 0|, &amp;lt;0 7 9 13 4 10|]&lt;br /&gt;
 
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;th&gt;Eigenmonzo&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Supermajor Third&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;9/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;435.084&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;437.829&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;15/14&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;439.814&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;18/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;440.846&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;15/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;441.290&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;6/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;442.179&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;14/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;442.766&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;442.924&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;16/15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;443.017&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;443.136&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;10/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;443.519&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;8/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;443.756&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;16/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;444.053&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;444.478&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;15/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;444.746&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;445.259&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;445.628&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13/12&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;446.191&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;14/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;446.390&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11/10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;446.999&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;448.202&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;12/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;450.212&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13/10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;454.214&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;br /&gt;
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Tuning Spectra-Spectrum of Sensus Tunings by Eigenmonzos"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Spectrum of Sensus Tunings by Eigenmonzos&lt;/h2&gt;
&lt;br /&gt;
Gencom: [2 9/7; 91/90 126/125 169/168 352/351]&lt;br /&gt;
Gencom map: [&amp;lt;1 -1 -1 -2 -8 0|, &amp;lt;0 7 9 13 31 10|]&lt;br /&gt;
 
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;th&gt;Eigenmonzo&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Supermajor Third&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;9/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;435.084&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;15/14&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;439.814&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;18/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;440.846&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;15/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;441.290&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;6/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;442.179&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;14/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;442.766&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;5/4&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;442.924&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;16/15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;443.017&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;4/3&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;443.136&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;443.371&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;14/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;443.472&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;10/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;443.519&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11/8&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;443.591&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;12/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;443.723&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;8/7&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;443.756&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11/10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;443.864&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;11/9&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;443.965&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;16/13&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;444.053&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;15/11&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;444.203&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7/6&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;444.478&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;7/5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;445.628&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13/12&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;446.191&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;13/10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;454.214&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;/body&gt;&lt;/html&gt;</pre></div>

Latest revision as of 12:04, 6 August 2025

< Sensi

Sensi has multiple competing extensions to the 11-limit. The simplest 7-limit commas of sensi are starling (126/125) and sensamagic (245/243), and it can be viewed as the merge of the two corresponding rank-3 temperaments. These rank-3 temperaments are associated with distinct paths to the 11-limit. On one hand, starling strongly suggests tempering out 176/175, leading to thrush ({126/125, 176/175}), since 126/125 = (176/175)(441/440). On the other, sensamagic strongly suggests tempering out 385/384, leading to undecimal sensamagic ({245/243, 385/384}), since 245/243 = (385/384)(896/891). Taking either path for sensi leads us to one of the following entries:

  • Sensor (19 & 27) – tempering out 126/125, 245/243, and 385/384
  • Sensus (19e & 27e) – tempering out 126/125, 176/175, and 245/243

The two unite in 46et, where both 176/175 and 385/384, as well as their sum, 121/120, are tempered out. They can be extended to the 13- and 17-limit naturally by adding 91/90 and 154/153 to the comma list in this order. Then the generator represents 9/7, 13/10, and 22/17.

In addition, there are some low-complexity low-accuracy entries:

  • Sensis (19 & 27e) – tempering out 56/55, 100/99, and 245/243
  • Sensa (19e & 27) – tempering out 55/54, 77/75, and 99/98

Another possible path which relates a sense of compromise is to temper out 121/120, leading to bisensi. This has the effect of slicing the octave in two, and is supported by 38df, 46, and 54c.

See Sensipent family #Sensor, #Sensus, #Sensis, and #Sensa for technical data.

Interval chain

In the following table, odd harmonics and subharmonics 1–21 are in bold.

# Cents* Approximate ratios
Sensi Sensor Sensis Sensus Sensa
0 0.0 1/1
1 443.4 9/7, 13/10, 22/17 14/11, 17/13
2 886.8 5/3 17/10, 18/11, 22/13, 28/17
3 130.2 13/12, 14/13, 15/14 12/11, 17/16 11/10, 18/17
4 573.6 7/5, 18/13 11/8, 24/17 15/11, 17/12
5 1017.0 9/5 20/11 11/6, 30/17
6 260.4 7/6, 15/13 13/11, 20/17
7 703.8 3/2 26/17
8 1147.2 27/14, 35/18
9 390.6 5/4 14/11
10 834.0 13/8, 21/13 18/11, 28/17
11 77.4 21/20, 25/24 18/17 17/16
12 520.8 27/20 15/11 11/8
13 964.2 7/4 30/17
14 207.6 9/8 17/15
15 651.0 35/24 16/11 22/15
16 1094.5 15/8 32/17 17/9
17 337.9 39/32 11/9, 17/14
18 781.3 25/16 11/7
19 24.7 49/48, 65/64, 81/80
20 468.1 21/16 17/13
21 911.5 27/16 17/10, 22/13
22 154.9 35/32 12/11 11/10
23 598.3 45/32 24/17 17/12
24 1041.7 117/64 20/11 11/6
25 285.1 75/64 13/11, 20/17
26 728.5 49/32 26/17
27 1171.9 63/32
28 415.3 81/64 14/11
29 858.7 105/64 18/11, 28/17
30 102.1 135/128 18/17 17/16
31 545.5 175/128 15/11 11/8
32 988.9 225/128 30/17

* In 2.3.5.7.13.17/11 subgroup CTE tuning

Tuning spectra

Sensor

Edo
generators 		Unchanged interval
(eigenmonzo)* 		Generator (¢) Comments
9/7 435.084
4\11 436.364 11cdefgg val
15/14 439.814
13/9 440.846
15/13 441.290
7\19 442.105
5/3 442.179
13/7 442.766
5/4 442.924 5-odd-limit minimax
15/8 443.017
11/10 443.125
15/11 443.127
3/2 443.136 15-odd-limit minimax
11/9 443.193
11/6 443.211
11/8 443.245
17\46 443.478
11/7 443.482 11-odd-limit minimax
9/5 443.519 9- and 13-odd-limit minimax
13/11 443.568
7/4 443.756 7-odd-limit minimax
13/8 444.053
10\27 444.444
7/6 444.478
117/70 444.649
7/5 445.628
13/12 446.191
3\8 450.000 8deg val
13/10 454.214

* Besides the octave

Sensis

Edo
generators 		Unchanged interval
(eigenmonzo)* 		Generator (¢) Comments
9/7 435.084
4\11 436.364 11cdf val
11/8 437.829
15/14 439.814
13/9 440.846
15/13 441.290
7\19 442.105
5/3 442.179
13/7 442.766
5/4 442.924 5-odd-limit minimax
15/8 443.017
3/2 443.136
17\46 443.478 46e val
9/5 443.519 9-odd-limit minimax
7/4 443.756 7- and 11-odd-limit minimax
13/8 444.053 13- and 15-odd-limit minimax
10\27 444.444 27e val
7/6 444.478
117/70 444.649
15/11 444.746
11/9 445.259
7/5 445.628
13/12 446.191
11/7 446.390
11/10 446.999
13/11 448.202
3\8 450.000 8d val
11/6 450.212
13/10 454.214

* Besides the octave

Sensus

Edo
generators 		Unchanged interval
(eigenmonzo)* 		Generator (¢) Comments
9/7 435.084
4\11 436.364 11cdeeeefggg val
15/14 439.814
13/9 440.846
15/13 441.290
7\19 442.105 19eg val
5/3 442.179
13/7 442.766
5/4 442.924 5-odd-limit minimax
15/8 443.017
3/2 443.136
13/11 443.371
11/7 443.472
17\46 443.478
9/5 443.519 9-odd-limit minimax
11/8 443.591
11/6 443.723
7/4 443.756 7- and 11-odd-limit minimax
11/10 443.864
11/9 443.965
13/8 444.053 13- and 15-odd-limit minimax
15/11 444.203
10\27 444.444 27eg val
7/6 444.478
117/70 444.649
7/5 445.628
13/12 446.191
3\8 450.000 8deegg val
13/10 454.214

* Besides the octave

Sensa

Edo
generators 		Unchanged interval
(eigenmonzo)* 		Generator (¢) Comments
11/7 417.508
11/9 426.296
15/11 434.238
9/7 435.084
4\11 436.364 11cdeef val
15/14 439.814
13/9 440.846
15/13 441.290
7\19 442.105 19e val
5/3 442.179
13/7 442.766
5/4 442.924 5-odd-limit minimax
15/8 443.017
3/2 443.136
17\46 443.478 46ee val
9/5 443.519 9-odd-limit minimax
7/4 443.756 7- and 11-odd-limit minimax
13/8 444.053 13- and 15-odd-limit minimax
10\27 444.444
7/6 444.478
117/70 444.649
7/5 445.628
11/8 445.943
13/12 446.191
11/6 449.873
3\8 450.000 8d val
13/10 454.214
11/10 455.001
13/11 455.395

* Besides the octave

Retrieved from "https://en.xen.wiki/index.php?title=Sensi_extensions&oldid=206424"