29edo - Revision history

Vector at 20:49, 8 March 2026

2026-03-08T20:49:01Z

← Older revision		Revision as of 20:49, 8 March 2026
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	=== 58edo ===		=== 58edo ===
	58edo splits the octave and perfect fifth in half, and provides a greatly improved 17th harmonic, along with improved mappings for 5, 11, 13, and especially 7, that all share a strong sharp tendency. It is a good tuning for temperaments such as [[Diaschismic]], based on the half-octave, and [[Hemififths]], based on the neutral third.		58edo splits the octave and perfect fifth in half, and provides a greatly improved 17th harmonic, along with improved mappings for 5, 11, 13, and especially 7, that all share a strong sharp tendency. It is a good tuning for temperaments such as [[Diaschismic]], based on the half-octave, and [[Hemififths]], based on the neutral third. It is also rather strong for the purposes of interval categorization schemes such as [[ADIN]] due to the fact that it has both [[neutral]] and [[interordinal]] intervals.
	{{Harmonics in ED\|58\|37\|0}}		{{Harmonics in ED\|58\|37\|0}}

Inthar at 16:11, 2 March 2026

2026-03-02T16:11:14Z

← Older revision		Revision as of 16:11, 2 March 2026
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	[[File:29edo.png\|thumb\|29edo as the division of the fourth into 3 and 4 equal parts.]]		[[File:29edo.png\|thumb\|29edo as the division of the fourth into 3 and 4 equal parts.]]
	'''29 equal divisions of the octave''', or '''29edo''', is the tuning system which divides the [[2/1]] ratio into 29 equal parts of approximately 41.3 cents each. It is notable for providing an accurate approximation of [[Pythagorean tuning]], and for tuning the [[perfect fourth]] to a highly divisible interval of 12 steps, allowing 29edo to support structures such as [[~~porcupine~~]] and [[~~negri~~]].		'''29 equal divisions of the octave''', or '''29edo''', is the tuning system which divides the [[2/1]] ratio into 29 equal parts of approximately 41.3 cents each. It is notable for providing an accurate approximation of [[Pythagorean tuning]], and for tuning the [[perfect fourth]] to a highly divisible interval of 12 steps, allowing 29edo to support structures such as [[Porcupine]] and [[Negri]].

	== General theory ==		== General theory ==

Lériendil: /* 87edo */

2026-03-02T04:21:59Z

87edo

← Older revision		Revision as of 04:21, 2 March 2026
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	=== 87edo ===		=== 87edo ===
	87edo splits the octave and perfect fifth into three. As 29edo's primes 5, 11, and 13 are close to 1/3 of a step off, 87edo tunes them near just, especially prime 5. Prime 7 receives the most damage out of the 13-limit, but it is dented flatwards precisely enough to provide an essentially ideal tuning for [[Rodan]]. Other important structures that 87edo supports include [[Kleismic]] and [[Didacus]].		87edo splits the octave and perfect fifth into three. As 29edo's primes 5, 11, and 13 are close to 1/3 of a step off, 87edo tunes them near just, especially prime 5. Prime 7 (and especially the interval [[9/7]]) receives the most damage out of the 13-limit, but it is dented flatwards precisely enough to provide an essentially ideal tuning for [[Rodan]] while still maintaining 15-odd-limit consistency. Other important structures that 87edo supports include [[Kleismic]] and [[Didacus]].
	{{Harmonics in ED\|87\|37\|0}}		{{Harmonics in ED\|87\|37\|0}}

	{{Navbox EDO}}		{{Navbox EDO}}
	{{Cat\|Edos}}		{{Cat\|Edos}}

Lériendil: added multiples

2026-03-02T04:20:36Z

added multiples

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	\|Tsohorayim		\|Tsohorayim
	\|}		\|}

			== Multiples ==
			As 29edo's chain of fifths is quite accurate, while improvement is to be desired on other prime harmonics, it makes sense to consider supersets of 29edo. The temperament they share is called Mystery, preserving 29edo's 2.3.7/5.11/5.13/5 subgroup.

			=== 58edo ===
			58edo splits the octave and perfect fifth in half, and provides a greatly improved 17th harmonic, along with improved mappings for 5, 11, 13, and especially 7, that all share a strong sharp tendency. It is a good tuning for temperaments such as [[Diaschismic]], based on the half-octave, and [[Hemififths]], based on the neutral third.
			{{Harmonics in ED\|58\|37\|0}}

			=== 87edo ===
			87edo splits the octave and perfect fifth into three. As 29edo's primes 5, 11, and 13 are close to 1/3 of a step off, 87edo tunes them near just, especially prime 5. Prime 7 receives the most damage out of the 13-limit, but it is dented flatwards precisely enough to provide an essentially ideal tuning for [[Rodan]]. Other important structures that 87edo supports include [[Kleismic]] and [[Didacus]].
			{{Harmonics in ED\|87\|37\|0}}

	{{Navbox EDO}}		{{Navbox EDO}}
	{{Cat\|Edos}}		{{Cat\|Edos}}

Lériendil at 02:37, 2 March 2026

2026-03-02T02:37:49Z

← Older revision		Revision as of 02:37, 2 March 2026
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	[[File:29edo.png\|thumb\|29edo as the division of the fourth into 3 and 4 equal parts.]]		[[File:29edo.png\|thumb\|29edo as the division of the fourth into 3 and 4 equal parts.]]
	'''29 equal divisions of the octave''', or '''29edo''', is the tuning system which divides the [[2/1]] ratio into 29 equal parts of approximately 41.3 cents each. It is notable for ~~its extremely~~ accurate tuning ~~of prime 3~~, and for ~~unique melodic properties that proponents~~ of ~~the system consider particularly desirable~~.		'''29 equal divisions of the octave''', or '''29edo''', is the tuning system which divides the [[2/1]] ratio into 29 equal parts of approximately 41.3 cents each. It is notable for providing an accurate approximation of [[Pythagorean tuning]], and for tuning the [[perfect fourth]] to a highly divisible interval of 12 steps, allowing 29edo to support structures such as [[porcupine]] and [[negri]].

	== General theory ==		== General theory ==

Lériendil: /* General theory */

2026-03-02T02:36:31Z

General theory

← Older revision		Revision as of 02:36, 2 March 2026
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	29edo excels at approximating prime 3, but it does not have good approximations to most of the following prime harmonics. However, while 5, 7, 11, and 13 are not tuned well individually, they are all flat by nearly the same amount, and therefore 29edo represents ratios between these primes, such as 13/11 or 7/5, very well. Looking further afield, we find primes 29 and 37 approximated rather well; it should be noted that [[32/29]] splits 4/3 in three, and [[37/32]] splits it in two, garnering these two primes important structural roles.		29edo excels at approximating prime 3, but it does not have good approximations to most of the following prime harmonics. However, while 5, 7, 11, and 13 are not tuned well individually, they are all flat by nearly the same amount, and therefore 29edo represents ratios between these primes, such as 13/11 or 7/5, very well. Looking further afield, we find primes 29 and 37 approximated rather well; it should be noted that [[32/29]] splits 4/3 in three, and [[37/32]] splits it in two, garnering these two primes important structural roles.

	If the patent approximations of 5, 7, 11, and 13 are used, since they are at least rather unambiguous, the accuracy of prime 3 and their shared flatness ~~makes~~ 29edo, in fact, [[consistent]] to the [[15-odd-limit]]. We find that 19 and 23 share a flat tendency in common with them as well, and if we skip prime 17, the 23-odd-limit is tuned consistently.		If the patent approximations of 5, 7, 11, and 13 are used, since they are at least rather unambiguous, the accuracy of prime 3 and their shared flatness make 29edo, in fact, [[consistent]] to the [[15-odd-limit]]. We find that 19 and 23 share a flat tendency in common with them as well, and if we skip prime 17, the 23-odd-limit is tuned consistently.
	{{Harmonics in ED\|29\|37\|0}}		{{Harmonics in ED\|29\|37\|0}}

Lériendil: /* Tuning Theory */

2026-03-02T02:33:24Z

Tuning Theory

← Older revision		Revision as of 02:33, 2 March 2026
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	'''29 equal divisions of the octave''', or '''29edo''', is the tuning system which divides the [[2/1]] ratio into 29 equal parts of approximately 41.3 cents each. It is notable for its extremely accurate tuning of prime 3, and for unique melodic properties that proponents of the system consider particularly desirable.		'''29 equal divisions of the octave''', or '''29edo''', is the tuning system which divides the [[2/1]] ratio into 29 equal parts of approximately 41.3 cents each. It is notable for its extremely accurate tuning of prime 3, and for unique melodic properties that proponents of the system consider particularly desirable.

	== ~~Tuning Theory~~ ==		== General theory ==
			=== JI approximation ===
			29edo excels at approximating prime 3, but it does not have good approximations to most of the following prime harmonics. However, while 5, 7, 11, and 13 are not tuned well individually, they are all flat by nearly the same amount, and therefore 29edo represents ratios between these primes, such as 13/11 or 7/5, very well. Looking further afield, we find primes 29 and 37 approximated rather well; it should be noted that [[32/29]] splits 4/3 in three, and [[37/32]] splits it in two, garnering these two primes important structural roles.

	~~=== JI Approximation ===~~		If the patent approximations of 5, 7, 11, and 13 are used, since they are at least rather unambiguous, the accuracy of prime 3 and their shared flatness makes 29edo, in fact, [[consistent]] to the [[15-odd-limit]]. We find that 19 and 23 share a flat tendency in common with them as well, and if we skip prime 17, the 23-odd-limit is tuned consistently.
	~~While 29edo excels at prime 3,~~ the ~~rest~~ of ~~the primes up to 31~~ are ~~relatively lacking. However~~, ~~primes 5 through 13 have roughly~~ the ~~same amount~~ of ~~error~~, and in ~~the same direction; difference tones such~~ as ~~7/5~~ and ~~11/7 are thus rather accurate~~.		{{Harmonics in ED\|29\|37\|0}}

	{{Harmonics in ED\|29\|31\|0}}

	== Intervals and Notation ==		== Intervals and Notation ==

Hotcrystal0 at 17:50, 26 February 2026

2026-02-26T17:50:54Z

@@ Line 6: / Line 6: @@
 === JI Approximation ===
 While 29edo excels at prime 3, the rest of the primes up to 31 are relatively lacking.  However, primes 5 through 13 have roughly the same amount of error, and in the same direction; difference tones such as 7/5 and 11/7 are thus rather accurate.
-{| class="wikitable"
-|+Approximations of prime harmonics in 29edo
+{{Harmonics in ED|29|31|0}}
 == Intervals and Notation ==

Unque at 16:30, 23 February 2026

2026-02-23T16:30:03Z

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	The [[Gramitonic]] scale takes the role of a diminished scale in 29edo: since four neominor thirds fall short of the octave, the chain of neominor thirds can be extended into this nine-note scale. Note how the four bright modes resemble the pattern of the familiar [[Tetrawood\|octatonic scale]], with one of the small steps duplicated, and the four darkest modes resemble the rotated variant of that scale; additionally, there is a symmetrical mode that is entirely new to 29edo. The mode names for this scale are given by Lilly Flores.		The [[Gramitonic]] scale takes the role of a diminished scale in 29edo: since four neominor thirds fall short of the octave, the chain of neominor thirds can be extended into this nine-note scale. Note how the four bright modes resemble the pattern of the familiar [[Tetrawood\|octatonic scale]], with one of the small steps duplicated, and the four darkest modes resemble the rotated variant of that scale; additionally, there is a symmetrical mode that is entirely new to 29edo. The mode names for this scale are given by Lilly Flores.

	''TODO: ~~switch to~~ ups and downs''		''TODO: add ups and downs''

	{\| class="wikitable sortable mw-collapsible"		{\| class="wikitable sortable mw-collapsible"
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	Similarly to the neominor third, the neomajor third of 29edo also does not close at the octave, allowing us to create an [[Checkertonic\|8-note augmented scale]]. Just like the previous "diminished" scale, notice how the three brightest modes resemble the bright mode of the [[Tcherepnin]] scale, with one of the nine steps omitted; the three darkest modes similarly resemble the dark mode of that scale; and the remaining two modes both resemble the symmetrical mode of Tcherepnin. The mode names for this scale are given by R-4981.		Similarly to the neominor third, the neomajor third of 29edo also does not close at the octave, allowing us to create an [[Checkertonic\|8-note augmented scale]]. Just like the previous "diminished" scale, notice how the three brightest modes resemble the bright mode of the [[Tcherepnin]] scale, with one of the nine steps omitted; the three darkest modes similarly resemble the dark mode of that scale; and the remaining two modes both resemble the symmetrical mode of Tcherepnin. The mode names for this scale are given by R-4981.

	''TODO: ~~switch to~~ ups and downs''		''TODO: add ups and downs''

	{\| class="wikitable sortable mw-collapsible"		{\| class="wikitable sortable mw-collapsible"

Inthar: /* Machinoid */

2026-02-23T04:58:15Z

Machinoid

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			{{Navbox EDO}}
	{{Cat\|Edos}}		{{Cat\|Edos}}