User:Godtone: Difference between revisions

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=== Favourite EDOs ===

12, 13, 16, 17, 19, 20, 22, 24, 31, 32, 34, 36, 50, 53, 58, 68, 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/>

Favourite EDOs best to worst, not listed = even worse, my opinion obviously, also my opinions are still in development about many of these:<br/>

* 12: The musical language. Also the first reasonable approximation of [[Pythagorean tuning]].

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** 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.

: 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.

* 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.

* 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.

* 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 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.

* 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.

* 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.

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.

* 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.

* 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.

=== Philosophy ===

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 === Favourite EDOs ===
 , 13, 16, 17, 19, 20, 22, 24, 31, 32, 34, 36, 50, 53, 58, 68, 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/>
 Favourite EDOs best to worst, not listed = even worse, my opinion obviously, also my opinions are still in development about many of these:<br/>
 * 12: The musical language. Also the first reasonable approximation of [[Pythagorean tuning]].
@@ Line 80: / Line 81: @@
 ** 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.
+: 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.
+* 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.
 * 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.
+* 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 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.
+* 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.
+* 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.
+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.
+* 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.
+* 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.
 === Philosophy ===