Godtone

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This is maybe the obvious place to start. I listen to a variety of dyads in order to judge and try to absorb their qualities and to figure out if and why I like them. My opinions of intervals have changed over time. Anyway, as all positive rationals are ratios of positive naturals (nonzero everyday numbers), I think superparticular intervals are a good place to start. I think the melodic Just Noticeable Difference is important here so that intervals have a reasonable chance at being singable, even if the harmonic JND is significantly lower (partly depending on timbre). For me a reasonable upper limit on the melodic JND is about 11 cents as more than that and I hear something as pretty definitively mistuned, although that doesn't necessarily imply unusabibility as an approximation in a low-complexity system (one with a small amount of average tones per octave). This means that in the series of superparticular intervals (of the form (n+1)/n), the first two that are too close in size to be comfortably distinguished are 14/13 and 13/12, whose difference is 169/168 or about 10.274c.

I also think that powers of 2 in the denominator of an interval, broadly/generally speaking, helps the interval feel less disorienting due to a stronger suggestion of the fundamental, so beyond 13/12, for a bit, the superparticulars of the form (2n+1)/(2n) should be prioritised. This concludes at the following superparticular intervals being of particular (no pun intended) importance to a 'general melodic semi-harmonic system':<br/>

=== Favourite EDOs ===

DISCLAIMER: The reasonings for the EDOs I note here are guaranteed to be incomplete; EDOs are fundamentally deep systems and the more I've learned the more reasons I've found to appreciate the various EDOs I speak of here. Therefore keep in mind that whatever I say is a rude oversimplification scratching the surface of its possibilities and deep elegances. Also, I've kept my old entries and reasonings here as they were based on somewhat different ways of thinking about these things so I believe still have value; for example, I now generally quite dislike [[22edo]] (and [[archy]] generally) as an approximation to harmony but I admit there is a lot of interesting music in it and it is something a beginner should consider and which has proven value to beginners (both listeners and musicians).<br/>

EDOs < 12 not included as usually better conceptualised in a superset of that EDO and because otherwise I'd list too many consecutive EDOs.<br/>

* 12: [[Pythagorean tuning|Pythagorean]] [[Meantone]]: the musical language. From a circle-of-nths relative-consistency point of view, it is very strong in the 2.3.5.19(.17) subgroup. Not to be underestimated. Has melodic hints of the 7-limit through the inaccuracy of its 5. Has been called "the [[311edo|EDO chosen by God]]" by some - I'm definitely inclined to agree in the context of casual non-xen Western music.

* 13: Distorted 12. As such, almost xenharmonic by definition, due to maximising opportunities for alienness. The next good EDO after 12. Dreamy scales that I like a lot but I'm not sure about if that alone means they're good to use. I hope it does as 13 has huge potential if so.

* 16: The first interesting superset of 4 other than 12. Also a [[Pelogic_family#Mavila|mavila]] tuning, not that I like Mavila too much.

* 17: Notable as the first step up from 12 in colour palette. Good fifths that are slightly worse than in 12 but in the sharp direction. Kinda a bright feel. It took me a while to deduce this, but its harmonic magic lies in its 2.3.25.13.17/15(.23) subgroup, especially in the glorious neogothic/neopythagorean pentads afforded by [[fiventeen]] which is tuned excellently. Also if someone tells you 17 has ~11 ask them to prove it with harmonic examples.

* 20: The first EDO to have both the 5L5s and 4L4s symmetrical scales, and significant for that reason alone. Can sound quite atonal, however:

** Its 10 EDO subset has a very strong circle of [[16/13]]'s and [[15/14]]'s.

** [[10/9]] is approximated well by 3\20 and [[14/11]] is approximated well by 7\20. Has a flattish approximation of [[7/4]] and some higher (octave-reduced) harmonics but I don't think I'd use it to approximate those higher harmonics.

: This gives it the (additional) remarkable property that all its flavours of seconds are arguably consonant other than 1\20, which is arguably an augmented unison anyway.

* 26: Neat for having very good 8/7's and 10/9's, both flavours of major second that I very much appreciate (while 9/8 can get pretty bland). Basically the only tuning of [[flattone]] that I'd consider using as its about as big as a flattone system should be. Note that while 19 EDO is technically also flattone, it represents the border between sharper meantones and flattone, so I do not consider it a proper (in the sense of typical/representative) example of such (for example 19 EDO trying to combine the mappings of 7 as augmented sixth and diminished seventh results in [[49/48|S7]] being tempered, which is to me harmonically almost as implausible as tempering [[25/24]], so definitely an [[exotemperament]]/"troll temperament"). Furthermore, in this tuning of flattone the minor seconds are 13/12's, thus representing a near-equal diatonic such that the minor seconds are subneutral seconds. Has the benefit of extending 13 EDO into a larger and more complete colour set and conceptual framework, creating some truly xenharmonic and xenmelodic opportunities with flattone acting as a rough roadmap back to the more familiar things. Very nice model of the 2.7.11 subgroup; in my eye, it is to the 2.7.11 subgroup as 31 EDO is to the 2.5.7 subgroup.

* 31: The next EDO that melodically approximates the 11-limit, and considerably better. Extremely nice arrangement of intervals that feels weirdly intuitive and ideal. Colourful EDO. Basically ideal meantone tuning as more notes than this is overkill for meantone if you don't specifically want meantone.

* 32: 16 EDO with a sharp fifth. I like it primarily because of it being a power of 2. Exploration into this EDO could be interesting. 80 EDO offers a reasonably good approximation of it through a 16L16s MOSS.

* 36: Because of being a superset of 12, quite overlooked. It is actually a very good subgroup temperament! A natural extension of 12 EDO's colour palette, preferring to avoid the neutral and semi- intervals of 24 EDO. I should note though that while both 24 and 36 are reasonably good systems, I do not think they should be used together, as there are preferable EDOs in the high end range, such as 80 EDO.

* 50: The last meantone EDO that should ever be considered because it is the last EDO to consistently map 9/8 and 10/9 to the same step and because 81/80 is a rather large comma to temper at this scale and thus costs you a lot of accuracy. It is surprisingly consistent in the higher limits, and that it is quite composite is appealing to me, especially given that it is a superset of 10 EDO.

* 58: Weirdly consistent tuning with a nice selection of colours. Also supports the important 53&58 temperament [[Buzzard]]. Record in [[Pepper ambiguity]] in the 13- and 15-odd-limit. The first EDO to be consistent in the 17-odd-limit. I haven't looked at this EDO very closely but suspect it may have some surprisingly accurate/good approximations hiding under its slightly meh prime error profile. It supports hemipyth thru 16/13 = 11/9, a questionable equivalence but arguably 58 is the only EDO to make it work/make sense. I like its organisation of intervals/colours a lot.

* 68: Superset of 34 that enables the 7-prime-limit. Not too remarkable for that reason alone, however my interest in this EDO was increased when I deduced that it has a step size that is close to half the size of 49/48 meaning a 7/6, an 8/7 and a semifourth can all be distinguished with accuracy. For that reason, this EDO is important as an EDO around which other EDOs have the potential for a good selection of colours which approximate these 3 intervals of interest. It also performs well as a no-11's no-29's [[31-limit]] temperament, although it shares the idiosyncracy of 80 EDO of splitting the [[81/80|syntonic comma]] into two [[64/63]]'s.

* 72: [[catakleismic]] [[miracle]] [[octopus]] (among other things). If you want the [[11-limit]] in a finite number of pitches, look no further, but it even does well in the no-13's 17-limit, the full 17-limit and the full 19-limit (with a few inconsistencies in the lattermost case). Added convenience of being a superset of 12 EDO and a very composite EDO. The first true EDO to represent [[ennealimmal]] (as [[27edo]] only makes sense if you want to use superpyth and [[45edo]] is a trollish flattone tuning). The page for the keenanisma, [[385/384]], has some explanation for why this is a theoretically interesting comma for extending the [[7-limit]] to the [[11-limit]], for which 72 does very logically (among the many very logical things it does).

* 77: Very good (and elegant) high-limit system. See [[#Important 23-limit EDOs]]. I overlooked this one. Supports the rudely-named [[absurdity]] temperament, which is ironically very reasonable if not in some senses ideal for modelling higher-limit harmony.

* 311: If you asked God what his favourite EDO was, he would say [[311edo|311 EDO]]. It is almost unsettling how much of the harmonic series this EDO approximates well considering its comparatively small size. Very recommendable alternative to cents for low-complexity (in the sense of integer- or odd-limited) JI, as this EDO is not only consistent in the ''full'' 41-odd-limit, but ''many'' (mainly non-prime) odd harmonics greater than 41 can be added to the set without causing inconsistencies between them and other odd harmonics. I wonder if a precise JI harmonic series singer would implicitly target notes of 311 EDO in both singing and in their conceptualisation of JI. I find describing the prime subgroup interpretation of this EDO rather amusing, so here it is: 2.3.5.7.11.13.17.19.23.29.31.37.41.73.89.109.113. Note that as 89, 109 and 113 aren't as accurate as 73, so they could arguably be omitted because of their combination of complexity and inaccuracy. Fun fact: in Group Theory (a subfield of Abstract Algebra), excepting 37, all the primes up to and including 41 appear in the prime factorisation of the order of the Monster Group. The largest prime to appear in its factorisation is 71, the prime just before 73, which is the first prime after 41 that 311 EDO approximates well.

==== Generalised colours of supertonics, subtonics, leads and contraleads ====

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inframinor (AKA 'ultraminor') second aka one quarter-tone: (many things)

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* [[1L 6s]] I find significant as a 7-note scale underlying all nL1s scales for n>=7. I have named it "onyx", which has a variety of aesthetic reasonings for it: "1Ln-ic's" and "nL1-ic's (like, the -ic suffix applied to MOSS names, collectivised for 1Lns and nL1s) sounds like "one-el-en-ics" or "en-el-one-ics" which abbreviated sort of sounds like "one-ics" => "onyx". Then "onyx" sounds sort of like "one-six". Furthermore the onyx mineral comes in many colours and types, which seems fitting given this is the parent scale for a wide variety of MOSSes; specifically of interest being [[7L 1s]] (pine), [[8L 1s]] (subneutralic) and [[9L 1s]] (sinatonic). Finally, the name "onyx" is also supposed to be vaguely reminiscent of "anti-archaeotonic" as "chi" (the greek letter) is written like an "x" (this is related to why "christmas" is abbreviated sometimes as "X-mas") and other than that, the letters "o" and "n" and their sounds are also present in "archaeotonic", and "x" is vaguely reminiscent of negation and multiplication. There is also something like a "y" sound in "archaeotonic" in the "aeo" part (depending partially on your pronounciation).

| [[2L 7s]] || joanatonic || jo- || jo || From [[joan]] temperament.

|-

|-

| [[5L 4s]] || semiquartal || sequar- || seq || From ''half-fourth''.

== My Python 3 code ==

<syntaxhighlight lang="python">

>>> sg = [2, 3, 7, 11, 13, 17, 19]

def sub_iv(r1, r2):

return iv( r1[0]*r2[1] - r2[0]*r1[1], r1[1]*r2[1] )

def as_float(x, p=None):

if type(x)==list: # monzo (factored)

# returns [] if cs or details of earliest counterexample if not;

# "earliest" is defined as "search all 1-steps from beginning to end, then all 2-steps, etc."

stepmap = dict()

scale = scale_in.copy()

stepmap[interval] = j

return []

def rotate(scale,n,p=(2,1)):

print(edo,'EDO interpretation')

if not sg:

odds = [odd for odd in range(1,ol+1,2) if in_subgroup(odd,sg) and odd not in no] + add

sedo = 0

styler=lambda x,relerr: hyperlink(x) if abs(relerr) < considered_bad else dec+hyperlink(x)+dec[::-1]

for x in sorted( odd_lim(1,[],odds), key=lambda x: steps(x,ed(edo)) ):

relerr = steps(x,ed(edo)) - sedoofx

if sedoofx!=sedo:

# the default badness is the sum of the squares of the errors with each interval's contribution weighted proportional to its odd-limit complexity,

# in terms of absolute (cent/octave) error (corresponding to the multiplication by et2) and divided by the sum of the weightings of the intervals.

# if the weighting is unspecified, use the default of:

# this is a compromise between theoretic sensitivity (approx. equal to the square of the odd-limit complexity)

# and not underweighting simpler intervals and accounting for that more complex intervals will tend to be used

# in a harmonic/chordal context which justifies their otherwise-higher effective felt error through templating

global ivs_cache

global ivs_int_cache

# alternatively, feed the real number of divisions of the octave that generates your val for each val you want to compare.   

global patent_vals

if v in patent_vals:

v = patent_vals[v]

elif type(v)==float:

v = val(lim(255),ed(v))

# deduce the number of arguments in the badness function to allow only using the first n arguments as needed

num_args = badness.__code__.co_argcount

et2 = 1/v[0]

# finally, return the result:

# if you just give a list or set of intervals (a,b), the default behaviour is to judge the badness of an edo as:

# * sum of squares of errors with each interval's contribution weighted proportional to its odd-limit complexity,

# IMPORTANT: on Jan 9 i corrected rel_err**2 * et2 to rel_err**2 * et2**2 in et_badness which optimal_edo_sequence depends on;

#            strict_optimal_edo_sequence is unaffected however.

et_badness_judger = ivs_or_edo_badness

if type(ivs_or_edo_badness) in [int,set,list]: # user gave intervals (default et_badness)

ivs = ivs_or_edo_badness

# else user gave et_badness manually (custom)

best_edo = et_badness_judger(1)

for edo in edo_set:

current = et_badness_judger(edo)

best_edo = current

best_edos.append(edo)

# this gives much sparser but also much more interesting lists.

# default weighting of an interval x is proportional to its odd-limit complexity iv_complexity(x).

[[15edo]] gets 9 wrong in the 25-odd-limit

[[311edo]] gets 0 wrong in the 27-odd-limit

=== Isomorphic keyboard code for launchpads with programmer mode ===

if j!=0 or i!=0: # equiv. if (i,j) != (0,0) (i think)

steps = i*x - i//every_x*x_reducer + j*y - j//every_y*y_reducer

print()

# for the launchpad pro MK3 specifically: