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Sensi extensions: Difference between revisions

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{{Breadcrumb|Sensi}}
[[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:  
[[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:  
* '''Sensor''' (19 & 27) – tempering out 126/125, 245/243, and 385/384
* '''Sensor''' ({{nowrap| 19 & 27 }}) – tempering out 126/125, 245/243, and 385/384
* '''Sensus''' (19e & 27e) – tempering out 126/125, 176/175, and 245/243
* '''Sensus''' ({{nowrap| 19e & 27e }}) – tempering out 126/125, 176/175, and 245/243


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]].
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]].


In addition, there are some low-complexity low-accuracy entries:  
In addition, there are some low-complexity low-accuracy entries:  
* '''Sensis''' (19 & 27e) – tempering out 56/55, 100/99, and 245/243
* '''Sensis''' ({{nowrap| 19 & 27e }}) – tempering out 56/55, 100/99, and 245/243
* '''Sensa''' (19e & 27) – tempering out 55/54, 77/75, and 99/98
* '''Sensa''' ({{nowrap| 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 [[38edo|38df]], 46, and [[54edo|54c]].  
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]].  
See [[Sensipent family #Sensor]], [[Sensipent family #Sensus|#Sensus]], [[Sensipent family #Sensis|#Sensis]], and [[Sensipent family #Sensa|#Sensa]] for technical data.


== Interval chain ==
== Interval chain ==
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{| class="wikitable right-1 right-2"
{| class="wikitable right-1 right-2"
|-
|-
! rowspan="2" | #
! rowspan="2" | #
! rowspan="2" | Cents*
! rowspan="2" | Cents*
! colspan="5" | Approximate Ratios
! colspan="5" | Approximate ratios
|-
|-
! Sensi
! Sensi
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|  
|  
|}
|}
<nowiki />* In 2.3.5.7.13.17/11 subgroup CTE tuning
<nowiki/>* In 2.3.5.7.13.17/11 subgroup CTE tuning


== Tuning spectra ==
== Tuning spectra ==
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{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br />generators
! Edo<br>generators
! [[Eigenmonzo|Eigenmonzo<br />(unchanged-interval)]]*
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
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| 7/6
| 7/6
| 444.478
| 444.478
|
|-
|
| 117/70
| 444.649
|  
|  
|-
|-
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{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br />generators
! Edo<br>generators
! Eigenmonzo<br />(unchanged-interval)*
! Unchanged interval<br>(eigenmonzo)*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
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| 7/6
| 7/6
| 444.478
| 444.478
|
|-
|
| 117/70
| 444.649
|  
|  
|-
|-
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|  
|  
|}
|}
<nowiki />* Besides the octave
<nowiki/>* Besides the octave


=== Sensus ===
=== Sensus ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br />generators
! Edo<br>generators
! Eigenmonzo<br />(unchanged-interval)*
! Unchanged interval<br>(eigenmonzo)*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
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| 7/6
| 7/6
| 444.478
| 444.478
|
|-
|
| 117/70
| 444.649
|  
|  
|-
|-
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{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br />generators
! Edo<br>generators
! Eigenmonzo<br />(unchanged-interval)*
! Unchanged interval<br>(eigenmonzo)*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
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| 7/6
| 7/6
| 444.478
| 444.478
|
|-
|
| 117/70
| 444.649
|  
|  
|-
|-
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|  
|  
|}
|}
<nowiki />* Besides the octave
<nowiki/>* Besides the octave


[[Category:Sensi]]
[[Category:Sensi]]
[[Category:Temperament extensions]]
[[Category:Temperament extensions]]
[[Category:Rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Sensi_extensions"