Xenharmonic Reference - Recent changes [en]

Oneirotonic

Inthar — Sat, 17 Oct 2026 05:44:41 GMT

Note on symbols

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	<math>\hat{1}, \hat{2}, \hat{3}, \hat{4}, \hat{\mathrm{t}}, \hat{5}, \hat{6}, \hat{7}.</math>		<math>\hat{1}, \hat{2}, \hat{3}, \hat{4}, \hat{\mathrm{t}}, \hat{5}, \hat{6}, \hat{7}.</math>

	When not modified, these denote the pitches of the Ilarnekian mode. Sharp and flat modifiers work as in standard functional diatonic chord notation; they denote chroma alterations from Ilarnekian.		When not modified, these denote the pitches of the Ilarnekian mode. Sharp and flat modifiers work as in standard functional diatonic chord notation; they denote chroma (L - s) alterations from Ilarnekian.

	The chord symbols used are ground's system, described above.		The chord symbols used are ground's system, described above.

Notation

Vector — Sat, 17 Oct 2026 04:33:07 GMT

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	* Wad Wizard characterizes degrees as positions in a structure (reserving 'points in space' for notes i.e. absolute pitches) and intervals as separations between notes or degrees.		* Wad Wizard characterizes degrees as positions in a structure (reserving 'points in space' for notes i.e. absolute pitches) and intervals as separations between notes or degrees.
	* Vector sees degrees and intervals as fundamentally the same kind of object, with intervals having unambiguous readings as degrees and vice versa when no further context is given.		* Vector sees degrees and intervals as fundamentally the same kind of object, with intervals having unambiguous readings as degrees and vice versa when no further context is given.
	* ~~Intha~~ treats the distinction between the two as analogous to that between points in 2D coordinate space and 2D vectors. Deciding on a "default mode/pitch-set" may be preferable for a functional harmony system, but in modal harmony contexts degrees should ideally be notated as isomorphic to intervals.		* Inthar treats the distinction between the two as analogous to that between points in 2D coordinate space and 2D vectors. Deciding on a "default mode/pitch-set" may be preferable for a functional harmony system, but in modal harmony contexts degrees should ideally be notated as isomorphic to intervals.

	== Microtonal notations ==		== Microtonal notations ==

Oneirotonic

Inthar — Sat, 17 Oct 2026 03:32:26 GMT

Note on symbols

Notation

Inthar — Sat, 17 Oct 2026 03:19:06 GMT

Scale degrees vs. intervals

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	* Wad Wizard characterizes degrees as positions in a structure (reserving 'points in space' for notes i.e. absolute pitches) and intervals as separations between notes or degrees.		* Wad Wizard characterizes degrees as positions in a structure (reserving 'points in space' for notes i.e. absolute pitches) and intervals as separations between notes or degrees.
	* Vector sees degrees and intervals as fundamentally the same kind of object, with intervals having unambiguous readings as degrees and vice versa when no further context is given.		* Vector sees degrees and intervals as fundamentally the same kind of object, with intervals having unambiguous readings as degrees and vice versa when no further context is given.
	* ~~Inthar~~ treats the distinction between the two as analogous to that between points in 2D coordinate space and 2D vectors.		* Intha treats the distinction between the two as analogous to that between points in 2D coordinate space and 2D vectors. Deciding on a "default mode/pitch-set" may be preferable for a functional harmony system, but in modal harmony contexts degrees should ideally be notated as isomorphic to intervals.

	== Microtonal notations ==		== Microtonal notations ==

Oneirotonic

Inthar — Sat, 17 Oct 2026 03:14:49 GMT

Note on symbols

Show changes

Chromatic semitone

Vector — Fri, 16 Oct 2026 03:15:44 GMT

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	{{Infobox interval\|2187/2048\|names=Chromatic semitone, augmented unison\|name=Chromatic semitone, augmented unison\|Name=Chromatic semitone, augmented unison, apotome}}		{{Infobox interval\|2187/2048\|names=Chromatic semitone, augmented unison\|name=Chromatic semitone, augmented unison\|Name=Chromatic semitone, augmented unison, apotome}}

	The 3-limit '''chromatic semitone''', also called the '''augmented unison''' ('''A1''') or the '''apotome''' (''ə-POT-ə-mee'') and represented by the ratio '''2187/2048,''' is the difference between the large and small steps of the MOS [[diatonic]] scale. It is generated by stacking 7 [[Perfect fifth\|fifths]] and octave-reducing. In [[Pythagorean tuning\|Pythagorean]] tuning (and thus purely-tuned [[just intonation]]), it is approximately {{Cents from ratio\|2187/2048\|ratio=2187/2048}} cents in size, but as an interval involved in the structure of the abstract diatonic scale it ranges from 0 to 240 cents depending on the tuning.		The 3-limit '''chromatic semitone''', also called the '''augmented unison''' ('''A1''') or the '''apotome''' (''ə-POT-ə-mee'') and represented by the ratio '''2187/2048,''' is the difference between the large and small steps of the MOS [[diatonic]] scale, therefore being a type of [[Interval arithmetic\|unison]] (in terms of interval arithmetic, at least) and a type of [[semitone]]. It is generated by stacking 7 [[Perfect fifth\|fifths]] and octave-reducing. In [[Pythagorean tuning\|Pythagorean]] tuning (and thus purely-tuned [[just intonation]]), it is approximately {{Cents from ratio\|2187/2048\|ratio=2187/2048}} cents in size, but as an interval involved in the structure of the abstract diatonic scale it ranges from 0 to 240 cents depending on the tuning. In 12edo, it is tuned to 100 cents (the same as the [[diatonic semitone]]); flatter tunings narrow it (leading to a "mellower" diatonic scale), and sharper tunings widen it.

	The chromatic semitone is significant in music notation. As it is the difference between, for example, the [[diatonic major third]] and [[diatonic minor third]], it is the interval represented by the # and b accidentals used in standard music notation. The chromatic semitone is distinct from the [[diatonic semitone]], which is the diatonic minor second and differs from the [[diatonic major second]] by a chromatic semitone.		The chromatic semitone is significant in music notation. As it is the difference between, for example, the [[diatonic major third]] and [[diatonic minor third]], it is the interval represented by the # and b accidentals used in standard music notation. The chromatic semitone is distinct from the [[diatonic semitone]], which is the diatonic minor second and differs from the [[diatonic major second]] by a chromatic semitone.

Whitewood

Hotcrystal0 — Fri, 16 Oct 2026 01:53:52 GMT

Redirected page to 7edo#Whitewood temperament

New page

#REDIRECT [[7edo#Whitewood temperament]]

7edo

Vector — Thu, 15 Oct 2026 02:08:41 GMT

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	=== JI approximation ===		=== JI approximation ===
	7edo is, very crudely, a 2.3.5 system, and strength in 2.3.5 is generally what carries into other equiheptatonic scales. It can also be viewed in various other subgroups, most notably 2.3.13 and 2.3.11, and equiheptatonic temperaments can be found that represent those subgroups as well. The diatonic scale in 7edo is equivalent to every note in the tuning system; sharps and flats are not meaningful and all intervals are perfect.		7edo is, very crudely, a 2.3.5 system, and strength in 2.3.5 is generally what carries into other equiheptatonic scales. It can also be viewed in various other subgroups, most notably 2.3.13 and 2.3.11, and equiheptatonic temperaments can be found that represent those subgroups as well. The diatonic scale in 7edo is equivalent to every note in the tuning system; sharps and flats are not meaningful and in Kite's ups and downs notation all intervals are perfect.
	{{Harmonics in ED\|7\|31}}		{{Harmonics in ED\|7\|31}}
	{\| class="wikitable"		{\| class="wikitable"

Tuning stability

Inthar — Wed, 14 Oct 2026 20:04:50 GMT

Inthar moved page Tuning stability to Tempering stability without leaving a redirect

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	'''~~Tuning~~ stability''' is, in a sense, the sensitivity of an interval to the ~~detuning~~ of the [[primes]] relative to its simplicity. It is distinct from other harmonic senses of "stability". Very unstable intervals usually have large powers of 3 in them (such as 9 or 27), while very stable intervals usually have mostly higher and more distinct primes in them. In specific, an interval with multiple of the same prime in it is less stable than an interval with multiple similarly sized primes in it, because in any arbitrary [[~~Temperament\|temperamen~~]]~~<nowiki/>t~~ it is more likely to have an extreme detuning.		'''Tempering stability''' is, in a sense, the sensitivity of an interval to the tempered values of the [[primes]] relative to its simplicity. It is distinct from other harmonic senses of "stability". Very tempering-unstable intervals usually have large powers of 3 in them (such as 9 or 27), while very stable intervals usually have mostly higher and more distinct primes in them. In specific, an interval with multiple of the same prime in it is less stable than an interval with multiple similarly sized primes in it, because in any arbitrary [[temperament]] it is more likely to have an extreme detuning.

	As a basic example, let's consider 11/9 vs. 15/14 (holding the octave fixed). The two have the same number of odd primes in them (11/(33) vs (35)/7) and are of similar complexity (Tenney height 6.6 vs 7.7). However, if you were to randomly detune all of the involved primes, 11/9 would be detuned on average by about 35% more. 28/27 gets detuned by over 60% more than 35/33 on average, because the former has three copies of 3 while the latter has four distinct primes, and so there are less chances for detuning to cancel out for 28/27, and detuning a single prime can have a much greater impact on the interval itself.		As a basic example, let's consider 11/9 vs. 15/14 (holding the octave fixed). The two have the same number of odd primes in them (11/(33) vs (35)/7) and are of similar complexity (Tenney height 6.6 vs 7.7). However, if you were to randomly detune all of the involved primes, 11/9 would be detuned on average by about 35% more. 28/27 gets detuned by over 60% more than 35/33 on average, because the former has three copies of 3 while the latter has four distinct primes, and so there are less chances for detuning to cancel out for 28/27, and detuning a single prime can have a much greater impact on the interval itself.
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	The most prominent unstable intervals have a factor of 9 in them; these are 9/7, 9/8, 10/9, 11/9, 13/9, and their complements. 27 and 25 also show up in more complex unstable intervals, with the interval 27/25 itself varying by 1.6 times as much as 143/105 (a stable interval with the same number of primes) and over 2.5 times as much as 35/32 (a stable interval of a similar complexity and size and with the same number of distinct primes).		The most prominent unstable intervals have a factor of 9 in them; these are 9/7, 9/8, 10/9, 11/9, 13/9, and their complements. 27 and 25 also show up in more complex unstable intervals, with the interval 27/25 itself varying by 1.6 times as much as 143/105 (a stable interval with the same number of primes) and over 2.5 times as much as 35/32 (a stable interval of a similar complexity and size and with the same number of distinct primes).

	The unstable intervals are an important consideration when considering the tunings of temperaments, as they are much more sensitive to detuning than the primes make apparent, and so the tunings of the primes must be carefully balanced if one wishes to preserve them; alternately they may be sacrificed to be tuned to intervals that are very different from them in just intonation, without much impacting the tuning of the primes themselves or of more stable intervals. The basic example of this is archy, where the unstable 9/8 is tuned to the stable 8/7; tunings around 27edo are common because they place the combined interval about 2/3 of the way between 9/8 and 8/7, because 8/7 is more stable and detuning it would directly affect prime 7 more than detuning 9/8 affects prime 3. However, tuning flat of 27edo moderates the detuning of the unstable interval 9/7, which is best tuned around 22edo by perfectly splitting the difference between 9/8 and 8/7. Hence, the best tunings of archy lie within those two extremes.		The tempering-unstable intervals are an important consideration when considering the tunings of temperaments, as they are much more sensitive to detuning than the primes make apparent, and so the tunings of the primes must be carefully balanced if one wishes to preserve them; alternately they may be sacrificed to be tuned to intervals that are very different from them in just intonation, without much impacting the tuning of the primes themselves or of more stable intervals. The basic example of this is archy, where the unstable 9/8 is tuned to the stable 8/7; tunings around 27edo are common because they place the combined interval about 2/3 of the way between 9/8 and 8/7, because 8/7 is more stable and detuning it would directly affect prime 7 more than detuning 9/8 affects prime 3. However, tuning flat of 27edo moderates the detuning of the unstable interval 9/7, which is best tuned around 22edo by perfectly splitting the difference between 9/8 and 8/7. Hence, the best tunings of archy lie within those two extremes.

	The instability of 11/9 also contributes to the difficulty of accurately tuning [[mohajira]] in the 11-limit. This temperament pins the tuning of the unstable 11/9 to half of 3/2; because it also flattens 3/2 by about 5 cents while tuning 11/9 nearly just, 11 itself must be flattened by about 10 cents, and only a few edos meet these criteria, lying between 24edo and 31edo or detuning 11 severely.		The instability of 11/9 also contributes to the difficulty of accurately tuning [[mohajira]] in the 11-limit. This temperament pins the tuning of the unstable 11/9 to half of 3/2; because it also flattens 3/2 by about 5 cents while tuning 11/9 nearly just, 11 itself must be flattened by about 10 cents, and only a few edos meet these criteria, lying between 24edo and 31edo or detuning 11 severely.

Schismic

Vector — Wed, 14 Oct 2026 07:35:05 GMT

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	==== Primes 11 and 13 ====		==== Primes 11 and 13 ====
	A reasonable extension to the 11-limit assuming Garibaldi is Cassandra (41 & 53), which sets 33/32 to twice the Garibaldi comma; alternatively there is Andromeda, which instead sets 33/32 to the difference between that and the chroma and is best tuned sharp of 41edo. In either case, 11/9 is set to the opposite neutral third to 16/13 to extend to the 13-limit.		A reasonable extension to the 11-limit assuming Garibaldi is Cassandra (41 & 53), which sets 33/32 to twice the Garibaldi comma; alternatively there is Andromeda, which instead sets 33/32 to the difference between that and the chroma and is best tuned sharp of 41edo. In either case, 16/13 is set to the opposite neutral third to 11/9 to extend to the 13-limit.

			In cassandra, the sequence of intervals 16/13 - 5/4 - 19/15 - 9/7 - 13/10 is made equidistant.

	=== Prime 17 ===		=== Prime 17 ===

37edo

Vector — Wed, 14 Oct 2026 07:17:48 GMT

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 === JI approximation ===
-edo's most notable feature is a very good representation of the no-3 13-limit and a reasonably good representation of the no-3 19-limit. When 3 is included, it is 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 passably accurate on their own in many contexts, such as approximating 4:5:6. The sharp fifth 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 scales 5L 1m 1s, resembling 38edo's diatonic but with an edostep removed from one of the small steps, and 5L 2m 8s "trackdye".{{Harmonics in ED|37|31|0}}
+edo's most notable feature is a very good representation of the no-3 13-limit and a reasonably good representation of the no-3 19-limit. When 3 is included, it is 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 passably accurate on their own in many contexts, such as approximating 4:5:6. The sharp fifth 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 scales 5L 1m 1s, resembling 38edo's diatonic but with an edostep removed from one of the small steps, and 5L 2m 8s "trackdye".
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 |616.2
 |Large Tritone
 |}

31edo

Vector — Wed, 14 Oct 2026 07:06:19 GMT

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	31edo as a whole contains a diverse palette of interval qualities and structures ranging from the very familiar to the quite exotic, and remarkably, almost all of these still have a reasonably simple harmonic interpretation. As a meantone system, 31edo's diatonic scale includes the basic qualities of the [[5-limit]], such as the [[perfect fourth]] and [[perfect fifth\|fifth]], and the classical minor and major thirds ([[6/5]] and [[5/4]]). But 31edo also includes subminor and supermajor intervals, identifiable with [[septal]] ratios such as [[7/6]] and [[9/7]], and [[neutral]] intervals, identifiable with [[11-limit]] ratios such as [[11/9]].		31edo as a whole contains a diverse palette of interval qualities and structures ranging from the very familiar to the quite exotic, and remarkably, almost all of these still have a reasonably simple harmonic interpretation. As a meantone system, 31edo's diatonic scale includes the basic qualities of the [[5-limit]], such as the [[perfect fourth]] and [[perfect fifth\|fifth]], and the classical minor and major thirds ([[6/5]] and [[5/4]]). But 31edo also includes subminor and supermajor intervals, identifiable with [[septal]] ratios such as [[7/6]] and [[9/7]], and [[neutral]] intervals, identifiable with [[11-limit]] ratios such as [[11/9]].
			[[File:Mohajira extensions in 31edo.png\|thumb\|31edo combines two forms of mohajira: mohajira with the ordinary mapping of 7 (left) and "migration" (i.e. septimal meantone, right). It can be demonstrated that this defines 7-limit 31 equal temperament (also found in 62edo).]]
	In terms of structures, or ways of organizing harmony, it should first be noted that 31edo's perfect fifth, of 18 steps, is quite divisible. The fifth can be split in two, giving us a neutral-third temperament, known in this case as [[Mohajira]], which emphasizes heptatonic structure, the 11th harmonic, and 2.3.5.11. Splitting the fifth in three gives us [[Slendric]] (or in this case [[Mothra]]), formed by stacking [[8/7]], and which emphasizes pentatonic structure and 2.3.7. Combining these gives us [[Miracle]], while splitting Slendric into three again gives us [[Valentine]]. In 31edo, each of these provides xenharmonic ways of accessing the 11-limit with more simplicity than Meantone. Yet another way to encompass the 11-limit is given by [[Orwell]], generated by 31edo's subminor third; and of course, as a prime EDO, 31edo contains several more structures unique to itself.		In terms of structures, or ways of organizing harmony, it should first be noted that 31edo's perfect fifth, of 18 steps, is quite divisible. The fifth can be split in two, giving us a neutral-third temperament, known in this case as [[Mohajira]], which emphasizes heptatonic structure, the 11th harmonic, and 2.3.5.11. Splitting the fifth in three gives us [[Slendric]] (or in this case [[Mothra]]), formed by stacking [[8/7]], and which emphasizes pentatonic structure and 2.3.7. Combining these gives us [[Miracle]], while splitting Slendric into three again gives us [[Valentine]]. In 31edo, each of these provides xenharmonic ways of accessing the 11-limit with more simplicity than Meantone. Yet another way to encompass the 11-limit is given by [[Orwell]], generated by 31edo's subminor third; and of course, as a prime EDO, 31edo contains several more structures unique to itself.

File:Mohajira extensions in 31edo.png

Vector — Wed, 14 Oct 2026 07:01:38 GMT

Vector uploaded File:Mohajira extensions in 31edo.png

New page

wrywryw

Mohajira

Vector — Wed, 14 Oct 2026 06:57:03 GMT

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	As a tuning generated by the [[Tuning stability\|unstable]] interval 11/9, it is perhaps viable to directly map 11/9 to edos and determine the tuning of 11 from there, instead of vice versa.		As a tuning generated by the [[Tuning stability\|unstable]] interval 11/9, it is perhaps viable to directly map 11/9 to edos and determine the tuning of 11 from there, instead of vice versa.

	Generally, Mohajira is also extended to include prime 7 as the semidiminished seventh, though an alternate mapping also available in 31edo shares septimal Meantone's mapping of 7 (and contributes much to 31edo's 7-limit structure).		Generally, Mohajira is also extended to include prime 7 as the semidiminished seventh, though an alternate mapping also available in 31edo shares septimal Meantone's mapping of 7 (and contributes much to 31edo's 7-limit structure). Combining both of these mappings results in 31edo, as it forces the diatonic semitone to be comprised of three chromatic quartertones, which defines 31edo.

	== Interval table ==		== Interval table ==

Tuning stability

Vector — Wed, 14 Oct 2026 06:55:58 GMT

Created page with "'''Tuning stability''' is, in a sense, the sensitivity of an interval to the detuning of the primes relative to its simplicity. It is distinct from other harmonic senses of "stability". Very unstable intervals usually have large powers of 3 in them (such as 9 or 27), while very stable intervals usually have mostly higher and more distinct primes in them. In specific, an interval with multiple of the same prime in it is less stable than an interval with multiple simil..."

New page

'''Tuning stability''' is, in a sense, the sensitivity of an interval to the detuning of the [[primes]] relative to its simplicity. It is distinct from other harmonic senses of "stability". Very unstable intervals usually have large powers of 3 in them (such as 9 or 27), while very stable intervals usually have mostly higher and more distinct primes in them. In specific, an interval with multiple of the same prime in it is less stable than an interval with multiple similarly sized primes in it, because in any arbitrary [[Temperament|temperamen]]<nowiki/>t it is more likely to have an extreme detuning.

As a basic example, let's consider 11/9 vs. 15/14 (holding the octave fixed). The two have the same number of odd primes in them (11/(3*3) vs (3*5)/7) and are of similar complexity (Tenney height 6.6 vs 7.7). However, if you were to randomly detune all of the involved primes, 11/9 would be detuned on average by about 35% more. 28/27 gets detuned by over 60% more than 35/33 on average, because the former has three copies of 3 while the latter has four distinct primes, and so there are less chances for detuning to cancel out for 28/27, and detuning a single prime can have a much greater impact on the interval itself.

The most prominent unstable intervals have a factor of 9 in them; these are 9/7, 9/8, 10/9, 11/9, 13/9, and their complements. 27 and 25 also show up in more complex unstable intervals, with the interval 27/25 itself varying by 1.6 times as much as 143/105 (a stable interval with the same number of primes) and over 2.5 times as much as 35/32 (a stable interval of a similar complexity and size and with the same number of distinct primes).

The unstable intervals are an important consideration when considering the tunings of temperaments, as they are much more sensitive to detuning than the primes make apparent, and so the tunings of the primes must be carefully balanced if one wishes to preserve them; alternately they may be sacrificed to be tuned to intervals that are very different from them in just intonation, without much impacting the tuning of the primes themselves or of more stable intervals. The basic example of this is archy, where the unstable 9/8 is tuned to the stable 8/7; tunings around 27edo are common because they place the combined interval about 2/3 of the way between 9/8 and 8/7, because 8/7 is more stable and detuning it would directly affect prime 7 more than detuning 9/8 affects prime 3. However, tuning flat of 27edo moderates the detuning of the unstable interval 9/7, which is best tuned around 22edo by perfectly splitting the difference between 9/8 and 8/7. Hence, the best tunings of archy lie within those two extremes.

The instability of 11/9 also contributes to the difficulty of accurately tuning [[mohajira]] in the 11-limit. This temperament pins the tuning of the unstable 11/9 to half of 3/2; because it also flattens 3/2 by about 5 cents while tuning 11/9 nearly just, 11 itself must be flattened by about 10 cents, and only a few edos meet these criteria, lying between 24edo and 31edo or detuning 11 severely.

37edo

Ground — Tue, 13 Oct 2026 22:28:51 GMT

Show changes

Ground's composition theory

Ground — Mon, 12 Oct 2026 23:42:51 GMT

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	=== Neutral ~~diachrome~~ ===		=== Neutral diatonic extensions ===

	Here's something I don't think we talk about enough: being raised on music that almost exclusively uses a diatonic framework makes that framework impossible to escape if we want to evoke any of the same emotions. I can use oneirotonic or smitonic for their melodic similarities or antidiatonic for its harmonic similarity, but none of them have a good approximation of 3/2 without using an extremely hard or soft tuning. Porcupine temperament is about as close as I can get to harmonically familiar yet melodically xen, but it's a tradeoff between sharp 6/5 and sharp 3/2. A solution to this that I've derived a few separate times is '''neutral diachrome''', which splits diatonic whole tones into quartertones and neutral seconds. This process can be extended to quasi-diatonic ternary scales like blackdye and diasem, especially when the neutral second splits the minor third equally.		Here's something I don't think we talk about enough: being raised on music that almost exclusively uses a diatonic framework makes that framework impossible to escape if we want to evoke any of the same emotions. I can use oneirotonic or smitonic for their melodic similarities or antidiatonic for its harmonic similarity, but none of them have a good approximation of 3/2 without using an extremely hard or soft tuning. Porcupine temperament is about as close as I can get to harmonically familiar yet melodically xen, but it's a tradeoff between sharp 6/5 and sharp 3/2. A solution to this that I've derived a few separate times is '''neutral diachrome''', which splits diatonic whole tones into quartertones and neutral seconds. This process can be extended to quasi-diatonic ternary scales like blackdye and diasem, especially when the neutral second splits the minor third equally.

			A simpler operation on ternary scales is splitting L into new L + s. The effectiveness of this varies widely depending on the aberrisma size.

			I believe xenharmony involves a balance of being unusual without being jarring. Neutral seconds around 150¢ are excellent for the former, but are more difficult to fit into a diatonic framework. When given the choice, I prefer to have neutral seconds around 135¢ to fit in place of a semitone and 165¢ to fit in place of a whole tone. Temperaments in small edos are Flattone for the former and Porcupine for the latter.

			When converting a simple subgroup to contain these intervals, it makes sense to use 2.3.5 because it's closer to neutral whereas 2.3.7 is closer to interordinal.
			* Making 16/15 larger requires a flatter 3 or 5, both of which are done by the more extreme tunings of Meantone.
			* Making 10/9 smaller requires a sharper 3 or flatter 5. Sharper 3 is better because it's squared, so the error is doubled.
			This is an excellent use case for straddle-3 temperaments. 37edo has mild Mavila and extreme Archy fifths, but their distance from 3/2 is somewhat masked in 5-limit chords due to the 5/4 anchor concordance.

	== Adaptive diatonic modification system ==		== Adaptive diatonic modification system ==

10-form

Vector — Mon, 12 Oct 2026 06:53:25 GMT

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	The 10-form has clean mappings of 3, 5, 7, and 13, and functions as an organization scheme for the 7-limit. There are also not fixed ranges, the boundaries may vary. The boundaries chosen here are loosely based on [[22edo]].		The 10-form has clean mappings of 3, 5, 7, and 13, and functions as an organization scheme for the 7-limit, however 11 is not cleanly mapped and may either be represented as a fourth or a tritone (the latter is preferred by Vector due to fourths being biased flat in the 10-form). There are also not fixed ranges, the boundaries may vary. The boundaries chosen here are loosely based on [[22edo]].
	{\| class="wikitable"		{\| class="wikitable"
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Tertian structure

Vector — Mon, 12 Oct 2026 02:54:17 GMT

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	'''Tertian structure''' is the relative spacing of the [[Third\|thirds]] in a tempered tuning system, such as an equal temperament. In general, it usually refers to the spacing of the 7-limit color-quality thirds 7/6, 6/5, 5/4, and 9/7, which can be expressed as the relative size of 25/24 and 36/35. Other thirds like 19/16, 19/15, or 11/9 may be invoked to describe more complex structures.		'''Tertian structure''' is the relative spacing of the [[Third\|thirds]] in a tempered tuning system, such as an equal temperament. In general, it usually refers to the spacing of the 7-limit color-quality thirds 7/6, 6/5, 5/4, and 9/7, which can be expressed as the relative size of 25/24 and 36/35. Other thirds like 19/16, 19/15, or 11/9 may be invoked to describe more complex structures.

	Common rank-3 temperaments can be expressed as tertian structures~~, and in fact, a tertian structure can always be thought of as a rank-3 temperament~~. These include Mint, Dicot.7, Keemic, and Mandos.		A tertian structure takes the form of a [[regular temperament]] which tempers out a single comma to create a rational relationship between 25/24 and 36/35. Common rank-3 temperaments can be expressed as tertian structures. These include Mint, Dicot.7, Keemic, and Mandos.

	In just intonation, there is no rational relationship in size between 25/24 and 36/35. The ratio of their sizes is log<sub>36/35</sub>(25/24) = 1.44908497.		In just intonation, there is no rational relationship in size between 25/24 and 36/35. The ratio of their sizes is log<sub>36/35</sub>(25/24) = 1.44908497.

	All tertian structures are supported by 7edo, which collapses all distinctions between the four thirds in question down to a single interval, outlining the structure of diatonic harmony.		All tertian structures are supported by 7edo, which tempers out 25/24 and 36/35 themselves and thus collapses all distinctions between the four thirds in question down to a single interval, outlining the structure of diatonic harmony.

	== Mint ==		== List of tertian structures ==
	Ratio (36/35:25/24): 0:1
			=== Mint ===
			Ratio (36/35:25/24): 0:1 (~inf)

	Comma: (36/35)^1 / (25/24)^0 = 36/35		Comma: (36/35)^1 / (25/24)^0 = 36/35
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	Supported edos: 5, 9, 12, 14, 16, 21, 23, 28, 33		Supported edos: 5, 9, 12, 14, 16, 21, 23, 28, 33

	Mint tempers out 36/35 entirely, equating the pental and septimal third qualities to some tuning around 300 and 400 cents~~. Unusually enough, as a temperament structure in and of itself, this is best optimized with a flat fifth~~. The most prominent mint temperaments are dominant (meantone, supported by 5edo and 12edo), mavila (supported by 9edo, 16edo, and 23edo), whitewood (supported by 14edo, 21edo, and 28edo), and dimisept (supported by 12edo and 16edo alongside 28edo). Interestingly, mint temperaments overwhelmingly have flat fifths, except for dominant. This makes sense, as mint essentially equally spaces 3/2 between 5/4 and 7/4, and if both are kept just then 3/2 is set to about 678 cents.		Mint tempers out 36/35 entirely, equating the pental and septimal third qualities to some tuning around 300 and 400 cents. The most prominent mint temperaments are dominant (meantone, supported by 5edo and 12edo), mavila (supported by 9edo, 16edo, and 23edo), whitewood (supported by 14edo, 21edo, and 28edo), and dimisept (supported by 12edo and 16edo alongside 28edo). Interestingly, mint temperaments overwhelmingly have flat fifths, except for dominant. This makes sense, as mint essentially equally spaces 3/2 between 5/4 and 7/4, and if both are kept just then 3/2 is set to about 678 cents.

	== Dicot ==		=== Dicot ===
	Ratio (36/35:25/24): 1:0		Ratio (36/35:25/24): 1:0 (0)

	Comma: (25/24)^1 / (36/35)^0 = 25/24		Comma: (25/24)^1 / (36/35)^0 = 25/24
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	Dicot is a much rarer archetype even in the small edos due to the fact that it can be described as a rank-2 temperament in a prime subgroup (although as a tertian structure 7 is given as an independent generator); additionally, its comma is the largest of any tertian structure discussed here. The temperaments supporting the main form of dicot, called dichotic, are 10, 13, 17, and 20. Unlike mint temperaments which prefer a flat fifth, dicot temperaments prefer a sharp fifth. Again, this makes sense if you examine the 5/4 eigenmonzo tuning, which jumps 3/2 all the way up to 773 cents; accurate tunings of dicot fall between the just tuning of 3/2 and that sharp tuning.		Dicot is a much rarer archetype even in the small edos due to the fact that it can be described as a rank-2 temperament in a prime subgroup (although as a tertian structure 7 is given as an independent generator); additionally, its comma is the largest of any tertian structure discussed here. The temperaments supporting the main form of dicot, called dichotic, are 10, 13, 17, and 20. Unlike mint temperaments which prefer a flat fifth, dicot temperaments prefer a sharp fifth. Again, this makes sense if you examine the 5/4 eigenmonzo tuning, which jumps 3/2 all the way up to 773 cents; accurate tunings of dicot fall between the just tuning of 3/2 and that sharp tuning.

	== Keemic ==		=== Keemic ===
	Ratio (36/35:25/24): 1:1		Ratio (36/35:25/24): 1:1 (1)

	Comma: (25/24)^1 / (36/35)^1 = 875/864		Comma: (25/24)^1 / (36/35)^1 = 875/864
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	Keemic is characterized by the narrowing of 25/24 and/or the widening of 36/35; eigenmonzo 54/49 (the distance between 7/6 and 9/7) and 3/2 leads to a 5/4 tuning of 379c; eigenmonzo 25/24 and 3/2 leads to a 9/7 tuning of 457c.		Keemic is characterized by the narrowing of 25/24 and/or the widening of 36/35; eigenmonzo 54/49 (the distance between 7/6 and 9/7) and 3/2 leads to a 5/4 tuning of 379c; eigenmonzo 25/24 and 3/2 leads to a 9/7 tuning of 457c.

	== Mandos ==		=== Mandos ===
	Ratio (36/35:25/24): 1:2		Ratio (36/35:25/24): 1:2 (2)

	Comma: (36/35)^2 / (25/24)^1 = 31104/30625		Comma: (36/35)^2 / (25/24)^1 = 31104/30625
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	''TODO: rank-3 name for amity? also cover "oddball" structures like sixix, 3-2-3, 1-4-1, 1-5-1, etc''		''TODO: rank-3 name for amity? also cover "oddball" structures like sixix, 3-2-3, 1-4-1, 1-5-1, etc''

			== Other mediant structures ==
			''todo: chthonic structure. 10/9 8/7 7/6 6/5. semaphore mint orwellismic?''

Keemic

Vector — Mon, 12 Oct 2026 02:37:27 GMT

Redirected page to Tertian structure

New page

#REDIRECT [[Tertian structure]]

Ground's composition theory

Ground — Sun, 11 Oct 2026 23:05:29 GMT

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	I believe the most aesthetically ideal and widely useful aberrismic scales are Meantone septal diasem and Archy pental blackdye. See [[Monarch]] for further explanation.		I believe the most aesthetically ideal and widely useful aberrismic scales are Meantone septal diasem and Archy pental blackdye. See [[Monarch]] for further explanation.

	== 9-note ~~diatonic analogue~~ ==		== Diatonic analogues ==

			=== 9-note "diatygic" ===

	ModMOSes and ternary scales based on them are the best. I haven't used them extensively since my first two albums (2014-2015), so I'm going to challenge myself to use one as often as possible. It's easier if the scale being modified is very familiar, such as quasi-diatonic. There's actually nothing about ILD that prefers MOSes over any other type of scale. This will need to be a whole section eventually.		ModMOSes and ternary scales based on them are the best. I haven't used them extensively since my first two albums (2014-2015), so I'm going to challenge myself to use one as often as possible. It's easier if the scale being modified is very familiar, such as quasi-diatonic. There's actually nothing about ILD that prefers MOSes over any other type of scale. This will need to be a whole section eventually.
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	\|1200		\|1200
	\|}		\|}

			=== Neutral diachrome ===

			Here's something I don't think we talk about enough: being raised on music that almost exclusively uses a diatonic framework makes that framework impossible to escape if we want to evoke any of the same emotions. I can use oneirotonic or smitonic for their melodic similarities or antidiatonic for its harmonic similarity, but none of them have a good approximation of 3/2 without using an extremely hard or soft tuning. Porcupine temperament is about as close as I can get to harmonically familiar yet melodically xen, but it's a tradeoff between sharp 6/5 and sharp 3/2. A solution to this that I've derived a few separate times is '''neutral diachrome''', which splits diatonic whole tones into quartertones and neutral seconds. This process can be extended to quasi-diatonic ternary scales like blackdye and diasem, especially when the neutral second splits the minor third equally.

	== Adaptive diatonic modification system ==		== Adaptive diatonic modification system ==

17edo

Inthar — Sun, 11 Oct 2026 19:44:25 GMT

Scales

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	* [[Diatonic#MOS diatonic\|MOS diatonic]] (5L2s, 3331331)		* [[Diatonic#MOS diatonic\|MOS diatonic]] (5L2s, 3331331)
	* [[Mosh]] (3L4s, 2323232)		* [[Mosh]] (3L4s, 2323232)
	* Lissotonic<sup>1</sup> (3223322)		* Lissotonic<sup>1</sup> (3223322): modes of the ascending mode of Maqam Rast
	* Trachytonic<sup>1</sup> (3313322)		* Trachytonic<sup>1</sup> or screamapillar<sup>2</sup> (3313322)

	<sup>1</sup> CellularAutomaton's names		<sup>1</sup> CellularAutomaton's names

			<sup>2</sup> Xen Wiki's name

	{{Navbox EDO}}		{{Navbox EDO}}

	{{cat\|Edos}}		{{cat\|Edos}}

Straddle primes

Hotcrystal0 — Sun, 11 Oct 2026 12:51:33 GMT

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	Straddle primes were created for the purpose of error cancellation and flexibility of tempering. For example:		Straddle primes were created for the purpose of error cancellation and flexibility of tempering. For example:
	* 37edo has a remarkably accurate 19-limit, with the exception of prime 3. It works, but it doesn't stack predictably. In order to represent intervals involving prime 3 more accurately, powers of 3 can involve stacking the more accurate sharp 3 with an occasional flat 3 to offset error.		* [[37edo]] has a remarkably accurate 19-limit, with the exception of prime 3. It works, but it doesn't stack predictably. In order to represent intervals involving prime 3 more accurately, powers of 3 can involve stacking the more accurate sharp 3 with an occasional flat 3 to offset error.
	* 86edo is larger and has two diatonic fifths, Meantone and Archy. This gives it two diatonic scales with different qualities useful as a base for quasi-diatonic ternary scales. It also clearly straddles the whole 11-limit, providing both the option to choose a version of a prime that results in a better approximation of a ratio, and doubling the probability of encountering a target prime in a given scale. In practice, this probability is lower than double due to preferring approximations with good error cancellation.		* [[86edo]] is larger and has two diatonic fifths, Meantone and Archy. This gives it two diatonic scales with different qualities useful as a base for quasi-diatonic ternary scales. It also clearly straddles the whole 11-limit, providing both the option to choose a version of a prime that results in a better approximation of a ratio, and doubling the probability of encountering a target prime in a given scale. In practice, this probability is lower than double due to preferring approximations with good error cancellation.

	With a sufficiently large edo, it's possible to have accurate approximations of primes while also using versions altered by an edostep to generate more interesting scales or result from stacking a different interval. This may be seen as better due to the lower error in using a different JI ratio to represent an altered prime, which begs the question, why use straddle primes at all? The answer is twofold:		With a sufficiently large edo, it's possible to have accurate approximations of primes while also using versions altered by an edostep to generate more interesting scales or result from stacking a different interval. This may be seen as better due to the lower error in using a different JI ratio to represent an altered prime, which begs the question, why use straddle primes at all? The answer is twofold:

Straddle primes

Ground — Sat, 10 Oct 2026 17:32:09 GMT

SRTT

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	'''SRTT''', short for Straddle Regular Temperament Theory, is a generalization of straddle temperaments with eracs that uses variables to represent small alterations of primes. Using the variable ''r'' to represent the erac alteration, the erac subgroup 2.<3.>3.5.7 may be written as 2.3/r.3r.5.7 or simplified to 2.3r.5.7.r^2. If error cancellation is not assumed, more variables may be used, such as 3m/2 for a Meantone fifth and 3a/2 for an Archy fifth as found in [[Monarch]].		'''SRTT''', short for Straddle Regular Temperament Theory, is a generalization of straddle temperaments with eracs that uses variables to represent small alterations of primes. Using the variable ''r'' to represent the erac alteration, the erac subgroup 2.<3.>3.5.7 may be written as 2.3/r.3r.5.7 or simplified to 2.3r.5.7.r^2. If error cancellation is not assumed, more variables may be used, such as 3m/2 for a Meantone fifth and 3a/2 for an Archy fifth as found in [[Monarch]].

			=== Cluster temperaments ===

			In temperaments with a generator very close to some equal temperament, intervals can be optimized by altering the generator in either direction. This flips the direction of eracs, which may be confusing. SRTT alterations may be positive or negative, and may also be stacked. For example, 4\5 is flat lf 7/4. Buzzard and Hemiseven are two well-known 5edo cluster temperaments, with the Buzzard generator flat of 2\5 and the Hemiseven generator sharp of 2\5.

			a and b are arbitrary. It doesn't matter whether they are multiplied or divided, just that both instances of a have the same operation and both instances of b have different operations.
			* -3 generators: 7/4 a b
			* 2 generatora: 7/4 a/b
			At 5edo, b is assumed to be about 1, so a is less than 1 and accounts for the flat tuning of 7/4.
			* In Buzzard, the accurate 7/4 is at -3 generators. This is where a and b are about reciprocal, cancelling their errors via multiplication.
			* In Hemiseven, the accurate 7/4 is at 2 generators. This is where a and b are about equal, cancelling their errors via division.
			* This also means that each cycle of 5 generators ends up at a pitch that differs by a ratio of (7/4 a b)/(7/4 a/b) = b^2.
			In temperaments further from optimal, the size of the offset a becomes less significant relative to the ratio between b and 1/b. This is the concept behind straddle cluster temperaments.

	== Straddle-2 ==		== Straddle-2 ==

← Older revision		Revision as of 03:32, 17 October 2026
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	<math>\hat{1}, \hat{2}, \hat{3}, \hat{4}, \hat{\mathrm{t}}, \hat{5}, \hat{6}, \hat{7}.</math>		<math>\hat{1}, \hat{2}, \hat{3}, \hat{4}, \hat{\mathrm{t}}, \hat{5}, \hat{6}, \hat{7}.</math>

	When not modified, these denote the pitches of the Ilarnekian mode. Sharp and flat modifiers work as in standard functional diatonic chord notation.		When not modified, these denote the pitches of the Ilarnekian mode. Sharp and flat modifiers work as in standard functional diatonic chord notation; they denote chroma alterations from Ilarnekian.

	The chord symbols used are ground's system, described above.		The chord symbols used are ground's system, described above.
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	In this cadence, the fifth on E is so narrow that it creates a leading tone relative to the root, and by playing the octaves above E, you can create a minor tritone that wants to resolve inwards to the tract-major chord on the root. The presence of the octaves above the major third helps drive this resolution, but can be omitted. Another neat effect is that given the dominant is now on the <math>\hat{4},</math> then the <math>\hat{2}</math> simic sixth chord would be exactly halfway to the dominant, making it the mediant. It also has a much nicer simic sixth chord on it compared to the <math>\hat{3}</math> 1st inversion tract-major chord, making it more akin to how the mediant works in diatonic.		In this cadence, the fifth on E is so narrow that it creates a leading tone relative to the root, and by playing the octaves above E, you can create a minor tritone that wants to resolve inwards to the tract-major chord on the root. The presence of the octaves above the major third helps drive this resolution, but can be omitted. Another neat effect is that given the dominant is now on the <math>\hat{4},</math> then the <math>\hat{2}</math> simic sixth chord would be exactly halfway to the dominant, making it the mediant. It also has a much nicer simic sixth chord on it compared to the <math>\hat{3}</math> 1st inversion tract-major chord, making it more akin to how the mediant works in diatonic.

	We can also relate other chords to the dominant, mediant and tonic. The relative minor is more or less exactly analogous to diatonic, being a minor third below the tonic (in the case of B Dylathian, it would be A Celephaïsian, the <math>\hat{6}</math> with the "minor" tonality). The mediant can also function as a secondary dominant for resolutions to the relative minor; the highest note in the <math>\hat{2}</math> ~~minor~~ chord is one semitone below the minor third of the <math>\hat{6}</math> simic chord. By playing the octave above certain notes, resolving between the two modes is pretty simple.		We can also relate other chords to the dominant, mediant and tonic. The relative minor is more or less exactly analogous to diatonic, being a minor third below the tonic (in the case of B Dylathian, it would be A Celephaïsian, the <math>\hat{6}</math> with the "minor" tonality). The mediant can also function as a secondary dominant for resolutions to the relative minor; the highest note in the <math>\hat{2}</math> simic chord is one semitone below the minor third of the <math>\hat{6}</math> simic chord. By playing the octave above certain notes, resolving between the two modes is pretty simple.
	# B-D-F-G-B		# B-D-F-G-B
	# `D-`E-`A-`C-C		# `D-`E-`A-`C-C
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	The ` denotes playing an octave lower than the root.		The ` denotes playing an octave lower than the root.

	The <math>\hat{2}</math> (D minor) and <math>\hat{6}</math> (A minor) would probably sound the smoothest when played in a lower register than the tonic (B major) as notated, but if you want to move upwards from the root it still works. Resolving from the relative minor (A minor) to the tonic (B major) is a pretty weak but still usable resolution. Another neat resolution is moving from the <math>\hat{3}</math> (inverted major) to the dominant <math>\hat{4}.</math>		The <math>\hat{2}</math> (D minor) and <math>\hat{6}</math> (A minor) would probably sound the smoothest when played in a lower register than the tonic (B major) as notated, but if you want to move upwards from the root it still works. Resolving from the relative minor (A minor) to the tonic (B major) is a pretty weak but still usable resolution. Another neat resolution is moving from the <math>\hat{3}</math> (inverted tract-major) to the dominant <math>\hat{4}.</math>

	If you play the 0-5-9 chord on <math>\hat{3},</math> the lowest note will be a semitone lower than the lowest in the dominant, and the highest note will be a semitone higher than the highest in the dominant. By either extending the 0-5-9 chord to 0-3-5-9, or simplifying the dominant chord to a 0-4-7 chord, you can drive this resolution very powerfully, and this could either create a chain of strong resolutions going <math>\hat{3}</math> (inverted major) - <math>\hat{4}</math> - <math>\hat{1},</math> or it could help drive resolutions to the Ilarnekian mode above (in this case, E Ilarnekian).		If you play the 0-5-9 chord on <math>\hat{3},</math> the lowest note will be a semitone lower than the lowest in the dominant, and the highest note will be a semitone higher than the highest in the dominant. By either extending the 0-5-9 chord to 0-3-5-9, or simplifying the dominant chord to a 0-4-7 chord, you can drive this resolution very powerfully, and this could either create a chain of strong resolutions going <math>\hat{3}</math> (inverted tract-major) - <math>\hat{4}</math> - <math>\hat{1},</math> or it could help drive resolutions to the Ilarnekian mode above (in this case, E Ilarnekian).

	Technically you wouldn't have to extend or simplify any of these chords, but the triad next to all the tetrads feels somewhat empty. All in all, using this technique you could probably simplify all the tetrads down to 0-4-7 and 0-3-8 for major and ~~minor~~, respectively. These would help since the 0-3-5-9 chord doesn't have much of a DR effect, while the simplified major and ~~minor~~ still have a DR effect, though a bit weaker than the tetrads. The <math>\hat{6}</math> inverted major (9-12-18-20) chord also has some neat features, as it functions as an inversion of the dominant <math>\hat{4}</math> chord. It also doesn't need any extensions with octaves to work well unlike the dominant chord, so it could be seen as a more tense version of dominant. Since it also drives the resolution up by a minor third, the same tetrad on the <math>\hat{3}</math> could be used to drive a resolution to a major <math>\sharp\hat{5},</math> helping to shift the key center from B to G#. If done twice, this resolution can shift your key center up a minor third from B -> G# -> F#, which gives the progression a really jazzy feel.		Technically you wouldn't have to extend or simplify any of these chords, but the triad next to all the tetrads feels somewhat empty. All in all, using this technique you could probably simplify all the tetrads down to 0-4-7 and 0-3-8 for tract-major and simic, respectively. These would help since the 0-3-5-9 chord doesn't have much of a DR effect, while the simplified tract-major and simic still have a DR effect, though a bit weaker than the tetrads. The <math>\hat{6}</math> inverted tract-major (9-12-18-20) chord also has some neat features, as it functions as an inversion of the dominant <math>\hat{4}</math> chord. It also doesn't need any extensions with octaves to work well unlike the dominant chord, so it could be seen as a more tense version of dominant. Since it also drives the resolution up by a minor third, the same tetrad on the <math>\hat{3}</math> could be used to drive a resolution to a tract-major <math>\sharp\hat{5},</math> helping to shift the key center from B to G#. If done twice, this resolution can shift your key center up a minor third from B -> G# -> F#, which gives the progression a really jazzy feel.

	The only chord we haven't covered now would be the simic <math>\hat{\mathrm{t}}</math> (namely 7-10-15-17). This chord has a much weaker relationship to the other chords, so it doesn't have any strong directionality. However, it does share some notes with a few important chords, notably the <math>\hat{1}</math> chord and the relative minor on the <math>\hat{6}.</math> A resolution to either of these will be similarly strong, that is to say, not very strong. In this case it could also be seen as a secondary mediant which is not the relative minor, about halfway between the <math>\hat{1}</math> chord and the <math>\hat{7}</math> chord an octave above it, and either of these resolutions would probably sound fine in most contexts. This gives it a role completely unlike any of the functions in traditional diatonic. It also works pretty well as a setup for the <math>\sharp\hat{5}</math> inverted major, so in a progression it can help add some flair or beef to the resolution.		The only chord we haven't covered now would be the simic <math>\hat{\mathrm{t}}</math> (namely 7-10-15-17). This chord has a much weaker relationship to the other chords, so it doesn't have any strong directionality. However, it does share some notes with a few important chords, notably the <math>\hat{1}</math> chord and the relative minor on the <math>\hat{6}.</math> A resolution to either of these will be similarly strong, that is to say, not very strong. In this case it could also be seen as a secondary mediant which is not the relative minor, about halfway between the <math>\hat{1}</math> chord and the <math>\hat{7}</math> chord an octave above it, and either of these resolutions would probably sound fine in most contexts. This gives it a role completely unlike any of the functions in traditional diatonic. It also works pretty well as a setup for the <math>\sharp\hat{5}</math> inverted tract-major, so in a progression it can help add some flair or beef to the resolution.
	}}		}}

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	With Ilarnekian being the second major mode (after Dylathian), we'd get the same <math>\hat{1}</math> chord, E>maj< One of the most immediate effects we'd see, however, is that the dominant <math>\hat{4}</math> is now IV>min7<. It still can function as a dominant, though only with the added octave above the 4th, and slightly weaker than the Dylathian dominant cadence. Another interesting thing is that the <math>\hat{4}-\hat{1}</math> cadence is now simultaneously a minor plagal and a dominant cadence, radically different from anything in diatonic.		With Ilarnekian being the second major mode (after Dylathian), we'd get the same <math>\hat{1}</math> chord, E>maj< One of the most immediate effects we'd see, however, is that the dominant <math>\hat{4}</math> is now IV>min7<. It still can function as a dominant, though only with the added octave above the 4th, and slightly weaker than the Dylathian dominant cadence. Another interesting thing is that the <math>\hat{4}-\hat{1}</math> cadence is now simultaneously a minor plagal and a dominant cadence, radically different from anything in diatonic.

	The <math>\hat{2}</math> chord is again simic sixth, with it now driving a secondary resolution to D Ultharian instead of D Celephaïsian. The <math>\hat{3}</math> inverted major chord is also still first inversion tract-major, and also still drives a pretty good resolution up to the simic sixth <math>\hat{4},</math> though admittedly weaker than Dylathian.		The <math>\hat{2}</math> chord is again simic sixth, with it now driving a secondary resolution to D Ultharian instead of D Celephaïsian. The <math>\hat{3}</math> inverted tract-major chord is also still first inversion tract-major, and also still drives a pretty good resolution up to the simic sixth <math>\hat{4},</math> though admittedly weaker than Dylathian.

	The <math>\hat{\mathrm{t}}</math> chord is still simic sixth, and still functions as a secondary mediant.		The <math>\hat{\mathrm{t}}</math> chord is still simic sixth, and still functions as a secondary mediant.
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	with ` again notating playing an octave lower than the starting chord.		with ` again notating playing an octave lower than the starting chord.

	Moving on, the <math>\hat{6}</math> chord would be simic sixth driving the resolution to the <math>\hat{4}</math> simic sixth, in the same way Dylathian's <math>\hat{2}</math> simic sixth drives the resolution to the relative minor. It would also function as the Ilarnekian relative minor, in this case D Ultharian. The <math>\hat{7}</math> chord would be an inverted major chord, and would drive a resolution to the tonic pretty well. This comes from the fact that the 0-3-9-11 chord would have the minor third become the major second of the tonic, the root move up a semitone to the tonic, and the perfect fifth move down a semitone to become the tritone of the tonic chord. The minor sixth in the chord could be omitted to make the resolution stronger, but the chord would sound much more dissonant.		Moving on, the <math>\hat{6}</math> chord would be simic sixth driving the resolution to the <math>\hat{4}</math> simic sixth, in the same way Dylathian's <math>\hat{2}</math> simic sixth drives the resolution to the relative minor. It would also function as the Ilarnekian relative minor, in this case D Ultharian. The <math>\hat{7}</math> chord would be an inverted tract-major chord, and would drive a resolution to the tonic pretty well. This comes from the fact that the 0-3-9-11 chord would have the minor third become the major second of the tonic, the root move up a semitone to the tonic, and the perfect fifth move down a semitone to become the tritone of the tonic chord. The minor sixth in the chord could be omitted to make the resolution stronger, but the chord would sound much more dissonant.
	}}		}}

	==== Oneirominor ====		==== Oneirominor ====
	{{UserTag\|KC\|Inthar\|000000\|The pitches of the oneirominor tonality are usually that of the Celephaisian mode, but the 7 degree may be either <math>\flat\hat 7</math> (for its relationship to other degrees) or <math>\hat 7</math> (mainly but not exclusively for a leading tone function).		{{UserTag\|KC\|Inthar\|000000\|The pitches of the oneirominor tonality are usually that of the Celephaisian mode, but the 7 degree may be either <math>\flat\hat 7</math> (for its relationship to other degrees) or <math>\hat 7</math> (mainly but not exclusively for a leading tone function).