Talk:Marvel
Subheadings for scales
I suggest this although I realize that an extensive table of contents has a certain repellent effect on some readers. What do you think? --Xenwolf (talk) 09:42, 1 June 2021 (UTC)
Challenge on optimality of 53edo for FloraC
53edo is consistent in the 7-limited 105-odd-limit except for two interval pairs (50/49 and 75/49 and their octave-complements). Can any other edo tuning of marvel even come close to this faithful of a representation of the 7-limit lattice? 72edo does better with only one inconsistent interval pair in the 125-odd-limit (128/125 and its octave-complement, unsurprisingly), but it's optimized for different things than just pure marvel. Similarly 41edo does even better in terms of consistency but it's clearly more overtempered than 72edo. I also don't believe that the inconsistency of 50/49 and 75/49 are particularly important, except for the damage on 7/5 and 10/7 which as far as I can tell is the only real flaw of 53edo's marvel.
Here is how every edo up to and including 240 which tempers out 225/224 with a consistent 9-odd-limit performs in the 105-odd-limit, which seems the largest 7-limited odd-limit that is reasonable to consider because 125 is obviously gonna cause inconsistencies in most tunings as 5 is the most tempered prime in marvel cuz of 32/25 = 9/7 and 7/6 = 75/64 among others.
>>> for edo in range(1,241): # using https://en.xen.wiki/w/User:Godtone#My_Python_3_code
... if inconsistent_ivs_by_val(odd_lim(9),val(lim(7),ed(edo)))==[] and pval(S(15),ed(edo))==0:
... print(edo,'EDO:',', '.join([ striv(x) for x in inconsistent_ivs_by_val( odd_lim(9,[],[15,21,25,27,35,45,49,63,75,81,105]), val(lim(7),ed(edo)) ) ])+'\n')
...
12 EDO: 49/48, 49/45, 54/49, 81/70, 98/81, 60/49, 35/27, 64/49, 49/36, 72/49, 49/32, 54/35, 49/30, 81/49, 140/81, 49/27, 90/49, 96/49
19 EDO: 64/63, 49/48, 128/105, 49/40, 64/49, 49/32, 80/49, 105/64, 96/49, 63/32
22 EDO: 81/80, 50/49, 27/25, 25/21, 49/40, 100/81, 63/50, 81/64, 80/63, 98/75, 75/49, 63/40, 128/81, 100/63, 81/50, 80/49, 42/25, 50/27, 49/25, 160/81
29 EDO: 49/48, 36/35, 28/27, 25/24, 27/25, 49/45, 35/32, 54/49, 81/70, 75/64, 98/81, 128/105, 60/49, 100/81, 32/25, 35/27, 64/49, 49/36, 48/35, 112/81, 25/18, 36/25, 81/56, 35/24, 72/49, 49/32, 54/35, 25/16, 81/50, 49/30, 105/64, 81/49, 128/75, 140/81, 49/27, 64/35, 90/49, 50/27, 48/25, 27/14, 35/18, 96/49
31 EDO: 81/80, 81/70, 100/81, 81/64, 112/81, 81/56, 128/81, 81/50, 140/81, 160/81
41 EDO:
50 EDO: 81/80, 64/63, 50/49, 21/20, 27/25, 81/70, 32/27, 128/105, 49/40, 100/81, 63/50, 81/64, 80/63, 64/49, 21/16, 27/20, 112/81, 45/32, 64/45, 81/56, 40/27, 32/21, 49/32, 63/40, 128/81, 100/63, 81/50, 80/49, 105/64, 27/16, 140/81, 50/27, 40/21, 49/25, 63/32, 160/81
53 EDO: 50/49, 98/75, 75/49, 49/25
60 EDO: 64/63, 49/48, 36/35, 25/24, 35/32, 54/49, 75/64, 128/105, 49/40, 32/25, 64/49, 21/16, 49/36, 48/35, 45/32, 64/45, 35/24, 72/49, 32/21, 49/32, 25/16, 80/49, 105/64, 128/75, 49/27, 64/35, 48/25, 35/18, 96/49, 63/32
72 EDO:
82 EDO: 81/80, 36/35, 25/24, 27/25, 35/32, 54/49, 28/25, 81/70, 75/64, 25/21, 98/81, 128/105, 100/81, 63/50, 32/25, 35/27, 75/56, 48/35, 25/18, 36/25, 35/24, 112/75, 54/35, 25/16, 100/63, 81/50, 105/64, 81/49, 42/25, 128/75, 140/81, 25/14, 49/27, 64/35, 50/27, 48/25, 35/18, 160/81
84 EDO: 81/80, 49/48, 28/27, 49/45, 54/49, 81/70, 98/81, 60/49, 81/64, 35/27, 98/75, 49/36, 112/81, 81/56, 72/49, 75/49, 54/35, 128/81, 49/30, 81/49, 140/81, 49/27, 90/49, 27/14, 96/49, 160/81
91 EDO: 81/80, 64/63, 49/48, 16/15, 35/32, 75/64, 32/27, 128/105, 49/40, 81/64, 80/63, 32/25, 64/49, 21/16, 48/35, 45/32, 64/45, 35/24, 32/21, 49/32, 25/16, 63/40, 128/81, 80/49, 105/64, 27/16, 128/75, 64/35, 15/8, 96/49, 63/32, 160/81
94 EDO: 50/49, 25/24, 27/25, 28/25, 75/64, 25/21, 100/81, 63/50, 32/25, 98/75, 75/56, 25/18, 36/25, 112/75, 75/49, 25/16, 100/63, 81/50, 42/25, 128/75, 25/14, 50/27, 48/25, 49/25
113 EDO: 25/24, 35/32, 28/25, 75/64, 128/105, 32/25, 75/56, 48/35, 25/18, 45/32, 64/45, 36/25, 35/24, 112/75, 25/16, 105/64, 128/75, 25/14, 64/35, 48/25
125 EDO: 50/49, 49/45, 54/49, 28/25, 75/64, 98/81, 60/49, 56/45, 98/75, 75/56, 112/81, 81/56, 112/75, 75/49, 45/28, 49/30, 81/49, 128/75, 25/14, 49/27, 90/49, 49/25
144 EDO: 81/80, 64/63, 16/15, 35/32, 75/64, 32/27, 128/105, 56/45, 81/64, 32/25, 64/49, 75/56, 112/81, 45/32, 64/45, 81/56, 112/75, 49/32, 25/16, 128/81, 45/28, 105/64, 27/16, 128/75, 64/35, 15/8, 63/32, 160/81
166 EDO: 50/49, 25/24, 16/15, 15/14, 27/25, 49/45, 28/25, 75/64, 25/21, 128/105, 60/49, 56/45, 63/50, 32/25, 98/75, 75/56, 25/18, 45/32, 64/45, 36/25, 112/75, 75/49, 25/16, 100/63, 45/28, 49/30, 105/64, 42/25, 128/75, 25/14, 90/49, 50/27, 28/15, 15/8, 48/25, 49/25
197 EDO: 81/80, 64/63, 50/49, 28/27, 25/24, 16/15, 15/14, 49/45, 54/49, 28/25, 75/64, 32/27, 25/21, 98/81, 128/105, 60/49, 56/45, 81/64, 32/25, 98/75, 75/56, 112/81, 45/32, 64/45, 81/56, 112/75, 75/49, 25/16, 128/81, 45/28, 49/30, 105/64, 81/49, 42/25, 27/16, 128/75, 25/14, 49/27, 90/49, 28/15, 15/8, 48/25, 27/14, 49/25, 63/32, 160/81
--Godtone (talk) 21:57, 15 January 2025 (UTC)
Because the consistency argument may not be sufficiently convincing, here is optimal_edo_sequences (minimising the mean square cent error on the tonality diamond, with cent error deviations weighted by the square-root of the odd-limit of each interval, which is the most forgiving tuning fidelity that can be reasonable) for edos tempering out S15:
>>> odds = [k for k in range(1,125,2) if len(fact_int(k))<=4]
>>> odds
[1, 3, 5, 7, 9, 15, 21, 25, 27, 35, 45, 49, 63, 75, 81, 105]
>>> for i in range(3,len(odds)): # 0th odd is 1, 1st odd is 3, 2nd odd is 5, 3rd odd is 7
... print('7-limited '+str(odds[i])+'-odd-limit:',optimal_edo_sequence(odds[:i+1],[edo for edo in range(1,312) if pval(S(15),ed(edo))==0]))
...
7-limited 7-odd-limit: [2, 9, 10, 12, 19, 22, 31, 72, 103, 175, 228]
7-limited 9-odd-limit: [2, 9, 10, 12, 19, 31, 41, 53, 72, 125, 166]
7-limited 15-odd-limit: [2, 9, 10, 12, 19, 22, 31, 41, 53, 72, 125, 166]
7-limited 21-odd-limit: [2, 9, 10, 12, 19, 22, 29, 31, 41, 72, 113, 125, 166, 197]
7-limited 25-odd-limit: [2, 9, 10, 12, 19, 31, 53, 72, 84, 156, 240]
7-limited 27-odd-limit: [2, 9, 10, 12, 19, 31, 41, 53, 72, 125, 197, 281]
7-limited 35-odd-limit: [2, 9, 10, 12, 19, 31, 41, 53, 72, 125]
7-limited 45-odd-limit: [2, 9, 10, 12, 19, 31, 41, 53, 72, 125]
7-limited 49-odd-limit: [2, 9, 10, 12, 19, 22, 31, 41, 72, 197]
7-limited 63-odd-limit: [2, 9, 10, 12, 19, 22, 31, 41, 72, 197]
7-limited 75-odd-limit: [2, 9, 10, 12, 19, 22, 31, 41, 72, 197]
7-limited 81-odd-limit: [2, 9, 10, 12, 19, 31, 41, 53, 72, 113, 166]
7-limited 105-odd-limit: [2, 9, 10, 12, 19, 22, 31, 41, 53, 72, 113, 125, 166]
Notice that we haven't put any constraints on tempering or consistency (other than tempering out 225/224 by patent val) and 53edo still shows up everywhere except the 7-limited 49-, 63- and 75-odd-limit. 31 shows up everywhere simply by absence of good enough smaller competitors; same with 41edo though it disappears from the 7-limited 25-odd-limit due to the overflat 25. 72edo is very good as it appears everywhere. I also want to point out that 240edo is not only arguably too many notes for marvel but also only appears a single time! I really doubt that 240edo is optimal in any meaningful sense (except being a nice composite number of notes I guess) because it has 4 inconsistent interval pairs in the 9-odd-limit already (which is almost half of all interval pairs of the 9-odd-limit). By contrast, 166edo and 197edo both appear 5 times so appear to be well justified in terms of absolute error at least. 125edo is even better and interestingly disappears in practically the same places that 53 disappears: in the 49- to 75- 7-limited odd-limits (though 53edo reappears in the 81-odd-limit while 125 appears one later in the 105-odd-limit). 84edo only appears once but it appears in a theoretically notable odd-limit for marvel: the 25-odd-limit, which is notable for being challenging because of marvel's inclination to temper 5 significantly flat (and this is also where 41edo disappears), so IMO isn't so bad either because we know it satisfies the strict requirements and because it appears in the optimal_edo_sequence for all full odd-limits 23 thru 51 and appears in the strict_optimal_edo_sequence (meaning identical except instead based on relative error instead of absolute, so that the list is a strict subset) for a lot of those too, so that it's a natural tuning to consider for high-limit marvel.
--Godtone (talk) 22:52, 15 January 2025 (UTC)
A clarification on why the 7-limited 25-odd-limit is important to analyse for marvel: it is the smallest odd-limit which introduces a tempered equivalence within the interval set of the odd-limit other than the trivial ~16/15~15/14, and the 25-odd-limit has significantly higher tuning fidelity than anything in the 9-odd-limit; the square root of 25/9 is 5/3, so the tuning fidelity required is almost double even if we use the very forgiving (nonstrict) "square root of the odd-limit" as a weighting for cent error, and is almost thrice otherwise, or even more if you are concerned with pure dyadic convincingness. Therefore, an optimized marvel tuning must clearly tune closer to 32/25 than to 9/7, because there is no good reason that the musically useful augmented fifth ~25/16 should be discarded as a target given how naturally marvel extends a 5-limit lattice into the 7-limit, giving rise to things like the marveldene. There is also [[~]28/25~9/8 in the 7-limited 25-odd-limit but the usefulness of that seems more dubious, but it does show why ideally prime 3 should be tuned flat, hence systems like 72edo and 84edo.
(I think the one thing I do agree with though is that 16/15 is obviously undertempered in 53edo, but it seems to come about as a result of other considerations so I'm not fully sure it can be evaded because only a single 3 and 5 are involved. If you ask me, the smallest edo that is obviously "more optimized" (in terms of tuning) than 53edo for marvel is 125edo, its tuning profile looks about exactly correct as far as I ca tell. But that is over double the notes! I wouldn't dare add a single note more because already there is a lot of inconsistencies in higher 7-limited odd-limits as I've shown; I'll elaborate a little in the next (and final) post.)
--Godtone (talk) 23:42, 15 January 2025 (UTC)
So here's one important reason why I think probably only 125edo is a more optimized marvel tuning:
>>> [edo for edo in range(1,312) if pval(S(15),ed(edo))==0 and len(inconsistent_ivs_by_val(odd_lim(9,[],[15,21,25,27,35,45]),val(lim(7),ed(edo)))) <= 6]
[12, 19, 22, 31, 41, 53, 72, 84, 125] # all of these have no more than 3 inconsistent interval pairs in the 7-limited 45-odd-limit
(The list here is the same if we instead use "no more than 2 inconsistent interval pairs in the 7-limited 45-odd-limit".)
Also just intuitively its tuning profile looks about as optimized as could be for marvel, and regardless, the step size is getting close to being the same size as 225/224 so that by the time we get to 130edo and 140edo we clearly wanna detemper it.