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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.
Line 7: Line 20:
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


<div class="toccolours mw-collapsible mw-collapsed" style="width: 800px; overflow: auto;">
=== As a detemperament of 8et ===
<div style="line-height: 1.6;">'''Intervals of Bohpier (8d &amp; 41)'''</div>
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="mw-collapsible-content">
 
{| class="wikitable"
{| class="wikitable center-all"
|-
|-
! 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 ==
Line 189: Line 352:


== 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
Line 364: Line 570:
* [[Lumatone mapping for bohpier]]
* [[Lumatone mapping for bohpier]]


[[Category:Temperaments]]
[[Category:Bohpier]] <!-- main article -->
[[Category:Bohpier]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Sensamagic clan]]
[[Category:Sensamagic clan]]
[[Category:Gariboh clan]]
[[Category:Gariboh clan]]
[[Category:Mirkwai clan]]
[[Category:Canopic clan]]
[[Category:Bohlen-Pierce]]
[[Category:Bohlen–Pierce]]

Latest revision as of 10:30, 6 June 2026

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);
100/99, 245/243, 1344/1331 (11-limit;
100/99, 144/143, 196/195, 275/273
(13-limit)
Reduced mapping ⟨1; 13 19 23 12 14]
ET join 41 & 49f
Generators (CWE) ~12/11 = 146.5 ¢
MOS scales 1L 7s, 8L 1s, 8L 9s, 8L 17s
Ploidacot alpha-13-cot
Minimax error 9-odd-limit: 6.53 ¢;
13-limit 21-odd-limit: 12.5 ¢
Target scale size 9-odd-limit: 25 notes;
13-limit 21-odd-limit: 41 notes

Bohpier is a rank-2 temperament which can be described as the Bohlen–Pierce scale with octaves. From this strong relation it derives its name. In this temperament, like in Bohlen–Pierce, 13 generator steps give the 3rd harmonic, 19 give the 5th harmonic, and 23 give the 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, gariboh, arcturus, and mirkwai clans. The extension to the 13-limit sees more involvement of the octave, with 14 steps giving the interval class of 11 and 12 steps giving the interval class of 13, tempering out 100/99, 144/143, 196/195, and 275/273.

Possible generators for bohpier include 1\13edt, 1\19ed5, and 1\23ed7. Another excellent tuning for the temperament is 41edo, with generator 5\41. Mos scales of 8, 9, 17, 25, or 33 notes are available.

See Sensamagic clan #Bohpier for technical data.

Interval chain

In the following table, odd harmonics 1–13 and their inverses are in bold.

# 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,
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

* 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.

Steps Double dim. Diminished Minor Major Augmented Double aug.
Gen. Cents* Ratios Gen. Cents* Ratios Gen. Cents* Ratios Gen. Cents* Ratios Gen. Cents* Ratios Gen. Cents* 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 17 90.9 21/20 9 118.7 14/13~15/14 1 146.5 12/11~13/12 −7 173.3 10/9~11/10 −15 202.2 28/25 −23 230.0 8/7
2 18 237.4 15/13 10 265.2 7/6 2 293.0 13/11 −6 320.9 6/5 −14 348.7 11/9~16/13 −22 376.5 26/21
3 19 383.9 5/4 11 411.8 14/11 3 439.6 9/7 −5 467.4 13/10 −13 495.2 4/3 −21 523.0 66/49
4 20 530.4 15/11 12 558.3 11/8~18/13 4 586.1 7/5 −4 613.9 10/7 −12 641.7 13/9~16/11 −20 669.5 22/15
5 21 677.0 49/33 13 704.8 3/2 5 732.6 20/13 −3 760.4 14/9 −11 788.2 11/7 −19 816.1 8/5
6 22 823.5 21/13 14 851.3 13/8~18/11 6 879.1 5/3 −2 907.0 22/13 −10 934.8 12/7 −18 962.6 26/15
7 23 970.0 7/4 15 997.8 25/14 7 1025.7 9/5~20/11 −1 1053.5 11/6~24/13 −9 1081.3 13/7~28/15 −17 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

* In 13-limit CWE tuning

Chords

Main article: Chords of bohpier

Tunings

7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~27/25 = 146.4741 ¢ CWE: ~27/25 = 146.4739 ¢ POTE: ~27/25 = 146.4741 ¢
11-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~12/11 = 146.4441 ¢ CWE: ~12/11 = 146.5009 ¢ POTE: ~12/11 = 146.5446 ¢
13-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~12/11 = 146.4006 ¢ CWE: ~12/11 = 146.5230 ¢ POTE: ~12/11 = 146.6027 ¢

TOP generators:

  • 7-limit: ~2 = 1200.00000 ¢, ~27/25 = 146.47407 ¢
  • 11-limit: ~2 = 1199.23623 ¢, ~12/11 = 146.45131 ¢
  • 13-limit: ~2 = 1198.55643 ¢, ~12/11 = 146.42630 ¢

Tuning spectrum

Edo
generator 		Unchanged interval
(eigenmonzo) 		Generator (¢) Comments
13/12 138.5727
3\25 144.0000 25bccddf val, lower bound of 5-odd-limit diamond monotone
13/11 144.6049
9/7 145.0280
9/5 145.3709
4\33 145.4545 33cd val, lower bound of 7-, 9-, 11-, and 13-odd-limit diamond monotone
7/5 145.6280
13/8 145.7520
11/8 145.9432
3/2 146.3042 9-odd-limit minimax
5\41 146.3415
7/4 146.4707
15/8 146.5084
15/14 146.6048
11/9 146.6137 11-odd-limit minimax
5/4 146.6481 5-, 7-, 13-, and 15-odd-limit minimax
7/6 146.6871
15/11 146.8475
6\49 146.9388 49f val, upper bound of 9-, 11-, and 13-odd-limit diamond monotone
13/9 146.9485
11/7 147.0462
15/13 147.0967
5/3 147.3931
13/7 147.5887
11/10 147.8565
13/10 149.1572
1\8 150.0000 8d val, upper bound of 5- and 7-odd-limit diamond monotone
11/6 150.6371

Scales

Music

Chris Vaisvil

See also

Retrieved from "https://en.xen.wiki/index.php?title=Bohpier&oldid=231668"