Father–3 equivalence continuum/Godtone's approach

< Father–3 equivalence continuum

The augmented–chromatic equivalence continuum is a continuum of 5-limit temperaments which equates a number of 128/125's (augmented commas) with the chroma, 25/24. As such, it represents the continuum of all 5-limit temperaments supported by 3edo.

This formulation has a specific reason: 128/125 is significantly smaller than 25/24, so that it makes sense to equate some number of 128/125's with 25/24, but because 25/24 = (25/16)/(3/2), this has the consequence of clearly relating the n in (128/125)ⁿ = 25/24 with how many 5/4's are used to reach 3/2 (when octave-reduced):

If n = 0, then it takes no 128/125's to reach 25/24, implying 25/24's size is 0 (so that it's tempered out), meaning that 3/2 is reached via (5/4)².

For integer n > 0, we always reach 25/24 via (25/16)/(128/125)ⁿ because of (128/125)ⁿ ~ 25/24 by definition, meaning that we reach 3/2 at 3n + 2 generators of ~5/4, octave-reduced.

The just value of n is log(25/24) / log(128/125) = 1.72125… where n = 2 corresponds to the Würschmidt comma.

Temperaments with integer n

n	Temperament	Comma
		Ratio	Monzo
−2	Smate (14 & 17c)	2048/1875	[11 -1 -4⟩
−1	Father (5 & 8)	16/15	[4 -1 -1⟩
0	Dicot (7 & 10)	25/24	[-3 -1 2⟩
1	Magic (19 & 22)	3125/3072	[-10 -1 5⟩
2	Würschmidt (31 & 34)	393216/390625	[17 1 -8⟩
3	Magus (43 & 46)	50331648/48828125	[24 1 -11⟩
4	Supermagus (55 & 58)	6442450944/6103515625	[31 1 -14⟩
5	Ultramagus (67 & 70)	824633720832/762939453125	[38 1 -17⟩
…	…	…	…
∞	Augmented (12 & 15)	128/125	[-7 0 3⟩

Notice that as n increases, we temper ~5/4 sharper and ~128/125 flatter (closer to unison), so that as n goes to infinity, ~5/4 goes to 1\3.

Temperaments with half-integer n aka
temperaments with a gen of ~sqrt(8/5)*

n	Temperament	Comma
		Ratio	Monzo
-1/2	Yo (2c & 5c)	10/9	[1 -2 1⟩
1/2	Wesley (26 & 29)	78125/73728	[13 2 -7⟩
3/2	Ditonic (50 & 53)	1220703125/1207959552	[-27 -2 13⟩
5/2	Novamajor** (77 & 80)	19791209299968/19073486328125	[41 2 -19⟩
7/2	3 & 101	(36 digits)	[55 2 -25⟩

* This corresponds to the denominator of 2 implying that 3 must be reached in a half-integer number of ~5/4's; the octave-complement of the generator is equal to ~sqrt(5/2).

** Note that "novamajor" (User:Godtone's name) is also called "isnes"; both names are based on the size of the generator being around 405 cents, but "isnes" was discovered as a point in the continuum while "novamajor" was discovered as one temperament in the fifth-chroma temperaments.

If we approximate the JIP with increasing accuracy, (that is, using n a rational that is an increasingly good approximation of 1.72125...) we find these high-accuracy temperaments:

Microtemperaments with fractional n

n	Temperament	Comma
		Ratio	Monzo
5/3	Mutt (84 & 87)	mutt comma	[-44 -3 21⟩
12/7	202 & 205 = 3 & 612	(70 digits)	[-105 -7 50⟩
7/4	3 & 118	(42 digits)	[61 4 -29⟩

The simplest of these is mutt which has interesting properties discussed there. In regards to mutt, the fact that the denominator of n is a multiple of 3 tells us that it has a 1\3 period because it's contained in 3edo. The fact that the numerator is 5 tells us that 25/24 is split into 5 parts. From (128/125)ⁿ = 25/24 we can thus deduce that each part is thus equal to ~cbrt(128/125) = (128/125)^1/3, so that ~5/4 is found at 1\3 minus a third of a diesis, so that ~125/64 is found at thrice that. This observation is more general, leading to consideration of temperaments of third-integer n.

The 3 & 118 microtemperament is at n = 7/4. Its generator is approximately 397 ¢ so that four generators reaches 5/2, corresponding to the denominator of 4. The number of generators of ~(5/2)^1/4 needed to find prime 3 is thus four times the result of plugging n = 7/4 into 3n + 2 , which is 3(7/4) + 2 = 21/4 + 8/4 = 29/4, that is, 29 generators.

Finally, the 3 & 612 microtemperament at n = 12/7 is extremely complex, because to find prime 5, you need 7 times 3(12/7) + 2 = 36/7 + 14/7 = 50/7, that is, 50 generators, and is noted only because of being extremely close to the JIP and being supported by the 5-limit microtemperament 612edo. The denominator of 7 indicates that 128/125 is split into 7 equal parts.

Temperaments with third-integer n
aka temperaments with 1\3 period

n	Temperament	Comma
		Ratio	Monzo
-2/3	32/27 (3 & 3c) (generator = father comma)	32/27	[5 0 -3⟩
-1/3	9c & 12c (generator = negative dicot comma)	125/108	[-2 -3 3⟩
1/3	33c & 36c (generator = ~dicot comma)	1953125/1769472	[-16 -3 9⟩
2/3	48 & 51 (generator = negative ~magic comma)	244140625/226492416	[-23 -3 12⟩
4/3	72 & 75 (generator = ~magic comma)	3814697265625/3710851743744	[41 2 -19⟩
5/3	Mutt (84 & 87) (generator = ~Würschmidt's comma)	mutt comma	[-44 -3 21⟩
7/3	108 & 111 (generator = negative ~Würschmidt's comma)	(38 digits)	[58 3 -27⟩
8/3	120 & 123 (generator = ~magus comma)	(42 digits)	[65 3 -30⟩
10/3	120 & 123 (generator = negative ~magus comma)	(52 digits)	[79 3 -36⟩
11/3	156c & 159c (generator = ~supermagus comma)	(56 digits)	[86 3 -39⟩

Notice the alternating pattern of comma offsets from 1\3, where those commas are themselves in the pattern present in the continuum of integer n.

Also notice that we always find ~5/4 in terms of 1\3 minus the generator, which is a tempered version of the aforementioned comma offset, which is either positive or negative, and that as n grows, the generator becomes smaller so that ~5/4 becomes sharper.