User:Godtone

I'll be putting basically all my microtonal thoughts, theories and even some coding here. It'll increase in organisation as I add stuff and figure out how to prettify stuff.

Simple ratios and where I think limits should be drawn

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':
2/1, 3/2, 4/3, 5/4, 6/5, 7/6, 8/7, 9/8, 10/9, 11/10, 12/11, 13/12, 15/14, 17/16, 19/18.
I stopped at 19/18 because (19/18)/(21/20) = 380/378 = 190/189 which is again under 11 cents. Note that this also corresponds to the 19-odd-limit, a subset of the 19-prime-limit, with 169/168 and 190/189 tempered. From there, we can choose to temper the missing superparticulars (the ones with odd denominators) to any of the adjacent superparticulars. Note that I am considering all of these intervals as intervals to move upwards with; 16/15 is an interval that to me works better for going downwards as it implies 15/8 = (3/2)(5/4) when you measure it relative to an imagined octave below the initial tone, and notice that 15/8 is expressible in terms of simple existing superparticulars (and thus as is 16/15).

Categories

Categories are about distinctions: the number of distinctions made and how they relate to other distinctions. As such, I will list various distinctions here, from the simplest to the most complex I consider interesting and based on their number and hierarchy, giving my thoughts and opinions along the way:

1. If we assume octave equivalence (meaning multiplying or dividing all intervals by 2 until they are within the range of 1/1 to 2/1 and so that all intervals that differ only by powers of 2 are considered equivalent), then performing an octave complement creates an interesting and nontrivial relation between intervals, splitting the octave into 2 pieces: from 0c to 600c and from 600c to 1200c.
2. If we use an xLys MOSS (Moment Of Symmetry Scale) to create interval categories, the most basic interesting MOSS is one with 5 notes in an octave, and the most consonant is the Pythagorean 2L3s MOSS, which extends naturally to a 5L2s MOSS while maintaining reasonable consonance in the Dorian mode, which I consider to be 5L2s Pyth's maximally consonant mode due to all intervals (measured relative to the root) containing no more than 3 factors of 3 in the numerator or denominator. These correspond to unequal 5-tone and 7-tone scales, which can be well approximated in 12 EDO as a good trade-off between accuracy and tone-efficiency. I also consider 5 to 9 notes to be the optimal number of notes for scales or they start to sound too much like chords (4) or start to sound too "chromatic" (10) although this doesn't at all mean scales outside of this range aren't useful and arguably 4 and 10 are edge cases. Furthermore, there are some more symmetric 12 EDO scales of great note: the diminished scale (4L4s), an 8-note scale, and the whole tone scale (6 EDO), a 6-note scale. Considering the diminished scale has 4 EDO, a 4-note scale, as its core, this about covers most of the scale sizes, and hence much of 12 EDO music implicitly makes use of combinations of these scales. Note however that I believe that scales themselves are defined by their pieces just as much or even more than their pieces define the scales, AKA that the "mode" of a song is often somewhat of a myth; only songs written in modes truly have a "mode" as opposed to impressions of modes.
3. If we examine the traditional 7 interval categories in 5L2s diatonic scales, then each interval comes in two basic "types": major and minor. If we merge these types as in 7 EDO, then we get intervals neither major nor minor, thus named "neutral", which can often be approximated using the 11th harmonic. This gives 3 "types" per category. Furthermore, if we push the categories to their extremes, we get supermajor and subminor, which can often be approximated using the 7th harmonic. This gives 5 "types" per category. There is a subtlety about neutral intervals though that I consider important which is they generally sound more "minor" and "sad" than "major", and thus should generally be slightly flat relative to an "exact midpoint" between minor and major, which begs the question, what would be the fifth-complement of a "minor-ish neutral"? A "major-ish neutral"? If we use 11/9 as a standard example, its fifth complement is 27/22, but it'd be useful if there was a simpler - perhaps more consonant - interval we can use in its place, and there is! It differs from 16/13 by (11/9)(16/13)/(3/2) = 352/351 which gives a sharp 3/2, and thus provides a nice link to the 13th harmonic. Meanwhile, if distinguishing two types of neutrals is too precise, we can instead consider (27/22)/(11/9) = (11/9)^2/(3/2) = 243/242 which gives a flat 3/2, which would thus be compatible with meantone.
4. In the case that we consider 11/9 and 16/13 as approximate fifth-complements, this clearly creates quite a precise distinction, and we thus must add at least 2 more distinctions in the spectrum. There is quite a large gap between subminor and minor or between major and supermajor, so adding some middleground subtly different from 12 EDO minor and major would be fitting, as the most basic minor and major are traditionally 5-limit and are closer together than in 12 EDO. For this purpose we will consider 12 EDO on its much more accurate "19-limit" (and in that sense "novemdecimal") 2.3.17.19 subgroup, thus creating a rather familiar "noveminor" and "novemajor" (short for novemdecimal), which, at least in the case of thirds, can be equated with "Pyth minor" and "Pyth major" due to identification by (81/64)/(24/19) = 513/512 (with 24/19 as the fifth-complement of the harmonic minor third of 19/16), leaving two options for the expression of this category depending on which makes more sense for a temperament. Then we will consider - if needed, optional additional/finer categories between noveminor and subminor and between novemajor and supermajor. These "new" and "subtly exaggerated from familiar" categories I think fit with the prefix of "neo" and can be considered represented by 13/11 and 14/11 which again can be equated with a sharp 3/2 through identification by (13/11)(14/11)/(3/2) = 364/363 and again creates an interesting link between the 11th and 13th harmonics. Then even subtler versions of the usual major and minor categories can be added - subtle in the sense of closer to but distinct from neutral - these are supraminor and submajor. Finally, for completeness, even more extreme versions of subminor and supermajor can be added that push into the "neither major nor minor at all" territory; these are ultraminor and ultramajor. The final list looks like this:
   ultraminor, subminor, neominor, (pyth) noveminor, (classic) minor, supraminor, (minor or sub-)neutral, (major or super-)neutral, submajor, (classic) major, (pyth) novemajor, neomajor, supermajor, ultramajor.
5. Note that overall, I think while measuring intervals relative to 12 EDO is useful initially, this should not be the final way of measuring them. Instead, I believe different intervals should be considered like "colours" or "flavours", of which 12 EDO's intervals are (approximately) one type, and that these new terms (or whatever terms you prefer) should eventually be more natural a musical language than comparison to 12 EDO which causes a variety of intervals to be used similarly due to a 12 EDO mindset, rather than distinguishing them as unique categories not subservient to other categories. This also makes me more open to larger EDOs which provide more distinctions, however, I have relatively high standards for large EDOs, as a large number of tones is something that needs to be quite seriously justified in my opinion. NOTE: THIS IS A TEMPORARY EDIT OF THIS PAGE AND IS FAR FROM COMPLETE. THE BELOW IS A SCRAP/THE START OF AN EXPERIMENTAL LIST.
   1. "comma", "chroma": <35c
   2. "fifth-tone", "quarter-tone", "subminor semitone", "ultraminor second", "tiny tone": 35c to 60c (includes 21/20 and 19/18)
   3. "third-tone", "minor semitone", "subminor second", "mini-tone": 60c to 75c
   4. "small semitone", "supraminor semitone", "minor second": 75c to 85c
   5. "medium semitone" (or "minor semitone"):