Bohpier
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
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]]
- 7-limit: ~27/25 = 146.47407
- 11-limit: ~12/11 = 146.54458
- 13-limit: ~12/11 = 146.60266
- 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]
- 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.
| 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
- Bohpier8 – 1L 7s scale
- Bohpier9 – 8L 1s scale
- Bohpier17 – 8L 9s scale
- Bohpier25 – 8L 17s scale
- Bohpier33 – 8L 25s scale