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Bohpier

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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, and the extension to the 13-limit tempers 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.

Temperament data

Bohpier temperament (8d & 41)

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 144/143, 196/195, 275/273

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

POTE generator:

  • 7-limit: ~27/25 = 146.47407
  • 11-limit: ~12/11 = 146.54458
  • 13-limit: ~12/11 = 146.60266

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

Diamond monotone ranges:

  • 5-odd-limit: ~27/25 = [144.00000, 150.00000] (3\25 to 1\8)
  • 7-odd-limit: ~27/25 = [145.45455, 150.00000] (4\33 to 1\8)
  • 9, 11, and 13-odd-limit: ~12/11 = [145.45455, 146.93878] (4\33 to 6\49)
  • 15-odd-limit: ~12/11 = [146.34146, 146.93878] (5\41 to 6\49)

Diamond tradeoff ranges:

  • 5-odd-limit: ~27/25 = [146.30423, 147.39312]
  • 7-odd-limit: ~27/25 = [145.62805, 147.39312]
  • 9 and 11-odd-limit: ~12/11 = [145.02803, 147.39312]
  • 13 and 15-odd-limit: ~12/11 = [138.57266, 150.63706]

Diamond monotone and tradeoff ranges:

  • 5-odd-limit: ~27/25 = [146.30423, 147.39312]
  • 7-odd-limit: ~27/25 = [145.62805, 147.39312]
  • 9, 11, and 13-odd-limit: ~12/11 = [145.45455, 146.93878]
  • 15-odd-limit: ~12/11 = [146.34146, 146.93878]

Optimal ET sequences:

Badness:

  • 7-limit: 0.068237
  • 11-limit: 0.033949
  • 13-limit: 0.024864

Intervals

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.

Intervals of Bohpier (8d & 41)
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 mapping: [⟨1 0 0 0 2 2], ⟨0 13 19 23 12 14]]

Eigenmonzo
(unchanged-interval) 		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

Scales

See also

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