BPS: Difference between revisions

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'''~~Bohlen–Pierce–Stearns~~''' (BPS) is a [[non-octave]] [[temperament]] in the 3.5.7 [[subgroup]] generated by a sharp [[~]][[9/7]] (or equivalently a flat ~[[7/3]]), [[tempering out]] the sensamagic comma ([[245/243]]) so that a stack of two generators represents [[5/3]] in addition to 81/49. This generates a [[MOS scale]] of [[4L 5s (3/1~~-equivalent)|4L 5s]]~~ against the tritave, known as the ~~Bohlen–Pierce~~ ''Lambda'' scale. The "canonical" tuning for the generator is [[13edt|3\13]]edt, representing the equal-tempered [[~~Bohlen–Pierce~~]] scale, but a range of other tunings are valid, including [[17edt|4\17]]edt, [[30edt|7\30]]edt, and [[43edt|10\43]]edt.

'''Bohlen–Pierce–Stearns''' (BPS) is a [[non-octave]] [[temperament]] in the 3.5.7 [[subgroup]] generated by a sharp [[~]][[9/7]] (or equivalently a flat ~[[7/3]]), [[tempering out]] the sensamagic comma ([[245/243]]) so that a stack of two generators represents [[5/3]] in addition to 81/49. This generates a [[MOS scale]] of {{sl|4L 5s|3/1}} against the tritave, known as the Bohlen–Pierce ''Lambda'' scale. The "canonical" tuning for the generator is [[13edt|3\13]]edt, representing the equal-tempered [[Bohlen–Pierce]] scale, but a range of other tunings are valid, including [[17edt|4\17]]edt, [[30edt|7\30]]edt, and [[43edt|10\43]]edt.

As the generator of the ~~Bohlen–Pierce~~ scale, and the simplest decently accurate temperament of the 3.5.7 subgroup, this temperament fulfills a niche similar to [[meantone]] of the 2.3.5 subgroup, allowing for the tetrad 3:5:7:9 to serve as the theory's primary consonant tetrad.

As the generator of the Bohlen–Pierce scale, and the simplest decently accurate temperament of the 3.5.7 subgroup, this temperament fulfills a niche similar to [[meantone]] of the 2.3.5 subgroup, allowing for the tetrad 3:5:7:9 to serve as the theory's primary consonant tetrad.

For technical data, see ''[[Sensamagic clan#BPS]]'' or ''[[No-twos subgroup temperaments#BPS]]'' (currently, extensions with 2 are stored on the former page and no-twos extensions are stored on the latter).

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Several extensions of this temperament are possible to incorporate additional harmonics.

In the [[11-limit]], [[1331/1323]] is the most convenient comma that can be tempered out, which produces ''Mintra'' temperament; this splits the 9/7 generator (plus a tritave) in three and therefore functions instead as a weak extension of BPS, and a strong add-5 extension of [[Mintaka]], which produces [[5L 2s (3/1~~-equivalent)~~|~~5L 2s]] and [[~~5L 7s (3/1~~-equivalent)|5L 7s]]~~ MOS scales (functioning as a macro-[[superpyth]]). Simple tunings include [[17edt]] and [[39edt]].

In the [[11-limit]], [[1331/1323]] is the most convenient comma that can be tempered out, which produces ''Mintra'' temperament; this splits the 9/7 generator (plus a tritave) in three and therefore functions instead as a weak extension of BPS, and a strong add-5 extension of [[Mintaka]], which produces {{sl|5L 2s|3/1}} and {{sl|5L 7s|3/1}} MOS scales (functioning as a macro-[[superpyth]]). Simple tunings include [[17edt]] and [[39edt]].

Another weak extension to add prime 17, known as ''[[Dubhe]]'', splits the 9/7 BPS generator in half, by tempering out [[2025/2023]] and equating two of [[17/15]] to 9/7. This produces [[8L 1s (3/1~~-equivalent)|8L 1s]]~~ enneatonic and [[9L 8s (3/1~~-equivalent)|9L 8s]]~~ chromatic MOS scales. Simple tunings include [[17edt]] and [[26edt]].

Another weak extension to add prime 17, known as ''[[Dubhe]]'', splits the 9/7 BPS generator in half, by tempering out [[2025/2023]] and equating two of [[17/15]] to 9/7. This produces {{sl|8L 1s|3/1}} enneatonic and {{sl|9L 8s|3/1}} chromatic MOS scales. Simple tunings include [[17edt]] and [[26edt]].

=== Strong extensions ===

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

It is also possible to incorporate octaves into ~~the~~ BPS ~~temperament~~. The logical choices for a mapping of 2 are 7 generators up (equating [[2/1]] to [[125/63]]), which produces [[sensi]] ~~temperament~~, and 6 generators down (equating 2/1 to [[49/25]]), which produces [[hedgehog]] ~~temperament~~.

It is also possible to incorporate octaves into BPS. The logical choices for a mapping of 2 are 7 generators up (equating [[2/1]] to [[125/63]]), which produces [[sensi]], and 6 generators down (equating 2/1 to [[49/25]]), which produces [[hedgehog]].

== Interval chains ==

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-'''Bohlen&ndash;Pierce&ndash;Stearns''' (BPS) is a [[non-octave]] [[temperament]] in the 3.5.7 [[subgroup]] generated by a sharp [[~]][[9/7]] (or equivalently a flat ~[[7/3]]), [[tempering out]] the sensamagic comma ([[245/243]]) so that a stack of two generators represents [[5/3]] in addition to 81/49. This generates a [[MOS scale]] of [[4L 5s (3/1-equivalent)|4L&nbsp;5s]] against the tritave, known as the Bohlen&ndash;Pierce ''Lambda'' scale. The "canonical" tuning for the generator is [[13edt|3\13]]edt, representing the equal-tempered [[Bohlen&ndash;Pierce]] scale, but a range of other tunings are valid, including [[17edt|4\17]]edt, [[30edt|7\30]]edt, and [[43edt|10\43]]edt.
+'''Bohlen–Pierce–Stearns''' (BPS) is a [[non-octave]] [[temperament]] in the 3.5.7 [[subgroup]] generated by a sharp [[~]][[9/7]] (or equivalently a flat ~[[7/3]]), [[tempering out]] the sensamagic comma ([[245/243]]) so that a stack of two generators represents [[5/3]] in addition to 81/49. This generates a [[MOS scale]] of {{sl|4L 5s|3/1}} against the tritave, known as the Bohlen–Pierce ''Lambda'' scale. The "canonical" tuning for the generator is [[13edt|3\13]]edt, representing the equal-tempered [[Bohlen–Pierce]] scale, but a range of other tunings are valid, including [[17edt|4\17]]edt, [[30edt|7\30]]edt, and [[43edt|10\43]]edt.
-As the generator of the Bohlen&ndash;Pierce scale, and the simplest decently accurate temperament of the 3.5.7 subgroup, this temperament fulfills a niche similar to [[meantone]] of the 2.3.5 subgroup, allowing for the tetrad 3:5:7:9 to serve as the theory's primary consonant tetrad.
+As the generator of the Bohlen–Pierce scale, and the simplest decently accurate temperament of the 3.5.7 subgroup, this temperament fulfills a niche similar to [[meantone]] of the 2.3.5 subgroup, allowing for the tetrad 3:5:7:9 to serve as the theory's primary consonant tetrad.
 For technical data, see ''[[Sensamagic clan#BPS]]'' or ''[[No-twos subgroup temperaments#BPS]]'' (currently, extensions with 2 are stored on the former page and no-twos extensions are stored on the latter).
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 Several extensions of this temperament are possible to incorporate additional harmonics.
-In the [[11-limit]], [[1331/1323]] is the most convenient comma that can be tempered out, which produces ''Mintra'' temperament; this splits the 9/7 generator (plus a tritave) in three and therefore functions instead as a weak extension of BPS, and a strong add-5 extension of [[Mintaka]], which produces [[5L 2s (3/1-equivalent)|5L&nbsp;2s]] and [[5L 7s (3/1-equivalent)|5L&nbsp;7s]] MOS scales (functioning as a macro-[[superpyth]]). Simple tunings include [[17edt]] and [[39edt]].
+In the [[11-limit]], [[1331/1323]] is the most convenient comma that can be tempered out, which produces ''Mintra'' temperament; this splits the 9/7 generator (plus a tritave) in three and therefore functions instead as a weak extension of BPS, and a strong add-5 extension of [[Mintaka]], which produces {{sl|5L 2s|3/1}} and {{sl|5L 7s|3/1}} MOS scales (functioning as a macro-[[superpyth]]). Simple tunings include [[17edt]] and [[39edt]].
-Another weak extension to add prime 17, known as ''[[Dubhe]]'', splits the 9/7 BPS generator in half, by tempering out [[2025/2023]] and equating two of [[17/15]] to 9/7. This produces [[8L 1s (3/1-equivalent)|8L&nbsp;1s]] enneatonic and [[9L 8s (3/1-equivalent)|9L&nbsp;8s]] chromatic MOS scales. Simple tunings include [[17edt]] and [[26edt]].
+Another weak extension to add prime 17, known as ''[[Dubhe]]'', splits the 9/7 BPS generator in half, by tempering out [[2025/2023]] and equating two of [[17/15]] to 9/7. This produces {{sl|8L 1s|3/1}} enneatonic and {{sl|9L 8s|3/1}} chromatic MOS scales. Simple tunings include [[17edt]] and [[26edt]].
 === Strong extensions ===
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 ''Main article: [[Relationship between Bohlen-Pierce and octave-ful temperaments#Relationship of rank-2 Bohlen-Pierce-Stearns temperament to octave-ful temperaments|Relationship between Bohlen-Pierce and octave-ful temperaments]].
-It is also possible to incorporate octaves into the BPS temperament. The logical choices for a mapping of 2 are 7 generators up (equating [[2/1]] to [[125/63]]), which produces [[sensi]] temperament, and 6 generators down (equating 2/1 to [[49/25]]), which produces [[hedgehog]] temperament.
+It is also possible to incorporate octaves into BPS. The logical choices for a mapping of 2 are 7 generators up (equating [[2/1]] to [[125/63]]), which produces [[sensi]], and 6 generators down (equating 2/1 to [[49/25]]), which produces [[hedgehog]].
 == Interval chains ==