Xenharmonic Wiki

Bohpier: Difference between revisions

← Older edit
VisualWikitext
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
5,013 edits
mNo edit summary
m Text replacement - "Category:Mirkwai clan" to "Category:Canopic clan"
 
(26 intermediate revisions by 6 users not shown)
Line 1: Line 1:
'''Bohpier temperament''' is a rank-two temperament for 7, 11, and 13 [[Harmonic limit|prime limits]]. It tempers out 100/99, 144/143, 196/195, and 275/273 in the 13-limit, as well as the [[245/243|sensamagic comma]] (245/243) and the gariboh comma (3125/3087). It is strongly related to [[Bohlen-Pierce]] scale, and from this it derives its name. Generator for the bohpier can be used either [[13edt|13ED3]], [[19ed5|19ED5]], or [[23ed7|23ED7]]. [[41edo|41EDO]] is an excellent tuning for the bohpier, with generator 5\41, and MOS of 8, 9, 17, 25, or 33 notes are available.
{{Infobox regtemp
| Title = Bohpier
| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
| Comma basis = [[245/243]], [[3125/3087]] (7-limit); <br>[[100/99]], [[245/243]], [[1344/1331]] (11-limit; <br>[[100/99]], [[144/143]], [[196/195]], [[275/273]]<br>(13-limit)
| Edo join 1 = 41 | Edo join 2 = 49f
| Mapping = 1; 13 19 23 12 14
| Generators = 12/11
| Generators tuning = 146.5
| Optimization method = CWE
| MOS scales = [[1L 7s]], [[8L 1s]], [[8L 9s]], [[8L 17s]]
| Odd limit 1 = 9 | Mistuning 1 = 6.53 | Complexity 1 = 25
| Odd limit 2 = 13-limit 21 | Mistuning 2 = 12.5 | Complexity 2 = 41
}}
'''Bohpier''' is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] which can be described as the [[Bohlen–Pierce]] scale with [[2/1|octaves]]. From this strong relation it derives its name. In this temperament, like in Bohlen–Pierce, 13 generator steps give the [[3/1|3rd harmonic]], 19 give the [[5/1|5th harmonic]], and 23 give the [[7/1|7th harmonic]], [[tempering out]] the sensamagic comma ([[245/243]]) and the gariboh comma ([[3125/3087]]). The only difference is the addition of the [[period]] of an octave.  


See [[Sensamagic clan #Bohpier]] for more technical data.
It is a member of [[sensamagic clan|sensamagic]], [[gariboh clan|gariboh]], [[arcturus clan|arcturus]], and [[mirkwai clan|mirkwai]] [[temperament families and clans|clans]]. The [[extension]] to the [[13-limit]] sees more involvement of the octave, with 14 steps giving the interval class of [[11/1|11]] and 12 steps giving the interval class of [[13/1|13]], tempering out [[100/99]], [[144/143]], [[196/195]], and [[275/273]].  


== Intervals ==
Possible generators for bohpier include [[13edt|1\13edt]], [[19ed5|1\19ed5]], and [[23ed7|1\23ed7]]. Another excellent tuning for the temperament is [[41edo]], with generator 5\41. [[Mos scale]]s of 8, 9, 17, 25, or 33 notes are available.
Bohpier is considered as a [[cluster temperament]] with eight clusters of notes in an octave. The chroma interval between adjacent notes in each cluster represents 40/39 ~ 50/49 ~ 55/54 ~ 56/55 ~ 66/65 ~ 78/77 all tempered together.


{| class="wikitable"
See [[Sensamagic clan #Bohpier]] for technical data.
! rowspan="2" |Steps
 
! colspan="3" |Double dim.
== Interval chain ==
! colspan="3" |Diminished
In the following table, odd harmonics 1–13 and their inverses are in '''bold'''.
! colspan="3" |Minor
{| class="wikitable center-1 right-2"
! colspan="3" |Major
|-
! colspan="3" |Augmented
! #
! colspan="3" |Double aug.
! Cents*
! Approximate ratios
|-
| 0
| 0.0
| '''1/1'''
|-
| 1
| 146.5
| 12/11, 13/12, 27/25
|-
| 2
| 293.0
| 13/11
|-
| 3
| 439.6
| 9/7
|-
| 4
| 586.1
| 7/5
|-
| 5
| 732.6
| 20/13
|-
| 6
| 879.1
| 5/3
|-
| 7
| 1025.7
| 9/5, 20/11
|-
| 8
| 1172.2
| 39/20, 49/25, 55/28, <br>65/33, 77/39, 108/55
|-
| 9
| 118.7
| 14/13, 15/14
|-
| 10
| 265.2
| 7/6
|-
| 11
| 411.8
| 14/11
|-
| 12
| 558.3
| '''11/8''', 18/13
|-
| 13
| 704.8
| '''3/2'''
|-
| 14
| 851.3
| '''13/8''', 18/11
|-
| 15
| 997.8
| 25/14
|-
| 16
| 1144.4
| 27/14, 35/18
|-
| 17
| 90.9
| 21/20
|-
| 18
| 237.4
| 15/13
|-
| 19
| 383.9
| '''5/4'''
|-
| 20
| 530.4
| 15/11, 27/20
|-
| 21
| 677.0
| 49/33
|-
| 22
| 823.5
| 21/13
|-
| 23
| 970.0
| '''7/4'''
|-
| 24
| 1116.6
| 21/11
|-
| 25
| 63.1
| 25/24, 27/26, 33/32
|}
<nowiki/>* In 13-limit CWE tuning
 
=== As a detemperament of 8et ===
Bohpier can be considered as a [[cluster temperament]] with eight clusters of notes in an octave. The chroma interval between adjacent notes in each cluster represents 40/39 ~ 50/49 ~ 55/54 ~ 56/55 ~ 66/65 ~ 78/77 ~ 91/90 all tempered together.
 
{| class="wikitable center-all"
|-
! rowspan="2" | Steps
! colspan="3" | Double dim.
! colspan="3" | Diminished
! colspan="3" | Minor
! colspan="3" | Major
! colspan="3" | Augmented
! colspan="3" | Double aug.
|-
|-
! Gen.
! Gen.
Line 33: Line 165:
! Cents*
! Cents*
! Ratios
! Ratios
|-
| 0
|
|
|
|
|
|
| 0
| 0.0
| 1/1
| −8
| 27.8
| 56/55~66/65
| −16
| 55.6
| 28/27~36/35
| −24
| 83.4
| 22/21
|-
|-
| 1
| 1
| 17
| 17
| 92.25
| 90.9
|  
| 21/20
| 9
| 9
| 119.42
| 118.7
| 15/14~14/13
| 14/13~15/14
| 1
| 1
| 146.60
| 146.5
| 13/12~12/11
| 12/11~13/12
| -7
| −7
| 173.78
| 173.3
| 11/10~10/9
| 10/9~11/10
| -15
| −15
| 200.96
| 202.2
|  
| 28/25
| -23
| −23
| 228.14
| 230.0
| 8/7
| 8/7
|-
|-
| 2
| 2
| 18
| 18
| 238.85
| 237.4
| 15/13
| 15/13
| 10
| 10
| 266.03
| 265.2
| 7/6
| 7/6
| 2
| 2
| 293.21
| 293.0
| 13/11
| 13/11
| -6
| −6
| 320.38
| 320.9
| 6/5
| 6/5
| -14
| −14
| 347.56
| 348.7
| 11/9~16/13
| 11/9~16/13
| -22
| −22
| 374.74
| 376.5
|  
| 26/21
|-
|-
| 3
| 3
| 19
| 19
| 385.45
| 383.9
| 5/4
| 5/4
| 11
| 11
| 412.63
| 411.8
| 14/11
| 14/11
| 3
| 3
| 439.81
| 439.6
| 9/7
| 9/7
| -5
| −5
| 466.99
| 467.4
| 13/10
| 13/10
| -13
| −13
| 494.17
| 495.2
| 4/3
| 4/3
| -21
| −21
| 521.34
| 523.0
|  
| 66/49
|-
|-
| 4
| 4
| 20
| 20
| 532.05
| 530.4
| 15/11
| 15/11
| 12
| 12
| 559.23
| 558.3
| 11/8~18/13
| 11/8~18/13
| 4
| 4
| 586.41
| 586.1
| 7/5
| 7/5
| -4
| −4
| 613.59
| 613.9
| 10/7
| 10/7
| -12
| −12
| 640.77
| 641.7
| 13/9~16/11
| 13/9~16/11
| -20
| −20
| 667.95
| 669.5
| 22/15
| 22/15
|-
|-
| 5
| 5
| 21
| 21
| 678.66
| 677.0
|  
| 49/33
| 13
| 13
| 705.83
| 704.8
| 3/2
| 3/2
| 5
| 5
| 733.01
| 732.6
| 20/13
| 20/13
| -3
| −3
| 760.19
| 760.4
| 14/9
| 14/9
| -11
| −11
| 787.37
| 788.2
| 11/7
| 11/7
| -19
| −19
| 814.55
| 816.1
| 8/5
| 8/5
|-
|-
| 6
| 6
| 22
| 22
| 825.26
| 823.5
|  
| 21/13
| 14
| 14
| 852.44
| 851.3
| 13/8~18/11
| 13/8~18/11
| 6
| 6
| 879.62
| 879.1
| 5/3
| 5/3
| -2
| −2
| 906.79
| 907.0
| 22/13
| 22/13
| -10
| −10
| 933.97
| 934.8
| 12/7
| 12/7
| -18
| −18
| 961.15
| 962.6
| 26/15
| 26/15
|-
|-
| 7
| 7
| 23
| 23
| 971.86
| 970.0
| 7/4
| 7/4
| 15
| 15
| 999.04
| 997.8
|  
| 25/14
| 7
| 7
| 1026.22
| 1025.7
| 9/5~20/11
| 9/5~20/11
| -1
| −1
| 1053.40
| 1053.5
| 11/6~24/13
| 11/6~24/13
| -9
| −9
| 1080.58
| 1081.3
| 13/7~28/15
| 13/7~28/15
| -17
| −17
| 1107.75
| 1109.1
| 40/21
|-
| 8
| 24
| 1116.6
| 21/11
| 16
| 1144.4
| 27/14~35/18
| 8
| 1172.2
| 55/28~65/33
| 0
| 1200.0
| 2/1
|
|
|
|
|  
|  
|
|}
<nowiki/>* In 13-limit CWE tuning
== Chords ==
{{Main| Chords of bohpier }}
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~27/25 = 146.4741{{c}}
| CWE: ~27/25 = 146.4739{{c}}
| POTE: ~27/25 = 146.4741{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~12/11 = 146.4441{{c}}
| CWE: ~12/11 = 146.5009{{c}}
| POTE: ~12/11 = 146.5446{{c}}
|}
|}
<nowiki>*</nowiki> in 13-limit POTE tuning


== Tuning spectrum ==
{| class="wikitable mw-collapsible mw-collapsed"
Gencom: [2 12/11; 100/99 144/143 196/195 275/273]
|+ style="font-size: 105%; white-space: nowrap;" | 13-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~12/11 = 146.4006{{c}}
| CWE: ~12/11 = 146.5230{{c}}
| POTE: ~12/11 = 146.6027{{c}}
|}


Gencom map: [{{val| 1 0 0 0 2 2 }}, {{val| 0 13 19 23 12 14 }}]
[[TOP tuning|TOP generators]]:  
* 7-limit: ~2 = 1200.00000{{c}}, ~27/25 = 146.47407{{c}}
* 11-limit: ~2 = 1199.23623{{c}}, ~12/11 = 146.45131{{c}}
* 13-limit: ~2 = 1198.55643{{c}}, ~12/11 = 146.42630{{c}}


{| class="wikitable center-1 right-2"
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
|-
! Eigenmonzo
! Edo<br>generator
! Neutral <br>second
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
! Generator (¢)
! Comments
! Comments
|-
|-
|
| 13/12
| 13/12
| 138.5727
| 138.5727
|  
|  
|-
|-
| 3\25
|
| 144.0000
| 25bccddf val, lower bound of 5-odd-limit diamond monotone
|-
|
| 13/11
| 13/11
| 144.6049
| 144.6049
|  
|  
|-
|-
|
| 9/7
| 9/7
| 145.0280
| 145.0280
|  
|  
|-
|-
| 10/9
|  
| 9/5
| 145.3709
| 145.3709
|  
|  
|-
|-
| 4\33
|
| 145.4545
| 33cd val, lower bound of 7-, 9-, 11-, and 13-odd-limit diamond monotone
|-
|
| 7/5
| 7/5
| 145.6280
| 145.6280
|  
|  
|-
|-
| 16/13
|  
| 13/8
| 145.7520
| 145.7520
|  
|  
|-
|-
|
| 11/8
| 11/8
| 145.9432
| 145.9432
|  
|  
|-
|-
| 4/3
|  
| 3/2
| 146.3042
| 146.3042
| 9-odd-limit minimax
| 9-odd-limit minimax
|-
|-
| 8/7
| 5\41
|
| 146.3415
|
|-
|
| 7/4
| 146.4707
| 146.4707
|  
|  
|-
|-
| 16/15
|  
| 15/8
| 146.5084
| 146.5084
|  
|  
|-
|-
|
| 15/14
| 15/14
| 146.6048
| 146.6048
|  
|  
|-
|-
|
| 11/9
| 11/9
| 146.6137
| 146.6137
| 11-odd-limit minimax
| 11-odd-limit minimax
|-
|-
|
| 5/4
| 5/4
| 146.6481
| 146.6481
| 5, 7, 13, and 15-odd-limit minimax
| 5-, 7-, 13-, and 15-odd-limit minimax
|-
|-
|
| 7/6
| 7/6
| 146.6871
| 146.6871
|  
|  
|-
|-
|
| 15/11
| 15/11
| 146.8475
| 146.8475
|  
|  
|-
|-
| 18/13
| 6\49
|
| 146.9388
| 49f val, upper bound of 9-, 11-, and 13-odd-limit diamond monotone
|-
|
| 13/9
| 146.9485
| 146.9485
|  
|  
|-
|-
| 14/11
|  
| 11/7
| 147.0462
| 147.0462
|  
|  
|-
|-
|
| 15/13
| 15/13
| 147.0967
| 147.0967
|  
|  
|-
|-
| 6/5
|  
| 5/3
| 147.3931
| 147.3931
|  
|  
|-
|-
| 14/13
|  
| 13/7
| 147.5887
| 147.5887
|  
|  
|-
|-
|
| 11/10
| 11/10
| 147.8565
| 147.8565
|  
|  
|-
|-
|
| 13/10
| 13/10
| 149.1572
| 149.1572
|  
|  
|-
|-
| 12/11
| 1\8
|
| 150.0000
| 8d val, upper bound of 5- and 7-odd-limit diamond monotone
|-
|
| 11/6
| 150.6371
| 150.6371
|  
|  
Line 281: Line 555:


== Scales ==
== Scales ==
* [[Bohpier8]] - [[1L 7s]] scale
* [[Bohpier8]] – [[1L 7s]] scale
* [[Bohpier9]] - [[8L 1s]] scale
* [[Bohpier9]] – [[8L 1s]] scale
* [[Bohpier17]] - [[8L 9s]] scale
* [[Bohpier17]] – [[8L 9s]] scale
* [[Bohpier25]] - [[8L 17s]] scale
* [[Bohpier25]] – [[8L 17s]] scale
* [[Bohpier33]] - [[8L 25s]] scale
* [[Bohpier33]] – [[8L 25s]] scale
 
== Music ==
; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127013822/http://micro.soonlabel.com/bophier/bophier-1.mp3 ''bophier-1'']
* [https://web.archive.org/web/20201127015147/http://micro.soonlabel.com/bophier/bophier-12equal-six-octaves.mp3 ''bophier-12equal-six-octaves'']
 
== See also ==
* [[Sensamagic chords]]
* [[Lumatone mapping for bohpier]]


[[Category:Temperaments]]
[[Category:Bohpier]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Sensamagic clan]]
[[Category:Sensamagic clan]]
[[Category:Bohlen-Pierce]]
[[Category:Gariboh clan]]
[[Category:Index of temperaments]]
[[Category:Canopic clan]]
[[Category:Bohlen–Pierce]]
Retrieved from "https://en.xen.wiki/w/Bohpier"