Jubilismic-augmented equivalence continuum
The jubilismic-augmented equivalence continuum is a continuum of 2.5.7 subgroup temperaments which equate a number of jubilismas (50/49) with the lesser diesis (128/125).
All temperaments in the continuum satisfy (50/49)n ~ 128/125. Varying n results in different temperaments listed in the table below. It converges to jubilic as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 2.5.7 subgroup temperaments supported by 6edo (due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them); due to 6edo representing this subgroup modestly well for its size, this continuum is structurally important to 2.5.7. The just value of n is 1.1739…, and temperaments near this tend to be the most accurate ones.
128/125 is the characteristic 2.5 comma tempered out in 6edo. In each case, we notice that n equals half the order of harmonic 7 in the corresponding comma (noting that 6edo's ring of 5/4s inherits from 3edo, 6edo therefore has two rings and any comma involving 7 therefore has 7 to an even power), and equals the number of generators to obtain a harmonic 5 in the MOS scale.
Note temperaments linked to in the below are generally 2.5.7 subgroup restrictions of full 7-limit temperaments.
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo (2.5.7 subgroup) | ||
| -2 | Ripple | 2560/2401 | [9 1 -4⟩ |
| -1 | Bapbo | 256/245 | [8 -1 -2⟩ |
| 0 | Augmented | 128/125 | [7 -3⟩ |
| 1/2 | Diaschismic | 401408/390625 | [13 -8 2⟩ |
| 1 | Didacus | 3136/3125 | [6 -5 2⟩ |
| 3/2 | Waage | 244140625/240945152 | [-11 12 -6⟩ |
| 2 | Superthird | 78125/76832 | [-5 7 -4⟩ |
| 3 | Fog | 1953125/1882384 | [-4 9 -6⟩ |
| … | … | … | |
| ∞ | Jubilic | 50/49 | [1 2 -2⟩ |
We may invert the continuum by setting m such that 1/m + 1/n = 1. This may be called the didacus-augmented equivalence continuum, as temperaments satisfy (3136/3125)m ~ 128/125. The just value of m is 6.7495…, and temperaments close to this value are the most accurate.
| m | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo (2.5.7 subgroup) | ||
| -1 | Diaschismic | 401408/390625 | [13 -8 2⟩ |
| 0 | Augmented | 128/125 | [7 -3⟩ |
| 1 | Jubilic | 50/49 | [1 2 -2⟩ |
| 2 | Superthird | 78125/76832 | [-5 7 -4⟩ |
| 3 | Waage | 244140625/240945152 | [-11 12 -6⟩ |
| 4 | Quintupole | 762939453125/755603996672 | [-17 17 -8⟩ |
| 5 | Undim | (32 digits) | [-23 22 -10⟩ |
| 6 | Term | (38 digits) | [-29 27 -12⟩ |
| 7 | 6 & 190 | (46 digits) | [35 -32 14⟩ |
| … | … | … | |
| ∞ | Didacus | 3136/3125 | [6 -5 2⟩ |
Graphs
Squared DKW error of temperaments in the equivalence continuum as a function of tuning of 5/4.
Squared DKW error of temperaments in the equivalence continuum as a function of tuning of 8/7.