Bohpier: Difference between revisions

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#redirect [[Sensamagic clan #Bohpier]]
'''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.
 
See [[Sensamagic clan #Bohpier]] for more technical data.
 
== Interval ==
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"
! rowspan="2" |Steps
! colspan="3" |Double dim.
! colspan="3" |Diminished
! colspan="3" |Minor
! colspan="3" |Major
! colspan="3" |Augmented
! colspan="3" |Double aug.
|-
! Gen.
! Cents*
! Ratios
! Gen.
! Cents*
! Ratios
! Gen.
! Cents*
! Ratios
! Gen.
! Cents*
! Ratios
! Gen.
! Cents*
! Ratios
! Gen.
! Cents*
! Ratios
|-
| text-align:"center" | 1
| 17
| 92.25
|
| 9
| 119.42
| 15/14~14/13
| 1
| 146.60
| 13/12~12/11
| -7
| 173.78
| 11/10~10/9
| -15
| 200.96
|
| -23
| 228.14
| 8/7
|-
| 2
| 18
| 238.85
| 15/13
| 10
| 266.03
| 7/6
| 2
| 293.21
| 13/11
| -6
| 320.38
| 6/5
| -14
| 347.56
| 11/9~16/13
| -22
| 374.74
|
|-
| 3
| 19
| 385.45
| 5/4
| 11
| 412.63
| 14/11
| 3
| 439.81
| 9/7
| -5
| 466.99
| 13/10
| -13
| 494.17
| 4/3
| -21
| 521.34
|
|-
| 4
| 20
| 532.05
| 15/11
| 12
| 559.23
| 11/8~18/13
| 4
| 586.41
| 7/5
| -4
| 613.59
| 10/7
| -12
| 640.77
| 13/9~16/11
| -20
| 667.95
| 22/15
|-
| 5
| 21
| 678.66
|
| 13
| 705.83
| 3/2
| 5
| 733.01
| 20/13
| -3
| 760.19
| 14/9
| -11
| 787.37
| 11/7
| -19
| 814.55
| 8/5
|-
| 6
| 22
| 825.26
|
| 14
| 852.44
| 13/8~18/11
| 6
| 879.62
| 5/3
| -2
| 906.79
| 22/13
| -10
| 933.97
| 12/7
| -18
| 961.15
| 26/15
|-
| 7
| 23
| 971.86
| 7/4
| 15
| 999.04
|
| 7
| 1026.22
| 9/5~20/11
| -1
| 1053.40
| 11/6~24/13
| -9
| 1080.58
| 13/7~28/15
| -17
| 1107.75
|
|}
<nowiki>*</nowiki> in 13-limit POTE tuning
 
== Tuning spectrum ==
Gencom: [2 12/11; 100/99 144/143 196/195 275/273]
 
Gencom map: [{{val| 1 0 0 0 2 2 }}, {{val| 0 13 19 23 12 14 }}]
 
{| class="wikitable center-1 right-2"
|-
! Eigenmonzo
! Neutral <br>second
! Comments
|-
| 13/12
| 138.5727
|
|-
| 13/11
| 144.6049
|
|-
| 9/7
| 145.0280
|
|-
| 10/9
| 145.3709
|
|-
| 7/5
| 145.6280
|
|-
| 16/13
| 145.7520
|
|-
| 11/8
| 145.9432
|
|-
| 4/3
| 146.3042
| 9-odd-limit minimax
|-
| 8/7
| 146.4707
|
|-
| 16/15
| 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
|
|-
| 18/13
| 146.9485
|
|-
| 14/11
| 147.0462
|
|-
| 15/13
| 147.0967
|
|-
| 6/5
| 147.3931
|
|-
| 14/13
| 147.5887
|
|-
| 11/10
| 147.8565
|
|-
| 13/10
| 149.1572
|
|-
| 12/11
| 150.6371
|
|}
 
== Scale ==
* [[Bohpier8]]
* [[Bohpier9]]
* [[Bohpier17]]
* [[Bohpier25]]
* [[Bohpier33]]


[[Category:Temperaments]]
[[Category:Temperaments]]
[[Category:Sensamagic clan]]
[[Category:Sensamagic clan]]
[[Category:Bohlen-Pierce]]
[[Category:Index of temperaments]]

Revision as of 14:43, 29 July 2021

Bohpier temperament is a rank-two temperament for 7, 11, and 13 prime limits. It tempers out 100/99, 144/143, 196/195, and 275/273 in the 13-limit, as well as the 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 13ED3, 19ED5, or 23ED7. 41EDO is an excellent tuning for the bohpier, with generator 5\41, and MOS of 8, 9, 17, 25 or 33 notes are available.

See Sensamagic clan #Bohpier for more technical data.

Interval

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.

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
1 17 92.25 9 119.42 15/14~14/13 1 146.60 13/12~12/11 -7 173.78 11/10~10/9 -15 200.96 -23 228.14 8/7
2 18 238.85 15/13 10 266.03 7/6 2 293.21 13/11 -6 320.38 6/5 -14 347.56 11/9~16/13 -22 374.74
3 19 385.45 5/4 11 412.63 14/11 3 439.81 9/7 -5 466.99 13/10 -13 494.17 4/3 -21 521.34
4 20 532.05 15/11 12 559.23 11/8~18/13 4 586.41 7/5 -4 613.59 10/7 -12 640.77 13/9~16/11 -20 667.95 22/15
5 21 678.66 13 705.83 3/2 5 733.01 20/13 -3 760.19 14/9 -11 787.37 11/7 -19 814.55 8/5
6 22 825.26 14 852.44 13/8~18/11 6 879.62 5/3 -2 906.79 22/13 -10 933.97 12/7 -18 961.15 26/15
7 23 971.86 7/4 15 999.04 7 1026.22 9/5~20/11 -1 1053.40 11/6~24/13 -9 1080.58 13/7~28/15 -17 1107.75

* in 13-limit POTE tuning

Tuning spectrum

Gencom: [2 12/11; 100/99 144/143 196/195 275/273]

Gencom map: [⟨1 0 0 0 2 2], ⟨0 13 19 23 12 14]]

Eigenmonzo Neutral
second 		Comments
13/12 138.5727
13/11 144.6049
9/7 145.0280
10/9 145.3709
7/5 145.6280
16/13 145.7520
11/8 145.9432
4/3 146.3042 9-odd-limit minimax
8/7 146.4707
16/15 146.5084
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
18/13 146.9485
14/11 147.0462
15/13 147.0967
6/5 147.3931
14/13 147.5887
11/10 147.8565
13/10 149.1572
12/11 150.6371

Scale

Retrieved from "https://en.xen.wiki/index.php?title=Bohpier&oldid=75230"