User:FloraC/Fumica's edo impressions - Revision history

FloraC: Implement a few fixes

2026-08-30T09:48:28Z

Implement a few fixes

← Older revision		Revision as of 09:48, 30 August 2026
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	* 34 – This is to 17edo what 24edo is to 12edo. While 17edo is often good enough, this offers some more sophisticated solutions such as tetracot. Even the harmonics 7 and 11, which come from 17edo and are commonly cited as relatively poor in this edo, are convincing enough to me, since when I worked with modus I never had a problem with the intonation at all, unlike with porcupine. The sound is better than the structure. B-tier.		* 34 – This is to 17edo what 24edo is to 12edo. While 17edo is often good enough, this offers some more sophisticated solutions such as tetracot. Even the harmonics 7 and 11, which come from 17edo and are commonly cited as relatively poor in this edo, are convincing enough to me, since when I worked with modus I never had a problem with the intonation at all, unlike with porcupine. The sound is better than the structure. B-tier.
	* 35 – Potentially useful as every other step of 70edo. D-tier.		* 35 – Potentially useful as every other step of 70edo. D-tier.
	* 36 – This edo is more difficult to use than it appears. The idea of adding sixth tones to plain 12edo music might be interesting, but I generally find them to sound forced. Added to this fact is that I think of 12edo as a complete 7-limit system, so the alternative mapping of harmonic 7 only serves to create mental dissonance. C-tier.		* 36 – This edo is more difficult to use than it appears. The idea of adding sixth tones to plain 12edo music might be interesting, but I generally find them to sound forced. Added to this fact is that I think of 12edo as a complete 7-limit system, so unfortunately the alternative mapping of harmonic 7 only serves to create mental dissonance for me. C-tier.
	* 37 – This edo offers good approximation to harmonics 5, 7, 11 and 13. The last two primes ~~adds~~ the much needed variety to the otherwise pretty dull sound of the 2.5.7 subgroup. C-tier.		* 37 – This edo offers good approximation to harmonics 5, 7, 11 and 13. The last two primes add the much needed variety to the otherwise pretty dull sound of the 2.5.7 subgroup. C-tier.
	* 38 – This is to 19edo what 24edo is to 12edo. On paper it adds decent approximation to harmonics 11, 17, and 19, but in practice I never had a situation where I felt I needed these additional notes when working with 19edo. C-tier.		* 38 – This is to 19edo what 24edo is to 12edo. On paper it adds decent approximation to harmonics 11, 17, and 19, but in practice I never had a situation where I felt I needed these additional notes when working with 19edo. C-tier.

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	=== Comma-size ===		=== Comma-size ===
	* 39 – 1/5-comma quasisuper, most notable for tuning the minor second to 28/27 (with the 39d val) which I feel is where the tension peaks for voice leading. Such an overlooked system despite the similarity to 27edo in many ways. B-tier.		* 39 – 1/5-comma quasisuper, most notable for tuning the minor second to 28/27 (with the 39d val) which I feel is where the tension peaks for voice leading. Such an overlooked system despite the similarity to 27edo in many ways. B-tier.
	* 41 – The first of the five essential comma-level edos, and the first edo to achieve 9-odd-limit distinction and consistency. This is most significant for providing three flavors for each chromatic category: classical, Pythagorean, and septimal. In this case it is a schismic and septischismic system, so that all three kinds are separated by the same comma step and can be found on a ~~stack~~ of fifths. The comma step is somewhat larger than just, making the differences more pronounced, which is part of why I think this edo is pretty deep – the step isn't only a comma, but many things at once, including but not limited to the septimal dieses, as well as the chroma of the archaeotonic scale, the scale of Tetracot[7]. The best subgroup of this edo is apparently 2.3.5.7.11.19.29.31 thanks to the unique uniform tunings for the harmonic segments 18::22 and 27::33 – a fact related to the vanish of 100/99 (S10) and 243/242 (S9/S11) – but prime 13 is just as plausible. I consider it the first edo to deal with the all-important 2.3.5.7.11.13.19.29 subgroup. The beauty of this edo goes beyond the structure, but to the intonation: it has a very slightly sharp 3 and a more noticeably flat 5, making a flat, more stable 15; that is ideal for my music. Finally, it's an ideal tuning for the magic temperament. I can't compliment it enough. S-tier.		* 41 – The first of the five essential comma-level edos, and the first edo to achieve 9-odd-limit distinction and consistency. This is most significant for providing three flavors for each chromatic category: classical, Pythagorean, and septimal. In this case it is a schismic and septischismic system, so that all three kinds are separated by the same comma step and can be found on a chain of perfect fifths. The comma step is somewhat larger than just, making the differences more pronounced, which is part of why I think this edo is pretty deep – the step isn't only a comma, but many things at once, including but not limited to the septimal dieses, as well as the chroma of the archaeotonic scale, the scale of Tetracot[7]. The best subgroup of this edo is apparently 2.3.5.7.11.19.29.31 thanks to the unique uniform tunings for the harmonic segments 18::22 and 27::33 – a fact related to the vanish of 100/99 (S10) and 243/242 (S9/S11) – but prime 13 is just as plausible. I consider it the first edo to deal with the all-important 2.3.5.7.11.13.19.29 subgroup. The beauty of this edo goes beyond the structure, but to the intonation: it has a very slightly sharp 3 and a more noticeably flat 5, making a flat, more stable 15; that is ideal for my music. Finally, it's an ideal tuning for the magic temperament. I can't compliment it enough. S-tier.
	* 43 – 1/5-comma meantone, not a bad meantone tuning in the 5-limit. The 3 and 5 are equally off, making up a beautifully pure 15. Unfortunately the diesis is too small to achieve good septimal and undecimal harmony. B-tier.		* 43 – 1/5-comma meantone, not a bad meantone tuning in the 5-limit. The 3 and 5 are equally off, making up a beautifully pure 15. Unfortunately the diesis is too small to achieve good septimal and undecimal harmony. B-tier.
	* 44 – This edo adds decent approximation to harmonic 13 on top of 22edo's 11-limit, which is pretty tense to start with. At this point it just all breaks down. D-tier.		* 44 – This edo adds decent approximation to harmonic 13 on top of 22edo's 11-limit, which is pretty tense to start with. At this point it just all breaks down. D-tier.

FloraC: Update

2026-08-30T09:36:27Z

Update

← Older revision		Revision as of 09:36, 30 August 2026
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	\|-		\|-
	! C-tier		! C-tier
	\| 26, 29, 36, 38		\| 26, 29, 36, 37, 38
	\|-		\|-
	! D-tier		! D-tier
	\| 13, 16, 18, 23, 25, 28, 30, 35~~, 37~~		\| 13, 16, 18, 23, 25, 28, 30, 35
	\|-		\|-
	! F-tier		! F-tier
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	=== Subsemitone-size ===		=== Subsemitone-size ===
	* 15 – The error of the fifth is getting quite large for its step size, particularly if you compare it with 10edo. Either this or 10edo can be viewed as the opposite of 14edo, so I think of this as the direct competitor of 10edo. As for which I prefer? I have no idea. A-tier.		* 15 – The error of the fifth is getting quite large for its step size, particularly if you compare it with 10edo. Either this or 10edo can be viewed as the opposite of 14edo, so I think of this as the direct competitor of 10edo. As for which I prefer? I have no idea. A-tier.
	* 16 – ~~Potentially useful~~ as ~~every other step~~ of ~~32edo or every third step of 48edo~~. Besides that, it has armodue, basically an extremely flat fifth that doesn't sound like the 3rd harmonic at all. "Fifthiness" is pointless if not for approximating the 3rd harmonic, so I'm afraid I don't consider this approach to ~~have~~ much ~~value~~. D-tier.		* 16 – Workable as a temperament of the 2.5.7 subgroup, a subgroup that calls for novel chord structures such as 4:7:10. Besides that, it has armodue, basically an extremely flat fifth that doesn't sound like the 3rd harmonic at all. "Fifthiness" is pointless if not for approximating the 3rd harmonic, so I'm afraid I don't consider this approach to hold much water. D-tier.
	* 17 – This edo is characterized by its hard diatonic scale, with more contrasting step sizes than 12edo's basic diatonic scale. This gives it more intense voice leading and more cathartic resolutions. Traditional tertian harmony works pretty well here, but semiquartal harmony, that is using the contrast between 7/4 and 12/7 as the basis of tonality, does even better. S-tier.		* 17 – This edo is characterized by its hard diatonic scale, with more contrasting step sizes than 12edo's basic diatonic scale. This gives it more intense voice leading and more cathartic resolutions. Traditional tertian harmony works pretty well here, but semiquartal harmony, that is using the contrast between 7/4 and 12/7 as the basis of tonality, does even better. S-tier.
	* 18 – Potentially useful as every other step of 36edo. D-tier.		* 18 – Potentially useful as every other step of 36edo. D-tier.
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	* 22 – The least evil solution to porcupine and less so to superpyth. I happen to have experience working with porcupine and it felt quite alright, except that I often found myself struggling to combat its out-of-tune nature. B-tier.		* 22 – The least evil solution to porcupine and less so to superpyth. I happen to have experience working with porcupine and it felt quite alright, except that I often found myself struggling to combat its out-of-tune nature. B-tier.
	* 23 – Potentially useful as every other step of 46edo. D-tier.		* 23 – Potentially useful as every other step of 46edo. D-tier.
	* 24 – This is kind of a normie's edo, but also the most accessible. Quartertones in my otherwise 12edo works often go unnoticed by the audience. A natural next step of 12edo with a structurally beautiful 2.3.5.11.17.19 subgroup interpretation. A-tier.		* 24 – This is kind of a normie's edo, but also the most accessible. Quartertones in my otherwise 12edo works often go unnoticed by the audience. A natural next step of 12edo with a structurally beautiful 2.3.5.11.17.19-subgroup interpretation. A-tier.
	* 25 – ~~Potentially useful~~ as ~~every other step~~ of ~~50edo~~. D-tier.		* 25 – Like 16edo, workable as a temperament of the 2.5.7 subgroup. D-tier.
	* 26 – Just as 19edo is the point separating meantone and flattone, this is the point separating flattone and a meantone extension that implies an even flatter fifth. Therefore it should share all the advantages of 12edo and 19edo, at least theoretically, that is if not for its poor intonation in the 5-limit. C-tier.		* 26 – Just as 19edo is the point separating meantone and flattone, this is the point separating flattone and a meantone extension that implies an even flatter fifth. Therefore it should share all the advantages of 12edo and 19edo, at least theoretically, that is if not for its poor intonation in the 5-limit. C-tier.

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	* 34 – This is to 17edo what 24edo is to 12edo. While 17edo is often good enough, this offers some more sophisticated solutions such as tetracot. Even the harmonics 7 and 11, which come from 17edo and are commonly cited as relatively poor in this edo, are convincing enough to me, since when I worked with modus I never had a problem with the intonation at all, unlike with porcupine. The sound is better than the structure. B-tier.		* 34 – This is to 17edo what 24edo is to 12edo. While 17edo is often good enough, this offers some more sophisticated solutions such as tetracot. Even the harmonics 7 and 11, which come from 17edo and are commonly cited as relatively poor in this edo, are convincing enough to me, since when I worked with modus I never had a problem with the intonation at all, unlike with porcupine. The sound is better than the structure. B-tier.
	* 35 – Potentially useful as every other step of 70edo. D-tier.		* 35 – Potentially useful as every other step of 70edo. D-tier.
	* 36 – The idea of adding sixth tones to plain 12edo music is interesting, but ~~none of my attempts have been successful as~~ I generally find them to sound forced. I think ~~this edo is more difficult~~ to ~~use than it appears~~. C-tier.		* 36 – This edo is more difficult to use than it appears. The idea of adding sixth tones to plain 12edo music might be interesting, but I generally find them to sound forced. Added to this fact is that I think of 12edo as a complete 7-limit system, so the alternative mapping of harmonic 7 only serves to create mental dissonance. C-tier.
	* 37 – ~~Potentially useful as every other step of 74edo or every third step~~ of ~~111edo. Besides that, it has a good~~ 2.5.7~~.11.13~~ subgroup ~~interpretation, tho I have no idea how harmony in this subgroup is supposed to work~~. D-tier.		* 37 – This edo offers good approximation to harmonics 5, 7, 11 and 13. The last two primes adds the much needed variety to the otherwise pretty dull sound of the 2.5.7 subgroup. C-tier.
	* 38 – This is to 19edo what 24edo is to 12edo. On paper it adds decent approximation to harmonics 11, 17, and 19, but in practice I never had a situation where I felt I needed these additional notes when working with 19edo. C-tier.		* 38 – This is to 19edo what 24edo is to 12edo. On paper it adds decent approximation to harmonics 11, 17, and 19, but in practice I never had a situation where I felt I needed these additional notes when working with 19edo. C-tier.

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	\|-		\|-
	! S-tier		! S-tier
	\| 41		\| 41, 58
	\|-		\|-
	! A-tier		! A-tier
	\| 46~~, 58~~, 62		\| 46, 62
	\|-		\|-
	! B-tier		! B-tier
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	=== Comma-size ===		=== Comma-size ===
	* 39 – 1/5-comma quasisuper, most notable for tuning the minor second to 28/27 (with the 39d val) which I feel is where the tension peaks for voice leading. Such an overlooked system despite the similarity to 27edo in many ways. B-tier.		* 39 – 1/5-comma quasisuper, most notable for tuning the minor second to 28/27 (with the 39d val) which I feel is where the tension peaks for voice leading. Such an overlooked system despite the similarity to 27edo in many ways. B-tier.
	* 41 – The first of the five essential comma-level edos, and the first edo to achieve 9-odd-limit distinction and consistency. This is most significant for providing three flavors for each chromatic category: classical, Pythagorean, and septimal. In this case it is a schismic and ~~garischismic~~ system, so that all three kinds are separated by the same comma step and can be found on a stack of fifths. The comma step is somewhat larger than just, making the differences more pronounced, which is part of why I think this edo is pretty deep – the step isn't only a comma, but many things at once, including but not limited to the septimal dieses, as well as the chroma of the archaeotonic scale, the scale of Tetracot[7]. The best subgroup of this edo is~~, actually,~~ 2.3.5.7.11.19. ~~Prime 13 is certainly plausible, but prime 19 fits way better~~. ~~There's a~~ unique uniform ~~tuning~~ for the harmonic ~~segment~~ 18::22, a fact related to the vanish of ~~s10 =~~ 100/99 and ~~s9/s11 =~~ 243/242. The beauty of this edo goes ~~even~~ beyond the structure, but ~~also~~ to the intonation: it has a very slightly sharp 3 and a more noticeably flat 5, making a flat, more stable 15; that is ideal for my music. Finally, it's an ideal tuning for the magic temperament. I can't compliment it enough. S-tier.		* 41 – The first of the five essential comma-level edos, and the first edo to achieve 9-odd-limit distinction and consistency. This is most significant for providing three flavors for each chromatic category: classical, Pythagorean, and septimal. In this case it is a schismic and septischismic system, so that all three kinds are separated by the same comma step and can be found on a stack of fifths. The comma step is somewhat larger than just, making the differences more pronounced, which is part of why I think this edo is pretty deep – the step isn't only a comma, but many things at once, including but not limited to the septimal dieses, as well as the chroma of the archaeotonic scale, the scale of Tetracot[7]. The best subgroup of this edo is apparently 2.3.5.7.11.19.29.31 thanks to the unique uniform tunings for the harmonic segments 18::22 and 27::33 – a fact related to the vanish of 100/99 (S10) and 243/242 (S9/S11) – but prime 13 is just as plausible. I consider it the first edo to deal with the all-important 2.3.5.7.11.13.19.29 subgroup. The beauty of this edo goes beyond the structure, but to the intonation: it has a very slightly sharp 3 and a more noticeably flat 5, making a flat, more stable 15; that is ideal for my music. Finally, it's an ideal tuning for the magic temperament. I can't compliment it enough. S-tier.
	* 43 – 1/5-comma meantone, not a bad meantone tuning in the 5-limit. The 3 and 5 are equally off, making up a beautifully pure 15. Unfortunately the diesis is too small to achieve good septimal and undecimal harmony. B-tier.		* 43 – 1/5-comma meantone, not a bad meantone tuning in the 5-limit. The 3 and 5 are equally off, making up a beautifully pure 15. Unfortunately the diesis is too small to achieve good septimal and undecimal harmony. B-tier.
	* 44 – This edo adds decent approximation to harmonic 13 on top of 22edo's 11-limit, which is pretty tense to start with. At this point it just all breaks down. D-tier.		* 44 – This edo adds decent approximation to harmonic 13 on top of 22edo's 11-limit, which is pretty tense to start with. At this point it just all breaks down. D-tier.
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	* 55 – This edo is out of the optimal range of meantone. Septimal meantone feels dumb here since we know 43edo is sharp enough. It can be used for mohaha, but that feels so similar to 24edo that I'd just go with the latter. D-tier.		* 55 – This edo is out of the optimal range of meantone. Septimal meantone feels dumb here since we know 43edo is sharp enough. It can be used for mohaha, but that feels so similar to 24edo that I'd just go with the latter. D-tier.
	* 56 – A hemicommatic edo with a rather messed-up tuning profile. Nothing notable about it. D-tier.		* 56 – A hemicommatic edo with a rather messed-up tuning profile. Nothing notable about it. D-tier.
	* 58 – The fourth essential comma-level edo. Being the first edo with full 11-odd-limit distinction, this one is easily adorable. Whereas 41edo tunes the fifth to 24 steps, this edo tunes the fourth to 24 steps, and the implication is its 2.3.5.7.13.29 subgroup is analogous to 41edo's 2.3.5.7.11.19 subgroup. ~~This edo~~ is ~~best~~ as a ~~2.3.5.7.~~11~~.13.29-subgroup system~~, but it has more to offer~~. Specifically,~~ the ~~harmonics~~ 19 and 23 in the 58hi val are surprisingly convincing, and although the 17 doesn't blend quite well it at least looks good on paper. ~~Everything considered~~, it ~~is virtually~~ the ~~first full 23-limit system~~. ~~All that bugs me is the minor fact that~~ 11~~/8 and 7/5 are~~ tuned ~~too close to each other, only one comma apart~~. A-tier.		* 58 – The fourth essential comma-level edo. Being the first edo with full 11-odd-limit distinction, this one is easily adorable. Whereas 41edo tunes the perfect fifth to 24 steps, this edo tunes the perfect fourth to 24 steps, and the implication is its 2.3.5.7.13.29 subgroup is analogous to 41edo's 2.3.5.7.11.19 subgroup. Due to this, it would seem like it is most useful as a temperament of above plus prime 11, but it has more to offer – the 19 and 23 in the 58hi val are surprisingly convincing, and although the 17 doesn't blend quite well it at least looks good on paper. So like 41edo, it deals adequately with everything in the 2.3.5.7.11.13.19.29 subgroup, making it essentially a dual. Best tuned with a pure perfect twelfth. S-tier.
	* 60 – This edo has a particularly problematic fifth, in that it closes after twelve steps, tempering out the Pythagorean comma. As the first thing I leave meantone is to look for a fifth that leads to a positive Pythagorean comma, this edo is clearly not my thing. C-tier.		* 60 – This edo has a particularly problematic fifth, in that it closes after twelve steps, tempering out the Pythagorean comma. As the first thing I leave meantone is to look for a fifth that leads to a positive Pythagorean comma, this edo is clearly not my thing. C-tier.

			=== Subcomma-size ===
	* 61 – Can be used to tune modus. Otherwise not bearing much utility. D-tier.		* 61 – Can be used to tune modus. Otherwise not bearing much utility. D-tier.
	* 62 – The ultimate 23-limit meantone tuning. It re-tunes harmonics 13, 17, and 19, and paves the path to the 23 from 31edo. I find these additions to 31edo's 11-limit very favorable. A-tier.		* 62 – The ultimate 23-limit meantone tuning. It re-tunes harmonics 13, 17, and 19, and paves the path to the 23 from 31edo. I find these additions to 31edo's 11-limit very favorable. A-tier.
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	* 65 – As every other step of 130edo, this edo is excellent in the 2.3.5.11.19.23-subgroup, but the contrast between that and the poor approximations to 7 and 13 is fatal. Still, it allows a dual-7 dual-13 approach, not very satisfying at this level but better than nothing. C-tier.		* 65 – As every other step of 130edo, this edo is excellent in the 2.3.5.11.19.23-subgroup, but the contrast between that and the poor approximations to 7 and 13 is fatal. Still, it allows a dual-7 dual-13 approach, not very satisfying at this level but better than nothing. C-tier.
	* 67 – A disastrous meantone tuning. F-tier.		* 67 – A disastrous meantone tuning. F-tier.

	~~=== Subcomma-size ===~~
	* 68 – While this edo contains 17edo and if we compare it with 51edo, it clearly hits more harmonic targets, it should be noted that linearly dividing the steps into four is a weaker move than into two or three, as quarter-step offsets don't tend to create new categories, unlike third- or half-step offsets. This is especially true of this edo, as 34edo hits lots of harmonic targets already. Another obvious flaw is the awkward situation with prime 11. C-tier.		* 68 – While this edo contains 17edo and if we compare it with 51edo, it clearly hits more harmonic targets, it should be noted that linearly dividing the steps into four is a weaker move than into two or three, as quarter-step offsets don't tend to create new categories, unlike third- or half-step offsets. This is especially true of this edo, as 34edo hits lots of harmonic targets already. Another obvious flaw is the awkward situation with prime 11. C-tier.
	* 70 – As every other step of 140edo, this edo's structure allows lots of niche uses. Unfortunately none of them is interesting enough to make me stay very long. C-tier.		* 70 – As every other step of 140edo, this edo's structure allows lots of niche uses. Unfortunately none of them is interesting enough to make me stay very long. C-tier.

FloraC: Update

2026-02-26T12:04:13Z

Update

← Older revision		Revision as of 12:04, 26 February 2026
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	=== Subsemitone-size ===		=== Subsemitone-size ===
	* 15 – The error of the fifth is getting quite large for its step size, particularly if you compare it with 10edo. Either this or 10edo can be viewed as the opposite of 14edo, so I think of this as the direct competitor of 10edo. As for which I prefer? I have no idea. A-tier.		* 15 – The error of the fifth is getting quite large for its step size, particularly if you compare it with 10edo. Either this or 10edo can be viewed as the opposite of 14edo, so I think of this as the direct competitor of 10edo. As for which I prefer? I have no idea. A-tier.
	* 16 – Potentially useful as every other step of 32edo. Besides that, it has armodue, basically an extremely flat fifth that doesn't sound like the 3rd harmonic at all. "Fifthiness" is ~~useless~~ if not for approximating the 3rd harmonic, so I'm afraid I don't consider this approach to have much value. D-tier.		* 16 – Potentially useful as every other step of 32edo or every third step of 48edo. Besides that, it has armodue, basically an extremely flat fifth that doesn't sound like the 3rd harmonic at all. "Fifthiness" is pointless if not for approximating the 3rd harmonic, so I'm afraid I don't consider this approach to have much value. D-tier.
	* 17 – This edo is characterized by its hard diatonic scale, with more contrasting step sizes than 12edo's basic diatonic scale. This gives it more intense voice leading and more cathartic resolutions. Traditional tertian harmony works pretty well here, but semiquartal harmony, that is using the contrast between 7/4 and 12/7 as the basis of tonality, does even better. S-tier.		* 17 – This edo is characterized by its hard diatonic scale, with more contrasting step sizes than 12edo's basic diatonic scale. This gives it more intense voice leading and more cathartic resolutions. Traditional tertian harmony works pretty well here, but semiquartal harmony, that is using the contrast between 7/4 and 12/7 as the basis of tonality, does even better. S-tier.
	* 18 – Potentially useful as every other step of 36edo. D-tier.		* 18 – Potentially useful as every other step of 36edo. D-tier.
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	* 35 – Potentially useful as every other step of 70edo. D-tier.		* 35 – Potentially useful as every other step of 70edo. D-tier.
	* 36 – The idea of adding sixth tones to plain 12edo music is interesting, but none of my attempts have been successful as I generally find them to sound forced. I think this edo is more difficult to use than it appears. C-tier.		* 36 – The idea of adding sixth tones to plain 12edo music is interesting, but none of my attempts have been successful as I generally find them to sound forced. I think this edo is more difficult to use than it appears. C-tier.
	* 37 – Potentially useful as every other step of 74edo. Besides that, it has a good 2.5.7.11.13 subgroup interpretation, tho I have no idea how harmony in this subgroup is supposed to work. D-tier.		* 37 – Potentially useful as every other step of 74edo or every third step of 111edo. Besides that, it has a good 2.5.7.11.13 subgroup interpretation, tho I have no idea how harmony in this subgroup is supposed to work. D-tier.
	* 38 – This is to 19edo what 24edo is to 12edo. On paper it adds decent approximation to harmonics 11, 17, and 19, but in practice I never had a situation where I felt I needed these additional notes when working with 19edo. C-tier.		* 38 – This is to 19edo what 24edo is to 12edo. On paper it adds decent approximation to harmonics 11, 17, and 19, but in practice I never had a situation where I felt I needed these additional notes when working with 19edo. C-tier.

FloraC at 20:04, 29 January 2026

2026-01-29T20:04:37Z

FloraC at 09:54, 18 January 2026

2026-01-18T09:54:22Z

← Older revision		Revision as of 09:54, 18 January 2026
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	== Large ==		== Large ==
	Starting from here, it'd be better to adopt a more systematic approach, for which I'm planning an essay where I'll investigate the 2.3.5.7.11.13.19-subgroup JI and temperaments.		Starting from here, it'd be better to adopt a more systematic approach, for which I'm planning an essay where I'll investigate the 2.3.5.7.11.13.19.29-subgroup JI and temperaments.

FloraC: Update

2026-01-05T10:06:10Z

Update

← Older revision		Revision as of 10:06, 5 January 2026
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	\|-		\|-
	! A-tier		! A-tier
	\| 10, 15, 24, 31~~, 34~~		\| 10, 15, 24, 27, 31
	\|-		\|-
	! B-tier		! B-tier
	\| 11, 14, 22, 27		\| 11, 14, 22, 34
	\|-		\|-
	! C-tier		! C-tier
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	=== Diesis-size ===		=== Diesis-size ===
	* 27 – The cyberpunk edo. Good sharp-tending tuning profile in the 2.3.5.7.13 subgroup with the sole exception of harmonic 15 tuned way too sharp, for I prefer a flat tuning of 15 or at least no sharper than 12edo's to improve its stability as a consonant major seventh. Other than that it's pretty good. B-tier.		* 27 – The cyberpunk edo. Good sharp-tending tuning profile in the 2.3.5.7.13 subgroup with the sole exception of harmonic 15 tuned way too sharp, for I prefer a flat tuning of 15 or at least no sharper than 12edo's to improve its stability as a consonant major seventh. Other than that it's pretty good. Octave compression gives better intonation. A-tier.
	* 28 – Potentially useful as every other step of 56edo. D-tier.		* 28 – Potentially useful as every other step of 56edo. D-tier.
	* 29 – The first edo that sounds like Pythagorean tuning with distinct chromatic and diatonic semitones, such that most contemporary 12edo music will sound alright if retempered to this through dominant. The patent-val interpretation is underwhelming. Otherwise it's a good framework as every other step of 58edo and every third step of 87edo. C-tier.		* 29 – The first edo that sounds like Pythagorean tuning with distinct chromatic and diatonic semitones, such that most contemporary 12edo music will sound alright if retempered to this through dominant. The patent-val interpretation is underwhelming. Otherwise it's a good framework as every other step of 58edo and every third step of 87edo. C-tier.
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	* 32 – 27edo but worse. F-tier.		* 32 – 27edo but worse. F-tier.
	* 33 – 26edo but worse. F-tier.		* 33 – 26edo but worse. F-tier.
	* 34 – This is to 17edo what 24edo is to 12edo. While 17edo is often good enough, this offers some more sophisticated solutions such as tetracot. Even the harmonics 7 and 11, commonly cited as poor in this edo, are convincing enough to me, since when I worked with modus I never ~~felt~~ the ~~out-of-tune~~ intonation at all, unlike with porcupine. A-tier.		* 34 – This is to 17edo what 24edo is to 12edo. While 17edo is often good enough, this offers some more sophisticated solutions such as tetracot. Even the harmonics 7 and 11, commonly cited as poor in this edo, are convincing enough to me, since when I worked with modus I never had a problem with the intonation at all, unlike with porcupine. The sound is better than the structure. B-tier.
	* 35 – Potentially useful as every other step of 70edo. D-tier.		* 35 – Potentially useful as every other step of 70edo. D-tier.
	* 36 – The idea of adding third tones to plain 12edo music is interesting, but none of my attempts have been successful as I generally find them to sound forced. I think this edo is more difficult to use than it appears. C-tier.		* 36 – The idea of adding third tones to plain 12edo music is interesting, but none of my attempts have been successful as I generally find them to sound forced. I think this edo is more difficult to use than it appears. C-tier.

FloraC: /* Small */

2025-08-29T09:59:21Z

Small

← Older revision		Revision as of 09:59, 29 August 2025
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	== Small ==		== Small ==
			These edos are where the steps tend to separate distinct interval categories rather than functioning as commatic inflections. I classify them into semitone-size, subsemitone-size (third-tone and quartertone), and diesis-size (fifth-tone and sixth-tone) edos.

	{\| class="wikitable"		{\| class="wikitable"
	\|+ Tier list for small edos (10–38)		\|+ Tier list for small edos (10–38)
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	* 36 – The idea of adding third tones to plain 12edo music is interesting, but none of my attempts have been successful as I generally find them to sound forced. I think this edo is more difficult to use than it appears. C-tier.		* 36 – The idea of adding third tones to plain 12edo music is interesting, but none of my attempts have been successful as I generally find them to sound forced. I think this edo is more difficult to use than it appears. C-tier.
	* 37 – Potentially useful as every other step of 74edo. Besides that, it has a good 2.5.7.11.13 subgroup interpretation, tho I have no idea how harmony in this subgroup is supposed to work. D-tier.		* 37 – Potentially useful as every other step of 74edo. Besides that, it has a good 2.5.7.11.13 subgroup interpretation, tho I have no idea how harmony in this subgroup is supposed to work. D-tier.
	* 38 – This is to 19edo what 24edo is to 12edo. On paper it adds decent approximation to harmonics 11, 17, and 19, but in practice I never had a situation where I felt I needed these additional notes when working with 19edo. C-tier.		* 38 – This is to 19edo what 24edo is to 12edo. On paper it adds decent approximation to harmonics 11, 17, and 19, but in practice I never had a situation where I felt I needed these additional notes when working with 19edo. C-tier.

	== Medium ==		== Medium ==

FloraC: Update

2025-05-09T16:09:37Z

Update

← Older revision		Revision as of 16:09, 9 May 2025
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	* 29 – The first edo that sounds like Pythagorean tuning with distinct chromatic and diatonic semitones, such that most contemporary 12edo music will sound alright if retempered to this through dominant. The patent-val interpretation is underwhelming. Otherwise it's a good framework as every other step of 58edo and every third step of 87edo. C-tier.		* 29 – The first edo that sounds like Pythagorean tuning with distinct chromatic and diatonic semitones, such that most contemporary 12edo music will sound alright if retempered to this through dominant. The patent-val interpretation is underwhelming. Otherwise it's a good framework as every other step of 58edo and every third step of 87edo. C-tier.
	* 30 – Potentially useful as every other step of 60edo. D-tier.		* 30 – Potentially useful as every other step of 60edo. D-tier.
	* 31 – This is a great edo, with a tuning profile close to what I consider the optimal tuning of meantone. ~~Too great it's a little unfun to work with~~. A-tier.		* 31 – This is a great edo. Too great it's a little unfun to work with. It has a tuning profile close to what I consider the optimal tuning of meantone, and migration, the meantone extension that maps harmonic 11 to the semi-augmented fourth, works almost perfectly in this system. Octave stretch gives better intonation. A-tier.
	* 32 – 27edo but worse. F-tier.		* 32 – 27edo but worse. F-tier.
	* 33 – 26edo but worse. F-tier.		* 33 – 26edo but worse. F-tier.
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	* 60 – This edo has a particularly problematic fifth, in that it closes after twelve steps, tempering out the Pythagorean comma. As the first thing I leave meantone is to look for a fifth that leads to a positive Pythagorean comma, this edo is clearly not my thing. C-tier.		* 60 – This edo has a particularly problematic fifth, in that it closes after twelve steps, tempering out the Pythagorean comma. As the first thing I leave meantone is to look for a fifth that leads to a positive Pythagorean comma, this edo is clearly not my thing. C-tier.
	* 61 – Can be used to tune modus. Otherwise not bearing much utility. D-tier.		* 61 – Can be used to tune modus. Otherwise not bearing much utility. D-tier.
	* 62 – The ultimate 23-limit meantone tuning. It re-tunes harmonics 13, 17, 19, and 23 ~~for~~ 31edo. I find these additions to 31edo's 11-limit very favorable. A-tier.		* 62 – The ultimate 23-limit meantone tuning. It re-tunes harmonics 13, 17, and 19, and paves the path to the 23 from 31edo. I find these additions to 31edo's 11-limit very favorable. A-tier.
	* 63 – Similar to 56edo, nothing notable about it. D-tier.		* 63 – Similar to 56edo, nothing notable about it. D-tier.
	* 65 – As every other step of 130edo, this edo is excellent in the 2.3.5.11.19.23-subgroup, but the contrast between that and the poor approximations to 7 and 13 is fatal. Still, it allows a dual-7 dual-13 approach, not very satisfying at this level but better than nothing. C-tier.		* 65 – As every other step of 130edo, this edo is excellent in the 2.3.5.11.19.23-subgroup, but the contrast between that and the poor approximations to 7 and 13 is fatal. Still, it allows a dual-7 dual-13 approach, not very satisfying at this level but better than nothing. C-tier.

FloraC: Some more finishing touch

2025-04-25T14:33:00Z

Some more finishing touch

← Older revision		Revision as of 14:33, 25 April 2025
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	* 55 – This edo is out of the optimal range of meantone. Septimal meantone feels dumb here since we know 43edo is sharp enough. It can be used for mohaha, but that feels so similar to 24edo that I'd just go with the latter. D-tier.		* 55 – This edo is out of the optimal range of meantone. Septimal meantone feels dumb here since we know 43edo is sharp enough. It can be used for mohaha, but that feels so similar to 24edo that I'd just go with the latter. D-tier.
	* 56 – A hemicommatic edo with a rather messed-up tuning profile. Nothing notable about it. D-tier.		* 56 – A hemicommatic edo with a rather messed-up tuning profile. Nothing notable about it. D-tier.
	* 58 – The fourth essential comma-level edo. Being the first edo with full 11-odd-limit distinction, this one is easily adorable. Whereas 41edo tunes the fifth to 24 steps, this edo tunes the fourth to 24 steps, and the implication is its 2.3.5.7.13.29 subgroup is analogous to 41edo's 2.3.5.7.11.19 subgroup. This edo is best as a 2.3.5.7.11.13.29-subgroup system, but it has more to offer. Specifically, the harmonics 19 and 23 in the 58hi val are surprisingly convincing, and although the 17 doesn't blend quite well it at least looks good on paper. Everything considered, it is virtually the first full 23-limit system. All that bugs me is the minor fact that 11/8 and 7/5 are tuned only one comma apart. A-tier.		* 58 – The fourth essential comma-level edo. Being the first edo with full 11-odd-limit distinction, this one is easily adorable. Whereas 41edo tunes the fifth to 24 steps, this edo tunes the fourth to 24 steps, and the implication is its 2.3.5.7.13.29 subgroup is analogous to 41edo's 2.3.5.7.11.19 subgroup. This edo is best as a 2.3.5.7.11.13.29-subgroup system, but it has more to offer. Specifically, the harmonics 19 and 23 in the 58hi val are surprisingly convincing, and although the 17 doesn't blend quite well it at least looks good on paper. Everything considered, it is virtually the first full 23-limit system. All that bugs me is the minor fact that 11/8 and 7/5 are tuned too close to each other, only one comma apart. A-tier.
	* 60 – This edo has a particularly problematic fifth, in that it closes after twelve steps, tempering out the Pythagorean comma. As the first thing I leave meantone is to look for a fifth that leads to a positive Pythagorean comma, this edo is clearly not my thing. C-tier.		* 60 – This edo has a particularly problematic fifth, in that it closes after twelve steps, tempering out the Pythagorean comma. As the first thing I leave meantone is to look for a fifth that leads to a positive Pythagorean comma, this edo is clearly not my thing. C-tier.
	* 61 – Can be used to tune modus. Otherwise not bearing much utility. D-tier.		* 61 – Can be used to tune modus. Otherwise not bearing much utility. D-tier.
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	* 68 – While this edo contains 17edo and if we compare it with 51edo, it clearly hits more harmonic targets, it should be noted that linearly dividing the steps into four is a weaker move than into two or three, as quarter-step offsets don't tend to create new categories, unlike third- or half-step offsets. This is especially true of this edo, as 34edo hits lots of harmonic targets already. Another obvious flaw is the awkward situation with prime 11. C-tier.		* 68 – While this edo contains 17edo and if we compare it with 51edo, it clearly hits more harmonic targets, it should be noted that linearly dividing the steps into four is a weaker move than into two or three, as quarter-step offsets don't tend to create new categories, unlike third- or half-step offsets. This is especially true of this edo, as 34edo hits lots of harmonic targets already. Another obvious flaw is the awkward situation with prime 11. C-tier.
	* 70 – As every other step of 140edo, this edo's structure allows lots of niche uses. Unfortunately none of them is interesting enough to make me stay very long. C-tier.		* 70 – As every other step of 140edo, this edo's structure allows lots of niche uses. Unfortunately none of them is interesting enough to make me stay very long. C-tier.
	* 72 – The last essential comma-level edo. Has the same problem ~~with~~ 60edo. Even tho it approximates JI way better and thus qualifies for an essential comma-level edo, most of its structural features have been provided by 41- and 58edo. B-tier.		* 72 – The last essential comma-level edo. Has the same problem as 60edo. Even tho it approximates JI way better and thus qualifies for an essential comma-level edo, most of its structural features have been provided by 41- and 58edo. B-tier.
	* 73 – A strange sharp-tending hemicommatic system that seems to allow some niche uses. C-tier.		* 73 – A strange sharp-tending hemicommatic system that seems to allow some niche uses. C-tier.
	* 74 – A good meantone tuning. Not much else to offer. D-tier.		* 74 – A good meantone tuning. Not much else to offer. D-tier.

FloraC at 14:15, 25 April 2025

2025-04-25T14:15:46Z

← Older revision		Revision as of 14:15, 25 April 2025
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	* 79 – A disastrous non-meantone tuning. F-tier.		* 79 – A disastrous non-meantone tuning. F-tier.

	== ~~Hemicomma-size~~ ==		== Large ==
	Starting from here, it'd be better to adopt a more systematic approach, for which I'm planning an essay where I'll investigate the 2.3.5.7.11.13.19-subgroup JI and temperaments.		Starting from here, it'd be better to adopt a more systematic approach, for which I'm planning an essay where I'll investigate the 2.3.5.7.11.13.19-subgroup JI and temperaments.

← Older revision		Revision as of 20:04, 29 January 2026
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	* 15 – The error of the fifth is getting quite large for its step size, particularly if you compare it with 10edo. Either this or 10edo can be viewed as the opposite of 14edo, so I think of this as the direct competitor of 10edo. As for which I prefer? I have no idea. A-tier.		* 15 – The error of the fifth is getting quite large for its step size, particularly if you compare it with 10edo. Either this or 10edo can be viewed as the opposite of 14edo, so I think of this as the direct competitor of 10edo. As for which I prefer? I have no idea. A-tier.
	* 16 – Potentially useful as every other step of 32edo. Besides that, it has armodue, basically an extremely flat fifth that doesn't sound like the 3rd harmonic at all. "Fifthiness" is useless if not for approximating the 3rd harmonic, so I'm afraid I don't consider this approach to have much value. D-tier.		* 16 – Potentially useful as every other step of 32edo. Besides that, it has armodue, basically an extremely flat fifth that doesn't sound like the 3rd harmonic at all. "Fifthiness" is useless if not for approximating the 3rd harmonic, so I'm afraid I don't consider this approach to have much value. D-tier.
	* 17 – This edo ~~contains an impressive~~ diatonic scale ~~which is nice for both melody~~ and harmony~~. Semiquartal~~ harmony, that is using the contrast between 7/4 and 12/7 as the basis of tonality, ~~works exceptionally well in this system~~. S-tier.		* 17 – This edo is characterized by its hard diatonic scale, with more contrasting step sizes than 12edo's basic diatonic scale. This gives it more intense voice leading and more cathartic resolutions. Traditional tertian harmony works pretty well here, but semiquartal harmony, that is using the contrast between 7/4 and 12/7 as the basis of tonality, does even better. S-tier.
	* 18 – Potentially useful as every other step of 36edo. D-tier.		* 18 – Potentially useful as every other step of 36edo. D-tier.
	* 19 – This edo is where my microtonal journey began. Extremely versatile yet friendly to beginners. Using it as a tuning of meantone, the tuning profile is sort of opposite to 12edo, but with seven more pitch classes, the expressive possibility explodes. Presence of an exact hemitwelfth sets it apart from many other meantone edos. Octave stretch solves the intonational problem to a large extent. S-tier.		* 19 – This edo is where my microtonal journey began. Extremely versatile yet friendly to beginners. Using it as a tuning of meantone, the tuning profile is sort of opposite to 12edo, but with seven more pitch classes, the expressive possibility explodes. Presence of an exact hemitwelfth sets it apart from many other meantone edos. Octave stretch solves the intonational problem to a large extent. S-tier.
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	* 32 – 27edo but worse. F-tier.		* 32 – 27edo but worse. F-tier.
	* 33 – 26edo but worse. F-tier.		* 33 – 26edo but worse. F-tier.
	* 34 – This is to 17edo what 24edo is to 12edo. While 17edo is often good enough, this offers some more sophisticated solutions such as tetracot. Even the harmonics 7 and 11, commonly cited as poor in this edo, are convincing enough to me, since when I worked with modus I never had a problem with the intonation at all, unlike with porcupine. The sound is better than the structure. B-tier.		* 34 – This is to 17edo what 24edo is to 12edo. While 17edo is often good enough, this offers some more sophisticated solutions such as tetracot. Even the harmonics 7 and 11, which come from 17edo and are commonly cited as relatively poor in this edo, are convincing enough to me, since when I worked with modus I never had a problem with the intonation at all, unlike with porcupine. The sound is better than the structure. B-tier.
	* 35 – Potentially useful as every other step of 70edo. D-tier.		* 35 – Potentially useful as every other step of 70edo. D-tier.
	* 36 – The idea of adding ~~third~~ tones to plain 12edo music is interesting, but none of my attempts have been successful as I generally find them to sound forced. I think this edo is more difficult to use than it appears. C-tier.		* 36 – The idea of adding sixth tones to plain 12edo music is interesting, but none of my attempts have been successful as I generally find them to sound forced. I think this edo is more difficult to use than it appears. C-tier.
	* 37 – Potentially useful as every other step of 74edo. Besides that, it has a good 2.5.7.11.13 subgroup interpretation, tho I have no idea how harmony in this subgroup is supposed to work. D-tier.		* 37 – Potentially useful as every other step of 74edo. Besides that, it has a good 2.5.7.11.13 subgroup interpretation, tho I have no idea how harmony in this subgroup is supposed to work. D-tier.
	* 38 – This is to 19edo what 24edo is to 12edo. On paper it adds decent approximation to harmonics 11, 17, and 19, but in practice I never had a situation where I felt I needed these additional notes when working with 19edo. C-tier.		* 38 – This is to 19edo what 24edo is to 12edo. On paper it adds decent approximation to harmonics 11, 17, and 19, but in practice I never had a situation where I felt I needed these additional notes when working with 19edo. C-tier.
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	=== Comma-size ===		=== Comma-size ===
	* 39 – 1/5-comma quasisuper, most notable for tuning the minor second to 28/27 (with the 39d val) which I feel is where the tension peaks for voice leading. Such an overlooked system despite ~~being similar~~ to 27edo in many ways. B-tier.		* 39 – 1/5-comma quasisuper, most notable for tuning the minor second to 28/27 (with the 39d val) which I feel is where the tension peaks for voice leading. Such an overlooked system despite the similarity to 27edo in many ways. B-tier.
	* 41 – The first of the five essential comma-level edos, and the first edo to achieve 9-odd-limit distinction and consistency. This is most significant for providing three flavors for each chromatic category: classical, Pythagorean, and septimal. In this case it is a schismic and garischismic system, so that all three kinds are separated by the same comma step and can be found on a stack of fifths. The comma step is somewhat larger than just, making the differences more pronounced, which is part of why I think this edo is pretty deep – the step isn't only a comma, but many things at once, including but not limited to the septimal dieses, as well as the chroma of the archaeotonic scale, the scale of Tetracot[7]. The best subgroup of this edo is, actually, 2.3.5.7.11.19. Prime 13 is certainly plausible, but prime 19 fits way better. There's a unique uniform tuning for the harmonic segment 18::22, a fact related to the vanish of s10 = 100/99 and s9/s11 = 243/242. The beauty of this edo goes even beyond the structure, but also to the intonation: it has a very slightly sharp 3 and a more noticeably flat 5, making a flat, more stable 15; that is ideal for my music. Finally, it's an ideal tuning for the magic temperament. I can't compliment it enough. S-tier.		* 41 – The first of the five essential comma-level edos, and the first edo to achieve 9-odd-limit distinction and consistency. This is most significant for providing three flavors for each chromatic category: classical, Pythagorean, and septimal. In this case it is a schismic and garischismic system, so that all three kinds are separated by the same comma step and can be found on a stack of fifths. The comma step is somewhat larger than just, making the differences more pronounced, which is part of why I think this edo is pretty deep – the step isn't only a comma, but many things at once, including but not limited to the septimal dieses, as well as the chroma of the archaeotonic scale, the scale of Tetracot[7]. The best subgroup of this edo is, actually, 2.3.5.7.11.19. Prime 13 is certainly plausible, but prime 19 fits way better. There's a unique uniform tuning for the harmonic segment 18::22, a fact related to the vanish of s10 = 100/99 and s9/s11 = 243/242. The beauty of this edo goes even beyond the structure, but also to the intonation: it has a very slightly sharp 3 and a more noticeably flat 5, making a flat, more stable 15; that is ideal for my music. Finally, it's an ideal tuning for the magic temperament. I can't compliment it enough. S-tier.
	* 43 – 1/5-comma meantone, not a bad meantone tuning in the 5-limit. The 3 and 5 are equally off, making up a beautifully pure 15. Unfortunately the diesis is too small to achieve good septimal and undecimal harmony. B-tier.		* 43 – 1/5-comma meantone, not a bad meantone tuning in the 5-limit. The 3 and 5 are equally off, making up a beautifully pure 15. Unfortunately the diesis is too small to achieve good septimal and undecimal harmony. B-tier.