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m FloraC moved page Bohlen–Pierce–Stearns to BPS: Official name is "BPS"
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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 {{sl|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.
'''BPS''' (for ''Bohlen–Pierce–Stearns'') is a [[non-octave]] [[regular temperament|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}} against the tritave, known as the Bohlen–Pierce ''Lambda'' scale. The "canonical" tuning for the generator is [[13edt|3\13edt]], representing the equal-tempered [[Bohlen–Pierce]] scale, but a range of other tunings are valid, including [[17edt|4\17edt]], [[30edt|7\30edt]], and [[43edt|10\43edt]].


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).
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).


== Extensions ==
== Extensions ==
Several extensions of this temperament are possible to incorporate additional harmonics.
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 {{sl|5L 2s}} and {{sl|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}} and {{sl|5L 7s}} 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 {{sl|8L 1s}} enneatonic and {{sl|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}} enneatonic and {{sl|9L 8s}} chromatic mos scales. Simple tunings include [[17edt]] and [[26edt]].


=== Strong extensions ===
=== Strong extensions ===
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=== Prime 2 ===
=== Prime 2 ===
''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]].
: ''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]].


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]].
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 ==
== Interval chains ==
These interval chains cover strong extensions of BPS. For Mintra, see [[Mintaka#Mintra]].
These interval chains cover strong extensions of BPS. For mintra, see [[Mintaka #Mintra]].


In the below, tritave-reduced harmonics below 243 are indicated in '''bold'''.
In the below, tritave-reduced harmonics below 243 are indicated in '''bold'''.
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|+ style="font-size: 105%;" | Basic BPS (extension-agnostic)
|+ style="font-size: 105%;" | Basic BPS (extension-agnostic)
|-
|-
! rowspan="2" | # !! rowspan="2" | Cents* !! colspan="2" | Approximate Ratios
! rowspan="2" | # !! rowspan="2" | Cents* !! colspan="2" | Approximate ratios
|-
|-
! 3.5.7 subgroup !! Add-19 extension
! 3.5.7 subgroup !! Add-19 extension
|-
|-
| −4 || 139.7 || 27/25, 49/45 || 21/19, 133/125
| −4 || 139.7 || 27/25, 49/45 || 21/19, 133/125
|-
|-
| −3 || 580.3 || 7/5, 243/175 || 27/19, 171/125
| −3 || 580.3 || 7/5, 243/175 || 27/19, 171/125
|-
|-
| −2 || 1020.8 || 9/5, '''49/27''' || 35/19, 133/75, 243/133
| −2 || 1020.8 || 9/5, '''49/27''' || 35/19, 133/75, 243/133
|-
|-
| −1 || 1461.4 || '''7/3''', 81/35 || 45/19, 57/25
| −1 || 1461.4 || '''7/3''', 81/35 || 45/19, 57/25
|-
|-
| 0 || 0.0 || '''1/1''', 245/243 || 135/133, 175/171, 375/361
| 0 || 0.0 || '''1/1''', 245/243 || 135/133, 175/171, 375/361
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| 9 || 161.1 || 375/343, 625/567 || 361/343
| 9 || 161.1 || 375/343, 625/567 || 361/343
|}
|}
<nowiki />* In 3.5.7-targeted [[DKW theory|DKW]] tuning
<nowiki/>* In 3.5.7-targeted [[DKW theory|DKW]] tuning


== Tuning spectrum ==
== Tuning spectrum ==
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
! Edt<br />Generator
! Edt<br>generator
! [[Eigenmonzo|Eigenmonzo<br />(Unchanged-interval)]]*
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
! Generator<br />(¢)
! Generator (¢)
! Comments
! Comments
|-
|-
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| Full comma
| Full comma
|}
|}
<nowiki />* Besides the [[3/1|tritave]]
<nowiki/>* Besides the [[3/1|tritave]]


=== Other tunings ===
=== Other tunings ===
* [[DKW theory|DKW]] (3.5.7): ~3 = 1\1, ~9/7 = 440.554
* [[DKW theory|DKW]] (3.5.7): ~3 = 1901.955, ~9/7 = 440.554


[[Category:Bohlen-Pierce]]
[[Category:Bohlen-Pierce]]
[[Category:Temperaments]]
[[Category:Temperaments]]
[[Category:Tritave-equivalent temperaments]]
[[Category:Tritave-equivalent temperaments]]
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