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<div>{{stub}}<br />
UNDER CONSTRUCTION <br />
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'''9edo''', or 9 equal divisions of the octave (sometimes called '''9-TET''' or '''9-tone equal temperament'''), is the [[Equal temperament|equal tuning]] featuring steps of (1200/9) = 133.333… [[Cent|cents]] exactly, 9 of which stack to the perfect octave [[2/1]].<br />
<br />
9edo is probably best known for its very flat, muddy-sounding [[fifth]] interval and [[antidiatonic]] (2L 5s) scale, a version of the diatonic scale with inverted harmonic properties (such as major and minor intervals being flipped) when you use the circle of fifths. It does not represent small [[harmonic series|harmonics]] that well, but it has extremely accurate renditions of the just intonation intervals [[27/25]] and [[7/6]], which forms the basis of an ultra-precise [[regular temperament]] called [[ennealimmal]]. <br />
<br />
== General theory ==<br />
<br />
=== Derivation ===<br />
9edo is the equal division corresponding to the 9-form, which may be understood from a polychordal point of view as dividing each perfect fourth into four, creating pentachords, while leaving the whole tone between them undivided (as opposed to the 10-form, which divides the whole tone in two). <br />
<br />
=== JI approximation ===<br />
[Overview of viable vals, tuning tendencies, and accurate/structurally interesting subgroups. Always cover the patent val alongside any notable non-patent vals.]{{Harmonics in ED|9|prime}}<br />
<br />
=== Edostep interpretations ===<br />
[Cover these in a list]<br />
<br />
=== Intervals and notation ===<br />
{| class="wikitable"<br />
|+<br />
! rowspan="2" |Edostep<br />
! rowspan="2" |Cents<br />
! colspan="2" |[[JI]] approximation<br />
! colspan="3" |Notation<br />
|-<br />
!7-limit [[Ennealimmal]]-based (accurate)<br />
!Coarse<br />
!Melodic antidiatonic<br />
!Harmonic antidiatonic<br />
![[36edo]] notation ([[ups and downs notation|ups and downs]])<br />
|-<br />
|0<br />
|0<br />
|[[1/1]]<br />
|<br />
|D<br />
|D<br />
|D<br />
|-<br />
|1<br />
|133.33<br />
|[[27/25]]<br />
|[[16/15]]<br />
|E<br />
|E<br />
|^Eb<br />
|-<br />
|2<br />
|266.67<br />
|[[7/6]]<br />
|[[8/7]]<br />
|E#, Fb<br />
|Eb, F#<br />
|vF<br />
|-<br />
|3<br />
|400<br />
|[[63/50]]<br />
|[[5/4]]<br />
|F<br />
|F<br />
|F#<br />
|-<br />
|5<br />
|533.33<br />
|[[49/36]]<br />
|[[4/3]], [[11/8]]<br />
|G<br />
|G<br />
|^G<br />
|-<br />
|6<br />
|666.67<br />
|[[72/49]]<br />
|[[3/2]], [[16/11]]<br />
|A<br />
|A<br />
|vA<br />
|-<br />
|6<br />
|800<br />
|[[100/63]]<br />
|[[8/5]]<br />
|B<br />
|B<br />
|Bb<br />
|-<br />
|7<br />
|933.33<br />
|[[12/7]]<br />
|[[7/4]]<br />
|B#, Cb<br />
|Bb, C#<br />
|^B<br />
|-<br />
|6<br />
|1066.67<br />
|[[50/27]]<br />
|[[15/8]]<br />
|C<br />
|C<br />
|vC#<br />
|-<br />
|9<br />
|1200<br />
|[[2/1]]<br />
|<br />
|D<br />
|D<br />
|D<br />
|}<br />
<br />
== Tempering properties ==<br />
<br />
=== Tempered commas ===<br />
[List commas with S-expressions and examples of what they equate]<br />
<br />
=== Arithmetic progressions ===<br />
<br />
=== Notable structural chains ===<br />
[Generator chains]<br />
<br />
== Compositional theory ==<br />
<br />
=== Tertian structure ===<br />
[Describe]<br />
{| class="wikitable"<br />
|+Thirds in Xedo<br />
!Quality ([[ADIN]])<br />
|'''Mosdiatonic Quality'''<br />
|Quality<br />
|-<br />
!Cents<br />
|'''XXX'''<br />
|XXX<br />
|-<br />
!Just interpretation<br />
|'''X/X'''<br />
|X/X<br />
|-<br />
!Steps<br />
|'''X'''<br />
|X<br />
|}<br />
Diatonic thirds are bolded.<br />
<br />
=== Scales ===<br />
[if edo is composite, link to subset edos and discuss scales of subset edos there]<br />
<br />
[List scales and scale descriptions including structure (generators if applicable), notable intervals available, associated temperaments, relationships to other scales, and logic for derivation.]<br />
<br />
[Explain more complex scale theory topics here]<br />
<br />
==== Tables of scales ====<br />
<br />
=== Harmony ===<br />
[explain notable JI and DR chords. TRY TO PROVIDE COHERENT CHORD SYSTEMS RATHER THAN JUST LISTING RANDOM CHORDS WITH NO RELATION. Explain tunings of familiar chords]<br />
<br />
[Explain systems of harmony here, this is how you put the chords together to make music]<br />
<br />
[Sections such as "functional harmony", "modal harmony", etc - varies based on the edo and the personal composition style]<br />
<br />
[Use individual voices maybe?]<br />
<br />
==== Tables of chords ====<br />
<br />
== Instruments ==<br />
[Put instruments and isomorphic layouts here]<br />
<br />
== Supersets and subsets ==<br />
<br />
== Comparisons to other tuning systems ==<br />
<br />
== Music in 9edo ==<br />
<br />
== See also ==</div>Vectorhttps://xenreference.com/wiki/index.php?title=Xenharmonic_Reference:EDO_page_skeleton&diff=7792Xenharmonic Reference:EDO page skeleton2026-07-19T06:46:47Z<p>Vector: Created page with "This page is to be used as a template for making EDO pages. '''Xedo''', or X equal divisions of the octave (sometimes called '''X-TET''' or '''X-tone equal temperament'''), is the equal tuning featuring steps of (1200/X) = Y cents exactly, X of which stack to the perfect octave 2/1. [Overview of tuning properties for key intervals] == General theory == === Derivation === [Derivation by equal division of various key intervals or of..."</p>
<hr />
<div>This page is to be used as a template for making EDO pages.<br />
<br />
<br />
'''Xedo''', or X equal divisions of the octave (sometimes called '''X-TET''' or '''X-tone equal temperament'''), is the [[Equal temperament|equal tuning]] featuring steps of (1200/X) = Y [[Cent|cents]] exactly, X of which stack to the perfect octave [[2/1]].<br />
<br />
[Overview of tuning properties for key intervals]<br />
<br />
== General theory ==<br />
<br />
=== Derivation ===<br />
[Derivation by equal division of various key intervals or of interval qualities]<br />
<br />
=== JI approximation ===<br />
[Overview of viable vals, tuning tendencies, and accurate/structurally interesting subgroups. Always cover the patent val alongside any notable non-patent vals.]{{Harmonics in ED|0|prime}}<br />
<br />
=== Edostep interpretations ===<br />
[Cover these in a list]<br />
<br />
=== Intervals and notation ===<br />
{| class="wikitable"<br />
|+<br />
! rowspan="2" |Edostep<br />
! rowspan="2" |Cents<br />
! colspan="2" |Intervals<br />
! colspan="3" |Notation<br />
! colspan="4" |Bame (optional)<br />
|-<br />
!Group 1<br />
!Additional groups<br />
!Notation 1 (usually ups and downs)<br />
!Notation 2 (usually neutral diatonic notation, if different))<br />
!Additional notations<br />
!System 1 (usually ADIN)<br />
!System 2 (usually ups and downs, if meaningfully different)<br />
!System 3 (usually neutral diatonic, if different)<br />
!Additional systems<br />
|-<br />
|0<br />
|XXX<br />
|XX/XX<br />
|XX/XX<br />
|D<br />
|D<br />
|J<br />
|...<br />
|...<br />
|...<br />
|...<br />
|-<br />
|...<br />
|...<br />
|...<br />
|...<br />
|...<br />
|...<br />
|...<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|}<br />
[Add solfege in a separate table, if applicable]<br />
<br />
== Tempering properties ==<br />
<br />
=== Tempered commas ===<br />
[List commas with S-expressions and examples of what they equate]<br />
<br />
=== Arithmetic progressions ===<br />
<br />
=== Notable structural chains ===<br />
[Generator chains]<br />
<br />
== Compositional theory ==<br />
<br />
=== Tertian structure ===<br />
[Describe]<br />
{| class="wikitable"<br />
|+Thirds in Xedo<br />
!Quality ([[ADIN]])<br />
|'''Mosdiatonic Quality'''<br />
|Quality<br />
|-<br />
!Cents<br />
|'''XXX'''<br />
|XXX<br />
|-<br />
!Just interpretation<br />
|'''X/X'''<br />
|X/X<br />
|-<br />
!Steps<br />
|'''X'''<br />
|X<br />
|}<br />
Diatonic thirds are bolded.<br />
<br />
=== Scales ===<br />
[if edo is composite, link to subset edos and discuss scales of subset edos there]<br />
<br />
[List scales and scale descriptions including structure (generators if applicable), notable intervals available, associated temperaments, relationships to other scales, and logic for derivation.]<br />
<br />
[Explain more complex scale theory topics here]<br />
<br />
==== Tables of scales ====<br />
<br />
=== Harmony ===<br />
[explain notable JI and DR chords. TRY TO PROVIDE COHERENT CHORD SYSTEMS RATHER THAN JUST LISTING RANDOM CHORDS WITH NO RELATION. Explain tunings of familiar chords]<br />
<br />
[Explain systems of harmony here, this is how you put the chords together to make music]<br />
<br />
[Sections such as "functional harmony", "modal harmony", etc - varies based on the edo and the personal composition style]<br />
<br />
[Use individual voices maybe?]<br />
<br />
==== Tables of chords ====<br />
<br />
== Instruments ==<br />
[Put instruments and isomorphic layouts here]<br />
<br />
== Supersets and subsets ==<br />
<br />
== Comparisons to other tuning systems ==<br />
<br />
== Music in Xedo ==<br />
<br />
== See also ==</div>Vectorhttps://xenreference.com/wiki/index.php?title=22edo&diff=779122edo2026-07-19T06:41:07Z<p>Vector: Restore moved theory sections</p>
<hr />
<div>[[File:Etj.png|thumb|307x307px|22edo visualization]]<br />
'''22edo''', or 22 equal divisions of the octave (sometimes called '''22-TET''' or '''22-tone equal temperament'''), is the [[equal tuning]] with a step size of 1200/22 ~= 54.5 [[cents]], dividing [[2/1]] into 22 steps.<br />
<br />
22edo is the fourth-smallest EDO with a diatonic ([[5L 2s]]) MOS scale formed by a [[chain of fifths]], which has a [[hardness]] of 4:1. It achieves this with a [[perfect fifth]] tuned sharpward (~709{{c}}) so that the same interval comprises [[9/8]] and [[8/7]]. Its logic is therefore that of [[Archy]] (or Superpyth) temperament, rather than [[Meantone]]: that is, the minor and major thirds available in the diatonic MOS approximate the [[2.3.7 subgroup|septal]] thirds, [[7/6]] and [[9/7]], often called "subminor" and "supermajor" (as in the [[ADIN]] system for melodic qualities, which will be used in much of this article). <br />
<br />
As an even EDO, 22edo includes the 600{{c}} tritone familiar from [[12edo]], but it divides neither the [[perfect fourth]] nor fifth in half, meaning that it does not include [[semifourth]]s or [[neutral third]]s. It divides the perfect fourth (9\22) in three, however, implying that a [[tetrachord]] of three equal intervals is possible in 22edo. 22edo also includes [[11edo]] as a subset, and similarly to [[6edo]] (the whole-tone scale)'s relation to 12edo, 11edo does not include a fifth; however, 22edo's approximations to intervals of 7, 9, 11, 15, and 17 come from 11edo.<br />
[[File:24edo 22edo comparison.png|thumb|The interval qualities found in 22edo vs. those found in 24edo.]]<br />
22edo distinguishes its native subminor and supermajor thirds from approximations to [[5-limit]] intervals, [[6/5]] and [[5/4]] (which can be called "nearminor" and "nearmajor" to disambiguate). As a result, 22 is perhaps the smallest EDO that can be considered to incorporate full [[7-limit]] harmony, as it is the first to distinctly (and [[consistent]]ly) represent the intervals 8/7, 7/6, 6/5, 5/4, 9/7, and 4/3, each one step apart. Additionally, 22edo contains a representation of the [[11/8|11th harmonic]], although many [[11-limit]] intervals are not distinguished from 5-limit intervals (e.g. [[11/9]] is mapped to the same interval as 6/5), as well as the 17th.<br />
[[File:22edo 1.mp3|thumb|22edo pajara scale and chords (0-3-6-8 major and minor)]]<br />
[[File:22edo 62.mp3|thumb|22edo porcupine scale and chords (0-1-3 and 0-2-3)]]<br />
22edo may be structurally understood as having four distinct interval qualities while 12edo has two - in fact, splitting each whole tone into four instead of two while keeping the semitones as one step each defines 22edo, although the split interval qualities are a more general feature of [[keemic]] temperaments such as septimal [[Porcupine]]. As such, two distinct qualities correspond to 12edo major (nearmajor and supermajor), and two distinct qualities correspond to 12edo minor (nearminor and subminor). This can be understood as an alternative approach relative to quarter-tone systems or other systems in which the chromatic semitone is halved; in those, the 12edo categories are retained while new categories are added in between them.<br />
<br />
== General theory ==<br />
=== Derivation of 22edo ===<br />
To fill out the structure of 22edo, we may start with the unison and the perfect fourth. Whereas in 12edo we have four intervals between them (the minor second, major second, minor third, and major third), in 22edo, each of these is doubled into a sharper and flatter counterpart, so that there is the subminor second, nearminor second, nearmajor second, supermajor second, subminor third, nearminor third, nearmajor third, and supermajor third. We may also view the thirds as the intervals encompassed by the perfect fourth and the whole tone (or supermajor second, which is the closest interval to the 12edo and Pythagorean 9/8 whole tones), which separates the fourth from the fifth. Flat of the whole tone, the remaining types of seconds function as three categories of semitone - the diatonic semitone is closer to a quarter-tone in size (about 55 cents), the equal semitone is half of the whole tone, and the chromatic semitone is three fourths of a whole tone. It may also be useful to think of the chromatic semitone as a "minor tone", separating 9/8 from 5/4.<br />
<br />
Because the whole tone now spans a wider portion of the perfect fourth, this implies that the distance between the fourth and fifth is widened, and thus that the fifth is sharper than in 12edo.<br />
<br />
From this point, we may fill out the rest of 22edo with a whole tone between the fourth and fifth, and another fourth to close the octave. We find that 22edo shares the perfect semi-octave tritone with 12edo, although because of its representation of intervals involving 7 it ends up having a much more fundamental harmonic role than it does in 12edo.<br />
<br />
=== JI approximation ===<br />
22edo's tuning of the 7-limit is marked by the sharpness of primes 3 and 7, and the slight flatness of prime 5. The combination of flat 5 and sharp 3, in particular, implies that [[25/24]], the chroma separating the classical major triad [[4:5:6]] and its complement, is considerably narrowed to the size of a quartertone. Meanwhile, as 7 is sharp, [[49/48]], the chroma separating [[6:7:8]] from its complement, is exaggerated, in fact to the same size as 25/24. This gives [[7/5]] the most damage out of the 7-[[odd-limit]], tuning it (and thus [[10/7]]) to the semioctave at 600{{c}}. One notable interval that 22edo (via 11edo) approximates very well, however, is 9/7, tuned only about 1.3{{c}} sharp, approximating quarter-comma superpyth.<br />
<br />
22edo also approximates the interval [[11/10]] to within 1.4{{c}}, as 3 steps. Thus prime 11 is tuned flatward, similarly to prime 5, and even though 22edo equates the intervals 6/5 and 11/9, its approximation to prime 11 still allows for convincingly smooth temperings of chords low in the harmonic series that contain the 11th harmonic. Characteristically of porcupine temperaments, there is no true "neutral third"; 13/8 must be approximated extremely inaccurately either as the nearmajor or nearminor sixth, a characteristic shared with 15edo. As such, it is best to avoid 13-limit harmony in 22edo, except for error-cancelling ratios (such as 52/49 or 19/13). <br />
<br />
Among the higher primes, 22edo approximates [[17/16]] as two steps and [[32/29]] as three steps, and one step of 22edo is extremely close to [[32/31]]. It is worth mentioning that prime 29 in particular allows for an interpretation of 22edo's nearminor third (6\22) as [[29/24]], which is only about 0.35{{c}} off. This leaves only 13, 19, and 23 out of the 31-limit as primes not approximated by 22edo in some way.<br />
{{Harmonics in ED|22|31|0}}<br />
<br />
=== Intervals of 22edo ===<br />
A list of intervals is available at [[22edo/Intervals]], which goes over each of the steps of 22edo in detail, as is done in the documentation for various other equal temperaments on various websites. For conciseness, the main page will present mainly general information.<br />
<br />
==== Edostep interpretations ====<br />
22edo's edostep has the following interpretations in the 7-limit:<br />
* 25/24 (the difference between 5/4 and 6/5)<br />
* 28/27 (the difference between 9/7 and 4/3, or 9/8 and 7/6)<br />
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)<br />
* 49/48 (the difference between 8/7 and 7/6)<br />
* 81/80 (the difference between [[10/9]] and 9/8)<br />
<br />
Including prime 11, it additionally serves as:<br />
* 22/21 (the difference between 7/6 and 11/9, or [[14/11]] and 4/3)<br />
* 33/32 (the difference between 4/3 and 11/8, or [[12/11]] and 9/8)<br />
* 45/44 (the difference between 11/9 and 5/4, or 11/10 and 9/8)<br />
* 56/55 (the difference between 5/4 and 14/11, or 11/8 and 7/5).<br />
* 80/77 (the difference between 11/10 and 8/7, or 11/8 and 10/7)<br />
22edo may be detempered as [28/27] [36/35-33/32-80/77] [49/48] [36/35-25/24-36/35] [28/27-33/32] [56/55-80/77] [33/32-28/27] [36/35-25/24-36/35] [49/48] [80/77-33/32-36/35] [28/27]<br />
<br />
==== Notation systems and a table of intervals ====<br />
[[File:Wryw.png|thumb|426x426px|Ascending whole tone in 22edo with normal chain-of-fifths and ups and downs notation (treble clef). (Ups and downs use strange symbols due to the limitations of MuseScore.)]]<br />
As 22edo is not a meantone system, the notes labeled with the standard diatonic names differ significantly in function from how these notes are treated in common-practice harmony. It is thus important to understand the many faces of each of 22edo's pitches (which some might consider as a downside of using the Pythagorean system, but can make notation easier to read when written on the staff, as there are no potential unpredictable wolf intervals).<br />
<br />
The "native-fifths" system mentioned here is the most commonly used system, and the one that most microtonal notation systems support by default. It is derived through stacking 22edo's tempered version of 3/2 and assigning names accordingly. As a result, the nominals (C, D, E, F, G, ...) follow the diatonic MOS, where the distances between the large steps (C-D, D-E, F-G, G-A, A-B) are 4 EDO steps, and the distances between the small steps (E-F, B-C) are 1 EDO step. Therefore, a sharp corresponds to +3 EDO steps (the difference between a large step and a small step, which is the difference between the MOS' major and minor) while a flat corresponds to -3 (representing the diatonic chroma in each case).<br />
<br />
Each sharp or flat can be split into three distinct notes, so we use the accidental ^ to raise by one EDO step and v to lower by one EDO step (ups and downs notation). Other accidentals that are identified with this in 22edo include any accidental representing the syntonic comma (such as in Ben Johnston, sagittal, SRS, or FJS notation), any accidental representing 25/24 (the accidentals to be used for porcupine[7] or pajara[10] notation, also Ben Johnston), and any accidental representing 33/32 (such as the FJS or HEJI accidentals for 11).<br />
<br />
That is, in the other notation systems proposed besides mosdiatonic, the difference between minor and major (as interval qualities in the scale being used for the notation) is 1 EDO step. To avoid ambiguity, these systems may use exclusively ups and downs, though it might be more natural to some to repurpose the diatonic # and b symbols (as is done here), especially if diatonic notation is not used simultaneously with these other schemes. The Zarlino notation here uses the [[ternary]] Zarlino scale (see [[#Zarlino diatonic]]), or Ptolemy's intense diatonic, as its basic scale (which prioritizes the 5-limit, whereas native fifths prioritize 2.3.7), and uses to its advantage the fact that one step of 22edo maps to both 25/24 and 81/80 (a property of Porcupine temperament). Pajara notation uses the 10-note Pajara scale (see [[#Pajara]]) as its basis, which is generated by a perfect fifth but splits the octave in two to reach intervals of the full 7-limit relatively easily; the scale has four 2-step and one 3-step intervals per half-octave, which differ in size by a diatonic minor second, which is 1 step in 22edo.<br />
<br />
Both degrees and notes in 10-form pajara should be notated with 0-indexed numerals in text: the tonic is always 0, and absolute pitch should be specified in relation to standard diatonic notation. 0-indexing is used so that systems such as figured bass that depend on numerals being a single symbol each still work (if 1-indexing was used, the number 10 would indicate a degree). Additionally, Roman numeral analysis in this case would use N for zero.<br />
<br />
As for notating the 10-form on the staff, there are a few different approaches. The first adds an extra line to each staff so that an octave can span 11 staff positions, but comes at the cost of losing intuition for people used to reading intervals from standard notation. The second uses the mosdiatonic notation for pentic, but uses an extra symbol to mark an alteration by a 109c semitone, allowing the full range of notes in pajara to be provided at the cost of potential overloading on symbols as opposed to visual distance to denote pitch. Meanwhile, the third option is simply to notate it starting from mosdiatonic as a base, with ups and downs notation.<br />
<br />
22edo's qualities correspond neatly to the basic color qualities proposed by Kite: red (ru) = supermajor, yellow (yo) = nearmajor, green (gu) = nearminor, and blue (zo) = subminor. That these are equally spaced shows that 22edo is a [[Tertian structure#Keemic|keemic]] temperament, a quality shared with tunings such as 41edo, and their adjacency in the edo preserves the intended mnemonic framework of a "rainbow" of qualities.<br />
<br />
=== On "major" vs. "supermajor" ===<br />
A large number of resources, including Unque's theory page on xen.wiki, the Lumatone introductory video to 22edo, and the 31et.com page on 22edo, reserve "major/minor" for 5/4 and 6/5, distinguishing an unmarked "major" from "supermajor" (and "minor" from "subminor"). There is no obvious reason to do this other than 5-limit preferentialism, and it leads to ambiguity where "major" could either refer to specifically classical major intervals or more generally to any major interval. Diatonic notation has the opposite behavior, leaving 9/7 and 7/6 unmarked, while labelling 6/5 and 5/4 as "upminor" and "downmajor" respectively; while this is not a problem on its own, it leads to a high degree of ambiguity with the other naming scheme and obscures the idea of 22edo splitting each 12edo quality into two (therefore, 5/4 and 9/7 should both be equally major).<br />
<br />
On this page, to resolve this and in accordance with the ADIN system of interval names, the two qualities of major are distinguished with "nearmajor" (5/4, 5/3) and "supermajor" (9/7, 12/7), and likewise the minor qualities are "nearminor" (6/5, 8/5) and "subminor" (7/6, 14/9, 7/4). Nearminor and nearmajor intervals may otherwise be called "classic(al)", "pental", or "ptolemaic" minor/major, which are terms used to describe the simple 5-limit intervals to which they correspond.<br />
<br />
On this page, unqualified "major" may refer to both nearmajor and supermajor collectively in cases like "either major key" or "the major thirds". However, when used to refer to a specific interval ("the major third"), it should refer to the supermajor intervals, as those are the major intervals of the diatonic MOS in 22edo. Likewise for minor.<br />
<br />
Short forms of interval names should use ups and downs notation, however, so that "M3" is always the supermajor third. But also, see [[#Table of chords]].<br />
<br />
JI approximations of steps in 22edo, as well as ways of notating 22edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.<br />
{| class="wikitable"<br />
|+<br />
! rowspan="2" | Edostep !! rowspan="2" | Cents !! rowspan="2" | 11-limit add-17 <br> JI approximation !! colspan="3" | Notation !! rowspan="2" | Interval category <br> (ADIN)<br />
|-<br />
! rowspan="1" | Native-fifths <br> (ups & downs) !! rowspan="1" | Blackdye/Zarlino <br> (Vector) !! rowspan="1" | Pajara <br> decatonic<br />
|-<br />
|0<br />
|0<br />
|1/1<br />
|C<br />
|C<br />
|0<br />
|Perfect unison<br />
|-<br />
|1<br />
|54.5<br />
|25/24, 28/27, ['''33/32'''], 36/35<br />
|^C, Db<br />
|C#<br />
|1b<br />
|(Sub)minor second<br />
|-<br />
|2<br />
|109.1<br />
|[16/15], 15/14, 18/17, ['''17/16''']<br />
|vC#, ^Db<br />
|Db<br />
|1<br />
|Nearminor second<br />
|-<br />
|3<br />
|163.6<br />
|10/9, [11/10], 12/11<br />
|C#, vD<br />
|D<br />
|1#<br />
|Nearmajor second<br />
|-<br />
|4<br />
|218.2<br />
|8/7, '''9/8''', [17/15]<br />
|D<br />
|D#<br />
|2<br />
|(Super)major second<br />
|-<br />
|5<br />
|272.7<br />
|7/6<br />
|^D, Eb<br />
|Ebb / Dx<br />
|2#<br />
|(Sub)minor third<br />
|-<br />
|6<br />
|327.3<br />
|6/5, 11/9, 17/14<br />
|vD#, ^Eb<br />
|Eb<br />
|3b<br />
|Nearminor third<br />
|-<br />
|7<br />
|381.8<br />
|'''[5/4]'''<br />
|D#, vE<br />
|E<br />
|3<br />
|Nearmajor third<br />
|-<br />
|8<br />
|436.4<br />
|[9/7], 14/11, 32/25<br />
|E<br />
|E#<br />
|4b<br />
|(Super)major third<br />
|-<br />
|9<br />
|490.9<br />
|4/3<br />
|F<br />
|F<br />
|4<br />
|Perfect fourth<br />
|-<br />
|10<br />
|545.5<br />
|'''11/8''', 15/11<br />
|^F, Gb<br />
|F#<br />
|4#<br />
|Near fourth<br />
|-<br />
|11<br />
|600<br />
|7/5, 10/7, [17/12]<br />
|vF#, ^Gb<br />
|Gbb / Fx<br />
|5<br />
|Tritone<br />
|-<br />
|12<br />
|654.5<br />
|16/11, 22/15<br />
|F#, vG<br />
|Gb<br />
|6b<br />
|Near fifth<br />
|-<br />
|13<br />
|709.1<br />
|'''3/2'''<br />
|G<br />
|G<br />
|6<br />
|Perfect fifth<br />
|-<br />
|14<br />
|763.6<br />
|[14/9], 11/7, '''25/16'''<br />
|^G, Ab<br />
|G#<br />
|6#<br />
|(Sub)minor sixth<br />
|-<br />
|15<br />
|818.2<br />
|[8/5]<br />
|vG#, ^Ab<br />
|Ab<br />
|7<br />
|Nearminor sixth<br />
|-<br />
|16<br />
|872.7<br />
|5/3, 18/11, 28/17<br />
|G#, vA<br />
|A<br />
|7#<br />
|Nearmajor sixth<br />
|-<br />
|17<br />
|927.3<br />
|12/7<br />
|A<br />
|A#<br />
|8b<br />
|(Super)major sixth<br />
|-<br />
|18<br />
|981.8<br />
|'''7/4''', 16/9, [30/17]<br />
|^A, Bb<br />
|Bbb / Ax<br />
|8<br />
|(Sub)minor seventh<br />
|-<br />
|19<br />
|1036.4<br />
|9/5, [20/11], 11/6<br />
|vA#, ^Bb<br />
|Bb<br />
|9b<br />
|Nearminor seventh<br />
|-<br />
|20<br />
|1090.9<br />
|['''15/8'''], 28/15, 17/9, [32/17]<br />
|A#, vB<br />
|B<br />
|9<br />
|Nearmajor seventh<br />
|-<br />
|21<br />
|1145.5<br />
|48/25, 27/14, [64/33], 35/18<br />
|B<br />
|Cb<br />
|9#<br />
|(Super)major seventh<br />
|-<br />
|22<br />
|1200<br />
|2/1<br />
|C<br />
|C<br />
|0<br />
|Octave<br />
|}<br />
<br />
=== Solfege ===<br />
Solfege may use the [[Porcupine#Solfege|porcupine]] solfege systems.<br />
<br />
== Tempering properties ==<br />
=== Tempered commas ===<br />
Important [[comma]]s tempered out by the 11-limit of 22et include:<br />
* [[50/49]] (jubilismic), equating 7/5 and 10/7 to exactly half an octave.<br />
* [[55/54]] (telepath), equating 6/5 with 11/9<br />
* [[64/63]] (archytas), equating 9/8 with 8/7 and a stack of two 4/3s to [[7/4]]<br />
* [[99/98]] (mothwellsmic), equating 14/11 with 9/7<br />
* [[100/99]] (ptolemismic), equating 10/9 with 11/10, and a stack of two 6/5s to [[16/11]]<br />
* [[121/120]] (biyatismic), splitting 6/5 into 11/10~12/11, and equating 11/8 with [[15/11]]<br />
* [[176/175]] (valinorsmic), equating a stack of two 5/4s to [[11/7]]<br />
* [[225/224]] (marvel), splitting 8/7 into 15/14~16/15 and equating a stack of two 5/4s to [[14/9]]<br />
* [[245/243]] (sensamagic), equating a stack of two 9/7s to [[5/3]]<br />
* [[250/243]] (porcupine), equating a stack of two 10/9s to 6/5 (splitting 4/3 in three)<br />
* [[385/384]] (keenanismic), equating the product of 7/6 and 5/4 to 16/11<br />
<br />
[[Regular temperament]]s associated with these are discussed in [[#Notable structural chains]]. In addition to the equivalences mentioned above, we can find that three 16/15s form 6/5 (diaschismic), three 6/5s form 7/4 (keemic), and three 7/6s form [[8/5]] (orwellismic). {{Adv|In terms of [[S-expression]]s, 22et equates S5, S6, S7, and S9 all to one step, and tempers out S8, S10, S11, and S15, as well as S16 and S17 if prime 17 is considered.}}<br />
<br />
=== Arithmetic progressions ===<br />
22et in the 2.3.5.7.11.17.29.31 subgroup can be specified entirely by equalizing an arithmetic division of 4/3: 27:28:29:30:31:32:33:34:35:36 is mapped to a chain of single steps of 22edo. Subsets of this division include 9:10:11:12 (porcupine) every 3 steps and 14:15:16:17:18 (pajara) every 2 steps.<br />
<br />
{{Adv|This chain can be extended further to 26::39, an arithmetic subdivision of 3/2 into 13 parts, which is mapped to a chain of single steps in the 22fh [[val]] (with primes 13 and 19 tuned over-critically sharp instead of near-critically flat). This is the largest arithmetic equal division of 3/2 that can be mapped onto a logarithmic equal division, and is the basis for forming '''Ringer 22fh''': 26:27:28:29:30:31:32:33:34:35:36:37:38:(39~40):41:42:44:45:46:48:(49~50):51:52.}}[[File:22edo.png|thumb|Porcupine and Pajara are the defining temperaments of 22edo.]]<br />
=== Notable structural chains ===<br />
22edo has five distinct intervals that [[generator|generate]] octave-periodic temperaments, not counting temperaments of 11edo. These are 1\22 (the subminor second), 3\22 (the nearmajor second), 5\22 (the subminor third), 7\22 (the nearmajor third), and 9\22 (the perfect fourth).<br />
<br />
3\22 serves as 10/9, 11/10, and 12/11 simultaneously, serving as a type of interval called a ''quill'' defined by those three simultaneous interpretations. The temperament associated with this equivalence is fittingly called '''Porcupine''', and the nearminor third (11/9~6/5) is found at two generators and the perfect fourth is found at three. Further on, the nearminor sixth (8/5) is found at five generators, and the minor seventh consisting of two stacked fourths is equated to 7/4. MOS scales produced by Porcupine include the equitetrachordal heptatonic (1L 6s) and its octatonic extension (7L 1s). This structure is shared with EDOs like [[15edo|15]] and [[37edo|37]], as well as [[29edo]] aside from the mapping of 7.<br />
<br />
5\22 represents a sharply tempered 7/6. Three of these represent 8/5 in '''[[Orwell]]''' temperament, while if stacked further, four 7/6s are made to reach [[15/8]], so that [[3/1]] is split into seven. Orwell also includes 11-limit equivalences by virtue of two generators forming 15/11 simultaneously with 11/8, and six generators forming 14/11 simultaneously with 9/7. MOS scales produced by Orwell include an enneatonic (4L 5s) and its tridecatonic extension to 9L 4s. This structure is shared with EDOs like [[31edo|31]] and [[53edo]], though note that the 11-limit is less accurate than the 7-limit component in general.<br />
<br />
7\22 represents a flattened 5/4, five of which stack to 3/1, which is '''[[Magic]]''' temperament. The deficit between the octave and three 5/4s, [[128/125]], is here equated to 25/24, which is tuned to half of 16/15. As far as the 7-limit goes, two generators reach the interval of 14/9, and its complement 9/7 divides 5/3 in two; the 7th harmonic itself is eventually found at 12 generators. This structure is shared with EDOs like [[19edo|19]] and [[41edo]].<br />
<br />
Finally, 9\22 represents 4/3, two of which stack to 7/4 in '''Archy/Superpyth''' temperament. The next two fourths give us 7/6 and 14/9, the subminor third and sixth. 22edo, by virtue of 9/7 being tuned nearly just, is close to the 1/4-comma tuning of Archy, with other important tunings generally having a sharper fifth than 22edo. The MOS scales produced by Archy include the native diatonic (5L 2s) and chromatic (5L 7s) scales. Note that 22edo tempers out 245/243, so that twice 9/7 gives 5/3, and this is how 5 is mapped in Superpyth as tuned also in [[27edo|27]] and [[49edo]]; this is not shared with even sharper tunings of Archy, such as 37edo.<br />
<br />
22edo also supports temperaments where the octave is split in half. The most notable one of these found in 22edo is '''[[Pajara]]''', generated by a perfect fifth or equivalently half a wholetone (identifiable as 16/15~17/16~18/17), against the half-octave. A wholetone (two generators) below the half octave gives 5/4. As the octave less a wholetone is 7/4 specifically in Archy, Pajara maps the half-octave to 7/5. Equivalently, 5/4 and 7/4 are separated by exactly a 600c tritone. MOS scales produced by Pajara include the decatonic (2L 8s) and dodecatonic (10L 2s) scales. This provides a very simple way of traversing the 7-limit, though it is rather high in damage as a temperament beyond 22edo specifically (and its trivial tunings [[10edo]] and 12edo). This general structure without prime 7, known as [[Diaschismic]], however, is supported by notable EDOs such as [[34edo|34]] and [[46edo]].<br />
<br />
In fact, pajara as a generator structure is able to reach the entire 7-odd-limit (see [[#Consonance and dissonance properties]]) in only a 14-note scale, the lowest out of any structure supported by 22edo (note that the 7-odd-limit consists of 12 intervals in 22edo, so only two intervals outside the set are even in the scale, namely ~109c and ~1090c). It also reaches the 9-odd-limit in 18 notes, again the lowest (the 9-odd-limit in 22edo has 16 intervals). The furthest number of generator steps from the unison to reach the most complex 9-odd-limit consonance in pajara (multiplied by 2 periods) is 8; for all other half-octave temperaments it is 10 and for the remainder it is 11 (due to 7/5 being at the tritone). And when considering only the prime harmonics, pajara reaches 3, 5, and 7 at an 8-note scale and at only 4 steps from the unison, again a greater simplicity than any other generator structure.<br />
[[File:Sensamagic.mp3|thumb|Sensamagic demonstration]]<br />
<br />
==== 11edo temperaments ====<br />
11edo serves as an analogue of the whole tone scale in 22edo, as 11edo happens to miss intervals based on 3/2 and 5/4 entirely, instead shifting focus to more complex intervals involving those primes (such as 5/3 or 9/7) or harmony based on solely the 7th and 11th harmonics. This is similar to how 12edo's whole tone scale skips over the perfect fifth, instead focusing on the major third. However, 22edo (and thus 11edo) is large enough that there are a couple notable relations that exist entirely within this subset. <br />
<br />
Firstly, one may note that in 22edo, the supermajor third stacks twice to reach a nearmajor sixth; in other words, the nearmajor sixth can be evenly split in two. This "semi-sixth" interval gives rise to the ''sensamagic'' category of temperaments, which in 11edo specifically becomes ''Sentry''. Another way to think of sentry is that 9/7 may be, as previously mentioned, found directly between 5/4 and 4/3. Without 4/3 or 3/2 themselves, however, our conventional scale-building anchors become absent. However, any interval may generate a scale simply by stacking it over and over. Sentry has an 8-note scale, constructed by this method, consisting of in 2-1-1-2-1-1-2-1 in 11edo, or 4-2-2-4-2-2-4-2 in 22edo. (The generator of sentry might also be considered to represent the shared function of 5/4 and 4/3 in a 3:4:5 system, structurally implying the inaccurate "Father" temperament, although that is not supported by 11edo patent.)<br />
<br />
Another temperament that resides in 11edo is called ''Orgone'', and splits 7/4 into three parts, two of which reach 16/11 (the octave complement of 11/8). One of these parts also functions as 6/5, or in a context without 3/2, more functionally as its octave complement 5/3 (perhaps to be further split in sentry). The scale generated by orgone is 2-4-2-4-2-4-4.<br />
<br />
== Compositional theory ==<br />
=== Tertian structure ===<br />
22edo is a keemic temperament, with four distinct types of thirds and in general four distinct interval qualities (which largely correspond to 7/, /5, 5/, and /7 modifications of the Pyth chain). As a result of supporting Porcupine, the interval qualities associated with /5 and 5/ are also associated with 11/ and /11, respectively.<br />
<br />
{| class="wikitable"<br />
|+Thirds in 22edo<br />
!Quality<br />
|'''Subminor'''<br />
|Nearminor<br />
|Nearmajor<br />
|'''Supermajor'''<br />
|-<br />
!Cents<br />
|'''272.7'''<br />
|327.3<br />
|381.8<br />
|'''436.4'''<br />
|-<br />
!Just interpretation<br />
|'''7/6''' (+5.9{{c}})<br />
|6/5 (+11.6{{c}})<br />
|5/4 (-4.5{{c}})<br />
|'''9/7''' (+1.3{{c}})<br />
|}<br />
Diatonic thirds are bolded.<br />
<br />
=== Scales ===<br />
==== Scale theory ====<br />
22edo has no one perfectly obvious counterpart to the diatonic scale found in 12edo. Instead, there are two heptatonic scales with diatonic-like behavior, the Pythagorean diatonic and the zarlino diatonic, coming from the fact that each 12edo quality is split into two distinct 22edo qualities. The distinction between the two diatonic scales arises from how the diatonic in 12edo is interpreted. 12edo's diatonic can be viewed as a simplification of 5-limit harmony, in which case 22edo, as a system that does not make the same simplifications, must make distinctions that 12edo does not. This gives rise to the distinction between the two sizes of whole tone, and the Zarlino diatonic of 4-3-2-4-3-4-2. Alternatively, one can choose to retain the MOS (moment of symmetry) structure of 12edo's diatonic, which yields the Pythagorean diatonic of 4-4-1-4-4-4-1. However, either you have to use the 5-limit accidental consistently, or notation gets irregular (as when you use Zarlino as your nominals).<br />
<br />
One way to resolve the issue is to ditch diatonic entirely, and instead use another scale as your base set of notes, which functions somewhat like, or is derived from, diatonic. These scales usually have more notes to account for the greater harmonic complexity of 22edo compared to 12edo.<br />
<br />
==== Pythagorean diatonic ====<br />
This is the diatonic scale most directly analogous in structure to the 12edo diatonic, given its MOS form. Advantages of using it include the fact that all steps are what they appear to be - for example, D-G and G-C are both perfect fifths - and that it appears as a subset of the chain of fifths itself. One key difference is that the major and minor thirds do not get mapped to the expected 5-limit interpretations, but rather to the supermajor and subminor thirds of 22edo. This is good for septimal harmony, not so much for 5-limit harmony (to get 5-limit thirds, you have to go 9 steps up!). A downside, or more generally a significant awkwardness, to using this system is the fact that due to the minor second being so small, the chromatic semitone is massive - closer to a whole tone than a proper semitone (in fact, it is enharmonic to the small whole-tone used in the Zarlino system). This can make interval classifications somewhat unintuitive if ups and downs are not used - for instance, a Nearmajor triad, 0-7-13, is C-D#-G (meaning that a third-sized interval is technically a second) - and in fact 22edo is the first edo other than 15edo (which is strange in its own right) to require ups and downs to notate all intervals without problems with intuition like this (Edos like 17 and 10 utilize semisharps and semiflats, which 22edo cannot use as it does not have neutral intervals.)<br />
<br />
It may be useful, perhaps for [[extraclassical tonality]], to use the full 12-note form of superpyth's scale. This yields a softer (yet still hard) scale, alongside making 5/4 accessible as the major 3-step.<br />
<br />
==== Zarlino diatonic ====<br />
A faithful way of representing the 5-limit diatonic structure in 22edo (which, unlike its just counterpart, keeps the 7-limit convenient to access) is to use 5/4 as the scale's major third, and likewise 5/3 and 15/8 as the major sixth and seventh respectively. The scale pattern for zarlino is 4-3-2-4-3-4-2. This encounters problems with existing familiarity with notation (for example, one of D-G and G-C must now be a flat 5th, called a "wolf fifth" and representing 16/11 as opposed to 3/2), however it does have one thing going for it: in 22edo in particular (and a family of edos including 15, 29, and 37), the interval which separates Pythagorean intervals from their 5-limit counterparts is the exact same as the interval separating 5-limit major and minor intervals. This means that, if # and b are used to represent this interval as an accidental, no additional accidentals are necessary. This is why the 3-step interval is a "Nearmajor second" along with being a chromatic semitone.<br />
<br />
==== Blackdye ====<br />
Blackdye is a rank-3 scale similar to zarlino diatonic, which attempts to compromise between zarlino and Pythagorean diatonic in a way, by including a few intervals from both at once. Blackdye can be thought of as dividing each 4-step whole tone into a 3-step tone and a single step called an [[aberrisma]], which separates zarlino and Pythagorean intervals. The blackdye scale pattern is 3-1-3-2-3-1-3-1-3-2. One way to use blackdye is to essentially treat it as multiple overlapping diatonics, which one can modulate between. Blackdye has the additional property of not having chirality, so that there is only a single form of the scale instead of a left- and right-handed version like with zarlino.<br />
<br />
In fact, blackdye or several characteristics of it are likely to naturally emerge in tonal harmony in the first place, given an overall desire to avoid wolf fifths in nearmajor and nearminor tonalities, resulting in certain intervals being doubled up. Most notably, it is reasonable to consider the two forms of major second equally part of a nearmajor tonal system, so that 5-2 and 2-6 can both be perfect fifths with different versions of the 2 degree (although unless 6 is additionally sharpened, some additional harmonic movements are needed to resolve the edostep offset that results from a pumped syntonic comma if you move from fifth-bounded triads on 5 to 2 in a single motion).<br />
<br />
==== Equiheptatonic ====<br />
The equable diatonic, 1/(18:20:22:24:27:30:33:36) (or in this case, equivalently its [[otonal]] counterpart) is represented as the MOS scale sssLsss (3-3-3-4-3-3-3) in 22edo, as a result of porcupine temperament. It is reasonable to, for that structural reason, consider 3-3-3-4-3-3-3 the default mode, with a nearminor chord on the tonic - it is the unique mode which possesses both a perfect fifth and a perfect fourth. 3-3-3-4-3-3-3 is more generally the MOS porcupine[7]; altering several notes of this MOS yields the Zarlino diatonic, explaining 22edo Zarlino's heavy reliance on porcupine's equivalences. Porcupine also has an 8-note scale 3-3-3-3-1-3-3-3, and a chromatic scale 1-2-1-2-1-2-1-2-1-2-1-2-1-2-1, a form of the Roklotian scale that may also be derived by dividing the intervals of superpyth pentatonic: 4-5-4-5-4 -> [1-2-1] [2-1-2] [1-2-1] [2-1-2] [1-2-1].<br />
<br />
==== Pajara ====<br />
Note that the 1-step interval which serves as the 3-limit diatonic semitone is the very same as the 5-limit chromatic semitone, and that the 3-step interval serving as the 3-limit chromatic semitone also happens to map to a certain kind of large 5-limit diatonic semitone (such that it and the chromatic semitone stack to the 4-step whole tone). This suggests that if there was a way to swap the diatonic and chromatic semitones, we could faithfully represent the full 7-limit, including 5.<br />
<br />
And as it turns out, there is a way to do that. We start with the pentatonic scale, not only because it's closer to even but because the interval between its small and large steps is exactly the 1-step interval we want to function as our chroma. Then, we add another pentatonic scale that is offset by a tritone from the first one. This results in the decatonic "Pajara[10]" scale (3-2-2-2-2-3-2-2-2-2), where every note has a corresponding note a tritone apart. It solves the problem of representing intervals of both 5 and 7 by introducing three new ordinal classes to provide space for 7-limit intervals to fit on their own degrees of the scale. That way, 7/4 isn't a subminor seventh, it's a major version of the Pajara 8-step. One can even define a notation system for Pajara, wherein the notes are numbered 0-9 and # and b represent alterations by a single step. Pajara retains the property where most notes have a fifth over them.<br />
<br />
To extend to the 11-limit, Pajara[12] (1-2-2-2-2-2-1-2-2-2-2-2) can be easily used, which has the same number of notes as 12edo's chromatic scale, but with two of the semitones from 12edo replaced with quartertones. This gives major and minor thirds separate interval categories (allowing both to be played on certain scale degrees), and the 11th harmonic can be found as the major 5-step. The MODMOS [2-2-2-1-2-2-2-2-1-2-2-2] of Pajara[12] is the Delkian scale; it may be derived again by splitting 4-5-4-5-4 ([2-2] [2-1-2] [2-2] [2-1-2] [2-2]), or by splitting each whole tone of MOS diatonic ([2-2] [2-2] 1 [2-2] [2-2] [2-2] 1), which makes for a more xenharmonic way of translating 12edo music into 22edo than simply retuning the chain of fifths. ("When the 12edo goes chromatic, equally divide the whole tone!")<br />
<br />
The following chart shows the modes of pajara[10]:<br />
{| class="wikitable"<br />
!<br />
!Chart<br />
!2<br />
!3<br />
!4<br />
!6<br />
!8<br />
!9<br />
|-<br />
|Dynamic minor<br />
|{{Interval ruler|22|0, 100, 200, 300, 450, 600, 700, 800, 900, 1050, 1200}}<br />
|minor<br />
|minor<br />
|dim<br />
|perfect<br />
|minor<br />
|minor<br />
|-<br />
|Static minor<br />
|{{Interval ruler|22|0, 100, 200, 300, 500, 600, 700, 800, 900, 1100, 1200}}<br />
|minor<br />
|minor<br />
|perfect<br />
|perfect<br />
|minor<br />
|major<br />
|-<br />
|Static major<br />
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 800, 1000, 1100, 1200}}<br />
|minor<br />
|major<br />
|perfect<br />
|perfect<br />
|major<br />
|major<br />
|-<br />
|Dynamic major<br />
|{{Interval ruler|22|0, 100, 250, 400, 500, 600, 700, 850, 1000, 1100, 1200}}<br />
|major<br />
|major<br />
|perfect<br />
|perfect<br />
|major<br />
|major<br />
|-<br />
|Augmented<br />
|{{Interval ruler|22|0, 150, 250, 400, 500, 600, 750, 850, 1000, 1100, 1200}}<br />
|major<br />
|major<br />
|perfect<br />
|aug<br />
|major<br />
|major<br />
|}<br />
And of a MODMOS of pajara[10], ssLsssssLs, the "pentachordal" pajara scale:<br />
{| class="wikitable"<br />
!<br />
!Chart<br />
!2<br />
!3<br />
!4<br />
!6<br />
!8<br />
!9<br />
|-<br />
|(Minor)<br />
|{{Interval ruler|22|0, 100, 200, 300, 450, 550, 700, 800, 900, 1050, 1200}}<br />
|minor<br />
|minor<br />
|dim<br />
|perfect<br />
|minor<br />
|minor<br />
|-<br />
|Alternate minor<br />
|{{Interval ruler|22|0, 100, 200, 300, 450, 600, 700, 800, 950, 1100, 1200}}<br />
|minor<br />
|minor<br />
|dim<br />
|perfect<br />
|minor<br />
|major<br />
|-<br />
|(Minor)<br />
|{{Interval ruler|22|0, 100, 200, 320, 500, 600, 700, 830, 990, 1100, 1200}}<br />
|minor<br />
|minor<br />
|perfect<br />
|perfect<br />
|major<br />
|major<br />
|-<br />
|Standard major<br />
|{{Interval ruler|22|0, 100, 200, 370, 500, 600, 700, 870, 990, 1100, 1200}}<br />
|minor<br />
|major<br />
|perfect<br />
|perfect<br />
|major<br />
|major<br />
|-<br />
|(Major)<br />
|{{Interval ruler|22|0, 100, 270, 370, 500, 600, 770, 870, 990, 1100, 1200}}<br />
|major<br />
|major<br />
|perfect<br />
|aug<br />
|major<br />
|major<br />
|-<br />
|(Major)<br />
|{{Interval ruler|22|0, 170, 270, 370, 500, 670, 770, 870, 990, 1100, 1200}}<br />
|major<br />
|major<br />
|perfect<br />
|aug<br />
|major<br />
|major<br />
|-<br />
|Standard minor<br />
|{{Interval ruler|22|0, 100, 200, 300, 500, 600, 700, 800, 900, 1050, 1200}}<br />
|minor<br />
|minor<br />
|perfect<br />
|perfect<br />
|minor<br />
|minor<br />
|-<br />
|(Major)<br />
|{{Interval ruler|22|0, 100, 200, 360, 500, 600, 700, 800, 920, 1110, 1200}}<br />
|minor<br />
|major<br />
|perfect<br />
|perfect<br />
|minor<br />
|major<br />
|-<br />
|Alternate major<br />
|{{Interval ruler|22|0, 100, 270, 360, 500, 600, 700, 800, 970, 1110, 1200}}<br />
|major<br />
|major<br />
|perfect<br />
|perfect<br />
|major<br />
|major<br />
|-<br />
|(Major)<br />
|{{Interval ruler|22|0, 170, 270, 360, 500, 600, 700, 870, 970, 1110, 1200}}<br />
|major<br />
|major<br />
|perfect<br />
|perfect<br />
|major<br />
|major<br />
|}<br />
Some names are from [https://web.archive.org/web/20180927081411/http://lumma.org/tuning/erlich/erlich-decatonic.pdf Paul Erlich].<br />
<br />
==== Generator sequences ====<br />
Sentry is an 11edo temperament which outlines 3:4:5-based harmony, but instead of having 4/3 or 5/4 it has a perfect "neutral" semisixth representing 9/7. Let's say we want to re-introduce the distinction between 5/4 and 4/3 to the sentry scale. We may do so by creating an alternating stack of 5/4 and 4/3, to produce [https://scaleworkshop.plainsound.org/scale/OlyP9eaUH a similar 8-note scale to the original], but with some added distinction in interval quality; all intervals except the step itself have 2 different qualities separated by the difference between 5/4 and 4/3, which also happens to be the chroma of the original 11edo scale due to being a single step of 11edo. Interestingly, this scale also happens to be a MODMOS of Porcupine[8].<br />
<br />
Another interesting property of this scale in particular is that each of its five re-acquired perfect fifths is found on an odd scale degree, meaning that dividing a fifth in two always results in a 2-step interval and a 3-step interval. Two specific degrees happen to have both a nearminor and nearmajor chord, allowing for some very unusual harmonic structures.<br />
<br />
==== Building scales from tetrachords ====<br />
A tetrachord is a series of four notes that span a perfect fourth (alongside a few other requirements). They are useful for building scales because two perfect fourths and a whole tone may be stacked in order to complete the octave with a heptatonic scale; when singing a scale, heptatonic or pentatonic forms are naturally common due to the fact that 4/3 and especially 3/2 can be easily sung without a reference, and tetrachords formalize this natural occurrence into scale-building. Tetrachords may be classified based on the size of their largest interval; there are nine tetrachords in 22edo that fall under this classification scheme.<br />
<br />
===== Diatonic tetrachords =====<br />
In diatonic tetrachords, the largest interval is a major second (less than half of the perfect fourth). There are four diatonic tetrachords in 22edo: 3-3-3, 3-4-2, 4-3-2, and 4-4-1 (remember that a perfect fourth totals 9 steps in 22edo). When these are built up into scales, we arrive at the 3-3-3-4-3-3-3 ("onyx", equable diatonic), 3-4-2-4-3-4-2 (zarlino), 4-3-2-4-4-3-2 (didymic), and 4-4-1-4-4-4-1 (MOS diatonic) scales. Onyx is an edge case for diatonic, but it is the tempered version of a historically relevant diatonic tetrachord 1/(9:10:11:12).<br />
<br />
===== Chromatic tetrachords =====<br />
Here, "chromatic" doesn't have to do with the modern sense of the chromatic scale, but instead indicates a tetrachord where the largest interval is some kind of minor third. In 22edo, there are also four chromatic tetrachords: 5-2-2, 5-3-1, 6-2-1, and 6-1-2. When built up into scales, we get scales with two adjacent semitones, which is a very different sound from a standard diatonic scale.<br />
<br />
===== Enharmonic tetrachords =====<br />
Similarly, "enharmonic" has little to do with enharmonic notes here, but instead with tetrachords where the largest interval is some kind of major third. In 22edo, one enharmonic tetrachord exists: 7-1-1, where the 7 is the nearmajor third. The scale ends up having two adjacent quarter tones, making for somewhat of a "shimmery" sound.<br />
<br />
Tetrachords were historically used in largely monophonic music, so that additional structural constraints didn't become relevant until additional harmonic complexity entered the scene, and music switched primarily to using diatonic. Therefore, tetrachord-based scales, especially chromatic and enharmonic ones, remain useful in a monophonic or homophonic context, where the harmonic relations between notes do not matter as much. One can think of tetrachordal music as entirely "degree-based", whereas modal music is to an extent "interval-based".<br />
<br />
However, tetrachords are often still theoretically relevant in discussing the construction of modern scales, for example in the examination of the conventional framing of the melodic minor scale, where the upper tetrachord varies when ascending vs. when descending; additionally the Ionian major scale can be thought of as a M2-M2-m2 tetrachord stacked into an octave-repeating scale.<br />
<br />
===== Other polychordal structures =====<br />
<br />
====== Trichord ======<br />
It's also possible to use trichords (subscales spanning a perfect fourth and consisting of only 3 notes) to build scales in 22edo; these scales will be pentatonic and vary entirely on what middle interval is used in the trichord. Standard MOS pentatonic is achieved by using a supermajor second or subminor third; the zarlino pentatonic is achieved with a nearmajor second or nearminor third, and other, more "enharmonic" scale forms may be achieved with either kind of major third or minor second. Therefore, there are four possible trichords, considering chiral variants the same.<br />
<br />
A trichord is structurally equivalent to a fourth-bounded chthonic triad, and so a trichordal scale may be conceptualized as a chthonic generator chain, especially since the whole tone separating the trichords is itself a form of chthonic. Interestingly, this implies that trichordal scales place no notes in between the notes of their chords, having the three notes of a chthonic chord fall on consecutive scale steps.<br />
<br />
====== Pentachord ======<br />
A pentachord consists of five notes spanning a perfect fourth. A pentachord may be constructed by dividing the steps of a trichord; therefore, a trichord or a chthonic triad consists of alternating notes of a pentachord. Because of this, it is also structurally useful to insert the perfect tritone (or augmented fourth/diminished fifth) between the fourth and the fifth to maintain a relatively even spacing of intervals. The most common pentachord is the pajara pentachord, consisting of 2-2-2-3, which extends out into the pentachordal scale 2-2-2-3-2-2-2-2-2-3.<br />
<br />
The largest interval that can exist between steps in a pentachord is a nearminor third, and as such, an "enharmonic" pentachord is impossible in 22edo (although it is at finer resolutions). It is a reasonable structural constraint for pentachords to need to divide the 4-5, 5-4, or possibly 3-6 or 6-3 trichords.<br />
<br />
Pentachordal extensions may be used to improve melodic cohesion in trichordal scales.<br />
<br />
[[File:Diatonic harmony demonstration.mp3|thumb|A demonstration of 22edo diatonic functional harmony.]]<br />
<br />
==== Tables of scales ====<br />
The following is a table of scales in 22edo.<br />
<br />
===== Porcupine scales =====<br />
MOS scales generated by a nearmajor second.<br />
{| class="wikitable"<br />
!Name<br />
!Chart<br />
!Notes<br />
|-<br />
|Onyx<br />
|{{Interval ruler|22|0, 160, 320, 480, 720, 880, 1040, 1200}}<br />
|The same as the "equable Dorian" discussed above.<br />
|-<br />
|Pine<br />
|{{Interval ruler|22|0, 160, 320, 480, 640, 720, 880, 1040, 1200}}<br />
|<br />
|-<br />
|Roklotic<br />
|{{Interval ruler|22|0, 80, 160, 240, 320, 400, 480, 560, 640, 720, 800, 880, 960, 1040, 1120, 1200}}<br />
|The "Roklotian" scale mentioned in the [[22edo#Equiheptatonic|#Equiheptatonic]] section; the MOS form is specifically exclusive to the porcupine/22edo-tempered version of the scale.<br />
|}<br />
<br />
===== Orwell scales =====<br />
MOS scales generated by a subminor third.<br />
{| class="wikitable"<br />
!Name<br />
!Chart<br />
!Notes<br />
|-<br />
|Manual<br />
|{{Interval ruler|22|0, 271, 543, 814, 1086, 1200}}<br />
|The basic pentatonic for Orwell, highlighting its basic structure of stacking subminor thirds. As there are less than seven steps other than the unison, there are no perfect fifths; the fourth degree of this scale may instead be either 8/5 or 16/11.<br />
|-<br />
|Gramitonic<br />
|{{Interval ruler|22|0, 157, 271, 429, 543, 700, 814, 971, 1086, 1200}}<br />
|The standard albitonic orwell scale, discussed extensively by Levi McClain (although in its 31edo tuning). As a 9-form scale, it features a contrast between major and minor thirds on the same degree. There are two perfect fifths in the scale.<br />
|-<br />
|Antiparagonic<br />
|{{Interval ruler|22|0, 50, 157, 271, 320, 429, 543, 600, 700, 814, 871, 971, 1086, 1200}}<br />
|A larger, more chromatic-esque orwell scale featuring additional perfect fifths to build chords around. This scale is 13-form, so the seven imperfect fifths are sharp rather than flat.<br />
|}<br />
<br />
===== Magic scales =====<br />
MOS scales generated by a nearmajor third.<br />
{| class="wikitable"<br />
!Name<br />
!Chart<br />
!Notes<br />
|-<br />
|Mosh<br />
|{{Interval ruler|22|0, 330, 380, 700, 760, 1090, 1150, 1200}}<br />
|Ultimately, Magic is 3-form, however that makes for an absurdly small scale; Magic is better conceptualized as not using MOSes themselves but rather inflecting from MOS-adjacent structures. Magic is additionally unusual in placing 3/2 on the sixth degree of a heptatonic scale, rather than on the fifth degree.<br />
|-<br />
|Sephiroid<br />
|{{Interval ruler|22|0, 280, 330, 380, 660, 700, 760, 1050, 1090, 1150, 1200}}<br />
|Magic may be conceptualized decatonically as well; however, 4/3 and 3/2 are placed on the same degree, unlike in a standard 10-form.<br />
|-<br />
|Antiluachoid<br />
|{{Interval ruler|22|0, 230, 280, 330, 380, 600, 660, 700, 760, 990, 1050, 1090, 1150, 1200}}<br />
|<br />
|}<br />
<br />
===== Superpyth scales =====<br />
MOS scales generated by a perfect fifth.<br />
{| class="wikitable"<br />
!Name<br />
!Chart<br />
!Notes<br />
|-<br />
|Pentic<br />
|{{Interval ruler|22|0, 210, 490, 710, 990, 1200}}<br />
|One of two tunings of pentic available in 22edo. Doubling this offset by the tritone yields pajara[10]; this form of pentic may debatably be considered "equipentatonic". Pentic in 22edo approximates the 12:14:16:18:21:24 "JI equable pentatonic".<br />
|-<br />
|Mosdiatonic<br />
|{{Interval ruler|22|0, 210, 270, 490, 710, 930, 990, 1200}}<br />
|A hard diatonic, with small steps too small to be leading tones yet that serves as the main basis of interval classification in 22edo.<br />
|-<br />
|P-chromatic<br />
|{{Interval ruler|22|0, 160, 210, 270, 430, 490, 660, 710, 880, 930, 990, 1150, 1200}}<br />
|The most important scale of Superpyth, Superpyth[12]. Replacing ss with m yields pental blackdye, the intersection of Superpyth and Porcupine.<br />
|}<br />
<br />
===== Half-octave scales =====<br />
MOS scales generated against the half-octave.<br />
{| class="wikitable"<br />
!Temperament<br />
!Name<br />
!Chart<br />
!Notes<br />
|-<br />
| rowspan="2" |Pajara<br />
|jaric<br />
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 800, 1000, 1100, 1200}}<br />
|<br />
|-<br />
|telluric<br />
|{{Interval ruler|22|0, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200}}<br />
|Adding two additional notes separates the 5-limit thirds onto different degrees, shared with the septimal ones, making for a much more traditional categorization of 22edo's interval space.<br />
|-<br />
| rowspan="3" |Hedgehog<br />
|malic<br />
|{{Interval ruler|22|0, 160, 320, 600, 760, 920, 1200}}<br />
|One of three tunings of malic available in 22edo.<br />
|-<br />
|ekic<br />
|{{Interval ruler|22|0, 160, 320, 480, 600, 760, 920, 1080, 1200}}<br />
|<br />
|-<br />
| -<br />
|{{Interval ruler|22|0, 50, 160, 210, 320, 370, 480, 600, 650, 760, 810, 920, 970, 1080, 1200}}<br />
|<br />
|-<br />
| rowspan="2" |Astrology<br />
|citric<br />
|{{Interval ruler|22|0, 160, 380, 600, 760, 980, 1200}}<br />
|One of two tunings of citric available in 22edo.<br />
|-<br />
|lemon<br />
|{{Interval ruler|22|0, 160, 320, 380, 540, 600, 760, 920, 980, 1140, 1200}}<br />
|<br />
|-<br />
| rowspan="2" |Doublewide<br />
|citric<br />
|{{Interval ruler|22|0, 50, 320, 600, 650, 920, 1200}}<br />
|One of two tunings of citric available in 22edo. Doublewide temperament makes apparent the fact that the subminor and nearminor thirds are equidistant from the 300c 12edo minor third, making the idea of 22edo splitting each of 12edo's qualities the most literally true in this particular case.<br />
|-<br />
|lime<br />
|{{Interval ruler|22|0, 50, 100, 320, 380, 600, 650, 700, 920, 980, 1200}}<br />
|<br />
|}<br />
<br />
===== Additional scales =====<br />
{| class="wikitable"<br />
!Name<br />
!Chart<br />
!Notes<br />
|-<br />
|Zarlino pentatonic<br />
|{{Interval ruler|22|0, 330, 500, 700, 1030, 1200}}<br />
|One possible pentatonic analog to the Zarlino diatonic.<br />
|-<br />
|Zarlino<br />
|{{Interval ruler|22|0, 100, 330, 500, 700, 800, 1030, 1200}}<br />
|The 5-limit diatonic in 22edo.<br />
|-<br />
|Pentachordal pajara<br />
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 850, 1000, 1100, 1200}}<br />
|<br />
|-<br />
|Tellurian<br />
|{{Interval ruler|22|0, 100, 200, 300, 400, 500, 600, 700, 800, 850, 1000, 1100, 1200}}<br />
|<br />
|}<br />
<br />
=== Harmony ===<br />
<br />
==== Consonance and dissonance properties ====<br />
The 12edo concept of consonance and dissonance as seen in modern harmony may be generalized via the membership of intervals to groups of intervals called odd-limits. The 3-odd-limit consists of intervals with 1, 2, 3, and 4 in their numerators or denominators, and is so called because the maximum value that either the numerator or denominator can have once all factors of two are removed is 3. The 3-odd-limit contains the perfect consonances - the unison, fourth, fifth, and octave. (Though note that the fourth may be considered a dissonance in some functional contexts, leading down to the major third). The 5-odd-limit expands the range to include ''imperfect consonances'', which are intervals that alongside 1, 2, 3, and 4, may also have numerators and denominators of 5, 6, and 8. These are 5/4, 6/5, 8/5, and 5/3 - the major and minor thirds, and the minor and major sixths, found in 12edo. They are also present in 22edo as the nearmajor and nearminor intervals. The diatonic intervals corresponding to these categories were considered dissonances historically, due to using the complex Pythagorean tunings instead of meantone-related ones.<br />
<br />
To expand the range of consonances further in 22edo, we may now consider the intervals of the 9-odd-limit. These include all the previous intervals, as well as intervals involving 7, 9, 10, 12, 14, and 16. In 22edo, this allows the whole tone, subminor third, supermajor third, and tritone to function as ''secondary consonances'', although because the tritone is tuned to the semioctave, that somewhat overwhelms its nominal consonance and makes it a dissonance; similarly, the nearmajor second and nearminor seventh's proximity to the unison and octave have a similar effect. The remaining intervals (the diminished fifth, augmented fourth, and the various semitones and sevenths not otherwise mentioned) are the rest of the dissonances.<br />
<br />
An alternative definition of consonance in 22edo is the 7-odd-limit, which contains the above except for 10/9, 9/7, and their octave complements; the 9-odd-limit is preferred due to 9/7's structural role as a third in chords.<br />
<br />
An important thing to note when it comes to 22edo is that intervals that serve as dissonances on their own may still play an important structural role in chords. For instance, the chords 5:6:7 (a kind of diminished chord) and 8:11:14 (an isoharmonic chord involving prime 11 that is represented by 22edo) prominently feature the tritone and diminished fifth, and yet are still somewhat consonant as chords. This is similar to the fact that the tritone is found in the dominant tetrad in 12edo, which is generally seen as the 'default' tetrad built on a major triad regardless of the tritone's presence. (In fact, the dominant tetrad in 22edo is best tuned to the harmonic seventh chord 4:5:6:7, which contains 5:6:7).<br />
<br />
However, at the same time, this means that when it comes to the supermajor and subminor thirds, the former is actually less stable in chords, due to sitting awkwardly between 5/4 and 4/3. We may loosely understand the stability of chords by examining their complexity when taken out of the harmonic series: the standard nearmajor triad is 4:5:6 (because its intervals are 5/4 and 3/2 (aka 6/4), but the standard nearminor triad is 10:12:15, which is somewhat more complex. Conversely, when considering supermajor and subminor, it is the subminor triad that is simpler at 6:7:9 (the subminor third is 7/6), meanwhile the supermajor triad is found all the way up at 14:18:21.<br />
<br />
==== Functional harmony ====<br />
There are two distinct approaches to harmony in 22edo: pajara and diatonic, each with tonal and modal approaches.<br />
<br />
First, we will explain 22edo leading tones: When considering Secor's supposed optimal leading tone at 70 cents, one may notice that 22edo skips this category entirely. However, 22edo instead matches with Aura's theory of functional harmony, which places the 70-cent leading tone at the intersection of two other functional categories at around 110 cents and 50 cents respectively - the collocant and gradient functions. The collocant functions as a conventional leading tone, whereas the gradient functions as a passing tone to either jump past the tonic or resolve to the collocant. 22edo represents both of these separately, and in doing do presents a distinct approach to leading tones from systems like 17edo and 31edo that have Secor's leading tone instead.<br />
<br />
===== Diatonic functional harmony =====<br />
[[File:Diatonic_harmony_demonstration.mp3|thumb|A demonstration of 22edo diatonic functional harmony.]]<br />
First of all, we will explore a heptatonic tonal approach to 22edo. Using tonal harmony solves our diatonic conundrum from earlier, as instead of having to remain in a scale, we can simply play over whatever the current chord is, or otherwise alter the scale to fit the harmony we're using. This generally requires, even more than in cases like 12edo minor, that a key not necessarily be considered as having a base scale.<br />
<br />
The standard qualities of major and minor retain their "bright" and "dark" feels respectively, which can be broken down into a combination of their complexity in triads and the actual width of the intervals. This suggests that the four qualities in 22edo should have more granularity in their feels, and can be broken down into stable/unstable and bright/dark.<br />
{| class="wikitable"<br />
|+<br />
!<br />
!Stable<br />
!Unstable<br />
|-<br />
!Bright<br />
|Nearmajor (warm, pleasant, comforting)<br />
|Supermajor (excited, animated, active)<br />
|-<br />
!Dark<br />
|Subminor (depressive, sad, bluesy)<br />
|Nearminor (angry, tense, stressful)<br />
|}<br />
Therefore, there are four distinct "keys" in 22edo, as compared to two in 12edo, where a key is defined as a system of tonal hierarchy based around a certain interval quality or tonic chord (independent of absolute pitch), which will be elaborated on below. When counting absolute pitch, there are 88 (4 x 22) keys. Note that relative major or minor depends on whether the key is near- or super/sub, and that, for instance, nearmajor and supermajor use different scales that are not rotations of one another. In specific, using ups and downs notation, C Nearmajor corresponds to vA Nearminor, meanwhile C Supermajor corresponds to A Subminor, and in general nearmajor-nearminor relative correspondences acquire an additional down accidental compared to standard MOSdiatonic correspondences.<br />
<br />
The chirality of the nearmajor or nearminor scale in question is ultimately of little relevance (see [[#Blackdye]]; the major second in nearmajor (and the fourth in nearminor) may be either note depending on context), but in general the right-handed version of nearmajor is assumed due to having a non-wolf V chord, and the left-handed version of nearminor is assumed due to having a non-wolf fourth over the tonic.<br />
<br />
The heptatonic interval functions remain as they are in 12edo, although with the caveat that the ideal leading tone ends up at the equal semitone rather than the semitone found in MOSdiatonic, which has implications for the subminor and supermajor keys and turns the use of the diatonic scale into a balancing act between the functional utility of MOSdiatonic and the tension of the leading tones in zarlino diatonic. (In particular, it suggests the use of a "harmonic supermajor" by flattening the seventh of supermajor by an edostep.)<br />
<br />
====== Nearmajor key ======<br />
[[File:Nearmajor.mp3|thumb|Natural nearmajor scale and tonic chord]]<br />
In nearmajor (the key with the nearmajor tonic chord), the fourth acts as it usually does in 12edo major, serving as a tendency tone towards the third. The basic tonal identity for nearmajor is 4:5:6, which extends generally to a nearmajor seventh chord, although a dominant (harmonic) seventh is also possible, and more justified in 22edo due to naturally extending the harmonic series segment corresponding to 4:5:6.<br />
<br />
====== Nearminor key ======<br />
[[File:Nearminor.mp3|thumb|Natural nearminor scale and tonic chord]]<br />
Nearminor harmony functions somewhat similarly to how you expect, with the nearminor sixth functioning as a leading tone down to the fifth and the seventh being able to be raised to a nearmajor seventh in order to give a more directed dominant resolution. The whole tone also provides a lead up to the minor third, like in standard diatonic.<br />
<br />
Melodic minor scales are somewhat interesting here as well, as there are a couple different reasonable ways to construct them, which would likely depend on the chords being used and the desired melodic contour.<br />
<br />
====== Supermajor key ======<br />
[[File:Supermajor.mp3|thumb|Natural supermajor scale and tonic chord]]<br />
In supermajor, a lead to the third would be a diminished fifth (11/8), perhaps justifying its inclusion in the scale over the fourth proper, or the functional alternation between the two in different contexts.<br />
<br />
The functionality of the seventh grows increasingly complicated in supermajor - while in 12edo, one may only see, for instance, a dominant chord replacing the I chord, in 22edo there are four different potential types of seventh, all with justifications. A fifth over the third would be a supermajor seventh (notably serving as the diatonic maj7, and distinguishing itself from the 12edo maj7 by not leading up to its own root), a tritone over the third would be a nearminor seventh, a lead up to the tonic would be a nearmajor seventh, and finally the MOS diatonic dominant chord utilizes a subminor seventh. Therefore, an alternate version of the supermajor scale usable in certain contexts makes the fourth wolf and the seventh nearmajor.<br />
<br />
This also means that the regular perfect fourth isn't as unstable an interval or as functionally dissonant in supermajor - in fact, the third is actually somewhat of a tension compared to it (though the step between them is half the size of a conventional leading tone).<br />
<br />
====== Subminor key ======<br />
The same kind of justification emerges for harmonic subminor, except that there is little reason to alter the seventh all the way up to a supermajor seventh if the objective is for it to function as a leading tone. In fact, the same logic can be used against the conventional dominant chord in nearmajor - leading inwards to a nearmajor third by equal semitones on either side requires that the initial interval be a perfect tritone, and that the chord to be used as a dominant is actually a harmonic 4:5:6:7 on the fifth. (Resolving to a supermajor chord actually wants a dom7 with a nearmajor third and nearminor seventh, if quartertones are not to be used).<br />
[[File:Subminor.mp3|thumb|Natural subminor scale and tonic chord]]<br />
In general, 22edo's functional harmony ends up a lot more context-bound and much less scale-bound than 12edo's, due to the multiple different qualities of intervals and notes doing different things, and the ideal leading tone not matching the standard diatonic structure.<br />
<br />
===== Alternative leading tones =====<br />
An alternative approach to simplify things is instead to discard Aura's theory of leading in favor of treating the quartertone as the optimal leading tone (as it is the diatonic major seventh), an entirely different paradigm emerges. Supermajor and subminor become definitive, stable diatonic tonality systems, with no awkwardness around leading tones and similar mechanics to their 12edo counterparts (albeit with the different, somewhat inverted "moods" presented by the supermajor and subminor intervals). Meanwhile, the nearmajor and nearminor scales acquire new "harmonic" variations, with the final note raised up to a quartertone below the tonic. In effect, supermajor/subminor and nearmajor/nearminor "switch" in regards to some functions. Instead of raising the fourth in supermajor, it is in this system viable to lower it in nearmajor. The best dom7 to resolve to a nearmajor triad on the tonic is not actually a seventh chord at all, but instead features the supermajor sixth and third, and can consequently be reanalyzed as a subminor seventh chord on the 9/7 over the tonic. The MOSdiatonic dominant seventh serves to resolve to a MOSdiatonic major triad, as in 12edo.<br />
<br />
===== 10-tone functional harmony =====<br />
The remainder of the discussion of functional harmony is simply the assignment of placements in the tonal hierarchy to the new degrees added by the 10-tone system. To put it simply, the antilatus and unilatus become the varicant and subvaricant, which sit between the mediant/submediant and supertonic/subtonic in terms of stability (and feature as elements of chthonic chords like 6:7:8, an alternative to standard diatonic chords available in the 10-form). Additionally, the tritone acquires the antitonic function. While the dominant serves as a stable "structural anchor" in diatonic, here the antitonic serves as an ''unstable'' structural anchor - the opposite of the tonic both in placement and stability. Note that in the 10-tone system, we return to having two distinct interval qualities down from four, so we go back to having two different keys. 10-tone harmony is also useful in modal music. Also, note that 109-cent leading tones comprise the majority of the intervals in pajara[10], so their impact may be reduced and in fact one might depend more on the few larger nearmajor seconds that exist in the scale, or skip steps entirely and use subsets.<br />
[[File:Chthonic_harmony_demonstration.mp3|thumb|10-form harmony demonstration in 22edo]]<br />
Another thing to note about the 10-tone system is that it is possible to constrain oneself entirely to chthonic harmony, in which case a lot of the familiar functional harmony language somewhat breaks. The role of the traditional dominant with respect to the tonic disappears completely (even if, for instance, the root position of a chord is assumed to be 4:6:7), with instead the chords on the tritone and the sixth including a leading tone up to the tonic (in fact, the dominant in this system becomes a ''stable'' chord rather than a tense one, assuming a 6:7:8 root position).<br />
<br />
We may contextualize these differences by examining the 3-function analysis of functional harmony, which in the 7-form (as in 22edo) places the tonic function on the degrees (1-indexed) 1, 3, and 6, the subdominant function on 2 and 4, and the dominant function on 5 and 7. In the chthonic 10-form, however, it requires some amount of reorganization. A theory for Vector's abandoned Earth#Pajara project utilizes four functional categories for chords, rather than three (0-indexed): tonic (0, 2, 8), dominant (4, 6), antitonic (3, 5, 7) and antidominant (1, 9) - the "dominant" function here acts similarly to the heptatonic subdominant (stable, dynamic), with the antitonic and antidominant serving as tense "static" and "dynamic" functions respectively. This essentially splits the 10-form into two pentatonic subscales, one built on the tonic and one built on the antitonic (which is actually how pajara[10] is constructed, but in this tonal system you can pretty strongly feel that construction determining how harmony is structured).<br />
<br />
If this theory is also altered to work with tertian harmony instead, the functions follow a chain of thirds rather than a chain of chthonics, so that tonic is (0, 3, 7), a more traditional subdominant is (4, 1), dominant is thus (6, 9), and the remaining degrees (2, 5, 8) constitute antitonic. In this case, dominant is the "tense dynamic" function and antitonic is the "tense static" function.<br />
<br />
==== Modal harmony ====<br />
[[File:Modes_in_22edo.png|thumb|252x252px|The modes presented here, arranged in a Tetrahedron.]]<br />
Modal harmony further emphasizes the qualities of the various intervals and chords found in the different scales used in music, as opposed to things like leading tendencies. It is within modal harmony that clear "supermajor", "nearmajor", "nearminor", and "subminor" diatonic scales can be defined, rather than used as context-dependent tonal systems. These mostly follow the interval qualities suggested above, except this time it becomes applicable to an entire scale rather than just to specific chords. (And of course, additional modes of zarlino or mosdiatonic may be used.)<br />
<br />
Given this, it's also useful to enumerate various modal scales, as a counterpart to the various non-Ionian/Aeolian modes used throughout standard modal harmony. These will not be exclusively "real" diatonic modes, but rather combinations of qualities loosely analogous to standard modes (and sharing the quality of having notes constrained to range in certain qualities), in six "series" comprising 22 unique modes, visible on the right. This setup overall aims to generalize the idea that diatonic modes exist on a gradation of "brightness" in 12edo, where successive alterations make a mode brighter or darker. Here, bright vs. dark isn't the only axis, however - there's near vs. super/sub and stable vs. unstable as well, so series of alterations along those allow for a much more complex selection of modes to choose from. Additionally, each series of modes has one quality in common, which I've labelled here, so all the "stable" modes have only stable intervals, even if they might contain both nearmajor and subminor ones, and all the "bright" modes contain only major intervals, even if they might be both nearmajor and supermajor. Holding the fourth and fifth constant (as Lydian and Locrian are rarely used in standard diatonic modal harmony) means that there are four "vertex" modes, corresponding to the pure nearmajor, nearminor, supermajor, and subminor qualities, as well as the Ionian and Phrygian modes of Zarlino and mosdiatonic.<br />
<br />
===== Choosing a mode =====<br />
Much as the choice of mode in 12edo largely depends on its position on the scale from bright to dark, you might choose a mode here by selecting a series based on the common sound you want your song or section to have, and then choosing a position on that series between its two extremes. For example, for something intense and somewhat uncanny, you might start by choosing the Unstable series, and then proceed to select a mode along that series between bright/super and near/dark that embodies the feel you want, such as Unstable Dorian. Alternatively, for an excited, cheerful sound, you might choose the Bright series and a mode between the near/stable and super/unstable extremes of it, such as Didymic Major.<br />
<br />
The "Equable" mode serves as a somewhat 'neutral' sound - despite the lack of neutral intervals in 22edo, it still occupies that somewhat soft position in between major and minor qualities, while at the same time being more equally distributed than any other version of Dorian available. Quality-wise, it has a mix of nearmajor (bright, stable) and nearminor (dark, unstable) intervals, serving as the opposite polarity to MOS Dorian, and its equidistant nature somewhat overrides other quality-based properties from a melodic perspective. On the opposite side of things, MOS Dorian can be seen as somewhat aggressively defined by its qualities, being a mix of subminor (stable, dark) and supermajor (unstable, bright), with a subminor third on the tonic.<br />
<br />
Other pairs of "opposing" modes include unstable Dorian vs. stable Dorian, and didymic major vs. didymic minor, both of which unlike equable vs. MOS Dorian form complementary pairs similar to Ionian and Phrygian in 12edo.<br />
<br />
Here they have been organized into two "loops"; bolded entries represent modes that differ along the loops, and italicized entries have had their positions flipped.<br />
{| class="wikitable"<br />
|+Class A<br />
! colspan="4" |Loop A<br />
! colspan="4" |Loop B<br />
|-<br />
!Series<br />
!Mode<br />
!Type<br />
!Name<br />
!Series<br />
!Mode<br />
!Type<br />
!Name<br />
|-<br />
|Super/Sub<br />
|{{Interval ruler|22|0, 200, 250, 500, 700, 760, 1000, 1200}}<br />
|Aeolian<br />
|Aeolian<br />
|Super/Sub<br />
|{{Interval ruler|22|0, 200, 250, 500, 700, 760, 1000, 1200}}<br />
|Aeolian<br />
|Aeolian<br />
|-<br />
|Super/Sub<br />
|{{Interval ruler|22|0, 200, 250, 500, 700, 930, 1000, 1200}}<br />
|Dorian<br />
|Dorian<br />
|Super/Sub<br />
|{{Interval ruler|22|0, 200, 250, 500, 700, 930, 1000, 1200}}<br />
|Dorian<br />
|Dorian<br />
|-<br />
|Super/Sub<br />
|{{Interval ruler|22|0, 200, 430, 500, 700, 930, 1000, 1200}}<br />
|Mixolydian<br />
|Mixolydian<br />
|Super/Sub<br />
|{{Interval ruler|22|0, 200, 430, 500, 700, 930, 1000, 1200}}<br />
|Mixolydian<br />
|Mixolydian<br />
|-<br />
|Bright, Super/Sub, Unstable<br />
|{{Interval ruler|22|0, 200, 430, 500, 700, 930, 1150, 1200}}<br />
|Ionian<br />
|Ionian ("Supermajor")<br />
|Bright, Super/Sub, Unstable<br />
|{{Interval ruler|22|0, 200, 430, 500, 700, 930, 1150, 1200}}<br />
|Ionian<br />
|Ionian ("Supermajor")<br />
|-<br />
|'''Bright'''<br />
|{{Interval ruler|22|0, 200, 430, 500, 700, 930, 1100, 1200}}<br />
|Ionian<br />
|'''Harmonic major'''<br />
|'''Unstable'''<br />
|{{Interval ruler|22|0, 220, 430, 500, 700, 930, 1050, 1200}}<br />
|Mixolydian<br />
|'''Unstable Mixolydian'''<br />
|-<br />
|'''Bright'''<br />
|{{Interval ruler|22|0, 200, 380, 500, 700, 930, 1100, 1200}}<br />
|Ionian<br />
|'''Didymic major'''<br />
|'''Unstable'''<br />
|{{Interval ruler|22|0, 220, 330, 500, 700, 930, 1050, 1200}}<br />
|Dorian<br />
|'''Unstable Dorian'''<br />
|-<br />
|'''Bright'''<br />
|{{Interval ruler|22|0, 200, 380, 500, 700, 880, 1100, 1200}}<br />
|Ionian<br />
|'''RH-Ionian'''<br />
|'''Unstable'''<br />
|{{Interval ruler|22|0, 220, 330, 500, 700, 830, 1050, 1200}}<br />
|Aeolian<br />
|'''LH-Aeolian'''<br />
|-<br />
|''Near, Bright, Stable''<br />
|{{Interval ruler|22|0, 160, 380, 500, 700, 880, 1100, 1200}}<br />
|Ionian<br />
|''LH-Ionian ("Nearmajor")''<br />
|''Near, Dark, Unstable''<br />
|{{Interval ruler|22|0, 100, 330, 500, 700, 830, 1050, 1200}}<br />
|Phrygian<br />
|''RH-Phrygian ("Nearminor")''<br />
|-<br />
|''Near''<br />
|{{Interval ruler|22|0, 160, 380, 500, 700, 880, 1050, 1200}}<br />
|Mixolydian<br />
|''Major equable''<br />
|''Near''<br />
|{{Interval ruler|22|0, 160, 330, 500, 700, 830, 1050, 1200}}<br />
|Aeolian<br />
|''Minor equable''<br />
|-<br />
|Near<br />
|{{Interval ruler|22|0, 160, 330, 500, 700, 880, 1050, 1200}}<br />
|Dorian<br />
|Equable<br />
|Near<br />
|{{Interval ruler|22|0, 160, 330, 500, 700, 880, 1050, 1200}}<br />
|Dorian<br />
|Equable<br />
|-<br />
|''Near''<br />
|{{Interval ruler|22|0, 160, 330, 500, 700, 830, 1050, 1200}}<br />
|Aeolian<br />
|''Minor equable''<br />
|''Near''<br />
|{{Interval ruler|22|0, 160, 380, 500, 700, 880, 1050, 1200}}<br />
|Mixolydian<br />
|''Major equable''<br />
|-<br />
|''Near, Dark, Unstable''<br />
|{{Interval ruler|22|0, 100, 330, 500, 700, 830, 1050, 1200}}<br />
|Phrygian<br />
|''RH-Phrygian ("Nearminor")''<br />
|''Near, Bright, Stable''<br />
|{{Interval ruler|22|0, 160, 380, 500, 700, 880, 1100, 1200}}<br />
|Ionian<br />
|''LH-Ionian ("Nearmajor")''<br />
|-<br />
|'''Dark'''<br />
|{{Interval ruler|22|0, 100, 330, 500, 700, 830, 1000, 1200}}<br />
|Phrygian<br />
|'''LH-Phrygian'''<br />
|'''Stable'''<br />
|{{Interval ruler|22|0, 160, 380, 500, 700, 880, 980, 1200}}<br />
|Mixolydian<br />
|'''RH-Mixolydian'''<br />
|-<br />
|'''Dark'''<br />
|{{Interval ruler|22|0, 100, 270, 500, 700, 830, 1000, 1200}}<br />
|Phrygian<br />
|'''Didymic minor'''<br />
|'''Stable'''<br />
|{{Interval ruler|22|0, 160, 270, 500, 700, 880, 980, 1200}}<br />
|Dorian<br />
|'''Stable Dorian'''<br />
|-<br />
|'''Dark'''<br />
|{{Interval ruler|22|0, 100, 270, 500, 700, 770, 1000, 1200}}<br />
|Phrygian<br />
|'''Subharmonic minor'''<br />
|'''Stable'''<br />
|{{Interval ruler|22|0, 160, 270, 500, 700, 760, 980, 1200}}<br />
|Aeolian<br />
|'''Stable Aeolian'''<br />
|-<br />
|Dark, Super/Sub, Stable<br />
|{{Interval ruler|22|0, 50, 270, 500, 700, 770, 1000, 1200}}<br />
|Phrygian<br />
|Phrygian ("Subminor")<br />
|Dark, Super/Sub, Stable<br />
|{{Interval ruler|22|0, 50, 270, 500, 700, 770, 1000, 1200}}<br />
|Phrygian<br />
|Phrygian ("Subminor")<br />
|}<br />
Note that Aeolian is not a vertex. Because of this, it might be prudent to construct a secondary, smaller tetrahedron that holds the major second constant alongside the fourth and fifth. Doing so yields six additional modes:<br />
<br />
* A set of two additional modes between RH-Ionian and LH-Aeolian, acting as alternative near forms of Mixolydian/major equable ({{Interval ruler|22|0, 220, 380, 500, 700, 880, 1050, 1200}}) and Dorian/equable ({{Interval ruler|22|0, 220, 330, 500, 700, 880, 1040, 1200}} )<br />
<br />
* A set of two additional modes between LH-Aeolian and mosdiatonic Aeolian, acting as alternative dark/minor scales ({{Interval ruler|22|0, 220, 330, 500, 700, 830, 1000, 1200}}, {{Interval ruler|22|0, 220, 270, 500, 700, 830, 1000, 1200}} ).<br />
<br />
* Alternative stable forms of Mixolydian ({{Interval ruler|22|0, 200, 380, 500, 700, 880, 980, 1200}}) and Dorian ({{Interval ruler|22|0, 200, 270, 500, 700, 880, 980, 1200}} ). These differ by one note varying by two steps; between them is in fact the simplest possible 5-limit Dorian, at ({{Interval ruler|22|0, 200, 330, 500, 700, 880, 980, 1200}} ), which is not a mode of zarlino due to distributing the large and medium steps differently. This appears to suggest that the sum total of all theoretically possible modes existing under this system is the complete volume of a tetrahedron with endpoints at near- Locrian and Lydian and at sub-Locrian and super-Lydian. There are 84 total modes in the scheme, which are the rotations of the following 8 base scales, including chirality. These are the set of scales that have the property that all instances of any diatonic interval between any two notes in the scale are either supermajor, nearmajor, nearminor, or subminor, which is the property that constrains the tetrahedron:<br />
<br />
{| class="wikitable"<br />
|+<br />
!Name<br />
!Scale<br />
!Note<br />
!Symmetrical?<br />
!Exists in the set of 22 modes?<br />
|-<br />
|mosdiatonic<br />
|{{Interval ruler|22|0, 200, 430, 500, 700, 930, 1150, 1200}}<br />
|<br />
|No<br />
|Yes<br />
|-<br />
|harmonic major<br />
|{{Interval ruler|22|0, 200, 430, 500, 700, 930, 1100, 1200}}<br />
|<br />
|Yes<br />
|Yes<br />
|-<br />
|didymic<br />
|{{Interval ruler|22|0, 200, 380, 500, 700, 930, 1100, 1200}}<br />
|<br />
|Yes<br />
|Yes<br />
|-<br />
|zarlino<br />
|{{Interval ruler|22|0, 160, 380, 500, 700, 880, 1100, 1200}}<br />
|<br />
|Yes<br />
|Yes<br />
|-<br />
|diatonyx-A<br />
|{{Interval ruler|22|0, 220, 380, 500, 700, 880, 1050, 1200}}<br />
|The upper tetrachord is a porcupine tetrachord.<br />
|No<br />
|No<br />
|-<br />
|diatonyx-B<br />
|{{Interval ruler|22|0, 160, 330, 500, 700, 830, 1050, 1200}}<br />
|The lower tetrachord is a porcupine tetrachord.<br />
|Yes<br />
|Yes<br />
|-<br />
|equable / onyx<br />
|{{Interval ruler|22|0, 160, 330, 500, 700, 880, 1050, 1200}}<br />
|<br />
|No<br />
|Yes<br />
|-<br />
|symmetrical dorian<br />
|{{Interval ruler|22|0, 200, 330, 500, 700, 880, 980, 1200}}<br />
|<br />
|No<br />
|No<br />
|}<br />
This reduces to a set of 35 if the fourth and fifth are held fixed, and 55 if only the fifth is.<br />
<br />
Every mode of one of these scales has a pattern of broadly major and minor intervals corresponding to one of the standard diatonic modes. For example, the equable scale in its primary mode is a form of Dorian, as its pattern is major-minor-perfect-perfect-major-minor (in this case, nearmajor and nearminor). However, there are not in fact 12 instances of each mode! The equable scale only has Mixolydian, Dorian, and Aeolian modes, and the symmetrical Dorian scale lacks a Locrian or Lydian mode.<br />
{| class="wikitable"<br />
|+<br />
!Mode type<br />
!84-set<br />
!35-set<br />
!22-set<br />
|-<br />
|Locrian<br />
|7<br />
| -<br />
| -<br />
|-<br />
|Phrygian<br />
|12<br />
|5<br />
|5<br />
|-<br />
|Aeolian<br />
|15<br />
|8<br />
|4<br />
|-<br />
|Dorian<br />
|16<br />
|9<br />
|4<br />
|-<br />
|Mixolydian<br />
|15<br />
|8<br />
|4<br />
|-<br />
|Ionian<br />
|12<br />
|5<br />
|5<br />
|-<br />
|Lydian<br />
|7<br />
| -<br />
| -<br />
|}<br />
Regardless, this is simply a mathematically complete enumeration - for actual modal music, it is best to stick to the list of 22 modes provided above, as those are the ones that have clear common qualities alongside the functionally important perfect fifth and fourth.<br />
<br />
Additional work needs to be done to determine if this can be generalized.<br />
<br />
===== Tetrachords in modal analysis =====<br />
Tetrachords, mentioned previously in an adaptation of their original Greek form, can be used in a different way, more in accordance with their use in modern 12edo theory. In modes where the fourth and fifth are perfect, the mode can always be thought of as being comprised of two tetrachords separated by a whole tone, although the constraints on these tetrachords are entirely different from the Greek versions. In short, a modal tetrachord must comprise the unison, the fourth, a second of one of the four qualities, and a third of one of the four qualities, such that the interval between two adjacent tones is never more than four steps. This is a generalization of the constraints on modal tetrachord patterns in 12edo, which must always contain either whole tones or semitones. By this constraint there are ten distinct tetrachords in 22edo. Considering all the possible scales constructed from these, there are 10x10 = 100 distinct possibilities, compared to the 3x3 = 9 options found in 12edo. This provides an extended set, including not only the 35 modes corresponding to diatonic but 65 additional scales corresponding in some regard to melodic minor or neapolitan major. Not all intervals are necessarily within their expected quality ranges.<br />
<br />
Loosening the constraint further to only necessitate that the two movable tones remain within their respective halves of the tetrachord allows for the generalization to a set of scales analogous to harmonic minor or double harmonic major, with 156 additional possibilities.<br />
<br />
The fact that while in 12edo there are 2^4=16 distinct tetrachordal scales under the extended definition, in 22edo there are 4^4=256, is not a coincidence: it is a direct result of the doubling of interval qualities within a tetrachord's span of the perfect fourth that leads to 22edo's construction in the first place. This extreme number of scales is overwhelming, but it simply shows the sheer inexhaustibility of 22edo modal harmony.<br />
<br />
===== In the 10-tone system =====<br />
As pajara[10] is a MOS, its modal harmony doesn't have as much of the same complexity as diatonic modal harmony does. However, it is useful to consider the previously mentioned MODMOSes as roughly on the same level as the MOS form of the scale (as the only additional variety they introduce is in the tritone), giving 15 distinct modes available to choose from, each with a degree of brightness or darkness to them, much more like conventional 12edo modal harmony. It is for this reason and the reasons given above with 10-tone functional harmony that pajara[10] can be considered a much more "familiar" approach to 22edo, despite being such a completely different scale.<br />
<br />
A pajara[10] pentachord may be considered to consist of five tones. To continue the theme of pajara mirroring conventional harmony with two qualities from 12edo, we may constrain this specific set of pentachords such that they must be comprised entirely of semitones and nearmajor seconds, which is an analogous constraint to the one stating that a 12edo tetrachord must be comprised entirely of tones and semitones, as it leads to four distinct pentachords.<br />
<br />
Alternatively in the more general interpretation, there are four additional "harmonic" pentachords. Note that in either case, no note in a pentachord may occupy the first or last step of the perfect fourth.<br />
<br />
==== Chromatic subsets ====<br />
In 22edo, multiple qualities may be combined together into a compound system. In 12edo, there is little reason to do this, because there are only two qualities available, so the scale combining them (the 12edo chromatic scale, or some other large scale like {{Interval ruler|12|0, 200, 300, 400, 500, 700, 800, 900, 1100, 1200}} ) is not particularly engaging from either a tonal or modal perspective. However, in 22edo, there are four different qualities, from which two may be selected to share characteristics.<br />
<br />
The standard chromatic scales combine nearmajor+nearminor and supermajor+subminor, which lead to somewhat of the same problem as 12edo chromatic; they are opposing pairs of qualities. However, if we make an ''asymmetric'' chromatic scale, with (for instance) supermajor and nearminor, we get a scale with the trait they have in common: being "unstable". Alternatively, you could get a generally "dark" system by combining subminor and nearminor qualities.<br />
<br />
The following are a few examples of these kinds of scales, including diatonic and chromatic variations. (Note that in tonal music, these become less distinct from standard counterparts, as degrees are already expected to be altered between different qualities depending on context.)<br />
<br />
Nearminor (harmonic): P1 - SM2 - nm3 - P4 - P5 - nm6 - NM7 - P8 ( {{Interval ruler|22|0, 210, 320, 500, 700, 810, 1100, 1200}} )<br />
<br />
Subminor (harmonic): P1 - SM2 - sm3 - P4 - P5 - sm6 - NM7 - P8 ( {{Interval ruler|22|0, 210, 270, 500, 700, 770, 1100, 1200}} )<br />
<br />
Compound minor (harmonic + natural): P1 - SM2 - sm3 - nm3 - P4 - P5 - sm6 - nm6 - NM7 - P8 ( {{Interval ruler|22|0, 210, 270, 330, 500, 700, 770, 830, 1100, 1200}} )<br />
<br />
Supermajor (harmonic): P1 - SM2 - SM3 - P4 - P5 - SM6 - NM7 - P8 ( {{Interval ruler|22|0, 210, 430, 500, 700, 930, 1100, 1200}} )<br />
<br />
Compound unstable (harmonic + natural): P1 - SM2 - nm3 - SM3 - P4 - P5 - nm6 - SM6 - nm7 - NM7 - P8 ( {{Interval ruler|22|0, 210, 330, 430, 500, 700, 830, 930, 1040, 1100, 1200}} )<br />
<br />
Aberrismic scales may also be leveraged for this purpose.<br />
<br />
==== Consonant vs. tense suspended chords ====<br />
The wider supermajor second and contrast with the supermajor third actually makes suspended chords somewhat of a point of resolution, rather than a point of tension like in 12edo. It's reasonable to have a suspended chord that doesn't resolve, perhaps making the term "suspended" inaccurate. These suspended chords can function like arto and tendo chords, with a 1-2-4-5 chord structure being plausible, or can be used in modal harmony as a form of "mode-agnostic" anchor point. The sus4 chord in particular is composed of the three octave-reduced perfect consonances, and thus can also be considered the most basic polychordal scale (perhaps a/the "dichordal" scale). The consonance of the supermajor second is additionally what allows chthonic harmony to function. However, suspensions that function more like 12edo ones in leading into the MOS diatonic intervals and being more tense can still be found with the ''nearmajor'' sus2 and ''wolf'' sus4, which lose some of the structural elegance of standard Pythagorean suspensions in favor of a more tense, crowded sound that can easily resolve to even the rather tense supermajor triad.<br />
<br />
==== Pajara vs. diatonic: a summary ====<br />
In general, it is ironically pajara that comes the closest to familiar diatonic structures from 12edo once you actually get to composing. There are two qualities of each interval, modes are ranked on a spectrum of brightness, and it feels like a logical extension of standard diatonic logic to the 7-limit. Pajara is the system to use if you just want to think of 22edo as "more notes", or simply as a more accurate JI tuning.<br />
<br />
However, diatonic allows for much more complex, dynamic harmonies, all because of the four distinct interval qualities it provides, taking full advantage of the structural characteristics of 22edo for new forms of both tonal and modal harmony, while having the advantage of being more superficially similar to the structures found in 12edo. However, it might be somewhat overwhelming or annoying to someone not used to working in it. This is simply a natural consequence of 22edo being a larger and more versatile system: as has been discussed extensively before, whereas in 12edo there's often only one way to do something, in larger systems like 22edo there are often many, each useful in its own little way.<br />
<br />
=== Tables of chords ===<br />
The following is a table of chords in 22edo.<br />
<br />
''The notation for chords here is an adaptation of conventional chord symbols; for a more systematic yet less backwards-compatible approach see [[User:Vector/Vector's chord names|Vector's chord names]]. For Roman numeral analysis, "M" and "m" are removed, all major chords receive an uppercase roman numeral (e.g. IV) and all minor chords receive a lowercase roman numeral (e.g. iv). For figured bass, the same conventions are used as in 12edo, with the addition of ups and downs as possible accidentals.''<br />
<br />
==== Fifth-bounded tertian triads ====<br />
Three-note chords built out of thirds, bounded by a perfect fifth.<br />
{| class="wikitable"<br />
!Name<br />
!Third<br />
!Fifth<br />
!Edostep<br />
|-<br />
|supermajor (M)<br />
|supermajor<br />
|perfect<br />
|[0 8 13]<br />
|-<br />
|nearmajor (P, unmarked)<br />
|nearmajor<br />
|perfect<br />
|[0 7 13]<br />
|-<br />
|nearminor (p)<br />
|nearminor<br />
|perfect<br />
|[0 6 13]<br />
|-<br />
|subminor (m)<br />
|subminor<br />
|perfect<br />
|[0 5 13]<br />
|}<br />
<br />
==== Other tertian triads ====<br />
Additional three-note chords built out of thirds.<br />
<br />
===== Augmented triads =====<br />
{| class="wikitable"<br />
!Name<br />
!Third<br />
!Fifth<br />
!Edostep<br />
!Notes<br />
|-<br />
|near augmented (z+)<br />
|nearmajor<br />
|up<br />
|[0 7 14]<br />
|Found by augmenting the fifth in zarlino diatonic by an edostep. Inverts to two other forms of augmented triad.<br />
|-<br />
|exo augmented (S+)<br />
|supermajor<br />
|augmented<br />
|[0 8 16]<br />
|"Neutral" counterpart of 5/3-bounded chords.<br />
|}<br />
<br />
===== Diminished triads =====<br />
{| class="wikitable"<br />
!Name<br />
!Third<br />
!Fifth<br />
!Edostep<br />
!Notes<br />
|-<br />
|near diminished (z°)<br />
|nearminor<br />
|down<br />
|[0 6 12]<br />
|Bounded by 16/11. Found by diminishing the fifth in zarlino by an edostep. Found in z7 chord.<br />
|-<br />
|major diminished (°)<br />
|nearminor<br />
|updiminished (tritone)<br />
|[0 6 11]<br />
|5:6:7. Found in harmonic 4:5:6:7.<br />
|-<br />
|minor diminished (m°)<br />
|subminor<br />
|updiminished (tritone)<br />
|[0 5 11]<br />
|<br />
|-<br />
|exo diminished (S°)<br />
|subminor<br />
|diminished<br />
|[0 5 10]<br />
|Equalized 16:19:22. Bounded by 11/8. Diminished triad in mosdiatonic. Found in x7 chord.<br />
|}<br />
<br />
==== Tetrads ====<br />
<br />
===== Supermajor tetrads =====<br />
{| class="wikitable"<br />
!Name<br />
!Third<br />
!Fifth<br />
!Seventh<br />
!Steps between 3 and 7<br />
!Edostep<br />
!Notes<br />
|-<br />
|exodominant seventh (S7)<br />
|supermajor<br />
|perfect<br />
|subminor<br />
|10<br />
|[0 8 13 18]<br />
|As a result of the symbol "7" going to the harmonic seventh chord, a couple new symbols had to be devised for the remaining types of dominant chord. "S" (super/sub) refers to chords involving supermajor/subminor interpretations of intervals, while "z" (zarlino) refers to chords involving nearmajor/nearminor interpretations of intervals.<br />
|-<br />
|supermajor seventh (M7, Δ7)<br />
|supermajor<br />
|perfect<br />
|supermajor<br />
|13 (P5)<br />
|[0 8 13 21]<br />
|Seventh chord of supermajor.<br />
|-<br />
|supermajor nearmajor seventh (MP7)<br />
|supermajor<br />
|perfect<br />
|nearmajor<br />
|12<br />
|[0 8 13 20]<br />
|Acts as a more directed version of a M7 chord.<br />
|}<br />
<br />
===== Nearmajor tetrads =====<br />
{| class="wikitable"<br />
!Name<br />
!Third<br />
!Fifth<br />
!Seventh<br />
!Steps between 3 and 7<br />
!Edostep<br />
!Notes<br />
|-<br />
|harmonic seventh (7), major harmonic (H)<br />
|nearmajor<br />
|perfect<br />
|subminor<br />
|11 (tritone)<br />
|[0 7 13 18]<br />
|There are a number of reasons to assign the unmarked "7" to the harmonic seventh chord. First of all is that it is backwards compatible with 12edo; the harmonic seventh chord is one possible 22edo generalization of the [0-4-7-10] dominant. Additionally, it is specifically this chord that functions as the dominant chord for a nearmajor chord on the tonic, presuming that 109c is used as the leading tone. Additionally, it uses the 600c tritone like the 12edo dominant does (MOSdiatonic dominants, alongside having the wrong leading tone, do not use the 600c tritone, making techniques like tritone substitution impossible). Also, this is the tonic chord in zarlino Mixolydian. Beyond standard chord symbol conventions, it also makes sense to allow the unmodified 7 to refer to what is arguably the simplest JI seventh chord.<br />
In pajara harmony, the symbol H should be preferred, to emphasize its contrast with the minor harmonic tetrad (Hm).<br />
|-<br />
|neardominant seventh (z7)<br />
|nearmajor<br />
|perfect<br />
|nearminor<br />
|12<br />
|[0 7 13 19]<br />
|<br />
|-<br />
|nearmajor seventh (P7)<br />
|nearmajor<br />
|perfect<br />
|nearmajor<br />
|13 (P5)<br />
|[0 7 13 20]<br />
|Seventh chord of nearmajor.<br />
|-<br />
|nearmajor supermajor seventh (PM7)<br />
|nearmajor<br />
|perfect<br />
|supermajor<br />
|14<br />
|[0 7 13 21]1]<br />
|Acts as a less directed version of a P7 chord.<br />
|}<br />
<br />
===== Nearminor tetrads =====<br />
{| class="wikitable"<br />
!Name<br />
!Third<br />
!Fifth<br />
!Seventh<br />
!Steps between 3 and 7<br />
!Edostep<br />
!Notes<br />
|-<br />
|minor harmonic (Hm)<br />
|nearminor<br />
|perfect<br />
|supermajor 6th<br />
|11 (tritone)<br />
|[0 6 13 17]<br />
|<br />
|-<br />
|nearminor seventh (p7)<br />
|nearminor<br />
|perfect<br />
|nearminor<br />
|13 (P5)<br />
|[0 6 13 19]<br />
|Seventh chord of nearminor.<br />
|-<br />
|nearminor nearmajor seventh (pP7)<br />
|nearminor<br />
|perfect<br />
|nearmajor<br />
|14<br />
|[0 6 13 20]<br />
|Seventh chord of harmonic nearminor.<br />
|-<br />
|nearminor subminor seventh (pm7)<br />
|nearminor<br />
|perfect<br />
|subminor<br />
|12<br />
|[0 6 13 18]<br />
|<br />
|}<br />
<br />
===== Subminor tetrads =====<br />
{| class="wikitable"<br />
!Name<br />
!Third<br />
!Fifth<br />
!Seventh<br />
!Steps between 3 and 7<br />
!Edostep<br />
!Notes<br />
|-<br />
|subminor seventh (m7)<br />
|subminor<br />
|perfect<br />
|subminor<br />
|13 (P5)<br />
|[0 5 13 18]<br />
|Seventh chord of subminor.<br />
|-<br />
|subminor nearminor seventh (mp7)<br />
|subminor<br />
|perfect<br />
|nearminor<br />
|14<br />
|[0 5 13 19]<br />
|<br />
|-<br />
|subminor nearmajor seventh (mP7)<br />
|subminor<br />
|perfect<br />
|nearmajor<br />
|15<br />
|[0 5 13 20]<br />
|Seventh chord of harmonic subminor.<br />
|}<br />
<br />
==== Non-tertian functional chords ====<br />
{| class="wikitable"<br />
!Name<br />
!Mediant<br />
!Bounding interval<br />
!Edostep<br />
!Notes<br />
|-<br />
|chthonic minor (Lm)<br />
|minor unilatus (whole tone)<br />
|perfect fourth<br />
|[0 4 9]<br />
|<br />
|-<br />
|chthonic major (LM)<br />
|major unilatus (subminor third)<br />
|perfect fourth<br />
|[0 5 9]<br />
|6:7:8 chord.<br />
|-<br />
|suspended 4th (sus4)<br />
|perfect 4th<br />
|perfect fifth<br />
|[0 9 13]<br />
|Suspension resolves to nearmajor. Alternately usable as a consonant 3-limit chord.<br />
|-<br />
|suspended up4th (sus^4)<br />
|up 4th<br />
|perfect fifth<br />
|[0 10 13]<br />
|Suspension resolves to supermajor. Uses the aforementioned supermajor up 4th.<br />
|-<br />
|suspended 2nd (sus2)<br />
|supermajor 2nd<br />
|perfect fifth<br />
|[0 4 13]<br />
|Suspension resolves to nearminor. Alternately usable as a consonant 3-limit or septal chord.<br />
|-<br />
|suspended down2nd (susv2)<br />
|nearmajor 2nd<br />
|perfect fifth<br />
|[0 3 13]<br />
|Suspension resolves to subminor<br />
|-<br />
|naiadic minor (S+m)<br />
|nearmajor third<br />
|nearmajor sixth<br />
|[0 7 16]<br />
|<br />
|-<br />
|naiadic major (S+M)<br />
|perfect fourth<br />
|nearmajor sixth<br />
|[0 9 16]<br />
|3:4:5 chord.<br />
|}<br />
<br />
== Isomorphic layouts and other instrument designs ==<br />
22edo approximates JI well enough to be playable on brass instruments, starting one octave higher than 12edo does, or taking advantage of an additional key to account for the extra intervals. For a keyboard, a layout which splits each black key into three is sufficient for mosdiatonic; alternatively, a layout can be used which places pajara[12] on the white keys and pajara[10] on the black keys, at the cost of a much wider octave and more difficult finger reaches. On a guitar, the standard guitar tuning works in 22edo and the edo is small enough to be fully fretted. However, as with all non-5n edos, the standard guitar tuning is not isomorphic. Tuning in nearmajor thirds on an 11edo-fretted guitar (similar to the Kite Guitar's nearmajor skip-fretting, but for a smaller edo) is isomorphic, however, and leads to a more comfortable spacing of frets at the cost of possibly a more difficult placement of certain notes. On an isomorphic keyboard, the [https://keyboard.snelgrove.science/?name=22&output=sample&instrument=piano&fundamental=254.5642522&rSteps=4&urSteps=1&hexSize=50&rotation=343&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=C%2C%5EC%2C%2CvD%2CD%2C%5ED%2C%2CvE%2CE%2CF%2C%5EF%2C%2CvG%2CG%2C%5EG%2C%2CvA%2CA%2C%5EA%2C%2CvB%2CB&note_colors=%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff%2C%23804040%2C%23800040%2C%238000ff%2C%238080ff standard diatonic layout] places the edostep moving down and to the right, as it is the diatonic semitone. As a superpyth temperament, this means that the nearmajor third is found a diatonic semitone below the major third of mosdiatonic. There is also a [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=2&urSteps=3&hexSize=50&rotation=343.897886248&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 pajara-based layout]. The [https://keyboard.snelgrove.science/?name=pabara&output=sample&instrument=piano&fundamental=254.5642522&rSteps=7&urSteps=13&hexSize=50&rotation=210&scale=54.54545%2C109.09091%2C163.63636%2C218.18182%2C272.72727%2C327.27273%2C381.81818%2C436.36364%2C490.90909%2C545.45455%2C600.00000%2C654.54545%2C709.09091%2C763.63636%2C818.18182%2C872.72727%2C927.27273%2C981.81818%2C1036.36364%2C1090.90909%2C1145.45455%2C2%2F1&key_labels=names&equivSteps=22&names=1%2C%5E1%2C2%2C%5E2%2C3%2C%5E3%2Cv4%2C4%2Cv5%2C5%2Cv6%2C6%2C%5E6%2C7%2C%5E7%2C8%2C%5E8%2Cv9%2C9%2Cv10%2C10%2Cv1&note_colors=%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff0000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23ff8000%2C%230000ff%2C%23400080%2C%23ff8000%2C%23400080%2C%23ff8000%2C%23400080 harmonic table] is also supported, though it is not as structurally critical as in 15edo.<br />
<br />
The standard diatonic layout follows:{{Lumatone edo mapping|n=22|start=6|xstep=4|ystep=-3}}<br />
<br />
== Supersets and subsets ==<br />
=== 44edo ===<br />
22edo is every other step of 44edo, which introduces a neutral third and semifourth while preserving 22edo's 11-limit structure; both of these naturally fit in with the insertion of prime 13. The price to pay is that intervals of 7 and 9 become particularly inaccurate (with 9/8 itself [[inconsistent]]) due to the addition of the alternative "neutral" ouranic, but using the latter leads to Semaphore temperament, not preserving the useful harmonic relations that 22edo gives to the 7-limit. Akin to 12edo's 5/4 in a system like 24edo, it remains structurally justified by the subset edo while losing relative accuracy. 44edo also contains accurate approximations of the 13th, 19th, and 23rd harmonics, all of which are nearly maximally inaccurate in 22edo.<br />
{{Harmonics in ED|44|31|0}}<br />
<br />
== Comparisons to other tuning systems ==<br />
* [[15edo]] shares Porcupine and various tuning tendencies associated with it (the sharp nearminor third, the sharp perfect fifth, and the flat 10/9). Because of this, it has a similar Zarlino structure to 22edo, so a lot of 22edo harmony that does not rely on Pajara's equivalences is preserved when moving to 15edo.<br />
* [[24edo]] essentially offers the "alternative" set of interval qualities to 22edo, with neutral/farmajor/ultramajor rather than nearmajor/supermajor.<br />
* [[26edo]] can be taken as the counterpart of 22edo with a flat fifth rather than sharp, as it preserves many of 22edo's other quirks, such as compressed 5-limit thirds, 7/5~10/7 being mapped to the semioctave (supporting jubilismic), and four versions of every 12edo interval quality.<br />
* [[27edo]] shares Superpyth, with a sharper tuning than 22edo where 5/4 is tuned to 400¢.<br />
* [[31edo]] shares Orwell, and is often taken as the smallest option for a representation of the 11-limit more faithful than 22, by virtue of making 11/9 a genuine neutral third.<br />
* [[32edo]] shares Pajara with a particularly sharp tuning.<br />
* [[41edo]] shares the keemic tertian structure, and more specifically Magic, while bringing the fifth close to just and distinguishing 11 from 5-limit intervals.<br />
<br />
== Music in 22edo ==<br />
Vector - [https://www.youtube.com/watch?v=DdJJu5tGCQs What Happens After]<br />
<br />
== See also ==<br />
* [[User:Vector/A rebuttal to 31et.com's interpretation of 22edo (common complaints)]]<br />
* [[22edo/Chords]]<br />
* [[22edo/Scales]]<br />
* [[22edo/Intervals]]<br />
* [[22edo/V/Exposition]] - an introduction to 22edo written by Vector<br />
<br />
{{Navbox EDO}}<br />
{{Cat|Edos}}</div>Vectorhttps://xenreference.com/wiki/index.php?title=Third&diff=7685Third2026-06-22T20:26:40Z<p>Vector: </p>
<hr />
<div>[[File:Thirds 2.png|thumb|Opposing qualities of third stack to a perfect fifth]]<br />
[[File:Thirds.mp3|thumb|Third qualities: subminor, farminor, nearminor, neutral, nearmajor, farmajor, supermajor]]<br />
[[File:Hwbnwr.png|thumb|Thirds on a diatonic staff. In [[mohajira]] temperament, these are 7/6, 6/5, 11/9, 5/4, and 9/7, representing the 5 basic qualities of third.]]<br />
''For intervals with a denominator of 3, see [[Perfect fourth]] and [[5/3]].''<br />
[[File:Thirsd spectrum.png|thumb|352x352px|A system of melodic qualities of third, with cent values on the left]]<br />
A '''third''' is an interval that spans, or could reasonably span, two steps of the 7-form. The most significant harmonic feature of thirds is that two thirds of opposite qualities stack to a [[perfect fifth]] and form a tertian [[triad]] (and therefore, qualities of thirds can be defined with respect to the size of the perfect fifth). Therefore, xenharmonic thirds are a way to extend familiar Western triadic concepts to new tunings. Basic just intonation thirds include [[9/7]], [[5/4]], [[6/5]], and [[7/6]]. Thirds are generally split into major, neutral, and minor. <br />
<br />
In tertian harmony, the quality of the third in a triad often determines the character of said triad, giving the third a central role in diatonic harmony. It is for this reason, and the reason that a set of diatonic intervals sharing a quality may be constructed trivially from a third of said quality and a perfect fifth (although this applies to all diatonic ordinals), that the structure and tuning of the thirds in a given tuning system, together with the tuning of the fifth, are seen as somewhat representative of the characteristics of the tuning system as a whole. <br />
<br />
== Name ==<br />
The term ''third'' comes from conventional music theory, wherein the degrees of the diatonic scale are numbered from 1 and their corresponding intervals are given the numbers' ordinals. Thus, a third encompasses three staff positions. This is somewhat counterintuitive, as thirds span two heptatonic scale steps.<br />
<br />
Several other coincidental ways in which "third" describes the interval region have been noted in the xenharmonic community:<br />
<br />
* A major third is approximately 1\3 ("one third" of an octave, logarithmically speaking).<br />
* A minor third is the third interval (zero-indexed) of the 12-form.<br />
** Consequently, the cent values of many thirds begin with the digit 3.<br />
* A third is the third interval (zero-indexed) of the 10-form.<br />
<br />
== In heptatonic scales ==<br />
The concept of a third may be defined in terms of a heptatonic [[MOS|moment-of-symmetry]] scale, which always has two varieties of thirds: <br />
<br />
* Diatonic has three major thirds ranging from 343 to 480 cents (basic tuning: 400 cents), and four minor thirds ranging from 240 to 343 cents (basic tuning: 300 cents). These are the [[diatonic major third]] and [[diatonic minor third]], and are often tuned specifically to approximate certain qualities of third in various [[Regular temperament|regular temperaments]], and in their just tunings of 81/64 = ~408c and 32/27 = ~296c respectively are usually the intervals meant when an unspecified "major" or "minor" is present in diatonic theory/notation or most (pyth-spine) JI notation systems. <br />
** In rank-3 diatonic scales under the [[scale quality]] scheme, the tuning of thirds is linked to that of sixths, sevenths, and optionally seconds, but independent of that of fifths. <br />
* Antidiatonic has four major thirds ranging from 343 to 600 cents (basic tuning: 400 cents), and three minor thirds ranging from 0 to 343 cents (basic tuning: 267 cents).<br />
* Onyx has two major thirds ranging from 343 to 1200 cents (basic tuning: 450 cents), and five minor thirds ranging from 0 to 343 cents (basic tuning: 300 cents).<br />
* Archaeotonic has five major thirds ranging from 343 to 400 cents (basic tuning: 369 cents), and two minor thirds ranging from 200 to 343 cents (basic tuning: 277 cents).<br />
<br />
* Mosh has six perfect thirds ranging from 343 to 400 cents (basic tuning: 360 cents), and one diminished third ranging from 0 to 343 cents (basic tuning: 240 cents).<br />
* Smitonic has six perfect thirds ranging from 300 to 343 cents (basic tuning: 327 cents), and one augmented third ranging from 343 to 600 cents (basic tuning: 436 cents).<br />
<br />
== As an interval region ==<br />
Thirds as an interval region generally range from about 260 to 440 cents, with major thirds in the larger portion of the range and minor thirds in the smaller portion, and [[interordinal]] intervals from 240-260 and 440-460 cents can also function as thirds. <br />
<br />
=== Major thirds ===<br />
Major thirds are the larger variety of third, at approximately 400 cents in size. 5/4 and 9/7 are examples of major thirds.<br />
<br />
=== Minor thirds ===<br />
Minor thirds serve as the fifth-complements of major thirds: for every major third, there is a unique corresponding minor third which stacks together with it to form a perfect fifth (given a particular tuning of said fifth). Minor thirds are approximately 300 cents in size.<br />
<br />
=== Neutral thirds ===<br />
Neutral thirds are "in-between" major and minor thirds. Two neutral thirds stack to a perfect fifth; pairs of neutral thirds are at most distinct from one another by a comma-sized interval, which may be tempered out to equate the thirds to the perfect semififth, so that two equal neutral thirds can produce a 3/2 fifth. In a matching pair of unequal neutral thirds, the larger is called tendoneutral, and the smaller is called artoneutral; this is not to be confused with the separate "arto" and "tendo" interval qualities. Neutral triads are often described as being ambiguous or "neutral" in quality between those of major and minor; in the absence of major and minor triads, they may function as both simultaneously (as in [[7edo]]).<br />
<br />
=== Thirds as scale generators ===<br />
<br />
* Thirds flat of 267 cents generate [[semiquartal]]. This is associated with [[semaphore]] temperament, although semaphore generators tend to be tempered closer to seconds than thirds.<br />
* Thirds between 267 and 300 cents generate [[gramitonic]]; this is the characteristic scale of [[orwell]] temperament (generator around 272 cents).<br />
* Thirds between 300 and 343 cents generate [[smitonic]]; this is the characteristic scale of [[amity]] (~336c), [[kleismic]] (~317c), and [[myna]] (~310c) temperaments, as well as many [[keemic]] temperaments with generators around 323 cents.<br />
* Thirds between 343 and 400 cents generate [[mosh]]; this is the characteristic scale of [[rastmic]] (and thus [[mohajira]]) (~348c) alongside [[magic]] (~380c) and [[wurschmidt]] (~387c).<br />
* Thirds sharp of 400 cents generate [[checkertonic]]; this is the characteristic scale of [[squares]] (~425c) and [[sensi]] (~445c) temperament.<br />
<br />
== In just intonation ==<br />
''TODO: explain comma inflections and Pythagorean thirds''<br />
<br />
In Pythagorean tuning, the perfect hemififth is sqrt(3/2), or about 351 cents. This is the "perfect" neutral third, and stacks twice to reach the perfect fifth. It is not a just interval, but is treated equivalently to one in systems such as [[Neutral FJS]].<br />
<br />
=== 5-limit ===<br />
The most common major third in 5-limit just intonation is [[5/4]], the classical major third (in JI notation systems, the 5-flat major third), which is tuned to approximately 386 cents in pure tuning. In a triad bounded by a fifth, 5/4 produces a 4:5:6 otonality, which functions as a major chord and explains the stability and consonance of major chords in general. 5/4 is also the octave-reduced 5th harmonic, giving it significance in the structure of the 5-limit. Building a scale by stacking 4:5:6 triads produces the Zarlino [[diatonic]] major scale. The 4:5:6 triad is well-represented in 15edo (which has a stretched triad), 19edo, 22edo, 31edo, 34edo, 41edo, 46edo, and 53edo.<br />
<br />
The minor third corresponding to [[5/4]] is [[6/5]]. In JI notation systems, 6/5 is the 5-sharp minor third. The two share the property of being superparticular, which is unique to them out of any pair of thirds, and produces a contrast between otonalities and utonalities involving them. For instance, the classical minor triad 10:12:15 is significantly more complex when expressed as an [[enumeration]], and may be written as a utonality /(4:5:6). Building a scale by stacking 10:12:15 triads produces the Zarlino minor scale. The difficulty of tuning 6/5 is somewhat in between that of 5/4 and 9/7, as it is in between them in terms of prime factorization complexity and does not have any repeated prime factors. For instance, while 22edo accurately tunes 4:5:6, its approximation of 6/5 (and thus especially 10:12:15) is insufficient for many people. 27edo has a similar tuning tendency for the fifth, and tunes 6/5 accurately at the cost of 5/4. Edos with accurate fifths, however, such as 12, 34, 41, and 53, are able to tune both thirds to comparable accuracy.<br />
<br />
=== 7-limit ===<br />
The characteristic major third of the 2.3.7 subgroup is [[9/7]], which is a septimal supermajor third (in JI notation systems, the 7-sharp major third), about 435 cents in size. Conversely to the classical major third, this is a very unstable interval when used in a triad (14:18:21), combining the melodic quality usually associated with major with the harmonic quality characteristic of minor. 9/7 contains a factor of 9, therefore it is very sensitive to tuning differences in the 3rd harmonic. Therefore, it is difficult to tune correctly in [[EDO|edos]] and regular temperaments without severely detuning the 7th harmonic. The 14:18:21 triad is well-tuned in 14edo (which has a compressed triad), 22edo, 27edo, 31edo, 36edo, 39edo, 41edo, 46edo, and 53edo; out of these, the only edos that derive an accurate 7th harmonic from their approximations are 31edo, 36edo, 41edo, 46edo, and 53edo. Notably, the first four of these support [[slendric]], making it the first 2.3.7 temperament that can tune both 7/4 and 9/7 well.<br />
<br />
The minor third corresponding to [[9/7]] is [[7/6]]. 7/6 shares a number of properties with 6/5 in terms of its ratio, being superparticular (and thus appearing between adjacent notes in the harmonic series, most notably in the tetradic form of the classical major chord, 4:5:6:7) and having a similar complexity in terms of its prime factorization, substituting 5 for 7. (In JI notation systems, 7/6 is the 7-flat minor third.) As such, many 2.3.7 edos that mistune 9/7 have a more accurate 7/6, such as 32edo. However, 7/6 is more analogous to 5/4 in its function, both in terms of being the first interval in the more stable and consonant of the two septal tertian triads (6:7:9), and being the "major" version of its own class of intervals called [[Chthonic|chthonics]] (loosely centered around interordinal inframinor thirds), which it shares with [[8/7|8/7.]]<br />
<br />
=== 11-limit ===<br />
<br />
==== Neutral thirds ====<br />
The 11-limit neutral thirds are 11/9 (~347c) and 27/22 (~355c). 11/9 may be obtained by taking the mediant of 5/4 and 6/5, and 27/22 is its fifth complement. 11/9 is the most common just intonation neutral third, and is found in the equable diatonic tetrachord 9:10:11:12 (between the notes of the utonal version, and over the root of the otonal version). It is so close to 27/22 that they are often tempered together without much damage to either, such as in 41edo. However, they are very unstable in terms of tuning due to the powers of 3 in their prime factorizations; porcupine (a temperament of reasonable accuracy overall) detunes 11/9 all the way to equate it with 6/5 (due to equalizing the aforementioned tetrachord), and even 53edo must treat it as supraminor. Therefore, the temperaments that tune 11/9 and 27/22 the best are often those that equate them to one another. These include 7edo, 17edo, 24edo, 27edo (with the second-best 11/8), 31edo, 34edo, and 41edo. However, of these, only 41edo, 17edo, and 24edo, and possibly 31edo, have reasonable tunings for 11/8.<br />
<br />
11/9 may be notated as the 11-(semi)flat neutral third or the 11-sharp minor third in JI notation systems, depending on the uninflected interval qualities available.<br />
<br />
==== Major and minor thirds ====<br />
The simplest pair of 11-limit major and minor thirds are:<br />
<br />
* 14/11 (~418c), an 11-limit interval that is the size of a major third but gets represented as an imperfect fourth in many systems of JI notation (as the 11-flat 7-flat fourth), and which is between 5/4 and 9/7 in size (serving as a [[neogothic]] major third),<br />
* 33/28 (~284c), the minor counterpart of 14/11, functioning as a neogothic minor third<br />
<br />
=== 13-limit ===<br />
<br />
==== Neutral thirds ====<br />
The 13-limit neutral thirds are 16/13 (~359c) and 39/32 (~343c). Unlike the 11-limit ones, these are very stable tuning-wise, as 16/13 is the octave complement of the 13th harmonic, so it is always mapped to its closest direct approximation (assuming pure octaves). These, therefore, show up as the most prominent neutral thirds generated by temperaments that detune 11/9, such as in porcupine. However, unless the fifth is sharpened considerably so that halving it approaches 16/13 (as in 10edo and 37edo), it is, unlike with 11/9 and 27/22, often best to leave the 13-limit neutral thirds separate from one another, such as in 36edo.<br />
<br />
In JI notation systems, 16/13 may be notated as the 13-flat major third (in FJS), the 13-sharp minor third (in HEJI), or the 13-sharp neutral third depending on the choice of formal comma and the uninflected interval qualities available.<br />
<br />
==== Arto and tendo thirds ====<br />
The tridecimal ultramajor (or tendo) third, 13/10 (~454c), a 13-limit interordinal interval that can function as a very sharp supermajor third, or "ultramajor third"; 13/10 is more stable in a triad than its corresponding minor third<br />
<br />
15/13 (~247c) is the tridecimal inframinor (or arto) third, and the minor counterpart of 13/10, which serves as the approximate center of the previously mentioned chthonic category, and as the less stable counterpart of 13/10 in a triad. 15/13 and 13/10 are far enough apart that they can be played simultaneously over a root without clashing; see [[Arto and tendo theory]].<br />
<br />
==== Other 13-limit thirds ====<br />
26/21 (~370c) is a 13-limit major third that is flatter than 5/4, and can be called a "submajor third".<br />
<br />
13/11 (~289c) is a 13-limit interval close to 33/28 which is often tempered together with 33/28 in neogothic harmony; this tunes the fifth sharply enough to generate these thirds as the mosdiatonic major and minor thirds.<br />
<br />
=== 17-limit ===<br />
17/14 (~336c) is a 17-limit minor third that is sharper than 6/5, and can be called a "supraminor third"<br />
<br />
=== 19-limit ===<br />
19/16 (~297c) is the octave-reduced 19th harmonic. Its triad, 16:19:24, is considered by some to be a major factor in the perceived rootedness of the 12edo minor triad, given how well 12edo approximates it compared to the simpler but less rooted 6/5. 24/19 is its major counterpart.<br />
<br />
There are also similarly tuned thirds of 19/15 (~409c) and 45/38 (~293c).<br />
<br />
{{Interval regions}}</div>Vectorhttps://xenreference.com/wiki/index.php?title=File:Thirds.mp3&diff=7684File:Thirds.mp32026-06-22T20:18:25Z<p>Vector: </p>
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<div>rywr</div>Vectorhttps://xenreference.com/wiki/index.php?title=Scale&diff=7652Scale2026-06-15T21:22:40Z<p>Vector: Removed redirect to Glossary#Scale</p>
<hr />
<div>{{WIP}}<br />
<br />
A scale is a set of pitches which are chosen from in making music. Scales are usually ''periodic'', i.e. the same pattern of intervals repeats at some interval called the ''equave''. On the Xenharmonic Reference, ''scales are periodic or repeating unless stated otherwise.'' (Though of course, finite but nonrepeating pitch material may be useful to consider in some contexts like voicing and register, especially in harmonic series or spectralist music.) A periodic scale can be visualized as a set of points in the circle of equave-equivalent pitch classes.<br />
<br />
== Descriptions ==<br />
{{UserTag|SS|Vector|cebaff|Scales are basically the same kind of object as chords, at least once an equave is chosen. There are a bunch of different equivalence relations between scales, which are somewhat confusingly all informally called things like "the same scale". The loosest such relation is step pattern equivalence, where "the same" means that two scales simply have the same step pattern (and often the same equave), and can vary in tuning, transposition, or rotation. This is the sense in which 5L 2s is "a scale". The strictest relation is wherein all three of these categories must be the same; in that two scales are only "the same" if they contain the exact same notes. This is the sense in which "C Ionian (12edo tuning)" is a scale.<br />
<br />
Other categories lie in between these:<br />
<br />
* "12edo diatonic" is a scale if only transposition and rotation may be varied;<br />
<br />
* "Ionian" is a scale if only transposition and tuning may be varied;<br />
<br />
* "C Ionian" is a scale if only tuning may be varied;<br />
<br />
* "2-2-1-2-2-2-1" is a scale if only transposition may be varied.<br />
<br />
"Scales" can, in practical music theory in certain cases, also have different forms to be used in different contexts. An example is melodic minor, which has a different ascending form and descending form.}}</div>Vectorhttps://xenreference.com/wiki/index.php?title=User:Vector/Xenlang&diff=7634User:Vector/Xenlang2026-06-14T07:31:56Z<p>Vector: Created page with "== Numerals == Numerals unless otherwise specified are in bijective seximal, alternating between consonant and vowel segments. {| class="wikitable" |+ ! !Vowel segment !Consonant segment |- |1 |a |tr |- |2 |e |dr |- |3 |o |pr |- |4 |i |br |- |5 |u |pl |- |6 |y |bl |} The spacer ''an'' or ''am'' separates a consonant numeral segment from a preceding consonant that is not part of the numeral. == Pythagorean ordinals == Pythagorean ordinals use the consonant roots d, r, m..."</p>
<hr />
<div>== Numerals ==<br />
Numerals unless otherwise specified are in bijective seximal, alternating between consonant and vowel segments. <br />
{| class="wikitable"<br />
|+<br />
!<br />
!Vowel segment<br />
!Consonant segment<br />
|-<br />
|1<br />
|a<br />
|tr<br />
|-<br />
|2<br />
|e<br />
|dr<br />
|-<br />
|3<br />
|o<br />
|pr<br />
|-<br />
|4<br />
|i<br />
|br<br />
|-<br />
|5<br />
|u<br />
|pl<br />
|-<br />
|6<br />
|y<br />
|bl<br />
|}<br />
The spacer ''an'' or ''am'' separates a consonant numeral segment from a preceding consonant that is not part of the numeral.<br />
<br />
== Pythagorean ordinals ==<br />
Pythagorean ordinals use the consonant roots d, r, m, f, s, l, t. Consonant-final numerals are added for additional octaves (although a lone -az- is sufficient to indicate a single added octave), and the final vowel segment indicates the quality, which is notated with -a for minor/perfect (unison, fourth) and -e for major/perfect (fifth). <br />
{| class="wikitable"<br />
|+<br />
!Interval<br />
!Name<br />
|-<br />
|P1<br />
|da<br />
|-<br />
|m2<br />
|ra<br />
|-<br />
|M2<br />
|re<br />
|-<br />
|m3<br />
|ma<br />
|-<br />
|M3<br />
|me<br />
|-<br />
|P4<br />
|fa<br />
|-<br />
|P5<br />
|se<br />
|-<br />
|M7<br />
|te<br />
|-<br />
|P8<br />
|dantra<br />
|}<br />
For additional accidentals, they progress -a, -o, -u for flatter intervals, and -e, -i, -y for sharper intervals. The y vowel is pronounced like /y~ɨ/. So a diminished 7th is ''to''. The consonants indicating additional sharps or flats (in groups of 3) are not the clusters from the Numeral section but -n, -g, -v, -th, -sh, -k, with -h- acting as a spacer to separate the possible initial vowel of a consonant-final numeral from the vowel indicating the direction being altered by. So, a 111-diminished compound fifth would be sazuhyk, with -yk- being the encoding of 36 groups of 3 (56<sub>bjsex</sub>), -u- being 3 additional steps and the indication that the direction is diminishing, -az- being +1 octave, and -s- being the 5th degree.</div>Vectorhttps://xenreference.com/wiki/index.php?title=Solfege&diff=7541Solfege2026-06-08T21:49:09Z<p>Vector: </p>
<hr />
<div>'''Solfege''' refers to any way of labelling notes with syllables to be sung, and conventionally one that utilizes syllables similar to the conventional ''do-re-mi-fa-so-la-ti'' system for labeling the notes of a heptatonic scale. <br />
<br />
Solfege is notoriously hard to generalize. A number of different approaches to solfege for microtonal tunings exist. The core conflict at the heart of generalizing solfege is the lack of uniformity in standard solfege. While solfege uses alternative vowels for qualities, these alternate vowels are not fully uniform (due to historical conflicts with the original diatonic solfege, which took its syllables from a Latin hymn), and so generalizing them becomes rather complicated and idiosyncratic. It becomes especially so if you want to be cross-linguistically compatible, with five vowels and ~13 widespread consonants constituting a total of 60-70 combinations, if codas are ignored, which is enough to notate most commonly-used edos if equivalent notes that are spelled differently are treated as the same thing, but completely fails for more complex rank-2 or JI systems.<br />
<br />
== Standard solfege (heptatonic) ==<br />
{| class="wikitable"<br />
|+<br />
!Degree (1-indexed)<br />
!Solfege<br />
|-<br />
|1<br />
|do<br />
|-<br />
|2<br />
|re<br />
|-<br />
|3<br />
|mi<br />
|-<br />
|4<br />
|fa<br />
|-<br />
|5<br />
|so / sol<br />
|-<br />
|6<br />
|la<br />
|-<br />
|7<br />
|ti<br />
|-<br />
|(8/1)<br />
|do<br />
|}<br />
This solfege generally represents the major [[diatonic]] scale. However, it may be applied to other 7-form scales as well.<br />
<br />
== Solfege for accidentals ==<br />
{| class="wikitable"<br />
!Degree (1-indexed)<br />
!Solfege<br />
|-<br />
|P1<br />
|do<br />
|-<br />
|A1<br />
|di<br />
|-<br />
|d2<br />
| -<br />
|-<br />
|m2<br />
|ra<br />
|-<br />
|M2<br />
|re<br />
|-<br />
|A2<br />
|ri<br />
|-<br />
|d3<br />
| -<br />
|-<br />
|m3<br />
|me<br />
|-<br />
|M3<br />
|mi<br />
|-<br />
|A3<br />
| -<br />
|-<br />
|d4<br />
|fe<br />
|-<br />
|P4<br />
|fa<br />
|-<br />
|A4<br />
|fi<br />
|-<br />
|d5<br />
|se<br />
|-<br />
|P5<br />
|so / sol<br />
|-<br />
|A5<br />
|si<br />
|-<br />
|d6<br />
| -<br />
|-<br />
|m6<br />
|le<br />
|-<br />
|M6<br />
|la<br />
|-<br />
|A6<br />
|li<br />
|-<br />
|d7<br />
| -<br />
|-<br />
|m7<br />
|te<br />
|-<br />
|M7<br />
|ti<br />
|-<br />
|A7<br />
| -<br />
|-<br />
|d8<br />
|de<br />
|-<br />
|P8/P1<br />
|do<br />
|}<br />
This is the standard solfege for accidentals. Note that is it is designed with 12edo in mind, it cannot represent d2, d3, A3, d6, d7, or A7 (in fact, it can at maximum extend itself to 19edo). <br />
<br />
=== Extended solfeges for accidentals ===<br />
Neutral intervals are generally indicated by the vowel ''u'', which goes unused in standard solfege and is prominently found in the word "neutral" and most of its cognates in other European languages.<br />
<br />
A 31edo solfege present on [https://31edo.com] that makes use of this is as follows, taken directly from the website (s = subminor, S = supermajor); "uh" is pronounced as in English "strut".<br />
{| class="wikitable"<br />
!<br />
!Solfege<br />
!Interval Names<br />
|-<br />
!1sns<br />
|Do Du<br />
|P1 S1<br />
|-<br />
!2nds<br />
|Ruh Re Ru Ra Ri<br />
|s2 m2 n2 M2 S2<br />
|-<br />
!3rds<br />
|Muh Me Mu Ma Mi<br />
|s3 m3 n3 M3 S3<br />
|-<br />
!4ths<br />
|Fuh Fo Fu<br />
|s4 P4 S4<br />
|-<br />
!Tritones<br />
|Fa/Suh Fi/Se<br />
|A4/sd5 SA4/d5<br />
|-<br />
!5ths<br />
|Su So Si<br />
|s5 P5 S5<br />
|-<br />
!6ths<br />
|Luh Le Lu La Li<br />
|s6 m6 n6 M6 S6<br />
|-<br />
!7ths<br />
|Tuh Te Tu Ta Ti<br />
|s7 m7 n7 M7 S7<br />
|-<br />
!8ves<br />
|Duh Do (Du)<br />
|s8 P8 (S8)<br />
|}<br />
<br />
== Solfege for scaleforms ==<br />
An alternative way to extend solfege by Vector is to ignore accidentals entirely, and let, for instance, ''fa'' stand for any note mapped to 3\7, regardless of quality and exact scale used.<br />
<br />
Instead alternative sets of syllables are provided to extend solfege to all forms up to and including the 15-form.<br />
<br />
To start with, the basic sequence ''do-re-na-mi-fa-zi-so-la-be-ti-do'' is constructed for the 10-form.<br />
<br />
Every other entry, that is ''do-na-fa-so-be-do'', is used for the 5-form, and the complementary ''do-re-mi-zi-la-ti-do'' for the 6-form. The standard solfege ''do-re-mi-fa-so-la-ti-do'' is used for the 7-form, and ''zi'' is added for the 8-form. The 9-form is equivalent to the 10-form but without ''zi''. Then, the 2- and 4-form are trivially constructible as subsets of the 8-form and similarly for the 3-form from the 6-form.<br />
<br />
Beyond the 10-form, ''re, mi, la,'' and ''ti'' bifurcate, and so does ''zi'' in odd forms: <br />
{| class="wikitable"<br />
|+<br />
!<br />
!7-form (1-indexed)<br />
!<br />
!<br />
!12-form (0-indexed)<br />
|-<br />
|re<br />
|2<br />
|ke<br />
|re<br />
|1, 2<br />
|-<br />
|mi<br />
|3<br />
|vi<br />
|mi<br />
|3, 4<br />
|-<br />
|la<br />
|6<br />
|pa<br />
|la<br />
|8, 9<br />
|-<br />
|ti<br />
|7<br />
|gi (hard "g")<br />
|ti<br />
|10, 11<br />
|-<br />
|zi<br />
| -<br />
|je ("ye")<br />
|wi<br />
|6<br />
|}<br />
The original syllables are assigned to the major versions, as that's what they represent in standard major-scale solfege.<br />
<br />
Therefore, the 11-form contains ''do-ke-re-vi-mi-je-wi-pa-la-gi-ti-do,'' the 12-form ''do-ke-re-vi-mi-fa-zi-so-pa-la-gi-ti-do,'' the 13-form ''do-ke-re-vi-mi-fa-je-wi-so-pa-la-gi-ti-do'', and the 14-form ''do-ke-re-na-vi-mi-fa-zi-so-pa-la-be-gi-ti-do'' which functions as a (bifurcated) superset of 7-form.<br />
<br />
The largest form which can be represented here is the 15-form, which functions as a superset of 5-form and is ''do-ke-re-na-vi-mi-fa-je-wi-so-pa-la-be-gi-ti-do''. Extending solfege beyond 15 would not only be out of the range of most things considered a "form" for a tuning system (note that the Roklotian scale follows the given 15-form rather well) but it would also run into the issue that it runs out of consonants; remaining available sounds are rare (such as ''th''). <br />
<br />
Additionally, these may also be used as fixed solfege syllables to name the notes of a scale.</div>Vectorhttps://xenreference.com/wiki/index.php?title=Template:Author_SS&diff=7540Template:Author SS2026-06-08T21:41:59Z<p>Vector: </p>
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<div><div class="ss">{{#ifeq: {{{2}}} | 1 | [[User:Vector|sylvanScreech]]: | SS: }} {{{1}}}</div><br />
<noinclude><br />
{{Template:Author_SS|blah|1}}<br />
</noinclude></div>Vectorhttps://xenreference.com/wiki/index.php?title=Pentagoth&diff=7539Pentagoth2026-06-08T21:41:19Z<p>Vector: </p>
<hr />
<div>'''Pentagoth''' is the rank-3 2.5.7.17(.11.13.19.23)[9 & 16 & 21] temperament and its variants, which can be used to extend existing 7-limit temperaments. It tempers out 2023/2000, the '''Pentagoth comma'''.<br />
<br />
{{Author SS|1=Pentagoth temperament, in the rank-3 form, splits 7/5 into two 20/17s (~287c * 2 = ~574c) and provides 5/4 (~390c) as a separate generator. This means 7/4 is found at two 20/17s stacked with a 5/4, and 17/16 itself is the ~100c semitone between the two generators. The minor third generator can also be seen as a flatly tuned 19/16 and a roughly in-tune 13/11. Combining the two generators gets a flat fifth of ~677c.|2=1}}<br />
<br />
''This portion of the page was written from the perspective of [[User:Ground|Ground]]. This is a new format being tested and the article is incomplete.''<br />
<br />
== 2.5.7: an introduction ==<br />
<br />
''Todo: add Ground author template''<br />
<br />
I've always been interested in the [[2.5.7 subgroup]] and its extensions. Prime 3 is so central to how we tend to understand harmony that removing it is always interesting, and 5 and 7 are the next two simplest, thus the best alternatives to create a new harmonic system.<br />
<br />
Here's a list of some of the best temperaments with their mappings of 5 and 7:<br />
* {{e|6}} & {{e|25}}: [[Didacus]] 2 5<br />
* {{e|16}} & {{e|21}}: Llywelyn 7 -1<br />
* {{e|16}} & {{e|25}}: [[Mabilic]] 3 -5<br />
* {{e|21}} & {{e|25}}: Sidewalk -7 -5<br />
* {{e|21}} & {{e|31}}: [[Miracle]] -7 -2<br />
* {{e|15}} & {{e|16}}: Rainy 5 -3<br />
* {{e|15}} & {{e|22}}: [[Porcupine]] -5 6<br />
All of these are pretty well-established names, except for Sidewalk, which I came up with.<br />
<br />
{{Author SS|2.5.7 is useful to explore in the context of temperaments, due to the fact that often times, 3/2 is sort of shoehorned into temperaments that don't tune it accurately. The most egregious example is Mabilic, which at its best tunes 5/4 and 7/4 to within 5 cents of just, but when extended to Mavila in the full 7-limit uses a much less accurate 3/2 with 29 cents of error. <br> 2.5.7 also generally takes the 6-form, with didacus serving a similar role for it as meantone does for 2.3.5 (in fact, didacus can be seen as a much more accurate restriction of septimal meantone, via the logic above). <br> Many apparent gaps in the temperament range are filled when 3/2 is not considered to be a target interval. }}<br />
<br />
== An asidewalk ==<br />
<br />
'''Sidewalk''' is likely the least well known of the basic 2.5.7 temperaments, given that I had a chance to coin the name for its comma, 823543/800000. It's one of the few temperaments with a generator in the neominor third region, half of 7/5 in this case.<br />
** 823543/800000 -> 2.5.7[21 & 25]; CWE 287.441¢, CE 287.185¢.<br />
<br />
The name Sidewalk comes from its edo join of 21 & 25, the ages to drink alcohol and rent a car in the USA. Instead of drinking and driving, you should use the sidewalk. I originally called it Gridacus for "Ground Didacus" as a half-joke, which became Gridwalker after a character, which became Sidewalk again. Sidewalk is also part of the ground, which is me. It reminds me of urbanism and the excessive number of walks I go on. I initially found it while looking through possible generators of 2.<5.7 or 2.>5.7 (see [[straddle primes]]) in [[67edo]] and being surprised by its low complexity, then again later while creating a 4L1M2s scale in the same tuning as a replacement for Gorgo.<br />
<br />
== Pentagoth temperament ==<br />
<br />
The current rank-3 version of Pentagoth began as an extension of Sidewalk, but I realized it could be applied to other 2.5.7 temperaments I've used in the past, showing up as early as my 2020 song Wallowing in Madness in 16edo. It works very well in several edos that I have a unique affinity for, including 25, {{e|37}}, {{e|46}}, and 67.<br />
<br />
The term was originally coined by UserMinusOne and me to refer to what is now called '''Vengeance''', a 2.5.17 Mavila-like temperament generated by the flat fifth 25/17. I described it in a [https://www.tumblr.com/groundfault/705198584894816256/the-best-mavila-probably 2022 Tumblr post]. We all independently discovered the temperament, but I agreed to let Vengeance stay even though ours came first because I had a feeling that Pentagoth was a broader category. My decision paid off. Pentagoth didn't just apply to Sidewalk, but to Mabilic (2.5.7 Mavila), Llywelyn (or Shoe or Gorgo or Laconic), and any other 2.5.7 temperament that split 7/5 in half.<br />
{| class="wikitable"<br />
|+2.5.7.11.13.17.19.23[9 & 16 & 21] Pentagoth Lattice (49-odd-limit) generated by 13/11-up and 5/4-right<br />
!Gens<br />
!-3<br />
!-2<br />
!-1<br />
!0<br />
!1<br />
!2<br />
!3<br />
|-<br />
!6<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|49/46<br />
|<br />
|-<br />
!5<br />
|<br />
|<br />
|<br />
|<br />
|49/34<br />
|<br />
|26/23<br />
|-<br />
!4<br />
|<br />
|<br />
|<br />
|49/25<br />
|28/23 49/40<br />
|26/17 35/23 49/32<br />
|44/23<br />
|-<br />
!3<br />
|<br />
|<br />
|<br />
|28/17 38/23<br />
|26/25 35/34<br />
|22/17 13/10 49/38<br />
|13/8<br />
|-<br />
!2<br />
|<br />
|<br />
|28/25 19/17 49/44<br />
|32/23 7/5<br />
|40/23 7/4 44/25<br />
|11/10 25/23 35/32<br />
|11/8 26/19<br />
|-<br />
!1<br />
|<br />
|38/25<br />
|32/17 19/10 49/26<br />
|20/17 13/11 19/16<br />
|28/19 52/35 34/23 25/17<br />
|13/7 35/19<br />
|22/19<br />
|-<br />
!0<br />
|<br />
|32/25 14/11<br />
|8/5 35/22<br />
|1/1<br />
|5/4 44/35<br />
|11/7 25/16<br />
|<br />
|-<br />
!-1<br />
|19/11<br />
|14/13 38/35<br />
|34/25 19/14 23/17 35/26<br />
|32/19 22/13 17/10<br />
|20/19 52/49 17/16<br />
|25/19<br />
|<br />
|-<br />
!-2<br />
|16/11 19/13<br />
|64/35 20/11 46/25<br />
|8/7 23/20 25/22<br />
|10/7 23/16<br />
|88/49 34/19 25/14<br />
|<br />
|<br />
|-<br />
!-3<br />
|16/13<br />
|20/13 17/11 76/49<br />
|68/35 25/13<br />
|17/14 23/19<br />
|<br />
|<br />
|<br />
|-<br />
!-4<br />
|23/22<br />
|64/49 17/13 46/35<br />
|80/49 23/14<br />
|50/49<br />
|<br />
|<br />
|<br />
|-<br />
!-5<br />
|23/13<br />
|<br />
|68/49<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
!-6<br />
|<br />
|92/49<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|}<br />
<br />
== Tempering process ==<br />
<br />
All temperaments are 9 & 16 & 21 unless otherwise mentioned.<br />
* Given a 2.5.7 temperament where 7/5 is split in half, 5/4 * sqrt(7/5) makes a flat fifth like 25/17. The supraminor third 49/40 is also close to 17/14. Equating these pairs tempers out 2023/2000, which I've decided to call the Pentagoth comma due to being the first and most obvious step.<br />
** 2023/2000 -> 2.5.7.17; CWE 388.049¢ 289.369¢, CE 390.556¢ 288.428¢.<br />
* Then, 7/5 will be reasonably biased flat due to being (20/17)^2, pulling it closer to 32/23. Also, the same supraminor third is close to 28/23 as well. This tempers out the 2.5.7.23 comma 161/160.<br />
** 161/160 -> 2.5.7.17.23; CWE 387.534¢ 288.767¢, CE 390.950¢ 287.244¢.<br />
* After that, things get messier. 13/11 can be easily equated to half of 7/5 by tempering out 847/845, and there are no better options than to do the same with 19/16, tempering out 1805/1792, even though this makes it very flat and the least accurate prime in the no-3 23-limit extension. It makes up for low accuracy with extremely low complexity, and makes 19/14 the octave complement of 25/17.<br />
** 847/845, 1805/1792 -> 2.5.7.13/11.17.19.23; CWE 386.133¢ 289.374¢, CE 390.581¢ 288.351¢.<br />
* The temperament ended up being rank-4 in the no-3 23-limit, so I looked for a good mapping for 11 and 13 with just the two important generators, and found one. I later learned that this equates 17/13 to 64/49, tempering out 833/832, which is a good choice. 11 and 13 are the most complex and may not be tuned as well, such as in 25edo and thus 50edo, but this temperament generally works.<br />
** 833/832 -> 2.5.7.11.13.17.19.23; CWE 389.217¢ 289.608¢, CE 391.425¢ 288.482¢.<br />
* So what do you do to add 3 and make it full 23-limit? It makes sense to either temper out 36/35 (Mint) for the low-complexity flat fifth or take advantage of the tuning range of 7 and temper out 1029/1024 (Slendric). Mint Pentagoth seems like it should be worse because of the very flat 3, but this allows 19/15 to be in tune. {{adv|5120/5103 ([[Aberschismic]]) tempering implies a [[gentle region|gentle-region]] fifth; in fact, adding 5120/5103 to 2.5.7 Sidewalk results in 2.3.5.7[{{e|29}} & 46], Leapday, of whose full 23-limit Pentagoth extension 46edo is the only reasonable patent-val tuning.}} It's also possible to add an accurate alternate 9 by tempering out 126/125 (Starling).<br />
** 36/35 -> 23-limit; CWE 682.871¢ 390.965¢, CE 681.014¢ 392.071¢.<br />
** 1029/1024 -> 23-limit[16 & 21 & 30]; CWE 677.955¢ 233.348¢, CE 679.315¢ 233.108¢.<br />
** 5120/5103 -> 23-limit[46 & 53[-17, -23] & 58]; CWE 703.389¢ 389.431¢, CE 703.820¢ 391.381¢.<br />
** 126/125 -> 2.9.5.7.11.13.17.19.23[9 & 21 & 37]; CWE 389.680¢ 100.487¢, CE 391.298¢ 102.695¢.<br />
<br />
== Sidewalk again ==<br />
<br />
As much as superfluous temperament names bother me, I'll propose these extensions to Sidewalk just to avoid resulting in 3-word names when combined with Pentagoth: Mint Sidewalk "Dandelion" and Slendric Pentagoth "Clover", after some of my favorite plants found near the sidewalk. The naming occurred immediately after a period in which I found dozens of clovers with at least 4 leaves. Coincidence? Yeah.<br />
<br />
Starling Sidewalk is unique in the 2.9.5.7 subgroup and is a weak restriction of the half-octave temperament [https://en.xen.wiki/w/Starling_temperaments#Vines Vines] in 2.3.5.7. Since it is also a plant, it fits perfectly into this new naming scheme.<br />
<br />
{{Navbox regtemp}}</div>Vectorhttps://xenreference.com/wiki/index.php?title=Template:Author_SS&diff=7538Template:Author SS2026-06-08T21:41:00Z<p>Vector: </p>
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<div><div class="ss">{{#ifeq: {{{2}}} | 1 | sylvanScreech: | SS: }} {{{1}}}</div><br />
<noinclude><br />
{{Template:Author_SS|blah|1}}<br />
</noinclude></div>Vectorhttps://xenreference.com/wiki/index.php?title=Template:Author_SS&diff=7537Template:Author SS2026-06-08T21:40:26Z<p>Vector: </p>
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<div><div class="ss">{{#ifeq: {{{2}}} | 1 | sylvanScreech: | SS: }} {{{1}}}</div></div>Vectorhttps://xenreference.com/wiki/index.php?title=Pentagoth&diff=7536Pentagoth2026-06-08T21:35:54Z<p>Vector: </p>
<hr />
<div>'''Pentagoth''' is the rank-3 2.5.7.17(.11.13.19.23)[9 & 16 & 21] temperament and its variants, which can be used to extend existing 7-limit temperaments. It tempers out 2023/2000, the '''Pentagoth comma'''.<br />
<br />
{{Author SS|1=Pentagoth temperament, in the rank-3 form, splits 7/5 into two 20/17s (~287c * 2 = ~574c) and provides 5/4 (~390c) as a separate generator. This means 7/4 is found at two 20/17s stacked with a 5/4, and 17/16 itself is the ~100c semitone between the two generators. The minor third generator can also be seen as a flatly tuned 19/16 and a roughly in-tune 13/11. Combining the two generators gets a flat fifth of ~677c.}}<br />
<br />
''This portion of the page was written from the perspective of [[User:Ground|Ground]]. This is a new format being tested and the article is incomplete.''<br />
<br />
== 2.5.7: an introduction ==<br />
<br />
''Todo: add Ground author template''<br />
<br />
I've always been interested in the [[2.5.7 subgroup]] and its extensions. Prime 3 is so central to how we tend to understand harmony that removing it is always interesting, and 5 and 7 are the next two simplest, thus the best alternatives to create a new harmonic system.<br />
<br />
Here's a list of some of the best temperaments with their mappings of 5 and 7:<br />
* {{e|6}} & {{e|25}}: [[Didacus]] 2 5<br />
* {{e|16}} & {{e|21}}: Llywelyn 7 -1<br />
* {{e|16}} & {{e|25}}: [[Mabilic]] 3 -5<br />
* {{e|21}} & {{e|25}}: Sidewalk -7 -5<br />
* {{e|21}} & {{e|31}}: [[Miracle]] -7 -2<br />
* {{e|15}} & {{e|16}}: Rainy 5 -3<br />
* {{e|15}} & {{e|22}}: [[Porcupine]] -5 6<br />
All of these are pretty well-established names, except for Sidewalk, which I came up with.<br />
<br />
{{Author SS|2.5.7 is useful to explore in the context of temperaments, due to the fact that often times, 3/2 is sort of shoehorned into temperaments that don't tune it accurately. The most egregious example is Mabilic, which at its best tunes 5/4 and 7/4 to within 5 cents of just, but when extended to Mavila in the full 7-limit uses a much less accurate 3/2 with 29 cents of error. <br> 2.5.7 also generally takes the 6-form, with didacus serving a similar role for it as meantone does for 2.3.5 (in fact, didacus can be seen as a much more accurate restriction of septimal meantone, via the logic above). <br> Many apparent gaps in the temperament range are filled when 3/2 is not considered to be a target interval. }}<br />
<br />
== An asidewalk ==<br />
<br />
'''Sidewalk''' is likely the least well known of the basic 2.5.7 temperaments, given that I had a chance to coin the name for its comma, 823543/800000. It's one of the few temperaments with a generator in the neominor third region, half of 7/5 in this case.<br />
** 823543/800000 -> 2.5.7[21 & 25]; CWE 287.441¢, CE 287.185¢.<br />
<br />
The name Sidewalk comes from its edo join of 21 & 25, the ages to drink alcohol and rent a car in the USA. Instead of drinking and driving, you should use the sidewalk. I originally called it Gridacus for "Ground Didacus" as a half-joke, which became Gridwalker after a character, which became Sidewalk again. Sidewalk is also part of the ground, which is me. It reminds me of urbanism and the excessive number of walks I go on. I initially found it while looking through possible generators of 2.<5.7 or 2.>5.7 (see [[straddle primes]]) in [[67edo]] and being surprised by its low complexity, then again later while creating a 4L1M2s scale in the same tuning as a replacement for Gorgo.<br />
<br />
== Pentagoth temperament ==<br />
<br />
The current rank-3 version of Pentagoth began as an extension of Sidewalk, but I realized it could be applied to other 2.5.7 temperaments I've used in the past, showing up as early as my 2020 song Wallowing in Madness in 16edo. It works very well in several edos that I have a unique affinity for, including 25, {{e|37}}, {{e|46}}, and 67.<br />
<br />
The term was originally coined by UserMinusOne and me to refer to what is now called '''Vengeance''', a 2.5.17 Mavila-like temperament generated by the flat fifth 25/17. I described it in a [https://www.tumblr.com/groundfault/705198584894816256/the-best-mavila-probably 2022 Tumblr post]. We all independently discovered the temperament, but I agreed to let Vengeance stay even though ours came first because I had a feeling that Pentagoth was a broader category. My decision paid off. Pentagoth didn't just apply to Sidewalk, but to Mabilic (2.5.7 Mavila), Llywelyn (or Shoe or Gorgo or Laconic), and any other 2.5.7 temperament that split 7/5 in half.<br />
{| class="wikitable"<br />
|+2.5.7.11.13.17.19.23[9 & 16 & 21] Pentagoth Lattice (49-odd-limit) generated by 13/11-up and 5/4-right<br />
!Gens<br />
!-3<br />
!-2<br />
!-1<br />
!0<br />
!1<br />
!2<br />
!3<br />
|-<br />
!6<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|49/46<br />
|<br />
|-<br />
!5<br />
|<br />
|<br />
|<br />
|<br />
|49/34<br />
|<br />
|26/23<br />
|-<br />
!4<br />
|<br />
|<br />
|<br />
|49/25<br />
|28/23 49/40<br />
|26/17 35/23 49/32<br />
|44/23<br />
|-<br />
!3<br />
|<br />
|<br />
|<br />
|28/17 38/23<br />
|26/25 35/34<br />
|22/17 13/10 49/38<br />
|13/8<br />
|-<br />
!2<br />
|<br />
|<br />
|28/25 19/17 49/44<br />
|32/23 7/5<br />
|40/23 7/4 44/25<br />
|11/10 25/23 35/32<br />
|11/8 26/19<br />
|-<br />
!1<br />
|<br />
|38/25<br />
|32/17 19/10 49/26<br />
|20/17 13/11 19/16<br />
|28/19 52/35 34/23 25/17<br />
|13/7 35/19<br />
|22/19<br />
|-<br />
!0<br />
|<br />
|32/25 14/11<br />
|8/5 35/22<br />
|1/1<br />
|5/4 44/35<br />
|11/7 25/16<br />
|<br />
|-<br />
!-1<br />
|19/11<br />
|14/13 38/35<br />
|34/25 19/14 23/17 35/26<br />
|32/19 22/13 17/10<br />
|20/19 52/49 17/16<br />
|25/19<br />
|<br />
|-<br />
!-2<br />
|16/11 19/13<br />
|64/35 20/11 46/25<br />
|8/7 23/20 25/22<br />
|10/7 23/16<br />
|88/49 34/19 25/14<br />
|<br />
|<br />
|-<br />
!-3<br />
|16/13<br />
|20/13 17/11 76/49<br />
|68/35 25/13<br />
|17/14 23/19<br />
|<br />
|<br />
|<br />
|-<br />
!-4<br />
|23/22<br />
|64/49 17/13 46/35<br />
|80/49 23/14<br />
|50/49<br />
|<br />
|<br />
|<br />
|-<br />
!-5<br />
|23/13<br />
|<br />
|68/49<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
!-6<br />
|<br />
|92/49<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|}<br />
<br />
== Tempering process ==<br />
<br />
All temperaments are 9 & 16 & 21 unless otherwise mentioned.<br />
* Given a 2.5.7 temperament where 7/5 is split in half, 5/4 * sqrt(7/5) makes a flat fifth like 25/17. The supraminor third 49/40 is also close to 17/14. Equating these pairs tempers out 2023/2000, which I've decided to call the Pentagoth comma due to being the first and most obvious step.<br />
** 2023/2000 -> 2.5.7.17; CWE 388.049¢ 289.369¢, CE 390.556¢ 288.428¢.<br />
* Then, 7/5 will be reasonably biased flat due to being (20/17)^2, pulling it closer to 32/23. Also, the same supraminor third is close to 28/23 as well. This tempers out the 2.5.7.23 comma 161/160.<br />
** 161/160 -> 2.5.7.17.23; CWE 387.534¢ 288.767¢, CE 390.950¢ 287.244¢.<br />
* After that, things get messier. 13/11 can be easily equated to half of 7/5 by tempering out 847/845, and there are no better options than to do the same with 19/16, tempering out 1805/1792, even though this makes it very flat and the least accurate prime in the no-3 23-limit extension. It makes up for low accuracy with extremely low complexity, and makes 19/14 the octave complement of 25/17.<br />
** 847/845, 1805/1792 -> 2.5.7.13/11.17.19.23; CWE 386.133¢ 289.374¢, CE 390.581¢ 288.351¢.<br />
* The temperament ended up being rank-4 in the no-3 23-limit, so I looked for a good mapping for 11 and 13 with just the two important generators, and found one. I later learned that this equates 17/13 to 64/49, tempering out 833/832, which is a good choice. 11 and 13 are the most complex and may not be tuned as well, such as in 25edo and thus 50edo, but this temperament generally works.<br />
** 833/832 -> 2.5.7.11.13.17.19.23; CWE 389.217¢ 289.608¢, CE 391.425¢ 288.482¢.<br />
* So what do you do to add 3 and make it full 23-limit? It makes sense to either temper out 36/35 (Mint) for the low-complexity flat fifth or take advantage of the tuning range of 7 and temper out 1029/1024 (Slendric). Mint Pentagoth seems like it should be worse because of the very flat 3, but this allows 19/15 to be in tune. {{adv|5120/5103 ([[Aberschismic]]) tempering implies a [[gentle region|gentle-region]] fifth; in fact, adding 5120/5103 to 2.5.7 Sidewalk results in 2.3.5.7[{{e|29}} & 46], Leapday, of whose full 23-limit Pentagoth extension 46edo is the only reasonable patent-val tuning.}} It's also possible to add an accurate alternate 9 by tempering out 126/125 (Starling).<br />
** 36/35 -> 23-limit; CWE 682.871¢ 390.965¢, CE 681.014¢ 392.071¢.<br />
** 1029/1024 -> 23-limit[16 & 21 & 30]; CWE 677.955¢ 233.348¢, CE 679.315¢ 233.108¢.<br />
** 5120/5103 -> 23-limit[46 & 53[-17, -23] & 58]; CWE 703.389¢ 389.431¢, CE 703.820¢ 391.381¢.<br />
** 126/125 -> 2.9.5.7.11.13.17.19.23[9 & 21 & 37]; CWE 389.680¢ 100.487¢, CE 391.298¢ 102.695¢.<br />
<br />
== Sidewalk again ==<br />
<br />
As much as superfluous temperament names bother me, I'll propose these extensions to Sidewalk just to avoid resulting in 3-word names when combined with Pentagoth: Mint Sidewalk "Dandelion" and Slendric Pentagoth "Clover", after some of my favorite plants found near the sidewalk. The naming occurred immediately after a period in which I found dozens of clovers with at least 4 leaves. Coincidence? Yeah.<br />
<br />
Starling Sidewalk is unique in the 2.9.5.7 subgroup and is a weak restriction of the half-octave temperament [https://en.xen.wiki/w/Starling_temperaments#Vines Vines] in 2.3.5.7. Since it is also a plant, it fits perfectly into this new naming scheme.<br />
<br />
{{Navbox regtemp}}</div>Vectorhttps://xenreference.com/wiki/index.php?title=Pentagoth&diff=7534Pentagoth2026-06-08T21:26:47Z<p>Vector: </p>
<hr />
<div>'''Pentagoth''' is the rank-3 2.5.7.17(.11.13.19.23)[9 & 16 & 21] temperament and its variants, which can be used to extend existing 7-limit temperaments. It tempers out 2023/2000, the '''Pentagoth comma'''.<br />
<br />
''This portion of the page was written from the perspective of [[User:Ground|Ground]]. This is a new format being tested and the article is incomplete.''<br />
<br />
== 2.5.7: an introduction ==<br />
<br />
''Todo: add Ground author template''<br />
<br />
I've always been interested in the [[2.5.7 subgroup]] and its extensions. Prime 3 is so central to how we tend to understand harmony that removing it is always interesting, and 5 and 7 are the next two simplest, thus the best alternatives to create a new harmonic system.<br />
<br />
Here's a list of some of the best temperaments with their mappings of 5 and 7:<br />
* {{e|6}} & {{e|25}}: [[Didacus]] 2 5<br />
* {{e|16}} & {{e|21}}: Llywelyn 7 -1<br />
* {{e|16}} & {{e|25}}: [[Mabilic]] 3 -5<br />
* {{e|21}} & {{e|25}}: Sidewalk -7 -5<br />
* {{e|21}} & {{e|31}}: [[Miracle]] -7 -2<br />
* {{e|15}} & {{e|16}}: Rainy 5 -3<br />
* {{e|15}} & {{e|22}}: [[Porcupine]] -5 6<br />
All of these are pretty well-established names, except for Sidewalk, which I came up with.<br />
<br />
{{Author SS|2.5.7 is useful to explore in the context of temperaments, due to the fact that often times, 3/2 is sort of shoehorned into temperaments that don't tune it accurately. The most egregious example is Mabilic, which at its best tunes 5/4 and 7/4 to within 5 cents of just, but when extended to Mavila in the full 7-limit uses a much less accurate 3/2 with 29 cents of error.<br />
<br />
2.5.7 also generally takes the 6-form, with didacus serving a similar role for it as meantone does for 2.3.5 (in fact, didacus can be seen as a much more accurate restriction of septimal meantone, via the logic above).}}<br />
<br />
== An asidewalk ==<br />
<br />
'''Sidewalk''' is likely the least well known of the basic 2.5.7 temperaments, given that I had a chance to coin the name for its comma, 823543/800000. It's one of the few temperaments with a generator in the neominor third region, half of 7/5 in this case.<br />
** 823543/800000 -> 2.5.7[21 & 25]; CWE 287.441¢, CE 287.185¢.<br />
<br />
The name Sidewalk comes from its edo join of 21 & 25, the ages to drink alcohol and rent a car in the USA. Instead of drinking and driving, you should use the sidewalk. I originally called it Gridacus for "Ground Didacus" as a half-joke, which became Gridwalker after a character, which became Sidewalk again. Sidewalk is also part of the ground, which is me. It reminds me of urbanism and the excessive number of walks I go on. I initially found it while looking through possible generators of 2.<5.7 or 2.>5.7 (see [[straddle primes]]) in [[67edo]] and being surprised by its low complexity, then again later while creating a 4L1M2s scale in the same tuning as a replacement for Gorgo.<br />
<br />
== Pentagoth temperament ==<br />
<br />
The current rank-3 version of Pentagoth began as an extension of Sidewalk, but I realized it could be applied to other 2.5.7 temperaments I've used in the past, showing up as early as my 2020 song Wallowing in Madness in 16edo. It works very well in several edos that I have a unique affinity for, including 25, {{e|37}}, {{e|46}}, and 67.<br />
<br />
The term was originally coined by UserMinusOne and me to refer to what is now called '''Vengeance''', a 2.5.17 Mavila-like temperament generated by the flat fifth 25/17. I described it in a [https://www.tumblr.com/groundfault/705198584894816256/the-best-mavila-probably 2022 Tumblr post]. We all independently discovered the temperament, but I agreed to let Vengeance stay even though ours came first because I had a feeling that Pentagoth was a broader category. My decision paid off. Pentagoth didn't just apply to Sidewalk, but to Mabilic (2.5.7 Mavila), Llywelyn (or Shoe or Gorgo or Laconic), and any other 2.5.7 temperament that split 7/5 in half.<br />
{| class="wikitable"<br />
|+2.5.7.11.13.17.19.23[9 & 16 & 21] Pentagoth Lattice (49-odd-limit) generated by 13/11-up and 5/4-right<br />
!Gens<br />
!-3<br />
!-2<br />
!-1<br />
!0<br />
!1<br />
!2<br />
!3<br />
|-<br />
!6<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|49/46<br />
|<br />
|-<br />
!5<br />
|<br />
|<br />
|<br />
|<br />
|49/34<br />
|<br />
|26/23<br />
|-<br />
!4<br />
|<br />
|<br />
|<br />
|49/25<br />
|28/23 49/40<br />
|26/17 35/23 49/32<br />
|44/23<br />
|-<br />
!3<br />
|<br />
|<br />
|<br />
|28/17 38/23<br />
|26/25 35/34<br />
|22/17 13/10 49/38<br />
|13/8<br />
|-<br />
!2<br />
|<br />
|<br />
|28/25 19/17 49/44<br />
|32/23 7/5<br />
|40/23 7/4 44/25<br />
|11/10 25/23 35/32<br />
|11/8 26/19<br />
|-<br />
!1<br />
|<br />
|38/25<br />
|32/17 19/10 49/26<br />
|20/17 13/11 19/16<br />
|28/19 52/35 34/23 25/17<br />
|13/7 35/19<br />
|22/19<br />
|-<br />
!0<br />
|<br />
|32/25 14/11<br />
|8/5 35/22<br />
|1/1<br />
|5/4 44/35<br />
|11/7 25/16<br />
|<br />
|-<br />
!-1<br />
|19/11<br />
|14/13 38/35<br />
|34/25 19/14 23/17 35/26<br />
|32/19 22/13 17/10<br />
|20/19 52/49 17/16<br />
|25/19<br />
|<br />
|-<br />
!-2<br />
|16/11 19/13<br />
|64/35 20/11 46/25<br />
|8/7 23/20 25/22<br />
|10/7 23/16<br />
|88/49 34/19 25/14<br />
|<br />
|<br />
|-<br />
!-3<br />
|16/13<br />
|20/13 17/11 76/49<br />
|68/35 25/13<br />
|17/14 23/19<br />
|<br />
|<br />
|<br />
|-<br />
!-4<br />
|23/22<br />
|64/49 17/13 46/35<br />
|80/49 23/14<br />
|50/49<br />
|<br />
|<br />
|<br />
|-<br />
!-5<br />
|23/13<br />
|<br />
|68/49<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
!-6<br />
|<br />
|92/49<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|}<br />
<br />
== Tempering process ==<br />
<br />
All temperaments are 9 & 16 & 21 unless otherwise mentioned.<br />
* Given a 2.5.7 temperament where 7/5 is split in half, 5/4 * sqrt(7/5) makes a flat fifth like 25/17. The supraminor third 49/40 is also close to 17/14. Equating these pairs tempers out 2023/2000, which I've decided to call the Pentagoth comma due to being the first and most obvious step.<br />
** 2023/2000 -> 2.5.7.17; CWE 388.049¢ 289.369¢, CE 390.556¢ 288.428¢.<br />
* Then, 7/5 will be reasonably biased flat due to being (20/17)^2, pulling it closer to 32/23. Also, the same supraminor third is close to 28/23 as well. This tempers out the 2.5.7.23 comma 161/160.<br />
** 161/160 -> 2.5.7.17.23; CWE 387.534¢ 288.767¢, CE 390.950¢ 287.244¢.<br />
* After that, things get messier. 13/11 can be easily equated to half of 7/5 by tempering out 847/845, and there are no better options than to do the same with 19/16, tempering out 1805/1792, even though this makes it very flat and the least accurate prime in the no-3 23-limit extension. It makes up for low accuracy with extremely low complexity, and makes 19/14 the octave complement of 25/17.<br />
** 847/845, 1805/1792 -> 2.5.7.13/11.17.19.23; CWE 386.133¢ 289.374¢, CE 390.581¢ 288.351¢.<br />
* The temperament ended up being rank-4 in the no-3 23-limit, so I looked for a good mapping for 11 and 13 with just the two important generators, and found one. I later learned that this equates 17/13 to 64/49, tempering out 833/832, which is a good choice. 11 and 13 are the most complex and may not be tuned as well, such as in 25edo and thus 50edo, but this temperament generally works.<br />
** 833/832 -> 2.5.7.11.13.17.19.23; CWE 389.217¢ 289.608¢, CE 391.425¢ 288.482¢.<br />
* So what do you do to add 3 and make it full 23-limit? It makes sense to either temper out 36/35 (Mint) for the low-complexity flat fifth or take advantage of the tuning range of 7 and temper out 1029/1024 (Slendric). Mint Pentagoth seems like it should be worse because of the very flat 3, but this allows 19/15 to be in tune. {{adv|5120/5103 ([[Aberschismic]]) tempering implies a [[gentle region|gentle-region]] fifth; in fact, adding 5120/5103 to 2.5.7 Sidewalk results in 2.3.5.7[{{e|29}} & 46], Leapday, of whose full 23-limit Pentagoth extension 46edo is the only reasonable patent-val tuning.}} It's also possible to add an accurate alternate 9 by tempering out 126/125 (Starling).<br />
** 36/35 -> 23-limit; CWE 682.871¢ 390.965¢, CE 681.014¢ 392.071¢.<br />
** 1029/1024 -> 23-limit[16 & 21 & 30]; CWE 677.955¢ 233.348¢, CE 679.315¢ 233.108¢.<br />
** 5120/5103 -> 23-limit[46 & 53[-17, -23] & 58]; CWE 703.389¢ 389.431¢, CE 703.820¢ 391.381¢.<br />
** 126/125 -> 2.9.5.7.11.13.17.19.23[9 & 21 & 37]; CWE 389.680¢ 100.487¢, CE 391.298¢ 102.695¢.<br />
<br />
== Sidewalk again ==<br />
<br />
As much as superfluous temperament names bother me, I'll propose these extensions to Sidewalk just to avoid resulting in 3-word names when combined with Pentagoth: Mint Sidewalk "Dandelion" and Slendric Pentagoth "Clover", after some of my favorite plants found near the sidewalk. The naming occurred immediately after a period in which I found dozens of clovers with at least 4 leaves. Coincidence? Yeah.<br />
<br />
Starling Sidewalk is unique in the 2.9.5.7 subgroup and is a weak restriction of the half-octave temperament [https://en.xen.wiki/w/Starling_temperaments#Vines Vines] in 2.3.5.7. Since it is also a plant, it fits perfectly into this new naming scheme.<br />
<br />
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color: black !important;<br />
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.acc1 {<br />
background-color: #BBEEEE !important;<br />
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background-color: #FFF8BB !important;<br />
color: black !important;<br />
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background-color: #FFD8BB !important;<br />
color: black !important;<br />
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background-color: #FFBBBB !important;<br />
color: black !important;<br />
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color: #fff !important;<br />
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background-color: #3F3F3F !important;<br />
color: white !important;<br />
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background-color: #8F0005 !important;<br />
color: white !important;<br />
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background-color: #669F36 !important;<br />
color: white !important;<br />
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background-color: #200066 !important;<br />
color: white !important;<br />
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background-color: #A07E12 !important;<br />
color: white !important;<br />
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color: white !important;<br />
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background-color: #0E1662 !important;<br />
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.prime31 {<br />
background-color: #0E3E6D !important;<br />
color: white !important;<br />
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background-color: #418172 !important;<br />
color: white !important;<br />
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background-color: #70813A !important;<br />
color: white !important;<br />
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background-color: #817532 !important;<br />
color: white !important;<br />
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background-color: #81391D !important;<br />
color: white !important;<br />
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.prime53 {<br />
background-color: #521D66 !important;<br />
color: white !important;<br />
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}</div>Vectorhttps://xenreference.com/wiki/index.php?title=Aberrisma&diff=7482Aberrisma2026-06-07T02:31:14Z<p>Vector: </p>
<hr />
<div>{{Proposed}}<br />
<br />
An '''aberrisma''' is an interval between roughly 20 and 55 cents representing some comma as an additional smaller type of melodic step (that is, a [[diesis]]). The aberrisma is used as one of the parameters in constructing an aberrismic scale, a type of ternary scale. For example, blackdye is a 10-note aberrismic superset of the 7-note nicetone, but with a more distinctive set of three step sizes and added opportunities to avoid pythagorean and wolf intervals.<br />
<br />
Aberrismas may also appear in MOS scales, such as garibaldi[17]. <br />
<br />
'''Aberrismic theory''' is the subset of microtonal theory pioneered by [[User:Ground|Ground]] and [[User:Inthar|Inthar]] that deals with aberrismas.<br />
<br />
== Example: The emergence of blackdye ==<br />
The Zarlino diatonic is chiral - there are two different, equally valid second degrees of the Ionian mode. Both are useful, as the sharp one forms a perfect fifth with the fifth degree but a wolf fifth with the sixth degree, and the flat one forms a perfect fifth with the sixth degree but a wolf fifth with the fifth degree.<br />
[[File:Blackdye.png|thumb|510x510px|The construction of blackdye from Zarlino diatonic]]<br />
One way to make it achiral is to temper out 81/80, the difference between these two steps, resulting in [[Meantone]] diatonic; intuitively this requires flattening the fifth and sharpening the sixth somewhat. However, an alternative way, if you wish to observe 81/80 or to use just intonation, is to include both varieties of whole tone over the unison, treating 81/80 as a melodic step between them. This can be thought of as dividing up a 9/8 into a 10/9 and an 81/80. It is then reasonable to extend this action to all instances of 9/8 in the scale (as, for instance, the Didymic diatonic has 27/16 as opposed to 5/3). The result is a 10-note ternary scale called '''blackdye'''.<br />
<br />
== The "why" of aberrismic theory ==<br />
This section will attempt to describe the principles and stylistic features of a specific style of music that justify aberrismic theory. It is not an attempt to present aberrismic theory as absolute truth.<br />
* A style of music that is melodic and heavy in modulations benefits from<br />
** Multiple step sizes for melodic interest, for example diesis-sized steps that are below conventional semitones, specifically ones large enough to be melodically distinct but small enough to represent intonational changes.<br />
** A set of modulatory intervals, including fifths.<br />
** A system that allows unlimited modulation. <br />
* It is widely agreed that lower primes are more robust to detuning. Hence for approximating JI with edos, we use lower prime temperaments, and which also represent either 81/80 or 64/63 steps for greater accuracy.<br />
The above suggests temperaments, in particular edos, that use tempered lower primes, and edos large enough to have small diesis-sized steps. In the context of fifth-based modulation, scales also benefit from having offset arcs of fifths. One simple way to have this is to detemper MOS scales into ternary scales with an additional smaller melodic step size, which have a generator arc with fifths or a generator arc that stacks to fifths via a detempered generator chain.<br />
<br />
== List of aberrismic scales ==<br />
* {{Adv|"GS(...)[n]" is [[generator sequence]] notation.}}<br />
* {{adv|"subst ax(bycz)" denotes [[MOS substitution]].}}<br />
* {{adv|"Almost" a cross-set means that one or two notes may be missing from the full cross-set and one note may have been added. Exact cross-sets are italicized.}}<br />
* {{adv|Under "Patterns", "C" is [[achiral]], and "R" and "L" denote two [[chiral]]ities of a chiral pair.}}<br />
=== Quasi-diatonic aberrismic scales ===<br />
{|class="wikitable"<br />
|-<br />
!|Name / Signature<br />
!|Pattern(s)<br />
!|Possible JI interp.<br />
!|{{adv|Almost a [[cross-set]] of...<br/>(interpreted)}}<br />
!|Notes<br />
|-<br />
!|pinedye / dia1s<br/>(5L2m1s)<br/>{{adv|{{nowrap|subst 5L(2m1s)}}}}<br />
||1sC: LLsLLmLm<br/>1sR: LLmLLmLs<br/>1sL: LLmLLsLm<br />
||2.3.5<br/>[L, m, s] = [10/9, 27/25, 81/80]<br />
||{{adv|GS(3/2)[3] and GS(10/9)[3]}}<br />
||1sC has 4 fifths and 1sR/1sL have 5<br />
|-<br />
!class="thl"|diasem / dia2s<br/>(5L2m2s)<br/>{{adv|{{nowrap|subst 5L(2m2s)}}}}<br />
||2sR: LmLsLmLsL<br/>2sL: LsLmLsLmL<br />
||2.3.7<br/>[L, m, s] = [9/8, 28/27, 64/63]<br />
||{{adv|GS(3/2)[5] and 7/6}}<br />
||Aggregate generator is 4/3, thus has fifth arcs of 5 and 4 notes respectively.<br />
''See also: [[Chthonic harmony#Diasem|Diasem]]''<br />
|-<br />
!class="thl"|blackdye / dia3s<br/>(5L2m3s)<br/>{{adv|{{nowrap|subst 5L(2m3s)}}}}<br />
||sLmLsLmLsL<br />
||2.3.5<br/>[L, m, s] = [10/9, 16/15, 81/80]<br />
||{{adv|''GS(3/2)[5] and 10/9''}}<br />
||Two interleaved 3-limit pentatonics<br />
''See also: [[10-form#Blackdye|Blackdye]]''<br />
|-<br />
!|diaslen / dia4s<br/>(5L2m4s)<br/>{{adv|{{nowrap|subst 5L(2m4s)}}}}<br />
||4sC: LmLsLsLmLss<br/>4sR: LsLmLsLsLms<br/>4sL: LsLsLmLsLsm<br />
||2.3.7<br/>[L, m, s] = [9/8, 49/48, 64/63]<br />
||{{adv|GS(3/2)[4] and GS(8/7)[3]}}<br />
||Fifth arcs with 4 notes, 4 notes, and 3 notes, with offset 8/7. Tempered to the slentonic {5L6s) MOS by [[Slendric]].<br/>{{adv|Detempered Slendric[11] generator structure, aggregate generator is 3/2}}<br />
|-<br />
!|diachrome / chromedye / dia5s<br/>(5L2m5s)<br />
||5sC: LsLsLmsLsLsm {{adv|{{nowrap|(subst 2m(5L5s))}}}}<br/>5sR: LmsLsLsLmsLs<br/>5sL: LsLsLsmLsLsm<br />
||5120/5103-tempered 2.3.5.7<br/>[L, m, s] = [10/9, 256/243, 81/80]<br />
||{{adv|5sC: ''GS(3/2)[6] and 40/27''}}<br />
||Fifth-generated but with a 6-step offset<br />
|-<br />
!|whitedye / dia7s<br/>(5L2m7s)<br/>{{adv|{{nowrap|subst 7s(5L2m)}}}}<br />
||LsLsLsmsLsLsms<br />
||5120/5103-tempered 2.3.5.7<br/>[L, m, s] = [10/9, 28/27, 81/80]<br />
||{{adv|''GS(3/2)[7] and 81/80''}}<br />
||Two interleaved diatonics<br />
|}<br />
<br />
=== Other aberrismic scales ===<br />
{|class="wikitable"<br />
|-<br />
!|Name / Signature<br />
!|Pattern(s)<br />
!|Possible JI/[[erac]] interp.<br />
!|{{adv|Almost a [[cross-set]] of...<br/>(interpreted)}}<br />
!|Notes<br />
|-<br />
!class="thl"|[[penslen]] / slen5m<br/>(5L5m6s)<br/>{{adv|{{nowrap|subst 6s(5L5m)}}}}<br />
||LmsLmsLsmLsmLsms<br />
||2.3.5.7.11[41 & 46]<br/>[L, m, s] = [12/11, 33/32, 64/63]<br />
||{{adv|''GS(8/7)[8] and 11/8''}}<br />
|| Has two aberrisma sizes, s and m.<br />
|-<br />
!|smi2m?<br/>(4L2m3s)<br/>{{adv|{{nowrap|subst 2m(4L3s)}}}}<br />
||C: LLsmLsLms<br/>R: LmLsLmsLs<br/>L: LmLsLsmLs<br />
||2.5.7<br/>[L, m, s] = [28/25, 35/32, 50/49]<br />
||{{adv|GS(5/4)[3] and GS(7/5)[3] (exact for C)}}<br />
||Didacus tempering makes L = m + s.<br />
|-<br />
!|arm5s<br/>(7L2m5s)<br/>{{adv|{{nowrap|subst 7L(2m5s)}}}}<br />
||LmLsLsLmLsLsLs<br />
||2.x<3.5.7.11.13[37edo] (4:2:1)<br />
||{{adv|''GS(<<3/2)[7] and 14/13''}}<br />
||An interleaving of two antidiatonic scales.<br />
|-<br />
!|mosh3s<br/>(3L4m3s)<br/>{{adv|{{nowrap|subst 4m(3L3s)}}}}<br />
||LmsLmsmLsm<br />
||2.x<3.7.11.13[37edo] (5:4:2)<br />
||{{adv|''GS(16/13)[5] and 11/8''}}<br />
||<br />
|-<br />
!|smi2s<br/>(4L3m2s)<br/>{{adv|{{nowrap|subst 2s(4L3m)}}}}<br />
||C: LLmsLmLsm<br/>R: LmLmsLmLs<br/>L: LmLsLmLsm<br />
||2.9.7.11.17[46edo] (8:4:1)<br />
||{{adv|GS(17/14)[3] and GS(11/8)[3] (exact for C)}}<br />
||<br />
|}<br />
<br />
=== Systematic naming ===<br />
Basic systematic names for aberrismic scales are of the form<br />
<br />
[mos_prefix]n[added_step_size] (e.g. dia2s for diasem),<br />
<br />
where the MOS prefix (a TAMNAMS prefix if one is available) is chosen based on the aberrismic-theoretical generator (as opposed to the offset), rather than from any particular mathematical construction. For example, penslen has MOS substitution type 6s(5L5m), but the systematic name is slen5m, not penwd6s, since the generator is conceived as a generator of 5L6s (Slendric[11]).<br />
<br />
This is subject to change as aberrismic theory notation is updated in the future.<br />
<br />
== Notation ==<br />
Aberrismic notation restricts to [[diatonic notation]] on the tempered 3-limit.<br />
<br />
Aberrismic/straddle-prime accidentals:<br />
* Plus / Minus +/- : These tune a note sharp or flat by a small aberrisma. They reference Johnston notation because 81/80 is a common aberrisma, but they can also represent 64/63 or any other interval of similar function. <!--They're also used to denote straddle primes, like 3+ and 3- (in a straddle-3 subgroup, these can be abbreviated as 3±). This results in some pretty heavy overloading, but +/- are only used on notes when they represent an aberrisma and only used on ratios when they represent straddle primes. --><br />
* Duplus / Duminus ≠/= : Short for ++/--, most often representing 36/35~33/32~1053/1024, which is the large aberrisma in scales like penslen, or two small aberrismas in Akea temperament. Of all options, this set of characters is the easiest to type, looks the best in various fonts, and is least likely to be confused for the similar semisharp accidental (although they happen to represent the same size of interval).<br />
<br />
== Aberrismic theory and RTT ==<br />
Aberrismic theory often applies RTT to ternary LCJI scales with comma steps. Certain scales with aberrismas may thus be endowed with JI interpretations via [[RTT]] temperaments, which may be used in suitable [[equal temperament]]s. Under groundfault's use of edos (usually patent vals) as RTT temperaments, the aberrisma tends to become a [[81/80]] in a 2.3.5 context and a [[64/63]] in a 2.3.7 context. Some scales such as 5L2m5s and 5L2m7s admit a more accurate 2.3.5.7 interpretation that tempers out neither 81/80 nor 64/63 but identifies the two commas, tempering out [[5120/5103]]. Tempering is important in aberrismic theory as a way to <!--simultaneously achieve sufficient accuracy to LCJI and -->improve the function of commas (frequently [[81/80]] or [[64/63]]) as aberrismas in ternary LCJI scales by tempering them larger than just.<br />
<br />
At times, a scale pattern has varying temperaments according to the tuning. For example, 5L2m3s may be given the temperament structure of either untempered 2.3.5 or [[Ultrapyth]] temperament.<br />
<br />
There are two choices involved in interpreting a given ternary scale, namely the choice of temperament and the choice of where to map the scale steps. The assignment of scale steps to tempered intervals is chosen to improve coverage of important LCJI intervals.<br />
<br />
=== Example: Blackdye ===<br />
The following table shows two different temperament interpretations for the same aberrismic scale pattern blackdye (sLmLsLmLsL), under untempered 2.3.5 and Ultrapyth respectively.<br />
* ''Untempered'' does not mean that the final tuning must be the JI tuning, but simply that there exists an exact JI tuning.<br />
* [[Ultrapyth]], 2.3.5.7.11.13[32 & 37], is a diatonic temperament generated by a fifth even sharper than in Superpyth. [[37edo]] provides a nearly optimal tuning. Note that we chose to regard the 3-step 2L + s as a 14/11 rather than as a 5/4, lest the interpretation merely be an extension of the untempered 2.3.5 one. groundfault terms the tuning of blackdye that makes aberrisma-altered Pyth thirds 13/11 and 14/11 ''Flutterpyth blackdye''.<br />
<br />
{| class="wikitable right-2 right-3 right-4 right-5"<br />
|+ style="font-size: 105%;" | Blackdye intervals in two temperaments<br />
|-<br />
! Interval class<br />
! Sizes<br />
! Untempered 2.3.5<br />
! 2.3.7.11.13 Flutterpyth (extended to 13-limit Ultrapyth)<br />
|-<br />
! 1-step<br />
| s<br/>m<br/>L <br />
| 81/80<br/>16/15<br/>10/9<br />
| 143/140<br/>22/21<br/>160/143<br />
|-<br />
! 2-step<br />
| L + s<br/>L + m <br />
| 9/8<br/>32/27<br />
| 8/7, 9/8<br/>7/6<br />
|- <br />
! 3-step<br />
| L + 2s<br/>L + m + s<br/>2L + s<br/>2L + m<br />
| 729/640<br/>6/5<br/>5/4<br/>320/243<br />
| 7/6<br/>13/11<br/>14/11<br/>13/10<br />
|- <br />
! 4-step<br />
| 2L + 2s<br/>2L + m + s<br />
| 81/64<br/>4/3<br />
| 13/10<br/>4/3<br />
|-<br />
! 5-step<br />
| 2L + m + 2s<br/>2L + 2m + s<br/>3L + 2s<br/>3L + m + s<br />
| 27/20<br/>64/45<br/>45/32<br/>40/27<br />
| 66/49<br/>11/8<br/>16/11<br/>49/33<br />
|-<br />
! 6-step<br />
| 3L + m + 2s<br/>3L + 2m + s<br />
| 3/2<br/>128/81<br />
| 3/2<br/>20/13<br />
|- <br />
! 7-step<br />
| 3L + m + 3s<br/>3L + 2m + 2s<br/>4L + m + 2s<br/>4L + 2m + s<br />
| 243/160<br/>8/5<br/>5/3<br/>1280/729<br />
| 20/13<br/>11/7<br/>22/13<br/>12/7<br />
|- <br />
! 8-step<br />
| 4L + m + 3s<br/>4L + 2m + 2s<br />
| 27/16<br/>16/9<br />
| 12/7<br/>7/4, 16/9<br />
|-<br />
! 9-step<br />
| 5L + 2m + s<br/>5L + m + 2s<br/>4L + 2m + 2s<br />
| 9/5<br/>15/8<br/>160/81<br />
| 143/80<br/>21/11<br/>280/143<br />
|}<br />
<br />
== Glossary ==<br />
* '''Aberration scale''': A scale made by interleaving aberrismas into a binary scale and stretching or compressing, usually a [[MOS substitution]] scale of type {{nowrap|[a+b+1]s(aLbm)}} (compression, called '''tractaberrated''') or {{nowrap|[a+b-1]s(aLbm)}} (stretching, called '''tensaberrated'''). For example, sLsmsLsLsLsmsLs is an aberration scale made from diatonic (a MOS substitution scale of type 8s(5L2m)).<br />
* '''Aberrisma''': The smallest interval region that melodically sounds like a step.<br />
* '''Magnitone''': The melodic function of L + s in quasi-diatonic aberrismic scales.<br />
* '''Monotone-MOS''': A ternary scale is ''monotone-MOS'' if it becomes a MOS under all three of the identifications L = M, M = s, and s = 0. If ''any'' (not necessarily all) of the identifications make the scale a MOS, the scale is said to ''satisfy a monotone-MOS subcondition''. For example, diasem (LmLsLmLsL) satisfies all three monotone-MOS subconditions, but blackdye (sLmLsLmLsL) satisfies only the m = s and s = 0 monotone-MOS subconditions. An aberrismic scale is required to satisfy the s = 0 monotone-MOS subcondition.<br />
* '''Solitone''': The melodic function of the L step in quasi-diatonic aberrismic scales.<br />
* '''Subaberrisma''': A step so small (smaller than an aberrisma) that its status as a melodic step is unclear.<br />
<br />
== Compositional examples ==<br />
Some compositional snippets using aberrismic scales:<br />
<br />
* The Art of It makes almost exclusive use of 31edo diasem: [[File:A New Dusk-02 The Art of It.mp3]]<br />
* A fugue using aberrismic scales: [[File:Inthar - Fugue in 32edo and 33edo.mp3]]<br />
* A 34edo blackdye fugue exposition: [[File:Blackdye-fugue-expo.mp3]]<br />
<br />
== External links ==<br />
* [https://turbofishcrow.github.io/tern Tern: aberrismic-focused ternary scale analysis]<br />
{{cat|<br />
Terms<br />
Aberrismic terms<br />
Ternary scales<br />
}}</div>Vectorhttps://xenreference.com/wiki/index.php?title=7edo&diff=74247edo2026-06-04T07:38:33Z<p>Vector: </p>
<hr />
<div>'''7edo''' is the basic equiheptatonic, where all the steps are tuned to be precisely equal. It features steps of (1200/7) ~= 171.4 cents.<br />
<br />
== Theory ==<br />
<br />
=== Edostep interpretations ===<br />
7edo's edostep has the following interpretations in the 2.3.5 subgroup:<br />
<br />
* 9/8 (the diatonic major second)<br />
* 10/9 (the interval separating 9/8 and 5/4)<br />
* 16/15 (the interval separating 5/4 and 4/3)<br />
<br />
=== JI approximation ===<br />
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. <br />
{{Harmonics in ED|7|31}}<br />
{| class="wikitable"<br />
|+Thirds in 7edo<br />
!Quality<br />
|'''Neutral'''<br />
|-<br />
!Cents<br />
|'''343'''<br />
|-<br />
!Just interpretation<br />
|'''11/9'''<br />
|}<br />
Diatonic thirds are bolded.<br />
<br />
=== Chords ===<br />
7edo features, for tertian triadic harmony, only a neutral chord [0 2 4] and the (rather discordant) sus chords [0 1 4] and [0 3 4]. Regardless, due to its triads and due to representing all seven degrees of the diatonic scale, it is the smallest edo where Western functional harmony works.<br />
<br />
=== Scales ===<br />
7edo is the first edo to distinguish the modes of the [[pentic]] scale. However, it is still small enough that it is well-temperable into scales (specifically, those of the 7-form discussed elsewhere in this article). In real world musical cultures which use near-equal 7-note scales, perfect 7edo is almost never used.<br />
<br />
=== Derivation ===<br />
7edo is derived by equalizing an equiheptatonic scale.<br />
<br />
== Notation ==<br />
In 7edo, pretty much all reasonable notation schemes collapse to ABCDEFG on A=440Hz. Accidentals are not used.<br />
<br />
== Polysomatic tuning ==<br />
Polysomatic tuning, coined by Cole Parker, refers to an octave stretch of 7edo (close to 11edt, about 6.929edo, step size 173.19c, octave 1212.33 cents) that has the property of approximating the first six harmonics of a standard harmonic instrument and of an unsupported bar ([http://hyperphysics.phy-astr.gsu.edu/hbase/Music/barres.html more context]), such as a glockenspiel, within 25% relative error (and in fact approximates them within 25 cents absolute error, except for the 6th frequency of an unsupported bar).<br />
{| class="wikitable"<br />
|+<br />
! rowspan="2" |Frequency #<br />
! colspan="5" |Unsupported bar<br />
! colspan="5" |Harmonic instrument<br />
|-<br />
!Decimal<br />
!Cents<br />
!Polysomatic tuning<br />
!Deviation<br />
!Steps<br />
!Decimal<br />
!Cents<br />
!Polysomatic tuning<br />
!Deviation<br />
!Steps<br />
|-<br />
|1<br />
|1<br />
|0<br />
|0<br />
|0<br />
|0<br />
|1<br />
|0<br />
|0<br />
|0<br />
|0<br />
|-<br />
|2<br />
|2.75625<br />
|1755.25<br />
|1731.91<br />
| -23.34<br />
|10<br />
|2<br />
|1200.00<br />
|1212.33<br />
|12.33<br />
|7<br />
|-<br />
|3<br />
|5.40225<br />
|2920.27<br />
|2944.24<br />
|23.97<br />
|17<br />
|3<br />
|1901.96<br />
|1905.10<br />
|3.14<br />
|11<br />
|-<br />
|4<br />
|8.93025<br />
|3790.44<br />
|3810.19<br />
|19.75<br />
|22<br />
|4<br />
|2400.00<br />
|2424.67<br />
|24.67<br />
|14<br />
|-<br />
|5<br />
|13.34025<br />
|4485.26<br />
|4502.95<br />
|17.70<br />
|26<br />
|5<br />
|2786.31<br />
|2771.05<br />
| -15.26<br />
|16<br />
|-<br />
|6<br />
|18.63225<br />
|5063.68<br />
|5022.53<br />
| -41.15<br />
|29<br />
|6<br />
|3101.96<br />
|3117.43<br />
|15.48<br />
|18<br />
|}<br />
Within the octave, polysomatic tuning also slightly improves the intervals 5/4 and 3/2, but makes 4/3 less accurate.<br />
<br />
== Whitewood temperament ==<br />
7edo may be interpreted as ''Whitewood'' temperament, which tempers out the Pythagorean [[chromatic semitone]]. The most obvious rank-2 extension is to add a free generator corresponding to 7/4, resulting in a system containing multiple copies of 7edo separated by the interval 7/4. This extension is supported by [[21edo]], which, along with 14edo, supports the [[Diatonic|omnidiatonic]] ternary diatonic scale.<br />
<br />
{{Cat|Edos}}{{Navbox EDO}}</div>Vectorhttps://xenreference.com/wiki/index.php?title=Sensamagic&diff=7423Sensamagic2026-06-03T03:15:13Z<p>Vector: </p>
<hr />
<div>[[File:Sentry8.png|thumb|Sentry octatonic MOS]]<br />
'''Sensamagic''' (b13 & b17), sometimes known in a tritave-equivalent context as '''Bohlen-Pierce-Stearns''' (BPS), is the temperament in the 3.5.7 subgroup equating a stack of two [[9/7]]<nowiki/>s with [[5/3]]; this means that the comma [[245/243]] is tempered out. 9/7 is tuned sharp (about 440 cents) and 5/3 is flattened (about 880 cents). It functions as a tritave analog of [[Meantone]], relating the two simplest prime harmonics after the equave with a medium accuracy.<br />
<br />
Sensamagic can be used as a temperament with octaves by one of several approaches:<br />
<br />
* simply taking the octave as the period instead of the tritave, resulting in a 2.9/7.5/3 subgroup temperament known as Sentry (11 & 19)<br />
* equating the octave to a false octave found on the Sensamagic generator chain, such as 125/63 (resulting in [[Sensi]] (19 & 27)) or 49/25 (resulting in an obscure [[Porcupine]] extension called "Hedgehog" that splits the octave into two 7/5~10/7 tritones)<br />
* adding the octave as an additional generator, resulting in rank-3 Sensamagic (41 & 19 & 27, or b65 & b30 & b43)<br />
<br />
This page will focus on tritave and rank-3 Sensamagic.<br />
[[File:Sensamagic9.png|thumb|Sensamagic enneatonic (3/1-equivalent) MOS]]<br />
''TODO: complete page''<br />
<br />
== Interval chain ==<br />
{| class="wikitable right-1 right-2"<br />
|-<br />
! #<br />
! Cents*<br />
! Approximate ratios<br />
|-<br />
| 0<br />
| 0.0<br />
| '''1/1'''<br />
|-<br />
| 1<br />
| 440.7<br />
| '''9/7'''<br />
|-<br />
| 2<br />
| 881.3<br />
| '''5/3'''<br />
|-<br />
| 3<br />
| 1322.0<br />
| 15/7<br />
|-<br />
| 4<br />
| 1762.7<br />
| '''25/9'''<br />
|-<br />
| 5<br />
| 301.4<br />
| 25/21<br />
|-<br />
| 6<br />
| 742.0<br />
| 75/49, 125/81<br />
|-<br />
| 7<br />
| 1182.7<br />
| <br />
|-<br />
| 8<br />
| 1623.4<br />
| <br />
|-<br />
| 9<br />
| 162.1<br />
|<br />
|-<br />
| 10<br />
| 602.7<br />
|<br />
|-<br />
| 11<br />
| 1043.4<br />
|<br />
|-<br />
| 12<br />
| 1484.1<br />
|<br />
|}<br />
<nowiki>*</nowiki> in 3.5.7-subgroup [[CWE]] tuning, tritave reduced. Intervals may be additionally octave-reduced in rank-3 sensamagic.<br />
<br />
{{Navbox regtemp}}</div>Vectorhttps://xenreference.com/wiki/index.php?title=File:Sensamagic9.png&diff=7422File:Sensamagic9.png2026-06-03T03:14:34Z<p>Vector: </p>
<hr />
<div>Sensamagic9</div>Vectorhttps://xenreference.com/wiki/index.php?title=Microtemperament&diff=7415Microtemperament2026-06-02T19:44:44Z<p>Vector: start page</p>
<hr />
<div>A microtemperament is a very accurate and usually very complex temperament. Due to their complexity, microtemperaments are usually not particularly useful for composition, although they can be in some cases and are often related to [[comma theory]]. Additionally, they usually have extremely narrow tuning ranges.<br />
<br />
Notable microtemperaments are schismic, alphatricot, and ennealimmal. Of these, schismic is the exception: while ennealimmal requires a 36-note scale in order to reach all of the prime harmonics in its target subgroup, consequently a 63-note scale if the prime subharmonics are to be included, and an 81-note scale to include 9/8 and 16/9, and while alphatricot reaches 5/4 after 29 generator steps leaving 53edo as the only reasonably small edo tuning, schismic finds 5/4 after only eight steps, and is viable as a 17-note scale in 29edo or 41edo, making it somewh</div>Vectorhttps://xenreference.com/wiki/index.php?title=Slendric&diff=7413Slendric2026-06-02T02:14:49Z<p>Vector: </p>
<hr />
<div>{{Infobox regtemp<br />
| Title = Slendric<br />
| Subgroups = 2.3.7<br />
| Comma basis = [[1029/1024]]<br />
| Edo join 1 = 5 | Edo join 2 = 21<br />
| Mapping = 1; 3 -1<br />
| Generators = 8/7 | Generators tuning = 233.7 | Optimization method = CWE<br />
| MOS scales = [[1L 4s]], [[5L 1s]], [[5L 6s]], [[5L 11s]], …<br />
| Odd limit 1 = 7 | Mistuning 1 = 2.11 | Complexity 1 = 11<br />
| Odd limit 2 = 2.3.7 27 | Mistuning 2 = 2.81 | Complexity 2 = 21<br />
}}<br />
'''Slendric''' (also known as "Wonder" or "Gamelic") is the basic harmonic interpretation as a [[regular temperament]] for the structure where the perfect fifth (~[[3/2]]) is split into three equal parts; each of these is taken to represent the interval [[8/7]]. Since the 7th [[harmonic]] is less than 3 [[cent]]s from just when 3/2 is pure, Slendric constitutes an exceptionally good [[rank-2 temperament|rank-2]] traversal of the [[2.3.7 subgroup|2.3.7]] tuning space for its simplicity. Its corresponding [[comma]] is the difference between 3/2 and (8/7)<sup>3</sup>, which is 1029/1024.<br />
<br />
Melodically, the Slendric generator stack forms a 5-note scale (1L 4s) that is [[equipentatonic]] (nearly 5edo). [[MOS]]es further down the hierarchy (6, 11, 16, ... notes) can be thought of as the notes of a basic pentatonic form, inflected by multiples of a characteristic small interval known as the ''quark'' (representing a third of a [[diatonic semitone]], and the commas [[49/48]] and [[64/63]] tempered together). As a result, these MOS scales tend to be extremely [[hard]].<br />
[[File:Slendric5.png|thumb|Slendric pentatonic MOS]]<br />
Slendric can exhibit a wide range of tunings, with fifths between those of [[26edo]] (692c) and [[56edo]] (707c), or generators roughly between 231 and 236c, while maintaining the recognizability of the 2.3.7 structure. Notable [[EDO]] tunings are in between these, and include EDOs that end in "1" or "6", i.e. [[31edo]], [[36edo]], [[41edo]], and [[46edo]]. Slendric is also supported by edos with 5edo's 3/2, since it is a [[subtemperament]] of 7-limit [[Blackwood]]. <br />
<br />
== Structural theory ==<br />
=== General theory ===<br />
==== Interval categories ====<br />
It is possible to define the intervals of Slendric in terms of diatonic categories, for at three steps is the perfect fifth, and at every three steps further are all of the standard fifth-generated intervals. For the remaining steps, a single pair of inflections suffices: "up"/"down", which can be abbreviated with the prefixes S and s, respectively (standing in for "super" and "sub", which can be used synonymously). An "up" is rigorously defined to be an inflection by the "quark" of 49/48~64/63. The slendric generator is then the upmajor second, and therefore the 2-generator interval is a downfourth (as a major second and a perfect fourth together reach a perfect fifth) as well as a double-upmajor third. Between a major third and perfect fourth is a minor second, which is therefore equivalent to three repetitions of "up"; because of this equivalence, it is never necessary to attach more than one "up"/"down" to a diatonic interval.<br />
<br />
Note that "up" intervals and "down" intervals can be represented as fractions with a single factor of 7 in the denominator and numerator (compactly, "/7" or "ru", and "7/" or "zo" intervals), respectively, with uninflected diatonic intervals representing the [[3-limit]]. Considering extensions to prime 5, Rodan maps 7/5 onto the [[chain of fifths]] so that "up" and "down" also comprise the /5 and 5/ classes of intervals, while Mothra maps 5 directly onto the chain of fifths. Each of these provides a very intuitive way to notate the full [[7-limit]].<br />
<br />
==== The pentatonic framework ====<br />
The intervals of Slendric can be organized according to how many steps of [[5edo]], or equivalently the 5-note MOS, they correspond to, since the MOS scales of Slendric up to at least 26 notes have 5 large steps and many small steps, each the size of a quark. The "major" interval of a class here is the one just larger than the corresponding 5edo interval, and the "minor" interval is just smaller. Below are the intervals of the symmetric mode of Slendric[21] (5L 16s). The generator tuning here is 3/10-comma, where the quark is exactly sqrt([[28/27]]), or about 31.5 cents.<br />
<br />
{| class="wikitable center-all left-1"<br />
|-<br />
! Steps of 5edo<br />
!0<br />
!1<br />
!2<br />
!3<br />
!4<br />
!5<br />
|- style="background-color: #003030;"<br />
! "Augmented" interval<br />
| 63.0<br />
| 296.7<br />
| 530.4<br />
| 764.1<br />
| 997.8<br />
| <br />
|-<br />
! JI intervals represented<br />
| 28/27<br />
| 32/27<br />
| 49/36<br />
| 14/9<br />
| 16/9<br />
| <br />
|- style="background-color: #003030;"<br />
! "Major" interval<br />
| 31.5<br />
| 265.2<br />
| 498.9<br />
| 732.6<br />
| 966.3<br />
| ''1200.0''<br />
|-<br />
! JI intervals represented<br />
| 49/48, 64/63<br />
| 7/6<br />
| 4/3<br />
| 32/21, 49/32<br />
| 7/4<br />
| ''2/1''<br />
|- style="background-color: #003030;"<br />
! "Minor" interval<br />
| ''0.0''<br />
| 233.7<br />
| 467.4<br />
| 701.1<br />
| 934.8<br />
| 1168.5<br />
|-<br />
! JI intervals represented<br />
| ''1/1''<br />
| 8/7<br />
| 21/16, 64/49<br />
| 3/2<br />
| 12/7<br />
| 63/32, 96/49<br />
|- style="background-color: #003030;"<br />
! "Diminished" interval<br />
| <br />
| 202.2<br />
| 435.9<br />
| 669.6<br />
| 903.3<br />
| 1137.0<br />
|-<br />
! JI intervals represented<br />
| <br />
| 9/8<br />
| 9/7<br />
| 72/49<br />
| 27/16<br />
| 27/14<br />
|}<br />
<br />
=== Notable features and related structures ===<br />
A distinctive feature of Slendric tuning systems is the subfourth of two generators, which represents [[21/16]]. Additionally, it serves as (8/7)<sup>2</sup> = 64/49, and thus is tempered a few cents flat of 21/16 in most tunings. Another interpretation then is [[17/13]], tempering out 273/272 and 833/832, into which 1029/1024 factors. (31edo's tuning comes particularly close to 17/13.) <br />
<br />
As a result of the ease of finding 55/32 and 17/13 along the Slendric chain, any extension to the full 7-limit can also find prime 11, and any extension to 2.3.7.13 can also find prime 17. This applies both to strong and weak extensions.<br />
==== A-Team ====<br />
Taking every other step of Slendric results in a subtemperament generated by this subfourth, which is known as '''A-Team''' (5 & 18; the name is from "18"). It is one of the main regular temperaments representing the [[oneirotonic]] (5L 3s) scale - specifically the hard tunings thereof such as in [[18edo|18]], [[23edo|23]], and [[31edo]]. The core [[subgroup]] interpretation of A-Team is 2.9.21.55: note that two A-Team generators, representing [[12/7]], come close to [[55/32]] and therefore [[385/384]] and [[441/440]], which again multiply to 1029/1024, can be tempered out. Different A-Team tunings can pick up other harmonic approximations; an interesting one is the 13:17:19 chord found in Mothra (and especially 31edo)'s version of A-Team.<br />
<br />
A-Team may be considered a [[straddle primes|straddle-3]] temperament; the >3 is the sharp 32/21 generator and the <3 is reached by +4 21/16 generators. Indeed, stacking >3 and <3 reaches 9 at +3 generators.<br />
<br />
Two subranges of A-Team are<br />
* 2.9.5.21[13 & 18], a weak restriction of Mothra, equating +6 oneirotonic generators with 5/4, thus tempering out 81/80<br />
* 2.9.15.21[18 & 23] (sometimes known as '''B-Team'''), a weak restriction of Rodan, equating the diminished 3-oneirostep (-7 oneirotonic generators) with 6/5<br />
<br />
==== Relationship with acoustic phi ====<br />
{{adv|The A-Team generator acquires the representations 21/16, 17/13, [[55/42]], and [[72/55]]. But if we look one octave higher, a pattern becomes clear: [[21/8]], [[34/13]], [[55/21]], and [[144/55]] are all ratios of two-apart Fibonacci numbers, which therefore closely approximate the square of [[acoustic phi]], entailing that acoustic phi squared over 2 is close to two Slendric generators. A single generator, therefore, is approximable by acoustic phi divided by [[sqrt(2)]]. This can also be explained by 18 being the 6th Lucas number, and therefore a close approximation to φ<sup>6</sup>; approximating 18<sup>1/6</sup> by φ gives us φ/√2 as an approximation of (3/2)<sup>1/3</sup>. This interval's precise value is about 233.09{{c}}, and using it as a generator produces a form of Slendric too sharp to be Mothra but flat of 36edo, with a fifth about 2.7 cents flat.}}<br />
<br />
=== Interval chain ===<br />
In the following tables, odd harmonics and subharmonics 1–27 are labeled in '''bold'''. Cent values reflect 3/10-comma tuning.<br />
<br />
<div><div style="display: inline-grid; margin-right: 25px;"><br />
<br />
{| class="wikitable sortable center-1 center-2 right-3"<br />
|-<br />
! #<br />
! class="unsortable" | Extended <br> diatonic <br> category<br />
! Cents<br />
! class="unsortable" | Approximate <br> 2.3.7 ratios<br />
|-<br />
| 0<br />
| P1<br />
| 0<br />
| '''1/1'''<br />
|-<br />
| 1<br />
| SM2<br />
| 234<br />
| '''8/7'''<br />
|-<br />
| 2<br />
| s4<br />
| 467<br />
| '''21/16''', 64/49<br />
|-<br />
| 3<br />
| P5<br />
| 701<br />
| '''3/2'''<br />
|-<br />
| 4<br />
| SM6<br />
| 935<br />
| 12/7<br />
|-<br />
| 5<br />
| s8<br />
| 1169<br />
| 63/32, 96/49<br />
|-<br />
| 6<br />
| M2<br />
| 202<br />
| '''9/8'''<br />
|-<br />
| 7<br />
| SM3<br />
| 436<br />
| 9/7<br />
|-<br />
| 8<br />
| s5<br />
| 670<br />
| 72/49<br />
|-<br />
| 9<br />
| M6<br />
| 903<br />
| '''27/16'''<br />
|-<br />
| 10<br />
| SM7<br />
| 1137<br />
| 27/14<br />
|-<br />
| 11<br />
| sM2<br />
| 171<br />
| 54/49<br />
|}<br />
</div><br />
<div style="display: inline-grid; margin-right: 25px;"><br />
{| class="wikitable sortable center-1 center-2 right-3"<br />
|-<br />
! #<br />
! class="unsortable" | Extended <br> diatonic <br> category<br />
! Cents<br />
! class="unsortable" | Approximate <br> 2.3.7 ratios<br />
|-<br />
| 0<br />
| P1<br />
| 0<br />
| '''1/1'''<br />
|-<br />
| −1<br />
| sm7<br />
| 966<br />
| '''7/4'''<br />
|-<br />
| −2<br />
| S5<br />
| 733<br />
| '''32/21''', 49/32<br />
|-<br />
| −3<br />
| P4<br />
| 499<br />
| '''4/3'''<br />
|-<br />
| −4<br />
| sm3<br />
| 265<br />
| 7/6<br />
|-<br />
| −5<br />
| S1<br />
| 31<br />
| 49/48, 64/63<br />
|-<br />
| −6<br />
| m7<br />
| 998<br />
| '''16/9'''<br />
|-<br />
| −7<br />
| sm6<br />
| 764<br />
| 14/9<br />
|-<br />
| −8<br />
| S4<br />
| 530<br />
| 49/36<br />
|-<br />
| −9<br />
| m3<br />
| 297<br />
| '''32/27'''<br />
|-<br />
| −10<br />
| sm2<br />
| 63<br />
| 28/27<br />
|-<br />
| −11<br />
| Sm7<br />
| 1029<br />
| 49/27<br />
|}<br />
</div><br />
<br />
== Tunings and extensions ==<br />
=== Tuning considerations ===<br />
The error induced by the comma 1029/1024, about 8.4{{c}}, has to be distributed between three factors of 7 and one factor of 3, and ideally both 3 and 7 should be flattened; we can define tunings of Slendric by the fraction of this comma by which 8/7 is sharpened. As representations of 2.3.7 intervals generally stack more factors of 3 than factors of 7, it can be argued 3 should be flattened less than 7. This occurs between 1/3-comma tuning (234.0{{c}}, just flat of 41edo) which sets 3/2, and thus the entire Pythagorean chain, just while 8/7 is sharpened by 2.8{{c}}; and 1/4-comma tuning (233.3{{c}}, very close to 36edo) which sets them equally flat, so that [[7/6]] is just. A notable EDO tuning in this range is [[77edo]].<br />
<br />
But, especially if [[6:7:8]] is considered the fundamental 2.3.7 harmony, it is reasonable to want a tuning where the error of [[4/3]] is split between that of 7/6 and 8/7. Furthermore, sharpward error is often considered more acceptable than flatward error on the interval 7/6, and these flatter tunings of Slendric are those which happen to tune 7/6 sharp. 1/5-comma tuning (232.9{{c}}, very near [[67edo]]) sets 7/6 and 8/7 equally sharp, by about 1.7{{c}} each.<br />
<br />
Based on the above, 36edo can be considered a practically optimal tuning, as it is an EDO of reasonable size in the best range for pure 2.3.7 subgroup accuracy; however, it is essentially [[straddle primes|straddle]]-5 and straddle-11, being between two full [[11-limit]] interpretations (the 36p and 36ce [[val]]s). Thus, other tunings of Slendric should be sought to improve the accuracy of [[5-limit]] and 11-limit harmony.<br />
<br />
Additional particularities of Slendric to consider include the tuning of the subfourth, and the size of the quark. The subfourth varies between nearly [[13/10]] in the flattest tunings, which has a potentially tertian function (e.g. in 10:13:15 [[triad]]s), and near-just 21/16 in the sharpest tunings, which much more closely resembles a fourth; of the intervals of Slendric, this is the one with the least clear independent role and the most variability in function between the different tunings. As for the quark, its size can vary between that of a comma and that of a quartertone. Tunings where the fifth is flattened significantly (specifiable as Mothra in the full 7-limit) have a more melodically salient quark that serves as an [[aberrisma]], and bring [[7/4|the 7th harmonic]] closer to purity.<br />
<br />
=== Extensions ===<br />
As Slendric is a structure present in very many EDOs of note and an obvious simplification of the 2.3.7 subgroup aside from that, it behooves us to consider how this structure interacts with other harmonies within the [[17-limit]]. Fortunately, there are a variety of choices for how each of the primes 5, 11, 13, and 17 fits into the Slendric framework.<br />
<br />
==== Prime 5 ====<br />
There are two most important strong extensions to reach prime 5 and complete the 7-limit, these being ''Mothra'' and ''Rodan''. Other mappings of 5 can be employed, such as the [[Schismic]] one (known as ''Guiron''), but these are significantly more complex and give prime 5 less structural presence.<br />
<br />
Mothra uses a [[meantone]] fifth in order to find [[5/4]] at the diatonic major third (12 generators up) and temper out [[81/80]]. The exaggerated quark now represents [[36/35]] in addition to 49/48 and 64/63. The most important Mothra tunings are 31edo, at the optimum for this temperament with a close-to-just 5/4, and 26edo, which approximates the tuning formed by stacking a pure 8/7. 36edo using the [[12edo]] major third of 400{{c}} as 5/4 also qualifies as Mothra.<br />
<br />
Rodan, meanwhile, slightly sharpens the fifth and can be constructed by equating 81/80 to the quark. This thereby tempers out the [[aberschisma]] (5120/5103), and furthermore implies the Sensamagic ([[245/243]]) equivalence, that [[9/7]] forms half of [[5/3]]. From this, it can be seen that 5/4 is found at a perfect fifth (3 generators) above twice 9/7 (7 generators each), or 17 generators in all: this is the downmajor third in [[#Interval categories|the system described earlier]]. 41edo and 46edo bound the main Rodan tuning range, but their sum, [[87edo]], is essentially optimal with a nearly just 5/4. 36edo using the flat major third of 367{{c}} as 5/4 also qualifies as Rodan.<br />
<br />
As regards weak extensions, notable ones include [[Miracle]], which splits 8/7 in two, and [[Valentine]], which splits it in three (therefore dividing the perfect fifth into 6 and 9 parts, respectively). They also include [[Superkleismic]] (15 & 26), which splits 7/4 into three intervals of [[6/5]]; this is supported by 26, 41, and 56edo. [[Lemba]] (16 & 26) is a less accurate weak extension that adds 1\2 interpreted as 7/5~10/7.<br />
<br />
Miracle's generator (known as the "secor") represents [[16/15]] and [[15/14]] simultaneously (tempering out [[225/224]], the marvel comma), so that [[8/5]] is placed at 7 secors. As a consequence of splitting 8/7 in half, Miracle also includes an exact [[neutral third]], interpretable in the 7-limit as [[49/40]]. Miracle is 10 & 21, and 31, 41, and [[72edo]] support it.<br />
<br />
Valentine places 6/5 and 5/4 at 4 and 5 steps respectively; the generator thus represents (5/4)/(6/5) = [[25/24]] and (6/5)/(8/7) = [[21/20]], and their ratio ([[126/125]], the starling comma) is tempered out. Valentine is 15 & 16, and 31, 46, and 77edo support it.<br />
<br />
==== Prime 11 ====<br />
As mentioned before, extensions to 11 can be created off of these by tempering out 385/384 and 441/440. This works almost perfectly in the 41 & 46 Rodan range, and the diatonic major third is identified with [[14/11]]. This also applies to the 26 & 31 Mothra range, yet the case with Mothra is slightly more complicated, as the interval formed from (sharpened) 7/6 stacked twice can reasonably represent either [[11/8]] or [[15/11]], depending on the tuning (note that in 31edo, it represents both). <br />
<br />
The former is supported by 26 & 31, and the latter by 31 & 36; the resulting extensions are called "Undecimal Mothra" and "Mosura" respectively. Undecimal Mothra equates 14/11 to 9/7 (tempering out [[99/98]]), and Mosura equates 14/11 to [[32/25]] (tempering out [[176/175]]). Of the two, the former is taken to be canonical primarily as 11/8 itself is reached by far fewer generators (-8, compared to +23).<br />
<br />
Miracle, Valentine, and Superkleismic all receive extensions to 11 in this manner as well. In Miracle's case, the neutral third is mapped to [[11/9]]~[[27/22]], while in Valentine's, the [[neutral second]] formed by two steps represents [[12/11]]~[[11/10]]. Superkleismic, in fact, tempers out [[100/99]], whereby [[16/11]] is reached at only two of its 6/5 generators, which produces the notable 2.7.11 subgroup structure known as [[Orgone]].<br />
<br />
==== Primes 13 and 17 ====<br />
While 36edo's representation of the 2.3.7 subgroup fails to provide comparably accurate harmonies of 5 and 11, it does somewhat better with the next higher primes: 13, 17, and 19 (though the latter two descend from 12edo). Looking at 36edo's mapping of 13, we see that it divides 7/6 into halves that can each be taken as [[14/13]]~[[13/12]], and further that two quarks represent 28/27 and [[27/26]] simultaneously.<br />
<br />
The former leads us to a weak extension, known as ''Baladic'', that tempers out [[169/168]] and splits the octave in two; equating 17/13 to the downfourth, we see [[9/8]] is also split into [[18/17]]~[[17/16]], and therefore that [[17/12]] is a semioctave.<br />
<br />
The latter leads us to a strong extension, called ''Euslendric'' (36 & 77), that reaches 13/8 after 19 generators, as the up-augmented fifth, and 17/16 after 21 generators, as the augmented unison. Euslendric is notable as its harmonies can extend to even higher limits, reaching [[19/16]] as the minor third (-9 generators), [[23/16]] as the up-diminished fifth (-23 generators), and [[29/16]] as the upminor seventh (-11 generators), all within the optimal tuning band for 2.3.7 accuracy.<br />
<br />
Revisiting the 11-limit extensions mentioned above, Rodan naturally obtains [[13/11]] as the minor third to find 13 at 22 generators down. Meanwhile, Mothra's 9/8 is flat enough that it is very close to 143/128 = (11/8)/([[16/13]]), so [[144/143]] can be tempered out as a way to extend each 11-limit extension of Mothra further to the [[13-limit]]. All of these take on the obvious mapping to reach the full 17-limit, though in the case of Rodan, 17 receives greater damage than any lower prime in the most accurate Rodan tunings (such as 87edo).<br />
<br />
=== Tuning spectrum ===<br />
[[File:Slendric Tuning Chart.png|thumb|alt=Slendric Tuning Chart.png|A chart of the tuning spectrum of Slendric, showing the offsets of odd harmonics 3, 7, 9, and 21, as a function of the generator. All EDO tunings are shown with vertical lines whose length indicates the EDO's tolerance, i.e. half of its step size in either direction of just, and some important EDOs supporting the temperament are labeled. Comma fractions with corresponding unchanged intervals are also labeled.]]<br />
<br />
{| class="wikitable center-all left-4 left-5"<br />
|-<br />
! Edo<br>generator<br />
! [[Eigenmonzo|Eigenmonzo<br>(unchanged interval)]]*<br />
! Generator (¢)<br />
! Mapping of 5<br />
! Comments<br />
|-<br />
| '''[[11edo|2\11]]'''<br />
| <br />
| '''218.182'''<br />
| <br />
| '''Lower bound of {1, 3, 7, 9} diamond monotone'''<br />
|-<br />
| [[16edo|3\16]]<br />
| <br />
| 225.000<br />
| ↓ +7 gens "Gorgo" <br> {36/35}<br />
| <br />
|-<br />
| [[37edo|7\37]]<br />
| <br />
| 227.027<br />
| <br />
| 37b val<br />
|-<br />
| [[21edo|4\21]]<br />
| <br />
| 228.571<br />
| ↑ Gorgo <br> ↓ -14 gens "Archaeotherium" <br> {405/392}<br />
|<br />
|-<br />
| [[47edo|9\47]]<br />
| <br />
| 229.787<br />
| <br />
|<br />
|-<br />
| [[26edo|5\26]]<br />
| <br />
| 230.769<br />
| ↑ Archaeotherium <br> ↓ +12 gens "[[Mothra#Tuning spectrum|Mothra]]" <br> {81/80}<br />
|<br />
|-<br />
| <br />
| [[8/7]]<br />
| 231.174<br />
| <br />
| Untempered tuning<br />
|-<br />
| [[57edo|11\57]]<br />
| <br />
| 231.579<br />
| <br />
|<br />
|-<br />
| <br />
| [[17/13]]<br />
| 232.214<br />
| <br />
| As s4, approx. 1/8-comma<br />
|-<br />
| [[31edo|6\31]]<br />
| <br />
| 232.258<br />
| <br />
|<br />
|-<br />
| [[67edo|13\67]]<br />
| <br />
| 232.836<br />
| <br />
|<br />
|-<br />
| <br />
| [[96/49]]<br />
| 232.861<br />
| <br />
| 1/5-comma<br />
|-<br />
| <br />
| φ/√2<br />
| 233.090<br />
| <br />
| As generator<br />
|-<br />
| <br />
| [[12/7]]<br />
| 233.282<br />
| <br />
| 1/4-comma; (2.3.7) 7-odd-limit minimax tuning<br />
|-<br />
| [[36edo|7\36]]<br />
| <br />
| 233.333<br />
| ↑ Mothra <br> ↓ -24 gens "Guiron" <br> {10976/10935}<br />
| <br />
|-<br />
| <br />
| [[9/7]]<br />
| 233.583<br />
| <br />
| 2/7-comma; (2.3.7) 9-odd-limit minimax tuning<br />
|-<br />
| [[113edo|22\113]]<br />
| <br />
| 233.628<br />
| <br />
| 113c val (guiron)<br />
|-<br />
| <br />
| [[27/14]]<br />
| 233.704<br />
| <br />
| 3/10-comma; 2.3.7 [[CEE]] tuning<br />
|-<br />
| [[77edo|15\77]]<br />
| <br />
| 233.766<br />
| <br />
| <br />
|-<br />
| [[118edo|23\118]]<br />
| <br />
| 233.898<br />
| <br />
| <br />
|-<br />
| <br />
| [[3/2]]<br />
| 233.985<br />
| <br />
| 1/3-comma; (2.3.7) 21- and 27-odd-limit minimax tuning<br />
|-<br />
| [[41edo|8\41]]<br />
| <br />
| 234.146<br />
| ↑ Guiron <br> ↓ +17 gens "[[Rodan#Tuning spectrum|Rodan]]" <br> {245/243}<br />
| <br />
|-<br />
| <br />
| [[55/32]]<br />
| 234.408<br />
| <br />
| As SM6, approx. 3/8-comma<br />
|-<br />
| [[87edo|17\87]]<br />
| <br />
| 234.483<br />
| <br />
| <br />
|-<br />
| <br />
| [[63/32]]<br />
| 234.547<br />
| <br />
| 2/5-comma<br />
|-<br />
| [[46edo|9\46]]<br />
| <br />
| 234.783<br />
| ↑ Rodan<br />
| <br />
|-<br />
| [[97edo|19\97]]<br />
| <br />
| 235.052<br />
| <br />
| <br />
|-<br />
| [[51edo|10\51]]<br />
| <br />
| 235.294<br />
| <br />
| <br />
|-<br />
| <br />
| [[21/16]]<br />
| 235.390<br />
| <br />
| 1/2-comma<br />
|-<br />
| [[56edo|11\56]]<br />
| <br />
| 235.714<br />
| <br />
| <br />
|-<br />
| [[61edo|12\61]]<br />
| <br />
| 236.066<br />
| <br />
| <br />
|-<br />
| [[66edo|13\66]]<br />
| <br />
| 236.364<br />
| <br />
| <br />
|-<br />
| [[71edo|14\71]]<br />
| <br />
| 236.620<br />
| <br />
| <br />
|-<br />
| '''[[5edo|1\5]]'''<br />
| <br />
| '''240.000'''<br />
| <br />
| '''Upper bound of {1, 3, 7, 9} diamond monotone'''<br />
|}<br />
<nowiki>*</nowiki> Besides the octave<br />
<br />
== Name ==<br />
"Slendric" comes from the slendro tuning used in gamelan, which creates a pentatonic framework like Slendric temperament does. (However, slendro tunings vary, often being closer to a very sharp Archy temperament with detuned octaves.) "Gamelic" (from "gamelisma") has a similar origin. <br />
<br />
This name has been criticized for its lack of relevance, and thus other names such as Wonder may be used instead.<br />
<br />
Pailiaq suggests "Tricot", after its ploidacot signature, but this runs into problems with continued reference to [[Microtemperament#Alphatricot|alphatricot]] as "tricot" instead.<br />
<br />
== List of patent vals ==<br />
{| class="wikitable sortable"<br />
|+<br />
!EDO<br />
!Generator tuning<br />
!Fifth tuning<br />
|-<br />
|11<br />
|218.18<br />
|654.55<br />
|-<br />
|16<br />
|225.00<br />
|675.00<br />
|-<br />
|21<br />
|228.57<br />
|685.71<br />
|-<br />
|47<br />
|229.79<br />
|689.36<br />
|-<br />
|26<br />
|230.77<br />
|692.31<br />
|-<br />
|57<br />
|231.58<br />
|694.74<br />
|-<br />
|88<br />
|231.82<br />
|695.45<br />
|-<br />
|31<br />
|232.26<br />
|696.77<br />
|-<br />
|129<br />
|232.56<br />
|697.67<br />
|-<br />
|98<br />
|232.65<br />
|697.96<br />
|-<br />
|67<br />
|232.84<br />
|698.51<br />
|-<br />
|170<br />
|232.94<br />
|698.82<br />
|-<br />
|103<br />
|233.01<br />
|699.03<br />
|-<br />
|139<br />
|233.09<br />
|699.28<br />
|-<br />
|175<br />
|233.14<br />
|699.43<br />
|-<br />
|211<br />
|233.18<br />
|699.53<br />
|-<br />
|247<br />
|233.20<br />
|699.60<br />
|-<br />
|36<br />
|233.33<br />
|700.00<br />
|-<br />
|257<br />
|233.46<br />
|700.39<br />
|-<br />
|221<br />
|233.48<br />
|700.45<br />
|-<br />
|185<br />
|233.51<br />
|700.54<br />
|-<br />
|149<br />
|233.56<br />
|700.67<br />
|-<br />
|113<br />
|233.63<br />
|700.88<br />
|-<br />
|190<br />
|233.68<br />
|701.05<br />
|-<br />
|77<br />
|233.77<br />
|701.30<br />
|-<br />
|195<br />
|233.85<br />
|701.54<br />
|-<br />
|118<br />
|233.90<br />
|701.69<br />
|-<br />
|159<br />
|233.96<br />
|701.89<br />
|-<br />
|200<br />
|234.00<br />
|702.00<br />
|-<br />
|41<br />
|234.15<br />
|702.44<br />
|-<br />
|169<br />
|234.32<br />
|702.96<br />
|-<br />
|128<br />
|234.38<br />
|703.13<br />
|-<br />
|87<br />
|234.48<br />
|703.45<br />
|-<br />
|133<br />
|234.59<br />
|703.76<br />
|-<br />
|46<br />
|234.78<br />
|704.35<br />
|-<br />
|143<br />
|234.97<br />
|704.90<br />
|-<br />
|97<br />
|235.05<br />
|705.15<br />
|-<br />
|148<br />
|235.14<br />
|705.41<br />
|-<br />
|51<br />
|235.29<br />
|705.88<br />
|-<br />
|107<br />
|235.51<br />
|706.54<br />
|-<br />
|56<br />
|235.71<br />
|707.14<br />
|-<br />
|61<br />
|236.07<br />
|708.20<br />
|-<br />
|66<br />
|236.36<br />
|709.09<br />
|-<br />
|71<br />
|236.62<br />
|709.86<br />
|-<br />
|5<br />
|240.00<br />
|720.00<br />
|}<br />
{{Navbox regtemp}}<br />
{{cat|Temperaments}}</div>Vectorhttps://xenreference.com/wiki/index.php?title=File:Slendric5.png&diff=7412File:Slendric5.png2026-06-02T02:14:37Z<p>Vector: </p>
<hr />
<div>Slendric5</div>Vectorhttps://xenreference.com/wiki/index.php?title=Orgone&diff=7411Orgone2026-06-02T02:12:13Z<p>Vector: </p>
<hr />
<div>{{Infobox regtemp<br />
| Title = Orgone<br />
| Subgroups = 2.7.11<br />
| Comma basis = [[65536/65219]] (2.7.11)<br />
| Edo join 1 = 11 | Edo join 2 = 15<br />
| Mapping = 1; 3 -2<br />
| Ploidacot = trimech<br />
| Generators = 77/64 | Generators tuning = 323.3 | Optimization method = CWE<br />
| MOS scales = [[4L 3s]], [[4L 7s]], [[11L 4s]]<br />
}}<br />
'''Orgone''', 11 & 15, is a highly efficient temperament of the [[2.7.11 subgroup]], tempering out [[65536/65219]], such that three intervals of [[11/8]] reach the same point as two intervals of [[8/7]]; the generator is therefore (11/8)/(8/7) = [[77/64]], two of which stack to 16/11 and three of which stack to 7/4.<br />
[[File:Orgone7.png|thumb|Orgone heptatonic MOS]]<br />
[[26edo]] is a good tuning of Orgone; [[41edo]] is on the flatter end of the generator spectrum and [[37edo]] is on the sharper end.<br />
<br />
== Extensions ==<br />
'''Superkleismic''' is the primary extension of Orgone to the full 11-limit. The generator is interpreted as 6/5, which stacks twice to make 16/11 and three times to make 7/4 ([[keemic]] tempering, which is also supported by porcupine and flattone); the 7/4 itself stacks three times (octave-reduced) to reach 4/3 ([[slendric]] tempering). Superkleismic is an important temperament in 15edo, 26edo, and 41edo. It may also be valuable to consider the 2.(5/3).7.11 version, which is supported by 11edo (and thus 22edo).<br />
<br />
== Interval chain ==<br />
In the following table, odd harmonics and subharmonics 1–11 are in '''bold'''.<br />
{| class="wikitable right-1 right-2"<br />
|-<br />
! #<br />
! Cents<br />
! Approximate ratios<br />
!Superkleismic (2.3.5.7.11)<br />
|-<br />
| 0<br />
| 0<br />
| '''1/1'''<br />
|<br />
|-<br />
| 1<br />
| 322<br />
| 77/64<br />
|6/5<br />
|-<br />
| 2<br />
| 644<br />
| '''16/11'''<br />
|<br />
|-<br />
| 3<br />
| 966<br />
| '''7/4'''<br />
|<br />
|-<br />
| 4<br />
| 88<br />
| 128/121<br />
|21/20, 22/21<br />
|-<br />
| 5<br />
| 410<br />
| 14/11<br />
|<br />
|-<br />
| 6<br />
| 732<br />
| 49/32<br />
|32/21<br />
|-<br />
| 7<br />
| 1054<br />
| 224/121<br />
|11/6<br />
|-<br />
|8<br />
|176<br />
|...<br />
|10/9<br />
|-<br />
|9<br />
|498<br />
|<br />
|4/3<br />
|-<br />
|10<br />
|820<br />
|<br />
|8/5<br />
|-<br />
|11<br />
|1142<br />
|<br />
|48/25, 36/35, 33/32<br />
|-<br />
|12<br />
|264<br />
|<br />
|7/6<br />
|-<br />
|13<br />
|586<br />
|<br />
|7/5<br />
|}<br />
<br />
<br />
== List of patent vals ==<br />
:''Main article: [[Orgone/Patent vals]]''<br />
{{navbox regtemp}}<br />
{{cat|temperaments}}</div>Vectorhttps://xenreference.com/wiki/index.php?title=File:Orgone7.png&diff=7410File:Orgone7.png2026-06-02T02:12:03Z<p>Vector: </p>
<hr />
<div>Orgone7</div>Vectorhttps://xenreference.com/wiki/index.php?title=Neutral_temperaments&diff=7409Neutral temperaments2026-06-02T02:10:59Z<p>Vector: </p>
<hr />
<div>'''Neutral temperaments''' are any temperaments represented by the edo join 7 & 10, or any reasonable extension of such a temperament, such that the generator is a neutral third of some kind which splits 3/2 into two. They are a subset of and largely cover the ''dicot'' temperament archetype, and impose upon it the condition that the neutral third must be mapped to 2\7 and 3\10. The two most well-known neutral temperaments are the 2.3.11 (Rastmatic) and 2.3.5 (Dicot) versions.<br />
<br />
10edo is a contorted 5edo in 2.3.7, hence 7 & 10 in that subgroup represents monocot [[Archy]] temperament.<br />
<br />
== Notation ==<br />
Neutral temperaments may be notated with neutral chain-of-fifths notation. <br />
{| class="wikitable"<br />
|+<br />
!Note<br />
!24edo<br />
!Notation<br />
!2.3...<br />
!5 (Dicot)<br />
!11 (Rastmatic)<br />
!13 (Namo)<br />
!13-limit (no-fives)<br />
!13-limit<br />
|-<br />
|A<br />
|0<br />
|P1<br />
|1/1<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|At<br />
|50<br />
|sA1<br />
|<br />
|81/80<br />
|33/32<br />
|<br />
|<br />
|<br />
|-<br />
|Bb<br />
|100<br />
|m2<br />
|256/243<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|Bd<br />
|150<br />
|n2<br />
|<br />
|10/9, 16/15<br />
|12/11<br />
|<br />
|<br />
|<br />
|-<br />
|B<br />
|200<br />
|M2<br />
|9/8<br />
|<br />
|<br />
|<br />
|8/7<br />
|11/10<br />
|-<br />
|C<br />
|300<br />
|m3<br />
|32/27<br />
|<br />
|<br />
|<br />
|7/6<br />
|<br />
|-<br />
|Ct<br />
|350<br />
|n3<br />
|<br />
|5/4, 6/5<br />
|11/9<br />
|16/13<br />
|<br />
|<br />
|-<br />
|C#<br />
|400<br />
|M3<br />
|81/64<br />
|<br />
|<br />
|<br />
|9/7<br />
|<br />
|-<br />
|Dd<br />
|450<br />
|sd4<br />
|<br />
|<br />
|<br />
|<br />
|14/11<br />
|<br />
|-<br />
|D<br />
|500<br />
|P4<br />
|4/3<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|Dt<br />
|550<br />
|sA4<br />
|<br />
|27/20<br />
|11/8<br />
|18/13<br />
|<br />
|10/7<br />
|-<br />
|Ed<br />
|650<br />
|sd5<br />
|<br />
|40/27<br />
|16/11<br />
|13/9<br />
|<br />
|7/5<br />
|-<br />
|E<br />
|700<br />
|P5<br />
|3/2<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|Et<br />
|750<br />
|sA5<br />
|<br />
|<br />
|<br />
|<br />
|11/7<br />
|<br />
|-<br />
|F<br />
|800<br />
|m6<br />
|128/81<br />
|<br />
|<br />
|<br />
|14/9<br />
|<br />
|-<br />
|Ft<br />
|850<br />
|n6<br />
|<br />
|5/3, 8/5<br />
|18/11<br />
|13/8<br />
|<br />
|<br />
|-<br />
|F#<br />
|900<br />
|M6<br />
|27/16<br />
|<br />
|<br />
|<br />
|12/7<br />
|<br />
|-<br />
|G<br />
|1000<br />
|m7<br />
|16/9<br />
|<br />
|<br />
|<br />
|7/4<br />
|20/11<br />
|-<br />
|Gt<br />
|1050<br />
|n7<br />
|<br />
|15/8, 9/5<br />
|11/6<br />
|<br />
|<br />
|<br />
|-<br />
|G#<br />
|1100<br />
|M7<br />
|243/128<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|-<br />
|Ad<br />
|1150<br />
|sd8<br />
|<br />
|160/81<br />
|64/33<br />
|<br />
|<br />
|<br />
|-<br />
|A<br />
|1200<br />
|P8<br />
|2/1<br />
|<br />
|<br />
|<br />
|<br />
|<br />
|}<br />
These intervals may additionally be arranged on a chart which explains their mappings to 7edo and 10edo:<br />
{| class="wikitable"<br />
! !! 0\7 !! 1\7 !! 2\7 !! 3\7 !! 4\7 !! 5\7 !! 6\7 !! 7\7<br />
|-<br />
! scope="row" | 0\10 <br />
| '''P1''' || m2|| d3|| || || || ||<br />
|-<br />
! scope="row" | 1\10 <br />
| sA1|| '''n2''' || sd3|| || || || ||<br />
|-<br />
! scope="row" | 2\10 <br />
| A1|| M2 || m3 || d4|| || || ||<br />
|-<br />
! scope="row" | 3\10 <br />
| || sA2|| '''n3''' || sd4 || || || ||<br />
|-<br />
! scope="row" | 4\10 <br />
| || A2|| M3 || '''P4''' || d5|| d6|| ||<br />
|-<br />
! scope="row" | 5\10 <br />
| || || sA3|| sA4 || sd5 || sd6|| ||<br />
|-<br />
! scope="row" | 6\10 <br />
| || || A3|| A4|| '''P5''' || m6 || d7||<br />
|-<br />
! scope="row" | 7\10 <br />
| || || || || sA5 || '''n6''' || sd7||<br />
|-<br />
! scope="row" | 8\10 <br />
| || || || || A5|| M6 || m7 ||d8<br />
|-<br />
! scope="row" | 9\10 <br />
| || || || || || sA6|| '''n7''' ||sd8<br />
|-<br />
! scope="row" | 10\10 <br />
| || || || || || A6|| M7|| '''P8'''<br />
|}<br />
<br />
== Rastmatic ==<br />
[[File:Mohajira7.png|thumb|Mohajira heptatonic MOS]]<br />
Rastmatic is the neutral temperament in the 2.3.11 subgroup, which equates 11-limit neutral intervals to their exact neutral (2.sqrt(3/2)) counterparts. The generator represents both 11/9 and 27/22; this is one of the most accurate prime subgroups for the temperament. Because the temperament tempers out 243/242, it is a rastmic temperament, hence "rastmic" may be used informally to refer to Rastmatic or its scales. The generator is best tuned around 350 cents, about 3 cents off from the justly tuned 11/9 and 4 cents off from just 27/22. Mohajira extends rastmatic to 2.3.5.11.<br />
<br />
=== Etymology ===<br />
Rastmatic is named after the rastma, the comma it tempers out, which is in turn named after the maqam ''Rast'' which utilizes a scale with several neutral intervals.<br />
<br />
== Dicot ==<br />
Dicot<ref group="note>The name ''Interpental'' has been proposed, however it currently is used by 43 & 53, a weak extension of [[Buzzard]].</ref>, not to be confused with the dicot archetype as a whole, is the neutral temperament in the 2.3.5 subgroup. an exotemperament that can be defined to temper out [[25/24]], the Dicot comma. The provided [[edo join]] also tempers out [[45/44]] and [[64/63]] in the 11-limit, representing the extension '''Dichotic''' and also tempering out [[55/54]]. Alternative extensions include 4 & 7 (which conflates 9/7~7/6~6/5~5/4). 7 & 10 and 10 & 17 are both reasonable edo joins, suggesting Dicot as a 3-, 7-, or 10-form temperament.<br />
<br />
Dicot makes 4:5:6 equidistant, suggesting the simplified structure of [[tertian]] harmony, the same way [[Semaphore]] does for [[chthonic harmony]]. As a result, the temperament archetype [[Neutral third scales|dicot]] is named after it.<br />
<br />
=== Etymology ===<br />
Dicot originates from the term "dicot" in botany, referring to plants with two embryonic leaves, perhaps by analogy with 3/2 being split into two generators. The name ''Dicot'' would also inspire [[Tetracot]], [[Alphatricot]], and by extension the [[ploidacot]] temperament archetype naming system as a whole.<br />
<br />
=== Tuning considerations ===<br />
A perfect ~351c tuning of the generator, while useful for understanding tertian harmony and suggested by some temperament tuning optimization systems, does not reasonably approximate either 5/4 or 6/5. The optimal tunings of Dicot are roughly bimodal, with ~360c (around 10edo) and ~343c (around 7edo) both being better tunings.<br />
<br />
=== History ===<br />
According to Unque, dicot temperament was discussed by Vicentino as a descriptive model of vocal music; the reason it is considered an exotemperament today is, if not members of the xenharmonic community tending to be sensitive to detuning, likely at least partially the fact that the tonality system motivates a distinction between 4:5:6 and 10:12:15. (Note that the tuning error on 5/4 in 10edo is not much greater than the error on 3/2 in the same tuning.)<br />
<br />
== Namo ==<br />
[[File:Suhajira7.png|thumb|Suhajira heptatonic MOS]]<br />
''Namo'', ''Intertridecimal'', or ''Harmoneutral'' is the temperament of 512/507, which is 7 & 10 in the 2.3.13 subgroup. It prefers a sharp tuning of the fifth.<br />
<br />
It is often framed as a (somewhat inaccurate) extension to Rastmatic. Its generator is best tuned around 355c.<br />
<br />
Namo extends to 7 with the temperament Suhajira, which takes the fifth as an [[archy]] fifth.<br />
<br />
== Patent vals ==<br />
<br />
=== List of patent vals ===<br />
{| class="wikitable"<br />
|+<br />
!EDO<br />
!Mappings supported<br />
!Generator tuning<br />
!3/2 tuning<br />
|-<br />
|10<br />
|5, 13, 11<br />
|360.0c<br />
|720.0c<br />
|-<br />
|37<br />
|13<br />
|356.8c<br />
|713.5c<br />
|-<br />
|27<br />
|13<br />
|355.6c<br />
|711.1c<br />
|-<br />
|71<br />
|13<br />
|354.9c<br />
|709.9c<br />
|-<br />
|44<br />
|13<br />
|354.5c<br />
|709.1c<br />
|-<br />
|61<br />
|13<br />
|354.1c<br />
|708.2c<br />
|-<br />
|78<br />
|13<br />
|353.8c<br />
|707.7c<br />
|-<br />
|95<br />
|13<br />
|353.7c<br />
|707.4c<br />
|-<br />
|17<br />
|5, 13, 11<br />
|352.9c<br />
|705.9c<br />
|-<br />
|75<br />
|13<br />
|352.0c<br />
|704.0c<br />
|-<br />
|58<br />
|13, 11<br />
|351.7c<br />
|703.4c<br />
|-<br />
|41<br />
|13, 11<br />
|351.2c<br />
|702.4c<br />
|-<br />
|147<br />
|11<br />
|351.0c<br />
|702.0c<br />
|-<br />
|106<br />
|11<br />
|350.9c<br />
|701.9c<br />
|-<br />
|171<br />
|11<br />
|350.9c<br />
|701.8c<br />
|-<br />
|65<br />
|13, 11<br />
|350.8c<br />
|701.5c<br />
|-<br />
|219<br />
|11<br />
|350.7c<br />
|701.4c<br />
|-<br />
|154<br />
|11<br />
|350.6c<br />
|701.3c<br />
|-<br />
|243<br />
|11<br />
|350.62c<br />
|701.23c<br />
|-<br />
|332<br />
|11<br />
|350.60c<br />
|701.20c<br />
|-<br />
|89<br />
|11<br />
|350.56c<br />
|701.12c<br />
|-<br />
|380<br />
|11<br />
|350.53c<br />
|701.05c<br />
|-<br />
|291<br />
|11<br />
|350.52c<br />
|701.03c<br />
|-<br />
|202<br />
|11<br />
|350.50c<br />
|700.99c<br />
|-<br />
|517<br />
|11<br />
|350.48c<br />
|700.97c<br />
|-<br />
|315<br />
|11<br />
|350.48c<br />
|700.95c<br />
|-<br />
|428<br />
|11<br />
|350.47c<br />
|700.93c<br />
|-<br />
|541<br />
|11<br />
|350.46c<br />
|700.92c<br />
|-<br />
|113<br />
|11<br />
|350.44c<br />
|700.88c<br />
|-<br />
|476<br />
|11<br />
|350.42c<br />
|700.84c<br />
|-<br />
|363<br />
|11<br />
|350.41c<br />
|700.83c<br />
|-<br />
|250<br />
|11<br />
|350.40c<br />
|700.80c<br />
|-<br />
|387<br />
|11<br />
|350.39c<br />
|700.78c<br />
|-<br />
|137<br />
|11<br />
|350.36c<br />
|700.73c<br />
|-<br />
|435<br />
|11<br />
|350.34c<br />
|700.69c<br />
|-<br />
|298<br />
|11<br />
|350.34c<br />
|700.67c<br />
|-<br />
|459<br />
|11<br />
|350.33c<br />
|700.65c<br />
|-<br />
|161<br />
|11<br />
|350.31c<br />
|700.62c<br />
|-<br />
|346<br />
|11<br />
|350.29c<br />
|700.58c<br />
|-<br />
|185<br />
|11<br />
|350.27c<br />
|700.54c<br />
|-<br />
|394<br />
|11<br />
|350.25c<br />
|700.51c<br />
|-<br />
|209<br />
|11<br />
|350.24c<br />
|700.48c<br />
|-<br />
|233<br />
|11<br />
|350.21c<br />
|700.43c<br />
|-<br />
|257<br />
|11<br />
|350.19c<br />
|700.39c<br />
|-<br />
|281<br />
|11<br />
|350.18c<br />
|700.36c<br />
|-<br />
|305<br />
|11<br />
|350.16c<br />
|700.33c<br />
|-<br />
|329<br />
|11<br />
|350.15c<br />
|700.30c<br />
|-<br />
|353<br />
|11<br />
|350.14c<br />
|700.28c<br />
|-<br />
|24<br />
|13, 11<br />
|350.00c<br />
|700.00c<br />
|-<br />
|247<br />
|11<br />
|349.80c<br />
|699.60c<br />
|-<br />
|223<br />
|11<br />
|349.78c<br />
|699.55c<br />
|-<br />
|199<br />
|11<br />
|349.7c<br />
|699.5c<br />
|-<br />
|175<br />
|11<br />
|349.7c<br />
|699.4c<br />
|-<br />
|151<br />
|11<br />
|349.7c<br />
|699.3c<br />
|-<br />
|127<br />
|11<br />
|349.6c<br />
|699.2c<br />
|-<br />
|103<br />
|11<br />
|349.5c<br />
|699.0c<br />
|-<br />
|79<br />
|11<br />
|349.4c<br />
|698.7c<br />
|-<br />
|55<br />
|13, 11<br />
|349.1c<br />
|698.2c<br />
|-<br />
|31<br />
|13, 11<br />
|348.4c<br />
|696.8c<br />
|-<br />
|38<br />
|13, 11<br />
|347.4c<br />
|694.7c<br />
|-<br />
|45<br />
|13<br />
|346.7c<br />
|693.3c<br />
|-<br />
|7<br />
|5, 13, 11<br />
|342.9c<br />
|685.7c<br />
|}<br />
<br />
== Other temperaments with neutral third generators ==<br />
<br />
=== Hemififths ===<br />
Hemififths, 41 & 58, is a temperament in the 2.3.5.7 subgroup which tempers out 2401/2400 = S49 (aka the breedsma) equating 49/40 to its 3/2-complement and additionally tempers out 5120/5103 making it an [[aberschismic]] temperament. Note that 7 & 10 in 2.3.5.7 is [[dichotic]].<br />
<br />
== Footnotes ==<br />
<references group="note"/><br />
<br />
{{Navbox regtemp}}</div>Vectorhttps://xenreference.com/wiki/index.php?title=File:Mohajira7.png&diff=7408File:Mohajira7.png2026-06-02T02:10:39Z<p>Vector: </p>
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<div>Mohajira7</div>Vectorhttps://xenreference.com/wiki/index.php?title=File:Suhajira7.png&diff=7407File:Suhajira7.png2026-06-02T02:09:24Z<p>Vector: </p>
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<div>Suhajira7</div>Vectorhttps://xenreference.com/wiki/index.php?title=Archy&diff=7406Archy2026-06-02T02:07:15Z<p>Vector: </p>
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<div>[[File:Archy5.png|thumb|The archy pentatonic scale. The step sizes are 7/6 and 8/7.]]<br />
'''Archy''' (22 & 27) is the temperament that tempers out the '''archytas comma,''' 64/63, equating [[2.3.7 subgroup|septal]] intervals with nearby [[Pythagorean tuning|diatonic]] ones. In Archy, the generator is a fourth, the period is an octave, and 2 flattened [[Perfect fourth|fourths]] of about 490 cents stack to a sharply tuned [[Septal subminor seventh|7th harmonic]] of about 980 cents. Equivalently, the pythagorean (9/8) major second is mapped to the same pitch as the septimal (8/7) major second.<br />
<br />
Archy is usually tuned such that the subminor (7/6) third is close to accurately tuned; flatter tunings of the fourth lead to more accurate tunings of the 7th harmonic, at the cost of the usability of the diatonic scale. The tuning that justly tunes the harmonic seventh places the perfect fourth at 484.4 cents, which leads to a diatonic scale with a small step at the uncomfortable size of 22 cents. <br />
<br />
As a monocot temperament (a temperament generated by a perfect fourth or fifth), Archy can be notated with standard [[Diatonic notation|diatonic]] notation. However, this is somewhat awkward, as Archy is more cleanly analyzed as a 5-form temperament, producing an [[equipentatonic]] scale, so perhaps diamond-MOS or KISS notation with [[pentic]] would be better suited for it.<br />
<br />
== Structural theory ==<br />
<br />
=== Extensions ===<br />
The following are extensions to prime 5 (i.e. ways to map intervals involving prime 5 onto the existing structure of 2.3.7 archy).<br />
<br />
Archy can be defined as 5 & 7, which is the [[meantone]] extension ''dominant'' in the full 7-limit. Other extensions follow:<br />
<br />
==== 5/4 as limma-flat major third (22 & 27), often called "Superpyth" ====<br />
The canonical extension, equates 5/4 with the diatonic augmented second, or an octave-reduced stack of 9 fifths, which can be seen in the 5-form as a major third flattened by a diatonic semitone representing the [[septimal quartertone]] (36/35, the interval between 5/4 and 9/7) and the [[Meantone|syntonic comma]]. It can be seen as the 22 & 27 temperament. The preferred tuning range for the fifth in this extension tends to be somewhat flatter than that of archy; tunings where both the supermajor (9/7) and subminor (7/6) thirds are somewhat accurate are preferred. It is a 5-cluster temperament, as indicated by the edo join (27 - 22 = 5).<br />
<br />
==== 5/4 as doubly limma-flat major third (5 & 37) ====<br />
This is an alternative extension, best tuned at least as sharp as [[32edo]]. Instead of flattening the major third by a diatonic semitone to reach the 5th harmonic, you flatten by two diatonic semitones. In diatonic notation, this means that 5/4 is the double-augmented unison.<br />
<br />
This range is often interpreted as Oceanfront or Ultrapyth, equating the diatonic major third (already interpreted as 9/7) to 13/10. A recommendable tuning for this temperament is [[37edo]].<br />
<br />
=== Machine temperament ===<br />
'''Machine''', 11 & 17, is a weak restriction to 2.9.7.11 of Supra, the [17 & 22] subrange of Archy. It is generated by 9/8~8/7, three of which make a 16/11. It's straddle-3 in that the ~14/9 is the >3 and the ~16/11 is the <3.<br />
<br />
* [[11edo]] nearly divides 9/7 in half<br />
* [[17edo]] provides a near-isodifferential tuning of ~8:11:14, actually much closer to 13:18:23<br />
* [[28edo]] (generator 5\28, 214.3c) provides a near-isodifferential ~7:9:11, which is actually much closer to 32:41:50.<br />
<br />
Machine generates machinoid (5L1s) and 6L5s.<br />
<br />
== Compositional theory ==<br />
<br />
=== Chords ===<br />
In Archy, the diatonic major and minor chords essentially have their roles swapped from in meantone, as they now represent the [[Collection of chords|supermajor triad]] and [[Collection of chords|subminor triad]] respectively, and the minor chord is the more stable of the two. This can be seen by how the supermajor third is, in the 5-form, a flat fourth, serving a somewhat similar role to the diminished fifth in diatonic. The triad [0 4/3 7/4~14/9] is an important [[Collection of chords#Essentially tempered chords|essentially tempered chord]], although HKM finds that its other closed-voice inversions do not sound as if they contain septimal intervals unless the fifth is tuned as sharp as that of 37edo.<br />
<br />
Due to existing in 2.3.7, Archy also supports the latal triads (bounded by a fourth, made from intervals near 250c, like 6:7:8), with 1/1-8/7-4/3 in particular appearing as part of the suspended tetrad. <br />
<br />
=== Full 7-limit harmony ===<br />
''This section assumes a reasonably accurate extension to prime 5 is used. The precise extension does not particularly matter, but an up / down symbol represents the difference between 9/7 and 5/4.'' <br />
<br />
''The primary characteristc'' of archy in a full 7-limit context is that a zarlino dominant chord (found in the zarlino tuning of Mixolydian) is a 4:5:6:7 harmonic seventh chord.<br />
<br />
==== Leading tones ====<br />
The semitones found in archy's MOS diatonic are too narrow to use as leading tones. The nearminor seconds provided by a 5-limit extension may be seen as too wide (usually exceeding the "optimal" size of a leading tone presented by George Secor at 70 cents, depending on the tuning). However, they align with Aura's system of functional harmony, which places the 70-cent leading tone at the intersection of two other functional categories at around 110 cents and 50 cents respectively - the collocant and gradient functions. The collocant functions as a conventional leading tone, whereas the gradient functions as a passing tone to either jump past the tonic or resolve to the collocant. In this case, the larger nearminor second represents the collocant, meanwhile the smaller subminor second represents the gradient.<br />
<br />
==== Further functional harmony ====<br />
There are four distinct "keys" in the 7-limit (nearmajor, supermajor, nearminor, subminor), as compared to two in MOS diatonic alone, where a key is defined as a system of tonal hierarchy based around a certain interval quality or tonic chord (independent of absolute pitch), which will be elaborated on below. Note that relative major or minor depends on whether the key is near- or super/sub, and that, for instance, nearmajor and supermajor use different scales that are not rotations of one another. In specific, using ups and downs notation, C Nearmajor corresponds to vA Nearminor, meanwhile C Supermajor corresponds to A Subminor, and in general nearmajor-nearminor relative correspondences acquire an additional down accidental compared to standard MOSdiatonic correspondences.<br />
<br />
The chirality of the nearmajor or nearminor scale in question is ultimately of little relevance (see [[blackdye]]; in short, the major second in nearmajor (and the fourth in nearminor) may be either note depending on context), but in general the right-handed version of nearmajor is assumed due to having a non-wolf V chord, and the left-handed version of nearminor is assumed due to having a non-wolf fourth over the tonic.<br />
<br />
The heptatonic interval functions remain as they are in 12edo, although with the caveat that the ideal leading tone ends up at the nearminor second rather than the semitone found in MOSdiatonic, which has implications for the subminor and supermajor keys and turns the use of the diatonic scale into a balancing act between the functional utility of MOSdiatonic and the tension of the leading tones in zarlino diatonic. (In particular, it suggests the use of a "harmonic supermajor" by flattening the seventh of supermajor by an edostep.)<br />
<br />
==== Nearmajor key ====<br />
[[File:Nearmajor.mp3|thumb|Natural nearmajor scale and tonic chord]]<br />
In nearmajor (the key with the nearmajor tonic chord), the fourth acts as it usually does in MOS major, serving as a tendency tone towards the third. The basic tonal identity for nearmajor is 4:5:6, which extends generally to a nearmajor seventh chord, although a dominant (harmonic in archy temperament) seventh is also possible, and more justified in archy due to naturally extending the harmonic series segment corresponding to 4:5:6.<br />
<br />
==== Nearminor key ====<br />
[[File:Nearminor.mp3|thumb|Natural nearminor scale and tonic chord]]<br />
Nearminor harmony functions somewhat similarly to how you expect, with the nearminor sixth functioning as a leading tone down to the fifth and the seventh being able to be raised to a nearmajor seventh in order to give a more directed dominant resolution. The whole tone also provides a lead up to the minor third, like in standard diatonic.<br />
<br />
Melodic minor scales are somewhat interesting here as well, as there are a couple different reasonable ways to construct them, which would likely depend on the chords being used and the desired melodic contour.<br />
<br />
==== Supermajor key ====<br />
[[File:Supermajor.mp3|thumb|Natural supermajor scale and tonic chord]]<br />
In supermajor, a lead to the third would be a wolf fourth (11/8), perhaps justifying its inclusion in the scale over the fourth proper, or the functional alternation between the two in different contexts.<br />
<br />
The functionality of the seventh grows increasingly complicated in supermajor - while in 12edo, one may only see, for instance, a dominant chord replacing the I chord, in the 7-limit there are four different potential types of seventh, all with justifications. A fifth over the third would be a supermajor seventh (notably serving as the MOSdiatonic maj7, and distinguishing itself from the 12edo maj7 by not leading up to its own root), a tritone (neardim 5; 7/5) over the third would be a nearminor seventh, a lead up to the tonic would be a nearmajor seventh, and finally the MOS diatonic dominant chord utilizes a subminor seventh. Therefore, an alternate version of the supermajor scale usable in certain contexts makes the fourth wolf and the seventh nearmajor.<br />
<br />
This also means that the regular perfect fourth isn't as unstable an interval or as functionally dissonant in supermajor - in fact, the third is actually somewhat of a tension compared to it (though the step between them is smaller than the size of a conventional leading tone).<br />
<br />
==== Subminor key ====<br />
The same kind of justification emerges for harmonic subminor, except that there is little reason to alter the seventh all the way up to a supermajor seventh if the objective is for it to function as a leading tone. In fact, the same logic can be used against a dominant chord with a nearminor seventh in nearmajor - leading inwards to a nearmajor third by equal semitones on either side requires that the initial interval be a neardiminished fifth, and that the chord to be used as a dominant is actually a harmonic 4:5:6:7 on the fifth. (Resolving to a supermajor chord actually wants a dom7 with a nearmajor third and nearminor seventh, if quartertones are not to be used).<br />
[[File:Subminor.mp3|thumb|Natural subminor scale and tonic chord]]<br />
In general, archy's functional harmony ends up a lot more context-bound and much less scale-bound than 12edo's, due to the multiple different qualities of intervals and notes doing different things, and the ideal leading tone not matching the standard diatonic structure.<br />
<br />
==== Alternative leading tones ====<br />
An alternative approach to simplify things is instead to discard Aura's theory of leading in favor of treating the quartertone as the optimal leading tone (as it is the diatonic major seventh), an entirely different paradigm emerges. Supermajor and subminor become definitive, stable diatonic tonality systems, with no awkwardness around leading tones, behaving identically to any MOSdiatonic temperament (albeit with the different, somewhat inverted "moods" presented by the supermajor and subminor intervals). Meanwhile, the nearmajor and nearminor scales acquire new "harmonic" variations, with the final note raised up to a quartertone below the tonic. In effect, supermajor/subminor and nearmajor/nearminor "switch" in regards to some functions. Instead of raising the fourth in supermajor, it is in this system viable to lower it in nearmajor. The best dom7 to resolve to a nearmajor triad on the tonic features a seventh lowered to one step below the subminor seventh, alongside the supermajor third, and can consequently be reanalyzed as a subminor seventh chord on the 9/7 over the tonic. The MOSdiatonic dominant seventh serves to resolve to a MOSdiatonic major triad, as in 12edo.<br />
<br />
==== Consonant vs. tense suspended chords ====<br />
The wider supermajor second and contrast with the supermajor third actually makes suspended chords somewhat of a point of resolution, rather than a point of tension like in 12edo. It's reasonable to have a suspended chord that doesn't resolve, perhaps making the term "suspended" inaccurate. These suspended chords can function like arto and tendo chords, with a 1-2-4-5 chord structure being plausible, or can be used in modal harmony as a form of "mode-agnostic" anchor point. The sus4 chord in particular is composed of the three octave-reduced perfect consonances, and thus can also be considered the most basic [[Tetrachord|polychordal scale]] (perhaps a/the "dichordal" scale). However, suspensions that function more like 12edo ones in leading into the MOS diatonic intervals and being more tense can still be found with the ''nearmajor'' sus2 and ''wolf'' sus4, which lose some of the structural elegance of standard Pythagorean suspensions in favor of a more tense, crowded sound that can easily resolve to even the rather tense supermajor triad.<br />
<br />
== Scales ==<br />
<br />
=== Superpyth diatonic ===<br />
This is the diatonic scale most directly analogous in structure to the 12edo diatonic, given its MOS form. Advantages of using it include the fact that all steps are what they appear to be - for example, D-G and G-C are both perfect fifths - and that it appears as a subset of the chain of fifths itself. One key difference is that the major and minor thirds do not get mapped to the expected 5-limit interpretations, but rather to the supermajor and subminor thirds of porcupine. A downside, or more generally a significant awkwardness, to using this system is the fact that due to the minor second being so small, the chromatic semitone is massive - closer to a whole tone than a proper semitone.<br />
<br />
It may be useful, perhaps for [[extraclassical tonality]], to use the full 12-note form of superpyth's scale.<br />
<br />
=== Superpyth pentatonic ===<br />
<br />
=== Zarlino ===<br />
''Similar to the previous section, this assumes a reasonably accurate extension to prime 5.''<br />
<br />
== List of patent vals ==<br />
{| class="wikitable"<br />
|+<br />
!EDO<br />
!Extension to 5<br />
!Generator tuning<br />
!Diatonic scale hardness<br />
!7/4 tuning<br />
|-<br />
|3<br />
|<br />
|800c<br />
|N/A<br />
|800c<br />
|-<br />
|8<br />
|<br />
|450c<br />
|N/A<br />
|900c<br />
|-<br />
|13<br />
|<br />
|461.5c<br />
|N/A<br />
|923.1c<br />
|-<br />
|5<br />
|[5 & 37], [22 & 27], dominant<br />
|480c<br />
|∞ (collapsed)<br />
|960c<br />
|-<br />
|42<br />
|[5 & 37]<br />
|485.7c<br />
|8<br />
|971.4c<br />
|-<br />
|37<br />
|[5 & 37]<br />
|486.5c<br />
|7<br />
|973c<br />
|-<br />
|32<br />
|[5 & 37]<br />
|487.5c<br />
|6<br />
|975c<br />
|-<br />
|59<br />
|<br />
|488.1c<br />
|5.5<br />
|976.3c<br />
|-<br />
|27<br />
|[22 & 27]<br />
|488.9c<br />
|5<br />
|977.8c<br />
|-<br />
|49<br />
|[22 & 27]<br />
|489.8c<br />
|4.5<br />
|979.6c<br />
|-<br />
|22<br />
|[22 & 27]<br />
|490.9c<br />
|4<br />
|981.8c<br />
|-<br />
|17<br />
|[22 & 27]<br />
|494.1c<br />
|3<br />
|988.2c<br />
|-<br />
|12<br />
|dominant<br />
|500c<br />
|2<br />
|1000c<br />
|-<br />
|7<br />
|dominant<br />
|514.3c<br />
|1 (equalized)<br />
|1028.6c<br />
|-<br />
|2<br />
|<br />
|600c<br />
|N/A<br />
|1200c<br />
|}<br />
<br />
{{cat|Temperaments}}{{Navbox regtemp}}</div>Vectorhttps://xenreference.com/wiki/index.php?title=File:Archy5.png&diff=7405File:Archy5.png2026-06-02T02:06:59Z<p>Vector: </p>
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<div>Archy5</div>Vectorhttps://xenreference.com/wiki/index.php?title=Gentle_tuning&diff=7404Gentle tuning2026-06-02T02:06:04Z<p>Vector: </p>
<hr />
<div>[[File:Gentle12.png|thumb|The chromatic MOS of a gentle-fifth temperament. Note the two step sizes.]]<br />
'''Gentle tuning''' (aka the '''gentle region''' of fifths) is the regular tuning of the perfect fifth somewhere between the tunings given by 29edo and 17edo (~703.4 to 705.9 cents), producing MOS [[diatonic]] scales with step size ratios between 5:2 and 3:1 (and consequently [[chromatic]] scales between 3:2 and 2:1, including the [[Golden generator|golden tuning]] of p-chromatic). As a result, the [[major third]] generated is between 413.8 and 423.5 cents, and approximates 14/11. It is so named because melodically it functions as a "gentle" version of [[Superpyth]]; conversely, it can instead be seen as exaggerating the characteristics of Pythagorean tuning as it compares with 12-tone equal temperament and is usable as a replacement for Pythagorean tuning. One notable characteristic of gentle tuning is that it narrows the diatonic semitone towards a size of 70 cents, described by some theorists as an ideal leading tone; Aura, however, considers it to be between a leading tone and a diesis or quarter tone in function, being able to either lead to the tonic or "skip over" it.<br />
<br />
== Regular temperament theory ==<br />
Gentle tuning is associated with the 2.3.(11/7).(13/7) temperament, 29 & 46, that finds 14/11 at the major third and 13/11 at the minor third, which has a number of names depending on the subgroup (see [[#Names]]). Therefore, the chromatic semitone is equated to 14/13 and the diatonic semitone is equated to 22/21; the whole tone is equated to 44/39.<br />
<br />
This implies the use of these thirds as elements of 13-limit harmony over the 11th and 13th harmonics themselves, similar to, but more intense and approach-specific than, the prioritization of 9/7 and 7/6 in Archy (and in septal harmony as a whole) as opposed to 7/4; both approaches are patterned off of [[Meantone]], which is often described outside of a regular temperament context as tempering the fifth to bring the major and minor third more in-tune with 5/4 and 6/5, where 5/4 is simply treated as "the major third" rather than a generator of just intonation in its own right.<br />
<br />
To extend to the 2.3.7.11.13 subgroup, 7/6 is equated to two chromatic semitones (so that 7/4 is C-G##). As a result, 11/8 is placed at the augmented third and 13/8 at the augmented fifth (this is the same interval that becomes 8/5 in [[Schismic]]). <br />
<br />
For flatter tunings (flat of 46edo), 5/4 is equated to three chromatic semitones (so that a major triad is C-C###-G). There is no stable mapping of 5 for sharper tunings due to 17edo's inaccuracy in approximating prime 5. This is similar to [[Archy]] tunings approaching 5edo.<br />
<br />
==== Chords ====<br />
While it is not particularly simple in just intonation, the primary chords implied by the temperament's structure are the major 22:28:33 and minor 22:26:33 which are complements of one another and also the diatonic major and minor chords (so that their union, P1-m3-M3-P5, is essentially tempered). Standard 5- and 7-limit chords are somewhat unviable when using standard diatonic logic, as a 4:5:6:7 chord is C-C###-G-G##, with all four notes overlapping on two nominals, and the supermajor and subminor triads are C-C##-G and C-Gbb-G, again overlapping.<br />
<br />
==== Names ====<br />
The most common name for this temperament, applying in the strict sense to the full 2.3.5.7.11.13 temperament, is ''Leapday.'' As a 2.3.(11/7).(13/7) temperament, however, the technical name is ''Pepperoni'' and as a 2.3.7.11.13 temperament the technical name is ''Leapfrog.'' Other names that may be used for this temperament or for gentle tuning as a whole are ''neogothic'' (in reference to the quality of the generated thirds), ''[[Parapyth]]'' (technically referring to a rank-3 temperament whose 3/2 generates Pepperoni), and simply ''gentle.''<br />
<br />
== Interval chain ==<br />
{| class="wikitable"<br />
|+<br />
! colspan="3" |Generators up<br />
! colspan="3" |Generators down<br />
|-<br />
!#<br />
!Cents<br />
!Regular interpretation (2.3.7.11.13)<br />
!#<br />
!Cents<br />
!Regular interpretation (2.3.7.11.13)<br />
|-<br />
|0<br />
|0.00<br />
|1/1<br />
|0<br />
|1,200.00<br />
|2/1<br />
|-<br />
|1<br />
|704.35<br />
|3/2<br />
|1<br />
|495.65<br />
|4/3<br />
|-<br />
|2<br />
|208.70<br />
|9/8<br />
|2<br />
|991.30<br />
|16/9<br />
|-<br />
|3<br />
|913.04<br />
|22/13<br />
|3<br />
|286.96<br />
|13/11<br />
|-<br />
|4<br />
|417.39<br />
|14/11<br />
|4<br />
|782.61<br />
|11/7<br />
|-<br />
|5<br />
|1,121.74<br />
|21/11<br />
|5<br />
|78.26<br />
|22/21<br />
|-<br />
|6<br />
|626.09<br />
|13/9<br />
|6<br />
|573.91<br />
|18/13<br />
|-<br />
|7<br />
|130.43<br />
|14/13<br />
|7<br />
|1,069.57<br />
|13/7<br />
|-<br />
|8<br />
|834.78<br />
|13/8<br />
|8<br />
|365.22<br />
|16/13<br />
|-<br />
|9<br />
|339.13<br />
|11/9<br />
|9<br />
|860.87<br />
|18/11<br />
|-<br />
|10<br />
|1,043.48<br />
|11/6, 20/11<br />
|10<br />
|156.52<br />
|11/10, 12/11<br />
|-<br />
|11<br />
|547.83<br />
|11/8, 15/11<br />
|11<br />
|652.17<br />
|16/11, 22/15<br />
|-<br />
|12<br />
|52.17<br />
|33/32, 40/39<br />
|12<br />
|1,147.83<br />
|64/33, 39/20<br />
|-<br />
|13<br />
|756.52<br />
|14/9, 20/13<br />
|13<br />
|443.48<br />
|9/7, 13/10<br />
|-<br />
|14<br />
|260.87<br />
|7/6<br />
|14<br />
|939.13<br />
|12/7<br />
|-<br />
|15<br />
|965.22<br />
|7/4<br />
|15<br />
|234.78<br />
|8/7<br />
|}<br />
{{Navbox regtemp}}</div>Vectorhttps://xenreference.com/wiki/index.php?title=Gentle_tuning&diff=7403Gentle tuning2026-06-02T02:05:42Z<p>Vector: </p>
<hr />
<div>[[File:Gentle12.png|thumb|The chromatic MOS of a gentle-fifth temperament.]]<br />
'''Gentle tuning''' (aka the '''gentle region''' of fifths) is the regular tuning of the perfect fifth somewhere between the tunings given by 29edo and 17edo (~703.4 to 705.9 cents), producing MOS [[diatonic]] scales with step size ratios between 5:2 and 3:1 (and consequently [[chromatic]] scales between 3:2 and 2:1, including the [[Golden generator|golden tuning]] of p-chromatic). As a result, the [[major third]] generated is between 413.8 and 423.5 cents, and approximates 14/11. It is so named because melodically it functions as a "gentle" version of [[Superpyth]]; conversely, it can instead be seen as exaggerating the characteristics of Pythagorean tuning as it compares with 12-tone equal temperament and is usable as a replacement for Pythagorean tuning. One notable characteristic of gentle tuning is that it narrows the diatonic semitone towards a size of 70 cents, described by some theorists as an ideal leading tone; Aura, however, considers it to be between a leading tone and a diesis or quarter tone in function, being able to either lead to the tonic or "skip over" it.<br />
<br />
== Regular temperament theory ==<br />
Gentle tuning is associated with the 2.3.(11/7).(13/7) temperament, 29 & 46, that finds 14/11 at the major third and 13/11 at the minor third, which has a number of names depending on the subgroup (see [[#Names]]). Therefore, the chromatic semitone is equated to 14/13 and the diatonic semitone is equated to 22/21; the whole tone is equated to 44/39.<br />
<br />
This implies the use of these thirds as elements of 13-limit harmony over the 11th and 13th harmonics themselves, similar to, but more intense and approach-specific than, the prioritization of 9/7 and 7/6 in Archy (and in septal harmony as a whole) as opposed to 7/4; both approaches are patterned off of [[Meantone]], which is often described outside of a regular temperament context as tempering the fifth to bring the major and minor third more in-tune with 5/4 and 6/5, where 5/4 is simply treated as "the major third" rather than a generator of just intonation in its own right.<br />
<br />
To extend to the 2.3.7.11.13 subgroup, 7/6 is equated to two chromatic semitones (so that 7/4 is C-G##). As a result, 11/8 is placed at the augmented third and 13/8 at the augmented fifth (this is the same interval that becomes 8/5 in [[Schismic]]). <br />
<br />
For flatter tunings (flat of 46edo), 5/4 is equated to three chromatic semitones (so that a major triad is C-C###-G). There is no stable mapping of 5 for sharper tunings due to 17edo's inaccuracy in approximating prime 5. This is similar to [[Archy]] tunings approaching 5edo.<br />
<br />
==== Chords ====<br />
While it is not particularly simple in just intonation, the primary chords implied by the temperament's structure are the major 22:28:33 and minor 22:26:33 which are complements of one another and also the diatonic major and minor chords (so that their union, P1-m3-M3-P5, is essentially tempered). Standard 5- and 7-limit chords are somewhat unviable when using standard diatonic logic, as a 4:5:6:7 chord is C-C###-G-G##, with all four notes overlapping on two nominals, and the supermajor and subminor triads are C-C##-G and C-Gbb-G, again overlapping.<br />
<br />
==== Names ====<br />
The most common name for this temperament, applying in the strict sense to the full 2.3.5.7.11.13 temperament, is ''Leapday.'' As a 2.3.(11/7).(13/7) temperament, however, the technical name is ''Pepperoni'' and as a 2.3.7.11.13 temperament the technical name is ''Leapfrog.'' Other names that may be used for this temperament or for gentle tuning as a whole are ''neogothic'' (in reference to the quality of the generated thirds), ''[[Parapyth]]'' (technically referring to a rank-3 temperament whose 3/2 generates Pepperoni), and simply ''gentle.''<br />
<br />
== Interval chain ==<br />
{| class="wikitable"<br />
|+<br />
! colspan="3" |Generators up<br />
! colspan="3" |Generators down<br />
|-<br />
!#<br />
!Cents<br />
!Regular interpretation (2.3.7.11.13)<br />
!#<br />
!Cents<br />
!Regular interpretation (2.3.7.11.13)<br />
|-<br />
|0<br />
|0.00<br />
|1/1<br />
|0<br />
|1,200.00<br />
|2/1<br />
|-<br />
|1<br />
|704.35<br />
|3/2<br />
|1<br />
|495.65<br />
|4/3<br />
|-<br />
|2<br />
|208.70<br />
|9/8<br />
|2<br />
|991.30<br />
|16/9<br />
|-<br />
|3<br />
|913.04<br />
|22/13<br />
|3<br />
|286.96<br />
|13/11<br />
|-<br />
|4<br />
|417.39<br />
|14/11<br />
|4<br />
|782.61<br />
|11/7<br />
|-<br />
|5<br />
|1,121.74<br />
|21/11<br />
|5<br />
|78.26<br />
|22/21<br />
|-<br />
|6<br />
|626.09<br />
|13/9<br />
|6<br />
|573.91<br />
|18/13<br />
|-<br />
|7<br />
|130.43<br />
|14/13<br />
|7<br />
|1,069.57<br />
|13/7<br />
|-<br />
|8<br />
|834.78<br />
|13/8<br />
|8<br />
|365.22<br />
|16/13<br />
|-<br />
|9<br />
|339.13<br />
|11/9<br />
|9<br />
|860.87<br />
|18/11<br />
|-<br />
|10<br />
|1,043.48<br />
|11/6, 20/11<br />
|10<br />
|156.52<br />
|11/10, 12/11<br />
|-<br />
|11<br />
|547.83<br />
|11/8, 15/11<br />
|11<br />
|652.17<br />
|16/11, 22/15<br />
|-<br />
|12<br />
|52.17<br />
|33/32, 40/39<br />
|12<br />
|1,147.83<br />
|64/33, 39/20<br />
|-<br />
|13<br />
|756.52<br />
|14/9, 20/13<br />
|13<br />
|443.48<br />
|9/7, 13/10<br />
|-<br />
|14<br />
|260.87<br />
|7/6<br />
|14<br />
|939.13<br />
|12/7<br />
|-<br />
|15<br />
|965.22<br />
|7/4<br />
|15<br />
|234.78<br />
|8/7<br />
|}<br />
{{Navbox regtemp}}</div>Vectorhttps://xenreference.com/wiki/index.php?title=File:Gentle12.png&diff=7402File:Gentle12.png2026-06-02T02:05:21Z<p>Vector: </p>
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<div>Gentle12</div>Vectorhttps://xenreference.com/wiki/index.php?title=Schismic&diff=7401Schismic2026-06-02T02:04:09Z<p>Vector: </p>
<hr />
<div>{{Infobox regtemp|Mapping=1; 1 -8 -3|Subgroup=2.3.5, 2.3.5.19|Edo join 1=12|Edo join 2=29|Generators=3/2|Comma basis=32805/32768 (2.3.5); <br> 361/360, 513/512 (2.3.5.19)|Odd limit 1=5|Complexity 1=12|Generators tuning=701.7|Subgroups=2.3.5, 2.3.5.19|Mistuning 1=0.217|Optimization method=CWE|MOS scales=[[5L 2s]], [[5L 7s]], [[12L 5s]]}}<br />
[[File:Schismic17.png|thumb|Schismic 17-note MOS. This is a sharp tuning; the Pythagorean comma has been inflated to function as an [[aberrisma]].]]<br />
'''Schismic''', '''Schismatic'''<sup>[a]</sup>, or '''Helmholtz''' [12 & 29], is the temperament that equates 5/4 to the Pythagorean diminished fourth. The difference between these intervals is 32805/32768, the ''schisma'', which is about 2 cents; this means that Schismic can be tuned to perfect [[Pythagorean tuning]] (and is considered by some to be the primary 5-limit interpretation of Pythagorean tuning), however it is technically optimal to flatten the fifth by a fraction of a cent. Schismic is one of the simplest microtemperaments.<br />
<br />
The Pythagorean and syntonic commas are thus equated to a single comma-sized step. Due to 3/2 being very close to just, it is also natural to equate 19/16 with the diatonic minor third, tempering out [[513/512]], which is sometimes called Boethius' comma. As a consequence, 19/15 is equated with the diatonic major third, tempering out 1216/1215, which is sometimes called Eratosthenes' comma; [[8/5]] is split into two, with 24/19 and 19/15 being equated, tempering out 361/360. 2.3.5.19 Schismic may be called ''Nestoria''.<br />
<br />
While Schismic can be considered a 12-form temperament, the primary MOS to use for Schismic is Schismic[17], which maps the 5-limit major and minor thirds to the same degree, ensuring like diatonic does in meantone that either one or the other is always accessible. 12edo and 29edo serve as the boundaries of the tuning range; in 12edo the Pythagorean comma and schisma (and thus the meantone comma) are tempered out, and in 29edo the Pythagorean/syntonic comma is inflated to the size of the 25/24 chromatic semitone, supporting [[porcupine]]. Functionally reasonable tunings of Schismic[17] lie broadly between 29 and 53edo and include argent on the sharper end (e.g. 70c val) and Pythagorean tuning on the flatter end; [[aberrismic]] theory suggests that narrowing the Pythagorean comma further than 53edo would lead to it losing its functional property as a step of the scale.<br />
<br />
48edo and 58edo are the first two edos to observe the schisma - that is, to not support Schismic - while tuning the fifth within the aforementioned Schismic tuning range.<br />
<br />
== Extensions ==<br />
<br />
=== Prime 7 ===<br />
The most important 7-limit extension of Schismic is Garibaldi (also 12 & 29, although 41 & 53 represents its tuning range better), which equates 64/63 with the syntonic comma. This is not considered canonical due to a significant loss in accuracy (tuned best with a fifth slightly sharp of just) - that is, Garibaldi is not a microtemperament - but it is still more accurate than [[Meantone]] as well as distinguishing 5-limit, 7-limit, and Pythagorean intervals in any given interval category. Garibaldi is an intuitive way of organizing just intonation as it tempers together the defining ~20-30c commas of the 7-limit. <br />
<br />
==== Primes 11 and 13 ====<br />
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. <br />
<br />
=== Prime 17 ===<br />
Schismic generally does not have 17; two options are setting 17/16 equal to 16/15 and 15/14 (preferring flatter schismic tunings) and tempering together 17/16 and 18/17 (which results in a weak extension including 17/12 as the semioctave).<br />
<br />
== Intervals ==<br />
These interpretations assume Cassandra (in Pythagorean tuning).<br />
{| class="wikitable"<br />
|+<br />
! colspan="3" |Up from the unison<br />
! colspan="3" |Down from the octave<br />
|-<br />
!#<br />
!Cents<br />
!JI<br />
!#<br />
!Cents<br />
!JI<br />
|-<br />
|'''0'''<br />
|'''0.00'''<br />
|'''1/1'''<br />
|'''0'''<br />
|'''1,200.00'''<br />
|'''2/1'''<br />
|-<br />
|'''1'''<br />
|'''701.96'''<br />
|'''3/2'''<br />
|1<br />
|498.04<br />
|4/3<br />
|-<br />
|2<br />
|203.91<br />
|9/8<br />
|2<br />
|996.09<br />
|16/9<br />
|-<br />
|3<br />
|905.87<br />
|32/19, 27/16<br />
|'''3'''<br />
|'''294.13'''<br />
|'''19/16'''<br />
|-<br />
|4<br />
|407.82<br />
|19/15<br />
|4<br />
|792.18<br />
|19/12<br />
|-<br />
|5<br />
|1,109.78<br />
|19/10<br />
|5<br />
|90.22<br />
|19/18, 20/19, 21/20<br />
|-<br />
|6<br />
|611.73<br />
|10/7<br />
|6<br />
|588.27<br />
|7/5<br />
|-<br />
|7<br />
|113.69<br />
|16/15, 15/14<br />
|7<br />
|1,086.31<br />
|15/8<br />
|-<br />
|8<br />
|815.64<br />
|8/5<br />
|'''8'''<br />
|'''384.36'''<br />
|'''5/4'''<br />
|-<br />
|9<br />
|317.60<br />
|5/3<br />
|9<br />
|882.40<br />
|6/5<br />
|-<br />
|10<br />
|1,019.55<br />
|9/5<br />
|10<br />
|180.45<br />
|10/9<br />
|-<br />
|11<br />
|521.51<br />
|27/20<br />
|11<br />
|678.49<br />
|40/27<br />
|-<br />
|12<br />
|23.46<br />
|50/49<br />
|12<br />
|1,176.54<br />
|49/25<br />
|-<br />
|13<br />
|725.42<br />
|32/21<br />
|13<br />
|474.58<br />
|21/16<br />
|-<br />
|14<br />
|227.37<br />
|8/7<br />
|'''14'''<br />
|'''972.63'''<br />
|'''7/4'''<br />
|-<br />
|15<br />
|929.33<br />
|12/7<br />
|15<br />
|270.67<br />
|7/6<br />
|-<br />
|16<br />
|431.28<br />
|9/7<br />
|16<br />
|768.72<br />
|14/9<br />
|-<br />
|17<br />
|1,133.24<br />
|28/27<br />
|17<br />
|66.76<br />
|27/14<br />
|-<br />
|18<br />
|635.19<br />
|13/9<br />
|18<br />
|564.81<br />
|18/13<br />
|-<br />
|19<br />
|137.15<br />
|13/12<br />
|19<br />
|1,062.85<br />
|24/13<br />
|-<br />
|'''20'''<br />
|'''839.10'''<br />
|'''13/8'''<br />
|20<br />
|360.90<br />
|16/13<br />
|-<br />
|21<br />
|341.06<br />
|11/9<br />
|21<br />
|858.94<br />
|18/11<br />
|-<br />
|22<br />
|1,043.01<br />
|11/6<br />
|22<br />
|156.99<br />
|12/11<br />
|-<br />
|'''23'''<br />
|'''544.97'''<br />
|'''11/8'''<br />
|23<br />
|655.03<br />
|16/11<br />
|-<br />
|24<br />
|46.92<br />
|33/32<br />
|24<br />
|1,153.08<br />
|64/33<br />
|-<br />
|25<br />
|748.88<br />
|54/35<br />
|25<br />
|451.12<br />
|35/27<br />
|-<br />
|26<br />
|250.83<br />
|81/70<br />
|26<br />
|949.17<br />
|140/81<br />
|-<br />
|27<br />
|952.79<br />
|26/15<br />
|27<br />
|247.21<br />
|15/13<br />
|-<br />
|28<br />
|454.74<br />
|13/10<br />
|28<br />
|745.26<br />
|20/13<br />
|-<br />
|29<br />
|1,156.70<br />
|39/20<br />
|29<br />
|43.30<br />
|40/39<br />
|}<br />
<br />
== Patent vals ==<br />
The following patent vals up to 272edo support Schismic.<br />
{| class="wikitable sortable mw-collapsible"<br />
|+<br />
!EDO<br />
!Generator tuning<br />
!Extension info<br />
|-<br />
|12<br />
|700.0000<br />
|Garibaldi<br />
|-<br />
|101<br />
|700.9901<br />
|<br />
|-<br />
|89<br />
|701.1236<br />
|<br />
|-<br />
|77<br />
|701.2987<br />
|<br />
|-<br />
|219<br />
|701.3699<br />
|<br />
|-<br />
|142<br />
|701.4085<br />
|<br />
|-<br />
|207<br />
|701.4493<br />
|<br />
|-<br />
|272<br />
|701.4706<br />
|<br />
|-<br />
|65<br />
|701.5385<br />
|<br />
|-<br />
|248<br />
|701.6129<br />
|<br />
|-<br />
|183<br />
|701.6393<br />
|<br />
|-<br />
|118<br />
|701.6949<br />
|<br />
|-<br />
|171<br />
|701.7544<br />
|<br />
|-<br />
|224<br />
|701.7857<br />
|(no Schismic 19)<br />
|-<br />
|53<br />
|701.8868<br />
|Garibaldi<br />
|-<br />
|253<br />
|701.9763<br />
|(no Schismic 19)<br />
|-<br />
|200<br />
|702.0000<br />
|(no Schismic 19)<br />
|-<br />
|147<br />
|702.0408<br />
|Garibaldi<br />
|-<br />
|94<br />
|702.1277<br />
|Garibaldi<br />
|-<br />
|135<br />
|702.2222<br />
|Garibaldi<br />
|-<br />
|41<br />
|702.4390<br />
|Garibaldi<br />
|-<br />
|29<br />
|703.4483<br />
|Garibaldi<br />
|-<br />
|17<br />
|705.8824<br />
|<br />
|}<br />
<br />
<br />
The full list of patent vals supporting schismic is below:<br />
{| class="wikitable sortable mw-collapsible mw-collapsed"<br />
|+<br />
!EDO<br />
!Generator tuning<br />
!Extension info<br />
|-<br />
|12<br />
|700.0000<br />
|Garibaldi<br />
|-<br />
|101<br />
|700.9901<br />
|<br />
|-<br />
|89<br />
|701.1236<br />
|<br />
|-<br />
|77<br />
|701.2987<br />
|<br />
|-<br />
|219<br />
|701.3699<br />
|<br />
|-<br />
|142<br />
|701.4085<br />
|<br />
|-<br />
|207<br />
|701.4493<br />
|<br />
|-<br />
|272<br />
|701.4706<br />
|<br />
|-<br />
|65<br />
|701.5385<br />
|<br />
|-<br />
|508<br />
|701.5748<br />
|(no Schismic 19)<br />
|-<br />
|443<br />
|701.5801<br />
|(no Schismic 19)<br />
|-<br />
|378<br />
|701.5873<br />
|(no Schismic 19)<br />
|-<br />
|313<br />
|701.5974<br />
|(no Schismic 19)<br />
|-<br />
|561<br />
|701.6043<br />
|(no Schismic 19)<br />
|-<br />
|248<br />
|701.6129<br />
|<br />
|-<br />
|679<br />
|701.6200<br />
|(no Schismic 19)<br />
|-<br />
|431<br />
|701.6241<br />
|(no Schismic 19)<br />
|-<br />
|614<br />
|701.6287<br />
|(no Schismic 19)<br />
|-<br />
|797<br />
|701.6311<br />
|(no Schismic 19)<br />
|-<br />
|183<br />
|701.6393<br />
|<br />
|-<br />
|1033<br />
|701.6457<br />
|(no Schismic 19)<br />
|-<br />
|850<br />
|701.6471<br />
|(no Schismic 19)<br />
|-<br />
|667<br />
|701.6492<br />
|(no Schismic 19)<br />
|-<br />
|1151<br />
|701.6507<br />
|(no Schismic 19)<br />
|-<br />
|484<br />
|701.6529<br />
|(no Schismic 19)<br />
|-<br />
|1269<br />
|701.6548<br />
|(no Schismic 19)<br />
|-<br />
|785<br />
|701.6561<br />
|(no Schismic 19)<br />
|-<br />
|1086<br />
|701.6575<br />
|(no Schismic 19)<br />
|-<br />
|1387<br />
|701.6583<br />
|(no Schismic 19)<br />
|-<br />
|301<br />
|701.6611<br />
|(no Schismic 19)<br />
|-<br />
|1322<br />
|701.6641<br />
|(no Schismic 19)<br />
|-<br />
|1021<br />
|701.6650<br />
|(no Schismic 19)<br />
|-<br />
|720<br />
|701.6667<br />
|(no Schismic 19)<br />
|-<br />
|1139<br />
|701.6681<br />
|(no Schismic 19)<br />
|-<br />
|1558<br />
|701.6688<br />
|(no Schismic 19)<br />
|-<br />
|419<br />
|701.6706<br />
|(no Schismic 19)<br />
|-<br />
|1794<br />
|701.6722<br />
|(no Schismic 19)<br />
|-<br />
|1375<br />
|701.6727<br />
|(no Schismic 19)<br />
|-<br />
|956<br />
|701.6736<br />
|(no Schismic 19)<br />
|-<br />
|1493<br />
|701.6745<br />
|(no Schismic 19)<br />
|-<br />
|2030<br />
|701.6749<br />
|(no Schismic 19)<br />
|-<br />
|537<br />
|701.6760<br />
|(no Schismic 19)<br />
|-<br />
|1729<br />
|701.6773<br />
|(no Schismic 19)<br />
|-<br />
|1192<br />
|701.6779<br />
|(no Schismic 19)<br />
|-<br />
|1847<br />
|701.6784<br />
|(no Schismic 19)<br />
|-<br />
|655<br />
|701.6794<br />
|(no Schismic 19)<br />
|-<br />
|2083<br />
|701.6803<br />
|(no Schismic 19)<br />
|-<br />
|1428<br />
|701.6807<br />
|(no Schismic 19)<br />
|-<br />
|773<br />
|701.6818<br />
|(no Schismic 19)<br />
|-<br />
|1664<br />
|701.6827<br />
|(no Schismic 19)<br />
|-<br />
|891<br />
|701.6835<br />
|(no Schismic 19)<br />
|-<br />
|1900<br />
|701.6842<br />
|(no Schismic 19)<br />
|-<br />
|1009<br />
|701.6848<br />
|(no Schismic 19)<br />
|-<br />
|2136<br />
|701.6854<br />
|(no Schismic 19)<br />
|-<br />
|1127<br />
|701.6859<br />
|(no Schismic 19)<br />
|-<br />
|1245<br />
|701.6867<br />
|(no Schismic 19)<br />
|-<br />
|1363<br />
|701.6875<br />
|(no Schismic 19)<br />
|-<br />
|1481<br />
|701.6880<br />
|(no Schismic 19)<br />
|-<br />
|1599<br />
|701.6886<br />
|(no Schismic 19)<br />
|-<br />
|1717<br />
|701.6890<br />
|(no Schismic 19)<br />
|-<br />
|1835<br />
|701.6894<br />
|(no Schismic 19)<br />
|-<br />
|1953<br />
|701.6897<br />
|(no Schismic 19)<br />
|-<br />
|2071<br />
|701.6900<br />
|(no Schismic 19)<br />
|-<br />
|2189<br />
|701.6903<br />
|(no Schismic 19)<br />
|-<br />
|118<br />
|701.6949<br />
|<br />
|-<br />
|2295<br />
|701.6993<br />
|(no Schismic 19)<br />
|-<br />
|2177<br />
|701.6996<br />
|(no Schismic 19)<br />
|-<br />
|2059<br />
|701.6999<br />
|(no Schismic 19)<br />
|-<br />
|1941<br />
|701.7002<br />
|(no Schismic 19)<br />
|-<br />
|1823<br />
|701.7005<br />
|(no Schismic 19)<br />
|-<br />
|1705<br />
|701.7009<br />
|(no Schismic 19)<br />
|-<br />
|1587<br />
|701.7013<br />
|(no Schismic 19)<br />
|-<br />
|1469<br />
|701.7018<br />
|(no Schismic 19)<br />
|-<br />
|1351<br />
|701.7024<br />
|(no Schismic 19)<br />
|-<br />
|1233<br />
|701.7032<br />
|(no Schismic 19)<br />
|-<br />
|2348<br />
|701.7036<br />
|(no Schismic 19)<br />
|-<br />
|1115<br />
|701.7040<br />
|(no Schismic 19)<br />
|-<br />
|2112<br />
|701.7045<br />
|(no Schismic 19)<br />
|-<br />
|997<br />
|701.7051<br />
|(no Schismic 19)<br />
|-<br />
|1876<br />
|701.7058<br />
|(no Schismic 19)<br />
|-<br />
|879<br />
|701.7065<br />
|(no Schismic 19)<br />
|-<br />
|1640<br />
|701.7073<br />
|(no Schismic 19)<br />
|-<br />
|2401<br />
|701.7076<br />
|(no Schismic 19)<br />
|-<br />
|761<br />
|701.7083<br />
|(no Schismic 19)<br />
|-<br />
|2165<br />
|701.7090<br />
|(no Schismic 19)<br />
|-<br />
|1404<br />
|701.7094<br />
|(no Schismic 19)<br />
|-<br />
|2047<br />
|701.7098<br />
|(no Schismic 19)<br />
|-<br />
|643<br />
|701.7107<br />
|(no Schismic 19)<br />
|-<br />
|2454<br />
|701.7115<br />
|(no Schismic 19)<br />
|-<br />
|1811<br />
|701.7118<br />
|(no Schismic 19)<br />
|-<br />
|1168<br />
|701.7123<br />
|(no Schismic 19)<br />
|-<br />
|1693<br />
|701.7129<br />
|(no Schismic 19)<br />
|-<br />
|2218<br />
|701.7133<br />
|(no Schismic 19)<br />
|-<br />
|525<br />
|701.7143<br />
|(no Schismic 19)<br />
|-<br />
|1982<br />
|701.7154<br />
|(no Schismic 19)<br />
|-<br />
|1457<br />
|701.7159<br />
|(no Schismic 19)<br />
|-<br />
|2389<br />
|701.7162<br />
|(no Schismic 19)<br />
|-<br />
|932<br />
|701.7167<br />
|(no Schismic 19)<br />
|-<br />
|2271<br />
|701.7173<br />
|(no Schismic 19)<br />
|-<br />
|1339<br />
|701.7177<br />
|(no Schismic 19)<br />
|-<br />
|1746<br />
|701.7182<br />
|(no Schismic 19)<br />
|-<br />
|2153<br />
|701.7185<br />
|(no Schismic 19)<br />
|-<br />
|407<br />
|701.7199<br />
|(no Schismic 19)<br />
|-<br />
|2324<br />
|701.7212<br />
|(no Schismic 19)<br />
|-<br />
|1917<br />
|701.7214<br />
|(no Schismic 19)<br />
|-<br />
|1510<br />
|701.7219<br />
|(no Schismic 19)<br />
|-<br />
|1103<br />
|701.7226<br />
|(no Schismic 19)<br />
|-<br />
|1799<br />
|701.7232<br />
|(no Schismic 19)<br />
|-<br />
|2495<br />
|701.7234<br />
|(no Schismic 19)<br />
|-<br />
|696<br />
|701.7241<br />
|(no Schismic 19)<br />
|-<br />
|2377<br />
|701.7249<br />
|(no Schismic 19)<br />
|-<br />
|1681<br />
|701.7252<br />
|(no Schismic 19)<br />
|-<br />
|985<br />
|701.7259<br />
|(no Schismic 19)<br />
|-<br />
|2259<br />
|701.7264<br />
|(no Schismic 19)<br />
|-<br />
|1274<br />
|701.7268<br />
|(no Schismic 19)<br />
|-<br />
|1563<br />
|701.7274<br />
|(no Schismic 19)<br />
|-<br />
|1852<br />
|701.7279<br />
|(no Schismic 19)<br />
|-<br />
|2141<br />
|701.7282<br />
|(no Schismic 19)<br />
|-<br />
|2430<br />
|701.7284<br />
|(no Schismic 19)<br />
|-<br />
|289<br />
|701.7301<br />
|(no Schismic 19)<br />
|-<br />
|2483<br />
|701.7318<br />
|(no Schismic 19)<br />
|-<br />
|2194<br />
|701.7320<br />
|(no Schismic 19)<br />
|-<br />
|1905<br />
|701.7323<br />
|(no Schismic 19)<br />
|-<br />
|1616<br />
|701.7327<br />
|(no Schismic 19)<br />
|-<br />
|1327<br />
|701.7332<br />
|(no Schismic 19)<br />
|-<br />
|2365<br />
|701.7336<br />
|(no Schismic 19)<br />
|-<br />
|1038<br />
|701.7341<br />
|(no Schismic 19)<br />
|-<br />
|1787<br />
|701.7348<br />
|(no Schismic 19)<br />
|-<br />
|2536<br />
|701.7350<br />
|(no Schismic 19)<br />
|-<br />
|749<br />
|701.7356<br />
|(no Schismic 19)<br />
|-<br />
|2707<br />
|701.7362<br />
|(no Schismic 19)<br />
|-<br />
|1958<br />
|701.7365<br />
|(no Schismic 19)<br />
|-<br />
|1209<br />
|701.7370<br />
|(no Schismic 19)<br />
|-<br />
|1669<br />
|701.7376<br />
|(no Schismic 19)<br />
|-<br />
|2129<br />
|701.7379<br />
|(no Schismic 19)<br />
|-<br />
|2589<br />
|701.7381<br />
|(no Schismic 19)<br />
|-<br />
|460<br />
|701.7391<br />
|(no Schismic 19)<br />
|-<br />
|2471<br />
|701.7402<br />
|(no Schismic 19)<br />
|-<br />
|2011<br />
|701.7404<br />
|(no Schismic 19)<br />
|-<br />
|1551<br />
|701.7408<br />
|(no Schismic 19)<br />
|-<br />
|1091<br />
|701.7415<br />
|(no Schismic 19)<br />
|-<br />
|1722<br />
|701.7422<br />
|(no Schismic 19)<br />
|-<br />
|2353<br />
|701.7425<br />
|(no Schismic 19)<br />
|-<br />
|631<br />
|701.7433<br />
|(no Schismic 19)<br />
|-<br />
|2064<br />
|701.7442<br />
|(no Schismic 19)<br />
|-<br />
|1433<br />
|701.7446<br />
|(no Schismic 19)<br />
|-<br />
|802<br />
|701.7456<br />
|(no Schismic 19)<br />
|-<br />
|1775<br />
|701.7465<br />
|(no Schismic 19)<br />
|-<br />
|973<br />
|701.7472<br />
|(no Schismic 19)<br />
|-<br />
|1144<br />
|701.7483<br />
|(no Schismic 19)<br />
|-<br />
|1315<br />
|701.7490<br />
|(no Schismic 19)<br />
|-<br />
|1486<br />
|701.7497<br />
|(no Schismic 19)<br />
|-<br />
|1657<br />
|701.7502<br />
|(no Schismic 19)<br />
|-<br />
|1828<br />
|701.7505<br />
|(no Schismic 19)<br />
|-<br />
|171<br />
|701.7544<br />
|<br />
|-<br />
|1421<br />
|701.7593<br />
|(no Schismic 19)<br />
|-<br />
|1250<br />
|701.7600<br />
|(no Schismic 19)<br />
|-<br />
|1079<br />
|701.7609<br />
|(no Schismic 19)<br />
|-<br />
|908<br />
|701.7621<br />
|(no Schismic 19)<br />
|-<br />
|737<br />
|701.7639<br />
|(no Schismic 19)<br />
|-<br />
|1303<br />
|701.7652<br />
|(no Schismic 19)<br />
|-<br />
|566<br />
|701.7668<br />
|(no Schismic 19)<br />
|-<br />
|961<br />
|701.7690<br />
|(no Schismic 19)<br />
|-<br />
|395<br />
|701.7722<br />
|(no Schismic 19)<br />
|-<br />
|1014<br />
|701.7751<br />
|(no Schismic 19)<br />
|-<br />
|619<br />
|701.7771<br />
|(no Schismic 19)<br />
|-<br />
|843<br />
|701.7794<br />
|(no Schismic 19)<br />
|-<br />
|1067<br />
|701.7807<br />
|(no Schismic 19)<br />
|-<br />
|224<br />
|701.7857<br />
|(no Schismic 19)<br />
|-<br />
|725<br />
|701.7931<br />
|(no Schismic 19)<br />
|-<br />
|501<br />
|701.7964<br />
|(no Schismic 19)<br />
|-<br />
|778<br />
|701.7995<br />
|(no Schismic 19)<br />
|-<br />
|277<br />
|701.8051<br />
|(no Schismic 19)<br />
|-<br />
|607<br />
|701.8122<br />
|(no Schismic 19)<br />
|-<br />
|330<br />
|701.8182<br />
|(no Schismic 19)<br />
|-<br />
|383<br />
|701.8277<br />
|(no Schismic 19)<br />
|-<br />
|436<br />
|701.8349<br />
|(no Schismic 19)<br />
|-<br />
|489<br />
|701.8405<br />
|(no Schismic 19)<br />
|-<br />
|542<br />
|701.8450<br />
|(no Schismic 19)<br />
|-<br />
|53<br />
|701.8868<br />
|Garibaldi<br />
|-<br />
|253<br />
|701.9763<br />
|(no Schismic 19)<br />
|-<br />
|200<br />
|702.0000<br />
|(no Schismic 19)<br />
|-<br />
|147<br />
|702.0408<br />
|Garibaldi<br />
|-<br />
|94<br />
|702.1277<br />
|Garibaldi<br />
|-<br />
|135<br />
|702.2222<br />
|Garibaldi<br />
|-<br />
|41<br />
|702.4390<br />
|Garibaldi<br />
|-<br />
|29<br />
|703.4483<br />
|Garibaldi<br />
|-<br />
|17<br />
|705.8824<br />
|<br />
|}<br />
<br />
== Footnotes ==<br />
<sup>[a]</sup> Despite ending in -isma, ''schisma'' is a [[Temperament naming#Comma declension categories|3rd-declension]] comma name, therefore its temperaments are ''Schismatic'' and ''Schismic'' (which, as it is a 2.3.5 comma, refer to the same temperament).<br />
{{Navbox regtemp}}</div>Vectorhttps://xenreference.com/wiki/index.php?title=File:Schismic17.png&diff=7400File:Schismic17.png2026-06-02T02:03:26Z<p>Vector: </p>
<hr />
<div>Schismic17</div>Vectorhttps://xenreference.com/wiki/index.php?title=Magic&diff=7399Magic2026-06-02T02:02:10Z<p>Vector: </p>
<hr />
<div>{{Infobox regtemp<br />
| Title = Magic<br />
| Subgroups = 2.3.5, 2.3.5.7<br />
| Comma basis = [[3125/3072]] (5-limit); <br>[[225/224]], [[245/243]] (7-limit)<br />
| Edo join 1 = 19 | Edo join 2 = 22<br />
| Mapping = 1; 5 1 12<br />
| Generators = 5/4<br />
| Generators tuning = 380.5<br />
| Optimization method = CWE<br />
| MOS scales = [[3L 4s]], [[3L 7s]], …, [[3L 16s]], [[19L 3s]]<br />
| Odd limit 1 = 5 | Mistuning 1 = 5.9 | Complexity 1 = 7<br />
| Odd limit 2 = 9 | Mistuning 2 = 5.9 | Complexity 2 = 13<br />
}}<br />
[[File:Magic10.png|thumb|Magic decatonic MOS]]<br />
'''Magic''' (19 & 22) is a 2.3.5 temperament that equates a stack of five 5/4 major thirds to one 3/1. It also equates 25/24 to 128/125, shrinking the difference between 5/4 and 6/5. It is a 3-cluster temperament, as indicated by the edo join (22 - 19 = 3).<br />
<br />
== Interval chain ==<br />
In the following table, odd harmonics 1–15 and their inverses are in '''bold'''.<br />
{| class="wikitable center-1 right-2"<br />
|-<br />
! #<br />
! Cents*<br />
! Approximate ratios<br />
|-<br />
| 0<br />
| 0.0<br />
| '''1/1'''<br />
|-<br />
| 1<br />
| 380.5<br />
| '''5/4'''<br />
|-<br />
| 2<br />
| 760.9<br />
| 14/9<br />
|-<br />
| 3<br />
| 1141.4<br />
| 27/14, 31/16<br />
|-<br />
| 4<br />
| 321.8<br />
| 6/5, 29/24<br />
|-<br />
| 5<br />
| 702.3<br />
| '''3/2'''<br />
|-<br />
| 6<br />
| 1082.7<br />
| '''15/8''', 28/15<br />
|-<br />
| 7<br />
| 263.2<br />
| 7/6<br />
|-<br />
| 8<br />
| 643.7<br />
| 36/25<br />
|-<br />
| 9<br />
| 1024.1<br />
| 9/5, 29/16<br />
|-<br />
| 10<br />
| 204.6<br />
| '''9/8'''<br />
|-<br />
| 11<br />
| 585.0<br />
| 7/5<br />
|-<br />
| 12<br />
| 965.5<br />
| '''7/4'''<br />
|-<br />
| 13<br />
| 145.9<br />
| 35/32<br />
|}<br />
<nowiki/>* In 7-limit CWE tuning<br />
<br />
== Extensions ==<br />
Magic divides ~16/15 in half (into two 25/24's), so it easily extends to prime 31 (at +3 generators) by tempering out S31 {{=}} 961/960. Since in 7-limit Magic, the 16/15 is also a 15/14, we can also extend 2.3.5.7.31 Magic to add prime 29 (at +9 generators) by equating 16/15 to 31/29.<br />
<br />
{{Adv|This can be seen from the following [[S-expression]] for the Magic comma:}}<br />
<br />
3125/3072<br />
= (25/24)^2/(16/15)<br />
= (25/24)*(25/24)/(32/31*31/30)<br />
= (25/24)S25*S26*S27*S28*S29*S30/(32/31)<br />
= (S25*S26*S27*S28*S29*S30)^2*S31<br />
= (S15*S28*S29*S30)^2*S31<br />
<br />
{{Adv|[[canonical extension|structurally inducing]] the above 2.3.5.7.29.31 extension.}}<br />
<br />
== Scales ==<br />
{{Wip}}<br />
<br />
=== MOS scales ===<br />
Because three generators just barely fall short of an octave, the [[MOS]] scales generated by Magic are quite commatic. The Magic Diesis (a small step simultaneously representing 25/24, 28/27, 36/35, and 128/125) is the s step of all MOS scales up to the 22-form<br />
<br />
==== 7-form ====<br />
[[File:22edo Magic Cadence.png|thumb|An example of the Magic Cadence (III - I) in 22edo, written in SATB format with native fifths notation.]]<br />
The 7-note Magic scale has the pattern 3L 4s, sometimes called [[Mosh]], with a large step of 6/5. The disparity in size between the two types of steps grants a quality to stepwise melodies that some find to sound awkward or lurching; while some composers may prefer or intend such a sound, those who do not are cautioned to avoid long stepwise runs for fear of creating an aimless and chromatic-sounding melodic core.<br />
<br />
The 4:5:6 triad can be found on the tonic in two of the modes: LsLsLss and LsLssLs; these modes place the triad on degrees 1, 3, and 4. Generalizing from this, we can see that the other main type of triad that occurs on these degrees is 1/1 - 5/4 - 9/7, with an additional 1/1 - 16/15 - 9/7 triad in the ssLsLsL mode.<br />
<br />
The LsLsLss mode has the clearest utility, with a clear cadence from the III chord to the I chord; we may consider this motion to be Magic temperament's analog to the Perfect Cadence from [[Meantone]][7]. This cadence creates contrary motion, with the 6 and 5 of the III chord resolving respectively to the 8 and 4 of the I chord, and the 3 sustained between both chords creates a strong sense of continuity.<br />
<br />
The LsLssLs mode contains an inverse version of this cadence, which can be seen as a Magic analog to Meantone's Plagal Cadence.<br />
<br />
=== Other scale forms ===<br />
Due to the juxtaposition of the Magic Diesis against significantly larger step sizes, it is often desirable to use a non-MOS structure in Magic, such as [[Blackdye]] or the [[5-odd-limit|5-odd Diamond]].<br />
<br />
== List of tunings ==<br />
<br />
=== EDO patent vals ===<br />
The following patent vals support 2.3.5 Magic. Vals that are contorted in 2.3.5 are not included. <br />
{| class="wikitable sortable"<br />
! Edo <br />
!Extension to 7|| Generator || Fifth<br />
|-<br />
||3<br />
| ||400.000||800.000<br />
|-<br />
||25<br />
| ||384.000||720.000<br />
|-<br />
||22<br />
|19 & 22||381.818||709.091<br />
|-<br />
||107<br />
| ||381.308||706.542<br />
|-<br />
||85<br />
|19 & 22||381.176||705.882<br />
|-<br />
||63<br />
|19 & 22||380.952||704.762<br />
|-<br />
||104<br />
| ||380.769||703.846<br />
|-<br />
||41<br />
|19 & 22||380.488||702.439<br />
|-<br />
||60<br />
|19 & 22||380.000||700.000<br />
|-<br />
||79<br />
| ||379.747||698.734<br />
|-<br />
||19<br />
|19 & 22||378.947||694.737<br />
|-<br />
||35<br />
| ||377.143||685.714<br />
|-<br />
||16<br />
| ||375.000||675.000<br />
|}<br />
<br />
=== Target-tunings ===<br />
The following are some examples of useful target tunings available in Magic temperament.<br />
{| class="wikitable"<br />
!Just interval<br />
!Comma fraction<br />
!Generator<br />
!Fifth<br />
|-<br />
|5/4<br />
|0-comma<br />
|386.314<br />
|731.569<br />
|-<br />
|15/8<br />
|1/6-comma<br />
|381.378<br />
|706.891<br />
|-<br />
|3/2<br />
|1/5-comma<br />
|380.391<br />
|701.955<br />
|-<br />
|9/5<br />
|2/9-comma<br />
|379.733<br />
|698.665<br />
|-<br />
|5/3<br />
|1/4-comma<br />
|378.910<br />
|1694.552<br />
|-<br />
|25/24<br />
|1/3-comma<br />
|376.443<br />
|682.213<br />
|}<br />
{{cat|Temperaments}}{{Navbox regtemp}}</div>Vectorhttps://xenreference.com/wiki/index.php?title=File:Magic10.png&diff=7398File:Magic10.png2026-06-02T02:02:01Z<p>Vector: </p>
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<div>Magic10</div>Vectorhttps://xenreference.com/wiki/index.php?title=Sensi&diff=7397Sensi2026-06-02T02:00:25Z<p>Vector: </p>
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<div>{{Infobox regtemp<br />
| Title = Sensi<br />
| Subgroups = 2.3.5.7<br />
| Comma basis = [[126/125]], [[245/243]] (7-limit)<br />
| Edo join 1 = 19 | Edo join 2 = 27<br />
| Mapping = 1; 7 9 13<br />
| Generators = 9/7 | Generators tuning = 443.3 | Optimization method = CWE<br />
| MOS scales = [[3L&nbsp;2s]], [[3L&nbsp;5s]], [[8L&nbsp;3s]], [[8L&nbsp;11s]]<br />
| Odd limit 1 = 7 | Mistuning 1 = 7.5 | Complexity 1 = 19<br />
}}<br />
[[File:Sentry11.png|thumb|Sensi 11-note MOS]]<br />
'''Sensi''', 2.3.5.7[{{e|19}} & {{e|27}}], is a rank-2 temperament generated by a sharpened 9/7 such that two of them are equated to 5/3. Distinctly from other sensamagic temperaments, 3/2 is also found at 7 generators, or equivalently the octave is found at 125/63. It is most accurately extended to add prime 31 (by equating the generator with 31/24), but less accurately prime 13 is canonically added (by equating the generator with 13/10).<br />
<br />
== Interval chain ==<br />
In the following table, odd harmonics and subharmonics 1–21 are in '''bold'''. <br />
<br />
{| class="wikitable right-1 right-2 sortable"<br />
|-<br />
! #<br />
! Cents*<br />
! class="unsortable" | Approximate ratios<br />
|-<br />
| 0<br />
| 0.0<br />
| '''1/1'''<br />
|-<br />
| 1<br />
| 443.4<br />
| 9/7, 13/10<br />
|-<br />
| 2<br />
| 886.7<br />
| 5/3, 42/25<br />
|-<br />
| 3<br />
| 130.1<br />
| 13/12, 14/13, 15/14, 27/25<br />
|-<br />
| 4<br />
| 573.4<br />
| 7/5, 18/13, 25/18<br />
|-<br />
| 5<br />
| 1016.8<br />
| 9/5<br />
|-<br />
| 6<br />
| 260.1<br />
| 7/6, 15/13<br />
|-<br />
| 7<br />
| 703.5<br />
| '''3/2'''<br />
|-<br />
| 8<br />
| 1146.9<br />
| 27/14, 35/18<br />
|-<br />
| 9<br />
| 390.2<br />
| '''5/4'''<br />
|-<br />
| 10<br />
| 833.6<br />
| '''13/8''', 21/13<br />
|-<br />
| 11<br />
| 76.9<br />
| 21/20, 25/24<br />
|-<br />
| 12<br />
| 520.3<br />
| 27/20<br />
|-<br />
| 13<br />
| 963.7<br />
| '''7/4'''<br />
|-<br />
| 14<br />
| 207.0<br />
| '''9/8'''<br />
|-<br />
| 15<br />
| 650.4<br />
| 35/24<br />
|-<br />
| 16<br />
| 1093.7<br />
| '''15/8'''<br />
|-<br />
| 17<br />
| 337.1<br />
| 39/32<br />
|-<br />
| 18<br />
| 780.4<br />
| 25/16<br />
|-<br />
| 19<br />
| 23.8<br />
| 49/48, 65/64, 81/80<br />
|-<br />
| 20<br />
| 467.2<br />
| '''21/16'''<br />
|}<br />
<nowiki/>* In 2.3.5.7.13 CWE tuning<br />
== Patent vals ==<br />
The following patent vals support 2.3.5.7 Sensi. Vals contorted in the 7-limit are not included.<br />
{| class="wikitable sortable"<br />
!|Edo!!13 extension!!Generator!!3/2 tuning!!5/4 tuning!!7/4 tuning!!13/8 tuning<br />
|-<br />
||19||19 & 27||442.105||694.737||378.947||947.368||821.053<br />
|-<br />
||65||||443.077||701.538||387.692||960.000||849.231<br />
|-<br />
||46||19 & 27||443.478||704.348||391.304||965.217||834.783<br />
|-<br />
||73||19 & 27||443.836||706.849||394.521||969.863||838.356<br />
|-<br />
||27||19 & 27||444.444||711.111||400.000||977.778||844.444<br />
|}<br />
{{Navbox regtemp}}<br />
{{Cat|temperaments}}</div>Vectorhttps://xenreference.com/wiki/index.php?title=File:Sentry11.png&diff=7396File:Sentry11.png2026-06-02T02:00:07Z<p>Vector: </p>
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<div>Sentry 11-note MOS</div>Vectorhttps://xenreference.com/wiki/index.php?title=Sensamagic&diff=7395Sensamagic2026-06-02T01:59:25Z<p>Vector: </p>
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<div>[[File:Sentry8.png|thumb|Sentry octatonic MOS]]<br />
'''Sensamagic''' (b13 & b17), sometimes known in a tritave-equivalent context as '''Bohlen-Pierce-Stearns''' (BPS), is the temperament in the 3.5.7 subgroup equating a stack of two [[9/7]]<nowiki/>s with [[5/3]]; this means that the comma [[245/243]] is tempered out. 9/7 is tuned sharp (about 440 cents) and 5/3 is flattened (about 880 cents). It functions as a tritave analog of [[Meantone]], relating the two simplest prime harmonics after the equave with a medium accuracy.<br />
<br />
Sensamagic can be used as a temperament with octaves by one of several approaches:<br />
<br />
* simply taking the octave as the period instead of the tritave, resulting in a 2.9/7.5/3 subgroup temperament known as Sentry (11 & 19)<br />
* equating the octave to a false octave found on the Sensamagic generator chain, such as 125/63 (resulting in [[Sensi]] (19 & 27)) or 49/25 (resulting in an obscure [[Porcupine]] extension called "Hedgehog" that splits the octave into two 7/5~10/7 tritones)<br />
* adding the octave as an additional generator, resulting in rank-3 Sensamagic (41 & 19 & 27, or b65 & b30 & b43)<br />
<br />
This page will focus on tritave and rank-3 Sensamagic.<br />
<br />
''TODO: complete page''<br />
<br />
== Interval chain ==<br />
{| class="wikitable right-1 right-2"<br />
|-<br />
! #<br />
! Cents*<br />
! Approximate ratios<br />
|-<br />
| 0<br />
| 0.0<br />
| '''1/1'''<br />
|-<br />
| 1<br />
| 440.7<br />
| '''9/7'''<br />
|-<br />
| 2<br />
| 881.3<br />
| '''5/3'''<br />
|-<br />
| 3<br />
| 1322.0<br />
| 15/7<br />
|-<br />
| 4<br />
| 1762.7<br />
| '''25/9'''<br />
|-<br />
| 5<br />
| 301.4<br />
| 25/21<br />
|-<br />
| 6<br />
| 742.0<br />
| 75/49, 125/81<br />
|-<br />
| 7<br />
| 1182.7<br />
| <br />
|-<br />
| 8<br />
| 1623.4<br />
| <br />
|-<br />
| 9<br />
| 162.1<br />
|<br />
|-<br />
| 10<br />
| 602.7<br />
|<br />
|-<br />
| 11<br />
| 1043.4<br />
|<br />
|-<br />
| 12<br />
| 1484.1<br />
|<br />
|}<br />
<nowiki>*</nowiki> in 3.5.7-subgroup [[CWE]] tuning, tritave reduced. Intervals may be additionally octave-reduced in rank-3 sensamagic.<br />
<br />
{{Navbox regtemp}}</div>Vectorhttps://xenreference.com/wiki/index.php?title=File:Sentry8.png&diff=7394File:Sentry8.png2026-06-02T01:59:14Z<p>Vector: </p>
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<div>{{Infobox regtemp<br />
| Title = Kleismic<br />
| Subgroups = 2.3.5, 2.3.5.13<br />
| Comma basis = [[15625/15552]] (2.3.5); <br>[[325/324]], [[625/624]] (2.3.5.13)<br />
| Edo join 1 = 15 | Edo join 2 = 19<br />
| Mapping = 1; 6 5 14<br />
| Generators = 6/5 | Generators tuning = 317.1 | Optimization method = CWE<br />
| MOS scales = [[3L 1s]], [[4L 3s]], [[4L 7s]], [[4L 11s]], [[15L 4s]]<br />
| Odd limit 1 = 5 | Mistuning 1 = 1.35 | Complexity 1 = 7<br />
| Odd limit 2 = 2.3.5.13 15 | Mistuning 2 = 2.35 | Complexity 2 = 15<br />
}}<br />
[[File:Kleismic11.png|thumb|Kleismic 11-note MOS]]<br />
'''Kleismic''', [15 & 19], is a high-accuracy temperament (usually seen in its basic form as a 2.3.5 temperament) that equates a stack of six 6/5 minor thirds to one 3/1. Via a [[canonical extension|structurally induced]] extension to 2.3.5.13, it equates three 6/5's to one semitwelfth [[26/15]] and equates 25/24 to 26/25 and 27/26.<br />
<br />
Kleismic harmony is naturally based on splitting 5/3 into 4/3 and 5/4 thus making 3:4:5 or 12:15:18 triads. Indeed, Kleismic[15] (4L11s) can be constructed from the [[generator sequence]] GS(3:4:5)[19] by tempering out four kleismas.<br />
<br />
Kleismic, despite generating a heptatonic scale, is not particularly usefully a 7-form temperament; this is because 3/2 is an imperfect sixth rather than a perfect fifth. In the 11-form, however, it is much better as 5/4 and 4/3 are mapped to the same degree, resulting in a dichotomy of [0 4 8]/11 triads similar to the [0 2 4]/7 ones found in fifth-centric temperaments.<br />
<br />
Kleismic can be seen as a counterpart to [[Diaschismic]], which as a 2.3.5.17 temperament splits 9/8 into two 16/15s that are also 17/16~18/17 - Diaschismic splits 9/8 into two; Kleismic splits it into three. The intersection of both temperaments in the 5-limit is [[34edo]], which splits 9/8 into six steps; for that reason the 34edo step is called the sextula.<br />
<br />
Kleismic contains [[interordinal]] intervals, because 3/1 is split into an even number of parts. Because it equates 9/8 to three 25/24s, the inframinor and ultramajor third in [[alpha-dicot]] tuning are separated by 9/8, such that in kleismic augmenting or diminishing a 5/4 or 6/5 respectively by 25/24 results in an interordinal.<br />
== Interval chain ==<br />
In the following table, odd harmonics 1–15 are labeled in '''bold'''. <br />
<br />
{| class="wikitable sortable center-1 right-2"<br />
! #<br />
! Cents*<br />
! class="unsortable" | Approximate ratios<br />
|-<br />
| 0<br />
| 0.0<br />
| '''1/1'''<br />
|-<br />
| 1<br />
| 317.1<br />
| 6/5<br />
|-<br />
| 2<br />
| 634.2<br />
| 13/9, 36/25<br />
|-<br />
| 3<br />
| 951.3<br />
| 26/15<br />
|-<br />
| 4<br />
| 68.4<br />
| 25/24, 26/25, 27/26<br />
|-<br />
| 5<br />
| 385.5<br />
| '''5/4'''<br />
|-<br />
| 6<br />
| 702.6<br />
| '''3/2'''<br />
|-<br />
| 7<br />
| 1019.6<br />
| 9/5<br />
|-<br />
| 8<br />
| 136.7<br />
| 13/12, 27/25<br />
|-<br />
| 9<br />
| 453.8<br />
| 13/10<br />
|-<br />
| 10<br />
| 770.9<br />
| 25/16, 39/25<br />
|-<br />
| 11<br />
| 1088.0<br />
| '''15/8'''<br />
|-<br />
| 12<br />
| 205.1<br />
| '''9/8'''<br />
|-<br />
| 13<br />
| 522.2<br />
| 27/20<br />
|-<br />
| 14<br />
| 839.3<br />
| '''13/8'''<br />
|-<br />
| 15<br />
| 1156.4<br />
| 39/20<br />
|-<br />
| 16<br />
| 273.5<br />
| 75/64<br />
|-<br />
| 17<br />
| 590.6<br />
| 45/32<br />
|-<br />
| 18<br />
| 907.7<br />
| 27/16<br />
|-<br />
| 19<br />
| 24.7<br />
| 65/64, 81/80<br />
|}<br />
<nowiki/>* In 2.3.5.13-subgroup [[CWE tuning]], octave reduced<br />
<br />
== List of patent vals ==<br />
:''Main article: [[Kleismic/Patent vals]]''<br />
<br />
{{Navbox regtemp}}<br />
{{cat|Temperaments}}</div>Vectorhttps://xenreference.com/wiki/index.php?title=File:Kleismic11.png&diff=7392File:Kleismic11.png2026-06-02T01:57:41Z<p>Vector: </p>
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<div>624624</div>Vectorhttps://xenreference.com/wiki/index.php?title=Semifourth-generated_scales&diff=7391Semifourth-generated scales2026-06-02T01:56:44Z<p>Vector: </p>
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<div>{{problematic}}<br />
[[File:Semaphore5.png|thumb|Semaphore pentatonic]]<br />
'''Semifourth-generated scales''' are scales in tuning systems where two generators of approximately 250 cents in size stack to a perfect fourth, defining a temperament archetype called '''alpha-dicot''' or '''omega-dicot'''{{Adv|, and notated [P8, P4/2] in pergen notation.}}<br />
<br />
The semifourth-generated scales are characteristically 4L 1s, [[semiquartal]] (5L 4s), and 5L 9s. 4L 1s is structurally significant because it is the [[Equipentatonic|equal trichordal pentatonic]].<br />
<br />
== Intervals and notation ==<br />
Alpha-dicot notation is complicated as it conventionally requires either the introduction of new "hemi-Pythagorean" ordinals or the use of scales other than the standard diatonic scale. As such, there is no universally accepted convention. An intuitive option, if one that takes getting used to, is to use [https://en.xen.wiki/w/KISS_notation KISS] notation or [https://en.xen.wiki/w/Diamond-mos_notation diamond-mos] notation for 5L 4s.<br />
<br />
== Temperaments ==<br />
Semifourth-generated temperaments have the edo join 5 & 9 (ignoring [[val#Wart notation|warts]]).<br />
* '''Interseptimal''' or '''Semaphore''' is a 2.3.7 temperament that tempers out 49/48, often considered inaccurate.<br />
* '''Island''' temperament is a 2.3.13/5 temperament that tempers out 676/675, equating the semifourth to 15/13.<br />
* '''Intergan''' (short for Interganassismic, a superset of 12edo's 2.3.17.19 Ganassismic) is a proposed 19&24 2.3.(5).11.(13 or 13/5).17.19 temperament by [[User:Ground]] that seeks to cover all of the notable concordances generated by a meantone-range semifourth.<br />
<br />
== See also ==<br />
* [[Chthonic harmony]]<br />
* [[Semiquartal]]</div>Vectorhttps://xenreference.com/wiki/index.php?title=File:Semaphore5.png&diff=7390File:Semaphore5.png2026-06-02T01:56:26Z<p>Vector: </p>
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<div>dgndg</div>Vectorhttps://xenreference.com/wiki/index.php?title=Meantone&diff=7389Meantone2026-06-02T01:54:33Z<p>Vector: </p>
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<div>[[File:Meantone.png|thumb|Meantone equates four 3/2s to 5/1 and generates pentic (2L 3s) and mosdiatonic (5L 2s) scales.]]<br />
'''Meantone''', or rarely '''Syntonic''' or '''Didymus,''' is a widespread historical [[temperament]] that forms the basis of Western music theory, where the [[Perfect fifth|fifths]] are flattened to about 696[[Cent|c]] to produce a [[diatonic major third]] tuned to roughly [[5/4]], enabling the use of 5-limit harmony in the diatonic scale. When all the fifths are tuned the same, Meantone is a [[regular temperament]], where the period is the [[octave]], the generator is 3/2, and four generators stack to reach the 5th harmonic, meaning that the '''syntonic comma''', which is the difference between Pythagorean intervals and nearby 5-limit intervals and has a ratio of 81/80, is tempered out.<br />
[[File:Meantone7.png|thumb|Meantone diatonic scale]]<br />
As a monocot temperament, Meantone can be notated with standard [[diatonic notation]], and in fact diatonic notation works the best for Meantone as the 5-limit 4:5:6 harmonic triad becomes simply C-E-G on C, and the chromatic semitone is usually smaller than the diatonic semitone. Meantone is a 7-form temperament, and is tuned well around the golden tuning of diatonic. Unsurprisingly, 7edo supports Meantone, and so does 12edo (which is the simplest ET to do so without exotempering the 5-limit), and so the best tunings of Meantone lie in between those two extremes. [7 & 12] is thus the ''edo join'' for meantone.<br />
<br />
== Extensions ==<br />
The edo join [7 & 12] results in an exotempered extension called ''dominant'' where 7/4 and 9/5 are equated, and is tuned best around Pythagorean tuning.<br />
<br />
If an unmapped (not equated to a stack of anything else, as [[Classical major third|5/4]] is in Blackwood) prime 7 is introduced as a second generator, then the result can be called Didymus.7. It is supported by 36edo's patent val. (This is not technically an extension.)<br />
<br />
More accurate extensions of Meantone's diatonic structure to include other primes follow.<br />
<br />
==== 7/4 as the augmented sixth (12 & 19) ====<br />
This is the primary extension of Meantone to the 7-limit, where 7/4 is the augmented sixth (C-A#, +10 fifths). It is best tuned with the generator around 696 cents. 5/4 is 384 cents, and 7/4 is 960 cents. It is notable for being the most accurate extension, as well as containing the [[Golden sequences and tuning|golden tuning]] of the diatonic scale, and thus a melodically convenient chromatic and enharmonic scale. This means that the [[augmented diesis]] 128/125, is equated with the septimal quartertone 36/35, and the 5-limit supermajor and subminor intervals are equated with their septimal counterparts.<br />
<br />
However, one drawback of this temperament is the large degree of complexity required to get to the 11th and 13th harmonics. In fact, there are two main options. In both cases, the [[Tridecimal neutral thirds|tridecimal neutral third]] 16/13 is conflated with the [[Undecimal neutral thirds|undecimal neutral third]] 11/9, representing a characteristic tendency to make 11/9 the sharper of the two 11-limit neutral thirds. (As a result, one might find it useful to irregularly map 11/9.)<br />
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===== 11-limit[12 & 19] =====<br />
The 11-limit form of 12 & 19 is an exotemperament called ''meanenneadecal'', which tunes 11/8 very sharp and conflates 14/11 with 5/4 (because both 12edo and 19edo do so). More accurate extensions are below.<br />
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===== 11/8 as the double-augmented third (12 & 31) =====<br />
This is best tuned around 697 cents, and places 11/9 as the double-augmented second (C-Dx, +16 fifths) and conflates 14/11 with [[Septimal supermajor third|9/7]] placed as the diminished fourth (C-Fb, -8 fifths). 13/8 is mapped to the double-diminished seventh (C-Bbb, -9 fifths).<br />
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===== 11/8 as the double-diminished fifth (19 & 31) =====<br />
This is best tuned around 696 cents, and places 11/9 as the double-diminished fourth (C-Fbb, -15 fifths). 13/8 is mapped to the double-augmented fifth (C-Gxx, +15 fifths).<br />
<br />
==== 7/4 as the diminished seventh (19 & 26) ====<br />
This temperament, often called "Flattone", sets 7/4 equal to the diminished seventh, and is best tuned with the generator 3/2 around 693 cents, 5/4 at 372 cents, and 7/4 at 963 cents. It is a melodically intuitive extension, as it creates an [[equiheptatonic]] scale with a quartertone-sized chroma, and interval sizes tend to match with their corresponding interval categories. For example, it can be easily extended to map prime 11 to the augmented fourth (C-F#, +6 fifths) and 13 to the minor sixth (C-Ab, -4 fifths) tuned to around 558 and 828 cents respectively. 26edo is the most commonly used tuning, though it can be tuned more accurately with 45edo. It is a 7-cluster temperament, as indicated by the edo join (26 - 19 = 7).<br />
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== Chords ==<br />
Meantone's main feature is its conflation of the standard harmonic triad 4:5:6 with the diatonic major triad P1–M3–P5, thus equating the [[Diatonic #MOS diatonic|MOS diatonic scale]] with the 5-limit tuning of [[Diatonic #Greek diatonic scales|Ancient Greek diatonic]] and allowing for 5-limit consonances to be easily accessed within a continuous circle of fifths. Modern Western music theory, which is derived in large part from meantone practice, treats triadic harmony (chords made by stacking two thirds over a root) as the basis of concordance, as the only way to fit three [[5-odd-limit]] intervals in one octave is via some permutation of 4/3, 5/4, and 6/5, which will always make some rotation or retroversion of 4:5:6.<br />
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The major and minor seventh chords in meantone diatonic can be enumerated as 8:10:12:15 and 10:12:15:18 respectively. The dominant seventh chord is 20:25:30:36, or in septimal meantone, the 1–5/4–3/2–9/5 [[collection of chords #Essentially tempered chords|essentially tempered chord]]; the half-diminished seventh chord is similarly 25:30:36:45, or in septimal meantone, the 1–6/5–10/7–9/5 essentially tempered chord. Additionally, the 5:6:7:9 chord is available as P1–m3–A4–m7.<br />
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During the late Renaissance era, septimal meantone tunings were the basis of Augmented Sixth chords. The Italian Sixth chord can be enumerated as 4:5:7, with the intervals of a root, a major third, and an augmented sixth; the German Sixth chord adds an additional interval 3/2 above the root, providing a full 4:5:6:7, whereas the French Sixth chord adds the augmented fourth of 7/5, making a 20:25:28:35 chord.<br />
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The septimal triads, 6:7:9 and 14:18:21, can additionally be found at P1-A2-P5 and P1-d4-P5 respectively. These can be further extended to the septimal seventh chords, 12:14:18:21 and 14:18:21:27, which are respectively P1-A2-P5-A6 and P1-d4-P5-d1 in septimal meantone.<br />
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5-limit Meantone also contains an essentially tempered chord, where 1-9/8-3/2-5/3-2 contains steps of 9/8, 4/3, 9/8, and 6/5. Note that in just intonation, the top interval would be 27/16, not 5/3, or the two whole tones would be different sizes (resulting in a 40/27 [[Wolf interval|wolf]] fifth).<br />
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== Tunings ==<br />
As essentially the only temperament that is both [[regular]] and attested outside [[xenharmony]], Meantone has a number of historical tunings that today correspond to various extensions and approximate edos. Here, "comma" refers to the syntonic comma.<br />
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=== 1/11-comma Meantone ===<br />
This tuning of Meantone is almost perfectly approximated by 12edo, having a fifth tuning of nearly exactly 700 cents and a step ratio of nearly exactly 2 (~basic). 12edo by definition lowers the fifth by 1/12 of a Pythagorean comma; setting 1/12 of a Pythagorean comma to 1/11 of a syntonic comma is done in edos such as [[34edo#612edo|612edo]]. <br />
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=== 1/5-comma Meantone ===<br />
This tuning of Meantone equalizes the error on 3/2 and 5/4, tuning the former to 697.65 cents and the latter to 390.61 cents. Equivalently, it tunes 16/15 purely. Its step ratio is 1.748 (minisoft), and consequently it is well approximated by 43edo. <br />
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=== Quarter-comma Meantone ===<br />
This tunes the fifth 1/4-comma flat, to a size of 696.57 cents. It has a just 5/4, and a step ratio of 1.65 (quasisoft). It approximates [[31edo]], and is often (rather insultingly to the rest of 31edo) seen as the latter's primary feature. It extends to 11-limit 19 & 31.<br />
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=== Golden meantone ===<br />
''Main article: [[Golden generator]]''<br />
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Golden meantone is the tuning of meantone wherein the large and small steps of the diatonic scale are in the golden ratio. It is the only meantone tuning which produces exclusively soft scales, and meantone's general proximity to the golden tuning captures the difficulty of representing it (and the 5-limit as a whole) within a specific form. <br />
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=== 2/7-comma Meantone ===<br />
This is the tuning of Meantone situated roughly between 50edo's and 69edo's tunings, with a fifth of 695.81 cents and a step ratio of 1.584 (quasisoft). Consequently, it extends to Septimal Meantone, but with a rather poor approximation of 7/4. However, it tunes other septimal intervals like 9/7 and 7/6 more accurately. It tunes 25/24 purely, and can thus be considered a compromise between 1/4-comma's perfect 5/4 and 1/3-comma's perfect 6/5.<br />
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=== 1/3-comma Meantone ===<br />
This is the tuning of the fifth to 694.78 cents, which has a just 6/5 and is extremely close to [[19edo]], having a step ratio of 1.503 (~monosoft). As a result, it does not cleanly extend to the 11-limit, although as it is slightly sharp of 19edo it does technically extend to 7-limit 12 & 19.<br />
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=== Silver flattone ===<br />
Silver flattone is the tuning of meantone such that the step size ratios of the diatonic and enharmonic (19-note) scale steps are the same, and that that ratio is the square root of 2. Alternatively, the step size ratio found in the chromatic scale is the silver ratio, sqrt(2)+1. It is somewhat sharp for flattone, tuning 7/4 flat of 960 cents. Silver flattone is the soft counterpart of [[argent]] tuning.<br />
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=== 2/5-comma Meantone ===<br />
This is very close to the [[45edo]] tuning of Meantone, tuning the fifth 693.35 cents and having a just 27/25 (note that 27/25 is tempered together with 16/15 in this system, resulting in a sharp minor second). As a result of the flat tuning, this extends to Flattone, rather than to Septimal Meantone. Its step ratio is 1.401 (parasoft) and is thus close to silver flattone.<br />
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=== 1/2-comma Meantone ===<br />
This is close to 33edo's diatonic tuning, which is not Meantone. As a result, it can be considered the lower bound of Meantone's tuning, where the tone is tuned to a just 10/9. It tunes the fifth to 691.2 cents. Its step ratio is 1.26 (ultrasoft).<br />
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=== (Half Comma) Cleantone ===<br />
Cleantone is Hans-Peter Deutsch's tuning of Meantone which tempers the octave to be sqrt(81/80) = 10.8c sharp and retains a just 4:5:6. It tunes 5/4, 6/5, 9/5, and 15/8 (in fact, any interval in the JI group (3/2).(5/4)) justly, but 2/1 complements of these intervals are detuned.<br />
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=== Lucy Tuning ([[88edo|88edo]]) ===<br />
This is nearly indistinguishable from the 88edo tuning of Meantone, with a fifth (600 + 300/π cents) just 0.038 cents higher than 88edo's 695.55-ish fifth. Its major third (1200/π = 381.97 cents) is flat 4.3 cents, but closer than 1/3 Meantone's. The proper extension to the 7-limit (and possible 11-limit) is Mothra (8/7 = 200+100/π = 231.83 cents), which yields a 7/4 flat by only 0.659 cents.<br />
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=== [[31edo]] ===<br />
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=== [[19edo]] ===<br />
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=== [[12edo]] ===<br />
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== List of patent vals ==<br />
{| class="wikitable"<br />
|+<br />
!EDO<br />
!7-limit strong extensions<br />
!11-limit strong extensions<br />
!Generator tuning<br />
|-<br />
|5<br />
|Dominant<br />
|<br />
|720c<br />
|-<br />
|12<br />
|Dominant, Septimal Meantone<br />
|[12 & 31], [12 & 19]<br />
|700c<br />
|-<br />
|67<br />
|<br />
|<br />
|698.5c<br />
|-<br />
|55<br />
|<br />
|<br />
|698.2c<br />
|-<br />
|98<br />
|<br />
|<br />
|698c<br />
|-<br />
|43<br />
|Septimal Meantone<br />
|[12 & 31]<br />
|697.7c<br />
|-<br />
|117<br />
|<br />
|[12 & 31]<br />
|697.4c<br />
|-<br />
|74<br />
|Septimal Meantone<br />
|[12 & 31]<br />
|697.3c<br />
|-<br />
|105<br />
|Septimal Meantone<br />
|[12 & 31]<br />
|697.1c<br />
|-<br />
|31<br />
|Septimal Meantone<br />
|[12 & 31], [19 & 31]<br />
|696.8c<br />
|-<br />
|81<br />
|Septimal Meantone<br />
|[19 & 31]<br />
|696.3c<br />
|-<br />
|50<br />
|Septimal Meantone<br />
|[19 & 31]<br />
|696c<br />
|-<br />
|69<br />
|<br />
|[19 & 31]<br />
|695.7c<br />
|-<br />
|88<br />
|<br />
|[26 & 31]<br />
|695.5c<br />
|-<br />
|19<br />
|Septimal Meantone, Flattone<br />
|[12 & 19], [19 & 31], (Flattone)<br />
|694.7c<br />
|-<br />
|45<br />
|Flattone<br />
|(Flattone)<br />
|693.3c<br />
|-<br />
|26<br />
|Flattone<br />
|(Flattone)<br />
|692.3c<br />
|-<br />
|7<br />
|Dominant, Flattone<br />
|(Flattone)<br />
|685.7c<br />
|}<br />
<br />
<br />
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{{Navbox regtemp}}<br />
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{{Cat|Temperaments}}</div>Vectorhttps://xenreference.com/wiki/index.php?title=File:Meantone7.png&diff=7388File:Meantone7.png2026-06-02T01:54:21Z<p>Vector: </p>
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<div>gnd</div>Vectorhttps://xenreference.com/wiki/index.php?title=Didacus&diff=7387Didacus2026-06-02T01:51:32Z<p>Vector: </p>
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<div>{{Infobox regtemp<br />
| Title = Didacus<br />
| Subgroups = 2.5.7<br />
| Comma basis = 3136/3125 (2.5.7)<br />
| Edo join 1 = 6 | Edo join 2 = 25<br />
| Mapping = 1; 2 5 9<br />
| Generators = 28/25 | Generators tuning = 194.4 | Optimization method = CWE<br />
| MOS scales = [[1L 5s]], [[6L 1s]], [[6L 7s]], [[6L 13s]], [[6L 19s]]<br />
| Odd limit 1 = 2.5.7 7 | Mistuning 1 = 1.22 | Complexity 1 = 13<br />
}}<br />
[[File:Didacus13.png|thumb|Didacus 13-note MOS]]<br />
'''Didacus''' is a highly efficient temperament of the [[2.5.7 subgroup]], tempering out 3136/3125, such that two intervals of [[7/5]] reach the same point as three intervals of [[5/4]]; the generator is therefore a slightly narrowed (7/5)/(5/4) = [[28/25]], two of which stack to 5/4 and three of which stack to 7/5, meaning that the [[4:5:7]] chord is "locked" to (0 2 5) in terms of logarithmic size and generator steps. In the full [[7-limit]], Didacus tempering has the consequence of making 28/27 - 25/24 - 21/20 equidistant and 16/15 - 15/14 - 27/25 equidistant.<br />
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[[31edo]] is a very good tuning of Didacus, with its generator 5\31 (which is the "mean tone" of 31edo); but [[25edo]], [[37edo]], and [[68edo]] among others are good tunings as well. As this generator tends to be slightly less than 1/6 of the octave, [[MOS]] scales of Didacus tend to consist of 6 long intervals interspersed by sequences of diesis-sized steps (representing [[50/49]]~[[128/125]]), therefore bearing similar properties to those of [[Slendric]].<br />
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Septimal [[Meantone]] is a relatively inaccurate weak extension of Didacus to prime 3. It may be used in the 2.9.5.7 subgroup as a strong extension. A more accurate but complex full-7-limit extension is Hemiwurschmidt ({{e|31}} & {{e|37}}) which adds the [[Wurschmidt]] relation (5/4)<sup>8</sup> ~= 3/2.<br />
== Interval chain ==<br />
In the following table, odd harmonics and subharmonics 1–35 are labeled in '''bold'''. <br />
<br />
{| class="wikitable sortable center-all right-2"<br />
|-<br />
! rowspan="3" | &#35; !! rowspan="3" | Cents* !! colspan="4" | Approximate ratios<br />
|-<br />
! rowspan="2" | 2.5.7 intervals !! colspan="3" | Intervals of extensions<br />
|-<br />
! Tridecimal Didacus !! [[Luna and hemithirds#Intervals|Hemithirds]] !! Hemiwürschmidt (L11.23)<br />
<br />
|-<br />
| 0<br />
| 0.0<br />
| '''1/1'''<br />
|<br />
|<br />
|<br />
|-<br />
| 1<br />
| 194.4<br />
| 28/25, 125/112<br />
| 49/44, 55/49<br />
|<br />
| <br />
|-<br />
| 2<br />
| 388.9<br />
| '''5/4'''<br />
| 44/35<br />
|<br />
| 144/115<br />
|-<br />
| 3<br />
| 583.3<br />
| 7/5<br />
| 128/91<br />
| <br />
|-<br />
| 4<br />
| 777.7<br />
| '''25/16'''<br />
| 11/7<br />
|<br />
| 36/23<br />
|-<br />
| 5<br />
| 972.1<br />
| '''7/4'''<br />
| 44/25, 160/91<br />
|<br />
| 184/105<br />
|-<br />
| 6<br />
| 1166.6<br />
| 49/25, 125/64<br />
| 55/28, 128/65<br />
|<br />
| 96/49, 45/23<br />
|-<br />
| 7<br />
| 161.0<br />
| '''35/32'''<br />
| 11/10, 100/91<br />
| <br />
| 23/21, 126/115<br />
|-<br />
| 8<br />
| 355.4<br />
| 49/40<br />
| '''16/13'''<br />
| 128/105<br />
| 11/9, 27/22, 60/49, 92/75<br />
|-<br />
| 9<br />
| 549.9<br />
| 175/128<br />
| '''11/8'''<br />
| <br />
| 48/35, 63/46, 115/84<br />
|-<br />
| 10<br />
| 744.3<br />
| 49/32<br />
| 20/13, 77/50<br />
| '''32/21'''<br />
| 23/15, 75/49<br />
|-<br />
| 11<br />
| 938.7<br />
| <br />
| 55/32, 112/65<br />
| 128/75<br />
| 12/7<br />
|-<br />
| 12<br />
| 1133.1<br />
| <br />
| 25/13, 77/40<br />
| 40/21<br />
| 23/12, 48/25<br />
|-<br />
| 13<br />
| 127.6<br />
| <br />
| 14/13<br />
| '''16/15'''<br />
| 15/14<br />
|-<br />
| 14<br />
| 322.0<br />
| <br />
| 77/64, 110/91<br />
| 25/21<br />
| 6/5<br />
|-<br />
| 15<br />
| 516.4<br />
| <br />
| 35/26, 88/65<br />
| '''4/3'''<br />
| 75/56<br />
|-<br />
| 16<br />
| 710.8<br />
| <br />
| 98/65<br />
| 112/75<br />
| '''3/2'''<br />
|-<br />
| 17<br />
| 905.3<br />
| <br />
| 22/13<br />
| 5/3<br />
| 42/25<br />
|-<br />
| 18<br />
| 1099.7<br />
| <br />
| 49/26<br />
| 28/15<br />
| '''15/8'''<br />
|-<br />
| 19<br />
| 94.1<br />
| <br />
| 55/52<br />
| 25/24<br />
| 21/20<br />
|}<br />
<nowiki/>* In [[CWE]] undecimal didacus<br />
<br />
== List of patent vals ==<br />
See [[Didacus/Patent vals]].<br />
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{{cat|temperaments}}</div>Vector