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

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.

FloraC: Missed 61edo. Some finishing touch

2025-04-25T14:14:32Z

Missed 61edo. Some finishing touch

← Older revision		Revision as of 14:14, 25 April 2025
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	== Exo ==		== Exo ==
			The exo-edos have a step size larger than a semitone, so they work more like equal chords and/or scales than tuning systems. For this reason I'm discussing but not rating them.

	* 1 – An exposition of pitch.		* 1 – An exposition of pitch.
	* 2 – An exposition of consonance and dissonance.		* 2 – An exposition of consonance and dissonance.
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	== Medium ==		== Medium ==
			The sense of uniqueness of each edo begins to diminish in this range. As the steps get smaller, more and more intervals start to be reasonably approximated, and the edos feel more and more like free pitch. For one thing, starting from here, all edos have at least one diatonic scale (35edo being the last nondiatonic edo). For this reason I'm only discussing and rating those which have a good fifth (relative error < 1/3).

	{\| class="wikitable"		{\| class="wikitable"
	\|+ Tier list for medium edos (39–79)		\|+ Tier list for medium edos (39–79)
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	\|-		\|-
	! D-tier		! D-tier
	\| 44, 45, 55, 56, 63, 74, 75		\| 44, 45, 55, 56, 61, 63, 74, 75
	\|-		\|-
	! F-tier		! F-tier
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	=== Comma-size ===		=== Comma-size ===
	The sense of uniqueness of each edo begins to diminish in this range. As the steps get smaller, more and more intervals start to be reasonably approximated, and the edos feel more and more like free pitch. For one thing, starting from here, all edos have at least one diatonic scale (35edo being the last nondiatonic edo). For this reason I'm only discussing those which have a good fifth (relative error < 1/3).

	* 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 being similar 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.
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	* 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 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.
	* 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, 19, and 23 for 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.
	* 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.
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	* 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.
	* 75 – A good tetracot tuning. Otherwise ~~there's~~ nothing notable. D-tier.		* 75 – A good tetracot tuning. Otherwise nothing notable. D-tier.
	* 77 – This edo is very sophisticated and hard to evaluate. It's an ideal tuning for the valentine temperament, obviously. It also seems to be capable of somewhat approximating the full 23-limit. Overall, the structure is a tight fit, with lots of quirks, but that's not too troublesome – they may as well be turned into advantages in the right circumstances. B-tier.		* 77 – This edo is very sophisticated and hard to evaluate. It's an ideal tuning for the valentine temperament, obviously. It also seems to be capable of somewhat approximating the full 23-limit. Overall, the structure is a tight fit, with lots of quirks, but that's not too troublesome – they may as well be turned into advantages in the right circumstances. B-tier.
	* 79 – A disastrous non-meantone tuning. F-tier.		* 79 – A disastrous non-meantone tuning. F-tier.

	== Hemicomma-size ==		== Hemicomma-size ==
	Starting from here, it'd be better to adopt a more systematic approach, for which I'm planning an essay ~~about~~ 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.

FloraC: Finish commatic edos

2025-04-25T13:47:31Z

Finish commatic edos

← Older revision		Revision as of 13:47, 25 April 2025
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	\|-		\|-
	! B-tier		! B-tier
	\| 39, 43, 53, 72		\| 39, 43, 53, 72, 77
	\|-		\|-
	! C-tier		! C-tier
	\| 48, 50, 51, 60, 68, 70		\| 48, 50, 51, 60, 65, 68, 70, 73
	\|-		\|-
	! D-tier		! D-tier
	\| 44, 45, 55, 56, 63, 74		\| 44, 45, 55, 56, 63, 74, 75
	\|-		\|-
	! F-tier		! F-tier
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	=== Comma-size ===		=== Comma-size ===
	The sense of uniqueness of each edo begins to diminish in this range. As the steps get smaller, more and more intervals start to be reasonably approximated, and the edos feel more and more like free pitch. For one thing, starting from here, all edos have at least one diatonic scale (35edo being the last nondiatonic edo). For this reason I'm only ~~listing~~ those which have a good fifth (relative error < 1/3)~~. For the rest, expect~~		The sense of uniqueness of each edo begins to diminish in this range. As the steps get smaller, more and more intervals start to be reasonably approximated, and the edos feel more and more like free pitch. For one thing, starting from here, all edos have at least one diatonic scale (35edo being the last nondiatonic edo). For this reason I'm only discussing those which have a good fifth (relative error < 1/3).

	~~: ''n''-edo – Potentially useful as every other step of 2''n''-edo. D-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 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 being similar to 27edo in many ways. B-tier.
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	* 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, 19, and 23 for 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.
	…

	* 67 – A disastrous meantone tuning. F-tier.		* 67 – A disastrous meantone tuning. F-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.		* 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 – ~~This~~ edo's structure allows lots of niche uses. Unfortunately none of them is interesting enough to make me stay very long ~~here~~. 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 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.
			* 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.
			* 75 – A good tetracot tuning. Otherwise there's nothing notable. D-tier.
			* 77 – This edo is very sophisticated and hard to evaluate. It's an ideal tuning for the valentine temperament, obviously. It also seems to be capable of somewhat approximating the full 23-limit. Overall, the structure is a tight fit, with lots of quirks, but that's not too troublesome – they may as well be turned into advantages in the right circumstances. B-tier.
			* 79 – A disastrous non-meantone tuning. F-tier.

	…		== Hemicomma-size ==
			Starting from here, it'd be better to adopt a more systematic approach, for which I'm planning an essay about the 2.3.5.7.11.13.19-subgroup JI and temperaments.
	* 79 – A disastrous non-meantone tuning. F-~~tier~~.

← 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.