User:Godtone/Augmented-chromatic equivalence continuum
This page details User:Godtone's subjectively ideal version of the page for the continuum of 5-limit temperaments supported by 3edo.
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 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 | ||
| 0 | Dicot (7 & 10) | 25/24 | [-3 -1 2⟩ |
| 1/2 | Wesley (26 & 29) | 78125/73728 | [13 2 -7⟩ |
| 1 | Magic (19 & 22) | 3125/3072 | [-10 -1 5⟩ |
| 4/3 | 72 & 75 (generator = ~magic comma) | 3814697265625/3710851743744 | [41 2 -19⟩ |
| 3/2 | Ditonic (50 & 53) | 1220703125/1207959552 | [-27 -2 13⟩ |
| 5/3 | Mutt (84 & 87) (generator = ~Würschmidt's comma) | mutt comma | [-44 -3 21⟩ |
| 2 | Würschmidt (31 & 34) | 393216/390625 | [17 1 -8⟩ |
| 7/3 | 108 & 111 (generator = negative ~Würschmidt's comma) | (38 digits) | [58 3 -27⟩ |
| 5/2 | Novamajor (77 & 80) | 19791209299968/19073486328125 | [41 2 -19⟩ |
| 8/3 | 120 & 123 (generator = ~magus comma) | (42 digits) | [65 3 -30⟩ |
| 3 | Magus (43 & 46) | 50331648/48828125 | [24 1 -11⟩ |
| 7/2 | 101 & 104c | (36 digits) | [55 2 -25⟩ |
| 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 of half-integer n correspond 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 thus equal to ~sqrt(5/2) for these temperaments.
Temperaments of third-integer n correspond to an 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 for these temperaments 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.
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 a sequence of increasingly accurate temperaments n = 2, 5/3, 7/4, 12/7:
| n | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| 3/2 | Ditonic (50 & 53) | 1220703125/1207959552 | [-27 -2 13⟩ |
| 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⟩ |
| 2 | Würschmidt (31 & 34) | 393216/390625 | [17 1 -8⟩ |
The simplest of these other than Würschmidt 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. (Note that ditonic at n = 3/2 is included as an alternative approximation of n = ~1.7... as it finds relevance in 53edo, whose 5-limit is exceptionally accurate for its note count, but also because its increased complexity relative to Würschmidt allows it to spread damage over more generators.)
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.