Sensi extensions: Difference between revisions

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[[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 ({126/125, 176/175})~~. Notice the factorization~~ 126/125 = (176/175)(441/440). On the other, [[sensamagic]] strongly suggests tempering out [[385/384]], leading to undecimal sensamagic ({245/243, 385/384})~~. Notice the factorization~~ 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 ({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 ({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, 385/384

* '''Sensor''' (19 & 27) – tempering out 126/125, 245/243, and 385/384

* '''Sensus''' (19e & 27e) – tempering out 126/125, 176/175, 245/243

* '''Sensus''' (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]].

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

* '''Sensis''' (19 & 27e) – tempering out 56/55, 100/99, 245/243

* '''Sensis''' (19 & 27e) – tempering out 56/55, 100/99, and 245/243

* '''Sensa''' (19e & 27) – tempering out 55/54, 77/75, 99/98

* '''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 [[38edo|38df]], 46, and [[54edo|54c]].

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{| class="wikitable right-1 right-2"

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! rowspan=2 | #

! rowspan="2" | #

! rowspan=2 | Cents*

! rowspan="2" | Cents*

! colspan=5 | Approximate Ratios

! colspan="5" | Approximate Ratios

|-

! Sensi

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<nowiki>*~~</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 ==

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! [[Eigenmonzo|Eigenmonzo (Unchanged-interval]])

! [[Eigenmonzo|Eigenmonzo (Unchanged-interval]])

! Generator (¢)

! Comments

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