Sensi: Difference between revisions

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| en = Sensi

| de = Sensi

~~| en = Sensi~~

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''~~This page~~ is ~~about~~ the ~~temperament where~~ 3/2 ~~is found at~~ 7 ~~steps~~. ~~For other structures sometimes called "sensi"~~, ~~see~~ [[~~BPS~~]] and [[~~sentry~~]].''

{{Infobox regtemp

| Title = Sensi

| Subgroups = 2.3.5.7, 2.3.5.7.13

| Comma basis = [[126/125]], [[245/243]] (7-limit); <br>[[91/90]], [[126/125]], [[169/168]] (2.3.5.7.13)

| Edo join 1 = 19 | Edo join 2 = 27

| Mapping = 1; 7 9 13 10

| Generators = 9/7 | Generators tuning = 443.3 | Optimization method = CWE

| MOS scales = [[3L 2s]], [[3L 5s]], [[8L 3s]], [[8L 11s]]

| Pergen = (P8, ccP5/7)

| Odd limit 1 = 7 | Mistuning 1 = 7.5 | Complexity 1 = 19

| Odd limit 2 = 2.3.5.7.13 21 | Mistuning 2 = 11.1 | Complexity 2 = 27

}}

'''Sensi''' is a [[rank-2 temperament|rank-2]] [[regular temperament]] that is [[generator|generated]] by an extremely sharp major third of between 442 and 445{{cent}}, which is taken in the [[7-limit]] to represent a sharpened [[9/7]]. The most important equivalence in sensi (i.e. [[tempering out]] the comma [[245/243]]) is known as ''sensamagic'', by which two of these thirds stack to a major sixth which approximates [[5/3]]. Sensi then makes the additional tempering of [[126/125]], through which three of these major sixths approximate [[7/6]], two octaves up. The [[6/1|6th harmonic]] is therefore split into seven, and [[5/4]] is divided into three parts, each identified with [[15/14]]. Furthermore, since the supermajor third is tempered so sharply, it makes sense to have it represent both 9/7 and [[13/10]], which means [[91/90]] is tempered out in the 2.3.5.7.13 [[subgroup]]. There the 15/14 interval also represents [[14/13]] and [[13/12]], which results in [[169/168]] and [[196/195]] being tempered out.

~~'''Sensi'''~~ is a [[~~rank-2 temperament|rank-2~~]] ~~[[regular temperament]] in~~ the ~~2.3.5.7(.13)~~ [[~~subgroup~~]] [[~~generator|generated~~]] ~~by an extremely sharp major third of between 440~~ and ~~446{{cent}} which represents both~~ [[~~9/7~~]] ~~and~~ [[~~13/10~~]], ~~such that two of these thirds stack to a major sixth which approximates~~ [[~~5/3~~]], ~~which cannot occur in~~ [[~~12edo~~]]~~. This results in~~ [[~~91/90~~]], [[~~126/125~~]], and [[~~245/243~~]] ~~being tempered out~~, ~~making it an [[extension]] of both [[sensipent]] and [[BPS]]~~.

The structure whereby 5/3 is split into two supermajor thirds is obviously xenharmonic as this cannot occur in [[12edo]]. But particularly, as the simplest [[EDO]]s with similar structures are [[8edo]] and [[11edo]] (hence the 8-note ([[3L 5s]], checkertonic) and 11-note ([[8L 3s]], flanatonic) [[MOS scale]]s), sensi has a very xenmelodic character compared to many other ways of organizing the 7-limit (such as [[superpyth]], which is based on the familiar [[chain of fifths]], and even [[porcupine]], which is fundamentally heptatonic).

Equal temperaments that support sensi include [[19edo]] (generator 7\19; [[soft]] [[checkertonic]]), [[27edo]] (generator 10\27; [[supersoft]] checkertonic), ~~and~~ [[46edo]] (generator 17\46; {{nowrap| L/s {{=}} 7/5 }}, more optimized for sensi temperament)~~. More obscure but significantly more accurate interpretations of its generator are given by~~ [[~~sensible~~]] ~~and especially~~ [[~~sensipent~~]]~~'s extension to the 2.3.5.31 subgroup~~.

Equal temperaments that support sensi include [[19edo]] (generator 7\19; [[soft]] checkertonic), [[27edo]] (generator 10\27; [[supersoft]] checkertonic), as well as [[46edo]] (generator 17\46; {{nowrap| L/s {{=}} 7/5 }}, more optimized for sensi temperament) and [[65edo]] (generator 24\65; {{nowrap| L/s {{=}} 10/7 }}) using the 65f [[val]] with a flat 13.

See [[Sensipent family #Sensi]] for more technical data, ~~and~~ [[~~Sensi~~ extensions]] for extensions of sensi ~~that~~ include the [[11/1|11th ~~harmonic~~]].

See [[Sensipent family #Sensi]] for more technical data, [[sensi extensions]] for extensions of sensi to include the [[11/1|11th]] and [[17/1|17th]] harmonics, and [[#Related temperaments]] for alternative interpretations of similar structures to sensi.

== ~~Intervals~~ ==

== Theory ==

=== Interval chain ===

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

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<nowiki/>* In 2.3.5.7.13 CWE tuning

=== In Sensi[8] ===

=== Intervals of Sensi[8] ===

Sensi[8] is a [[mos scale]] with a [[3L 5s]] pattern ~~(or [[5L 3s]] in extreme cases where the generator is larger than 450{{c}})~~. See [[3L 5s #Modes]] ~~(resp. [[5L 3s #Modes]])~~ to see which modes have which qualities for each interval size.

Sensi[8] is a [[mos scale]] with a [[3L 5s]] pattern. See [[3L 5s #Modes]] to see which modes have which qualities for each interval size.

Sortable table of Sensi[8]'s major and minor intervals in various sensi tunings:

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|}

== Chords ==

=== Related temperaments ===

It is worth noting that sensi distinguishes itself from other structures, the [[sensamagic clan|sensamagic temperaments]], based around 245/243 (whose basic form in the 2.9/7.5/3 subgroup is known as [[sentry]]) by virtue of its minor third (6/5) being ''flattened'' from just rather than sharpened. This results in the supermajor third being sharpened even more than is typical, so much so that it is tuned [[interseptimal]]ly and may not fulfill all the functions that [[~]]9/7 is intended to have.

One way around this is to eschew the generator's interpretation as 9/7 altogether, and focus on the [[5-limit]] part of sensi, which is known as [[sensipent]] (whose comma is [[78732/78125]]). From there, an interpretation of the generator as {{nowrap|[[31/24]]~[[40/31]]}} is apparent. Beyond the 2.3.5.31 subgroup, more accurate interpretations (in comparison to sensi) of sensipent's extended harmony are given by [[sensible]] (adding primes 11, 17, and 23) and [[sendai]] (adding 23 and 29). There are also alternative mappings of 7, including [[sensei]] (+32 generators, with a tuning flat of 65edo) and [[warrior]] (−33 generators, with a tuning between 65edo and 46edo); warrior combines well with the mapping of sensible, and sensei with sendai.

==== BPS ====

: ''Main article: [[Relationship between Bohlen–Pierce and octave-ful temperaments#Relationship of rank-2 Bohlen.E2.80.93Pierce.E2.80.93Stearns temperament to octave-ful temperaments|Relationship between Bohlen–Pierce and octave-ful temperaments]].

Since the sensamagic comma, 245/243, contains no 2 in its [[monzo|factorization]], only primes 3, 5, and 7, it can be tempered out in a [[3/1|tritave (3/1)]]-repeating, [[3.5.7 subgroup]] context, where the generator (9/7) is now the tritave-reduced 7th subharmonic, two of which give the 5th harmonic. This is known as [[BPS|Bohlen–Pierce–Stearns (BPS)]] temperament, and it generates a [[4L 5s (3/1-equivalent)|4L 5s]] scale against the tritave (sometimes known as ''Lambda''). Where this temperament connects to sensi is that, at 7 generators, BPS reaches an interval that it identifies with [[125/63]], which is rather close to the octave; sensi is obtained by treating this interval as the mapping of 2/1, which provides the interesting notion of using sensi in a 3/1-periodic 3.5.7.2 setting.

== Chords and harmony ==

The fundamental otonal consonance of sensi is 4:5:6:7:9:13. However, the full chord is only available in the 19-note mos.

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|

|-

| 4\11

| [[11edo|4\11]]

|

| 436.364

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|

|-

| 7\19

| [[19edo|7\19]]

|

| 442.105

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| 443.025

|

|-

| [[65edo|24\65]]

|

| 443.077

| 65f val

|-

|

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| 2.3.5.7.13-subgroup 15- and 21-odd-limit minimax

|-

| 17\46

| [[46edo|17\46]]

|

| 443.478

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| 443.756

| 7-odd-limit minimax

|-

| [[73edo|27\73]]

|

| 443.836

|

|-

|

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|

|-

| 10\27

| [[27edo|10\27]]

|

| 444.444

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|

|-

| 3\8

| [[8edo|3\8]]

|

| 450.000

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; [[Budjarn Lambeth]]

* [https://www.youtube.com/watch?v=qc0CkUKj7t4 Music in Sensi Temperament (+ Tempered Octaves) ~~– Mar~~ 2024]

* [https://www.youtube.com/watch?v=qc0CkUKj7t4 ''Music in Sensi Temperament (+ Tempered Octaves)''] (2024)

; [[Claudi Meneghin]]

* [https://www.youtube.com/watch?v=rmgWC_jruSg ''Sensi Fugue''] (2024) – fugue for two organs, in sensi, 46edo tuning

[[Category:Sensi| ]]

@@ Line 1: / Line 1: @@
 {{interwiki
+| en = Sensi
 | de = Sensi
-| en = Sensi
 }}
-''This page is about the temperament where 3/2 is found at 7 steps. For other structures sometimes called "sensi", see [[BPS]] and [[sentry]].''
+{{Infobox regtemp
+| Title = Sensi
+| Subgroups = 2.3.5.7, 2.3.5.7.13
+| Comma basis = [[126/125]], [[245/243]] (7-limit); <br>[[91/90]], [[126/125]], [[169/168]] (2.3.5.7.13)
+| Edo join 1 = 19 | Edo join 2 = 27
+| Mapping = 1; 7 9 13 10
+| Generators = 9/7 | Generators tuning = 443.3 | Optimization method = CWE
+| MOS scales = [[3L&nbsp;2s]], [[3L&nbsp;5s]], [[8L&nbsp;3s]], [[8L&nbsp;11s]]
+| Pergen = (P8, ccP5/7)
+| Odd limit 1 = 7 | Mistuning 1 = 7.5 | Complexity 1 = 19
+| Odd limit 2 = 2.3.5.7.13 21 | Mistuning 2 = 11.1 | Complexity 2 = 27
+}}
+'''Sensi''' is a [[rank-2 temperament|rank-2]] [[regular temperament]] that is [[generator|generated]] by an extremely sharp major third of between 442 and 445{{cent}}, which is taken in the [[7-limit]] to represent a sharpened [[9/7]]. The most important equivalence in sensi (i.e. [[tempering out]] the comma [[245/243]]) is known as ''sensamagic'', by which two of these thirds stack to a major sixth which approximates [[5/3]]. Sensi then makes the additional tempering of [[126/125]], through which three of these major sixths approximate [[7/6]], two octaves up. The [[6/1|6th harmonic]] is therefore split into seven, and [[5/4]] is divided into three parts, each identified with [[15/14]]. Furthermore, since the supermajor third is tempered so sharply, it makes sense to have it represent both 9/7 and [[13/10]], which means [[91/90]] is tempered out in the 2.3.5.7.13 [[subgroup]]. There the 15/14 interval also represents [[14/13]] and [[13/12]], which results in [[169/168]] and [[196/195]] being tempered out.
-'''Sensi''' is a [[rank-2 temperament|rank-2]] [[regular temperament]] in the 2.3.5.7(.13) [[subgroup]] [[generator|generated]] by an extremely sharp major third of between 440 and 446{{cent}} which represents both [[9/7]] and [[13/10]], such that two of these thirds stack to a major sixth which approximates [[5/3]], which cannot occur in [[12edo]]. This results in [[91/90]], [[126/125]], and [[245/243]] being tempered out, making it an [[extension]] of both [[sensipent]] and [[BPS]].
+The structure whereby 5/3 is split into two supermajor thirds is obviously xenharmonic as this cannot occur in [[12edo]]. But particularly, as the simplest [[EDO]]s with similar structures are [[8edo]] and [[11edo]] (hence the 8-note ([[3L&nbsp;5s]], checkertonic) and 11-note ([[8L&nbsp;3s]], flanatonic) [[MOS scale]]s), sensi has a very xenmelodic character compared to many other ways of organizing the 7-limit (such as [[superpyth]], which is based on the familiar [[chain of fifths]], and even [[porcupine]], which is fundamentally heptatonic).
-Equal temperaments that support sensi include [[19edo]] (generator 7\19; [[soft]] [[checkertonic]]), [[27edo]] (generator 10\27; [[supersoft]] checkertonic), and [[46edo]] (generator 17\46; {{nowrap| L/s {{=}} 7/5 }}, more optimized for sensi temperament). More obscure but significantly more accurate interpretations of its generator are given by [[sensible]] and especially [[sensipent]]'s extension to the 2.3.5.31 subgroup.
+Equal temperaments that support sensi include [[19edo]] (generator 7\19; [[soft]] checkertonic), [[27edo]] (generator 10\27; [[supersoft]] checkertonic), as well as [[46edo]] (generator 17\46; {{nowrap| L/s {{=}} 7/5 }}, more optimized for sensi temperament) and [[65edo]] (generator 24\65; {{nowrap| L/s {{=}} 10/7 }}) using the 65f [[val]] with a flat 13.
-See [[Sensipent family #Sensi]] for more technical data, and [[Sensi extensions]] for extensions of sensi that include the [[11/1|11th harmonic]].
+See [[Sensipent family #Sensi]] for more technical data, [[sensi extensions]] for extensions of sensi to include the [[11/1|11th]] and [[17/1|17th]] harmonics, and [[#Related temperaments]] for alternative interpretations of similar structures to sensi.
-== Intervals ==
+== Theory ==
 === Interval chain ===
 In the following table, odd harmonics and subharmonics 1–21 are in '''bold'''.
@@ Line 107: / Line 119: @@
 <nowiki/>* In 2.3.5.7.13 CWE tuning
-=== In Sensi[8] ===
+=== Intervals of Sensi[8] ===
-Sensi[8] is a [[mos scale]] with a [[3L&nbsp;5s]] pattern (or [[5L&nbsp;3s]] in extreme cases where the generator is larger than 450{{c}}). See [[3L&nbsp;5s #Modes]] (resp. [[5L&nbsp;3s #Modes]]) to see which modes have which qualities for each interval size.
+Sensi[8] is a [[mos scale]] with a [[3L&nbsp;5s]] pattern. See [[3L&nbsp;5s #Modes]] to see which modes have which qualities for each interval size.
 Sortable table of Sensi[8]'s major and minor intervals in various sensi tunings:
@@ Line 227: / Line 239: @@
 |}
-== Chords ==
+=== Related temperaments ===
-{{Main| Chords of sensus }}
+It is worth noting that sensi distinguishes itself from other structures, the [[sensamagic clan|sensamagic temperaments]], based around 245/243 (whose basic form in the 2.9/7.5/3 subgroup is known as [[sentry]]) by virtue of its minor third (6/5) being ''flattened'' from just rather than sharpened. This results in the supermajor third being sharpened even more than is typical, so much so that it is tuned [[interseptimal]]ly and may not fulfill all the functions that [[~]]9/7 is intended to have.
+One way around this is to eschew the generator's interpretation as 9/7 altogether, and focus on the [[5-limit]] part of sensi, which is known as [[sensipent]] (whose comma is [[78732/78125]]). From there, an interpretation of the generator as {{nowrap|[[31/24]]~[[40/31]]}} is apparent. Beyond the 2.3.5.31 subgroup, more accurate interpretations (in comparison to sensi) of sensipent's extended harmony are given by [[sensible]] (adding primes 11, 17, and 23) and [[sendai]] (adding 23 and 29). There are also alternative mappings of 7, including [[sensei]] (+32 generators, with a tuning flat of 65edo) and [[warrior]] (−33 generators, with a tuning between 65edo and 46edo); warrior combines well with the mapping of sensible, and sensei with sendai.
+==== BPS ====
+: ''Main article: [[Relationship between Bohlen–Pierce and octave-ful temperaments#Relationship of rank-2 Bohlen.E2.80.93Pierce.E2.80.93Stearns temperament to octave-ful temperaments|Relationship between Bohlen–Pierce and octave-ful temperaments]].
+Since the sensamagic comma, 245/243, contains no 2 in its [[monzo|factorization]], only primes 3, 5, and 7, it can be tempered out in a [[3/1|tritave (3/1)]]-repeating, [[3.5.7 subgroup]] context, where the generator (9/7) is now the tritave-reduced 7th subharmonic, two of which give the 5th harmonic. This is known as [[BPS|Bohlen–Pierce–Stearns (BPS)]] temperament, and it generates a [[4L 5s (3/1-equivalent)|4L&nbsp;5s]] scale against the tritave (sometimes known as ''Lambda''). Where this temperament connects to sensi is that, at 7 generators, BPS reaches an interval that it identifies with [[125/63]], which is rather close to the octave; sensi is obtained by treating this interval as the mapping of 2/1, which provides the interesting notion of using sensi in a 3/1-periodic 3.5.7.2 setting.
+== Chords and harmony ==
+{{See also| Chords of sensus }}
 The fundamental otonal consonance of sensi is 4:5:6:7:9:13. However, the full chord is only available in the 19-note mos.
@@ Line 342: / Line 364: @@
 |
 |-
-| 4\11
+| [[11edo|4\11]]
 |
 | 436.364
@@ Line 362: / Line 384: @@
 |
 |-
-| 7\19
+| [[19edo|7\19]]
 |
 | 442.105
@@ Line 391: / Line 413: @@
 | 443.025
 |
+|-
+| [[65edo|24\65]]
+|
+| 443.077
+| 65f val
 |-
 |
@@ Line 397: / Line 424: @@
 | 2.3.5.7.13-subgroup 15- and 21-odd-limit minimax
 |-
-| 17\46
+| [[46edo|17\46]]
 |
 | 443.478
@@ Line 416: / Line 443: @@
 | 443.756
 | 7-odd-limit minimax
+|-
+| [[73edo|27\73]]
+|
+| 443.836
+|
 |-
 |
@@ Line 427: / Line 459: @@
 |
 |-
-| 10\27
+| [[27edo|10\27]]
 |
 | 444.444
@@ Line 452: / Line 484: @@
 |
 |-
-| 3\8
+| [[8edo|3\8]]
 |
 | 450.000
@@ Line 496: / Line 528: @@
 ; [[Budjarn Lambeth]]
-* [https://www.youtube.com/watch?v=qc0CkUKj7t4 Music in Sensi Temperament (+ Tempered Octaves) – Mar 2024]
+* [https://www.youtube.com/watch?v=qc0CkUKj7t4 ''Music in Sensi Temperament (+ Tempered Octaves)''] (2024)
+; [[Claudi Meneghin]]
+* [https://www.youtube.com/watch?v=rmgWC_jruSg ''Sensi Fugue''] (2024) – fugue for two organs, in sensi, 46edo tuning
 [[Category:Sensi| ]] <!-- Main article -->