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'''Bohlen-Pierce-Stearns''' (BPS) is a [[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, which 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.
{{Infobox regtemp
| Title = BPS
| Subgroups = 3.5.7
| Comma basis = [[245/243]]
| Edo join 1 = b13 | Edo join 2 = b17
| Mapping = 1; 2 -1
| Generators = 9/7
| Generators tuning = 440.7
| Optimization method = CWE
| MOS scales = [[4L 1s (3/1-equivalent)|4L 1s <3/1>]], [[4L 5s (3/1-equivalent)|4L 5s <3/1>]]
| Color name =
| Odd limit 1 = 7 | Mistuning 1 = 4.73 | Complexity 1 = 4
| Odd limit 2 = 3.5.7 49 | Mistuning 2 = 9.46 | Complexity 2 = 9
}}
{{Wikipedia|Bohlen–Pierce scale}}
'''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 {{mos scalesig|4L 5s<3/1>|link=1}}, 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 3:5:7:9 tetrad to serve as BPS' primary consonance, similar to how the 4:5:6 triad serves as meantone's primary consonance.


For technical data, see ''[[Sensamagic clan#BPS]]''.
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 {{mos scalesig|5L 2s<3/1>|link=1}} and {{mos scalesig|5L 7s<3/1>|link=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 {{mos scalesig|8L 1s<3/1>|link=1}} enneatonic and {{mos scalesig|9L 8s<3/1>|link=1}} chromatic mos scales. Simple tunings include [[17edt]] and [[26edt]].
=== Strong extensions ===
While strong 11-limit extensions can be proposed, tempering out [[77/75]] in the sharper range (i.e. sharp of [[13edt|3\13edt]]) and [[1375/1323]] in the flatter range, neither of these are of particular accuracy; more accurate extensions would be of considerably higher complexity. However, one could argue for the canonicity of the latter extension by being the no-twos retraction of 11-limit [[hedgehog]] temperament (which, as a member of the [[porcupine family]], makes more sense to consider with prime 11 in mind than without it).
In the 13-limit, sharp tunings can generally map the 13th harmonic by tempering out [[637/625]] and identifying ([[25/21]])<sup>2</sup> with [[13/9]], which is optimal near the 30edt tuning. For flat tunings, it is more accurate to temper out [[65/63]] instead.
One harmonic that can be placed on the generator chain with some accuracy, compared to other primes, is the 19th. Sharp of 13edt, it is best to temper out [[11907/11875]] and equate (25/21)<sup>2</sup> to [[27/19]], thereby having the 19th harmonic 10 generators down. But on the flat side of the spectrum, it is less complex and more accurate flat of 13edt to temper out [[6561/6517]], or equivalently [[135/133]], so that [[19/9]] is equated to (9/7)<sup>3</sup>, or otherwise [[15/7]], though this mapping of 19 is exact ''flat'' of 22edt.
=== 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]].
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]].
In the below, tritave-reduced harmonics below 243 are indicated in '''bold'''.


<div><div style="display: inline-grid; margin-right: 25px;">
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|+ style="font-size: 105%;" | Basic 3.5.7 temperament
|+ style="font-size: 105%;" | Basic BPS (extension-agnostic)
|-
! rowspan="2" | # !! rowspan="2" | Cents* !! colspan="2" | Approximate ratios
|-
! 3.5.7 subgroup !! Add-19 extension
|-
| −4 || 139.7 || 27/25, 49/45 || 21/19, 133/125
|-
| −3 || 580.3 || 7/5, 243/175 || 27/19, 171/125
|-
| −2 || 1020.8 || 9/5, '''49/27''' || 35/19, 133/75, 243/133
|-
| −1 || 1461.4 || '''7/3''', 81/35 || 45/19, 57/25
|-
| 0 || 0.0 || '''1/1''', 245/243 || 135/133, 175/171, 375/361
|-
| 1 || 440.6 || 9/7, '''35/27''' || 19/15, 25/19
|-
| 2 || 881.1 || '''5/3''', 81/49 || 57/35, '''133/81''', 225/133
|-
| 3 || 1321.7 || 15/7, '''175/81''' || '''19/9''', 125/57
|-
|-
! &#35; !! Cents&#42; !! Approximate Ratios
| 4 || 1762.2 || '''25/9''', 135/49 || 19/7, 375/133
|-
|-
| 0 || 0.0 || '''1/1'''
| 5 || 300.8 || 25/21, 405/343 || 57/49, '''95/81'''
|-
|-
| 1 || 440.6 || 9/7, 35/27
| 6 || 741.4 || 75/49, '''125/81''' || 95/63, 361/243
|-
|-
| 2 || 881.1 || 5/3, 81/49
| 7 || 1181.9 || 125/63, 675/343 || 95/49, 361/189
|-
|-
| 3 || 1321.7 || 15/7, 175/81
| 8 || 1622.5 || 125/49, 625/243 || 361/147, 475/189
|-
| 9 || 161.1 || 375/343, 625/567 || 361/343
|}
<nowiki/>* In 3.5.7-targeted [[DKW theory|DKW]] tuning
 
== Tunings ==
=== Norm-based tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 3.5.7-subgroup norm-based tunings
|-
|-
| 4 || 1762.2 || 25/9, 135/49
! rowspan="2" |  
! colspan="3" | Euclidean
|-
|-
| 5 || 300.8 || 25/21, 405/343, 875/729
! Constrained
! Constrained & skewed
! Destretched
|-
|-
| 6 || 741.4 || 75/49, 125/81
! Tenney
{{table notes|cols=3
| CTE: ~9/7 = 441.1431{{c}}
| In 3.5.7-subgroup [[DKW theory|DKW]] tuning
| CWE: ~9/7 = 440.6646{{c}}
}}
| POTE: ~9/7 = 440.4881{{c}}
|}
|}
</div>


== Tuning spectrum ==
=== Other tunings ===
* [[DKW theory|DKW]] (3.5.7): ~3 = 1901.955{{c}}, ~9/7 = 440.554{{c}}
 
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
! Edt<br>Generator
|-
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
! Edt<br>generator
! Generator<br>(¢)
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Comments
! Comments
|-
|-
Line 45: Line 113:
|  
|  
| 432.263
| 432.263
|
|-
| [[35edt|8\35]]
|
| 434.733
|  
|  
|-
|-
|  
|  
| 7/3
| [[9/7]]
| 435.084
| 435.084
| 0-comma
| Untempered
|-
| [[48edt|11\48]]
|
| 435.865
|
|-
|-
| [[61edt|14\61]]
| [[61edt|14\61]]
Line 58: Line 136:
|-
|-
|  
|  
| 3645/2401
| 437.449
| 1/6-comma
|-
| [[100edt|23\100]]
|
| 437.450
|  
|  
| 438.038
| [[Delta-rational chord|DR]] 5:7:9, close to 5/24-comma
|-
|-
|  
|  
| [[49/45]]
| [[135/49]]
| 438.632
| 438.632
| 1/4-comma
| 1/4-comma
Line 75: Line 148:
|  
|  
| 438.913
| 438.913
|  
| Equal-tempered [[Bohlen–Pierce]]
|-
|-
|  
|  
| [[7/5]]
| [[15/7]]
| 439.814
| 439.814
| 1/3-comma
| 1/3-comma
|-
| [[108edt|25\108]]
|
| 440.267
|
|-
|-
|  
|  
Line 105: Line 173:
| [[25/21]]
| [[25/21]]
| 440.760
| 440.760
| 2/5-comma
| 2/5-comma; [[CEE]] tuning
|-
|-
| [[69edt|16\69]]
| [[69edt|16\69]]
Line 131: Line 199:
| 442.921
| 442.921
|  
|  
|-
|
| [[2/1]]
| 443.136
| [[Sensi]] mapping of 2/1 to ~125/63
|-
|-
| [[30edt|7\30]]
| [[30edt|7\30]]
|  
|  
| 443.790
| 443.790
|
|-
|
| 175/81
| 444.544
| 2/3-comma
|-
| [[77edt|18\77]]
|
| 444.613
|  
|  
|-
|-
Line 162: Line 225:
| Full comma
| Full comma
|}
|}
<nowiki/>* Besides the [[3/1|tritave]]


=== Other tunings ===
[[Category:BPS| ]] <!-- main article -->
* [[DKW theory|DKW]] (3.5.7): ~3 = 1\1, ~9/7 = 440.554
[[Category:Rank-2 temperaments]]
 
[[Category:Non-octave temperaments]]
[[Category:Temperaments]]
[[Category:Sensamagic clan]]
[[Category:Tritave-equivalent temperaments]]
[[Category:Bohlen–Pierce]]
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