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Godtone

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→Favourite EDOs: try to make this section less out of date
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=== Favourite EDOs ===
=== 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/>
12, 13, 16, 17, 19, 20, 22, 24, 26, 31, 32, 34, 36, 50, 53, 58, 68, 72, 80, 87, 270, 311.<br/>
12, 13, 16, 17, 19, 20, 22, 24, 26, 31, 32, 34, 36, 50, 53, 58, 68, 72, 80, 87, 270, 311.<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/>
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/>
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* 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.
* 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.
* 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.
* 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.
* 19: Flattish/solemn [[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.
* 19: Flattish/solemn [[meantone]] tuning with xenmelodic potential. 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:
* 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.
** Its 10 EDO subset has a very strong circle of [[16/13]]'s and [[15/14]]'s.
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** [[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.
** [[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.
: 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.
* 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.
* 22: The first EDO that melodically approximates the 11-limit, and very tone efficient for that purpose. Sounds harmonically complex. [[Superpyth]] + [[Orwell]] tuning. Really not a fan of porcupine; it's an [[exotemperament]] (AKA "troll temperament") IMO.
* 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.
* 24: I think neutral intervals and semifourths are kinda cool as an addition 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 if what you are looking for is the ''microtonal''. But I wouldn't recommend it to a beginner as there are more approachable systems that offer a healthier introduction to microtonality and (especially) xenharmony, such as (especially?) 31 EDO. I also include it because I like highly composite EDOs, and this is very clearly one. Represents the 2.3.11.17.19.31.37 subgroup particularly well.
* 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. 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.
* 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.
* 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.
* 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.
* 34: The first good approximation of the 5-prime-limit due to being the first reasonably accurate tuning of [[Kleismic family|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.
* 34: The first good approximation of the 5-prime-limit due to being the first reasonably accurate tuning of [[Kleismic family|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. Also is a very logical "completion" of 17 due to giving a very logical 2.3.5.13.17(.23)-subgroup interpretation of the sqrt(2).sqrt(3) subgroup with some really intriguing possibilities. If you're lacking in inspiration and its wide array of supported MOSSes aren't inspiration enough, try taking a look at the diaschismic-tetracot continuum ([[2048/2025]])<sup>n</sup> / ([[20000/19683]]).
* 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.
* 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.
* 37: Truly an excellent no-3's [[13-limit]] system. Ridiculously overlooked just because of its not so great approximation of prime 3 (which is at least [[5L 2s|diatonic]] and sounds convincing enough especially in context). A logical system for building on it is [[111edo]] which keeps this 13-limit mapping (but improves the 3).
* 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.
* 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.
* 53: [[Kleismic family#Catakleismic|Catakleismic]] [[Pythagorean tuning|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 [[3-limit|2.3]] subgroup.
* 53: [[Kleismic family#Catakleismic|Catakleismic]] [[Pythagorean tuning|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 [[3-limit|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.
* 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.
* 65: Very cool. Underappreciated. Good [[nestoria]] + [[wurschmidt]] tuning, but more importantly, it is in some surprisingly exact sense the "dual" to what 53 EDO's schismic offers harmonically, including the fact that its subgroup is larger and involves larger primes at the cost of some accuracy (depending on how strict you wanna be about which primes you consider approximated for the purposes of interpreting harmony). Also has a very cool superset, [[130edo]], but I implore people to explore what 65 EDO has to offer first, being 5 * 13 with lots of cool xen stuff deriving from the implications of that.
* 72: ([[catakleismic]]) [[miracle]] [[Ragismic_microtemperaments#Octoid|octoid]] [[hemiennealimmal]] temperament, and thus a very nice [[11-limit]] (and to a lesser extent (due to inaccuracy) [[13-limit]]) temperament with the added convenience of being a superset of 12 EDO and a very composite EDO. The first true EDO to represent [[ennealimmal]], and better yet, it extends it into the 11-limit well. Note it can also be described as [[385/384|keenanismic]] hemiennealimmal, and 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]].
* 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.
* 80: My favourite EDO. In the past, my 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. Tunes [[Tolermic family|17-limit Tolermic]], a strange temperament which tempers many commas I'm interested in tempering.
* 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).
* 87: A good approximation of the 13-limit, with the 5-limit also good, and an alternative Tolermic tuning, so it's closely related to 80 EDO but with better fifths and harmonic sevenths. Compared to 80 EDO, 7/4 is still the worst prime but lower in both absolute and relative error, and it is tuned flatly instead of sharply. The final and most colourful EDO, but 80 EDO is more than enough colours for me. Has an interesting conceptualisation as 29 EDO representing an approximate 2.3 subgroup with 5, 7, 11 and 13 all being 1\87 flat of the 29 EDO circle, providing an elegant model of navigation. 29 EDO is itself not bad as something that sounds like a brighter 12 EDO, but it feels more elegantly and interestingly conceptualised in this superset.
* 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.
Beyond 99 EDO (which is interesting in its own right, especially the 99 ED4 subset) I don't see much point for using an EDO as opposed to JI, with the exception of 2 truly exceptional EDOs which may be used for simplified models of JI itself.
* 80: My former favourite EDO. In the past, my favourite was 53 EDO. Now I am leaning again to 53 EDO as my favourite but honestly there are so many seriously good EDOs that it feels unfair to single one out. 80 EDO may be a surprising choice for former favourite at first but there are a lot of reasons feeding into it which means if you don't get it it means you're probably underestimating it and haven't looked closely enough (I don't mean this as some way of trying to impose my opinion; there is a lot of exceptional properties that 80 EDO is hiding of many natures). Tunes [[Tolermic family|17-limit Tolermic]], a strange temperament which tempers many commas I'm interested in tempering; in fact there is a strange intuitiveness to 80 EDO's tempering. The highly complex 80&311 temperament [[superlimmal]] is of note as being essentially a no-31's 37- or 41-limit temperament.
* 270: At the moment I don't really have anything to add which isn't already on [[270edo|the page for 270 EDO]]. Ridiculously strong approximation of the 11 and 13 prime limits and with the nice property of being very composite. However, I don't take too much interest in it as, at this scale, I prefer higher prime limits than 13.
* 87: A good approximation of the 13-limit, with the 5-limit also good, and an alternative Tolermic tuning, so it's closely related to 80 EDO but with better fifths and harmonic sevenths. Compared to 80 EDO, 7/4 is still the worst prime but lower in both absolute and relative error, and it is tuned flatly instead of sharply. My former "final and most colourful EDO", but 80 EDO is more than enough colours for me in that it covers all the colours I'd want from 87 and in ways I prefer and find more intuitive. Has an interesting conceptualisation as 29 EDO representing an approximate 2.3 subgroup with 5, 7, 11 and 13 all being 1\87 flat of the 29 EDO circle, providing an elegant model of navigation. 29 EDO is itself not bad as something that sounds like a brighter 12 EDO, but it feels more elegantly and interestingly conceptualised in this superset.
* 111: an absurdly elegant system from a tempering perspective in the sheer wealth and intuitiveness of equivalences it affords you; so much so that it sacrifices tuning accuracy for temperamental beauty and efficiency. One of a kind EDO.
* 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.
* 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.


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