37edo - Revision history

Ground at 02:32, 28 August 2026

2026-08-28T02:32:35Z

← Older revision		Revision as of 02:32, 28 August 2026
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	== Multiples ==		== Multiples ==
	=== 74edo ===		=== 74edo ===
	74edo is best known as a nearly optimal tuning of Grosstone, an extension of Septimal Meantone. Its flat 3 and 37edo's sharp 3 stack to make a sharp 9, which fills some gaps in 37edo's sharp-tending 13-limit. It ~~adds~~ a half-octave period which turns Porcupine into Stearn, a very efficient 2.3.7 restriction of Hedgehog. It is a decent [[Monarch]] tuning.		74edo is best known as a nearly optimal tuning of Grosstone, an extension of Septimal Meantone. Its flat 3 and 37edo's sharp 3 stack to make a sharp 9, which fills some gaps in 37edo's sharp-tending 13-limit. It is an excellent tuning of [[Diaschismic straddle temperaments\|Diaschismic straddle temperaments]], adding a half-octave period which turns Porcupine into Stearn, a very efficient 2.3.7 restriction of Hedgehog. It is a decent [[Monarch]] tuning.
	{{Harmonics in ED\|74\|37\|0}}		{{Harmonics in ED\|74\|37\|0}}
	=== 111edo ===		=== 111edo ===
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	{{UserTag\|g_\|Ground\|7766ff\|With no regard for structure, 37edo is an extremely special set of notes, which transcends structure. If you assume that approximating successive primes is good, this is a good tuning. If you think straddling 3 causes things to bend in a desirable way, this is the perfect tuning. After years of experience, I think 37edo is best described as 2.<<3.>3.5.7.11.13.17.19.		{{UserTag\|g_\|Ground\|7766ff\|With no regard for structure, 37edo is an extremely special set of notes, which transcends structure. If you assume that approximating successive primes is good, this is a good tuning. If you think straddling 3 causes things to bend in a desirable way, this is the perfect tuning. After years of experience, I think 37edo is best described as 2.<<3.>3.5.7.11.13.17.19.

	74 doesn't change much over 37, just adding a flat 3 and a reasonable 23, but the sharp 9 makes a big difference in available intervals, especially 9/7, which I think is the most obvious gap in 37edo. A more granular erac subgroup is 2.<3.>>>3.<<<5.>5.<<<7.>7.11.<<<13.>13.<<17.>>17.<19.>>>19.<<<23.>23 (this distribution is about optimal for overall accuracy of erac representations, but an interval directly between <<<<3 and >>3 as in 37edo would normally be <3).		74 doesn't change much over 37, just adding a flat 3 and a reasonable 23, but the sharp 9 makes a big difference in available intervals, especially 9/7, which I think is the most obvious gap in 37edo. I think that Diaschismic Porcupine/Semioceanfront are by far the most important new structures, not just to 74 as a multiple of 37, but 74 in general.

			A more granular erac subgroup in 74edo is 2.<3.>>>3.<<<5.>5.<<<7.>7.11.<<<13.>13.<<17.>>17.<19.>>>19.<<<23.>23 (this distribution is about optimal for overall accuracy of erac representations, but an interval directly between <<<<3 and >>3 as in 37edo would normally be <3).

	I use 37edo in the piano roll with six channels to reach 222, the superset of 74 and 111, which are a weird combination. 74 is a similar extension in spirit of 37, filling in missing intervals and continuing to straddle 3. On the other hand, 111 is a microtempered Buzzard edo (53+58) where the 17-limit JI lattice is more useful than straddling in most cases. 222 just adds extra intervals including a better flat 5 and 7 which I don't expect to use often.		I use 37edo in the piano roll with six channels to reach 222, the superset of 74 and 111, which are a weird combination. 74 is a similar extension in spirit of 37, filling in missing intervals and continuing to straddle 3. On the other hand, 111 is a microtempered Buzzard edo (53+58) where the 17-limit JI lattice is more useful than straddling in most cases. 222 just adds extra intervals including a better flat 5 and 7 which I don't expect to use often.

Ground at 13:01, 27 August 2026

2026-08-27T13:01:12Z

← Older revision		Revision as of 13:01, 27 August 2026
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	</math>}}		</math>}}

	{{Adv\|I don’t know if this is audibly significant, but at least it’s a cool justification for specifically 37edo as a neogothic blackdye tuning.}}		{{Adv\|I don’t know if this is audibly significant, but at least it’s a cool justification for specifically 37edo as a neogothic blackdye tuning. It is also a good tuning of "[[Ground's composition theory#Quasi-diatonic scales\|hiberidian]]"}}

			74edo scales:
			* 12:5:2 Diasem (eridian meantone)
			* 12:4:3 Diasem (Animist detemper of semiquartal, supermajor third is ~9/7 and subfourth is ~13/10)
			* Half-octave Porcupine (9/7)
			* Half-octave Oceanfront (17/14)

			111edo scales:
			* 5L2m3s4t
			* Vengeance

	== Multiples ==		== Multiples ==

Ground at 12:24, 27 August 2026

2026-08-27T12:24:41Z

← Older revision		Revision as of 12:24, 27 August 2026
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	{{UserTag\|g_\|Ground\|7766ff\|With no regard for structure, 37edo is an extremely special set of notes, which transcends structure. If you assume that approximating successive primes is good, this is a good tuning. If you think straddling 3 causes things to bend in a desirable way, this is the perfect tuning. After years of experience, I think 37edo is best described as 2.<<3.>3.5.7.11.13.17.19.		{{UserTag\|g_\|Ground\|7766ff\|With no regard for structure, 37edo is an extremely special set of notes, which transcends structure. If you assume that approximating successive primes is good, this is a good tuning. If you think straddling 3 causes things to bend in a desirable way, this is the perfect tuning. After years of experience, I think 37edo is best described as 2.<<3.>3.5.7.11.13.17.19.

	74 doesn't change much over 37, just adding a flat 3 and a reasonable 23, but the sharp 9 makes a big difference in available intervals, especially 9/7, which I think is the most obvious gap in 37edo. A more granular erac subgroup is 2.<3.>>>3.<<<5.>5.<<<7.>7.11.<<<13.>13.<<17.>>17.<19.>>>19.<<<23.>23 (this distribution is about optimal for overall accuracy).		74 doesn't change much over 37, just adding a flat 3 and a reasonable 23, but the sharp 9 makes a big difference in available intervals, especially 9/7, which I think is the most obvious gap in 37edo. A more granular erac subgroup is 2.<3.>>>3.<<<5.>5.<<<7.>7.11.<<<13.>13.<<17.>>17.<19.>>>19.<<<23.>23 (this distribution is about optimal for overall accuracy of erac representations, but an interval directly between <<<<3 and >>3 as in 37edo would normally be <3).

	I use 37edo in the piano roll with six channels to reach 222, the superset of 74 and 111, which are weird ~~together~~. 74 is a ~~pretty~~ similar extension of 37, ~~whereas~~ 111 is a microtempered Buzzard edo, 53+58.		I use 37edo in the piano roll with six channels to reach 222, the superset of 74 and 111, which are a weird combination. 74 is a similar extension in spirit of 37, filling in missing intervals and continuing to straddle 3. On the other hand, 111 is a microtempered Buzzard edo (53+58) where the 17-limit JI lattice is more useful than straddling in most cases. 222 just adds extra intervals including a better flat 5 and 7 which I don't expect to use often.
	}}		}}

	{{Navbox EDO}}		{{Navbox EDO}}
	{{Cat\|edos}}		{{Cat\|edos}}

Ground at 05:10, 27 August 2026

2026-08-27T05:10:08Z

← Older revision		Revision as of 05:10, 27 August 2026
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	== Multiples ==		== Multiples ==
			=== 74edo ===
			74edo is best known as a nearly optimal tuning of Grosstone, an extension of Septimal Meantone. Its flat 3 and 37edo's sharp 3 stack to make a sharp 9, which fills some gaps in 37edo's sharp-tending 13-limit. It adds a half-octave period which turns Porcupine into Stearn, a very efficient 2.3.7 restriction of Hedgehog. It is a decent [[Monarch]] tuning.
			{{Harmonics in ED\|74\|37\|0}}
	=== 111edo ===		=== 111edo ===
	111edo is a ~~significant~~ multiple of 37edo that adds an accurate prime 3. It is notable for being an [[Aberschismic]] system (tempering out 5120/5103, equating 81/80 and 64/63)~~; it is also~~ a [[Buzzard]] tuning with exact [[interordinal]] intervals. It is consistent in the 21-odd-limit ~~though~~ sharp-tending.		111edo is a multiple of 37edo that adds an accurate prime 3 to 37edo's 13-limit, making it significant to RTT. It is notable for being an [[Aberschismic]] system (tempering out 5120/5103, equating 81/80 and 64/63) and a [[Buzzard]] tuning with exact [[interordinal]] intervals. It is highly accurate in the 17-limit and consistent in the 21-odd-limit, although notably sharp-tending. It contains good tunings of [[Pentagoth\|Cohemimabila]] and [[Hibernal]].
	{{Harmonics in ED\|111\|37\|0}}		{{Harmonics in ED\|111\|37\|0}}

			{{UserTag\|g_\|Ground\|7766ff\|With no regard for structure, 37edo is an extremely special set of notes, which transcends structure. If you assume that approximating successive primes is good, this is a good tuning. If you think straddling 3 causes things to bend in a desirable way, this is the perfect tuning. After years of experience, I think 37edo is best described as 2.<<3.>3.5.7.11.13.17.19.

			74 doesn't change much over 37, just adding a flat 3 and a reasonable 23, but the sharp 9 makes a big difference in available intervals, especially 9/7, which I think is the most obvious gap in 37edo. A more granular erac subgroup is 2.<3.>>>3.<<<5.>5.<<<7.>7.11.<<<13.>13.<<17.>>17.<19.>>>19.<<<23.>23 (this distribution is about optimal for overall accuracy).

			I use 37edo in the piano roll with six channels to reach 222, the superset of 74 and 111, which are weird together. 74 is a pretty similar extension of 37, whereas 111 is a microtempered Buzzard edo, 53+58.
			}}

	{{Navbox EDO}}		{{Navbox EDO}}
	{{Cat\|edos}}		{{Cat\|edos}}

Inthar at 15:51, 6 June 2026

2026-06-06T15:51:02Z

← Older revision		Revision as of 15:51, 6 June 2026
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	== Multiples ==		== Multiples ==
	=== 111edo ===		=== 111edo ===
	111edo is a significant multiple of 37edo that adds an accurate prime 3. It is notable for being ~~a Hemifamity/~~Aberschismic system (tempering out 5120/5103, equating 81/80 and 64/63); it is also a [[Buzzard]] tuning with exact [[interordinal]] intervals. It is consistent in the 21-odd-limit though sharp-tending.		111edo is a significant multiple of 37edo that adds an accurate prime 3. It is notable for being an [[Aberschismic]] system (tempering out 5120/5103, equating 81/80 and 64/63); it is also a [[Buzzard]] tuning with exact [[interordinal]] intervals. It is consistent in the 21-odd-limit though sharp-tending.
	{{Harmonics in ED\|111\|37\|0}}		{{Harmonics in ED\|111\|37\|0}}
	{{Navbox EDO}}		{{Navbox EDO}}
	{{Cat\|edos}}		{{Cat\|edos}}

Inthar at 02:40, 19 April 2026

2026-04-19T02:40:41Z

← Older revision		Revision as of 02:40, 19 April 2026
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	=== Scales ===		=== Scales ===
	As mentioned previously, 37edo supports not only the very hard mosdiatonic generated by its sharp fifth, and the soft antidiatonic generated by its flat fifth, but a "dual-3" diatonic resembling 38edo's mosdiatonic, generated by alternating the two fifths. It also has a Zarlino diatonic where, like in 15 and 22edo, the three different step sizes are equidistant, meaning a single accidental can be used to cover the Zarlino scale. (In fact, this makes it a MODMOS of ~~porcupine~~[7].		As mentioned previously, 37edo supports not only the very hard mosdiatonic generated by its sharp fifth, and the soft antidiatonic generated by its flat fifth, but a "dual-3" diatonic resembling 38edo's mosdiatonic, generated by alternating the two fifths. It also has a Zarlino diatonic where, like in 15 and 22edo, the three different step sizes are equidistant, meaning a single accidental can be used to cover the Zarlino scale. (In fact, this makes it a MODMOS of Porcupine[7].

	[Add groundfault scales here]		[Add groundfault scales here]

	=== Regular temperaments ===		=== Regular temperaments ===
	If the two versions of the third harmonic are treated as stacking to the 9th harmonic, then 37edo supports a form of "2.<3.>3.5" meantone, where two sharp fifths and two flat fifths stack to create a 5/1. Another important temperament 37edo supports using only the sharp fifth is ~~porcupine~~, which it shares with 15edo and 22edo and for which it is the first edo that features a reasonable extension to the 13-limit.		If the two versions of the third harmonic are treated as stacking to the 9th harmonic, then 37edo supports a form of "2.<3.>3.5" meantone, where two sharp fifths and two flat fifths stack to create a 5/1. Another important temperament 37edo supports using only the sharp fifth is Porcupine, which it shares with 15edo and 22edo and for which it is the first edo that features a reasonable extension to the 13-limit.

	== Notation ==		== Notation ==
	As an archy edo, 37edo may be notated with diatonic notation, or with KISS or diamond-mos notation using its flat antidiatonic fifth (which also has the advantage of accidentals only raising or lowering by one step). It may also be notated with a straddle-3 erac notation system, taking advantage of its straddle-3 diatonic — this has an advantage in that 5/4 is a major third — or by [[~~porcupine~~]] zarlino notation, which has the same advantage.		As an archy edo, 37edo may be notated with diatonic notation, or with KISS or diamond-mos notation using its flat antidiatonic fifth (which also has the advantage of accidentals only raising or lowering by one step). It may also be notated with a straddle-3 erac notation system, taking advantage of its straddle-3 diatonic — this has an advantage in that 5/4 is a major third — or by [[Porcupine]] zarlino notation, which has the same advantage.

	== Ground's notes ==		== Ground's notes ==

Inthar at 02:40, 19 April 2026

2026-04-19T02:40:21Z

← Older revision		Revision as of 02:40, 19 April 2026
Line 83:		Line 83:

	== Notation ==		== Notation ==
	As an archy edo, 37edo may be notated with diatonic notation, or with KISS or diamond-mos notation using its flat antidiatonic fifth (which also has the advantage of accidentals only raising or lowering by one step). It may also be notated with a straddle-3 erac notation system, taking advantage of its straddle-3 diatonic - this has an advantage in that 5/4 is a major third - or by [[porcupine]] zarlino notation, which has the same advantage.		As an archy edo, 37edo may be notated with diatonic notation, or with KISS or diamond-mos notation using its flat antidiatonic fifth (which also has the advantage of accidentals only raising or lowering by one step). It may also be notated with a straddle-3 erac notation system, taking advantage of its straddle-3 diatonic — this has an advantage in that 5/4 is a major third — or by [[porcupine]] zarlino notation, which has the same advantage.

	== Ground's notes ==		== Ground's notes ==

Vector at 02:10, 19 April 2026

2026-04-19T02:10:22Z

← Older revision		Revision as of 02:10, 19 April 2026
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	=== JI approximation ===		=== JI approximation ===
	37edo's most notable ~~feature~~ is its status as a "dual-3" system, meaning that not only are there two tunings of 3 that can be combined to create an accurate 9/8, but those tunings of 3 are also rather accurate on their own. As a dual-3 system, it features the dual-3 diatonic scale 5L 1m 1s, resembling 38edo's diatonic but with an edostep removed from one of the small steps.{{Harmonics in ED\|37\|31\|0}}		One of 37edo's most notable features is its status as a "dual-3" system, meaning that not only are there two tunings of 3 that can be combined to create an accurate 9/8, but those tunings of 3 are also rather accurate on their own, especially its sharp fifth which functions as a diatonic fifth and the primary fifth in a regular temperament interpretation of the EDO. As a dual-3 system, it features the dual-3 diatonic scale 5L 1m 1s, resembling 38edo's diatonic but with an edostep removed from one of the small steps.{{Harmonics in ED\|37\|31\|0}}
	{\| class="wikitable"		{\| class="wikitable"
	\|+Thirds in 37edo		\|+Thirds in 37edo
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	\|14		\|14
	\|}		\|}
	Fifth-generated thirds are bolded.		Fifth-generated thirds (mosdiatonic and antidiatonic) are bolded.

	=== Chords ===		=== Chords ===
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	=== Scales ===		=== Scales ===
	As mentioned previously, 37edo supports not only the very hard mosdiatonic generated by its sharp fifth, and the soft antidiatonic generated by its flat fifth, but a "dual-3" diatonic resembling 38edo's mosdiatonic, generated by alternating the two fifths. It also has a Zarlino diatonic, ~~where as~~ in 15 and 22edo the three different step sizes are equidistant.		As mentioned previously, 37edo supports not only the very hard mosdiatonic generated by its sharp fifth, and the soft antidiatonic generated by its flat fifth, but a "dual-3" diatonic resembling 38edo's mosdiatonic, generated by alternating the two fifths. It also has a Zarlino diatonic where, like in 15 and 22edo, the three different step sizes are equidistant, meaning a single accidental can be used to cover the Zarlino scale. (In fact, this makes it a MODMOS of porcupine[7].

	[Add groundfault scales here]		[Add groundfault scales here]
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	== Notation ==		== Notation ==
	As an archy edo, 37edo may be notated with diatonic notation, or with KISS or diamond-mos notation using its flat antidiatonic fifth (which also has the advantage of accidentals only raising or lowering by one step). It may also be notated with a straddle-3 erac notation system, taking advantage of its straddle-3 diatonic~~. This~~ has an advantage in that 5/4 is a major third, or by [[porcupine]] zarlino notation.		As an archy edo, 37edo may be notated with diatonic notation, or with KISS or diamond-mos notation using its flat antidiatonic fifth (which also has the advantage of accidentals only raising or lowering by one step). It may also be notated with a straddle-3 erac notation system, taking advantage of its straddle-3 diatonic - this has an advantage in that 5/4 is a major third - or by [[porcupine]] zarlino notation, which has the same advantage.

	== Ground's notes ==		== Ground's notes ==

Inthar: /* 111edo */

2026-04-13T04:27:24Z

111edo

← Older revision		Revision as of 04:27, 13 April 2026
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	== Multiples ==		== Multiples ==
	=== 111edo ===		=== 111edo ===
	111edo is a significant multiple of 37edo that adds an accurate prime 3~~; it~~ is notable for being a Hemifamity/Aberschismic system (tempering out 5120/5103, equating 81/80 and 64/63); it is also a [[Buzzard]] tuning with exact [[interordinal]] intervals. It is consistent in the 21-odd-limit though sharp-tending.		111edo is a significant multiple of 37edo that adds an accurate prime 3. It is notable for being a Hemifamity/Aberschismic system (tempering out 5120/5103, equating 81/80 and 64/63); it is also a [[Buzzard]] tuning with exact [[interordinal]] intervals. It is consistent in the 21-odd-limit though sharp-tending.
	{{Harmonics in ED\|111\|37\|0}}		{{Harmonics in ED\|111\|37\|0}}
	{{Navbox EDO}}		{{Navbox EDO}}
	{{Cat\|edos}}		{{Cat\|edos}}

Inthar: /* 111edo */

2026-03-09T03:05:25Z

111edo

← Older revision		Revision as of 03:05, 9 March 2026
Line 167:		Line 167:
	== Multiples ==		== Multiples ==
	=== 111edo ===		=== 111edo ===
	111edo is a significant multiple of 37edo that adds an accurate prime 3; it is notable for being a Hemifamity/Aberschismic system (tempering out 5120/5103, equating 81/80 and 64/63); it is also a [[Buzzard]] tuning. It is consistent in the 21-odd-limit though sharp-tending.		111edo is a significant multiple of 37edo that adds an accurate prime 3; it is notable for being a Hemifamity/Aberschismic system (tempering out 5120/5103, equating 81/80 and 64/63); it is also a [[Buzzard]] tuning with exact [[interordinal]] intervals. It is consistent in the 21-odd-limit though sharp-tending.
	{{Harmonics in ED\|111\|37\|0}}		{{Harmonics in ED\|111\|37\|0}}
	{{Navbox EDO}}		{{Navbox EDO}}
	{{Cat\|edos}}		{{Cat\|edos}}