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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]]'' 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 [[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 {{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).


Another weak extension to add prime 17, known as ''Rigil'', 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]].
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.


While strong 11-limit extensions can be proposed, tempering out [[77/75]] in the flat range and [[1375/1323]] in the sharp range, neither of these are of particular accuracy; more accurate extensions would be of considerably higher complexity.
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.  


Sharp tunings generally possess a more convenient 13th harmonic than 11th, by tempering out [[637/625]] and identifying (25/21)<sup>2</sup> with [[13/9]], which is optimal near the 30edt tuning. It is then very easy to insert in 19 by tempering [[247/245]], and identifying [[13/9]] with [[27/19]], therefore placing the 19th harmonic 10 generators down; this extension is optimal near the 56edt tuning.
=== 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'''.
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)
|-
|-
! &#35; !! Cents&#42; !! Approximate Ratios
! rowspan="2" | # !! rowspan="2" | Cents* !! colspan="2" | Approximate ratios
|-
|-
| -4 || 139.7 || 27/25, 49/45
! 3.5.7 subgroup !! Add-19 extension
|-
|-
| -3 || 580.3 || 7/5, 243/175
| −4 || 139.7 || 27/25, 49/45 || 21/19, 133/125
|-
|-
| -2 || 1020.8 || 9/5, '''49/27'''
| −3 || 580.3 || 7/5, 243/175 || 27/19, 171/125
|-
|-
| -1 || 1461.4 || '''7/3''', 81/35
| −2 || 1020.8 || 9/5, '''49/27''' || 35/19, 133/75, 243/133
|-
|-
| 0 || 0.0 || '''1/1'''
| −1 || 1461.4 || '''7/3''', 81/35 || 45/19, 57/25
|-
|-
| 1 || 440.6 || 9/7, '''35/27'''
| 0 || 0.0 || '''1/1''', 245/243 || 135/133, 175/171, 375/361
|-
|-
| 2 || 881.1 || '''5/3''', 81/49
| 1 || 440.6 || 9/7, '''35/27''' || 19/15, 25/19
|-
|-
| 3 || 1321.7 || 15/7, '''175/81'''
| 2 || 881.1 || '''5/3''', 81/49 || 57/35, '''133/81''', 225/133
|-
|-
| 4 || 1762.2 || '''25/9''', 135/49
| 3 || 1321.7 || 15/7, '''175/81''' || '''19/9''', 125/57
|-
|-
| 5 || 300.8 || 25/21, 405/343
| 4 || 1762.2 || '''25/9''', 135/49 || 19/7, 375/133
|-
|-
| 6 || 741.4 || 75/49, '''125/81'''
| 5 || 300.8 || 25/21, 405/343 || 57/49, '''95/81'''
|-
|-
| 7 || 1181.9 || 125/63, 675/343
| 6 || 741.4 || 75/49, '''125/81''' || 95/63, 361/243
|-
|-
| 8 || 1622.5 || 125/49, 625/243
| 7 || 1181.9 || 125/63, 675/343 || 95/49, 361/189
|-
|-
| 9 || 161.1 || 375/343, 625/567
| 8 || 1622.5 || 125/49, 625/243 || 361/147, 475/189
{{table notes|cols=3
|-
| In 3.5.7-targeted [[DKW theory|DKW]] tuning
| 9 || 161.1 || 375/343, 625/567 || 361/343
}}
|}
|}
</div>
<nowiki/>* In 3.5.7-targeted [[DKW theory|DKW]] tuning


<div style="display: inline-grid; margin-right: 25px;">
== Tunings ==
{| class="wikitable center-1 right-2 right-4"
=== Norm-based tunings ===
|+ style="font-size: 105%;" | 3.5.7.13.19 extension
{| class="wikitable mw-collapsible mw-collapsed"
|-
|+ style="font-size: 105%; white-space: nowrap;" | 3.5.7-subgroup norm-based tunings
! &#35; !! Cents&#42; !! Approximate Ratios
|-
| -4 || 131.4 || 27/25, 49/45, 125/117
|-
| -3 || 574.0 || 7/5, 243/175, 125/91
|-
| -2 || 1016.7 || 9/5, '''49/27'''
|-
| -1 || 1459.3 || '''7/3''', 81/35
|-
| 0 || 0.0 || '''1/1'''
|-
| 1 || 442.6 || 9/7, '''35/27'''
|-
| 2 || 885.3 || '''5/3''', 81/49
|-
| 3 || 1327.9 || 15/7, '''175/81''', 273/125
|-
| 4 || 1770.5 || '''25/9''', 135/49, 351/125
|-
|-
| 5 || 311.2 || 91/75, 25/21, 343/285, 405/343
! rowspan="2" |  
! colspan="3" | Euclidean
|-
|-
| 6 || 753.8 || 39/25, 75/49, 147/95, '''125/81'''
! Constrained
! Constrained & skewed
! Destretched
|-
|-
| 7 || 1196.4 || 91/45, 189/95, 125/63
! Tenney
|-
| CTE: ~9/7 = 441.1431{{c}}
| 8 || 1639.1 || 13/5, 637/243, 243/95, 125/49
| CWE: ~9/7 = 440.6646{{c}}
|-
| POTE: ~9/7 = 440.4881{{c}}
| 9 || 179.7 || 39/35, '''91/81''', 21/19, 375/343
|-
| 10 || 622.4 || '''13/9''', 27/19, 625/441
|-
| 11 || 1065.0 || 13/7, 455/243, 243/133, 35/19
|-
| 12 || 1507.6 || '''65/27''', 117/49, 45/19
|-
| 13 || 48.3 || 65/63, 135/133, 175/171
{{table notes|cols=3
| In 3.5.7.13.19-subgroup [[CWE]] tuning
}}
|}
|}
</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 116: 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 129: 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 146: Line 148:
|  
|  
| 438.913
| 438.913
| Equal-tempered [[Bohlen-Pierce]]
| Equal-tempered [[Bohlen–Pierce]]
|-
|-
|  
|  
| [[7/5]]
| [[15/7]]
| 439.814
| 439.814
| 1/3-comma
| 1/3-comma
|-
| [[108edt|25\108]]
|
| 440.267
|
|-
|-
|  
|  
Line 176: Line 173:
| [[25/21]]
| [[25/21]]
| 440.760
| 440.760
| 2/5-comma
| 2/5-comma; [[CEE]] tuning
|-
|-
| [[69edt|16\69]]
| [[69edt|16\69]]
|  
|  
| 441.033
| 441.033
|
|-
|
| 19/9
| 441.226
|  
|  
|-
|-
Line 207: 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
|
|-
|
| 13/9
| 444.053
|
|-
|
| 175/81
| 444.544
| 2/3-comma
|-
| [[77edt|18\77]]
|
| 444.613
|  
|  
|-
|-
Line 243: 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]]

Latest revision as of 10:04, 9 February 2026

BPS
Subgroups 3.5.7
Comma basis 245/243
Reduced mapping ⟨1; 2 -1]
ET join b13 & b17
Generators (CWE) ~9/7 = 440.7 ¢
MOS scales 4L 1s <3/1>, 4L 5s <3/1>
Ploidacot monogem
Minimax error 7-throdd-limit: 4.73 ¢;
3.5.7 49-throdd-limit: 9.46 ¢
Target scale size 7-throdd-limit: 4 notes;
3.5.7 49-throdd-limit: 9 notes
English Wikipedia has an article on:

BPS (for Bohlen–Pierce–Stearns) 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⟩, known as the Bohlen–Pierce Lambda scale. The "canonical" tuning for the generator is 3\13edt, representing the equal-tempered Bohlen–Pierce scale, but a range of other tunings are valid, including 4\17edt, 7\30edt, and 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 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 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

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⟩ and 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⟩ enneatonic and 9L 8s⟨3/1⟩ 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 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)2 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)2 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)3, 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.

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

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.

Basic BPS (extension-agnostic)
# Cents* 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
4 1762.2 25/9, 135/49 19/7, 375/133
5 300.8 25/21, 405/343 57/49, 95/81
6 741.4 75/49, 125/81 95/63, 361/243
7 1181.9 125/63, 675/343 95/49, 361/189
8 1622.5 125/49, 625/243 361/147, 475/189
9 161.1 375/343, 625/567 361/343

* In 3.5.7-targeted DKW tuning

Tunings

Norm-based tunings

3.5.7-subgroup norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~9/7 = 441.1431 ¢ CWE: ~9/7 = 440.6646 ¢ POTE: ~9/7 = 440.4881 ¢

Other tunings

  • DKW (3.5.7): ~3 = 1901.955 ¢, ~9/7 = 440.554 ¢

Tuning spectrum

Edt
generator 		Unchanged interval
(eigenmonzo)* 		Generator (¢) Comments
5\22 432.263
8\35 434.733
9/7 435.084 Untempered
11\48 435.865
14\61 436.514
438.038 DR 5:7:9, close to 5/24-comma
135/49 438.632 1/4-comma
3\13 438.913 Equal-tempered Bohlen–Pierce
15/7 439.814 1/3-comma
440.340 DR 3:5:7, close to 10/27-comma
22\95 440.453
19\82 440.697
25/21 440.760 2/5-comma; CEE tuning
16\69 441.033
13\56 441.525
5/3 442.179 1/2-comma
10\43 442.315
17\73 442.921
2/1 443.136 Sensi mapping of 2/1 to ~125/63
7\30 443.790
11\47 445.138
4\17 447.519
35/27 449.275 Full comma

* Besides the tritave

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