Father–3 equivalence continuum/Godtone's approach
The augmented–dicot equivalence continuum is a continuum of 5-limit temperaments which equates a number of 128/125's (augmented commas) with the dicot comma (or chroma), 25/24. As such, it represents the continuum of all 5-limit temperaments supported by 3edo.
This formulation has a number of specific reasons:
- 128/125 is significantly smaller than 25/24, so that it makes sense to equate some number of 128/125's with 25/24.
- 128/125 is fundamental because it uniquely defines the relatively-very-accurate (strongly form-fitting) representation of the 2.5 subgroup in 3edo.
- 25/24 is fundamental because it gives the trivial way to relate ~5/4 = 1\3 to ~3/2 = 2\3 as 2 generators in 3edo. (By contrast, using 16/15 requires taking the octave-complement of one of the generators. There is also another stronger argument against using 16/15 detailed later in this list.)
- Using 25/24 is also useful because we then know how many intervals between ~6/5 and ~5/4 are guaranteed in a nontrivial tuning; because 25/24 is divided into n equal parts, the answer is n - 1. Meanwhile, if n = a/b is not an integer (meaning b > 1), then 25/24 is divided into a equal parts of ~(128/125)1/b, giving a clear meaning to the numerator and denominator (though more meanings are discussed later).
- 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. In other words, for natural n, the way to reach ~3/2 (up to octave equivalence) is always by flattening ((5/4)2 =) 25/16 (by n dieses) into 3/2, where flattening by a diesis is equivalent to multiplying by (5/4)3 (up to octave-equivalence).
- * Therefore, if n = a/b is a rational with b > 1 and b not a multiple of 3 (so that 3a/b + 2 doesn't simplify), we reach prime 3 in a fractional number of generators of ~5/4, which means that the generator is not ~5/4 but rather b equal divisions of some octave-equivalent of ~5/4 or ~8/5, which as a result means that we reach prime 3 in b(3a/b + 2) = 3a + 2b generators, and also means that ~128/125 is split into b equal parts.
- 16/15 = 25/24 * 128/125, so that tempering out 16/15 (father) is found at n = -1. The reason it shouldn't be found at n = 0 instead is because n = -1 has an absurdly sharp tuning of ~5/4 because of being equated with ~4/3, which breaks the pattern from dicot at n = 0 having an absurdly flat tuning of ~5/4 because of being equated with ~6/5, and from n = 0 onwards, ~5/4 is tuned increasingly sharp. This observation is important enough for its own point:
- For n a nonnegative integer, half-integer or third-integer(*), increasing n corresponds to increasingly sharp tunings of ~5/4. In the limit, as n goes to infinity, these all approach ~5/4 = 1\3, corresponding to augmented temperament.
- (* It is conjectured by User:Godtone that for a given choice of denominator b in n = a/b, a larger value of a always corresponds to a sharper tuning of ~5/4, where the sharpness in a pure-octaves tuning is always strictly flat of 1\3, so that (more trivially) taking the limit as a goes to infinity, ~5/4 = 1\3. The intuition for why we might expect this to be true is that in a pure-2's pure-3's tuning, we are always constraining ~128/125's size to be equal to the appropriate relation to ~25/24, where as 2 and 3 are fixed, the ~5 is the only free variable and depending only on n, with that n essentially indirectly specifying the degree of tempering.)
- Also, if one is interested in what intervals are present between ~5/4 and ~4/3, it is simple to observe because (16/15)/(25/24) = 128/125, meaning for a nonnegative integer n there is exactly one more interval between ~5/4 and ~4/3 as between ~6/5 and ~5/4. As there is n - 1 intervals between ~6/5 and ~5/4 (because of splitting ~25/24 into n parts), that means that (for nonnegative integer n) there is exactly n intervals between ~5/4 and ~4/3. More generally, for rational n = a/b, we have a - 1 intervals between ~6/5 and ~5/4 and because there is another ~128/125 between ~5/4 and ~4/3 we have a/b + 1 = a/b + b/b for the translated coordinates so that we have a + b - 1 intervals between ~5/4 and ~4/3, corresponding to splitting ~16/15 into a + b equal parts.
Therefore, if n = a/b is a rational with b > 1 and b not a multiple of 3 (so that 3a/b + 2 doesn't simplify), we reach prime 3 in a fractional number of generators of ~5/4, which means that the generator is not ~5/4 but rather b equal divisions of some octave-equivalent of ~5/4 or ~8/5, which as a result means that we reach prime 3 in b(3a/b + 2) = 3a + 2b generators, and also means that ~128/125 is split into b equal parts.
The just value of n is log(25/24) / log(128/125) = 1.72125… where n = 2 corresponds to the Würschmidt comma.
| 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 mentioned), 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.
| 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:
| 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 | Squarschmidt (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 squarschmidt 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 (7) indicates that 128/125 is split into 7 equal parts, while the numerator indicates that each (128/125)1/7 part represents (25/24)1/12, that is, a twelfth of 25/24.
| 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 | 147c & 150c = 291cc & 297cc (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.