Xenharmonic Wiki

BPS: Difference between revisions

← Older edit
VisualWikitext
Lériendil (talk | contribs)
8,033 edits
m Cleanup on infobox
 
(38 intermediate revisions by 4 users not shown)
Line 1: Line 1:
'''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 [[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 {{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, as by tempering out [[6561/6517]], or equivalently [[135/133]], [[19/9]] is equated to (9/7)<sup>3</sup>, or otherwise [[15/7]]. However, this mapping of 19 requires a generator close to just (in fact, it is exact ''flat'' of 22edt) and is therefore feasible mainly on the flat side of the spectrum.
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 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]].
: ''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 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 ==
== 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'''.


<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 BPS (extension-agnostic)
|+ style="font-size: 105%;" | Basic BPS (extension-agnostic)
|-
|-
! rowspan="2" | &#35; !! rowspan="2" | Cents&#42; !! 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
Line 64: Line 78:
|-
|-
| 9 || 161.1 || 375/343, 625/567 || 361/343
| 9 || 161.1 || 375/343, 625/567 || 361/343
{{table notes|cols=4
| In 3.5.7-targeted [[DKW theory|DKW]] tuning
}}
|}
|}
</div>
<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>Generator
|-
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]]
! Edt<br>generator
! Generator<br>(¢)
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]*
! Generator (¢)
! Comments
! Comments
|-
|-
Line 82: Line 115:
|  
|  
|-
|-
| [[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 93: 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 110: 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 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
|
|-
|
| 175/81
| 444.544
| 2/3-comma
|-
| [[77edt|18\77]]
|
| 444.613
|  
|  
|-
|-
Line 197: 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:Bohlen-Pierce]]
[[Category:Sensamagic clan]]
[[Category:Temperaments]]
[[Category:Bohlen–Pierce]]
[[Category:Tritave-equivalent temperaments]]
Retrieved from "https://en.xen.wiki/w/BPS"