Xenharmonic Reference - User contributions [en]

5-odd-limit

2026-07-31T16:58:31Z

Overthink: start section

{{Odd-limit navigation}}
The '''5-odd-limit''' is the set of intervals where the largest allowable odd factor in the numerator and denominator is 5. It is the smallest odd-limit containing intervals of the 5-limit. In general, the intervals of the 5-odd-limit are also those considered [[Consonance|consonances]] in standard Western music theory, and include as a subset the intervals of the 3-odd-limit, which are the perfect consonances, and of the 1-odd-limit (or 2-prime-limit), which are the unison, octave, and its multiples. This is where the xenharmonic generalization of a set of intervals considered 'consonances' comes from, and is why odd-limits are used as a complexity measure for JI intervals.

The 5-odd-limit is equivalent to the intervals considered to be consonant by Zarlino, constructed from the numbers 1, 2, 3, 4, 5, 6, and 8. (Note the absence of 7.)

== Table of 5-odd-limit intervals ==
Reduced to an octave, the intervals of the 5-odd-limit are:
{| class="wikitable"
!Interval
!Cents
!Name
!Type
|-
|1/1
|0.0
|Unison
|Equivalence
|-
|6/5
|315.6
|Classical minor 3rd
|Imperfect consonance
|-
|5/4
|386.4
|Classical major 3rd
|Imperfect consonance
|-
|4/3
|498.0
|Perfect 4th
|Perfect consonance
|-
|3/2
|702.0
|Perfect 5th
|Perfect consonance
|-
|8/5
|813.6
|Classical minor 6th
|Imperfect consonance
|-
|5/3
|884.4
|Classical major 6th
|Imperfect consonance
|-
|2/1
|1200.0
|Octave
|Equivalence
|}

== Approximation by edos ==
The first edo consistent to the 5-odd-limit is 3edo, outlining the structure of triadic harmony with the augmented triad. The minor 3rd, major 3rd, and perfect 4th are mapped to 400c, while the perfect fifth, minor sixth, and major sixth are mapped to 800 cents. Beyond this, 7edo is often the first edo to be seriously considered as approximating the 5-odd-limit, as its most damaging 5-limit temperament, [[Dicot]], does not lead to categorical conflicts the same way 3edo's [[Father]] does. However, for all the intervals of the 5-odd-limit to be distinctly represented, the smallest viable edo is 9edo. Although 9edo severely damages the perfect fifth and minor third, it does make all the categorical distinctions necessary to support some form of triadic harmony based on the contrast between major and minor, which is characteristic of the use of 5-odd-limit consonances.

The first edo to distinguish all of the 5-odd-limit intervals while tuning them all reasonably accurately is 12edo — this is one factor that led to 12edo's worldwide standardization. The second edo to do so, [[19edo]], is a [[Meantone]] tuning like 12edo though more accurate; the arithmetic of 5-limit intervals may lead to non-12edo results, such as [[Magic|five major thirds stacking to a fifth]]. [[22edo]], a non-Meantone tuning, has the opposite tuning tendencies to 12edo.

== Intervals of the 5-odd-limit ==

=== Perfect consonances ===

==== Perfect fourth (4/3) ====
''Main article: [[Perfect fourth]]''

The perfect fourth is a perfect consonance and the bounding interval for [[chthonic]] harmony. It also exists between the fifth over the root and the root an octave up. It is the dark generator of the diatonic scale; stacking it produces the Locrian [[mode]].

In certain triadic musical traditions that use 4:5:6 as a consonant chord, the perfect fourth over the root can be considered dissonant, as it resolves downwards to the major third.

==== Perfect fifth (3/2) ====
''Main article: [[Perfect fifth]]''

The perfect fifth is an unambiguous perfect consonance. It appears in musical systems worldwide, and can be easily tuned by ear. This is the reason behind the prevalence of the [[Pythagorean tuning|Pythagorean]] system of tuning.

In the context of 5-limit consonances, the perfect fifth serves as the bounding interval of the triads 10:12:15 (minor) and 4:5:6 (major), which utilize the other 5-odd-limit consonances 5/4 and 6/5.

=== Imperfect consonances ===

==== Classical major third (5/4) ====
The major third 5/4 serves primarily as a component in tertian chords like 4:5:6. It is the most consonant "third" interval. 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.

5/4 is also the octave-reduced generator of the [[5-limit|prime 5]] axis in [[lattice-based just intonation]].

==== Classical minor third (6/5) ====
The classical minor third 6/5 is the fifth complement of 5/4. The distinction between the two leads to the paradigm of major vs. minor in interval classification and in triadic harmony; it is why the "third" category of intervals exists at all, and additionally why thirds are often considered the "default" example of interval qualities.

One quality of 6/5 worth noting is that chords with 6/5 as a lower interval are, as a rule, not "rooted" (in that their root note is not a power of 2 in the harmonic series). The significance of this is debated by xenharmonic theorists; Lamplight uses it as a model for the different "feels" of the chords 4:5:6 and 10:12:15.

==== Classical major sixth (5/3) ====
The classical major sixth is the octave complement of 6/5. It is according to some the next most consonant interval within the octave after 4/3; Leriendil sees it as an important target interval on the level of 4/3 and it is also the bounding interval of the chord 3:4:5, which may be seen as an inversion of 4:5:6 or as the primary focus of 5-limit "/3" harmony (such as in [[Kleismic]]).

==== Classical minor sixth (8/5) ====
The classical minor sixth is, while consonant on its own, unusually dissonant for a 5-odd-limit consonance in certain contexts; the result is likely a combination of factors. First is its complex ratio - it is the only 5-odd-limit interval in the octave that uses 8 in the numerator. Second is its proximity to the golden ratio, which serves as a distinctly dissonant target (similar to the [[semioctave]]'s influence on 7/5). Third is its proximity to 3/2, which produces a 'zone' of dissonance around it. Also of relevance to the discussion is the 12edo augmented triad, which contains a note tuned the same way as the classical minor sixth (and which may be voiced as it in JI depending on interpretation) yet is considered a dissonant chord.

This and the major sixth mainly show up in chords in Western harmony as the bounding intervals of triads in certain inversions.

== Intervals separating adjacent 5-odd-limit ratios ==
{{WIP}}
=== 9/8 ===
{{See also|9-odd-limit #9/8|Pythagorean tuning #Major second}}
9/8 is the difference between 3/2 and 4/3, and is known as the Pythagorean major 2nd, or (major) whole tone. It is traditionally considered an imperfect dissonance, being less dissonant than the semitones and the tritone, but less consonant than the 5-odd-limit consonances.

=== 16/15 ===
=== 25/24 ===

Pythagorean tuning

2026-07-31T16:12:14Z

Overthink: /* Schismic and Garibaldi */ template

'''Pythagorean tuning''' is the tuning system in which only '''3-limit''' just intonation intervals are used - that is, intervals generated by stacking [[Perfect fifth|perfect fifths]] of 3/2 and [[Octave|octaves]] of 2/1 up and down. Pythagorean tuning is a rank-2 system that does not include any tempering, and is thus useful as a basis for notation. When accounting for octave equivalence, Pythagorean tuning mirrors the structure of the [[chain of fifths]]. Simple Pythagorean intervals include 3/2, 9/8, 32/27, 81/64, and their octave complements.

== Pythagorean tuning and temperaments ==
Note that Pythagorean tuning often refers to the ''tuning'', not the interpretation, and this is its distinction from the 3-limit - that is, some people consider [[Regular temperament|regular temperaments]] that are well-tuned in Pythagorean tuning to, themselves, count as Pythagorean.

==== Schismic and Garibaldi ====
{{Main|Schismic}}

The most notable example of this is '''Schismic temperament''', which equates the moderately complex Pythagorean interval 8192/6561, the diatonic diminished fourth, to [[5/4]], which when tuned to just Pythagorean tuning has only 2 cents of error, and its extension '''Garibaldi''', which further equates the double-diminished octave to [[7/4]], with only 4 cents of error when the former is tuned just. There is also no reason, if you are using Schismic, to not further equate 19/16 to 32/27, tempering out 513/512, at only 3 cents of error.

Garibaldi in particular has the advantage of equating both 81/80 and 64/63 to the same interval, in particular the Pythagorean comma. Thus Garibaldi supports Hemifamitt temperament.

== Monocot ==
Monocot is the [[temperament archetype]] where an octave is the period and a perfect fifth is the generator. Monocot is equivalent to the standard chain of fifths, going ... B♭ - F - C - G - D - A - E - B - F♯ ... , and is strongly associated with the [[diatonic]] scale as the MOS form of diatonic is generated by a perfect fifth and octave. Common monocot temperaments include the aforementioned Schismic, as well as [[Meantone]] and [[Archy]].

Monocot is the only [[ploidacot]] to have an agreed-upon, fully unambiguous scheme for interval and note names.

Generally, "monocot" is broader than "Pythagorean", as Pythagorean implies that the fifth is tuned to a perfect 3/2, while monocot temperaments tune the fifth to a wide range of tunings.

== Intervals of Pythagorean diatonic ==
=== Minor second ===
{{Infobox interval|256/243|Name=diatonic semitone, diatonic minor second, limma}}The 3-limit '''diatonic semitone''', also called the '''diatonic minor second (m2)''' or the '''limma''', and represented by the ratio 256/243, is the smaller of the two seconds (1-step intervals) in the MOS diatonic scale. It is generated by stacking 5 fourths and octave-reducing. In [[Pythagorean tuning|Pythagorean]] tuning, and thus purely-tuned [[just intonation]], it is approximately 90.2 cents in size, but as an interval in the abstract diatonic scale it may range between 0 and 171 cents, depending on the tuning.

It functions as the small step of the diatonic MOS, and along with the [[diatonic major second]] may be used to construct other diatonic intervals. For example, the [[diatonic minor third]] is a major second stacked with a minor second. The [[chromatic semitone]] is the difference between these two intervals. Note that the chromatic semitone itself is distinct from the diatonic semitone; they are separated by the [[12edo#Compton temperament|Pythagorean comma]], which separates all [[enharmonic]] intervals in Pythagorean tuning, and which, if tempered out, yields 12edo.

As a harmonic interval, the diatonic semitone is usually considered a dissonance, due to its small size and complex ratio.

The diatonic scale contains two minor seconds. In the Ionian mode, minor seconds are found on the third and seventh scale degrees; the others have major seconds. The small number of minor seconds compared to major seconds ensures that thirds that include minor seconds (that is, minor thirds) are roughly evenly distributed with major thirds; in a scale with three small steps and four large steps, for example, six out of the seven thirds are minor.

The diatonic semitone is a product of square superparticulars, denoted [[Interseptimal diesis|S7]] * [[Archy|S8]]2. When tempered out, it leads to Blackwood temperament, which tunes the [[2.3.7 subgroup]] to [[Equipentatonic#5edo|5edo]].

=== Major second ===
{{Infobox interval|9/8|names=Chromatic semitone, augmented unison|name=Chromatic semitone, augmented unison|Name=Diatonic major second}}

The '''diatonic major second''' ('''M2'''), represented by the frequency ratio '''9/8''', is the larger of the two seconds (1-step intervals) in the MOS form of the diatonic scale. It is generated by stacking 2 fifths and octave-reducing. In [[Pythagorean tuning]] (and thus purely-tuned [[just intonation]]), it is approximately 203.9 cents in size, but as an interval in the abstract diatonic scale it may range from 171 to 240 cents, depending on the tuning.

It functions as the large step of diatonic, and along with the [[diatonic semitone]] (diatonic minor second) may be used to construct other diatonic intervals. For example, the [[diatonic major third]] is two major seconds stacked, and the [[diatonic minor third]] is a major second stacked with a minor second. The [[chromatic semitone]] is the difference between these two intervals.

As a harmonic interval, the diatonic major second is considered a dissonance in most contexts, due to its small size, but can in some contexts (such as [[arto and tendo theory]]) be considered a consonance or ambisonance. In 5edo, it is a consonant 8/7 interval much like the [[chromatic semitone]].

The diatonic scale contains five major seconds. In the Ionian mode, major seconds are found on the 1st, 2nd, 4th, 5th, and 6th degrees of the scale; the other two degrees have minor seconds. The large number of major seconds compared to minor seconds ensures that thirds that include minor seconds (that is, minor thirds) are roughly evenly distributed with major thirds; in a scale with three small steps and four large steps, for example, six out of the seven thirds are minor.
=== Major third ===
{{Infobox interval|81/64|Name=Diatonic major third}}
The '''diatonic major third (M3)''', represented by the frequency ratio '''81/64''', is the larger of the two thirds (2-step intervals) in the MOS form of the diatonic scale. It is generated by stacking 4 fifths octave-reduced. In [[Pythagorean tuning]] (and thus purely-tuned [[just intonation]]), it is approximately {{Cents from ratio|81/64|ratio=81/64}} cents in size, but as an interval in the abstract diatonic scale it may range from 343 to 480 cents, depending on the tuning.

It can be constructed by stacking two [[Diatonic major second|diatonic major seconds]], and as such may be called the '''ditone'''.

As a harmonic interval, the diatonic major third may be considered either a consonance or a dissonance depending on its tuning. Important tuning targets for the diatonic major third are 5/4 ([[Meantone]] temperament), 14/11 (Pentacircle temperament), 9/7 ([[Archy|Archytas]] temperament) and 13/10 (Oceanfront temperament).

The diatonic scale contains three major thirds. In the Ionian mode, major thirds are found on the first, fourth, and fifth degrees of the scale; the other four degrees have minor thirds. This roughly equal distribution leads to diatonic tonality being largely based on the distinction between major and minor thirds and triads.

{{Interval regions}}
{{Cat|JI groups}}

Adaptive diatonic interval names

2026-07-04T21:30:19Z

Overthink: /* Interval regions */ fix error

The system of '''adaptive diatonic interval names (ADIN)''', developed by Vector and Leriendil, is a way to (mostly) uniquely label the intervals in an EDO based on size and relation to that EDO's patent fifth. It is ''diatonic'' because it attempts to behave predictably relative to MOSdiatonic staff notation, and it is ''adaptive'' because the differing qualities of diatonic intervals in different tunings are reflected in the interval names (that is to say, it "adapts" to different diatonic tunings). Finally, it is an ''interval naming system'', not a notation system, because it provides no way to write notes and labels intervals based on "what they are", not "what they do". (The creator of the ADIN system endorses [[Modified ups and downs notation|ups and downs notation]] for the latter.)

It is an attempt at formalizing the systems of interval qualities used by various xenharmonic resources on the internet.

=== On "major" vs. "supermajor" ===
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, alongside resources for temperaments such as 41edo (31et.com) and 27edo (Lumatone), reserve "major/minor" for 5/4 and 6/5, distinguishing an unmarked "major" from terms like "supermajor" (and "minor" from terms like "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. This is especially egregious as in [[Archy|Archytas]] temperaments (a major form of structural temperament for the 7-limit), diatonic notation has the opposite behavior, leaving 9/7 and 7/6 unmarked while 5/4 and 6/5 get the extra prefix, meanwhile standard diatonic notation in [[just intonation]], [[Schismic]], [[Aberschismic]], etc. uses "major" to refer to neither, instead denoting an interval in between the two, leaving only Meantone temperaments unambiguous.

While the use of "major" in standard diatonic notation is not a problem on its own, it leads to a large degree of ambiguity with naming schemes wherein 5/4 is prioritized outside of meantone temperaments. To resolve this, ADIN provides the original label "nearmajor" for intervals with a similar quality to 5/4, and "nearminor" for intervals with a similar quality to 6/5. Additionally, "farmajor" and "farminor" are used to refer to intervals in the standard JI diatonic range, regardless of the actual tuning of the diatonic scale. 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.

Unqualified "major" may refer to the range of major qualities 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 specifically the MOS diatonic intervals. Likewise for minor.

== Premise ==
ADIN names qualities, and then applies those names to intervals based on their distance from the nearest (possibly imaginary) diatonic neutral interval. The diatonic neutral intervals are as follows:

* Semidiminished unison (-3.5 fifths)
* Neutral second (-1.5 fifths)
* Neutral third (+0.5 fifths)
* Semiaugmented fourth (+2.5 fifths)
* Semidiminished fifth (-2.5 fifths)
* Neutral sixth (-0.5 fifths)
* Neutral seventh (+1.5 fifths)
* Semiaugmented octave (+3.5 fifths)

Intervals are named on a per-octave basis (that is, by octave-reducing, naming the interval, and adding back octaves according to conventional interval arithmetic), so the semidiminished unison and semiaugmented octave (which are lesser than and greater than the unison and octave respectively) do not actually appear in any interval names. Instead, they are chosen to ensure that the boundary between "unison" and "second" always falls precisely halfway between the perfect unison and the minor second.

These intervals may not exist in an edo (for instance, if it maps the fifth to an odd number of steps). This is okay, as they are being used as points of reference to compare to, not as actual necessary steps in the edo.

== Interval regions ==
Each neutral interval defines a series of regions (or "qualities") extending outwards from it, which are defined in terms of equal divisions of [[15/14]]. The use of 15/14 was proposed by [[User:Lériendil|Lériendil]] for threefold reasons:
* Firstly, 15/14 is a mapping of the [[apotome]] in [[aberschismic]] tunings: that is, it is the interval between [[7/6]] and [[5/4]] and between [[6/5]] and [[9/7]], and therefore the interval between the midpoint of 7/6 and 6/5, and the midpoint of 5/4 and 9/7;
* Secondly, it is close to 120 cents, which is the maximum amount of separation an interval can have from a diatonic neutral (assuming the fifth does indeed generate a diatonic scale), ensuring all intervals can be named;
* Finally, it is not itself an equal division of the octave, ensuring that no EDO intervals (aside from the true neutrals) land on region boundaries.
{| class="wikitable"
|+
!\25ed(15/14)
!Cents
!Major
!Minor
|-
|0
|0
| colspan="2" |neutral
|-
|0-2
|0-9.6
|tendoneutral
|artoneutral
|-
|2-5
|9.6-23.9
|submajor
|supraminor
|-
|5-10
|23.9-47.8
|nearmajor
|nearminor
|-
|10-15
|47.8-71.7
|farmajor
|farminor
|-
|15-20
|71.7-95.6
|supermajor
|subminor
|-
|20+
|95.6+
|ultramajor
|inframinor
|}
For instance, assuming a fifth is tuned to JI, the categories of thirds are found at <255c (inframinor), 256-279c (subminor), 280-303c (farminor), 304-327c (nearminor), 327-341c (supraminor), 342-360c (neutral, arto/tendo-), 361-375c (submajor), 376-398c (nearmajor), 399-422c (farmajor), 423-446c (supermajor), and >446c (ultramajor).

With these, the complete sets of intervals of each edo may be given a name. When an interval is an equal distance from two neutrals, thirds are always given precedence over fourths (so that an interval equidistant between the neutral third and neutral fourth is always a kind of third), and over seconds, which take precedence over unisons (except for the perfect unison and octave). The same rules apply to the complementary region of the octave. Fourths always take precedence below the tritone, and fifths always take precedence above it.

The exception is when the diatonic intervals coincide, in which case the conflated interval belongs to the category corresponding to its simplest diatonic interpretation (i.e. 240c is a second, not a third, and 480c is a fourth, not a third or (diminished) fifth). The same applies to oneirotonic and antidiatonic structures.

If there is only one kind of major or minor, drop all prefixes on major and minor. For example, if the only interval qualities found are "farminor", "neutral", and "farmajor", then rename "farminor" to "minor" and "farmajor" to "major". As a result, skip step 3.

== Disambiguation ==
In large edos, multiple intervals may be assigned the same name at the current point. This is where the disambiguation scheme comes into play. Based on the number of intervals in each category, a fixed set of names is assigned in order of size.
{| class="wikitable"
|+
!Quality
!2
!3
|-
|inframinor
|arto, inframinor
|
|-
|subminor
|sensaminor, gothminor
|sensaminor, septiminor, gothminor
|-
|farminor
|neominor, novaminor
|neominor, triminor, novaminor
|-
|nearminor
|valaminor, magiminor
|valaminor, pentaminor, magiminor
|-
|supraminor
|daemominor, aurominor
|
|-
|artoneutral
|subneutral, artoneutral
|
|-
|tendoneutral
|tendoneutral, supraneutral
|
|-
|submajor
|auromajor, daemomajor
|
|-
|nearmajor
|magimajor, valamajor
|magimajor, pentamajor, valamajor
|-
|farmajor
|novamajor, neomajor
|novamajor, trimajor, neomajor
|-
|supermajor
|gothmajor*, sensamajor
|gothmajor*, septimajor, sensamajor
|-
|ultramajor
|ultramajor, tendo
|
|}

In cases where there are two intervals belonging to the nearminor/major, farminor/major, and subminor/supermajor qualities, "pentamajor", "trimajor", and "septimajor" are substituted in for major thirds within 4.78{{c}} (1\25ed(15/14)) of the characteristic just intervals 5/4, 19/15, and 9/7 respectively**. If any major third acquires one of these subqualities, it is then propagated to its complement and other interval degrees.

=== Alternative system ===
Primarily in the case of tuning systems other than EDOs, or large EDOs where more than 3 intervals exist within the space of a single quality band, another fallback system can be used to assign subqualities to specific intervals.

"Trimajor" is defined as a radius of 1\25ed(15/14) around 19/15**, the same way as it is above. "Septimajor" then directly occupies the band 1\5ed(15/14) sharp of trimajor, while "pentamajor" occupies the band 9\50ed(15/14) flat of trimajor. The sharp edge of pentamajor is then taken to be the edge between auromajor and daemomajor. Subneutral and supraneutral intervals are not distinguished in this system.

Pentamajor and septimajor can variantly be defined to center around 5/4 and 9/7 as above, for the sake of consistency with the system generally employed for EDOs.

In cent values, with a justly tuned 3/2, the subqualities sharpward of the neutral third are then bounded as follows:
* 350.978 <- tendoneutral -> 360.533 <- auromajor -> 368.634 <- daemomajor -> 374.866
* 374.866 <- magimajor -> 382.967 <- pentamajor -> 392.522 <- valamajor -> 398.755
* 398.755 <- novamajor -> 404.467 <- trimajor -> 414.022 <- neomajor -> 422.643
* 422.643 <- shrubmajor* -> 428.355 <- septimajor -> 437.911 <- sensamajor -> 446.532

In that case, two intervals falling within the same subquality can then be disambiguated as "small" and "large", or three as "small", "mid", and "large".

* "Shrub-" can be replaced with "goth-".

** A variation would be for 5/4, 19/15, and 9/7 to be substituted here with sqrt(25/24), sqrt(722/675), and sqrt(54/49) above the neutral third, snapping all subqualities to the same positions relative the neutral third.

== Final steps ==
There are some additional replacements to be done:

1) Examine the diatonic fourth and whether it is major or minor. Remove the corresponding quality from all fourth names (for example, if the diatonic fourth is a farminor fourth, replace all instances of "minor fourth" with simply "fourth". Rename the opposing quality from "major" to "augmented", or "minor" to "diminished". If the fourth is any kind of neutral, no change is necessary to any interval names.

2) Label the diatonic fourth "perfect fourth" regardless of its quality.

3) Repeat for the unison, fifth, and octave.

3a) The result may create ambiguities with terms like "far octave" in some edos (the smallest edo to feature this problem being 26edo, between 25\26 and 27\26). In that case, restore "major" to octaves, fifteenths, etc above their perfect counterparts and which have ambiguous labels, and "minor" to fifteenths and above.

4) If quality is not necessary to distinguish intervals at all, remove it entirely (i.e. if there are only neutral intervals, do not specify "neutral").

== Qualities in small diatonic EDOs ==
Below lists the palettes of neutral and major qualities (noting that minor qualities always exist as the complements of major qualities) that can be found in diatonic EDOs below about 60, that is, the EDOs that do not require the disambiguation step. A few EDOs have two diatonic fifths, one which is divisible in two and one which is not. Both fifths are kept track of, but non-patent fifths are in parentheses.

Ultramajor qualities are treated separately, since they are ambiguous in degree. However, for EDOs with flat fifths ([[19edo]] or flatter) and which divide the perfect fourth in two, subminor and supermajor qualities are in fact interordinal (e.g. supermajor thirds are the same as sub(minor) fourths). These EDOs will be marked with an asterisk. Some EDOs with sharp fifths have ultramajor (and inframinor) intervals which are, however, not interordinal; these will be marked with a superscript plus sign.

=== Without a neutral third ===
EDOs without a neutral third have:
* with a step size 21.25-27.3{{c}} -> '''submajor, nearmajor, farmajor, supermajor'''
** diatonic fifths: [[46edo|46]], [[47edo|47]]*, [[49edo|49]], [[50edo|50]], [[53edo|53]], [[56edo|56]]+ ([[52edo|52b]]+, [[54edo|54b]])

* with a step size 27.3-28.65{{c}} -> '''submajor, nearmajor, farmajor'''
** diatonic fifths: [[42edo|42]]+, [[43edo|43]]

* with a step size 28.65-31.85{{c}} -> '''submajor, nearmajor, supermajor'''
** diatonic fifths: ''[[39edo|39]]'', [[40edo|40]]

* with a step size 31.85-38.2{{c}} -> '''submajor, farmajor, supermajor'''
** diatonic fifths: [[32edo|32]], [[33edo|33]]*, ''[[36edo|36]]''

* with a step size 38.2-47.8{{c}} -> '''submajor, farmajor'''
** diatonic fifths: [[26edo|26]], [[29edo|29]]

* with a step size 47.8-63.7{{c}} -> '''nearmajor, supermajor'''
** diatonic fifths: [[19edo|19]]*, [[22edo|22]]

* with a step size > 63.7{{c}} -> '''major'''
** diatonic fifths: [[12edo|12]]

=== With a neutral third ===
EDOs with a neutral third have:
* with a step size 19.1-23.9c -> '''neutral, submajor, nearmajor, farmajor, supermajor'''
** diatonic fifths: [[51edo|51]], [[52edo|52]]*, [[54edo|54]], [[55edo|55]], ''[[58edo|58]]'', [[61edo|61]]+, [[62edo|62]] (''[[57edo|57b]]''+, [[59edo|59b]])

* with a step size 23.9-31.85c -> '''neutral, nearmajor, farmajor, supermajor'''
** diatonic fifths: [[38edo|38]]*, [[41edo|41]], [[44edo|44]], [[45edo|45]], [[48edo|48]] ([[47edo|47b]]+)

* with a step size 31.85-35.85c -> '''neutral, nearmajor, farmajor'''
** diatonic fifths: [[34edo|34]], [[37edo|37]]+

* with a step size 35.85-47.8c -> '''neutral, nearmajor, supermajor'''
** diatonic fifths: [[27edo|27]], [[31edo|31]]

* with a step size 47.8-71.65c -> '''neutral, (far)major'''
** diatonic fifths: [[17edo|17]], [[24edo|24]]

== Notes ==
The first EDO this system fails to name the intervals for is currently 159edo, as it has four intervals within each supermajor range.

== Extensions ==

=== Oneirotonic ===
Add an extra ordinal for "tritone" rather than just treating it as a special case for even edos. The chroma is the moschroma of oneirotonic.

=== Antidiatonic ===
The chroma is the moschroma of antidiatonic. Note that the pythagorean semidiminished unison is still the center of the unison range, despite being larger than 0c.

7-odd-limit

2026-06-21T18:25:44Z

Overthink: /* Chords dividing the perfect fourth */ example

{{Odd-limit navigation}}
The '''7-[[odd-limit]]''' consists of all intervals where the largest allowable odd factor in the numerator and denominator is 7. It is the smallest odd-limit containing intervals of the [[7-limit|7-prime-limit]], thus creating xenharmonic categories not found in traditional music theory. In a 7-prime-limit system, all the ratios of the 7- or [[9-odd-limit]] can be treated as consonances.

== Table of 7-odd-limit intervals ==
Reduced to an octave, the intervals of the 7-odd-limit are:

{| class="wikitable"
! Interval
! Cents
! Name
|-
| 1/1
| 0.0
| Unison
|-
| '''8/7'''
| '''231.2'''
| '''Septimal major 2nd'''
|-
| '''7/6'''
| '''266.9'''
| '''Septimal minor 3rd'''
|-
| 6/5
| 315.6
| Classical minor 3rd
|-
| 5/4
| 386.4
| Classical major 3rd
|-
| 4/3
| 498.0
| Perfect 4th
|-
| '''7/5'''
| '''582.5'''
| '''Lesser septimal tritone'''
|-
| '''10/7'''
| '''617.5'''
| '''Greater septimal tritone'''
|-
| 3/2
| 702.0
| Perfect 5th
|-
| 8/5
| 813.6
| Classical minor 6th
|-
| 5/3
| 884.4
| Classical major 6th
|-
| '''12/7'''
| '''933.1'''
| '''Septimal major 6th'''
|-
| '''7/4'''
| '''968.8'''
| '''Septimal minor 7th'''
|-
| 2/1
| 1200.0
| Octave
|}

== Approximation by edos ==
[[File:7-odd-limit in edos.png|thumb|right|A diagram showing the approximation of the 7-odd-limit by various edos.]]
The first [[edo]] consistent to the 7-odd-limit is [[4edo]], which maps 5/4 to 1 step, 3/2 to 2 steps, and 7/4 to 3 steps, laying down a rough framework of tetradic harmony. Then, [[10edo]] approximates the 7-odd-limit relatively accurately for size, though it conflates several interval pairs: 5/4~6/5, 7/6~8/7, and 7/5~10/7. As such, the [[10-form]] is useful for classifying the 7-limit. After that, [[12edo]] distinguishes 5/4 from 6/5 and 7/6 from 8/7, though it has 6/5~7/6 and 7/5~10/7, and the 7th harmonic is tuned very sharply. The [[15edo]] and [[19edo]] tunings distinguish 5/4, 6/5, and 7/6, as well as 7/5 and 10/7, but 7/6 is equated to 8/7, an equivalence known as [[Interseptimal (temperament)|Interseptimal or Semaphore temperament]]. The first to distinguish all of 5/4, 6/5, 7/6, and 8/7 is [[22edo]], though 7/5 is still equated with 10/7. The first edo to distinguish the entire 7-odd-limit is [[27edo]], but one may prefer [[31edo]] for a more accurate approximation.

== Intervals of the 7-odd-limit ==
=== 8/7 ===
The '''8/7''' interval can be considered the '''septimal major second''', or '''supermajor second''', by diatonic interval classification, in the sense that it is slightly wider than the [[9/8]] major second at 231.2 cents. Due to its larger size compared to 9/8, it does not cause as much crowding, and is thus more consonant. It is also approximately 1/3 of the [[perfect fifth]], and it is mapped as such in the [[Slendric]] temperament.

It can also be considered in an ambiguous category of "semifourths" between seconds and thirds. Here, it is a minor interval, with 7/6, its fourth complement, being the corresponding major interval. The 7/6 and 8/7 intervals can be stacked to make fourth-bound triads; see [[#Triads dividing the perfect fourth]].

=== 7/6 ===
The '''7/6''' interval is known as the '''septimal minor third''' or '''subminor third''', since it is narrower than the Pythagorean minor third [[32/27]] and the classical minor third [[6/5]], being 266.9 cents in size. We can build a triad bounded by the [[perfect fifth]], that being 1–7/6–3/2. The interval between 7/6 and 3/2 is [[9/7]], which can be considered the supermajor third, being the fifth complement of 7/6. (However, note that 9/7 is a [[9-odd-limit]] interval, not a 7-odd-limit one.) We can also stack 7/6 on top of a triad to get a seventh chord; for example, stacking 7/6 on top of the 1–5/4–3/2 major triad gives us 1–5/4–3/2–7/4, the harmonic seventh chord.

As described in [[#Triads dividing the perfect fourth]], 7/6 can also be seen as contrasting with [[#8/7|8/7]] in triads such as 1–7/6–4/3, with 7/6 being considered the major counterpart of 8/7.

=== 7/5 ===
The 7/5 interval can be called the '''lesser septimal tritone''', having a size of 582.5 cents. It is called the ''lesser'' septimal tritone because the "greater septimal tritone" is [[#10/7|10/7]], its octave complement, from which it differs by [[50/49]], the jubilisma. Unlike the tritone found in [[12edo]], it is a ''consonant'' tritone, having a more restful sound than the half-octave. It is found between the third and the seventh of the 1–5/4–3/2–7/4 harmonic seventh chord. It is also the outer interval of the 1–6/5–7/5 diminished triad, which is the simplest and most consonant diminished triad in JI.

In systems such as [[HEJI]] and the [[FJS]], it is a diminished fifth, being the difference between [[5/4]], which is a major third, and [[#7/4|7/4]], which is a minor seventh. However, since it is less than a half-octave, it can also be classified as an augmented fourth, and it is mapped as such in septimal [[Meantone]] temperament. As such, interval categories in the 7-limit are rather ambiguous, and 7/4 has qualities of both a sixth and a seventh, instead of simply being a subminor seventh.

It is fairly close to the Pythagorean diminished fifth [[1024/729]], being flat of it by an [[Aberschisma]], or about 5.8 cents. It is also rather close to the [[5-limit]] tritone [[45/32]], being flat of it by the [[Marvel]] comma 225/224.

=== 10/7 ===
The 10/7 interval can be named the '''greater septimal tritone''', being 617.5 cents in size, analogous to how [[#7/5|7/5]] is called the lesser septimal tritone. It is somewhat less consonant than 7/5 due to its more complex ratio, though it is still considerably more consonant than the half-octave. It can be seen as a stack of 5/4 and 8/7, appearing in chords such as 1–7/4–5/2 and 1–6/5–3/2–12/7.

=== 12/7 ===
The interval 12/7, known as the '''septimal major sixth''' or '''supermajor sixth''', measures at 933.1 cents in size. It is the octave complement of [[#7/6|7/6]], and the twelfth complement of [[#7/4|7/4]]. Chords using it include 1–9/7–3/2–12/7 and 1–6/5–3/2–12/7.

It is somewhat ambiguous and can be considered in a category of "hemitwelfths" between sixths and sevenths, where it is a minor interval, and 7/4 is its major counterpart.

=== 7/4 ===
The 7/4 interval is often called the '''septimal minor seventh''', '''subminor seventh''', or '''harmonic seventh''', being 968.8 cents in size. Being the octave-reduced seventh harmonic, it can naturally be added to a 1–5/4–3/2 major triad to get 1–5/4–3/2–7/4, often called the ''harmonic seventh chord''. Due to its simpler ratio, it is more consonant than [[16/9]], the Pythagorean minor seventh, and [[9/5]], the classical minor seventh.

Though often considered a minor seventh, it also has some qualities of a sixth. For example, the [[#7/5|7/5]] interval can be considered an augmented fourth due to being smaller than the semioctave, so 7/4 would accordingly be an augmented sixth. Thus 7/4 is somewhat ambiguous by diatonic classification, and can be considered to be in a category between a sixth and a seventh, a "hemitwelfth". Here 7/4 is a major interval, with the corresponding minor interval being [[#12/7|12/7]].

== Harmony in the 7-odd-limit ==
=== Tetradic harmony ===
The 7-odd-limit is where tetrads start to get more prevalent. For example, we can build the 1–5/4–3/2–7/4 "harmonic seventh chord" by adding 7/4 on top of a 1–5/4–3/2 major triad. The harmonic seventh chord sounds somewhat similar to the dominant seventh chord, except it is more consonant and resolved. It can also be called the "major tetrad", similarly to how 1–5/4–3/2 is called the major triad.

The 5-limit minor triad 1–6/5–3/2 can be derived by reflecting every note about the midpoint of the root and the fifth. If we do the same for the harmonic seventh chord and reduce the steps to an octave, then we get 1–6/5–3/2–12/7 "subharmonic seventh chord" or "minor tetrad".

=== Chords dividing the perfect fourth ===
{{Main|Chthonic harmony}}
The 7/6 and 8/7 intervals can be seen as contrasting with each other, differing from each other by 49/48 (35.7 cents). We can build triads by stacking 7/6 and 8/7 that span the [[perfect fourth]], such as the 1–7/6–4/3 triad, which may also be voiced as 1–3/2–7/4. The minor version of this triad is 1–8/7–4/3, which can also be voiced as 1–3/2–12/7. This is analogous to how [[5/4]] and [[6/5]] contrast each other in the 1–5/4–3/2 and 1–6/5–3/2 triads, but these septimal triads split the perfect fourth, rather than splitting the [[perfect fifth]] like 5-limit triads do. As such, it can be considered a form of [[chthonic harmony|"semiquartal" or "chthonic" harmony]], which is one approach to septimal harmony.

Here, 7/6 is a type of major interval, and 8/7 is a type of minor interval. Their octave complements can be classified accordingly, with 12/7 being a minor interval, and 7/4 being a major interval. This is different from diatonic, where 8/7 is a supermajor second, 7/6 a subminor third, 12/7 a supermajor sixth, and 7/4 a subminor seventh.

Larger chthonic chords, such as tetrads bound by a [[cocytic]] (fifth-sixth), exist as well; details are given in the main article.

Template:Intervals

2026-06-21T18:23:35Z

Overthink: redirect

#redirect [[Template:Interval regions]]

Interordinal

2026-06-21T18:22:04Z

Overthink: /* Some JI interordinals */ add a space

[[File:Ytwy.png|thumb|502x502px|The interval regions surrounding neutral (blue) and mosdiatonic (red/yellow) intervals. Interordinal regions are the "gaps" in this scheme.]]
'''Interordinals''' are interval categories halfway between adjacent [[ordinal]]s, or interval classes, of the diatonic scale. For example, 250c is an interordinal because it falls between 200c (the 12edo major second) and 300c (the 12edo minor third). Interordinals may sometimes be called '''interseptimals''', as interordinals are midpoints of septimal intervals (such as 8/7 and 7/6) separated by 49/48; however, the term "interseptimal" on this wiki may refer to 49/48 itself.

There are usually considered to be four interordinal regions:
# '''semifourth''' (between major 2nd and minor 3rd)
# '''semisixth''' (between major 3rd and perfect 4th)
# '''semitenth''' (between perfect 5th and minor 6th)
# '''semitwelfth''' (between major 6th and minor 7th).
Sometimes the '''interizer'''/'''semisecond''', and its octave-complement, the '''antiinterizer'''/'''semifourteenth''', are included. The ''interizer'' is defined as the interval that separates interordinals from adjacent diatonic ordinals; it is half of the diatonic small step.

[[19edo]], [[24edo]], [[29edo]], and [[53edo]] are notable edos with a complete set of interordinals; the MOS scales manual (5L1s) and semiquartal (5L4s) in certain tunings have all four interordinal regions as well. Notable JI interordinals include 15/13 (247.7c, a semifourth) and 13/10 (454.2c, a semisixth); thus 10:13:15 is a fairly low-complexity JI triad with a semisixth. See [[Chthonic harmony]] for a compositional theory using interordinals.

== Naming ==
There is no unified nomenclature for interordinal regions. The following table shows various ways to name interordinals:
{| class="wikitable"
|+ Interordinal nomenclatures
!|24edo interval
!|"semi" names
!|"inter" names
!|"ultra"/"infra"
!|Greek-derived names
!|[[Latal]] (2L3s-based) names
|-
!|50c, 1\24
||semisecond
||interizer, unison-inter-second
||ultraunison inframinor second
||''-''
||latal semichroma
|-
!|250c, 5\24
||semifourth
||second-inter-third
||ultramajor second inframinor third
||chthonic
||neutral (uni)latus
|-
!|450c, 9\24
||semisixth
||third-inter-fourth
||ultramajor third infrafourth
||naiadic
||neutral/semidiminished bilatus
|-
!|750c, 15\24
||semitenth
||fifth-inter-sixth
||ultrafifth inframinor sixth
||cocytic
||neutral/semiaugmented trilatus
|-
!|950c, 19\24
||semitwelfth
||sixth-inter-seventh
||ultramajor sixth inframinor seventh
||ouranic
||neutral antilatus
|-
!|1150c, 23\24
||semifourteenth
||antiinterizer, seventh-inter-octave
||ultramajor seventh infraoctave
||''-''
||latal antisemichroma
|}

== Pythagorean-based interordinals ==
The Pythagorean-based sizes for interordinals are the logarithmic midpoints (mathematically, geometric means) of the corresponding Pythagorean diatonic intervals, hence being the canonical sizes for interordinals in some sense. These are also the mathematically exact "interseptimals", separated from the surrounding septimal intervals by sqrt(49/48).
* interizer: sqrt(256/243) = 45.1c = midpoint of 64/63 and 28/27
* semifourth: sqrt(9/8 * 32/27) = sqrt(4/3) = 249.0c = midpoint of 8/7 and 7/6
* semisixth: sqrt(81/64 * 4/3) = sqrt(27/16) = 452.9c = midpoint of 9/7 and 21/16
* semitenth: sqrt(3/2 * 128/81) = sqrt(64/27) = 747.1c = midpoint of 32/21 and 14/9
* semitwelfth: sqrt(27/16 * 16/9) = sqrt(3/1) = 951.0c = midpoint of 12/7 and 7/4
* semifourteenth: sqrt(243/128 * 2/1) = sqrt(243/64) = 1154.9c = midpoint of 27/14 and 63/32

== Some JI interordinals ==
* semifourth: 15/13; 22/19; 37/32
* semisixth: 13/10; 22/17; 31/24; 35/27
* semitenth: 17/11; 20/13; 37/24; 99/64
* semitwelfth: 19/11; 26/15; 45/26

{{Interval regions}}

7-odd-limit

2026-06-21T18:21:13Z

Overthink: /* Triads dividing the perfect fourth */ larger chords too

{{Odd-limit navigation}}
The '''7-[[odd-limit]]''' consists of all intervals where the largest allowable odd factor in the numerator and denominator is 7. It is the smallest odd-limit containing intervals of the [[7-limit|7-prime-limit]], thus creating xenharmonic categories not found in traditional music theory. In a 7-prime-limit system, all the ratios of the 7- or [[9-odd-limit]] can be treated as consonances.

== Table of 7-odd-limit intervals ==
Reduced to an octave, the intervals of the 7-odd-limit are:

{| class="wikitable"
! Interval
! Cents
! Name
|-
| 1/1
| 0.0
| Unison
|-
| '''8/7'''
| '''231.2'''
| '''Septimal major 2nd'''
|-
| '''7/6'''
| '''266.9'''
| '''Septimal minor 3rd'''
|-
| 6/5
| 315.6
| Classical minor 3rd
|-
| 5/4
| 386.4
| Classical major 3rd
|-
| 4/3
| 498.0
| Perfect 4th
|-
| '''7/5'''
| '''582.5'''
| '''Lesser septimal tritone'''
|-
| '''10/7'''
| '''617.5'''
| '''Greater septimal tritone'''
|-
| 3/2
| 702.0
| Perfect 5th
|-
| 8/5
| 813.6
| Classical minor 6th
|-
| 5/3
| 884.4
| Classical major 6th
|-
| '''12/7'''
| '''933.1'''
| '''Septimal major 6th'''
|-
| '''7/4'''
| '''968.8'''
| '''Septimal minor 7th'''
|-
| 2/1
| 1200.0
| Octave
|}

== Approximation by edos ==
[[File:7-odd-limit in edos.png|thumb|right|A diagram showing the approximation of the 7-odd-limit by various edos.]]
The first [[edo]] consistent to the 7-odd-limit is [[4edo]], which maps 5/4 to 1 step, 3/2 to 2 steps, and 7/4 to 3 steps, laying down a rough framework of tetradic harmony. Then, [[10edo]] approximates the 7-odd-limit relatively accurately for size, though it conflates several interval pairs: 5/4~6/5, 7/6~8/7, and 7/5~10/7. As such, the [[10-form]] is useful for classifying the 7-limit. After that, [[12edo]] distinguishes 5/4 from 6/5 and 7/6 from 8/7, though it has 6/5~7/6 and 7/5~10/7, and the 7th harmonic is tuned very sharply. The [[15edo]] and [[19edo]] tunings distinguish 5/4, 6/5, and 7/6, as well as 7/5 and 10/7, but 7/6 is equated to 8/7, an equivalence known as [[Interseptimal (temperament)|Interseptimal or Semaphore temperament]]. The first to distinguish all of 5/4, 6/5, 7/6, and 8/7 is [[22edo]], though 7/5 is still equated with 10/7. The first edo to distinguish the entire 7-odd-limit is [[27edo]], but one may prefer [[31edo]] for a more accurate approximation.

== Intervals of the 7-odd-limit ==
=== 8/7 ===
The '''8/7''' interval can be considered the '''septimal major second''', or '''supermajor second''', by diatonic interval classification, in the sense that it is slightly wider than the [[9/8]] major second at 231.2 cents. Due to its larger size compared to 9/8, it does not cause as much crowding, and is thus more consonant. It is also approximately 1/3 of the [[perfect fifth]], and it is mapped as such in the [[Slendric]] temperament.

It can also be considered in an ambiguous category of "semifourths" between seconds and thirds. Here, it is a minor interval, with 7/6, its fourth complement, being the corresponding major interval. The 7/6 and 8/7 intervals can be stacked to make fourth-bound triads; see [[#Triads dividing the perfect fourth]].

=== 7/6 ===
The '''7/6''' interval is known as the '''septimal minor third''' or '''subminor third''', since it is narrower than the Pythagorean minor third [[32/27]] and the classical minor third [[6/5]], being 266.9 cents in size. We can build a triad bounded by the [[perfect fifth]], that being 1–7/6–3/2. The interval between 7/6 and 3/2 is [[9/7]], which can be considered the supermajor third, being the fifth complement of 7/6. (However, note that 9/7 is a [[9-odd-limit]] interval, not a 7-odd-limit one.) We can also stack 7/6 on top of a triad to get a seventh chord; for example, stacking 7/6 on top of the 1–5/4–3/2 major triad gives us 1–5/4–3/2–7/4, the harmonic seventh chord.

As described in [[#Triads dividing the perfect fourth]], 7/6 can also be seen as contrasting with [[#8/7|8/7]] in triads such as 1–7/6–4/3, with 7/6 being considered the major counterpart of 8/7.

=== 7/5 ===
The 7/5 interval can be called the '''lesser septimal tritone''', having a size of 582.5 cents. It is called the ''lesser'' septimal tritone because the "greater septimal tritone" is [[#10/7|10/7]], its octave complement, from which it differs by [[50/49]], the jubilisma. Unlike the tritone found in [[12edo]], it is a ''consonant'' tritone, having a more restful sound than the half-octave. It is found between the third and the seventh of the 1–5/4–3/2–7/4 harmonic seventh chord. It is also the outer interval of the 1–6/5–7/5 diminished triad, which is the simplest and most consonant diminished triad in JI.

In systems such as [[HEJI]] and the [[FJS]], it is a diminished fifth, being the difference between [[5/4]], which is a major third, and [[#7/4|7/4]], which is a minor seventh. However, since it is less than a half-octave, it can also be classified as an augmented fourth, and it is mapped as such in septimal [[Meantone]] temperament. As such, interval categories in the 7-limit are rather ambiguous, and 7/4 has qualities of both a sixth and a seventh, instead of simply being a subminor seventh.

It is fairly close to the Pythagorean diminished fifth [[1024/729]], being flat of it by an [[Aberschisma]], or about 5.8 cents. It is also rather close to the [[5-limit]] tritone [[45/32]], being flat of it by the [[Marvel]] comma 225/224.

=== 10/7 ===
The 10/7 interval can be named the '''greater septimal tritone''', being 617.5 cents in size, analogous to how [[#7/5|7/5]] is called the lesser septimal tritone. It is somewhat less consonant than 7/5 due to its more complex ratio, though it is still considerably more consonant than the half-octave. It can be seen as a stack of 5/4 and 8/7, appearing in chords such as 1–7/4–5/2 and 1–6/5–3/2–12/7.

=== 12/7 ===
The interval 12/7, known as the '''septimal major sixth''' or '''supermajor sixth''', measures at 933.1 cents in size. It is the octave complement of [[#7/6|7/6]], and the twelfth complement of [[#7/4|7/4]]. Chords using it include 1–9/7–3/2–12/7 and 1–6/5–3/2–12/7.

It is somewhat ambiguous and can be considered in a category of "hemitwelfths" between sixths and sevenths, where it is a minor interval, and 7/4 is its major counterpart.

=== 7/4 ===
The 7/4 interval is often called the '''septimal minor seventh''', '''subminor seventh''', or '''harmonic seventh''', being 968.8 cents in size. Being the octave-reduced seventh harmonic, it can naturally be added to a 1–5/4–3/2 major triad to get 1–5/4–3/2–7/4, often called the ''harmonic seventh chord''. Due to its simpler ratio, it is more consonant than [[16/9]], the Pythagorean minor seventh, and [[9/5]], the classical minor seventh.

Though often considered a minor seventh, it also has some qualities of a sixth. For example, the [[#7/5|7/5]] interval can be considered an augmented fourth due to being smaller than the semioctave, so 7/4 would accordingly be an augmented sixth. Thus 7/4 is somewhat ambiguous by diatonic classification, and can be considered to be in a category between a sixth and a seventh, a "hemitwelfth". Here 7/4 is a major interval, with the corresponding minor interval being [[#12/7|12/7]].

== Harmony in the 7-odd-limit ==
=== Tetradic harmony ===
The 7-odd-limit is where tetrads start to get more prevalent. For example, we can build the 1–5/4–3/2–7/4 "harmonic seventh chord" by adding 7/4 on top of a 1–5/4–3/2 major triad. The harmonic seventh chord sounds somewhat similar to the dominant seventh chord, except it is more consonant and resolved. It can also be called the "major tetrad", similarly to how 1–5/4–3/2 is called the major triad.

The 5-limit minor triad 1–6/5–3/2 can be derived by reflecting every note about the midpoint of the root and the fifth. If we do the same for the harmonic seventh chord and reduce the steps to an octave, then we get 1–6/5–3/2–12/7 "subharmonic seventh chord" or "minor tetrad".

=== Chords dividing the perfect fourth ===
{{Main|Chthonic harmony}}
The 7/6 and 8/7 intervals can be seen as contrasting with each other, differing from each other by 49/48 (35.7 cents). We can build triads by stacking 7/6 and 8/7 that span the [[perfect fourth]], such as the 1–7/6–4/3 triad, which may also be voiced as 1–3/2–7/4. The minor version of this triad is 1–8/7–4/3, which can also be voiced as 1–3/2–12/7. This is analogous to how [[5/4]] and [[6/5]] contrast each other in the 1–5/4–3/2 and 1–6/5–3/2 triads, but these septimal triads split the perfect fourth, rather than splitting the [[perfect fifth]] like 5-limit triads do. As such, it can be considered a form of [[chthonic harmony|"semiquartal" or "chthonic" harmony]], which is one approach to septimal harmony.

Here, 7/6 is a type of major interval, and 8/7 is a type of minor interval. Their octave complements can be classified accordingly, with 12/7 being a minor interval, and 7/4 being a major interval. This is different from diatonic, where 8/7 is a supermajor second, 7/6 a subminor third, 12/7 a supermajor sixth, and 7/4 a subminor seventh.

Larger chthonic chords exist as well; details are given in the main article.

5-odd-limit

2026-06-21T18:19:53Z

Overthink: specify

6/5

2026-06-21T18:17:55Z

Overthink: redirect

#redirect [[5-odd-limit #Classical minor third (6/5)]]

5/4

2026-06-21T18:15:45Z

Overthink: redirect

#REDIRECT [[5-odd-limit #Classical major third (5/4)]]

9-odd-limit

2026-06-21T18:11:55Z

Overthink: /* Table of 9-odd-limit intervals */ description

{{WIP}}
{{Odd-limit navigation}}
The '''9-[[odd-limit]]''' consists of all intervals where the largest allowable odd factor in the numerator and denominator is 9. Reduced to an octave, these are:

==Table of 9-odd-limit intervals==
Reduced to an octave, the intervals of the 9-odd-limit are:

{| class="wikitable"
! Interval
! Cents
! Name
|-
| 1/1
| 0.0
| Unison
|-
| '''10/9'''
| '''182.4'''
| '''Minor whole tone, Ptolemaic major 2nd'''
|-
| '''9/8'''
| '''203.9'''
| '''Major whole tone, Pythagorean major 2nd'''
|-
| 8/7
| 231.2
| Septimal major 2nd
|-
| 7/6
| 266.9
| Septimal minor 3rd
|-
| 6/5
| 315.6
| Classical minor 3rd
|-
| 5/4
| 386.4
| Classical major 3rd
|-
| '''9/7'''
| '''435.1'''
| '''Septimal major 3rd'''
|-
| 4/3
| 498.0
| Perfect 4th
|-
| 7/5
| 582.5
| Lesser septimal tritone
|-
| 10/7
| 617.5
| Greater septimal tritone
|-
| 3/2
| 702.0
| Perfect 5th
|-
| '''14/9'''
| '''764.9'''
| '''Septimal minor 6th'''
|-
| 8/5
| 813.6
| Classical minor 6th
|-
| 5/3
| 884.4
| Classical major 6th
|-
| 12/7
| 933.1
| Septimal major 6th
|-
| 7/4
| 968.8
| Septimal minor 7th
|-
| '''16/9'''
| '''996.1'''
| '''Pythagoran minor 7th'''
|-
| '''9/5'''
| '''1017.6'''
| '''Classical minor 7th'''
|-
| 2/1
| 1200.0
| Octave
|}

== Approximation by edos ==
The first edo to be consistent to the 9-odd-limit is [[5edo]], giving a rough outline for harmony with its relatively accurate 3/2 and 7/4 of 720 and 960 cents respectively, while very sharply mapping 5/4 to 480 cents. The first edo to be distinctly consistent in this limit is [[41edo]].

== Intervals of the 9-odd-limit ==
Here are the intervals in the 9-odd-limit which are not part of any lower odd-limit. Note that the intervals of the 9-odd-limit are all contained within the [[7-limit|7-prime-limit]], as 9 factors as 3 × 3.

=== 10/9 ===
'''10/9''', often called the '''minor whole tone''' or '''ptolemaic whole tone''', is an interval of 182.4 cents. It is often considered a counterpart to 9/8, the major whole tone, as 9/8 and 10/9 add up to [[5/4]], the classical major third. [[Meantone]] temperament eliminates the distinction between 10/9 and 9/8 by tempering out 81/80, the syntonic comma. Due to its smaller size and more complex ratio, it is generally considered to be somewhat more dissonant than 9/8, and has a darker quality than 9/8. It is notably the octave complement of 9/5, the classical minor seventh.

In terms of [[chthonic harmony]], it can be considered a subminor latus, as the fourth complement of [[6/5]].

[[Porcupine]] temperament uses a very flat (~163c) 10/9 as a generator, where it is equated with [[11/10]] and [[12/11]].

=== 9/8 ===
{{See also|Pythagorean tuning #Major second}}
'''9/8''' can be called the '''Pythagorean whole tone''' or '''major whole tone''', and sometimes simply the '''whole tone'''. It is reached by stacking up 2 [[perfect fifth]]s and down an octave. It is a very important melodic interval in the [[diatonic]] scale, being a major second. In temperaments generated by the fifth, it is often equated with nearby intervals; for example, [[Meantone]] equates it with 10/9, while [[Archy]] equates it with [[8/7]]. An octave above 9/8 is the major ninth 9/4, which often appears in chords such as 1–5/4–3/2–9/4 (4:5:6:9).

=== 9/7 ===
=== 14/9 ===
=== 16/9 ===
=== 9/5 ===

9-odd-limit

2026-06-21T18:11:18Z

Overthink: /* Intervals */ header

{{WIP}}
{{Odd-limit navigation}}
The '''9-[[odd-limit]]''' consists of all intervals where the largest allowable odd factor in the numerator and denominator is 9. Reduced to an octave, these are:

==Table of 9-odd-limit intervals==

{| class="wikitable"
! Interval
! Cents
! Name
|-
| 1/1
| 0.0
| Unison
|-
| '''10/9'''
| '''182.4'''
| '''Minor whole tone, Ptolemaic major 2nd'''
|-
| '''9/8'''
| '''203.9'''
| '''Major whole tone, Pythagorean major 2nd'''
|-
| 8/7
| 231.2
| Septimal major 2nd
|-
| 7/6
| 266.9
| Septimal minor 3rd
|-
| 6/5
| 315.6
| Classical minor 3rd
|-
| 5/4
| 386.4
| Classical major 3rd
|-
| '''9/7'''
| '''435.1'''
| '''Septimal major 3rd'''
|-
| 4/3
| 498.0
| Perfect 4th
|-
| 7/5
| 582.5
| Lesser septimal tritone
|-
| 10/7
| 617.5
| Greater septimal tritone
|-
| 3/2
| 702.0
| Perfect 5th
|-
| '''14/9'''
| '''764.9'''
| '''Septimal minor 6th'''
|-
| 8/5
| 813.6
| Classical minor 6th
|-
| 5/3
| 884.4
| Classical major 6th
|-
| 12/7
| 933.1
| Septimal major 6th
|-
| 7/4
| 968.8
| Septimal minor 7th
|-
| '''16/9'''
| '''996.1'''
| '''Pythagoran minor 7th'''
|-
| '''9/5'''
| '''1017.6'''
| '''Classical minor 7th'''
|-
| 2/1
| 1200.0
| Octave
|}

== Approximation by edos ==
The first edo to be consistent to the 9-odd-limit is [[5edo]], giving a rough outline for harmony with its relatively accurate 3/2 and 7/4 of 720 and 960 cents respectively, while very sharply mapping 5/4 to 480 cents. The first edo to be distinctly consistent in this limit is [[41edo]].

== Intervals of the 9-odd-limit ==
Here are the intervals in the 9-odd-limit which are not part of any lower odd-limit. Note that the intervals of the 9-odd-limit are all contained within the [[7-limit|7-prime-limit]], as 9 factors as 3 × 3.

=== 10/9 ===
'''10/9''', often called the '''minor whole tone''' or '''ptolemaic whole tone''', is an interval of 182.4 cents. It is often considered a counterpart to 9/8, the major whole tone, as 9/8 and 10/9 add up to [[5/4]], the classical major third. [[Meantone]] temperament eliminates the distinction between 10/9 and 9/8 by tempering out 81/80, the syntonic comma. Due to its smaller size and more complex ratio, it is generally considered to be somewhat more dissonant than 9/8, and has a darker quality than 9/8. It is notably the octave complement of 9/5, the classical minor seventh.

In terms of [[chthonic harmony]], it can be considered a subminor latus, as the fourth complement of [[6/5]].

[[Porcupine]] temperament uses a very flat (~163c) 10/9 as a generator, where it is equated with [[11/10]] and [[12/11]].

=== 9/8 ===
{{See also|Pythagorean tuning #Major second}}
'''9/8''' can be called the '''Pythagorean whole tone''' or '''major whole tone''', and sometimes simply the '''whole tone'''. It is reached by stacking up 2 [[perfect fifth]]s and down an octave. It is a very important melodic interval in the [[diatonic]] scale, being a major second. In temperaments generated by the fifth, it is often equated with nearby intervals; for example, [[Meantone]] equates it with 10/9, while [[Archy]] equates it with [[8/7]]. An octave above 9/8 is the major ninth 9/4, which often appears in chords such as 1–5/4–3/2–9/4 (4:5:6:9).

=== 9/7 ===
=== 14/9 ===
=== 16/9 ===
=== 9/5 ===

9-odd-limit

2026-06-21T18:10:58Z

Overthink: start intervals section

{{WIP}}
{{Odd-limit navigation}}
The '''9-[[odd-limit]]''' consists of all intervals where the largest allowable odd factor in the numerator and denominator is 9. Reduced to an octave, these are:

==Table of 9-odd-limit intervals==

{| class="wikitable"
! Interval
! Cents
! Name
|-
| 1/1
| 0.0
| Unison
|-
| '''10/9'''
| '''182.4'''
| '''Minor whole tone, Ptolemaic major 2nd'''
|-
| '''9/8'''
| '''203.9'''
| '''Major whole tone, Pythagorean major 2nd'''
|-
| 8/7
| 231.2
| Septimal major 2nd
|-
| 7/6
| 266.9
| Septimal minor 3rd
|-
| 6/5
| 315.6
| Classical minor 3rd
|-
| 5/4
| 386.4
| Classical major 3rd
|-
| '''9/7'''
| '''435.1'''
| '''Septimal major 3rd'''
|-
| 4/3
| 498.0
| Perfect 4th
|-
| 7/5
| 582.5
| Lesser septimal tritone
|-
| 10/7
| 617.5
| Greater septimal tritone
|-
| 3/2
| 702.0
| Perfect 5th
|-
| '''14/9'''
| '''764.9'''
| '''Septimal minor 6th'''
|-
| 8/5
| 813.6
| Classical minor 6th
|-
| 5/3
| 884.4
| Classical major 6th
|-
| 12/7
| 933.1
| Septimal major 6th
|-
| 7/4
| 968.8
| Septimal minor 7th
|-
| '''16/9'''
| '''996.1'''
| '''Pythagoran minor 7th'''
|-
| '''9/5'''
| '''1017.6'''
| '''Classical minor 7th'''
|-
| 2/1
| 1200.0
| Octave
|}

== Approximation by edos ==
The first edo to be consistent to the 9-odd-limit is [[5edo]], giving a rough outline for harmony with its relatively accurate 3/2 and 7/4 of 720 and 960 cents respectively, while very sharply mapping 5/4 to 480 cents. The first edo to be distinctly consistent in this limit is [[41edo]].

== Intervals ==
Here are the intervals in the 9-odd-limit which are not part of any lower odd-limit. Note that the intervals of the 9-odd-limit are all contained within the [[7-limit|7-prime-limit]], as 9 factors as 3 × 3.

=== 10/9 ===
'''10/9''', often called the '''minor whole tone''' or '''ptolemaic whole tone''', is an interval of 182.4 cents. It is often considered a counterpart to 9/8, the major whole tone, as 9/8 and 10/9 add up to [[5/4]], the classical major third. [[Meantone]] temperament eliminates the distinction between 10/9 and 9/8 by tempering out 81/80, the syntonic comma. Due to its smaller size and more complex ratio, it is generally considered to be somewhat more dissonant than 9/8, and has a darker quality than 9/8. It is notably the octave complement of 9/5, the classical minor seventh.

In terms of [[chthonic harmony]], it can be considered a subminor latus, as the fourth complement of [[6/5]].

[[Porcupine]] temperament uses a very flat (~163c) 10/9 as a generator, where it is equated with [[11/10]] and [[12/11]].

=== 9/8 ===
{{See also|Pythagorean tuning #Major second}}
'''9/8''' can be called the '''Pythagorean whole tone''' or '''major whole tone''', and sometimes simply the '''whole tone'''. It is reached by stacking up 2 [[perfect fifth]]s and down an octave. It is a very important melodic interval in the [[diatonic]] scale, being a major second. In temperaments generated by the fifth, it is often equated with nearby intervals; for example, [[Meantone]] equates it with 10/9, while [[Archy]] equates it with [[8/7]]. An octave above 9/8 is the major ninth 9/4, which often appears in chords such as 1–5/4–3/2–9/4 (4:5:6:9).

=== 9/7 ===
=== 14/9 ===
=== 16/9 ===
=== 9/5 ===

Template:See also

2026-06-21T18:08:25Z

Overthink: copy from WP

<includeonly>{{#invoke:Labelled list hatnote|labelledList|See also|ifexists={{{ifexists|true}}}}}</includeonly><noinclude>
{{documentation}}
</noinclude>

62edo

2026-06-09T03:57:36Z

Overthink: redirect

#redirect [[31edo #62edo]]

31edo

2026-06-09T03:57:10Z

Overthink: /* Multiples */ add 62edo

[[File:31edo whr.png|thumb|418x418px|31edo supports Valentine and Miracle, alongside supporting Meantone.]]
'''31edo''', or 31 equal divisions of the octave, is an equal tuning with a step size of approximately 39 [[cent]]s. It is most commonly known as a tuning of [[Meantone]], and for its accurate approximation of the 2.5.7 [[subgroup]].

31edo as a whole contains a diverse palette of interval qualities and structures ranging from the very familiar to the quite exotic, and remarkably, almost all of these still have a reasonably simple harmonic interpretation. As a meantone system, 31edo's diatonic scale includes the basic qualities of the [[5-limit]], such as the [[perfect fourth]] and [[perfect fifth|fifth]], and the classical minor and major thirds ([[6/5]] and [[5/4]]). But 31edo also includes subminor and supermajor intervals, identifiable with [[septal]] ratios such as [[7/6]] and [[9/7]], and [[neutral]] intervals, identifiable with [[11-limit]] ratios such as [[11/9]].

In terms of structures, or ways of organizing harmony, it should first be noted that 31edo's perfect fifth, of 18 steps, is quite divisible. The fifth can be split in two, giving us a neutral-third temperament, known in this case as [[Mohajira]], which emphasizes heptatonic structure, the 11th harmonic, and 2.3.5.11. Splitting the fifth in three gives us [[Slendric]] (or in this case [[Mothra]]), formed by stacking [[8/7]], and which emphasizes pentatonic structure and 2.3.7. Combining these gives us [[Miracle]], while splitting Slendric into three again gives us [[Valentine]]. In 31edo, each of these provides xenharmonic ways of accessing the 11-limit with more simplicity than Meantone. Yet another way to encompass the 11-limit is given by [[Orwell]], generated by 31edo's subminor third; and of course, as a prime EDO, 31edo contains several more structures unique to itself.

== Theory ==
==== Edostep interpretations ====
31edo's edostep has the following interpretations in the 11-limit:

* 128/125 (the residue between three stacked 5/4s and the octave)
* 36/35 (the difference between 7/6 and 6/5, or 5/4 and 9/7)
* 49/48 (the difference between 8/7 and 7/6)
* 50/49 (the difference between [[7/5]] and [[10/7]])
* 64/63 (the difference between 8/7 and [[9/8]])
* 33/32 (the difference between [[12/11]] and 9/8)
* 45/44 (the difference between 11/9 and 5/4, or [[11/10]] and 9/8)
* 55/54 (the difference between 6/5 and 11/9, or 12/11 and [[10/9]])
* 56/55 (the difference between 5/4 and [[14/11]])

==== JI approximation ====
31edo can be understood as a 7-limit system with a somewhat flat 3/2 but nearly-perfect 5th and 7th harmonics. In particular, the product of 5 and 7, [[35/32]], is approximated to within about 0.3{{c}}. 31edo also has an approximation to the 11th harmonic that, while tuned flat, has the property that the flatness of harmonics 9 and 11 mostly cancel out, producing a close-to-pure ~11/9 neutral third. The harmonic 23 turns out to be flat in a very similar way to 11.

The intervening harmonics - 13, 17, and 19 - are tuned rather sharp, but by almost exactly the same amount; therefore the chord 13:17:19 is extremely well-approximated by 31edo, with the interval [[17/13]] tuned less than 0.1{{c}} off.

31edo's fifth generates a functional diatonic scale. Its whole tone, of 5 steps, is split into semitones of 2 and 3; as 31edo's fifth is flatter than that of [[12edo]], the chromatic semitone, comprised by 2 steps, is smaller than the diatonic semitone, which is 3 steps.
{{Harmonics in ED|31|31|0}}
{| class="wikitable"
|+Thirds in 31edo
!Quality ([[ADIN]])
|Subminor
|'''Nearminor'''
|Neutral
|'''Nearmajor'''
|Supermajor
|-
!Cents
|271.0
|'''309.7'''
|348.4
|'''387.1'''
|425.8
|-
!Just interpretation
|7/6 (+4.1{{c}})
|'''6/5''' (-6.0{{c}})
|11/9 (+1.0{{c}})
|'''5/4''' (+0.8{{c}})
|9/7 (-9.2{{c}})
|-
!Steps
|7
|'''8'''
|9
|'''10'''
|11
|}
Diatonic thirds are bolded.

If one wants to stretch the octave to improve JI approximation, the [https://en.xen.wiki/w/Zeta_peak_index zeta peak] octave stretch and the CWE optimization of 7-limit 31tet give an octave of 1200.8 cents, while the CWE optimization of 11-limit 31tet gives an octave of 1201.2 cents.

=== Chords ===
Along with its diatonic major and minor chords which approximate 5-limit harmony, 31edo also has a narrow but functional supermajor triad, and a well-tuned subminor triad. It also supports arto and tendo chords, with its slendric chords of [0 6 18] and [0 12 18], and has a neutral triad [0 9 18] which represents both artoneutral and tendoneutral triads in the 11- and 13-limit.

=== Regular temperaments ===
Besides [[Meantone]] (for which it provides an excellent tuning and which is shared with 19edo), 31edo also supports variations of [[Rastmic]] temperament (like 24edo), [[Slendric]] (like 36edo), [[Miracle]] (like 41edo), and [[Orwell]] (like 22edo).

== Scales ==
31edo does not temper out 64/63, meaning that it can be used to tune [[Diasem]] while representing some simpler 5-limit intervals. 31edo's step is called a [[diesis]], and can function as an [[aberrisma]]. Due to being a prime number, every MOS scale in 31edo has a full-octave period, and 31edo has a large number of them. Orwell[9] ([[gramitonic]]) is one example, so is Mohajira[7] ([[mosh]]).

31edo also has a usable 12-note chromatic scale, approximating [[Golden sequences and tuning|golden]] Meantone/monocot.

=== MOS scales ===
A key aspect of 31edo noted by several sources is the diversity of MOS scales represented. The 15 edo-distinct regular temperaments of 31edo are divided into three loops that are traversed by halving or doubling their generators.
{| class="wikitable"
|+
!
! colspan="5" |Loop 1
|-
!Temperament
|[[Didacus]]
|[[Wurschmidt]]
|[[Squares]]
|[[Mohajira]]
|[[Meantone]]
|-
!Complexity of 3/2
|15
|8
|4
|2
|1
|-
!Scale (albitonic)
|5-5-5-5-5-6
|8-1-1-8-1-1-8-1-1-1
|2-2-7-2-2-7-2-7
|5-4-5-4-5-4-4
|5-5-3-5-5-5-3
|-
!Generator
|5, 26
|10, 21
|20, 11
|22, 9
|18, 13
|-
!
! colspan="5" |Loop 2
|-
!Temperament
|[[Miracle]]
|[[Slendric]]
|[[A-Team]]
|[[Orwell]]
|[[Casablanca]]
|-
!Complexity of 3/2
|6
|3
|14
|7
|12
|-
!Scale (albitonic)
|3-3-3-3-3-3-3-3-3-4
|6-6-6-6-7
|2-5-2-5-5-2-5-5
|4-3-4-3-4-3-4-3-3
|3-3-3-3-5-3-3-3-5
|-
!Generator
|3, 28
|6, 25
|12, 19
|24, 7
|17, 14
|-
!
! colspan="5" |Loop 3
|-
!Temperament
|[[Slender]]
|[[Carlos Alpha|Valentine]]
|[[Nusecond]]
|[[Myna]]
|[[Tritonic]]
|-
!Complexity of 3/2
|13
|9
|11
|10
|5
|-
!Scale (albitonic)
| - (11-note scale has >10 steps interval)
| - (11-note scale has >10 steps interval)
|4-4-4-4-4-4-4-3
|1-1-6-1-1-6-1-1-6-1-6
| - (11-note scale has >10 steps interval)
|-
!Generator
|1, 30
|2, 29
|4, 27
|8, 23
|15, 16
|}

== Notation ==
31edo, as one of the more popular edos, has a somewhat agreed-upon notation system. This notation is simply neutral [[diatonic notation]] applied to the edo, where a half-# or half-b represents an alteration by one diesis. In this manner, all notes can be spelled in a way that does not require multiple sharps or flats.

{| class="wikitable"
|+
!Step
!Cents
!ADIN*
!Neutral diatonic
!Notation
!Just intervals represented
|-
|0
|0.00
|unison
|unison
|A
|1/1
|-
|1
|38.71
|superunison
|semiaugmented unison
|At
|49/48, 50/49, 128/125
|-
|2
|77.42
|subminor second
|semidiminished second
|A#
|25/24
|-
|3
|116.13
|nearminor second
|minor second
|Bb
|16/15
|-
|4
|154.84
|neutral second
|neutral second
|Bd
|11/10, 12/11
|-
|5
|193.55
|nearmajor second
|major second
|B
|10/9, 9/8
|-
|6
|232.26
|supermajor second
|semiaugmented second
|Bt
|8/7
|-
|7
|270.97
|subminor third
|semidiminished third
|Cd
|7/6
|-
|8
|309.68
|nearminor third
|minor third
|C
|6/5
|-
|9
|348.39
|neutral third
|neutral third
|Ct
|11/9, 16/13
|-
|10
|387.10
|nearmajor third
|major third
|C#
|5/4
|-
|11
|425.81
|supermajor third
|semiaugmented third
|Db
|9/7
|-
|12
|464.52
|subfourth
|semidiminished fourth
|Dd
|21/16, 13/10
|-
|13
|503.23
|perfect fourth
|perfect fourth
|D
|4/3
|-
|14
|541.94
|neutral fourth
|semiaugmented fourth
|Dt
|11/8, 15/11
|-
|15
|580.65
|nearaugmented fourth
|augmented fourth
|D#
|7/5
|-
|16
|619.35
|neardiminished fifth
|diminished fifth
|Eb
|10/7
|-
|17
|658.06
|neutral fifth
|semidiminished fifth
|Ed
|16/11, 22/15
|-
|18
|696.77
|perfect fifth
|perfect fifth
|E
|3/2
|-
|19
|735.48
|superfifth
|semiaugmented fifth
|Et
|32/21, 20/13
|-
|20
|774.19
|subminor sixth
|semidiminished sixth
|Fd
|14/9
|-
|21
|812.90
|nearminor sixth
|minor sixth
|F
|8/5
|-
|22
|851.61
|neutral sixth
|neutral sixth
|Ft
|13/8, 18/11
|-
|23
|890.32
|nearmajor sixth
|major sixth
|F#
|5/3
|-
|24
|929.03
|supermajor sixth
|semiaugmented sixth
|Gb
|12/7
|-
|25
|967.74
|subminor seventh
|semidiminished seventh
|Gd
|7/4
|-
|26
|1006.45
|nearminor seventh
|minor seventh
|G
|9/5, 16/9
|-
|27
|1045.16
|neutral seventh
|neutral seventh
|Gt
|11/6, 20/11
|-
|28
|1083.87
|nearmajor seventh
|major seventh
|G#
|15/8
|-
|29
|1122.58
|supermajor seventh
|semiaugmented seventh
|Ab
|48/25
|-
|30
|1161.29
|suboctave
|semidiminished octave
|Ad
|49/25, 125/64, 96/49
|-
|31
|1200.00
|octave
|octave
|A
|2/1
|}
<nowiki>*</nowiki>Aligned with the consensus agreed upon by various 31edo resources. ADIN in specific clarifies the unspecified "major" and "minor" qualities found in these resources as "nearmajor" and "nearminor" to distinguish them from supermajor/subminor.

== SCL files ==
See [[31edo/SCL files]].

== Multiples ==
=== 62edo ===
62edo uses flat approximations of harmonics 13, 17, and 19 instead of sharp ones, and is arguably usable as a full 23-limit Meantone system with an overall flat tendency.
{{Harmonics in ED|62|37|0}}

=== 217edo ===
217edo is a theoretically strong system which keeps 31edo's tuning of 2.5.7.(13:17:19). 217edo is strong in the 19-limit and the smallest edo distinctly consistent in the 19-odd-limit.
{{Harmonics in ED|217|37|0}}

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

Pajara

2026-06-02T16:28:42Z

Overthink: 7-limit simplicity

'''Pajara''', 10 & 12, is a regular temperament wherein the octave is split into two tritone periods, and the generator is a fifth (3/2). A fifth minus a tritone is 16/15 ([[Diaschismic]] tempering), and therefore the 5/4 major third is found two generators below the tritone. Pajara makes the further equivalence of 5/4 plus a period to 7/4 ([[Jubilismic]] tempering) and therefore twice 4/3 is 7/4 ([[Archy]] tempering). The result is a 10-form system generated by a fifth tuned somewhere around 710 cents.

There are five [[patent]] tunings of Pajara: [[12edo|12]], [[22edo|22]], [[54edo|54]], [[32edo|32]], and [[10edo|10]] (which is also the [[20edo]] [[val]] for the [[7-limit]]); of these 22, 54, and 32 are considered non-trivial, and 22edo is the generally assumed "canonical" tuning. As regards non-patent vals, [[34edo|34d]] (using [[17edo]]'s mapping of 7) and [[56edo|56d]] are commonly considered as well.

Pajara is arguably the most accurate temperament that contains full 7-limit tetrads in small (5—10 note) [[MOS]] scales.

== Extensions ==
There are two main extensions of Pajara to the [[11-limit]]: Pajarous (10 & 22) and Undecimal Pajara (12 & 22, which is supported by only those two patent vals). Undecimal Pajara is best flat of 22edo; Pajarous is best sharp of 22edo. Note that although Pajarous is supported patently by 32 and 54edo, no extension of Pajara can be [[monotonic]] in the [[11-odd-limit]] sharp of 22edo, which calls into question the utility of involving prime 11 in Pajara at all.

As with all Diaschismic temperaments, the generating semitone can be identified as 17/16~18/17 and prime 17 hence comes for free.

== Tuning considerations ==
Most optimization methods place the optimal tuning of Pajara's perfect fifth at around 707 cents (hence the focus on 12 & 22 as an extension). However, interpretations more accurate than Pajara exist for those structures (although they may well be more complex). EDOs with tunings of fifths flat of that of 22edo do not support Pajara in the patent val (except for 12edo), and 12edo is generally extremely inaccurate in the 7-limit due to the fact that Archy temperament forces 9/8 and 8/7 together.

Conversely, systems with fifths sharp of 32edo narrow the distinction between 5/4 and 6/5 considerably, detuning 6/5 to a neutral sound. Therefore, the tuning range of Pajara can be considered to lie between about 709 to 712.5 cents.

== Compositional theory ==
Every note has a corresponding note a tritone apart. Pajara[10] solves the problem of representing intervals of both 5 and 7 in diatonic 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 (or ^ and v in 22edo or 32edo) represent alterations by a single step. Pajara retains several desirable properties of MOS diatonic:

- soft-of-basic tunings in the main harmonic temperament

- most notes have a fifth over them

- structurally similar to diatonic, with strings of steps of one size broken by two steps of a different size

- the fifth is divided into two thirds

Additionally, there is a MODMOS sssLsssssL, which is considered the "pentachordal" pajara MOS due to being constructed from an sssL [[Tetrachord#Pentachord|pentachord]].

To extend to the 11-limit, Pajara[12] (sLLLLLsLLLLL) 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 [LLLsLLLLsLLL] of Pajara[12] is the Tellurian scale; it may be derived by splitting a pentatonic scale ([LL][LsL][LL][LsL][LL]), or by splitting each whole tone of MOS diatonic ([LL][LL]s[LL][LL][LL]s), which makes for a more xenharmonic way of generalizing 12edo music than simply retuning the chain of fifths. ("When the 12edo goes chromatic, equally divide the whole tone!")

==== Degrees of pajara[10] ====
Bolded names are preferred.
{| class="wikitable"
|+
!Degree
!Name (Leriendil)
!Name (standard)
!Function
!Minor
!Perfect
!Major
|-
|0
| colspan="2" |'''Unison'''
|Tonic
| -
|1/1
| -
|-
|1
|Grade
|'''Second'''
|Supervicinant
|16/15 (perfect)
| -
|10/9
|-
|2
|'''Unilatus'''
|Chthonic
|Subvaricant
|8/7
| -
|7/6
|-
|3
|Semitres
|'''Third'''
|Mediant
|6/5
| -
|5/4
|-
|4
|Bilatus
|'''Fourth'''
|Subdominant
|9/7
| -
|4/3 (perfect)
|-
|5
|Median
|'''Tritone'''
|Antitonic
| -
|7/5, 10/7
| -
|-
|6
|Trilatus
|'''Fifth'''
|Dominant
|3/2 (perfect)
| -
|14/9
|-
|7
|Semisept
|'''Sixth'''
|Submediant
|8/5
| -
|5/3
|-
|8
|'''Antilatus'''
|Ouranic
|Varicant
|12/7
| -
|7/4
|-
|9
|Degrade
|'''Seventh'''
|Subvicinant
|9/5
| -
|15/8 (perfect)
|-
|10
|Duplance
|'''Octave'''
|Tonic
| -
|2/1
| -
|}

==== Notation ====
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.

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.

==== 10-tone functional harmony ====
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 ~100-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.
[[File:Chthonic_harmony_demonstration.mp3|thumb|10-form harmony demonstration in 22edo]]
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).

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).

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.

=== Modal harmony ===
The following chart shows the modes of pajara[10], in 22edo tuning:
{| class="wikitable"
!
!Chart
!2
!3
!4
!6
!8
!9
|-
|Dynamic minor
|{{Interval ruler|22|0, 100, 200, 300, 450, 600, 700, 800, 900, 1050, 1200}}
|minor
|minor
|dim
|perfect
|minor
|minor
|-
|Static minor
|{{Interval ruler|22|0, 100, 200, 300, 500, 600, 700, 800, 900, 1100, 1200}}
|minor
|minor
|perfect
|perfect
|minor
|major
|-
|Static major
|{{Interval ruler|22|0, 100, 200, 400, 500, 600, 700, 800, 1000, 1100, 1200}}
|minor
|major
|perfect
|perfect
|major
|major
|-
|Dynamic major
|{{Interval ruler|22|0, 100, 250, 400, 500, 600, 700, 850, 1000, 1100, 1200}}
|major
|major
|perfect
|perfect
|major
|major
|-
|Augmented
|{{Interval ruler|22|0, 150, 250, 400, 500, 600, 750, 850, 1000, 1100, 1200}}
|major
|major
|perfect
|aug
|major
|major
|}
And of a MODMOS of pajara[10], ssLsssssLs, the "pentachordal" pajara scale:
{| class="wikitable"
!
!Chart
!2
!3
!4
!6
!8
!9
|-
|(Minor)
|{{Interval ruler|22|0, 100, 200, 300, 450, 550, 700, 800, 900, 1050, 1200}}
|minor
|minor
|dim
|perfect
|minor
|minor
|-
|Alternate minor
|{{Interval ruler|22|0, 100, 200, 300, 450, 600, 700, 800, 950, 1100, 1200}}
|minor
|minor
|dim
|perfect
|minor
|major
|-
|(Minor)
|{{Interval ruler|22|0, 100, 200, 320, 500, 600, 700, 830, 990, 1100, 1200}}
|minor
|minor
|perfect
|perfect
|major
|major
|-
|Standard major
|{{Interval ruler|22|0, 100, 200, 370, 500, 600, 700, 870, 990, 1100, 1200}}
|minor
|major
|perfect
|perfect
|major
|major
|-
|(Major)
|{{Interval ruler|22|0, 100, 270, 370, 500, 600, 770, 870, 990, 1100, 1200}}
|major
|major
|perfect
|aug
|major
|major
|-
|(Major)
|{{Interval ruler|22|0, 170, 270, 370, 500, 670, 770, 870, 990, 1100, 1200}}
|major
|major
|perfect
|aug
|major
|major
|-
|Standard minor
|{{Interval ruler|22|0, 100, 200, 300, 500, 600, 700, 800, 900, 1050, 1200}}
|minor
|minor
|perfect
|perfect
|minor
|minor
|-
|(Major)
|{{Interval ruler|22|0, 100, 200, 360, 500, 600, 700, 800, 920, 1110, 1200}}
|minor
|major
|perfect
|perfect
|minor
|major
|-
|Alternate major
|{{Interval ruler|22|0, 100, 270, 360, 500, 600, 700, 800, 970, 1110, 1200}}
|major
|major
|perfect
|perfect
|major
|major
|-
|(Major)
|{{Interval ruler|22|0, 170, 270, 360, 500, 600, 700, 870, 970, 1110, 1200}}
|major
|major
|perfect
|perfect
|major
|major
|}
Some names are from [https://web.archive.org/web/20180927081411/http://lumma.org/tuning/erlich/erlich-decatonic.pdf Paul Erlich].

It is useful to consider the MODMOS as roughly on the same level as the MOS form of the scale (as the only additional variety it introduces is in the tritone), giving 15 distinct modes available to choose from, each with a degree of brightness or darkness to them, much like conventional MOSdiatonic modal harmony.

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.

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.

== Generator chain ==
The following chart assumes Pajarous.
{| class="wikitable"
|+
! colspan="5" |Period 1
! colspan="5" |Period 2
|-
! colspan="3" |Up
! colspan="2" |Down
! colspan="3" |Up
! colspan="2" |Down
|-
!#
!Cents
!JI
!Cents
!JI
!#
!Cents
!JI
!Cents
!JI
|-
|0
|0
|1/1
|600
|7/5
|0
|600
|12/7
|1200
|2/1
|-
|1
|110
|18/17, 17/16, 16/15, 15/14
|490
|4/3
|1
|710
|3/2
|1090
|28/15, 15/8, 32/17, 17/9
|-
|2
|220
|8/7, 9/8
|380
|5/4
|2
|820
|8/5
|980
|7/4, 16/9
|-
|3
|330
|6/5, 11/9
|270
|7/6
|3
|930
|12/7
|870
|5/3, 18/11
|-
|4
|440
|9/7
|160
|12/11, 10/9
|4
|1040
|9/5, 11/6
|760
|14/9
|-
|5
|550
|11/8
|50
|25/24, 49/48
|5
|1150
|96/49, 48/25
|650
|16/11
|}

== List of patent vals ==
Due to being a weak extension of [[Archy]], the patent vals of Pajara are a subset of the Archy edos. Specifically, they must be even (and therefore, some edos are doubled compared to the Archy table).
{| class="wikitable"
!EDO
!Extension to 11
!Generator tuning
!7/4 tuning
!25/24 tuning
|-
|10
|Pajarous
|480c
|960c
|0c
|-
|32
|Pajarous
|487.5c
|975c
|37.5c
|-
|54
|Pajarous
|488.9c
|977.8c
|44.4c
|-
|22
|Pajarous, Undecimal Pajara
|490.9c
|981.8c
|54.5c
|-
|12
|Undecimal Pajara
|500c
|1000c
|100c
|}
While Pajarous may appear to be canonical from this chart, one important consideration is the historical prevalence of non-patent Pajara tunings and the fact that Pajarous necessarily tunes 10/9~12/11 flatter than 11/10, while undecimal Pajara instead equates 10/9 and 11/10. 22edo, the intersection of the two systems, equates all three and consequently supports [[Porcupine]].{{Navbox regtemp}}
{{Cat|Temperaments}}

Aberrisma

2026-05-25T23:50:58Z

Overthink: /* Example: The emergence of blackdye */ capitalize temperament name

{{Proposed}}

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.

'''Aberrismic theory''' is the subset of microtonal theory pioneered by [[User:Ground|Ground]] and [[User:Inthar|Inthar]] that deals with aberrismas.

== Example: The emergence of blackdye ==
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.
[[File:Blackdye.png|thumb|510x510px|The construction of blackdye from Zarlino diatonic]]
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'''.

== The "why" of aberrismic theory ==
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.
* A style of music that is melodic and heavy in modulations benefits from
** 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.
** A set of modulatory intervals, including fifths.
** A system that allows unlimited modulation.
* 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.
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.

== List of aberrismic scales ==
* {{Adv|"GS(...)[n]" is [[generator sequence]] notation.}}
* {{adv|"subst ax(bycz)" denotes [[MOS substitution]].}}
* {{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.}}
* {{adv|Under "Patterns", "C" is [[achiral]], and "R" and "L" denote two [[chiral]]ities of a chiral pair.}}
=== Quasi-diatonic aberrismic scales ===
{|class="wikitable"
|-
!|Name / Signature
!|Pattern(s)
!|Possible JI interp.
!|{{adv|Almost a [[cross-set]] of... (interpreted)}}
!|Notes
|-
!|pinedye / dia1s (5L2m1s) {{adv|{{nowrap|subst 5L(2m1s)}}}}
||1sC: LLsLLmLm 1sR: LLmLLmLs 1sL: LLmLLsLm
||2.3.5 [L, m, s] = [10/9, 27/25, 81/80]
||{{adv|GS(3/2)[3] and GS(10/9)[3]}}
||1sC has 4 fifths and 1sR/1sL have 5
|-
!class="thl"|diasem / dia2s (5L2m2s) {{adv|{{nowrap|subst 5L(2m2s)}}}}
||2sR: LmLsLmLsL 2sL: LsLmLsLmL
||2.3.7 [L, m, s] = [9/8, 28/27, 64/63]
||{{adv|GS(3/2)[5] and 7/6}}
||Aggregate generator is 4/3, thus has fifth arcs of 5 and 4 notes respectively.
''See also: [[Chthonic harmony#Diasem|Diasem]]''
|-
!class="thl"|blackdye / dia3s (5L2m3s) {{adv|{{nowrap|subst 5L(2m3s)}}}}
||sLmLsLmLsL
||2.3.5 [L, m, s] = [10/9, 16/15, 81/80]
||{{adv|''GS(3/2)[5] and 10/9''}}
||Two interleaved 3-limit pentatonics
''See also: [[10-form#Blackdye|Blackdye]]''
|-
!|diaslen / dia4s (5L2m4s) {{adv|{{nowrap|subst 5L(2m4s)}}}}
||4sC: LmLsLsLmLss 4sR: LsLmLsLsLms 4sL: LsLsLmLsLsm
||2.3.7 [L, m, s] = [9/8, 49/48, 64/63]
||{{adv|GS(3/2)[4] and GS(8/7)[3]}}
||Fifth arcs with 4 notes, 4 notes, and 3 notes, with offset 8/7. Tempered to the slentonic {5L6s) MOS by [[Slendric]]. {{adv|Detempered Slendric[11] generator structure, aggregate generator is 3/2}}
|-
!|diachrome / chromedye / dia5s (5L2m5s)
||5sC: LsLsLmsLsLsm {{adv|{{nowrap|(subst 2m(5L5s))}}}} 5sR: LmsLsLsLmsLs 5sL: LsLsLsmLsLsm
||5120/5103-tempered 2.3.5.7 [L, m, s] = [10/9, 256/243, 81/80]
||{{adv|5sC: ''GS(3/2)[6] and 40/27''}}
||Fifth-generated but with a 6-step offset
|-
!|whitedye / dia7s (5L2m7s) {{adv|{{nowrap|subst 7s(5L2m)}}}}
||LsLsLsmsLsLsms
||5120/5103-tempered 2.3.5.7 [L, m, s] = [10/9, 28/27, 81/80]
||{{adv|''GS(3/2)[7] and 81/80''}}
||Two interleaved diatonics
|}

=== Other aberrismic scales ===
{|class="wikitable"
|-
!|Name / Signature
!|Pattern(s)
!|Possible JI/[[erac]] interp.
!|{{adv|Almost a [[cross-set]] of... (interpreted)}}
!|Notes
|-
!class="thl"|[[penslen]] / slen5m (5L5m6s) {{adv|{{nowrap|subst 6s(5L5m)}}}}
||LmsLmsLsmLsmLsms
||2.3.5.7.11[41 & 46] [L, m, s] = [12/11, 33/32, 64/63]
||{{adv|''GS(8/7)[8] and 11/8''}}
|| Has two aberrisma sizes, s and m.
|-
!|smi2m? (4L2m3s) {{adv|{{nowrap|subst 2m(4L3s)}}}}
||C: LLsmLsLms R: LmLsLmsLs L: LmLsLsmLs
||2.5.7 [L, m, s] = [28/25, 35/32, 50/49]
||{{adv|GS(5/4)[3] and GS(7/5)[3] (exact for C)}}
||Didacus tempering makes L = m + s.
|-
!|arm5s (7L2m5s) {{adv|{{nowrap|subst 7L(2m5s)}}}}
||LmLsLsLmLsLsLs
||2.x<3.5.7.11.13[37edo] (4:2:1)
||{{adv|''GS(<<3/2)[7] and 14/13''}}
||An interleaving of two antidiatonic scales.
|-
!|mosh3s (3L4m3s) {{adv|{{nowrap|subst 4m(3L3s)}}}}
||LmsLmsmLsm
||2.x<3.7.11.13[37edo] (5:4:2)
||{{adv|''GS(16/13)[5] and 11/8''}}
||
|-
!|smi2s (4L3m2s) {{adv|{{nowrap|subst 2s(4L3m)}}}}
||C: LLmsLmLsm R: LmLmsLmLs L: LmLsLmLsm
||2.9.7.11.17[46edo] (8:4:1)
||{{adv|GS(17/14)[3] and GS(11/8)[3] (exact for C)}}
||
|}

=== Systematic naming ===
Basic systematic names for aberrismic scales are of the form

[mos_prefix]n[added_step_size] (e.g. dia2s for diasem),

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]).

This is subject to change as aberrismic theory notation is updated in the future.

== Notation ==
Aberrismic notation restricts to [[diatonic notation]] on the tempered 3-limit.

Aberrismic/straddle-prime accidentals:
* 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. 
* 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).

== Aberrismic theory and RTT ==
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 improve the function of commas (frequently [[81/80]] or [[64/63]]) as aberrismas in ternary LCJI scales by tempering them larger than just.

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.

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.

=== Example: Blackdye ===
The following table shows two different temperament interpretations for the same aberrismic scale pattern blackdye (sLmLsLmLsL), under untempered 2.3.5 and Ultrapyth respectively.
* ''Untempered'' does not mean that the final tuning must be the JI tuning, but simply that there exists an exact JI tuning.
* [[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''.

{| class="wikitable right-2 right-3 right-4 right-5"
|+ style="font-size: 105%;" | Blackdye intervals in two temperaments
|-
! Interval class
! Sizes
! Untempered 2.3.5
! 2.3.7.11.13 Flutterpyth (extended to 13-limit Ultrapyth)
|-
! 1-step
| s m L
| 81/80 16/15 10/9
| 143/140 22/21 160/143
|-
! 2-step
| L + s L + m
| 9/8 32/27
| 8/7, 9/8 7/6
|-
! 3-step
| L + 2s L + m + s 2L + s 2L + m
| 729/640 6/5 5/4 320/243
| 7/6 13/11 14/11 13/10
|-
! 4-step
| 2L + 2s 2L + m + s
| 81/64 4/3
| 13/10 4/3
|-
! 5-step
| 2L + m + 2s 2L + 2m + s 3L + 2s 3L + m + s
| 27/20 64/45 45/32 40/27
| 66/49 11/8 16/11 49/33
|-
! 6-step
| 3L + m + 2s 3L + 2m + s
| 3/2 128/81
| 3/2 20/13
|-
! 7-step
| 3L + m + 3s 3L + 2m + 2s 4L + m + 2s 4L + 2m + s
| 243/160 8/5 5/3 1280/729
| 20/13 11/7 22/13 12/7
|-
! 8-step
| 4L + m + 3s 4L + 2m + 2s
| 27/16 16/9
| 12/7 7/4, 16/9
|-
! 9-step
| 5L + 2m + s 5L + m + 2s 4L + 2m + 2s
| 9/5 15/8 160/81
| 143/80 21/11 280/143
|}

== Glossary ==
* '''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)).
* '''Aberrisma''': The smallest interval region that melodically sounds like a step.
* '''Magnitone''': The melodic function of L + s in quasi-diatonic aberrismic scales.
* '''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.
* '''Solitone''': The melodic function of the L step in quasi-diatonic aberrismic scales.
* '''Subaberrisma''': A step so small (smaller than an aberrisma) that its status as a melodic step is unclear.

== Compositional examples ==
Some compositional snippets using aberrismic scales:

* The Art of It makes almost exclusive use of 31edo diasem: [[File:A New Dusk-02 The Art of It.mp3]]
* A fugue using aberrismic scales: [[File:Inthar - Fugue in 32edo and 33edo.mp3]]
* A 34edo blackdye fugue exposition: [[File:Blackdye-fugue-expo.mp3]]

== External links ==
* [https://turbofishcrow.github.io/ternary Ternary: aberrismic-focused ternary scale analysis]
{{cat|
Terms
Aberrismic terms
Ternary scales
}}

Archy

2026-05-24T00:49:31Z

Overthink: /* List of patent vals */ Technically they do patently support archy. I don't know if they should be included, but you know, WP:BEBOLD!

'''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.

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.

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.

== Structural theory ==

=== Extensions ===
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).

Archy can be defined as 5 & 7, which is the [[meantone]] extension ''dominant'' in the full 7-limit. Other extensions follow:

==== 5/4 as limma-flat major third (22 & 27), often called "Superpyth" ====
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).

==== 5/4 as doubly limma-flat major third (5 & 37) ====
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.

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]].

=== Machine temperament ===
'''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.

* [[11edo]] nearly divides 9/7 in half
* [[17edo]] provides a near-isodifferential tuning of ~8:11:14, actually much closer to 13:18:23
* [[28edo]] (generator 5\28, 214.3c) provides a near-isodifferential ~7:9:11, which is actually much closer to 32:41:50.

Machine generates machinoid (5L1s) and 6L5s.

== Compositional theory ==

=== Chords ===
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.

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.

=== Full 7-limit harmony ===
''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.''

''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.

==== Leading tones ====
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.

==== Further functional harmony ====
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.

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.

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.)

==== Nearmajor key ====
[[File:Nearmajor.mp3|thumb|Natural nearmajor scale and tonic chord]]
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.

==== Nearminor key ====
[[File:Nearminor.mp3|thumb|Natural nearminor scale and tonic chord]]
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.

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.

==== Supermajor key ====
[[File:Supermajor.mp3|thumb|Natural supermajor scale and tonic chord]]
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.

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.

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).

==== Subminor key ====
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).
[[File:Subminor.mp3|thumb|Natural subminor scale and tonic chord]]
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.

==== Alternative leading tones ====
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.

==== Consonant vs. tense suspended chords ====
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.

== Scales ==

=== Superpyth diatonic ===
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.

It may be useful, perhaps for [[extraclassical tonality]], to use the full 12-note form of superpyth's scale.

=== Superpyth pentatonic ===

=== Zarlino ===
''Similar to the previous section, this assumes a reasonably accurate extension to prime 5.''

== List of patent vals ==
{| class="wikitable"
|+
!EDO
!Extension to 5
!Generator tuning
!Diatonic scale hardness
!7/4 tuning
|-
|3
|
|800c
|N/A
|800c
|-
|8
|
|450c
|N/A
|900c
|-
|13
|
|461.5c
|N/A
|923.1c
|-
|5
|[5 & 37], [22 & 27], dominant
|480c
|∞ (collapsed)
|960c
|-
|42
|[5 & 37]
|485.7c
|8
|971.4c
|-
|37
|[5 & 37]
|486.5c
|7
|973c
|-
|32
|[5 & 37]
|487.5c
|6
|975c
|-
|59
|
|488.1c
|5.5
|976.3c
|-
|27
|[22 & 27]
|488.9c
|5
|977.8c
|-
|49
|[22 & 27]
|489.8c
|4.5
|979.6c
|-
|22
|[22 & 27]
|490.9c
|4
|981.8c
|-
|17
|[22 & 27]
|494.1c
|3
|988.2c
|-
|12
|dominant
|500c
|2
|1000c
|-
|7
|dominant
|514.3c
|1 (equalized)
|1028.6c
|-
|2
|
|600c
|N/A
|1200c
|}

{{cat|Temperaments}}{{Navbox regtemp}}

Meantone

2026-05-23T23:22:19Z

Overthink: /* 11-limit[12 & 19] */ add an equivalence

[[File:Meantone.png|thumb|Meantone equates four 3/2s to 5/1 and generates pentic (2L 3s) and mosdiatonic (5L 2s) scales.]]
'''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.

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.

== Extensions ==
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.

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.)

More accurate extensions of Meantone's diatonic structure to include other primes follow.

==== 7/4 as the augmented sixth (12 & 19) ====
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.

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.)

===== 11-limit[12 & 19] =====
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.

===== 11/8 as the double-augmented third (12 & 31) =====
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).

===== 11/8 as the double-diminished fifth (19 & 31) =====
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).

==== 7/4 as the diminished seventh (19 & 26) ====
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).

== Chords ==
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.

The major and minor seventh chords in meantone diatonic can be enumerated as 8:10:12:15 and 10:12:15:18 respectively, and the half-diminished seventh chord as 25:30:36:45, or 35:42:50:63 in septimal meantone. Additionally, the 5:6:7:9 chord is available as P1-m3-A4-m7.

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.

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.

Meantone also contains an [[Collection of chords#Essentially tempered chords|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).

== Tunings ==
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.

=== 1/11-comma Meantone ===
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]].

=== 1/5-comma Meantone ===
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.

=== Quarter-comma Meantone ===
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.

=== Golden meantone ===
''Main article: [[Golden generator]]''

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.

=== 2/7-comma Meantone ===
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.

=== 1/3-comma Meantone ===
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.

=== Silver flattone ===
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.

=== 2/5-comma Meantone ===
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.

=== 1/2-comma Meantone ===
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).

=== (Half Comma) Cleantone ===
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.

=== Lucy Tuning ([[88edo|88edo]]) ===
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.

=== [[31edo]] ===

=== [[19edo]] ===

=== [[12edo]] ===

== List of patent vals ==
{| class="wikitable"
|+
!EDO
!7-limit strong extensions
!11-limit strong extensions
!Generator tuning
|-
|5
|Dominant
|
|720c
|-
|12
|Dominant, Septimal Meantone
|[12 & 31], [12 & 19]
|700c
|-
|67
|
|
|698.5c
|-
|55
|
|
|698.2c
|-
|98
|
|
|698c
|-
|43
|Septimal Meantone
|[12 & 31]
|697.7c
|-
|117
|
|[12 & 31]
|697.4c
|-
|74
|Septimal Meantone
|[12 & 31]
|697.3c
|-
|105
|Septimal Meantone
|[12 & 31]
|697.1c
|-
|31
|Septimal Meantone
|[12 & 31], [19 & 31]
|696.8c
|-
|81
|Septimal Meantone
|[19 & 31]
|696.3c
|-
|50
|Septimal Meantone
|[19 & 31]
|696c
|-
|69
|
|[19 & 31]
|695.7c
|-
|88
|
|[26 & 31]
|695.5c
|-
|19
|Septimal Meantone, Flattone
|[12 & 19], [19 & 31], (Flattone)
|694.7c
|-
|45
|Flattone
|(Flattone)
|693.3c
|-
|26
|Flattone
|(Flattone)
|692.3c
|-
|7
|Dominant, Flattone
|(Flattone)
|685.7c
|}

{{Navbox regtemp}}

{{Cat|Temperaments}}

Archy

2026-05-23T05:29:36Z

Overthink: /* List of patent vals */ add these.

'''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.

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.

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.

== Structural theory ==

=== Extensions ===
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).

Archy can be defined as 5 & 7, which is the [[meantone]] extension ''dominant'' in the full 7-limit. Other extensions follow:

==== 5/4 as limma-flat major third (22 & 27), often called "Superpyth" ====
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).

==== 5/4 as doubly limma-flat major third (5 & 37) ====
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.

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]].

=== Machine temperament ===
'''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.

* [[11edo]] nearly divides 9/7 in half
* [[17edo]] provides a near-isodifferential tuning of ~8:11:14, actually much closer to 13:18:23
* [[28edo]] (generator 5\28, 214.3c) provides a near-isodifferential ~7:9:11, which is actually much closer to 32:41:50.

Machine generates machinoid (5L1s) and 6L5s.

== Compositional theory ==

=== Chords ===
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.

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.

=== Full 7-limit harmony ===
''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.''

''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.

==== Leading tones ====
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.

==== Further functional harmony ====
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.

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.

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.)

==== Nearmajor key ====
[[File:Nearmajor.mp3|thumb|Natural nearmajor scale and tonic chord]]
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.

==== Nearminor key ====
[[File:Nearminor.mp3|thumb|Natural nearminor scale and tonic chord]]
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.

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.

==== Supermajor key ====
[[File:Supermajor.mp3|thumb|Natural supermajor scale and tonic chord]]
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.

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.

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).

==== Subminor key ====
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).
[[File:Subminor.mp3|thumb|Natural subminor scale and tonic chord]]
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.

==== Alternative leading tones ====
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.

==== Consonant vs. tense suspended chords ====
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.

== Scales ==

=== Superpyth diatonic ===
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.

It may be useful, perhaps for [[extraclassical tonality]], to use the full 12-note form of superpyth's scale.

=== Superpyth pentatonic ===

=== Zarlino ===
''Similar to the previous section, this assumes a reasonably accurate extension to prime 5.''

== List of patent vals ==
{| class="wikitable"
|+
!EDO
!Extension to 5
!Generator tuning
!Diatonic scale hardness
!7/4 tuning
|-
|8
|
|450c
|N/A
|900c
|-
|13
|
|461.5c
|N/A
|923.1c
|-
|5
|[5 & 37], [22 & 27], dominant
|480c
|∞ (collapsed)
|960c
|-
|42
|[5 & 37]
|485.7c
|8
|971.4c
|-
|37
|[5 & 37]
|486.5c
|7
|973c
|-
|32
|[5 & 37]
|487.5c
|6
|975c
|-
|59
|
|488.1c
|5.5
|976.3c
|-
|27
|[22 & 27]
|488.9c
|5
|977.8c
|-
|49
|[22 & 27]
|489.8c
|4.5
|979.6c
|-
|22
|[22 & 27]
|490.9c
|4
|981.8c
|-
|17
|[22 & 27]
|494.1c
|3
|988.2c
|-
|12
|dominant
|500c
|2
|1000c
|-
|7
|dominant
|514.3c
|1 (equalized)
|1028.6c
|}

{{cat|Temperaments}}{{Navbox regtemp}}

19edo

2026-05-11T23:04:48Z

Overthink: /* 665edo */ wait that kinda breaks it

'''19edo''' is an equal division of 2/1 into 19 steps of 1200c/19 ~= 63.2c each.

19edo is interesting as a flatter [[Meantone]] system; it is in fact very close to 1/3-comma Meantone (i.e. Meantone with exact 6/5). It has [[interordinal]]s and supports [[Semaphore]], which means that it treats half of its perfect fourth as a septimal (8/7) major second, leading to a 9-note MOS semiquartal (5L4s) that can be created from stacking its 8/7. 19edo is also a tuning of the following temperaments which provide non-12edo [[comma pump|comma loop]] progressions:
* [[Kleismic]] temperament (six 6/5's = 3/1) but with lower accuracy due to its 6/5 being barely sharpened at all
* [[Magic]] temperament (five 5/4's = 3/1) with the generator 5/4 on the flat end for Magic
* [[Negri]], equating four 16/15's to one 4/3.

An early use of 19edo was in the mid-1500s by French composer Guillaume Costeley in his chanson "Seigneur Dieu ta pitie", in which he uses mostly the standard consonances from the diatonic scale (perfect fifth, major third, etc.) as contrapuntal consonances, but incorporates motions by a third of a tone that are outside the diatonic scale. A keyboard rendition of the chanson can be heard on [https://www.youtube.com/watch?v=wT6-Ndx1EbM YouTube].

== Basic theory ==
=== Intervals and notation ===
19edo can be notated entirely with standard [[diatonic notation]], with #/b = 1\19 and x/bb = 2\19, and equivalences E# = Fb and B# = Cb.

=== Prime harmonic approximations ===
{{Harmonics in ED|19|23|0}}
{{Cat|Edos}}19edo has a reasonable approximation of 9/7 and is a good 2.3.5.7.13 temperament for its size. Its actual mappings of 7 and 13 are inaccurate, however, and so it can be seen as a 2.<3.<5.<<<7.<<<13 tuning, where 28/27, 13/7, and 125/112 are approximated very accurately, and 9/7, 15/13, and 28/25 are approximated somewhat accurately.

As most of 19edo's errors on primes are approximately a third of an edostep, that suggests that 57edo is a good tuning in 2.7.11.13.17.19.

=== Edostep interpretations ===
19edo's edostep has the following interpretations in the 2.3.5.7.13 subgroup:

* 25/24 (the interval between 6/5 and 5/4)
* 26/25 (the interval between 25/16 and 13/8)
* 27/26 (the interval between 13/8 and 27/16)
* 28/27 (the interval between 9/8 and 7/6)

25/24 ~ 26/25 ~ 27/26 is an equivalence that defines 2.3.5.13 Kleismic.

=== Scales ===
Basic or soft MOSes in 19edo include:

* diatonic
* checkertonic (3-2-3-2-2-3-2-2)
* manual (4-4-4-4-3)
* antimachinoid (3-3-3-3-3-4)
* antisubneutralic (2-2-2-2-2-2-2-2-3)

Due to being a meantone tuning with an interval in between 5/4 and 4/3, 19edo has a 7-limit [[omnidiatonic]] scale (3-4-1-3-4-3-1).

The semiquartal (5L4s) MOS (3-1-3-1-3-1-3-1-3) is also commonly used in 19edo. Also note the Kleismic MOSes:
* 4L7s, interpreted as Kleismic[11]: 1 1 3 1 1 3 1 1 3 1 4
* 4L11s, Kleismic[15]: 1 1 1 2 1 1 1 2 1 1 1 2 1 1 2

== Multiples ==

=== 57edo ===
57edo is a good no-threes no-fives 19-limit or 2.5/3.7.11.13.19 subgroup system.
{{Harmonics in ED|57}}

=== 171edo ===
Combining 19edo's accurate 28/27 and 6/5 with [[9edo]]'s accurate 7/6 results in 171edo which is a very accurate model of 7-limit JI, even more accurate than [[99edo]]. Since its 14/13 is inherited from 19edo, its 13th harmonic is acceptable too. 171edo supports [[Schismic]] and [[Ennealimmal]].
{{Harmonics in ED|171}}

=== 665edo ===
665edo = 19 * 35 is mostly known for its absurdly accurate Pythagorean tuning, supporting the "Satanic" temperament equating a stack of 666 perfect fifths (octave reduced) to a single perfect fifth, with an accuracy within 1/1000 of a cent. It additionally shares 19edo's tunings of 6/5 and 28/27, supporting Enneadecal temperament, and is a strong no-11 23-limit tuning.
{{Harmonics in ED|665}}

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

19edo

2026-05-11T23:04:26Z

Overthink: /* 665edo */ more info

'''19edo''' is an equal division of 2/1 into 19 steps of 1200c/19 ~= 63.2c each.

19edo is interesting as a flatter [[Meantone]] system; it is in fact very close to 1/3-comma Meantone (i.e. Meantone with exact 6/5). It has [[interordinal]]s and supports [[Semaphore]], which means that it treats half of its perfect fourth as a septimal (8/7) major second, leading to a 9-note MOS semiquartal (5L4s) that can be created from stacking its 8/7. 19edo is also a tuning of the following temperaments which provide non-12edo [[comma pump|comma loop]] progressions:
* [[Kleismic]] temperament (six 6/5's = 3/1) but with lower accuracy due to its 6/5 being barely sharpened at all
* [[Magic]] temperament (five 5/4's = 3/1) with the generator 5/4 on the flat end for Magic
* [[Negri]], equating four 16/15's to one 4/3.

An early use of 19edo was in the mid-1500s by French composer Guillaume Costeley in his chanson "Seigneur Dieu ta pitie", in which he uses mostly the standard consonances from the diatonic scale (perfect fifth, major third, etc.) as contrapuntal consonances, but incorporates motions by a third of a tone that are outside the diatonic scale. A keyboard rendition of the chanson can be heard on [https://www.youtube.com/watch?v=wT6-Ndx1EbM YouTube].

== Basic theory ==
=== Intervals and notation ===
19edo can be notated entirely with standard [[diatonic notation]], with #/b = 1\19 and x/bb = 2\19, and equivalences E# = Fb and B# = Cb.

=== Prime harmonic approximations ===
{{Harmonics in ED|19|23|0}}
{{Cat|Edos}}19edo has a reasonable approximation of 9/7 and is a good 2.3.5.7.13 temperament for its size. Its actual mappings of 7 and 13 are inaccurate, however, and so it can be seen as a 2.<3.<5.<<<7.<<<13 tuning, where 28/27, 13/7, and 125/112 are approximated very accurately, and 9/7, 15/13, and 28/25 are approximated somewhat accurately.

As most of 19edo's errors on primes are approximately a third of an edostep, that suggests that 57edo is a good tuning in 2.7.11.13.17.19.

=== Edostep interpretations ===
19edo's edostep has the following interpretations in the 2.3.5.7.13 subgroup:

* 25/24 (the interval between 6/5 and 5/4)
* 26/25 (the interval between 25/16 and 13/8)
* 27/26 (the interval between 13/8 and 27/16)
* 28/27 (the interval between 9/8 and 7/6)

25/24 ~ 26/25 ~ 27/26 is an equivalence that defines 2.3.5.13 Kleismic.

=== Scales ===
Basic or soft MOSes in 19edo include:

* diatonic
* checkertonic (3-2-3-2-2-3-2-2)
* manual (4-4-4-4-3)
* antimachinoid (3-3-3-3-3-4)
* antisubneutralic (2-2-2-2-2-2-2-2-3)

Due to being a meantone tuning with an interval in between 5/4 and 4/3, 19edo has a 7-limit [[omnidiatonic]] scale (3-4-1-3-4-3-1).

The semiquartal (5L4s) MOS (3-1-3-1-3-1-3-1-3) is also commonly used in 19edo. Also note the Kleismic MOSes:
* 4L7s, interpreted as Kleismic[11]: 1 1 3 1 1 3 1 1 3 1 4
* 4L11s, Kleismic[15]: 1 1 1 2 1 1 1 2 1 1 1 2 1 1 2

== Multiples ==

=== 57edo ===
57edo is a good no-threes no-fives 19-limit or 2.5/3.7.11.13.19 subgroup system.
{{Harmonics in ED|57}}

=== 171edo ===
Combining 19edo's accurate 28/27 and 6/5 with [[9edo]]'s accurate 7/6 results in 171edo which is a very accurate model of 7-limit JI, even more accurate than [[99edo]]. Since its 14/13 is inherited from 19edo, its 13th harmonic is acceptable too. 171edo supports [[Schismic]] and [[Ennealimmal]].
{{Harmonics in ED|171}}

=== 665edo ===
665edo = 19 * 35 is mostly known for its absurdly accurate Pythagorean tuning, supporting the "Satanic" temperament equating a stack of 666 perfect fifths (octave reduced) to a single perfect fifth, with an accuracy within 1/1000 of a cent. It additionally shares 19edo's tunings of 6/5 and 28/27, supporting Enneadecal temperament, and is a strong no-11 23-limit tuning.

{{Harmonics in ED|665}}

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

Golden generator

2026-04-22T01:36:08Z

Overthink: /* Example, Golden meantone */ specify interpretation

''Not to be confused with [[31edo#Scales 2|MOS scales in 31edo]].''

A '''golden generator''' is an interval that, when taken as a [[generator]] against some [[period]] (usually the [[octave]]), produces [[MOS]]<nowiki/>es that all have the same step size ratio. This ratio is the golden ratio, an irrational number equaling approximately 1.61803 and falling within the "quasisoft" [[TAMNAMS]] hardness category. Therefore, temperaments that are well-tuned at golden generators are a kind of "anti-cluster" temperament, where all notes are as evenly spaced as possible at all MOS sizes. Temperaments with golden generators may be desirable because cluster temperaments, especially corresponding to small edos, are melodically inconvenient for the purposes for which Western composers and many xenharmonic composers use scales, and (ironically) difficult to tune in an edo of reasonable size. As a result, however, subgroups that naturally simplify into temperaments with golden generators (such as the 5-limit with meantone) are difficult to consider in terms of any given [[form]].

Soft scales are a natural tendency for musical cultures around the world; Leriendil suggests that having a soft scale was a subconscious motivation behind the choice of meantone as opposed to another tuning. The soft children of MOSes are also musically convenient for having few enharmonic intervals.

As a result of this connection to the golden ratio, MOS theory has several connections to the Fibonacci sequence and other sequences generated in a similar way (which will be called golden sequences in this article).

== MOSes and golden sequences ==
The best way to derive a generator tuning with the desired properties is by repeatedly taking the soft child of some MOS. The tuning ranges of the successive daughter MOSes will approach a value, which if used as a generator will generate soft MOSes infinitely.

If we repeatedly take the soft child of any MOS, this is equivalent to taking the MOS' scale pattern (for instance, LLsLLsLs) and replacing each "L" with "st" (ststsststssts). We then relabel s as L and t as s, resulting in LsLsLLsLsLLsL. Note that we have taken the number of large steps and added it to the total size of the MOS, resulting in a MOS with a number of large steps equal to the previous MOS size. Here, we went from 5L 3s (8 notes total) to 8L 5s (with 13 notes total). This is a representation of the same operation that produces the numbers of the Fibonacci sequence! If we kept going here, we'd get MOSes with 21, 34, 55, and 89 notes. If we start with a MOS whose step counts aren't in the Fibonacci sequence, we get an analogous sequence with those step sizes as the starting pair of numbers. For example, diatonic generates 2, 5, 7, 12, 19, 31, 50, 81...

This also serves as an intuitive explanation for how the golden ratio is the "most irrational" number.

== Root MOSes ==
But what happens when you take a hard child? The result has the step sizes flipped from what the "golden child" would be, and therefore becomes a new golden MOS chain with its own corresponding sequence. For example, if you take the Fibonacci sequence, (1, 1, 2, 3...) and stop it at 2 and 3, then flip them, you get (3, 2, 5, 7, 12, 19, 31...). Seem familiar? In fact, a common property of all hard child MOSes is that the number of large steps is smaller than the number of small steps. This means that any golden generator chain may be simply identified by an unordered pair of numbers, and allows us to unambiguously identify the root MOS of any given chain. Additionally, if we take the step sizes of any "root" MOS and generate a golden sequence with them, we get exactly the step sizes of MOSes generated by the golden tuning of that MOS.
{| class="wikitable"
!Root MOS
!Sequence
!MOSes
!Golden generator
!Related temperaments
!Notes
|-
|0L 1s
|1, 0, 1, 1, 2, 3, 5, 8, 13…
|[[1edo|1L 0s]], [[1L 1s]], [[2L 1s]], [[3L 2s]], [[5L 3s]], [[8L 5s]], [[13L 8s]]…
|458.36, 741.64
|Aurora
|Logarithmic phi MOSes
|-
|1L 2s
|2, 1, 3, 4, 7, 11, 18…
|[[3L 1s]], [[4L 3s]], [[7L 4s]], [[11L 7s]], [[18L 11s]]…
|331.67, 868.33
|
|
|-
|1L 3s
|3, 1, 4, 5, 9, 14, 23…
|[[4L 1s]], [[5L 4s]], [[9L 5s]], [[14L 9s]], [[23L 14s]]…
|259.85, 940.15
|
|
|-
|2L 3s
|3, 2, 5, 7, 12, 19, 31…
|[[Diatonic|5L 2s]], [[Collection of scales#Mellow chromatic|7L 5s]], [[12L 7s]], [[19L 12s]], [[31L 19s]]…
|503.79, 696.21
|Meantone
|Golden pentic
|-
|1L 4s
|4, 1, 5, 6, 11, 17, 28…
|[[5L 1s]], [[6L 5s]], [[11L 6s]], [[17L 11s]], [[28L 17s]]…
|213.60, 986.40
|Machine
|
|-
|3L 4s
|4, 3, 7, 10, 17, 27…
|[[7L 3s]], [[10L 7s]], [[17L 10s]], [[27L 17s]]…
|354.82, 845.18
|Rastmic
|
|-
|2L 5s
|5, 2, 7, 9, 16, 25, 41…
|[[7L 2s]], [[9L 7s]], [[16L 9s]], [[25L 16s]], [[41L 25s]]…
|527.15, 672.85
|Trismegistus
|
|-
|3L 5s
|5, 3, 8, 11, 19, 30…
|[[8L 3s]], [[11L 8s]], [[19L 11s]], [[30L 19s]]…
|440.59, 759.41
|Sentry
|
|-
|2L 7s
|7, 2, 9, 11, 20, 31, 51…
|[[9L 2s]], [[11L 9s]], [[20L 11s]], [[31L 20s]], [[51L 30s]]…
|541.38, 658.62
|Joan
|
|}

== Finding the generator ==
For any given MOS pattern, one can easily determine what generator creates it, by the power of golden sequences. This is done by working back from the MOS to 1L 1s (which has the generator of 1L + 0s) and then working forward with the generator size as if it is a MOS itself.

Non-octave-periodic MOSes can be seen as their reduced patterns with a fractional-octave period (for example, 5L 5s can be seen as 240c-periodic 1L 1s); for this trick to work the numbers of large and small steps must be coprime.

For example, let's take the MOS 11L 6s, and interpret it as the Fibonacci sequence fragment [6, 11]. Next, we will "step" backwards, moving our two-number window back so that 6 becomes the second element and the previous entry (which is trivial to calculate as 11-6 = 5) is the first element. So, we reach [5, 6]. Then, we proceed to [1, 5], and then [4, 1]. At this point, we've reached the "beginning" of a sequence, where the two elements are descending. So, we flip: [1, 4], then proceed back to [3, 1]. Continue on until you reach [1, 1], and log the steps you took:
{| class="wikitable"
|+
!Step
!Sequence fragment
!Sequence
|-
|(Start)
|6, 11
|(4, 1)
|-
|Step
|5, 6
|(4, 1)
|-
|Step
|1, 5
|(4, 1)
|-
|Step
|4, 1
|(4, 1)
|-
|Flip
|1, 4
|(3, 1)
|-
|Step
|3, 1
|(3, 1)
|-
|Flip
|1, 3
|Lucas
|-
|Step
|2, 1
|Lucas
|-
|Flip
|1, 2
|Fibonacci
|-
|Step
|1, 1
|Fibonacci
|}
Then, we work back through our steps, starting with [0, 1] instead of [1, 1].
{| class="wikitable"
!Step
!Sequence fragment
!Sequence
|-
|(Start)
|0, 1
|Fibonacci
|-
|Step
|1, 1
|Fibonacci
|-
|Flip
|1, 1
|Fibonacci
|-
|Step
|1, 2
|Fibonacci
|-
|Flip
|2, 1
|Lucas
|-
|Step
|1, 3
|Lucas
|-
|Flip
|3, 1
|(3, 1)
|-
|Step
|1, 4
|(3, 1)
|-
|Step
|4, 5
|(3, 1)
|-
|Step
|5, 9
|(3, 1)
|}
Note that one flip operation leaves the ordered pair unchanged as it is [1, 1].

If we take our result, [5, 9] as a number of small and large steps, we get 9L + 5s, which is the generator.

For a simpler example, let's try diatonic, 5L 2s:
{| class="wikitable"
!Step
!Sequence fragment
!Sequence
|-
|(Start)
|2, 5
|(3, 2)
|-
|Step
|3, 2
|(3, 2)
|-
|Flip
|2, 3
|Fibonacci
|-
|Step
|1, 2
|Fibonacci
|-
|Step
|1, 1
|Fibonacci
|}
And then to find the generator:
{| class="wikitable"
!Step
!Sequence fragment
!Sequence
|-
|(Start)
|0, 1
|Fibonacci
|-
|Step
|1, 1
|Fibonacci
|-
|Step
|1, 2
|Fibonacci
|-
|Flip
|2, 1
|Lucas
|-
|Step
|1, 3
|Lucas
|}
And the generator is 3 large steps and 1 small step (which is correct).

Note that the large steps are read from the ''second'' entry, which is opposite to the convention used on the wiki where the number of large steps comes first.

Now we have the generator in steps, now how do we get to a range in cents? Well, for a generator (A)L + (B)s, and a scale (C)L + (D)s, {{Adv|1=the soft boundary (equalized tuning) is (A+B)\(C+D), and the hard boundary (collapsed tuning) is A\C. For example, the range for our first scale is between 9\11 (982 cents) and (9+5)\(11+6) = 14\17 (988 cents), and the range for diatonic is, as expected, between (3+1)\(5+2) = 4\7 (686 cents) and 3\5 (720 cents). Additionally,}} for any hardness L/s = k, the tuning for the generator is (kA+B)/(kC+D). For the golden tuning, k is equal to the golden ratio.

== Golden generators for common temperaments ==
Here are some golden generators for many temperaments with simple golden MOS scales, consequently leaving out many cluster temperaments.

Names in italic are not rank-2 regular temperaments.
{| class="wikitable sortable"
!Temperament
!Generator
!Golden tuning
!MOS
|-
|[[Aurora]]
|32/21
|741.6
|1L 1s
|-
|[[A-team]]
|32/21
|734.9
|5L 8s
|-
|''Sensamagic''
| -
|759.4
|3L 5s
|-
|
|
|768.9
|3L 8s
|-
|[[Meantone]]
|3/2
|696.2
|2L 3s
|-
|[[Leapday]]
| -
|704.1
|5L 7s
|-
|[[Trismegistus]]
|28/19
|672.9
|2L 5s
|-
|[[Joan]]
|16/11
|658.6
|2L 7s
|-
|
|
|649.0
|2L 9s
|-
|''Daemotonic''
| -
|331.7
|1L 2s
|-
|[[Orgone]]
|77/64
|322.3
|4L 7s
|-
|[[Rastmatic]]
|11/9
|354.8
|3L 4s
|-
|''Acoustic phi''
| -
|366.3
|3L 7s
|-
|[[Submerged]]
|5/4
|373.1
|3L 10s
|-
|
|
|259.9
|1L 3s
|-
|[[Semaphore]]
|8/7
|254.0
|5L 9s
|-
|[[Orwell]]
|7/6
|273.8
|4L 5s
|-
|
|
|280.6
|4L 9s
|-
|[[Machine]]
|9/8
|213.6
|1L 4s
|-
|[[Shoe]]
|8/7
|223.0
|5L 6s
|-
|
|
|181.3
|1L 5s
|-
|[[Jugular]]
|10/9
|188.0
|6L 7s
|-
|
|
|157.5
|1L 6s
|-
|[[Porcupine]]
|11/10
|162.6
|7L 8s
|-
|
|
|139.2
|1L 7s
|-
|[[Negri]]
|16/15
|124.8
|1L 8s
|-
|[[Miracle]]
|16/15
|113.0
|1L 9s
|}
2.3.7 is often exotempered or of very high complexity, this is because of the fact that 2.3.7 just intonation itself functions as a sort of cluster temperament.

== Argent tuning ==
A number with similar properties to the golden ratio is the square root of 2; 1+sqrt(2) is the "silver ratio". The "silver generator" is a near-[[perfect fifth]] of about 703 cents, and is the tuning of the diatonic generator such that the ratio between the large and small steps of the pentic scale is the same as between the large and small steps of p-chromatic, and that ratio is the square root of 2. Additionally, the ratio between the large and small steps of diatonic is the same as between the large and small steps of p-enharmonic. This tuning range is closely associated with [[Hemifamity]] temperament and is approximated by 29edo, 41edo, and 70edo.

Each scale actually has two silver generators, a hard one and a soft one; as such, silver generators naturally bifurcate in a somewhat similar way to how I've forced golden ones to. Soft silver [[diatonic]] is a flattone tuning.

== Example, Golden meantone ==
[[File:Meantone isomorphic table.png|thumb|Golden meantone intervals, with 13-limit 19 & 31 interpretation]]
Golden meantone is the golden tuning of [[pentic]], being a good tuning of [[meantone]] (specifically with 11/8 as the double-diminished fifth).

34edo

2026-04-21T23:56:01Z

Overthink: /* 68edo */ 64/63 and 81/80 mapping

[[File:34edo chart.png|thumb|377x377px|The structure of the 5-limit in 34et, visualized.]]
'''34edo''' is the equal tuning system which splits the octave into 34 equal steps, of about (1200/34) ~= 35.3 cents each. It is a 5-limit and 2.3.5.13 system with a number of melodically intuitive structures.

== Derivation ==

=== From doubling 17edo ===
One can observe that 17edo's step is a nearly perfectly tuned 25/24, and also that 5/4 and 6/5 are almost exactly halfway in-between notes of 17edo. Thus, 17edo can be doubled to improve the tunings of 5-limit intervals.

===== From the usage of Pythagorean diatonic semitones as classical chromatic semitones =====
Forcing 17edo's near-just 25/24, which is a Pythagorean diatonic semitone, to surround a neutral third and function as a chromatic semitone, requires offsetting the chain of fifths by a perfect semioctave, effectively allowing one to 'swap' the tunings of diatonic and chromatic semitones. This results in the 34edo tuning of Diaschismic.

=== From the DKW step sizes ===
[https://en.xen.wiki/w/DKW_theory DKW theory] suggests that the core step sizes of the 5-limit are 9/8, 16/15, and 25/24. It can be observed that 16/15 stacks twice to approximate 9/8, and that 25/24 stacks 3 times to approximate 9/8. Tempering these equivalences together results in 34edo. Because the latter (kleismic) equalizes 24:25:26:27, and the former (diaschismic) equalizes 15:16:17:18, 34edo can be seen as a 2.3.5.13.17 system. 34edo can, thus, be broken up as 6-3-2-3-6-3-2-3-6, with 6 representing 9/8, 3 representing 16/15, and 2 representing 25/24. (In the Delkian system, the 9/8 intervals are further split into two 16/15s, and in the Roklotian, the 9/8 intervals are split into three 25/24s.)

== Theory ==

==== Edostep interpretations ====
34edo's edostep, the ''sextula'', has the following interpretations in the 2.3.5.13.17 subgroup, and equal divisions thereof:

* [[81/80]], the difference between the fifth-generated major third and the classical major third
* [[128/125]], the difference between the 5-limit enharmonic intervals
* [[40/39]], the difference between [[13/10]] and [[4/3]] and between [[15/13]] and [[9/8]]; also between [[6/5]] and [[16/13]], among others
* [[65/64]], the difference between [[8/5]] and [[13/8]]
* One half of [[25/24]], the difference between [[5/4]] and [[6/5]], such that the two are inflected from the neutral third by a sextula
* One sixth of [[9/8]] (from whence derives the name ''sextula'')

''TODO: add to list''

==== JI approximation ====
34edo is straddle-7, straddle-11, and straddle-19, but has good accuracy on the 2.3.5.13.17.23 subgroup. 34edo inherits 17edo's mosdiatonic scale, 6-6-2-6-6-6-2, with the "optimally tuned" leading tone approximating 25/24. It also supports the zarlino scale, but because it does not support [[Porcupine]], the zarlino scale requires 2 sets of accidentals to notate, making it awkward to use as the basis of notation. (The best option is to use 5-sharp and 5-flat accidentals from 17edo's diatonic, as if you are notating 5-limit JI, which may in 34edo be represented as ups and downs.)
{{Harmonics in ED|34|31}}
{| class="wikitable"
|+Thirds in 31edo
!Quality
|Inframinor
|'''Farminor'''
|Nearminor
|Neutral
|Nearmajor
|'''Farmajor'''
|Ultramajor
|-
!Cents
|247
|'''282'''
|318
|353
|388
|'''424'''
|459
|-
!Just interpretation
|15/13
|'''20/17'''
|6/5
|16/13
|5/4
|'''32/25'''
|13/10
|}
MOS diatonic thirds are bolded.

=== Chords ===
34edo supports [[arto and tendo theory]] with its inframinor and ultramajor thirds. Being a Diaschismic edo, it has a series of tetrads wherein the third and seventh are separated by 600 cents, but due to not supporting [[Pajara]], these do not approximate simple 7-limit chords.

=== Scales ===
34edo contains 17edo's diatonic scale and alongside it a zarlino scale. Other scales it includes are:

* the blackdye scale, with steps 1-5-3-5-1-5-3-5-1-5 (sLmLsLmLsL)
* the 5-odd-limit tonality diamond 9-2-3-6-3-2-9 (LsmMmsL)
** the 5-limit [[DKW theory|DKW]] signature, 6-3-2-3-6-3-2-3-6 (LmsmLmsmL)
* Diaschismic[12], with steps 3-3-3-3-3-2-3-3-3-3-3-2 (LLLLLsLLLLLs)
** the Delkian scale, a MODMOS of diaschismic[12] with steps 3-3-3-2-3-3-3-3-3-3-2-3 (LLLsLLLLLLsL)
* the hemipythagorean decatonic, with steps 3-4-3-4-3-3-4-3-4-3 (sLsLssLsLs)
** the Sixanian scale, a MODMOS of the above with steps 3-4-3-4-3-3-3-4-3-4 (sLsLsssLsL)
* the Roklotian scale, 2-2-2-3-2-3-2-2-2-3-2-3-2-2-2 (sssLsLsssLsLsss)
* the MOS pentatonic, pythagorean[5], 6-8-6-8-6 (sLsLs)
* the equable pentatonic, Semaphore[5], 7-7-6-7-7 (LLsLL)
* the vertical pentatonic, 5-9-6-5-9 (sLmsL)

=== Additional regular temperaments ===
Alongside [[Kleismic]] (shared with [[15edo]] and [[19edo]]), and [[Diaschismic]] (shared with [[12edo]]), 34edo supports the following temperaments:

* [[Tetracot]] (splitting 3/2 into four 10/9s), shared with [[Equiheptatonic|7edo]] and [[27edo]])
* [[Gammic]] (setting 25/24 to a tenth of 3/2), shared with 103edo

== Notation ==

=== Notation ===
34edo may use ups and downs notation. It may also use the diaschismic notation detailed on 22edo's page. Because it splits the perfect fifth exactly in half, it may use neutral diatonic notation for a 17edo subset, relying on ups and downs to notate the rest. Note that 5/4 is simultaneously an upneutral 3rd and a downmajor 3rd; while downmajor may seem like the obvious choice, 34edo's emphasis on dividing intervals makes it somewhat logical to compare the 5-limit thirds to the neutral third instead, especially if it is understood that the semisharp represents a 25/24 semitone or one third of a whole tone.

== Multiples ==
=== 68edo ===
68edo is the double of 34edo, and improves its mapping of 7 much as 34edo improves 17edo's mapping of 5. This improves the mappings of 11 and 19 as well, making 68edo function as a general 19-limit system. While its 13 is inaccurate in relative error, it is 34edo's kleismic tuning of 13 and shares all the structural properties thereof, which justifies it. Due to its sharp fifth, the septimal comma 64/63 is mapped to one step while 81/80 is two steps, thus mapping the [[aberschisma]] to a negative step, and making 68edo a Bidic temperament.

The new 7/4 supports [[Sensamagic]], doubling 9/7 to reach 5/3, and 2.5.7 [[Didacus]], splitting 5/4 into two wholetones that stack 5 times to reach 7/4. Additionally, the new 11/8 makes 14/11 equal to 81/64, supporting [[Pentacircle]] (and various gentle/neogothic temperaments).

To notate 68edo, there is a rather neat scheme wherein semisharps and semiflats represent 17edo steps, ups and downs represent 34edo steps, and lifts and drops represent 68edo steps.

{{Harmonics in ED|68|31|0}}

=== 306edo ===
306 is the decominator of a continued fraction convergent to log2(3/2), and as such 306edo has a nearly perfectly accurate 3/2 representation. Its step is the difference between 34edo's 3/2 and the near-just one. It also has a 7/4 accurate to within 0.2 cents.

{{Harmonics in ED|306|31|0}}

=== 612edo ===
612edo doubles 306edo, adding the perfect fifth from [[12edo]] and a nearly perfect [[5/4]]. Its main utility is as a fine-grained interval size measurement system for the 11-limit, wherein 3/2 is 358 steps and 5/4 is 197 steps, as its step size is almost exactly a (consistently represented) [[schisma]]. The 12edo perfect fifth is 357 steps, 34edo's is 360 steps. Thus, 34edo is about twice as inaccurate as 12edo in its tuning of 3/2.

{{Harmonics in ED|612|31|0}}

{{Navbox EDO}}

{{Cat|Edos}}

5-odd-limit

2026-04-14T01:11:01Z

Overthink: add navigation

{{Odd-limit navigation}}
The '''5-odd-limit''' is the set of intervals where the largest allowable odd factor in the numerator and denominator is 5. It is the smallest odd-limit containing intervals of the 5-limit. In general, the intervals of the 5-odd-limit are also those considered [[Consonance|consonances]] in standard Western music theory, and include as a subset the intervals of the 3-limit, which are the perfect consonances. This is where the xenharmonic generalization of a set of intervals considered 'consonances' comes from, and is why odd-limits are used as a complexity measure for JI intervals.

The 5-odd-limit is equivalent to the intervals considered to be consonant by Zarlino.

== Table of 5-odd-limit intervals ==
Reduced to an octave, the intervals of the 5-odd-limit are:
{| class="wikitable"
!Interval
!Cents
!Name
!Type
|-
|1/1
|0.0
|Unison
|Equivalence
|-
|6/5
|315.6
|Classical minor 3rd
|Imperfect consonance
|-
|5/4
|386.4
|Classical major 3rd
|Imperfect consonance
|-
|4/3
|498.0
|Perfect 4th
|Perfect consonance
|-
|3/2
|702.0
|Perfect 5th
|Perfect consonance
|-
|8/5
|813.6
|Classical minor 6th
|Imperfect consonance
|-
|5/3
|884.4
|Classical major 6th
|Imperfect consonance
|-
|2/1
|1200.0
|Octave
|Equivalence
|}

== Approximation by edos ==
The first edo consistent to the 5-odd-limit is 3edo, outlining the structure of triadic harmony with the augmented triad. The minor 3rd, major 3rd, and perfect 4th are mapped to 400c, while the perfect fifth, minor sixth, and major sixth are mapped to 800 cents. Beyond this, 7edo is often the first edo to be seriously considered as approximating the 5-odd-limit, as its most damaging 5-limit temperament, [[Dicot]], does not lead to categorical conflicts the same way 3edo's [[Father]] does. However, for all the intervals of the 5-odd-limit to be distinctly represented, the smallest viable edo is 9edo. Although 9edo severely damages the perfect fifth and minor third, it does make all the categorical distinctions necessary to support some form of triadic harmony based on the contrast between major and minor, which is characteristic of the use of 5-odd-limit consonances.

The first edo to distinguish all of the 5-odd-limit intervals while tuning them all reasonably accurately is 12edo — this is one factor that led to 12edo's worldwide standardization. The second edo to do so, [[19edo]], is a [[Meantone]] tuning like 12edo though more accurate; the arithmetic of 5-limit intervals may lead to non-12edo results, such as [[Magic|five major thirds stacking to a fifth]]. [[22edo]], a non-Meantone tuning, has the opposite tuning tendencies to 12edo.

== Intervals of the 5-odd-limit ==

=== Perfect consonances ===

==== Perfect fourth (4/3) ====
''Main article: [[Perfect fourth]]''

The perfect fourth is a perfect consonance and the bounding interval for [[chthonic]] harmony. It also exists between the fifth over the root and the root an octave up. It is the dark generator of the diatonic scale; stacking it produces the Locrian [[mode]].

In certain triadic musical traditions that use 4:5:6 as a consonant chord, the perfect fourth over the root can be considered dissonant, as it resolves downwards to the major third.

==== Perfect fifth (3/2) ====
''Main article: [[Perfect fifth]]''

The perfect fifth is an unambiguous perfect consonance. It appears in musical systems worldwide, and can be easily tuned by ear. This is the reason behind the prevalence of the [[Pythagorean]] system of tuning.

In the context of 5-limit consonances, the perfect fifth serves as the bounding interval of the triads 10:12:15 (minor) and 4:5:6 (major), which utilize the other 5-odd-limit consonances 5/4 and 6/5.

=== Imperfect consonances ===

==== Classical major third (5/4) ====
The major third 5/4 serves primarily as a component in tertian chords like 4:5:6. It is the most consonant "third" interval. 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.

5/4 is also the octave-reduced generator of the [[5-limit|prime 5]] axis in [[lattice-based just intonation]].

==== Classical minor third (6/5) ====
The classical minor third 6/5 is the fifth complement of 5/4. The distinction between the two leads to the paradigm of major vs. minor in interval classification and in triadic harmony; it is why the "third" category of intervals exists at all, and additionally why thirds are often considered the "default" example of interval qualities.

One quality of 6/5 worth noting is that chords with 6/5 as a lower interval are, as a rule, not "rooted" (in that their root note is not a power of 2 in the harmonic series). The significance of this is debated by xenharmonic theorists; Lamplight uses it as a model for the different "feels" of the chords 4:5:6 and 10:12:15.

==== Classical major sixth (5/3) ====
The classical major sixth is the octave complement of 6/5. It is according to some the next most consonant interval within the octave after 4/3; Leriendil sees it as an important target interval on the level of 4/3 and it is also the bounding interval of the chord 3:4:5, which may be seen as an inversion of 4:5:6 or as the primary focus of 5-limit "/3" harmony (such as in [[Kleismic]]).

==== Classical minor sixth (8/5) ====
The classical minor sixth is, while consonant on its own, unusually dissonant for a 5-odd-limit consonance in certain contexts; the result is likely a combination of factors. First is its complex ratio - it is the only 5-odd-limit interval in the octave that uses 8 in the numerator. Second is its proximity to the golden ratio, which serves as a distinctly dissonant target (similar to the [[semioctave]]'s influence on 7/5). Third is its proximity to 3/2, which produces a 'zone' of dissonance around it. Also of relevance to the discussion is the 12edo augmented triad, which contains a note tuned the same way as the classical minor sixth (and which may be voiced as it in JI depending on interpretation) yet is considered a dissonant chord.

This and the major sixth mainly show up in chords in Western harmony as the bounding intervals of triads in certain inversions.

Perfect fourth

2026-04-12T05:03:48Z

Overthink: start page

{{Stub}}
The '''perfect fourth''' is the octave complement of the [[perfect fifth]]. It generally has a frequency ratio of '''4/3''', and is 498.0 cents in size when justly tuned.

While it has a relatively simple ratio, in classical theory, the perfect fourth above the root is often treated as a dissonance to resolve down to the major third.

7-odd-limit

2026-04-12T04:59:57Z

Overthink: start harmony section

{{Odd-limit navigation}}
The '''7-[[odd-limit]]''' consists of all intervals where the largest allowable odd factor in the numerator and denominator is 7. It is the smallest odd-limit containing intervals of the [[7-limit|7-prime-limit]], thus creating xenharmonic categories not found in traditional music theory. In a 7-prime-limit system, all the ratios of the 7- or [[9-odd-limit]] can be treated as consonances.

== Table of 7-odd-limit intervals ==
Reduced to an octave, the intervals of the 7-odd-limit are:

{| class="wikitable"
! Interval
! Cents
! Name
|-
| 1/1
| 0.0
| Unison
|-
| '''8/7'''
| '''231.2'''
| '''Septimal major 2nd'''
|-
| '''7/6'''
| '''266.9'''
| '''Septimal minor 3rd'''
|-
| 6/5
| 315.6
| Classical minor 3rd
|-
| 5/4
| 386.4
| Classical major 3rd
|-
| 4/3
| 498.0
| Perfect 4th
|-
| '''7/5'''
| '''582.5'''
| '''Lesser septimal tritone'''
|-
| '''10/7'''
| '''617.5'''
| '''Greater septimal tritone'''
|-
| 3/2
| 702.0
| Perfect 5th
|-
| 8/5
| 813.6
| Classical minor 6th
|-
| 5/3
| 884.4
| Classical major 6th
|-
| '''12/7'''
| '''933.1'''
| '''Septimal major 6th'''
|-
| '''7/4'''
| '''968.8'''
| '''Septimal minor 7th'''
|-
| 2/1
| 1200.0
| Octave
|}

== Approximation by edos ==
[[File:7-odd-limit in edos.png|thumb|right|A diagram showing the approximation of the 7-odd-limit by various edos.]]
The first [[edo]] consistent to the 7-odd-limit is [[4edo]], which maps 5/4 to 1 step, 3/2 to 2 steps, and 7/4 to 3 steps, laying down a rough framework of tetradic harmony. Then, [[10edo]] approximates the 7-odd-limit relatively accurately for size, though it conflates several interval pairs: 5/4~6/5, 7/6~8/7, and 7/5~10/7. As such, the [[10-form]] is useful for classifying the 7-limit. After that, [[12edo]] distinguishes 5/4 from 6/5 and 7/6 from 8/7, though it has 6/5~7/6 and 7/5~10/7, and the 7th harmonic is tuned very sharply. The [[15edo]] and [[19edo]] tunings distinguish 5/4, 6/5, and 7/6, as well as 7/5 and 10/7, but 7/6 is equated to 8/7, an equivalence known as [[Interseptimal (temperament)|Interseptimal or Semaphore temperament]]. The first to distinguish all of 5/4, 6/5, 7/6, and 8/7 is [[22edo]], though 7/5 is still equated with 10/7. The first edo to distinguish the entire 7-odd-limit is [[27edo]], but one may prefer [[31edo]] for a more accurate approximation.

== Intervals of the 7-odd-limit ==
=== 8/7 ===
The '''8/7''' interval can be considered the '''septimal major second''', or '''supermajor second''', by diatonic interval classification, in the sense that it is slightly wider than the [[9/8]] major second at 231.2 cents. Due to its larger size compared to 9/8, it does not cause as much crowding, and is thus more consonant. It is also approximately 1/3 of the [[perfect fifth]], and it is mapped as such in the [[Slendric]] temperament.

It can also be considered in an ambiguous category of "semifourths" between seconds and thirds. Here, it is a minor interval, with 7/6, its fourth complement, being the corresponding major interval. The 7/6 and 8/7 intervals can be stacked to make fourth-bound triads; see [[#Triads dividing the perfect fourth]].

=== 7/6 ===
The '''7/6''' interval is known as the '''septimal minor third''' or '''subminor third''', since it is narrower than the Pythagorean minor third [[32/27]] and the classical minor third [[6/5]], being 266.9 cents in size. We can build a triad bounded by the [[perfect fifth]], that being 1–7/6–3/2. The interval between 7/6 and 3/2 is [[9/7]], which can be considered the supermajor third, being the fifth complement of 7/6. (However, note that 9/7 is a [[9-odd-limit]] interval, not a 7-odd-limit one.) We can also stack 7/6 on top of a triad to get a seventh chord; for example, stacking 7/6 on top of the 1–5/4–3/2 major triad gives us 1–5/4–3/2–7/4, the harmonic seventh chord.

As described in [[#Triads dividing the perfect fourth]], 7/6 can also be seen as contrasting with [[#8/7|8/7]] in triads such as 1–7/6–4/3, with 7/6 being considered the major counterpart of 8/7.

=== 7/5 ===
The 7/5 interval can be called the '''lesser septimal tritone''', having a size of 582.5 cents. It is called the ''lesser'' septimal tritone because the "greater septimal tritone" is [[#10/7|10/7]], its octave complement, from which it differs by [[50/49]], the jubilisma. Unlike the tritone found in [[12edo]], it is a ''consonant'' tritone, having a more restful sound than the half-octave. It is found between the third and the seventh of the 1–5/4–3/2–7/4 harmonic seventh chord. It is also the outer interval of the 1–6/5–7/5 diminished triad, which is the simplest and most consonant diminished triad in JI.

In systems such as [[HEJI]] and the [[FJS]], it is a diminished fifth, being the difference between [[5/4]], which is a major third, and [[#7/4|7/4]], which is a minor seventh. However, since it is less than a half-octave, it can also be classified as an augmented fourth, and it is mapped as such in septimal [[Meantone]] temperament. As such, interval categories in the 7-limit are rather ambiguous, and 7/4 has qualities of both a sixth and a seventh, instead of simply being a subminor seventh.

It is fairly close to the Pythagorean diminished fifth [[1024/729]], being flat of it by an [[Aberschisma]], or about 5.8 cents. It is also rather close to the [[5-limit]] tritone [[45/32]], being flat of it by the [[Marvel]] comma 225/224.

=== 10/7 ===
The 10/7 interval can be named the '''greater septimal tritone''', being 617.5 cents in size, analogous to how [[#7/5|7/5]] is called the lesser septimal tritone. It is somewhat less consonant than 7/5 due to its more complex ratio, though it is still considerably more consonant than the half-octave. It can be seen as a stack of 5/4 and 8/7, appearing in chords such as 1–7/4–5/2 and 1–6/5–3/2–12/7.

=== 12/7 ===
The interval 12/7, known as the '''septimal major sixth''' or '''supermajor sixth''', measures at 933.1 cents in size. It is the octave complement of [[#7/6|7/6]], and the twelfth complement of [[#7/4|7/4]]. Chords using it include 1–9/7–3/2–12/7 and 1–6/5–3/2–12/7.

It is somewhat ambiguous and can be considered in a category of "hemitwelfths" between sixths and sevenths, where it is a minor interval, and 7/4 is its major counterpart.

=== 7/4 ===
The 7/4 interval is often called the '''septimal minor seventh''', '''subminor seventh''', or '''harmonic seventh''', being 968.8 cents in size. Being the octave-reduced seventh harmonic, it can naturally be added to a 1–5/4–3/2 major triad to get 1–5/4–3/2–7/4, often called the ''harmonic seventh chord''. Due to its simpler ratio, it is more consonant than [[16/9]], the Pythagorean minor seventh, and [[9/5]], the classical minor seventh.

Though often considered a minor seventh, it also has some qualities of a sixth. For example, the [[#7/5|7/5]] interval can be considered an augmented fourth due to being smaller than the semioctave, so 7/4 would accordingly be an augmented sixth. Thus 7/4 is somewhat ambiguous by diatonic classification, and can be considered to be in a category between a sixth and a seventh, a "hemitwelfth". Here 7/4 is a major interval, with the corresponding minor interval being [[#12/7|12/7]].

== Harmony in the 7-odd-limit ==
=== Tetradic harmony ===
The 7-odd-limit is where tetrads start to get more prevalent. For example, we can build the 1–5/4–3/2–7/4 "harmonic seventh chord" by adding 7/4 on top of a 1–5/4–3/2 major triad. The harmonic seventh chord sounds somewhat similar to the dominant seventh chord, except it is more consonant and resolved. It can also be called the "major tetrad", similarly to how 1–5/4–3/2 is called the major triad.

The 5-limit minor triad 1–6/5–3/2 can be derived by reflecting every note about the midpoint of the root and the fifth. If we do the same for the harmonic seventh chord and reduce the steps to an octave, then we get 1–6/5–3/2–12/7 "subharmonic seventh chord" or "minor tetrad".

=== Triads dividing the perfect fourth ===
{{Main|Chthonic harmony}}
The 7/6 and 8/7 intervals can be seen as contrasting with each other, differing from each other by 49/48 (35.7 cents). We can build triads by stacking 7/6 and 8/7 that span the [[perfect fourth]], such as the 1–7/6–4/3 triad, which may also be voiced as 1–3/2–7/4. The minor version of this triad is 1–8/7–4/3, which can also be voiced as 1–3/2–12/7. This is analogous to how [[5/4]] and [[6/5]] contrast each other in the 1–5/4–3/2 and 1–6/5–3/2 triads, but these septimal triads split the perfect fourth, rather than splitting the [[perfect fifth]] like 5-limit triads do. As such, it can be considered a form of [[chthonic harmony|"semiquartal" or "chthonic" harmony]], which is one approach to septimal harmony.

Here, 7/6 is a type of major interval, and 8/7 is a type of minor interval. Their octave complements can be classified accordingly, with 12/7 being a minor interval, and 7/4 being a major interval. This is different from diatonic, where 8/7 is a supermajor second, 7/6 a subminor third, 12/7 a supermajor sixth, and 7/4 a subminor seventh.

7-odd-limit

2026-04-12T04:43:31Z

Overthink: rework intro

{{Odd-limit navigation}}
The '''7-[[odd-limit]]''' consists of all intervals where the largest allowable odd factor in the numerator and denominator is 7. It is the smallest odd-limit containing intervals of the [[7-limit|7-prime-limit]], thus creating xenharmonic categories not found in traditional music theory. In a 7-prime-limit system, all the ratios of the 7- or [[9-odd-limit]] can be treated as consonances.

== Table of 7-odd-limit intervals ==
Reduced to an octave, the intervals of the 7-odd-limit are:

{| class="wikitable"
! Interval
! Cents
! Name
|-
| 1/1
| 0.0
| Unison
|-
| '''8/7'''
| '''231.2'''
| '''Septimal major 2nd'''
|-
| '''7/6'''
| '''266.9'''
| '''Septimal minor 3rd'''
|-
| 6/5
| 315.6
| Classical minor 3rd
|-
| 5/4
| 386.4
| Classical major 3rd
|-
| 4/3
| 498.0
| Perfect 4th
|-
| '''7/5'''
| '''582.5'''
| '''Lesser septimal tritone'''
|-
| '''10/7'''
| '''617.5'''
| '''Greater septimal tritone'''
|-
| 3/2
| 702.0
| Perfect 5th
|-
| 8/5
| 813.6
| Classical minor 6th
|-
| 5/3
| 884.4
| Classical major 6th
|-
| '''12/7'''
| '''933.1'''
| '''Septimal major 6th'''
|-
| '''7/4'''
| '''968.8'''
| '''Septimal minor 7th'''
|-
| 2/1
| 1200.0
| Octave
|}

== Approximation by edos ==
[[File:7-odd-limit in edos.png|thumb|right|A diagram showing the approximation of the 7-odd-limit by various edos.]]
The first [[edo]] consistent to the 7-odd-limit is [[4edo]], which maps 5/4 to 1 step, 3/2 to 2 steps, and 7/4 to 3 steps, laying down a rough framework of tetradic harmony. Then, [[10edo]] approximates the 7-odd-limit relatively accurately for size, though it conflates several interval pairs: 5/4~6/5, 7/6~8/7, and 7/5~10/7. As such, the [[10-form]] is useful for classifying the 7-limit. After that, [[12edo]] distinguishes 5/4 from 6/5 and 7/6 from 8/7, though it has 6/5~7/6 and 7/5~10/7, and the 7th harmonic is tuned very sharply. The [[15edo]] and [[19edo]] tunings distinguish 5/4, 6/5, and 7/6, as well as 7/5 and 10/7, but 7/6 is equated to 8/7, an equivalence known as [[Interseptimal (temperament)|Interseptimal or Semaphore temperament]]. The first to distinguish all of 5/4, 6/5, 7/6, and 8/7 is [[22edo]], though 7/5 is still equated with 10/7. The first edo to distinguish the entire 7-odd-limit is [[27edo]], but one may prefer [[31edo]] for a more accurate approximation.

== Intervals of the 7-odd-limit ==
=== 8/7 ===
The '''8/7''' interval can be considered the '''septimal major second''', or '''supermajor second''', by diatonic interval classification, in the sense that it is slightly wider than the [[9/8]] major second at 231.2 cents. Due to its larger size compared to 9/8, it does not cause as much crowding, and is thus more consonant. It is also approximately 1/3 of the [[perfect fifth]], and it is mapped as such in the [[Slendric]] temperament.

==== Triads dividing the perfect fourth ====
{{Main|Chthonic harmony}}
Harmonically, 8/7 can be seen as contrasting with [[#7/6|7/6]], differing from it by 49/48 (35.7 cents). As such, we can build triads by stacking 7/6 and 8/7, such as the 1–7/6–4/3 triad, which may also be voiced as 1–3/2–7/4. The minor version of this triad is 1–8/7–4/3, which can also be voiced as 1–3/2–12/7. This is analogous to how [[5/4]] and [[6/5]] contrast each other in the 1–5/4–3/2 and 1–6/5–3/2 triads, but these septimal triads split the [[perfect fourth]], rather than splitting the [[perfect fifth]] like the pental triads do. As such, it can be considered a form of [[chthonic harmony|"semiquartal" or "chthonic" harmony]], which is one approach to septimal harmony.

Here, 8/7 is a type of minor interval, and 7/6 is a type of major interval. Their octave complements can be classified accordingly, with 12/7 being a minor interval, and 7/4 being a major interval. This contrasts with diatonic, where 8/7 is a supermajor second, 7/6 a subminor third, 12/7 a supermajor sixth, and 7/4 a subminor seventh.

=== 7/6 ===
The '''7/6''' interval is known as the '''septimal minor third''' or '''subminor third''', since it is narrower than the Pythagorean minor third [[32/27]] and the classical minor third [[6/5]], being 266.9 cents in size. We can build a triad bounded by the [[perfect fifth]], that being 1–7/6–3/2. The interval between 7/6 and 3/2 is [[9/7]], which can be considered the supermajor third, being the fifth complement of 7/6. (However, note that 9/7 is a [[9-odd-limit]] interval, not a 7-odd-limit one.) We can also stack 7/6 on top of a triad to get a seventh chord; for example, stacking 7/6 on top of the 1–5/4–3/2 major triad gives us 1–5/4–3/2–7/4, the harmonic seventh chord.

As described above in [[#Triads dividing the perfect fourth]], 7/6 can also be seen as contrasting with [[#8/7|8/7]] in triads such as 1–7/6–4/3, with 7/6 being considered the major counterpart of 8/7.

=== 7/5 ===
The 7/5 interval can be called the '''lesser septimal tritone''', having a size of 582.5 cents. It is called the ''lesser'' septimal tritone because the "greater septimal tritone" is [[#10/7|10/7]], its octave complement, from which it differs by [[50/49]], the jubilisma. Unlike the tritone found in [[12edo]], it is a ''consonant'' tritone, having a more restful sound than the half-octave. It is found between the third and the seventh of the 1–5/4–3/2–7/4 harmonic seventh chord. It is also the outer interval of the 1–6/5–7/5 diminished triad, which is the simplest and most consonant diminished triad in JI.

In systems such as [[HEJI]] and the [[FJS]], it is a diminished fifth, being the difference between [[5/4]], which is a major third, and [[#7/4|7/4]], which is a minor seventh. However, since it is less than a half-octave, it can also be classified as an augmented fourth, and it is mapped as such in septimal [[Meantone]] temperament. As such, interval categories in the 7-limit are rather ambiguous, and 7/4 has qualities of both a sixth and a seventh, instead of simply being a subminor seventh.

It is fairly close to the Pythagorean diminished fifth [[1024/729]], being flat of it by an [[Aberschisma]], or about 5.8 cents. It is also rather close to the [[5-limit]] tritone [[45/32]], being flat of it by the [[Marvel]] comma 225/224.

=== 10/7 ===
The 10/7 interval can be named the '''greater septimal tritone''', being 617.5 cents in size, analogous to how [[#7/5|7/5]] is called the lesser septimal tritone. It is somewhat less consonant than 7/5 due to its more complex ratio, though it is still considerably more consonant than the half-octave. It can be seen as a stack of 5/4 and 8/7, appearing in chords such as 1–7/4–5/2 and 1–6/5–3/2–12/7.

=== 12/7 ===
The interval 12/7, known as the '''septimal major sixth''' or '''supermajor sixth''', measures at 933.1 cents in size. It is the octave complement of [[#7/6|7/6]], and the twelfth complement of [[#7/4|7/4]]. Chords using it include 1–9/7–3/2–12/7 and 1–6/5–3/2–12/7.

It is somewhat ambiguous and can be considered in a category of "hemitwelfths" between sixths and sevenths, where it is a minor interval, and 7/4 is its major counterpart.

=== 7/4 ===
The 7/4 interval is often called the '''septimal minor seventh''', '''subminor seventh''', or '''harmonic seventh''', being 968.8 cents in size. Being the octave-reduced seventh harmonic, it can naturally be added to a 1–5/4–3/2 major triad to get 1–5/4–3/2–7/4, often called the ''harmonic seventh chord''. Due to its simpler ratio, it is more consonant than [[16/9]], the Pythagorean minor seventh, and [[9/5]], the classical minor seventh.

Though often considered a minor seventh, it also has some qualities of a sixth. For example, the [[#7/5|7/5]] interval can be considered an augmented fourth due to being smaller than the semioctave, so 7/4 would accordingly be an augmented sixth. Thus 7/4 is somewhat ambiguous by diatonic classification, and can be considered to be in a category between a sixth and a seventh, a "hemitwelfth". Here 7/4 is a major interval, with the corresponding minor interval being [[#12/7|12/7]].

Template:Odd-limit navigation

2026-04-12T04:34:19Z

Overthink: don't think this margin is really needed

<includeonly><div class="wikitable" style="float: right; min-width: 8.5em; padding: 0.6em 0.5em; margin: 0em 0em 0.75em 1em;"> '''[[Odd-limit]]s:'''
----
* [[5-odd-limit]]
* [[7-odd-limit]]
* [[9-odd-limit]]
* [[11-odd-limit]]
* [[13-odd-limit]]
* [[15-odd-limit]]
* [[17-odd-limit]]
* [[19-odd-limit]]
* [[21-odd-limit]]
* [[23-odd-limit]]
* [[25-odd-limit]]
* [[27-odd-limit]]
[[Category:Odd-limits]]</div></includeonly><noinclude>
{{Documentation}}
</noinclude>

7-odd-limit

2026-04-12T04:32:22Z

Overthink: /* Intervals of the 7-odd-limit */ links, slight rewording

{{Odd-limit navigation}}
The '''7-[[odd-limit]]''' consists of all intervals where the largest allowable odd factor in the numerator and denominator is 7. Reduced to an octave, these are:

{| class="wikitable"
! Interval
! Cents
! Name
|-
| 1/1
| 0.0
| Unison
|-
| '''8/7'''
| '''231.2'''
| '''Septimal major 2nd'''
|-
| '''7/6'''
| '''266.9'''
| '''Septimal minor 3rd'''
|-
| 6/5
| 315.6
| Classical minor 3rd
|-
| 5/4
| 386.4
| Classical major 3rd
|-
| 4/3
| 498.0
| Perfect 4th
|-
| '''7/5'''
| '''582.5'''
| '''Lesser septimal tritone'''
|-
| '''10/7'''
| '''617.5'''
| '''Greater septimal tritone'''
|-
| 3/2
| 702.0
| Perfect 5th
|-
| 8/5
| 813.6
| Classical minor 6th
|-
| 5/3
| 884.4
| Classical major 6th
|-
| '''12/7'''
| '''933.1'''
| '''Septimal major 6th'''
|-
| '''7/4'''
| '''968.8'''
| '''Septimal minor 7th'''
|-
| 2/1
| 1200.0
| Octave
|}

== Approximation by edos ==
[[File:7-odd-limit in edos.png|thumb|right|A diagram showing the approximation of the 7-odd-limit by various edos.]]
The first [[edo]] consistent to the 7-odd-limit is [[4edo]], which maps 5/4 to 1 step, 3/2 to 2 steps, and 7/4 to 3 steps, laying down a rough framework of tetradic harmony. Then, [[10edo]] approximates the 7-odd-limit relatively accurately for size, though it conflates several interval pairs: 5/4~6/5, 7/6~8/7, and 7/5~10/7. As such, the [[10-form]] is useful for classifying the 7-limit. After that, [[12edo]] distinguishes 5/4 from 6/5 and 7/6 from 8/7, though it has 6/5~7/6 and 7/5~10/7, and the 7th harmonic is tuned very sharply. The [[15edo]] and [[19edo]] tunings distinguish 5/4, 6/5, and 7/6, as well as 7/5 and 10/7, but 7/6 is equated to 8/7, an equivalence known as [[Interseptimal (temperament)|Interseptimal or Semaphore temperament]]. The first to distinguish all of 5/4, 6/5, 7/6, and 8/7 is [[22edo]], though 7/5 is still equated with 10/7. The first edo to distinguish the entire 7-odd-limit is [[27edo]], but one may prefer [[31edo]] for a more accurate approximation.

== Intervals of the 7-odd-limit ==
=== 8/7 ===
The '''8/7''' interval can be considered the '''septimal major second''', or '''supermajor second''', by diatonic interval classification, in the sense that it is slightly wider than the [[9/8]] major second at 231.2 cents. Due to its larger size compared to 9/8, it does not cause as much crowding, and is thus more consonant. It is also approximately 1/3 of the [[perfect fifth]], and it is mapped as such in the [[Slendric]] temperament.

==== Triads dividing the perfect fourth ====
{{Main|Chthonic harmony}}
Harmonically, 8/7 can be seen as contrasting with [[#7/6|7/6]], differing from it by 49/48 (35.7 cents). As such, we can build triads by stacking 7/6 and 8/7, such as the 1–7/6–4/3 triad, which may also be voiced as 1–3/2–7/4. The minor version of this triad is 1–8/7–4/3, which can also be voiced as 1–3/2–12/7. This is analogous to how [[5/4]] and [[6/5]] contrast each other in the 1–5/4–3/2 and 1–6/5–3/2 triads, but these septimal triads split the [[perfect fourth]], rather than splitting the [[perfect fifth]] like the pental triads do. As such, it can be considered a form of [[chthonic harmony|"semiquartal" or "chthonic" harmony]], which is one approach to septimal harmony.

Here, 8/7 is a type of minor interval, and 7/6 is a type of major interval. Their octave complements can be classified accordingly, with 12/7 being a minor interval, and 7/4 being a major interval. This contrasts with diatonic, where 8/7 is a supermajor second, 7/6 a subminor third, 12/7 a supermajor sixth, and 7/4 a subminor seventh.

=== 7/6 ===
The '''7/6''' interval is known as the '''septimal minor third''' or '''subminor third''', since it is narrower than the Pythagorean minor third [[32/27]] and the classical minor third [[6/5]], being 266.9 cents in size. We can build a triad bounded by the [[perfect fifth]], that being 1–7/6–3/2. The interval between 7/6 and 3/2 is [[9/7]], which can be considered the supermajor third, being the fifth complement of 7/6. (However, note that 9/7 is a [[9-odd-limit]] interval, not a 7-odd-limit one.) We can also stack 7/6 on top of a triad to get a seventh chord; for example, stacking 7/6 on top of the 1–5/4–3/2 major triad gives us 1–5/4–3/2–7/4, the harmonic seventh chord.

As described above in [[#Triads dividing the perfect fourth]], 7/6 can also be seen as contrasting with [[#8/7|8/7]] in triads such as 1–7/6–4/3, with 7/6 being considered the major counterpart of 8/7.

=== 7/5 ===
The 7/5 interval can be called the '''lesser septimal tritone''', having a size of 582.5 cents. It is called the ''lesser'' septimal tritone because the "greater septimal tritone" is [[#10/7|10/7]], its octave complement, from which it differs by [[50/49]], the jubilisma. Unlike the tritone found in [[12edo]], it is a ''consonant'' tritone, having a more restful sound than the half-octave. It is found between the third and the seventh of the 1–5/4–3/2–7/4 harmonic seventh chord. It is also the outer interval of the 1–6/5–7/5 diminished triad, which is the simplest and most consonant diminished triad in JI.

In systems such as [[HEJI]] and the [[FJS]], it is a diminished fifth, being the difference between [[5/4]], which is a major third, and [[#7/4|7/4]], which is a minor seventh. However, since it is less than a half-octave, it can also be classified as an augmented fourth, and it is mapped as such in septimal [[Meantone]] temperament. As such, interval categories in the 7-limit are rather ambiguous, and 7/4 has qualities of both a sixth and a seventh, instead of simply being a subminor seventh.

It is fairly close to the Pythagorean diminished fifth [[1024/729]], being flat of it by an [[Aberschisma]], or about 5.8 cents. It is also rather close to the [[5-limit]] tritone [[45/32]], being flat of it by the [[Marvel]] comma 225/224.

=== 10/7 ===
The 10/7 interval can be named the '''greater septimal tritone''', being 617.5 cents in size, analogous to how [[#7/5|7/5]] is called the lesser septimal tritone. It is somewhat less consonant than 7/5 due to its more complex ratio, though it is still considerably more consonant than the half-octave. It can be seen as a stack of 5/4 and 8/7, appearing in chords such as 1–7/4–5/2 and 1–6/5–3/2–12/7.

=== 12/7 ===
The interval 12/7, known as the '''septimal major sixth''' or '''supermajor sixth''', measures at 933.1 cents in size. It is the octave complement of [[#7/6|7/6]], and the twelfth complement of [[#7/4|7/4]]. Chords using it include 1–9/7–3/2–12/7 and 1–6/5–3/2–12/7.

It is somewhat ambiguous and can be considered in a category of "hemitwelfths" between sixths and sevenths, where it is a minor interval, and 7/4 is its major counterpart.

=== 7/4 ===
The 7/4 interval is often called the '''septimal minor seventh''', '''subminor seventh''', or '''harmonic seventh''', being 968.8 cents in size. Being the octave-reduced seventh harmonic, it can naturally be added to a 1–5/4–3/2 major triad to get 1–5/4–3/2–7/4, often called the ''harmonic seventh chord''. Due to its simpler ratio, it is more consonant than [[16/9]], the Pythagorean minor seventh, and [[9/5]], the classical minor seventh.

Though often considered a minor seventh, it also has some qualities of a sixth. For example, the [[#7/5|7/5]] interval can be considered an augmented fourth due to being smaller than the semioctave, so 7/4 would accordingly be an augmented sixth. Thus 7/4 is somewhat ambiguous by diatonic classification, and can be considered to be in a category between a sixth and a seventh, a "hemitwelfth". Here 7/4 is a major interval, with the corresponding minor interval being [[#12/7|12/7]].

7/4

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Overthink: Redirected page to 7-odd-limit#7/4

#redirect [[7-odd-limit #7/4]]

12/7

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Overthink: Redirected page to 7-odd-limit#12/7

#redirect [[7-odd-limit #12/7]]

10/7

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Overthink: Redirected page to 7-odd-limit#10/7

#redirect [[7-odd-limit #10/7]]

7/5

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Overthink: Redirected page to 7-odd-limit#7/5

#redirect [[7-odd-limit #7/5]]

7/6

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Overthink: Redirected page to 7-odd-limit#7/6

#redirect [[7-odd-limit #7/6]]

8/7

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Overthink: redirect page

#redirect [[7-odd-limit #8/7]]

7-limit

2026-04-12T02:05:48Z

Overthink: odd limits

The '''7-limit''' or the '''2.3.5.7 subgroup''' is the subgroup of [[just intonation]] consisting of the intervals reachable by stacking [[2/1]], [[3/2]], [[5/4]], and [[7/4]]. Important subsets of the 7-limit include the [[7-odd-limit]] and [[9-odd-limit]].

Rank-3 subgroups:
* [[5-limit]]
* [[2.3.7 subgroup]]
* [[2.5.7 subgroup]]
* [[3.5.7 subgroup]]

== Full 7-limit JI scales ==
The scales are shown in [https://sw3.lumipakkanen.com/ Scale Workshop 3] format.
=== Mode 5 ===
<pre>
8:9:10:12:14:16
</pre>
The simplest full 7-limit JI scale. This scale is notably used in the music of the Wagogo people in Tanzania.

=== Rooted Mixolydian ===
<pre>
16:18:20:21:24:27:28:32
</pre>

=== Rooted Ionian ===
<pre>
16:18:20:21:24:27:30:32
</pre>
=== Zil ===
Zil (from the temperament Godzilla which the zil series serves as a detempering of) is a series of 7-limit JI scales created from a [[generator sequence]] GS(8/7, 7/6, 8/7, 7/6, 8/7, 7/6, 8/7, 189/160).
==== Zil[14] ====
The most discussed of the zil scales is zil[14] which is chiral depending on the chirality of the [[interleaving|interleaved]] 5-limit [[zarlino]] copies:

RH zil[14]
<pre>
35/32; 9/8; 315/256; 5/4; 21/16; 45/32; 189/128; 3/2; 105/64; 27/16; 7/4; 15/8; 63/32; 2/1
</pre>

LH zil[14]
<pre>
21/20; 9/8; 7/6; 6/5; 21/16; 4/3; 7/5; 3/2; 63/40; 8/5; 7/4; 9/5; 63/32; 2/1
</pre>

==== Zil[24] ====
Zil[24] is achiral. It has a 4×3×2 structure in the 7-limit lattice.
<pre>
525/512; 135/128; 35/32; 9/8; 4725/4096; 75/64; 315/256; 5/4; 21/16; 675/512; 175/128; 45/32; 189/128; 3/2; 1575/1024; 25/16; 105/64; 27/16; 7/4; 225/128; 945/512; 15/8; 63/32; 2/1
</pre>

=== 12:14:16:18:21:24 by 5/4 ===
A 10-note scale with an analogous structure to zil[14] (note that these are subsets of both zil[14] chiralities):

RH
<pre>
35/32; 9/8; 5/4; 21/16; 45/32; 3/2; 105/64; 7/4; 15/8; 2/1
</pre>

LH
<pre>
16/15; 8/7; 6/5; 4/3; 48/35; 3/2; 8/5; 12/7; 64/35; 2/1
</pre>

=== A Mothra[36] detemper ===
GS(8/7 8/7 147/128 8/7 8/7 147/128 8/7 8/7 147/128 8/7 8/7 245/216)[36]; 4×3×3 generator structure

<pre>
33075/32768; 525/512; 135/128; 2205/2048; 35/32; 9/8; 147/128; 4725/4096; 75/64; 1225/1024; 315/256; 5/4; 1323/1024; 21/16; 675/512; 11025/8192; 175/128; 45/32; 735/512; 189/128; 3/2; 49/32; 1575/1024; 25/16; 6615/4096; 105/64; 27/16; 441/256; 7/4; 225/128; 3675/2048; 945/512; 15/8; 245/128; 63/32; 2/1
</pre>

== Full 7-limit tempered scales ==
=== Superpyth[12] ===
Superpyth[12] is constructed by applying [[Superpyth]] temperament (2.3.5.7[22 & 27]; equivalently tempering out 64/63 and 245/243) to a 12-note chain of fifths. It contains Superpyth-tempered 5-limit [[blackdye]].
=== Pajara ===
[[Pajara]] can be used as an interpretation of 2L8s and 10L2s or their modifications. Pajara works best in [[22edo]].
=== 7-limit diachrome ===
7-limit diachrome, an [[aberrismic]] scale, is constructed by taking a 6+6 or 7+5 fifth chain structure and tempering out [[5120/5103]].

=== Aberschismic whitedye ===
Aberschismic whitedye is constructed by taking a diatonic scale and offsetting it by 64/63~81/80, tempering out [[5120/5103]].
{{Cat|JI groups}}

Category:Pages with unsourced statements

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Overthink: create category

[[Category:Maintenance categories]]
__HIDDENCAT__

7-odd-limit

2026-04-11T04:25:17Z

Overthink: remove WIP banner

{{Odd-limit navigation}}
The '''7-[[odd-limit]]''' consists of all intervals where the largest allowable odd factor in the numerator and denominator is 7. Reduced to an octave, these are:

{| class="wikitable"
! Interval
! Cents
! Name
|-
| 1/1
| 0.0
| Unison
|-
| '''8/7'''
| '''231.2'''
| '''Septimal major 2nd'''
|-
| '''7/6'''
| '''266.9'''
| '''Septimal minor 3rd'''
|-
| 6/5
| 315.6
| Classical minor 3rd
|-
| 5/4
| 386.4
| Classical major 3rd
|-
| 4/3
| 498.0
| Perfect 4th
|-
| '''7/5'''
| '''582.5'''
| '''Lesser septimal tritone'''
|-
| '''10/7'''
| '''617.5'''
| '''Greater septimal tritone'''
|-
| 3/2
| 702.0
| Perfect 5th
|-
| 8/5
| 813.6
| Classical minor 6th
|-
| 5/3
| 884.4
| Classical major 6th
|-
| '''12/7'''
| '''933.1'''
| '''Septimal major 6th'''
|-
| '''7/4'''
| '''968.8'''
| '''Septimal minor 7th'''
|-
| 2/1
| 1200.0
| Octave
|}

== Approximation by edos ==
[[File:7-odd-limit in edos.png|thumb|right|A diagram showing the approximation of the 7-odd-limit by various edos.]]
The first [[edo]] consistent to the 7-odd-limit is [[4edo]], which maps 5/4 to 1 step, 3/2 to 2 steps, and 7/4 to 3 steps, laying down a rough framework of tetradic harmony. Then, [[10edo]] approximates the 7-odd-limit relatively accurately for size, though it conflates several interval pairs: 5/4~6/5, 7/6~8/7, and 7/5~10/7. As such, the [[10-form]] is useful for classifying the 7-limit. After that, [[12edo]] distinguishes 5/4 from 6/5 and 7/6 from 8/7, though it has 6/5~7/6 and 7/5~10/7, and the 7th harmonic is tuned very sharply. The [[15edo]] and [[19edo]] tunings distinguish 5/4, 6/5, and 7/6, as well as 7/5 and 10/7, but 7/6 is equated to 8/7, an equivalence known as [[Interseptimal (temperament)|Interseptimal or Semaphore temperament]]. The first to distinguish all of 5/4, 6/5, 7/6, and 8/7 is [[22edo]], though 7/5 is still equated with 10/7. The first edo to distinguish the entire 7-odd-limit is [[27edo]], but one may prefer [[31edo]] for a more accurate approximation.

== Intervals of the 7-odd-limit ==
=== 8/7 ===
The '''8/7''' interval can be considered the '''septimal major second''', or '''supermajor second''', by diatonic interval classification, in the sense that it is slightly wider than the [[9/8]] major second at 231.2 cents. Due to its larger size compared to 9/8, it does not cause as much crowding, and is thus more consonant. It is also approximately 1/3 of the perfect fifth [[3/2]], and it is mapped as such in the [[Slendric]] temperament.

==== Triads dividing the perfect fourth ====
{{Main|Chthonic harmony}}
Harmonically, 8/7 can be seen as contrasting with 7/6, differing from it by 49/48 (35.7 cents). As such, we can build triads by stacking 7/6 and 8/7, such as the 1–7/6–4/3 triad, which may also be voiced as 1–3/2–7/4. The minor version of this triad is 1–8/7–4/3, which can also be voiced as 1–3/2–12/7. This is analogous to how [[5/4]] and [[6/5]] contrast each other in the 1–5/4–3/2 and 1–6/5–3/2 triads, but the septimal triads split the [[perfect fourth]], rather than splitting the [[perfect fifth]] like pental triads do. As such, it can be considered a form of [[chthonic harmony|"semiquartal" or "chthonic" harmony]], which is one approach to septimal harmony.

Here, 8/7 is a type of minor interval, and 7/6 is a type of major interval. Their octave complements can be classified accordingly, with 12/7 being a minor interval, and 7/4 being a major interval. This contrasts with diatonic, where 8/7 is a supermajor second, 7/6 a subminor third, 12/7 a supermajor sixth, and 7/4 a subminor seventh.

=== 7/6 ===
The '''7/6''' interval is known as the '''septimal minor third''' or '''subminor third''', since it is narrower than the Pythagorean minor third [[32/27]] and the classical minor third [[6/5]], being 266.9 cents in size. We can build a triad bounded by the perfect fifth, that being 1–7/6–3/2. The interval between 7/6 and 3/2 is [[9/7]], which can be considered the supermajor third, being the fifth complement of 7/6. (However, note that 9/7 is a [[9-odd-limit]] interval, not a 7-odd-limit one.) We can also stack 7/6 on top of a triad to get a seventh chord; for example, stacking 7/6 on top of the 1–5/4–3/2 major triad gives us 1–5/4–3/2–7/4, the harmonic seventh chord.

As described above in [[#Triads dividing the perfect fourth]], 7/6 can also be seen as contrasting with 8/7 in triads such as 1–7/6–4/3, with 7/6 being considered the major counterpart of 8/7.

=== 7/5 ===
The 7/5 interval can be called the '''lesser septimal tritone''', having a size of 582.5 cents. It is called the ''lesser'' septimal tritone because the "greater septimal tritone" is 10/7, its octave complement, from which it differs by [[50/49]], the jubilisma. Unlike the tritone found in [[12edo]], it is a ''consonant'' tritone, having a more restful sound than the half-octave. It is found between the third and the seventh of the 1–5/4–3/2–7/4 harmonic seventh chord. It is also the outer interval of the 1–6/5–7/5 diminished triad, which is the simplest and most consonant diminished triad in JI.

In systems such as [[HEJI]] and the [[FJS]], it is a diminished fifth, being the difference between a major third 5/4 and a minor seventh 7/4. However, since it is less than a half-octave, it can also be classified as an augmented fourth, and it is mapped as such in septimal [[Meantone]] temperament. As such, interval categories in the 7-limit are rather ambiguous, and 7/4 has qualities of both a sixth and a seventh, instead of simply being a subminor seventh.

It is fairly close to the Pythagorean diminished fifth [[1024/729]], being flat of it by an [[Aberschisma]], or about 5.8 cents. It is also rather close to the [[5-limit]] tritone [[45/32]], being flat of it by the [[Marvel]] comma 225/224.

=== 10/7 ===
The 10/7 interval can be named the '''greater septimal tritone''', being 617.5 cents in size, analogous to how 7/5 is called the lesser septimal tritone. It is somewhat less consonant than 7/5 due to its more complex ratio, though it is still considerably more consonant than the half-octave. It can be seen as a stack of 5/4 and 8/7, appearing in chords such as 1–7/4–5/2 and 1–6/5–3/2–12/7.

=== 12/7 ===
The interval 12/7, known as the '''septimal major sixth''' or '''supermajor sixth''', measures at 933.1 cents in size. It is the octave complement of 7/6, and the twelfth complement of 7/4. Chords using it include 1–9/7–3/2–12/7 and 1–6/5–3/2–12/7.

It is somewhat ambiguous and can be considered in a category of "hemitwelfths" between sixths and sevenths, where it is a minor interval, and 7/4 is its major counterpart.

=== 7/4 ===
The 7/4 interval is often called the '''septimal minor seventh''', '''subminor seventh''', or '''harmonic seventh''', being 968.8 cents in size. Being the octave-reduced seventh harmonic, it can naturally be added to a 1–5/4–3/2 major triad to get 1–5/4–3/2–7/4, often called the ''harmonic seventh chord''. Due to its simpler ratio, it is more consonant than [[16/9]], the Pythagorean minor seventh, and [[9/5]], the classical minor seventh.

Though often considered a minor seventh, it also has some qualities of a sixth. For example, the 7/5 interval can be considered an augmented fourth due to being smaller than the semioctave, so 7/4 would accordingly be an augmented sixth. Thus 7/4 is somewhat ambiguous by diatonic classification, and can be considered to be in a category between a sixth and a seventh, a "hemitwelfth". Here 7/4 is a major interval, with the corresponding minor interval being 12/7.

7-odd-limit

2026-04-11T04:24:44Z

Overthink: /* Intervals of the 7-odd-limit */ fill in later sections

{{WIP}}
{{Odd-limit navigation}}
The '''7-[[odd-limit]]''' consists of all intervals where the largest allowable odd factor in the numerator and denominator is 7. Reduced to an octave, these are:

{| class="wikitable"
! Interval
! Cents
! Name
|-
| 1/1
| 0.0
| Unison
|-
| '''8/7'''
| '''231.2'''
| '''Septimal major 2nd'''
|-
| '''7/6'''
| '''266.9'''
| '''Septimal minor 3rd'''
|-
| 6/5
| 315.6
| Classical minor 3rd
|-
| 5/4
| 386.4
| Classical major 3rd
|-
| 4/3
| 498.0
| Perfect 4th
|-
| '''7/5'''
| '''582.5'''
| '''Lesser septimal tritone'''
|-
| '''10/7'''
| '''617.5'''
| '''Greater septimal tritone'''
|-
| 3/2
| 702.0
| Perfect 5th
|-
| 8/5
| 813.6
| Classical minor 6th
|-
| 5/3
| 884.4
| Classical major 6th
|-
| '''12/7'''
| '''933.1'''
| '''Septimal major 6th'''
|-
| '''7/4'''
| '''968.8'''
| '''Septimal minor 7th'''
|-
| 2/1
| 1200.0
| Octave
|}

== Approximation by edos ==
[[File:7-odd-limit in edos.png|thumb|right|A diagram showing the approximation of the 7-odd-limit by various edos.]]
The first [[edo]] consistent to the 7-odd-limit is [[4edo]], which maps 5/4 to 1 step, 3/2 to 2 steps, and 7/4 to 3 steps, laying down a rough framework of tetradic harmony. Then, [[10edo]] approximates the 7-odd-limit relatively accurately for size, though it conflates several interval pairs: 5/4~6/5, 7/6~8/7, and 7/5~10/7. As such, the [[10-form]] is useful for classifying the 7-limit. After that, [[12edo]] distinguishes 5/4 from 6/5 and 7/6 from 8/7, though it has 6/5~7/6 and 7/5~10/7, and the 7th harmonic is tuned very sharply. The [[15edo]] and [[19edo]] tunings distinguish 5/4, 6/5, and 7/6, as well as 7/5 and 10/7, but 7/6 is equated to 8/7, an equivalence known as [[Interseptimal (temperament)|Interseptimal or Semaphore temperament]]. The first to distinguish all of 5/4, 6/5, 7/6, and 8/7 is [[22edo]], though 7/5 is still equated with 10/7. The first edo to distinguish the entire 7-odd-limit is [[27edo]], but one may prefer [[31edo]] for a more accurate approximation.

== Intervals of the 7-odd-limit ==
=== 8/7 ===
The '''8/7''' interval can be considered the '''septimal major second''', or '''supermajor second''', by diatonic interval classification, in the sense that it is slightly wider than the [[9/8]] major second at 231.2 cents. Due to its larger size compared to 9/8, it does not cause as much crowding, and is thus more consonant. It is also approximately 1/3 of the perfect fifth [[3/2]], and it is mapped as such in the [[Slendric]] temperament.

==== Triads dividing the perfect fourth ====
{{Main|Chthonic harmony}}
Harmonically, 8/7 can be seen as contrasting with 7/6, differing from it by 49/48 (35.7 cents). As such, we can build triads by stacking 7/6 and 8/7, such as the 1–7/6–4/3 triad, which may also be voiced as 1–3/2–7/4. The minor version of this triad is 1–8/7–4/3, which can also be voiced as 1–3/2–12/7. This is analogous to how [[5/4]] and [[6/5]] contrast each other in the 1–5/4–3/2 and 1–6/5–3/2 triads, but the septimal triads split the [[perfect fourth]], rather than splitting the [[perfect fifth]] like pental triads do. As such, it can be considered a form of [[chthonic harmony|"semiquartal" or "chthonic" harmony]], which is one approach to septimal harmony.

Here, 8/7 is a type of minor interval, and 7/6 is a type of major interval. Their octave complements can be classified accordingly, with 12/7 being a minor interval, and 7/4 being a major interval. This contrasts with diatonic, where 8/7 is a supermajor second, 7/6 a subminor third, 12/7 a supermajor sixth, and 7/4 a subminor seventh.

=== 7/6 ===
The '''7/6''' interval is known as the '''septimal minor third''' or '''subminor third''', since it is narrower than the Pythagorean minor third [[32/27]] and the classical minor third [[6/5]], being 266.9 cents in size. We can build a triad bounded by the perfect fifth, that being 1–7/6–3/2. The interval between 7/6 and 3/2 is [[9/7]], which can be considered the supermajor third, being the fifth complement of 7/6. (However, note that 9/7 is a [[9-odd-limit]] interval, not a 7-odd-limit one.) We can also stack 7/6 on top of a triad to get a seventh chord; for example, stacking 7/6 on top of the 1–5/4–3/2 major triad gives us 1–5/4–3/2–7/4, the harmonic seventh chord.

As described above in [[#Triads dividing the perfect fourth]], 7/6 can also be seen as contrasting with 8/7 in triads such as 1–7/6–4/3, with 7/6 being considered the major counterpart of 8/7.

=== 7/5 ===
The 7/5 interval can be called the '''lesser septimal tritone''', having a size of 582.5 cents. It is called the ''lesser'' septimal tritone because the "greater septimal tritone" is 10/7, its octave complement, from which it differs by [[50/49]], the jubilisma. Unlike the tritone found in [[12edo]], it is a ''consonant'' tritone, having a more restful sound than the half-octave. It is found between the third and the seventh of the 1–5/4–3/2–7/4 harmonic seventh chord. It is also the outer interval of the 1–6/5–7/5 diminished triad, which is the simplest and most consonant diminished triad in JI.

In systems such as [[HEJI]] and the [[FJS]], it is a diminished fifth, being the difference between a major third 5/4 and a minor seventh 7/4. However, since it is less than a half-octave, it can also be classified as an augmented fourth, and it is mapped as such in septimal [[Meantone]] temperament. As such, interval categories in the 7-limit are rather ambiguous, and 7/4 has qualities of both a sixth and a seventh, instead of simply being a subminor seventh.

It is fairly close to the Pythagorean diminished fifth [[1024/729]], being flat of it by an [[Aberschisma]], or about 5.8 cents. It is also rather close to the [[5-limit]] tritone [[45/32]], being flat of it by the [[Marvel]] comma 225/224.

=== 10/7 ===
The 10/7 interval can be named the '''greater septimal tritone''', being 617.5 cents in size, analogous to how 7/5 is called the lesser septimal tritone. It is somewhat less consonant than 7/5 due to its more complex ratio, though it is still considerably more consonant than the half-octave. It can be seen as a stack of 5/4 and 8/7, appearing in chords such as 1–7/4–5/2 and 1–6/5–3/2–12/7.

=== 12/7 ===
The interval 12/7, known as the '''septimal major sixth''' or '''supermajor sixth''', measures at 933.1 cents in size. It is the octave complement of 7/6, and the twelfth complement of 7/4. Chords using it include 1–9/7–3/2–12/7 and 1–6/5–3/2–12/7.

It is somewhat ambiguous and can be considered in a category of "hemitwelfths" between sixths and sevenths, where it is a minor interval, and 7/4 is its major counterpart.

=== 7/4 ===
The 7/4 interval is often called the '''septimal minor seventh''', '''subminor seventh''', or '''harmonic seventh''', being 968.8 cents in size. Being the octave-reduced seventh harmonic, it can naturally be added to a 1–5/4–3/2 major triad to get 1–5/4–3/2–7/4, often called the ''harmonic seventh chord''. Due to its simpler ratio, it is more consonant than [[16/9]], the Pythagorean minor seventh, and [[9/5]], the classical minor seventh.

Though often considered a minor seventh, it also has some qualities of a sixth. For example, the 7/5 interval can be considered an augmented fourth due to being smaller than the semioctave, so 7/4 would accordingly be an augmented sixth. Thus 7/4 is somewhat ambiguous by diatonic classification, and can be considered to be in a category between a sixth and a seventh, a "hemitwelfth". Here 7/4 is a major interval, with the corresponding minor interval being 12/7.

9-odd-limit

2026-04-11T03:33:41Z

Overthink: add navigation

{{WIP}}
{{Odd-limit navigation}}
The '''9-[[odd-limit]]''' consists of all intervals where the largest allowable odd factor in the numerator and denominator is 9. Reduced to an octave, these are:

{| class="wikitable"
! Interval
! Cents
! Name
|-
| 1/1
| 0.0
| Unison
|-
| '''10/9'''
| '''182.4'''
| '''Minor whole tone, Ptolemaic major 2nd'''
|-
| '''9/8'''
| '''203.9'''
| '''Major whole tone, Pythagorean major 2nd'''
|-
| 8/7
| 231.2
| Septimal major 2nd
|-
| 7/6
| 266.9
| Septimal minor 3rd
|-
| 6/5
| 315.6
| Classical minor 3rd
|-
| 5/4
| 386.4
| Classical major 3rd
|-
| '''9/7'''
| '''435.1'''
| '''Septimal major 3rd'''
|-
| 4/3
| 498.0
| Perfect 4th
|-
| 7/5
| 582.5
| Lesser septimal tritone
|-
| 10/7
| 617.5
| Greater septimal tritone
|-
| 3/2
| 702.0
| Perfect 5th
|-
| '''14/9'''
| '''764.9'''
| '''Septimal minor 6th'''
|-
| 8/5
| 813.6
| Classical minor 6th
|-
| 5/3
| 884.4
| Classical major 6th
|-
| 12/7
| 933.1
| Septimal major 6th
|-
| 7/4
| 968.8
| Septimal minor 7th
|-
| '''16/9'''
| '''996.1'''
| '''Pythagoran minor 7th'''
|-
| '''9/5'''
| '''1017.6'''
| '''Classical minor 7th'''
|-
| 2/1
| 1200.0
| Octave
|}

Category:Odd limits

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Overthink: redirect category

Category:Templates

2026-04-11T03:32:26Z

Overthink: create templates category

This category contains [[mw:Help:Templates|templates]], which are meant to be [[mw:Help:Transclusion|transcluded]] onto other pages, so that the same content can be added to multiple pages, such that they share the template's style, and are updated along with the template.

Template:Category redirect

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Overthink: add that back

<includeonly>'''This category is located at [[:{{#ifeq: {{NAMESPACE:{{{1}}}}}|Category||Category:}}{{{1}}}]].'''
: '''Note:''' This category should be empty.
See the [[Template:Category redirect#Instructions|instructions]] for more information.
'''Administrators''': If this category name is unlikely to be entered on new pages, and all [[Special:WhatLinksHere/{{FULLPAGENAME}}|incoming links]] have been cleaned up, [{{fullurl:{{FULLPAGENAMEE}}|wpReason={{urlencode:[[wikipedia:WP:SPEEDY#G6|G6]]: This category is located at [[{{#ifeq: {{NAMESPACE:{{{1}}} }}|Category||Category:}}{{{1}}}]].}}}}&action=delete click here to delete].
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{{documentation}}

[[Category:Templates]]
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Template:Category redirect

2026-04-11T03:28:15Z

Overthink: create template

7-odd-limit

2026-04-11T03:26:58Z

Overthink: - redundant category

Template:Odd-limit navigation

2026-04-11T03:26:12Z

Overthink: auto-categorize

<includeonly><div class="wikitable" style="float: right; min-width: 8.5em; padding: 0.6em 0.5em; margin: 0.9em 0em 0.75em 1em;"> '''[[Odd-limit]]s:'''
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* [[5-odd-limit]]
* [[7-odd-limit]]
* [[9-odd-limit]]
* [[11-odd-limit]]
* [[13-odd-limit]]
* [[15-odd-limit]]
* [[17-odd-limit]]
* [[19-odd-limit]]
* [[21-odd-limit]]
* [[23-odd-limit]]
* [[25-odd-limit]]
* [[27-odd-limit]]
[[Category:Odd-limits]]</div></includeonly><noinclude>
{{Documentation}}
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Category:Odd limits

2026-04-11T03:25:58Z

Overthink: Overthink moved page Category:Odd limits to Category:Odd-limits: standardize with rest of wiki

#REDIRECT [[:Category:Odd-limits]]

Category:Odd-limits

2026-04-11T03:25:58Z

Overthink: Overthink moved page Category:Odd limits to Category:Odd-limits: standardize with rest of wiki

This category lists pages for individual [[odd-limit]]s.