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''' | {{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 | 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 | 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 | 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 '' | 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 === | === 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). | 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. | 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, | 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 === | === Prime 2 === | ||
''Main article: [[Relationship between | : ''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 | 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 | 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'''. | ||
{| class="wikitable center-1 right-2" | {| class="wikitable center-1 right-2" | ||
|+ style="font-size: 105%;" | Basic BPS (extension-agnostic) | |+ style="font-size: 105%;" | Basic BPS (extension-agnostic) | ||
|- | |- | ||
! rowspan="2" | | ! 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 | ||
|- | |- | ||
| | | −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 | | 0 || 0.0 || '''1/1''', 245/243 || 135/133, 175/171, 375/361 | ||
| Line 64: | Line 78: | ||
|- | |- | ||
| 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 | ||
== Tunings == | |||
=== Norm-based tunings === | |||
{| class="wikitable mw-collapsible mw-collapsed" | |||
|+ style="font-size: 105%; white-space: nowrap;" | 3.5.7-subgroup norm-based tunings | |||
|- | |||
! rowspan="2" | | |||
! colspan="3" | Euclidean | |||
|- | |||
! Constrained | |||
! Constrained & skewed | |||
! Destretched | |||
|- | |||
! Tenney | |||
| CTE: ~9/7 = 441.1431{{c}} | |||
| CWE: ~9/7 = 440.6646{{c}} | |||
| POTE: ~9/7 = 440.4881{{c}} | |||
|} | |||
== 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> | |- | ||
! [[Eigenmonzo| | ! Edt<br>generator | ||
! Generator | ! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]* | ||
! Generator (¢) | |||
! Comments | ! Comments | ||
|- | |- | ||
| Line 82: | Line 115: | ||
| | | | ||
|- | |- | ||
| [[35edt|8\35]] | |||
| | |||
| 434.733 | |||
| | | | ||
| [[7 | |- | ||
| | |||
| [[9/7]] | |||
| 435.084 | | 435.084 | ||
| | | Untempered | ||
|- | |||
| [[48edt|11\48]] | |||
| | |||
| 435.865 | |||
| | |||
|- | |- | ||
| [[61edt|14\61]] | | [[61edt|14\61]] | ||
| Line 93: | Line 136: | ||
|- | |- | ||
| | | | ||
| | | | ||
| 438.038 | |||
| [[Delta-rational chord|DR]] 5:7:9, close to 5/24-comma | |||
|- | |- | ||
| | | | ||
| [[49 | | [[135/49]] | ||
| 438.632 | | 438.632 | ||
| 1/4-comma | | 1/4-comma | ||
| Line 110: | Line 148: | ||
| | | | ||
| 438.913 | | 438.913 | ||
| Equal-tempered [[ | | Equal-tempered [[Bohlen–Pierce]] | ||
|- | |- | ||
| | | | ||
| [[7 | | [[15/7]] | ||
| 439.814 | | 439.814 | ||
| 1/3-comma | | 1/3-comma | ||
|- | |- | ||
| | | | ||
| Line 140: | 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 166: | 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 | ||
| | | | ||
|- | |- | ||
| Line 197: | Line 225: | ||
| Full comma | | Full comma | ||
|} | |} | ||
<nowiki/>* Besides the [[3/1|tritave]] | |||
[[Category:BPS| ]] <!-- main article --> | |||
[[Category:Rank-2 temperaments]] | |||
[[Category:Non-octave temperaments]] | |||
[[Category: | [[Category:Sensamagic clan]] | ||
[[Category: | [[Category:Bohlen–Pierce]] | ||
[[Category: | |||
Latest revision as of 10:04, 9 February 2026
| BPS |
3.5.7 49-throdd-limit: 9.46 ¢
3.5.7 49-throdd-limit: 9 notes
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
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.
| # | 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
| 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