29edo/Unque's compositional approach - Revision history

Unque: Major rearrangement of interval table

2026-02-01T04:48:48Z

Major rearrangement of interval table

Unque: prime 37

2026-01-23T22:59:25Z

prime 37

← Older revision		Revision as of 22:59, 23 January 2026
Line 315:		Line 315:
	(7)		(7)
	\|}		\|}
	Additionally, 4\29 can be interpreted as 32/29, adding prime 29 to the subgroup; this allows the chromatic semitone to be interpreted as ~~the simple and accurate~~ 29/27, the supraminor third as 29/24, and the submajor third as 36/29.		Because, for example, 11/7 can be found at (11/5)/(7/5), we only need to add three fractions to the subgroup; here, I'll use 7/5, 11/5, and 13/5.

			Additionally, 4\29 can be interpreted as 32/29, adding prime 29 to the subgroup; this allows the chromatic semitone to be interpreted as 29/27, the supraminor third as 29/24, and the submajor third as 36/29.

			Finally, the arto third / semifourth at 6\29 can be interpreted as 37/32, adding prime 37 to the subgroup; this allows the upfourth to be interpreted as 37/27, the tendo third as 48/37, and the diesis as 37/36.

			So in total, our accurate subgroup for 29edo is '''2.3.7/5.11/5.13/5.29.37'''. Not bad for a tuning that supposedly isn't useful beyond the 3-limit.

	== Chords of 29edo ==		== Chords of 29edo ==

Unque: Enharmonic notes

2026-01-23T16:36:34Z

Enharmonic notes

← Older revision		Revision as of 16:36, 23 January 2026
Line 835:		Line 835:

	=== 4L 3s ===		=== 4L 3s ===
	The [[4L 3s]] scale can be thought of as an alteration of the Harmonic Minor scale, which is unique to 29edo. If we notice that the augmented second is precisely three steps larger than a major second, we can distribute this error amongst the three semitones that occur in the scale, which reduces the scale to a maximum variety of two. We may also notice that this scale's pattern creates a circle of ~~augmented seconds~~, which can be used to quantify the brightness of the seven modes.		The [[4L 3s]] scale can be thought of as an alteration of the Harmonic Minor scale, which is unique to 29edo. If we notice that the augmented second is precisely three steps larger than a major second, we can distribute this error amongst the three semitones that occur in the scale, which reduces the scale to a maximum variety of two. We may also notice that this scale's pattern creates a circle of upminor thirds, which can be used to quantify the brightness of the seven modes.

	See [[User:Unque/Dietic Minor\|Dietic Minor]] for a more in-depth discussion of how the Harmonic Minor structure can be treated in 29edo, and how this idea generalizes to other tuning systems.		See [[User:Unque/Dietic Minor\|Dietic Minor]] for a more in-depth discussion of how the Harmonic Minor structure can be treated in 29edo, and how this idea generalizes to other tuning systems.

Unque at 14:44, 23 January 2026

2026-01-23T14:44:12Z

← Older revision		Revision as of 14:44, 23 January 2026
Line 268:		Line 268:

	One benefit of using extraclassical intervals is that the art and tendo thirds (unlike their diatonic counterparts) do not crowd one another when used in a chord together, being separated by a whole tone rather than a chroma.		One benefit of using extraclassical intervals is that the art and tendo thirds (unlike their diatonic counterparts) do not crowd one another when used in a chord together, being separated by a whole tone rather than a chroma.

			=== Harmonic Approximations ===
			Critics of 29edo have often said that it fails to approximate most ratios beyond the 3-limit; primes 5, 7, 11, and 13 are all of relatively high error, and as such the tuning does not provide the necessary colors to suffice as an introductory system to tunings beyond 12edo.

			Assuming for the sake of the argument that approximating JI is somehow the only thing that one may want of a tuning, and further assuming that one even knows what that entails on their first venture into xenharmonic tunings, 29edo does not fall quite as flat as some would have you believe. Notably, the aforementioned error on primes 5, 7, 11, and 13 are all in the same direction and of roughly the same amount, which leads their difference tones to be tuned quite well.
			{\| class="wikitable"
			\|+13-limit difference tones
			! colspan="2" \|Interval
			!7/5
			!11/5
			!13/5
			!11/7
			!13/7
			!13/11
			\|-
			! rowspan="2" \|Error
			!Absolute (¢)
			\| -3.2
			\| +0.5
			\| +1.0
			\| +3.7
			\| +4.2
			\| +0.5
			\|-
			!Relative (%)
			\| -7.7
			\| +1.2
			\| +2.3
			\| +8.9
			\| +10.0
			\| +1.1
			\|-
			! colspan="2" \|Steps
			(Reduced)
			\|14
			(14)
			\|33
			(4)
			\|40
			(11)
			\|19
			(19)
			\|26
			(26)
			\|7
			(7)
			\|}
			Additionally, 4\29 can be interpreted as 32/29, adding prime 29 to the subgroup; this allows the chromatic semitone to be interpreted as the simple and accurate 29/27, the supraminor third as 29/24, and the submajor third as 36/29.

	== Chords of 29edo ==		== Chords of 29edo ==

Unque at 01:15, 17 January 2026

2026-01-17T01:15:35Z

← Older revision		Revision as of 01:15, 17 January 2026
Line 1,586:		Line 1,586:
	== See Also ==		== See Also ==

	* [[User:Unque/29edo Counterpoint Treatise]]		* For my treatise on counterpoint in particular: [[User:Unque/29edo Counterpoint Treatise\|29edo Counterpoint Treatise]]
			* For my discussion of divisions of the fourth and other intervals: [[User:Unque/On Voice Leading\|On Voice Leading]]

	[[Category:Approaches to tuning systems]]		[[Category:Approaches to tuning systems]]
	[[Category:29edo]]		[[Category:29edo]]

Unque: Noted "division of the fourth" scales

2025-11-18T19:19:43Z

Noted "division of the fourth" scales

← Older revision		Revision as of 19:19, 18 November 2025
Line 1,048:		Line 1,048:
	\|C - E𝄫 - F♭ - G♭ - A♭ - B♭ - C		\|C - E𝄫 - F♭ - G♭ - A♭ - B♭ - C
	\|Tsohorayim		\|Tsohorayim
			\|}

			=== 7L 1s ===
			One interesting structural property of 29edo is that the perfect fourth, at twelve steps wide, is highly divisible. This allows it to be split into various divisors that can be used to access harmonies that are all at once familiar and alien. Specifically, splitting the fourth into three steps, we find a downmajor second and upminor third leading to it, and we can extend this pattern to create [[7L 1s\|an 8-note scale]].

			The mode names for this scale are given by Willian Lynch.
			{\| class="wikitable sortable mw-collapsible"
			\|+Modes of 7L 1s
			!Gens Up
			!Step Pattern
			!Notation
			!Name (Lynch)
			\|-
			\|7
			\|LLLLLLLs
			\|C - vD - ^E♭ - F - vG - ^A♭ - B♭ - vC - C
			\|Octopus
			\|-
			\|6
			\|LLLLLLsL
			\|C - vD - ^E♭ - F - vG - ^A♭ - B♭ - ^B♭ - C
			\|Mantis
			\|-
			\|5
			\|LLLLLsLL
			\|C - vD - ^E♭ - F - vG - ^A♭ - vA - ^B♭ - C
			\|Dolphin
			\|-
			\|4
			\|LLLLsLLL
			\|C - vD - ^E♭ - F - vG - G - vA - ^B♭ - C
			\|Crab
			\|-
			\|3
			\|LLLsLLLL
			\|C - vD - ^E♭ - F - ^F - G - vA - ^B♭ - C
			\|Tuna
			\|-
			\|2
			\|LLsLLLLL
			\|C - vD - ^E♭ - vE - ^F - G - vA - ^B♭ - C
			\|Salmon
			\|-
			\|1
			\|LsLLLLLL
			\|C - vD - D - vE - ^F - G - vA - ^B♭ - C
			\|Starfish
			\|-
			\|0
			\|sLLLLLLL
			\|C - ^C - D - vE - ^F - G - vA - ^B♭ - C
			\|Whale
	\|}		\|}

	=== 5L 4s ===		=== 5L 4s ===
	The [[5L 4s]] scale is the ~~first truly unusual scale~~ in ~~29edo, being created via a stack of perfect chthonic intervals~~. ~~This means~~ that ~~every second interval in~~ the ~~chain will represent an interval from the familiar circle~~ of ~~fifths, whereas each interval between them will be an entirely alien~~ interordinal.		The [[5L 4s]] scale follows from similar principles to 7L 1s, this time splitting the perfect fourth in two parts rather than three. Continuing this sequence provides a 9-note scale that is significantly more alien than the previous, due to its repeated inclusion of true interordinal intervals.

	The mode names for this scale are given by [[User:Inthar\|Inthar]].		The mode names for this scale are given by [[User:Inthar\|Inthar]].
Line 1,063:		Line 1,115:
	\|8		\|8
	\|LLsLsLsLs		\|LLsLsLsLs
	\|C - D - E - ^E - ^F♯ - G - A - ~~vB♭~~ - ^B - C		\|C - D - E - ^E - ^F♯ - G - A - ^A - ^B - C
	\|Cristacan		\|Cristacan
	\|-		\|-
	\|7		\|7
	\|LsLLsLsLs		\|LsLLsLsLs
	\|C - D - ^D - ^E - ^F♯ - G - A - ~~vB♭~~ - ^B - C		\|C - D - ^D - ^E - ^F♯ - G - A - ^A - ^B - C
	\|Pican		\|Pican
	\|-		\|-
	\|6		\|6
	\|LsLsLLsLs		\|LsLsLLsLs
	\|C - D - ^D - ^E - F - G - A - ~~vB♭~~ - ^B - C		\|C - D - ^D - ^E - F - G - A - ^A - ^B - C
	\|Stellerian		\|Stellerian
	\|-		\|-
	\|5		\|5
	\|LsLsLsLLs		\|LsLsLsLLs
	\|C - D - ^D - ^E - F - G - ^G - ~~vB♭~~ - ^B - C		\|C - D - ^D - ^E - F - G - ^G - ^A - ^B - C
	\|Podocian		\|Podocian
	\|-		\|-
	\|4		\|4
	\|LsLsLsLsL		\|LsLsLsLsL
	\|C - D - ^D - ^E - F - G - ^G - ~~vB♭~~ - B♭ - C		\|C - D - ^D - ^E - F - G - ^G - ^A - B♭ - C
	\|Nucifragan		\|Nucifragan
	\|-		\|-
	\|3		\|3
	\|sLLsLsLsL		\|sLLsLsLsL
	\|C - ^C - ^D - ^E - F - G - ^G - ~~vB♭~~ - B♭ - C		\|C - ^C - ^D - ^E - F - G - ^G - ^A - B♭ - C
	\|Coracian		\|Coracian
	\|-		\|-
	\|2		\|2
	\|sLsLLsLsL		\|sLsLLsLsL
	\|C - ^C - ^D - E♭ - F - G - ^G - ~~vB♭~~ - B♭ - C		\|C - ^C - ^D - E♭ - F - G - ^G - ^A - B♭ - C
	\|Frugilegian		\|Frugilegian
	\|-		\|-
	\|1		\|1
	\|sLsLsLLsL		\|sLsLsLLsL
	\|C - ^C - ^D - E♭ - F - ^F - ^G - ~~vB♭~~ - B♭ - C		\|C - ^C - ^D - E♭ - F - ^F - ^G - ^A - B♭ - C
	\|Temnurial		\|Temnurial
	\|-		\|-

Unque at 17:12, 31 August 2025

2025-08-31T17:12:26Z

← Older revision		Revision as of 17:12, 31 August 2025
Line 1,531:		Line 1,531:

	In this example, the circle of submajor thirds is extremely relevant to our scale, as this is the circle that defines the scale; if we were to visualize this circle, we would see that C - vE - ^G - B forms a continuous sequence, and thus we can move through ''vmaj ^5'' chords on each of these roots to smoothly move from C to B. If we treat the submajor third as an approximation of 5/4, this can also be seen as drifting by the comma [[128/125]].		In this example, the circle of submajor thirds is extremely relevant to our scale, as this is the circle that defines the scale; if we were to visualize this circle, we would see that C - vE - ^G - B forms a continuous sequence, and thus we can move through ''vmaj ^5'' chords on each of these roots to smoothly move from C to B. If we treat the submajor third as an approximation of 5/4, this can also be seen as drifting by the comma [[128/125]].

			== See Also ==

			* [[User:Unque/29edo Counterpoint Treatise]]

	[[Category:Approaches to tuning systems]]		[[Category:Approaches to tuning systems]]
	[[Category:29edo]]		[[Category:29edo]]

Unque at 17:32, 20 August 2025

2025-08-20T17:32:46Z

← Older revision		Revision as of 17:32, 20 August 2025
Line 1,525:		Line 1,525:
	These changes can also be performed in the opposite direction, giving us the progression ''C vmaj - C sus2 - G vmaj'', or equivalently ''C vmaj - G sus4 - G vmaj''; the former version of the progression, with all tertiary triads, creates a more blended sound due to the chords all being of similar quality, while the latter creates a more expressive sound with the ''G sus4'' chord creating a resolution to ''G vmaj''.		These changes can also be performed in the opposite direction, giving us the progression ''C vmaj - C sus2 - G vmaj'', or equivalently ''C vmaj - G sus4 - G vmaj''; the former version of the progression, with all tertiary triads, creates a more blended sound due to the chords all being of similar quality, while the latter creates a more expressive sound with the ''G sus4'' chord creating a resolution to ''G vmaj''.

	=== Moving ~~Modes~~ ===		=== Moving Transpositions ===
	For this example, presume that we are working with the 3L 4s "neutral" scale, beginning on the Kleeth mode on C. Let's then presume that we want to transition into a section that is rooted in the Kleeth move on B, a transposition of the original scale. Because the two scales contain the same relative pitches on different degrees, we need a way to make this change without sounding like a jarring leap.		For this example, presume that we are working with the 3L 4s "neutral" scale, beginning on the Kleeth mode on C. Let's then presume that we want to transition into a section that is rooted in the Kleeth move on B, a transposition of the original scale. Because the two scales contain the same relative pitches on different degrees, we need a way to make this change without sounding like a jarring leap.

Unque: Added section on mode and scale changes, plus a bit of minor cleanup elsewhere

2025-08-20T16:41:31Z

Added section on mode and scale changes, plus a bit of minor cleanup elsewhere

@@ Line 1,215: / Line 1,215: @@
 === 5L 2M 3s ===
-The [[Blackdye|5L 2M 3s]] scale is an extension of 3L 2M 2s that adds three additional tones, created by inserting a dietic step into each of the large steps, and unifies the two chiralities of the modes.  It can also be constructed using two pentatonic scales of [[2L 3s]], where the roots of the scales differ by an interval of 4\29.  This construction allows us to separate the ten modes into five "acute" modes and five "grave" modes: the grave modes place the root somewhere in the flatter pentatonic, and the acute modes place the root on the sharper one.
+The [[Blackdye|5L 2M 3s]] scale, often called Blackdye, is an extension of 3L 2M 2s that adds three additional tones, created by inserting a dietic step into each of the large steps, and unifies the two chiralities of the modes.  It can also be constructed using two pentatonic scales of [[2L 3s]], where the roots of the scales differ by an interval of 4\29.  This construction allows us to separate the ten modes into five "acute" modes and five "grave" modes: the grave modes place the root somewhere in the flatter pentatonic, and the acute modes place the root on the sharper one.
 By noticing that the spacer 4\29 differs from the pentatonic large step by a chroma, we can most effectively notate the scale by treating that spacer as a diminished third; while the usage of double-flats in the grave modes and double-sharps in the acute modes makes this notation seem a bit unruly at first, it bypasses the additional ups and downs that would be necessitated if we were to treat the spacer as a submajor second or an upchroma.
@@ Line 1,513: / Line 1,513: @@
 Thus, our final progression is ''C ^min ^5 - ^F ^min v5 - D ^min v5 - vA vct ^4''.  This progression uses a combination of voice leading and circle of fifths movement to create a sound that is both dynamic and functional.
 [[Category:Approaches to tuning systems]]

Unque: Spelling, grammar, and slight wording changes to make everything slightly more coherent.

2025-07-03T20:43:55Z

Spelling, grammar, and slight wording changes to make everything slightly more coherent.

← Older revision		Revision as of 20:43, 3 July 2025
Line 657:		Line 657:

	===5L 2s===		===5L 2s===
	The [[5L 2s]] scale is generated by taking seven adjacent tones from the Circle of Fifths, just as it is in 12edo. Melodies and chords made using this scale will sound nearly identical to those that can be made using 12edo.		The [[5L 2s]] scale is generated by taking seven adjacent tones from the Circle of Fifths, just as it is in 12edo. Melodies and chords made using this scale will sound nearly identical to those that can be made using 12edo, with the notable exception of the tritone (which comes in two distinct forms depending on which mode it's found in).
	{\| class="wikitable sortable mw-collapsible"		{\| class="wikitable sortable mw-collapsible"
	\|+ style="font-size: 105%;" \|Modes of 5L 2s		\|+ style="font-size: 105%;" \|Modes of 5L 2s
Line 843:		Line 843:
	\|}		\|}
	===4L 5s===		===4L 5s===
	The [[4L 5s]] scale takes the role of a diminished scale in 29edo~~. Since~~ four minor thirds fall short of the octave, the chain of minor thirds can be extended into this enneatonic form. Note how the four bright modes resemble the pattern of the familiar 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 [[4L 5s]] scale takes the role of a diminished scale in 29edo: since four minor thirds fall short of the octave, the chain of minor thirds can be extended into this enneatonic form. Note how the four bright modes resemble the pattern of the familiar 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 mode names for this scale are given by Lilly Flores.
Line 900:		Line 900:
	\|}		\|}
	===3L 5s===		===3L 5s===
	Similarly to the minor third, the major third of 29edo also does not close at the octave, allowing us to create an octatonic augmented scale. Just like the diminished scale, notice how the three brightest modes resemble the bright mode of the [[3L 6s\|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.		Similarly to the minor third, the major third of 29edo also does not close at the octave, allowing us to create an [[3L 5s\|octatonic augmented scale]]. Just like the diminished scale, notice how the three brightest modes resemble the bright mode of the [[3L 6s\|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 [[User:R-4981\|R-4981]].		The mode names for this scale are given by [[User:R-4981\|R-4981]].
Line 953:		Line 953:

	=== 3L 4s ===		=== 3L 4s ===
	~~The first truly unheard-of scale~~ that ~~29edo pulls off~~ is its approximation of the neutral scale ~~by stacking the downmajor third seven times~~. Like 4L 3s, this scale uses harmony based on upfifths and downfifths rather than normal perfect fifths, which makes its harmony more distant from familiar structures. Just like 5L 3s, it can be compared to the Tcherepnin scale, and as such it relies on augmented ~~triads~~ as its source of harmony; however, this scale pattern removes two of the nine Tcherepnin steps rather than three, reducing it to a more standard heptatonic form.		29edo additionally offers scales that 12edo cannot remotely approximate; among these is its approximation of the [[3L 4s\|neutral scale]], made via a stack of submajor sevenths. Like 4L 3s, this scale uses harmony based on upfifths and downfifths rather than normal perfect fifths, which makes its harmony more distant from familiar structures. Just like 5L 3s, it can be compared to the Tcherepnin scale, and as such it relies on a kind of augmented triad as its main source of harmony; however, this scale pattern removes two of the nine Tcherepnin steps rather than three, reducing it to a more standard heptatonic form.

	The modes names for this scale are given by [[Andrew Heathwaite]]. They can also be named by comparing two diatonic modes.		The modes names for this scale are given by [[Andrew Heathwaite]]. They can also be named by comparing two diatonic modes.
Line 1,008:		Line 1,008:

	=== 5L 1s ===		=== 5L 1s ===
	Just like the thirds, we can notice that the whole tones in 29edo do not close at the octave; instead, we see that five whole tones exceed the minor seventh by an edostep. However, the octave can still be closed by employing a diminished third to act as a "wolf" version of the whole tone; this leads to ~~the~~ scale ~~having~~ six distinct modes, rather than having an identical pattern on every degree as 12edo had.		Just like the thirds, we can notice that the whole tones in 29edo do not close at the octave; instead, we see that five whole tones exceed the minor seventh by an edostep. However, the octave can still be closed by employing a diminished third to act as a "wolf" version of the whole tone; this leads to [[5L 1s\|a whole tone scale that has six distinct modes]], rather than having an identical pattern on every degree as 12edo had.

	The mode names for this scale are given by Lilly Flores.		The mode names for this scale are given by Lilly Flores.
Line 1,110:		Line 1,110:

	=== 3L 2M 2s ===		=== 3L 2M 2s ===
	In terms of its harmonies and melodies, [[Nicetone\|3L 2M 2s]] acts as a half-way point between 5L 2s and 3L 4s. The scale is generated using an alternating stack of wide and narrow neutral thirds; as such, the thirds are neutral like 3L 4s, but its fifths are perfect like 5L 2s, which makes it a very interesting extension of ideas from both MOS scales. Additionally, the wide neutral third is closer to 5/4 than the diatonic major third is (though admittedly not by a lot), which makes the ~~downmajor~~ triad a decent analog to the [[4:5:6]] triad that is so prevalent in 5-limit music.		In terms of its harmonies and melodies, [[Nicetone\|3L 2M 2s]] acts as a half-way point between 5L 2s and 3L 4s. The scale is generated using an alternating stack of wide and narrow neutral thirds; as such, the thirds are neutral like 3L 4s, but its fifths are perfect like 5L 2s, which makes it a very interesting extension of ideas from both MOS scales. Additionally, the wide neutral third is closer to 5/4 than the diatonic major third is (though admittedly not by a lot), which makes the submajor triad a decent analog to the [[4:5:6]] triad that is so prevalent in 5-limit music.

	There are two chiralities of the 3L 2M 2s scale based on which of the two neutral thirds you stack first; using the wider third first yields the right-hand version of the scale, while using the narrower third first yields the left-hand version.		There are two chiralities of the 3L 2M 2s scale based on which of the two neutral thirds you stack first; using the wider third first yields the right-hand version of the scale, while using the narrower third first yields the left-hand version.
Line 1,215:		Line 1,215:

	=== 5L 2M 3s ===		=== 5L 2M 3s ===
	The [[Blackdye\|5L 2M 3s]] scale is an extension of 3L 2M 2s that adds three additional tones, created by inserting a dietic step into each of the large steps, and unifies the two chiralities of the modes. It can also be constructed using two pentatonic scales of 2L 3s, where the roots of the scales differ by an interval of 4\29~~; this~~ construction allows us to separate the ten modes into five "acute" modes and five "grave" modes~~, with~~ the grave modes place the root somewhere in the flatter pentatonic, and the acute modes place the root on the sharper one.		The [[Blackdye\|5L 2M 3s]] scale is an extension of 3L 2M 2s that adds three additional tones, created by inserting a dietic step into each of the large steps, and unifies the two chiralities of the modes. It can also be constructed using two pentatonic scales of [[2L 3s]], where the roots of the scales differ by an interval of 4\29. This construction allows us to separate the ten modes into five "acute" modes and five "grave" modes: the grave modes place the root somewhere in the flatter pentatonic, and the acute modes place the root on the sharper one.

	By noticing that the spacer 4\29 differs from the pentatonic large step by a chroma, we can most effectively notate the scale by treating that spacer as a diminished third; while the usage of double-flats in the grave modes and double-sharps in the acute modes makes this notation seem a bit unruly at first, it bypasses the additional ups and downs that would be necessitated if we were to treat the spacer as a ~~downmajor~~ second or an upchroma.		By noticing that the spacer 4\29 differs from the pentatonic large step by a chroma, we can most effectively notate the scale by treating that spacer as a diminished third; while the usage of double-flats in the grave modes and double-sharps in the acute modes makes this notation seem a bit unruly at first, it bypasses the additional ups and downs that would be necessitated if we were to treat the spacer as a submajor second or an upchroma.
	{\| class="wikitable"		{\| class="wikitable"
	\|+Grave Modes of 5L 2M 3s		\|+Grave Modes of 5L 2M 3s
Line 1,500:		Line 1,500:
	29edo has three unique types of leading tones: from narrowest to widest, they are the [[Pythagorean comma\|diesis]] (1\29), the [[256/243\|semitone]] (2\29), and the [[2187/2048\|chroma]] (3\29). Of the three, the semitone has the strongest pull; it is narrow enough to create tension (whereas the wider chroma is often more recognizable as a regular melodic small step) while being wide enough to be recognized as a distinct interval (whereas the diesis acts more like an enharmonic alteration of the same note).		29edo has three unique types of leading tones: from narrowest to widest, they are the [[Pythagorean comma\|diesis]] (1\29), the [[256/243\|semitone]] (2\29), and the [[2187/2048\|chroma]] (3\29). Of the three, the semitone has the strongest pull; it is narrow enough to create tension (whereas the wider chroma is often more recognizable as a regular melodic small step) while being wide enough to be recognized as a distinct interval (whereas the diesis acts more like an enharmonic alteration of the same note).

	Finally, it is important to recognize certain tense intervals that resolve via contrary motion to certain perfect consonances. Notably, 14th century composer and theorist [[wikipedia:Marchetto_da_Padova\|Marchetto de Padova]] used the interordinal intervals as counterpoint dissonances: two notes a semisixth apart (11\29) can resolve outwards by a chroma (or more accurately, the enharmonically equivalent upminor second) to create a perfect fifth, and two notes a semifourth apart (6\29) can resolve outwards by a chroma to reach a perfect fourth, or ~~outwards~~ to reach a unison. These paradigms can be reversed to account for the octave complements of those ~~notes~~.		Finally, it is important to recognize certain tense intervals that resolve via contrary motion to certain perfect consonances. Notably, 14th century composer and theorist [[wikipedia:Marchetto_da_Padova\|Marchetto de Padova]] used the interordinal intervals as counterpoint dissonances: two notes a semisixth apart (11\29) can resolve outwards by a chroma (or more accurately, the enharmonically equivalent upminor second) to create a perfect fifth, and two notes a semifourth apart (6\29) can resolve outwards by a chroma to reach a perfect fourth, or inwards to reach a unison. These paradigms can be reversed to account for the octave complements of those intervals.

	=== Example: Progression in C Vivecan ===		=== Example: Progression in C Vivecan ===
	The Vivecan mode of 4L 3s does not contain a perfect fifth over the root, which may make it difficult to root the mode; however, it does contain an upfifth over the root, whereas five of the other six degrees have downfifths instead, so we may be able to create believable resolutions by using harmonic patterns to "convince" ourselves that the upfifth is more resolved than the downfifth.		The Vivecan mode of 4L 3s does not contain a perfect fifth over the root, which may make it difficult to root the mode; however, it does contain an upfifth over the root, whereas five of the other six degrees have downfifths instead, so we may be able to create believable resolutions by using harmonic patterns to "convince" ourselves that the upfifth is more resolved than the downfifth.

	Firstly, we can notice that vA and vB ~~are separated by~~ a downchthonic, which can resolve by contrary motion to ^G and C; this mimics the perfect chthonic's tendency to resolve to the perfect fifth in a similar fashion. Thus, we can use the ''vA vct ^4'' chord (with degrees vA, vB, and D) as a useful lead into the ''C ^min ^5'' tonic (with degrees C, ^E♭, and ^G).		Firstly, we can notice that the major second between vA and vB is enharmonically equivalent to a downchthonic, which can resolve by contrary motion to ^G and C; this mimics the perfect chthonic's tendency to resolve to the perfect fifth in a similar fashion. Thus, we can use the ''vA vct ^4'' chord (with degrees vA, vB, and D) as a useful lead into the ''C ^min ^5'' tonic (with degrees C, ^E♭, and ^G).

	By noticing that the vA and D also occur in the ''D ^min v5'' supertonic, we can use that triad as a predominant that leads nicely into the vA chord. The movement by an upfourth from the dyad ^F-vA to vB-D creates a pseudo circle-of-fifths rotation, making this progression feel more coherent than it might look at first.		By noticing that the vA and D also occur in the ''D ^min v5'' supertonic, we can use that triad as a predominant that leads nicely into the vA chord. The movement by an upfourth from the dyad ^F-vA to vB-D creates a pseudo circle-of-fifths rotation, making this progression feel more coherent than it might look at first.