Sensamagic-gariboh equivalence continuum
The sensamagic-gariboh equivalence continuum is a continuum of 3.5.7 subgroup temperaments which equate a number of sensamagic commas (245/243) with the greater BP diesis (3125/3087).
All temperaments in the continuum satisfy (245/243)n ~ 3125/3087. Varying n results in different temperaments listed in the table below. It converges to BPS as n approaches infinity and Sirius as n approaches 0. If we allow non-integer and infinite n, the continuum describes the set of all 3.5.7 subgroup temperaments supported by 13edt (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them); additionally, if prime 2 is added, this describes the set of all rank-3 7-limit temperaments supported by bohpier. Due to 13edt, the equal-tempered Bohlen-Pierce scale, being a very good representation of the subgroup, and many named temperaments within the subgroup being supported by 13edt, this continuum is structurally important. The just value of n is 1.4926…, and temperaments near this tend to be the most accurate ones.
An alternative definition of this equivalence continuum satisfies (245/243)k ~ 713/323 = 96889010407/94143178827, which sets k = 5 - n. This definition might be advantageous as k gives the order of harmonic 5 in the corresponding comma, and equals the number of generators to obtain a harmonic 7 in the MOS scale.
| n | k | Temperament | Comma | |
|---|---|---|---|---|
| Ratio | Monzo (3.5.7 subgroup) | |||
| -2 | 7 | Muphrid (b13 & b2d) | 546875/531441 | [-12 7 1⟩ |
| -1 | 6 | Arcturus | 15625/15309 | [-7 6 -1⟩ |
| 0 | 5 | Sirius | 3125/3087 | [-2 5 -3⟩ |
| 1 | 4 | Canopus | 16875/16807 | [3 4 -5⟩ |
| 2 | 3 | Procyon (b13 & b144) | 823543/820125 | [-8 -3 7⟩ |
| 3 | 2 | Betelgeuse (b13 & b148) | 40353607/39858075 | [-13 -2 9⟩ |
| 4 | 1 | Pollux (b13 & b139) | 1977326743/1937102445 | [-18 -1 11⟩ |
| 5 | 0 | 13-7-commatic | 96889010407/94143178827 | [-23 0 13⟩ |
| … | … | … | ||
| ∞ | ∞ | BPS | 245/243 | [-5 1 2⟩ |
We may invert the continuum by setting m such that 1/m + 1/n = 1. This may be called the canopus-sirius equivalence continuum, as temperaments satisfy (16875/16807)m ~ 3125/3087. The just value of m is 3.0300…, and temperaments close to this value are the most accurate (note how accurate the Izar temperament then must be!)
| m | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo (3.5.7 subgroup) | ||
| -1 | Miaplacidus | 5859375/5764801 | [1 9 -8⟩ |
| 0 | Sirius | 3125/3087 | [-2 5 -3⟩ |
| 1 | BPS | 245/243 | [-5 1 2⟩ |
| 2 | Procyon (b13 & b144) | 823543/820125 | [-8 -3 7⟩ |
| 3 | Izar | 13841287201/13839609375 | [-11 -7 12⟩ |
| 4 | b13 & b307 | 233543408203125/232630513987207 | [14 11 -17⟩ |
| … | … | … | |
| ∞ | Canopus | 16875/16807 | [3 4 -5⟩ |
Projective tuning space
Below is a depiction of the temperaments of this continuum in 3.5.7 projective tuning space.
|
| Labeled by name |
|
| Labeled by comma |