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. 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
Tunings
- 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
Tuning spectrum
| Edo generator |
Eigenmonzo (unchanged-interval) |
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
- Bohpier8 – 1L 7s scale
- Bohpier9 – 8L 1s scale
- Bohpier17 – 8L 9s scale
- Bohpier25 – 8L 17s scale
- Bohpier33 – 8L 25s scale