Sensi: Difference between revisions

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'''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.

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).

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), 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.

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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 [[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.

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.

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 ==

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 '''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.
-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).
+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), 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.
@@ Line 242: / Line 242: @@
 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 [[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.
+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.
+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 ==