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.

Temperaments in the continuum

n	Temperament	Comma
		Ratio	Monzo (2.5.7 subgroup)
-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.

Temperaments in the continuum

n	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⟩

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.