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Bohpier: Difference between revisions

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{{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.  
'''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.  


It is a member of [[sensamagic clan|sensamagic]], [[gariboh clan|gariboh]], [[arcturus clan|arcturus]], and [[mirkwai clan|mirkwai]] [[temperament families and clans|clans]], and the [[extension]] to the 13-limit tempers out 100/99, 144/143, 196/195, and 275/273.  
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]].  


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.
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.
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See [[Sensamagic clan #Bohpier]] for technical data.  
See [[Sensamagic clan #Bohpier]] for technical data.  


== Intervals ==
== Interval chain ==
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 ~ 91/90 all tempered together.
In the following table, odd harmonics 1–13 and their inverses are in '''bold'''.
{| class="wikitable center-1 right-2"
|-
! #
! 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.


<div class="toccolours mw-collapsible mw-collapsed" style="width: 800px; overflow: auto;">
{| class="wikitable center-all"
<div style="line-height: 1.6;">'''Intervals of Bohpier (8d &amp; 41)'''</div>
<div class="mw-collapsible-content">
{| class="wikitable"
|-
|-
! rowspan="2" | Steps
! rowspan="2" | Steps
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! 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 POTE tuning
<nowiki/>* In 13-limit CWE tuning
</div></div>


== Chords ==
== Chords ==
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== Tunings ==
== Tunings ==
[[POTE generator]]:  
{| class="wikitable mw-collapsible mw-collapsed"
* 7-limit: ~27/25 = 146.47407
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
* 11-limit: ~12/11 = 146.54458
|-
* 13-limit: ~12/11 = 146.60266
! 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}}
|}
 
{| class="wikitable mw-collapsible mw-collapsed"
|+ 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}}
|}


[[TOP tuning|TOP generators]]:  
[[TOP tuning|TOP generators]]:  
* 7-limit: ~2 = 1200.00000, ~27/25 = 146.47407
* 7-limit: ~2 = 1200.00000{{c}}, ~27/25 = 146.47407{{c}}
* 11-limit: ~2 = 1199.23623, ~12/11 = 146.45131
* 11-limit: ~2 = 1199.23623{{c}}, ~12/11 = 146.45131{{c}}
* 13-limit: ~2 = 1198.55643, ~12/11 = 146.42630
* 13-limit: ~2 = 1198.55643{{c}}, ~12/11 = 146.42630{{c}}


=== Tuning spectrum ===
=== Tuning spectrum ===
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|-
|-
! Edo<br>generator
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]
! [[Eigenmonzo|Unchanged interval<br>(eigenmonzo)]]
! Generator (¢)
! Generator (¢)
! Comments
! Comments
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* [[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 ==
== See also ==
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* [[Lumatone mapping for bohpier]]
* [[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:Canopic clan]]
[[Category:Bohlen–Pierce]]
Retrieved from "https://en.xen.wiki/w/Bohpier"