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, as well as because superparticulars beyond 19/18 aren't really musically/melodically interesting to me. 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). Also note that this does not mean I think the 19-limit should be where we stop. I think if we apply the idea of prime limit to Just Intonation music, the 23-limit is a far better stopping point due to being just before a record prime gap, and as there are various intervals using the prime 23 and more generally using the 27-odd-limit that I like.

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 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 scales themselves are defined by their pieces just as much as 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 (9/8)^2/(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. Note that using the "nove(m)-" prefix for something that could be either "novemdecimal" or "Pyth" may be confusing and so I propose altering slightly to "novaminor" and "novamajor", using the analogy that novemdecimal and Pyth is exaggerated from classical major and minor and builds from both it and from itself, like a star creating new, stronger elements, and creating a "brighter" (as opposed to "sweet"/"solemn") sound to triads, and just as a star's nova is initially bright and fades over time, this "brightness" of Pyth/novemdecimal major/minor has faded over time due to familiarisation/desensitisation. Next, we will consider - if needed - optional additional/finer categories between novaminor and subminor and between novamajor 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:
   
   1. ultraminor
   2. subminor
   3. neominor
   4. novaminor
   5. (classic) minor
   6. supraminor
   7. subneutral (or "minor neutral" if you prefer)
   8. superneutral (or "major neutral" if you prefer)
   9. submajor
   10. (classic) major
   11. novamajor
   12. neomajor
   13. supermajor
   14. ultramajor
5. More specifically though, the order in which "types" are introduced is approximately:
   1. minor, major (2 types)
   2. minor, neutral, major (3 types)
   3. (ultraminor or) subminor, (supra)minor, (sub)major, supermajor (or ultramajor) (4 types)
   4. (ultraminor or) subminor, minor, neutral, major, supermajor (or ultramajor) (5 types)
   5. (ultraminor,) subminor, novaminor/neominor, minor, neutral, major, novamajor/neomajor, supermajor(, ultramajor) (7 (or 9) types)
   6. [same as 7 (or 9) types but with neutral split into subneutral & superneutral] (8 (or 10) types)
   7. [same as 8 (or 10) types but with either neo- & nova- distinguished (+2) or with supraminor & submajor added (+2)] (10 (or 12) types)
   8. [both neo- & nova- and supraminor & submajor distinguished] (12 (or 14) types)
   Note that in the above, ultraminor and ultramajor are not necessarily to be included as they represent the points at which the basic "minor vs major" distinction ceases to make sense and becomes unclear/too dependent on context, so they represent absolute limits. Also note that these categories are mainly centered on thirds and to a lesser extent seconds as I consider thirds and seconds to be the most important intervals in terms of providing musical meaning, however this doesn't mean other interval sizes should be ignored or taken less seriously.
6. To better understand what these interval types mean, study of a variety of intervals and their qualities is required. Due to the in-depth-ness of this study, this is something that is far from complete, and which I will dedicate its own large section to: the "Intervals" section. This study is also something that is subjective just as 12 EDO music is subjective, but that doesn't mean we can't try to blaze a trail for an approximate framework for the interpretation of microtonal music by trying to stick to some guiding principles.

Colourful EDOs

My above progression of "types"/"colours" can be used as a perhaps-interesting alternative to finding "good" EDOs for music by judging them not based on approximation of rationals of interest but instead based on their "colour palette"; not that these two methods are contradictory, and in fact I believe a combination of both is desirable. However, as a demonstration and a starting point, we will look at EDOs providing progressively more complex colour palettes, starting from a broad equalised 7-note approximation of the 5L2s diatonic scale (AKA 7 EDO) and considering only the 'seconds' and 'thirds' (and thus by octave complement, their inversions), with fourths and fifths not considered except to the extent that they should ideally not be too "out of tune", with "out-of-tune-ness" judged relative to approximating either 4/3 or 11/8 (but not both), as these are the simplest J.I fourths. Furthermore, we want the colour palettes to be generally symmetric about the "neutral" type (whether a system has a neutral type or not), so this excludes a large number of EDOs; this is intentional, as otherwise we would end up listing every EDO, and as it is a symmetry which I think is important or at least an interesting restriction.

1. 7 EDO is the simplest/"trivial" EDO as it provides only the (at times very approximate) "neutral" colour. Note that its fourths are very out-of-tune; this EDO is mainly included as a trivial case. This corresponds to "1 type".
2. 12 EDO is the next simplest as it provides (nova)major and (nova)minor seconds and thirds. Also a tone-efficient pure Pyth approximation so very good fourths. This corresponds to "2 types".
3. 15 EDO, in terms of colours, is similar to 12 EDO except that its minor third is a little sharper and that it now has 3 types of second which are roughly subminor, neutral and supermajor. The "subminor" and "supermajor" designations are used due to symmetry; in actuality the subminor second is closer to a neominor second and the supermajor second is closer to an ultramajor second. The prefixes may be omitted, or more exact colour terms may be used, making it have a "superneutral second". Note this EDO is more of an honourable mention due both to somewhat significant asymmetry and due to very off fifths, and also to explain why I won't include it in the "final" list..
4. 17 EDO is the first EDO to truly have minor, neutral and major for both seconds and thirds, and is thus quite significant as a potential next step up from 12 EDO. More exactly, these come in the flavours of neominor, neutral and neomajor.
5. 19 EDO is the next step up, having seconds and thirds of the ultraminor, minor, major and ultramajor varieties. It does this by conflating an ultramajor second as an ultraminor third, creating quite a distinct interval that escapes 5L2s categorisation. Note that 18 EDO does this too, but is less symmetric. Thus 19 (or 18 - if you are so inclined - which represents a sharpening of seconds and thirds) is the next step up as corresponding to "4 types".

Note that 17 and 19 have an interesting symmetry with each-other: while 17's new type is per 5L2s category and represents something between the types of major and minor, 19's new type is between 5L2s categories. Thus 17 respects diatonic interval categories more, which actually makes 19 the more novel system to me, for example I quite like the very distinct sound of the 5L4s "semaphore" scale; dyads and triads alike.
Furthermore, note that after this point I focus on EDOs with 'good enough' fourths.

1. 22 EDO is next, having subminor, supraminor, submajor and supermajor seconds and thirds, although only approximately, and in this spirit "supraminor and submajor" can be shortened to "minor and major" for brevity. 22 is distinct from 19 in that it does not have any of its types overlap between categories.
2. 24 EDO is next, being easy to categorise as a colour extension of 12, and thus it has seconds and thirds of the ultraminor, novaminor, neutral, novamajor and ultramajor varieties, with the ultramajor second equal to the ultraminor third.
3. 26 EDO is next and mirrors 22 in its thirds but has one more type of second, and so the seconds are approximately: subminor, minor, neutral, major, supermajor.
4. 29 EDO next, (approximately) with thirds of types ("fifth-tone" =) ~ultraminor, neominor, supraminor, submajor, neomajor, ("semifourth" =) ~ultramajor, and with seconds of types ("semifourth" =) ultraminor, neominor, supraminor, submajor, neomajor, ("semisixth" =) ulramajor.
5. 31 EDO is then quite ideal, as it gives us seconds and thirds of subminor, minor, neutral, major and supermajor types!

Beyond this point, it really becomes about narrowing down EDOs based on approximative or other considerations than just 'types'/'colours'.

Favourite EDOs

12, 13, 16, 17, 19, 20, 22, 24, 31, 32, 34, 36, 50, 53, 58, 68, 80, 87, 270, 311.
Favourite EDOs best to worst, not listed = even worse, my opinion obviously, also my opinions are still in development about many of these:

• 12: The musical language. Also the first reasonable approximation of Pythagorean tuning.
• 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 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.
• 19: Flattish meantone tuning. The semifourth in semaphore has a very neat sound but I wouldn't say it approximates the 7-limit. If anything, 19 is 2.3.5.37 with it representing a circle of 37/32's, thus also being the first good approximation of the 2.37 subgroup, and thus of 37/32, which represents probably my favourite interval of 19.
• 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 a strong circle of 15/14's.
  • Its 5 EDO subset has a strong (and remarkably small) circle of 23/20's.
  • Its 4 EDO subset has a strong (and remarkably small) circle of 19/16'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.

• 22: The first EDO that melodically approximates the 11-limit, and very tone efficient for that purpose. Sounds harmonically complex. Superpyth + Orwell tuning. Not a fan of porcupine to be honest.
• 24: I think neutral intervals and semifourths are kinda cool and unexpected root movement is cool, so acts as a nice stepping stone into microtonality with a strong base of familiarity to build off of. Alternate tuning for semaphore. I also include it because I like highly composite EDOs, and this is very clearly one. Represents the 2.3.11.19.37 subgroup particularly well.
• 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.
• 34: The first good approximation of the 5-prime-limit due to being the first reasonably accurate tuning of Hanson AKA kleismic. 19 is also a tuning for kleismic but feels like it doesn't do justice to the accuracy and pristineness of kleismic to me. Has the sharp 3/2's of 17 EDO, and as 17 EDO is a good colour system, 34 EDO is a natural extension.
• 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, 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.
• 53: Catakleismic Pythagorean Orwell. If that description doesn't sound epic I don't really know what will. Very colourful EDO. Near-perfect 5-limit JI with good 7-limit, passable 11-limit through Orwell and good no-17's 19-limit. Normally I wouldn't like large prime EDOs but this is a rare exception as in this case it's a practically perfect representation of the 2.3 subgroup.
• 58: Weirdly consistent tuning with a nice selection of colours. 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.
• 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.
• 80: My favourite EDO. My previous favourite was 53 EDO. 80 EDO may be a surprising choice for favourite at first but there are a lot of reasons feeding into it which also make it unlikely to become my second favourite any time soon. I will write in depth about it and about my theories for microtonal music based on 80 EDO in the future.

Philosophy

Firstly, this will obviously be heavily influenced by my opinions, so I may state subjective things or theories in a rather matter-of-fact way, but feel free to disagree with me/provide criticisms on my user page. Having said that... 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 (corresponding generally to novaminor and novamajor), and that these new terms (or whatever terms you prefer) should eventually be more natural a musical language than comparison to 12 EDO, which often causes a variety of intervals to be used similarly rather than distinguishing them as unique categories not subservient to (but related to) other categories. This is primarily due to a stark lack of theory in the musical and especially emotional meanings of these new intervals. This also makes me more open to the idea of using larger EDOs - which provide more distinctions - as frameworks in which many colours are possible, however, I have relatively high standards for large EDOs, as a large number of tones is something that needs to be quite seriously justified. For example, I generally dislike prime or not-very-composite EDOs unless they have exceedingly pure intervals and/or are very remarkable for other reasons. Sub-EDOs or extremely pure intervals in an EDO generally help to give it a feeling of griddedness and thus orientation in. I think the most important takeaway from 12 EDO is the effectiveness of simplicity and that you shouldn't underestimate how much can be built and how much nuance can be achieved by carefully combining simple things, and that the most objective possible interpretations of music come from the aggregation and combination of many patterns. This also leads me to believe scales are interpreted in terms of their pieces more than vice versa, as aforementioned.
Secondly, I think in new and unfamiliar territory, it will be helpful to use symbolism and analogies in order to get a rough initial understanding for potential emotional/symbolic meanings as the usual music theory isn't enough.

Number symbolism

1 : Unity, simplicity & wholeness (obviously). The ultimate source/root. The Formless. The first Form.
2 : Duality (obviously), contrast, distinction; the first prime number, and thus the first pure essence (assuming we don't count 1 as a pure essence). The source of the most basic/primitive/trivial kinds of order.
3 : Trinity (obviously) and stability - due to each element helping define and stabilise the other two in relation to each-other; think of how the triangle is the strongest shape for scaffolding. Thus represents a place of completeness and strength, and sometimes, rest. (It is a pure essence due to being prime.)
4 : The tetrahedron (AKA 3-simplex) has 4 vertices/corners, 4 sides/faces and 6 edges/connections with 3 edges/connections per vertex/corner. Thus represents the 3 spatial dimensions or the 3 dimensions of space paired with 1 dimension of time. The tetrahedron is a strong shape due to being made of 4 triangles, thus 4 similarly represents a place of completeness, strength and sometimes rest, but can be spiky/hazardous partly due to containing repeated duality (this applies (at least to small extent) to all numbers with powers of 2 in their prime factorisations) and thus pitfalls/negations of the ideal (as duality creates negation). Thus represents the Earth/its likeness (AKA the material world).
5 : The number of sides of a square-based pyramid, which has 4 triangular sides and thus similar stability. 6 square based pyramids form a cube which tiles 3D space, and where squares themselves tile 2D space. Additionally, pentagons are the last 2D regular convex polygon for which you can join 3 to each-other and get something 3D, and thus 5 is also related to the dodecahedron, and thus the number 12. Also the 4 points of a tetrahedron with an added/unifying centre, thus representing Earthly dualities that have had their pitfalls tempered through unification through an ideal, or alternatively a triangle-based pyramid with its highest corner projected onto the ground, thus again representing construction upwards to something better, using what is available . Thus represents the pure essence of The Transcendent's grace on the Earth, AKA the first/simplest form of transcendence from/beyond the Earth (as it is 4+1). (It is a pure essence due to being prime.)

Intervals

For now, I will leave tables for the interval/colour/type namings of the seconds and thirds of my two favourite highly-chromatic EDOs, both of which related to 17-limit Tolermic due to this family tempering many commas I am interested in tempering, and due to the resulting intervals providing a good framework for thinking about interval colours:

80 EDO interval colours/types (seconds and thirds)

 45c ultraminor (AKA ~quarter-tone)
 60c subminor (AKA ~third-tone)
 75c neominor
 90c novaminor
105c minor
120c supraminor
135c subneutral (AKA minor neutral)
150c superneutral (AKA major neutral)
165c submajor
180c major
195c novamajor
210c neomajor
225c supermajor
240c ultramajor
255c ultraminor (AKA ~semifourth)
270c subminor
285c neominor
300c novaminor
315c minor
330c supraminor
345c subneutral
360c superneutral
375c submajor
390c major
405c novamajor
420c neomajor
435c supermajor
450c ultramajor (AKA ~semisixth)

87 EDO interval colours/types (seconds and thirds)

 41.4c fifth-tone
 55.2c ultraminor (AKA quarter-tone)
 69.0c subminor (AKA third-tone)
 82.8c neominor
 96.6c novaminor
110.3c minor
124.1c supraminor
137.9c subneutral (AKA minor neutral)
151.7c superneutral (AKA major neutral)
165.5c submajor
179.3c major
193.1c novamajor
206.9c neomajor
220.7c supermajor
234.5c ultramajor
248.3c semifourth
262.1c ultraminor
275.9c subminor
289.7c neominor
303.4c novaminor
317.2c minor
331.0c supraminor
344.8c subneutral
358.6c superneutral
372.4c submajor
386.2c major
400.0c novamajor
413.8c neomajor
427.6c supermajor
441.4c ultramajor
455.2c semisixth