User:Lhearne/Extra-Diatonic Intervals
Today a small under of competing interval naming schemes exist for the description of microtonal music. More common than any particular defined standard are certain tendencies for microtonal interval naming, or names for specific intervals. While risking the creation of simply another competing standard, an effort is made to develop a scheme that is able to take the best aspects of the existing standards and apply them in a formal interval naming system built on common undefined practice. Such a system is developed, where in addition to the standard diatonic interval name qualifiers - 'M', 'm', 'P', 'A' and 'd', only the three most commonly used microtonal qualifies, 'N', 'S' and 's' are used, along with interval-class degrees. Using this system all intervals in three fifths of all edos up to 50 can be named such that 'S' and 's' correspond to a displacement of an interval up or down a single interval of the edo, respectively. Many commonly used MOS scales may also be described using this scheme such that these scales' interval names are consistent expression in any tuning that supports them. The resultant scheme can also be easily mapped to any of the current naming standards, and may even facilitate translation between. The resulting scheme should improve pedagogy and communication in microtonal music.
Background
The origin of diatonic interval names
Music theory describing the use of heptatonic-diatonic scales, including interval names, has been traced back as far as 2000BC, deciphered from a Sumerian cuneiform tablet from Nippur by Kilmer (1986). From Kummel (1970) we know that 'the names given to the seven tunings/scales were derived from the specific intervals on which the tuning procedure started' (Kilmer, 1986). This formed the basis of their musical notation (Kilmer, 2016). The table to the right following table displays the Ancient Mesopotamian interval names accompanied by their modern names.
Kilmer also writes that 'the ancient Mesopotamian musicians/“musicologists” knew what we call today the Pythagorean series of fifths, and that the series could be accomplished within a single octave by means of “inversion.” '. The Mesopotamian's music and theory was passed down through the Babylonians and the Assyrians to the Ancient Greeks, as well as their mathematics, particularly concerning musical and acoustical sound ratios (Ibid, Hodgekin, 2005).
Such mathematical and musical ideas are attributed to Pythagoras, who undoubtedly made them popular., although many scholars suggest he may have learned these ideas from his Babylonian and Egyptian mentors. None the less, Pythagoras' idea that that by dividing the length of a string into ratios of halves, thirds, quarters and fifths created the musical intervals of an octave, a perfect fifth, an octave again, and a major third form the basis of Ancient Greek music theory (http://www.historyofmusictheory.com/?page_id=20). His tuning of the diatonic scale by only octaves and perfect fifths (Pythagorean tuning) is influential through to today.
Ancient Greek interval names
Intervals in Ancient Greek music were written either as frequency ratios, after Pythagoras, or as positions in a tetrachord. Some ratios/intervals were also given names:
2/1, the octave, was named diapason meaning 'through all [strings]'
3/2, the perfect fifth was labelled diapente, meaning 'through 5 [strings]'
4/3, the perfect fourth, was labelled diatessaron, meaning 'through 4 [strings]'
dieses, 'sending through', refers to any interval smaller than about 1/3 of a perfect fourth
tonos referred both to the interval of a whole tone, and something more akin to mode or key in the modern sense (Chalmers, 1993)
ditone referred to the interval made by stacking two 9/8 whole tones, was referred to as tonos, resulting in 81/64, the Pythagorean major third, as a ditone (Monzo, http://www.tonalsoft.com)
256/243 - the limma, which is the ratio between left over after subtracting two 9/8 tones (together making a ditone) a perfect fourth, the diatonic semitone of the Pythagorean diatonic scale
2187/2048 - the apotome, which is the ratio between the tone and the limma, the chromatic semitone of the Pythagorean diatonic scale
The Ancient greek term diatonon, meaning 'through tones', refers to the genus with two whole tones and a semitone, or any genus in which no interval is greater than one half of the fourth (Chalmers, 1993). The Pythaogrean diatonic scale is the scale that may be built from one two Pythagorean tetrachords, and the left over interval of 9/8.
Our current diatonic interval names
2/1, 3/2 and 4/3 were labelled 'perfect' because they were seen to be the perfect consonances. The two sizes of second, third, sixth and seventh in the diatonic scale are labelled major or minor, the larger sizes of each labelled major. Any perfect or major interval raised by a chromatic semitone (the difference, for e.g. between C and C#) is referred to as 'Augmented' and any minor or perfect interval lowered by a chromatic semitone is referred to as 'diminished'.
Common microtonal interval names
Dating back at least to 1880, after Alexander Ellis and John Land, the interval 7/6 has been associated with the label subminor third. in a generalisation of this idea, 9/7 is most commonly reffered to as a supermajor third, 12/7 a supermajor sixth, 14/9 a subminor sixth, 8/7 a supermajor second, 7/4 a subminor seventh, 27/14 a supermajor seventh and 28/27 a subminor second. This system was further generalised by some theorists and musicians such that an interval a bit smaller than a major is referred to as a subminor third, and an interval a bit larger than a minor third as a supraminor third. Notice that 'supra' is used instead of 'super', but 'sub' is still used. Similarly defined are submajor and supraminor seconds, sixths and sevenths. 'Sub' and 'super' prefixes have also seen occasional application to the perfect scale degrees. Like in the case of the submajor 3rd, etc., super and sub unisons, fourths, fifths and octaves are not associated with particular frequency ratio by all or most microtonal musicians and theorists. In the case of seconds, thirds, sixths and sevenths, intervals half way between major and minor are often called neutral. Finally, in limited use are 'intermediates', where an interval in-between a major 3rd and a perfect fourth, for example, is referred to as a 'third-fourth'. There are undoubtedly other interval naming practice that exists that are not as well known to the author, which are likely to be less commonly used. Although all these microtonal interval naming concepts are in common use, there is not yet a complete system that defined them, only complete systems that depart from them.
Sagittal - sagispeak
One such system, sagispeak was developed by Dave Keenan and others as an interval naming system that maps 1-1 with the Sagittal microtonal music notation system. Sagittal notation was developed as a generalised diatonic-based notation system applicable equally to just intonation, equal tunings and rank-n temperaments. Dozens of different accidentals can be used on a regular diatonic staff to notate up to extremely fine divisions, however in most cases only a handful are needed. In sagispeak, each accidental is presented by a prefix, made up of a single letter, in most cases, followed by either 'ai' if the accidental raises a note, or 'ao' if it lowers a note. Pythagorean intonation is assumed as a basis, like in previous microtonal ideas by Ellis, Helmholtz, Wolf, Monzo and others (combined into HEWM notation). Then the prefixes depart from Pythaogrean intonation, altering by commas and introducing other primes. Noting that the set of 'sub' and 'super' 2nds, 3rds, 4ths and 5ths above all involve ratios of 7, Sagittal features an accidental of 64/63, the Archytas comma, which may be used to take a Pythagorean major third, or minor third to a supermajor third or subminor third respectively, for example. The prefix 'tao' takes Pythagorean minor 2nds, 3rds, 6ths and 7ths to subminor 2nds, 3rds, 6ths and 7ths respectively, and the prefix 'tai' takes Pythagorean major 2nds, 3rds, 6ths and 7ths to supermajor 2nds, 3rds, 6ths and 7ths respectively. Other prefixes include 'pai' and 'pao' (where 'p' is for 'pental', as in, involving prime 5), which raise or lower a note by 81/80, the meantone comma, respectively, and 'vai' and 'vao', which raise or lower a note by 33/32 respectively, leading to ratios of 11. The ratio 5/4, the just major third is considered almost universally to be a major third, however, so is the Pythagorean major third, 81/64. In Meantone temperament, with which we are very familiar, 81/80, the difference between these notes is tempered out, however in Sagittal, and arguably in any effective notation or interval naming system, these two major thirds need to be distinguished from each other. In Sagispeak 5/4 is a 'pao-major 3rd' (and 6/5 a 'pai-minor third), and 81/64, a major third. In meantone tunings they are both represented by the same interval, the major third.
Because it is built off of the diatonic scale, sagispeak conserves diatonic interval arithmetic, i.e. familiar relations in the diatonic scale, i.e. M2 + m3 = P4. Diatonic interval arithmetic is also extended, where, for example, tai-major 2 + tao-minor 3 = P4 (8/7 + 7/6 = 4/3), where opposite alterations cancel each other out, and diatonic interval arithmetic is conserved . This is a very helpful property for a microtonal interval naming system to possess. Another helpful property of sagispeak is the generalised applicability to edos, just intonation and other tunings, where the same intervals maintain their spelling across different tunings. Despite these benefits however, many see Sagittal and Sagispeak as overly complex (even though the entire extended system need hardly ever be applied), and requiring too many new terms to be learnt.
We have seen that there are two competing definitions of a major third, the ratio '5/4' or the interval built from fourth stacked fifths, that may or may not correspond to 5/4. Interval naming systems wherein the major third is defined as an approximation to 5/4 rather than as four fifths minus two octaves may benefit from a familiar name for 5/4, but they are unable to conserve diatonic interval arithmetic.
Dave Keenan's system
In 2016 Dave Keenan proposed an alternative generalised microtonal interval naming system for edos. He uses the familiar prefixes 'sub', 'super', 'supra' and 'neutral'. His scheme is based on the diatonic scale, however the diatonic interval names are not defined by their position in a cycle of fifths like is Sagispeak. In Keenan's system the ET's best 3/2 is first labelled P5, and the fourth P4. The interval half-way between the tonic and fifth is labelled the neutral third, or 'N3', and halfway between the fourth and the octave N6. Then the interval a perfect fifth larger than N3 is labelled N7, and the interval a fifth smaller than N6 labelled N2. The neutral intervals then lie either at a step of the ET, or between two steps. After this the remaining interval names are decided based on the distance they lie in pitch from the 7 labelled intervals, which make up the Neutral scale, P1 N2 N3 P4 P5 N6 N7, which, like the diatonic, is an MOS scale, which may be labelled Neutral[7] 3|3 using Modal UPD notation. To name an interval in an edo, the number of steps of 72edo that most closely approximate the size of the interval difference from a note of the neutral scale is first found. Then the prefix corresponding to that number of steps of 72edo is applied to the interval name. The following chart details this process (can't load the chart :( ). An interval just smaller than a major third in Keenan's system is labelled a narrow major third, and an interval just wider than a 6/5 minor third a wide minor third, however he notes that 'narrow' and 'wide' are only necessary in edos greater than 31.
Keenan's system is an elegant way to keep the 'major 3rd' label for 5/4, where labels depend on the size of the best fifth, however it suffers from it's applicability only to edos, and that it does not conserve interval arithmetic. Another potentially undesirable result of the system is that the major second approximates 10/9, and a wide major second 9/8, where as 9/8 is almost always considered a major second, and 10/9 often a narrow or small major second. One such system that considers 10/9 a narrow major second, is that of Aaron Hunt.
Size based systems
Microtonal theorist Aaron Andrew Hunt devised the Hunt system, which includes interval name assignments for JI (just intonation) and edos based on 41edo. Compared to Keenan's 72 interval names, Aaron's system includes 41. His system is based directly on 41edo, and unlike Keenan's system, interval are given the name of the closest step of 41edo, and no account is taken of the size of the edos fifth. In 41edo, Major, minor, augmented and diminished intervals are those obtained through the approximately Pythagorean cycle of fifths. Intervals one step of 41edo above these are given the prefix 'small', one step larger are given the prefix 'large', two steps smaller the prefix 'narrow' and two larger the prefix 'wide'. As a result, 5/4 is labelled a 'small major 3rd', or SM3 (not to be confused with a super major third, a label that does not exist in this system).
Neo-medieval musicians and early music historian and theorist Margo Schulter described her own interval naming scheme built on approximations to JI intervals. Each interval names corresponds to an approximate size, and no particular edo is referenced. In her scheme middle major thirds range in size from 400-423 cents, and small major thirds from 372-400c. 5/4 is labelled a small major third, 81/64 a middle major third and 9/7 a large major third. Margo's scheme includes small, middle and large varieties of major, minor and neutral 2nds, 3rds, 6ths, 7ths; perfect fourth and fifths; and tritones, as well as a sub fifth and super fourth a dieses and comma and an octave less dieses and comma and interseptimals, which correspond to intermediates, her name referencing the fact that they may each approximate two ratios of 7.
In Hunt's system when used in 41edo or JI diatonic interval arithmetic is conserved, but in other tunings it may not be, and Margo's system may not conserve diatonic interval arithmetic either. Both systems may be applied to arbitrary tunings, but the same intervals (defined, perhaps by a MOS scale) may not be given the same interval names across different tunings.
- User:PiotrGrochowski/Extra-Diatonic Intervals gives each 43edo interval a name, then maps each desired interval to a 43edo interval. PiotrGrochowski (info, talk, contribs) 14:59, 7 October 2018 (UTC)
Ups and Downs
One final interval naming system, associated with the Ups and Downs Notation system, belonging to microtonal theorist and musicians Kite Giedraitis, like Sagittal is based on deviations from the diatonic scale. In this system however, deviations (from major, minor, perfect, augmented and diminished) are notated simply by the addition of up or down arrows: '^' or 'v', corresponding to raising or lowering of a single step of an edo. In some tunings (12edo, 19edo or 31edo for example) 5/4 may be a M3, and in others a vM3 (downmajor 3rd) (e.g. 15edo, 22edo, 41edo, 72edo), or even an up-major 3rd (e.g. 21edo). Ups and downs also includes neutrals, which lay exactly in-between major and minor intervals of the same degree, labelled '~' (mid). 'Up' and 'down' prefixes may be used before mid also, i.e. 'v~ 3). This system benefits from it's simplicity as well as it's conservation of interval arithmetic. It can be used for some MOS scales where one of the generators is a perfect fifth or a fraction of a perfect fifth, but not all of these (e.g. Diminished[8]), and not all MOS scales (if such scales are to be described, an additional pair of accidentals/qualifiers is used. Although the scales then are described, their intervals still are not given the same names in Ups and Downs' edo names). Another criticism of Kite's system that does not apply to the others is the fact that when an edo is doubled or multiplied by some simple fraction, and the best fifth is constant across the two edos, the same intervals may be be given different names.
Igliashon Jones is a supporter of this system, but for the relabeling of 'down' as 'sub' and 'up' as 'super' (or supra) and 'mid' as 'neutral', so that more common names are used, wherein 'super' infers a raise of 1 step of the edo, and 'sub' a lowering of one step. In this 'Extra-diatonic' system 'super' and 'sub' may be doubly applied, as in Ups and Downs, but they may not be applied before 'neutral' where in Ups and Downs they may be applied before 'mid'. The author's own extra-diatonic system is developed as a departure with caveat that 'S' and 's' prefixes are defined not as alterations by a single step of the edo, but by comma alterations as in Sagittal, in order that interval of MOS scales may be represented consistently across different tunings. Throughout the rest of the article the development is detailed, and the system defined.
Premise:
Extra-diatonic names should be simple, generalisable, widely applicable, backwards compatible with standard diatonic notation and reflecting current informal practice as closely as possible. Extra-diatonic interval names are fifth based; extended from the familiar major, minor and perfect interval names so that diatonic interval arithmetic is conserved. ‘M’, ‘m’, and ‘P’ remain the short-hand for major, minor and perfect. ‘A’ and ‘d’ for Augmented and diminished may also be used in the familiar way. In cases where the chroma (the chromatic semitone, or augmented unison) is represented by multiple steps in the tuning the prefix ‘super’ raises major and perfect intervals by a single step while ‘sub’ lowers minor and perfect intervals, with short-hand ‘S’ and ‘s’. ‘S’ and ‘s’ may also be used to raise minor and lower major intervals respectively, reflecting occasion practice. In this case ‘S’ is short-hand for ‘supra’, and 's' for 'small'. They may also be used to raise or lower diminished and augmented intervals similarly. In this way this scheme is equivalent thus far to Ups and Downs notation, where ‘^’ or ‘up’ corresponds to ‘S’, ‘super’ or ‘supra’ and ‘v’ or ‘down’ to ‘sub’ or 'small' .
Additions and examples:
Neutrals and intermediates are also included, where neutrals occur between opposing sizes of a single generic interval the intermediates between each generic interval and the next.
Interval names for equal tunings are ranked in five tiers.
Interval names for equal tunings are ranked in eight tiers.
- Perfect
- Neutral
- Major, minor, A4 and d5 and, if the chroma is subtended by a single (positive) step of the edo, other augmented and diminished intervals
- Super, sub, supra and small prefixes to major, minor, perfect interval and to A4 and d5
- Intermediates
- Augmented and diminished intervals (for when the chroma is subtended by more than a single (positive) step of the edo)
- Super, sub, supra and small prefixes to augmented and diminished intervals
- Intervals augmented and diminished more than singularly
When more than one interval name corresponds to a specific interval, the names are privileged in order of the tiers. By this ordering, the first available name is the ‘primary’ for that interval, the second available ‘secondary’ and third 'tertiary'.
I initially intended for Intermediates to lay in the second tier with neutrals, but that led to primary intervals results that were not preferred by my colleagues over other options and I was persuaded to avoid using intermediates unless they are necessary.
Neutrals
N2, N3, N6 and N7, i.e. neutral 2nds, 3rds, 6ths and 7ths, falling exactly in-between the major and minor intervals of the same interval class, add native support for neutral-thirds temperament, where the N3 divides the P5 in exact halves and N2 divides the m3 is exact halves. In ups and downs neutrals indicated with '~' and said 'mid'.
Extending this familiar application to provide support for larger neutral scales, we add that neutrals occur also between P4 and A4; P5 and d5; P1 and A1; and P8 and d8.
Then Neutral[7] 3|3 can then be written:
P1 N2 N3 P4 P5 N6 N7 P8.
The same names give the primary interval names for 7edo, whose secondary intervals names are:
N1 m2/M2 m3/M3 N4 N5 m6/M6 m7/M7 N8.
The secondary interval names show that the chroma is equivalent to a unison in 7edo.
Neutral[10] 5|4 may then be written as
P1 N2 M2 N3 P4 N4 P5 N6 m7 N7 P8
Neutral[17] 8|8 may be written as
P1 N1 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 N8 P8,
which is equivalent to the primary interval names of 17edo.
Intermediates
To provide native support for Barbados and Diminishes temperaments, intermediates are also added to the system. It should be noted immediately that intermediates are not as common to common microtonal interval naming as neutrals and though are a useful addition to this scheme, may be left out if desired. The appendix includes the MOS scales and edos from 'lists of all edos and MOS Scales', but without any intermediates.
‘2-3’ lies exactly half-way between M2 and m3 and divides the P4 in half. It may be read ‘second-third’ or ‘serd’. ‘6-7’, it’s octave-inverse lies exactly half-way between M6 and m7 and may be read ‘sixth-seventh’ or ‘sinth’.
‘1-2’ lies exactly half-way between P1 and m2, dividing the m2 in half. It may be read ‘unison-second’ or ‘unicond’. Its octave-inverse, ‘7-8’, lies exactly half-way between M7 and P8 and may be read ‘seventh-octave’ or ‘sevtave’.
‘3-4’ lies exactly half-way between M3 and P4, dividing the M6 in half. It may be read ‘third-fourth’ or ‘thourth’. It’s octave-inverse, ‘5-6’, lies exactly half-way between P5 and m6 and may be read ‘fifth-sixth’ or ‘fixth’.
5edo can be spelled with the list of only these intermediates:
1-2 2-3 3-4 5-6 6-7 7-8.
The primary interval names for 5edo do not include them however, as they may be described by diatonic intervals:
P1 M2/m3 P4 P5 M6/m7 P8
Barbados[5] 2|2 may be described as
P1 2-3 P4 P5 6-7 P8,
and Barbados[9] 4|4 as
P1 M2 2-3 3-4 P4 P5 5-6 6-7 m7 P8.
The final intermediate, ‘4-5’ lies exactly between P4 and P5 and divides the octave in half. It may be read ‘fourth-fifth’ or ‘firth’. It is necessary for Diminished temperament, where the half-octave cannot be represented as any alteration of A4 or d5. Diminished temperament has a period of 1/4 of an octave, an approximation of 6/5. Therefore, the difference between four 6/5's and 2, 648/625, is tempered out.
Diminished[8] 1|0 (4) can be written as P1 sM2 Sm3 P4 4-5 Sm6 sM6 sM7 P8, and
Diminished[12] 1|1 (4): P1 Sm2 sM2 Sm3 sM3 P4 4-5 P5 Sm6 sM6 Sm7 sM7 P8,
12edo and a problem
The primary interval names for 12edo are as we are familiar:
P1 m2 M2 m3 M3 P4 A4/d5 P5 m6 M6 m7 M7 P8,
but the interval labeled A4/d5 has the secondary name '4-5'.
We know that 12edo supports Diminished temperament, but we can't see that 12edo as equivalent to Diminished[12]: If we follow the rules from our premise, the notes of Diminished[12] in 12edo would give: P1 M2 m2 M3 m3 P4 A4/d5 P5 M6 m6 M7 m7 P8, with the majors and minor flipped. To be equivalent, sM must equal M and Sm must equal m.
10edo, Pajara and a problem
10edo may be written as only neutrals and intermediates:
N1/1-2 N2 2-3 N3 3-4 N4/4-5/N5 5-6 N6 6-7 N7 7-8/N8
However it much more sense to write is using the primary interval name set:
P1 N2 M2/m3 N3 P4 N4/N5 P5 N6 M6/m7 N7 P8
It's secondary intervals as follows:
m2 Sm2/sM2 2-3 Sm3/sM3 M3 S4/s5 m6 Sm6/sM6 6-7 Sm7/sM7 M7
We can see that 10edo supports Neutral third scales, given that we can make the interval names for Neutral[10] using the primary interval names for 10edo.
We know 10edo and 12edo both support Pajara temperament. Pajara[10] 2|2 (2) consists of:
P1 Sm2 M2 sM3 P4 sA4/Sd5 P5 Sm6 m7 sM7 P8,
and Pajara[12] 3|2 (2) of
P1 Sm2 M2 Sm3 sM3 P4 sA4/Sd5 P5 Sm6 sM6 m7 sM7 P8.
We can see Pajara[10] in 10edo, but in 12edo, wouldn’t sM3 be m3?
This is where my system diverges from Igliashon Jones’. We have to break our first rule here, or at least add some conditions to it.
Comma association
To address this problem of consistency, we now state that when 81/80 is tempered out, M=sM and m=Sm, and when 64/63 is tempered out, M=SM and m=sm. In the case of sm and SM, ‘S’ and ‘s’ raise and lower by 64/63, and in the case of Sm and sM, ‘S’ and ‘s’ raise and lower by 81/80. In this way extra-diatonic interval names are equivalent to Sagispeak interval names, where for sm and SM ‘S’ and ‘s’ are equivalent to ‘tai’ and ‘pao’ and for Sm and sM ‘S’ and ‘s’ are equivalent to ‘pai’ and ‘pao’.
It is important to note that given this change, 'S' and 's' may alter an interval by a different number of steps in an edo depending on which interval names they prefix. This may seem confusing, but it seems to reflect existing informal practice.
In 12edo, which represents the union between the two, where both 64/63 and 81/80 are tempered out, ‘S’ and ‘s’ do not raise or lower intervals at all. We can now easily see that 12edo supports Pajara, where simply removing all the ‘s’s and ‘S’s from Pajara[12] gives us our primary interval names of 12edo.
14edo and Injera
Like 10edo, 14edo may be written using all the neutrals and intermediates, but without any intervals described both as a neutral and as an intermediate:
N1 1-2 N2 2-3 N3 3-4 N4 4-5 N5 5-6 N6 6-7 N7 7-8 N8.
The primary interval names for 14edo are as follows:
P1 sm2 N2 SM2/sm3 N3 SM3/s4 P4 SA4/sd5 P5 S5/sm6 N6 SM6/sm7 N7 SM7 P1
We know that 12edo and 14edo support Injera, where Injera[12] 3|2 (2) may be labelled
P1 sm2 M2 sm3 SM3 P4 SA4/sd5 P5 sm6 M6 m7 SM7 P8,
and Injera[14] 3|3 (2) labelled
P1 sm2 M2 sm3 m3 SM3 P4 SA4/sd5 P5 sm6 M6 SM6 m7 SM7 P8.
We can see both in 14edo, and to get 12edo from Injera[12], as with Pajara, we remove all the ‘s’ and ‘S’ prefixes.
Blacksmith and further extension
10edo also supports Blacksmith temperament, and we may think to write Blacksmith[10] 1|0 (5) as:
P1 sM2 M2/m3 sM3 P4 s5 P5 sM6 M6/m7 sM7 P8
But we have now added mappings, but are yet to define the use of ‘S’ and ‘s’ for perfect intervals. In Blacksmith, the interval we might call ‘s5’ is 81/80 below P5, however, more commonly ‘s5’ is used to refer to 16/11, and S4 11/8. Since these intervals have above been labelled N4 and N5 above however, we do not need to worry about that, and can add that s5, a 'small 5th', is 81/80 below 3/2, and S4, a 'supra 4th' lies 81/80 above 4/3. where ‘s4’ has been typically been used to refer to 21/16, and ‘S5’ to 32/21, we add that s4 is lower than P4 by 64/63 and that S5 is higher than P5 by 64/63.
The primary intervals names of Blacksmith[15] 1|1 (5) then are as follows:
P1 Sm2 sM2 M2/m3 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 M6/m7 Sm7 sM7 P8,
these being identical to the the primary interval names of 15edo.
We may further add that ‘S’ (supra) and ‘s’ (small) may raise diminished, and lower augmented intervals by 81/80 as they do to minor and major respectively and that when ‘S’ (super) raised an augmented interval, or ‘s’ (sub) lowers it, the change is by 64/63.
In edos where 81/80 or 64/63 are represented by a single step, or when the apotome is represented by a single step, Ups and Downs with 'S', 's', and 'N' can name the intervals equivalently to this system. This can be seen in add edos considered thus far, as well as in those listed directly below. What this system adds is that it may also describe the intervals of MOS scales (as well as JI scales), such that these interval can be identically named in edos that approximate the scales they belong to. This is true for all MOS scales mentioned so far, as well as those listed below.
Further application in edos
The primary interval names are shown below for some larger edos.
22edo: P1 m2 Sm2 sM2 M2 m3 Sm3 sM3 M3 P4 S4 4-5 s5 P5 m6 Sm6 sM6 M6 m7 Sm7 sM7 M7 P8
24edo: P1 N1 m2 N2 M2 SM2/sm3 m3 N3 M3 SM3/s4 P4 N4 A4/d5 N5 P5 S5/sm6 m6 N6 M6 SM6/sm7 m7 N7 M7 N8 P8
26edo: P1 A2 d2 m2 M2 A2 d3 m3 M3 A3 d4 P4 A4 SA4/sd5 d5 P5 A5 d6 m6 M6 A6 d7 m7 M7 A7 d8 P8
27edo: P1 m2 Sm2 N2 sM2 M2 m3 Sm3 N3 sM3 M3 P4 N4 d6 A3 N5 P5 m6 Sm6 N6 sM6 M6 m7 Sm7 N7 sM7 M7 P8
29edo: P1 sm2 m2 Sm2 sM2 M2 SM2/sm3 m3 Sm3 sM3 M3 SM3/s4 P4 S4 d5 A4 s5 P5 sm6 m6 Sm6 sM6 M6 SM6/sm7 m7 Sm7 sM7 M7 SM7 P8
31edo: P1 N1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 N4 A4 d5 N4 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 N8 P8
34edo: P1 1-2 m2 Sm2 N2 sM2 M2 2-3 m3 Sm3 N3 sM3 M3 3-4 P4 S4 N4 4-5 N5 s5 P5 5-6 m6 Sm6 N6 sM6 M6 6-7 m7 Sm7 N7 sM7 M7 7-8 P8
38edo: P1 S1 1-2 sm2 m2 N2 M2 SM2 2-3 sm3 m3 N3 M3 SM3 3-4 s4 P4 N4 A4 SA4/sd5 d5 N5 P5 S5 5-6 sm6 m6 N6 M6 SM6 6-7 sm7 m7 N7 M7 SM7 7-8 s8 P8
41edo: P1 N1 sm2 m2 Sm2 N2 sM2 M2 SM2 sm3 m3 Sm3 N3 sM3 M3 SM3 s4 P4 S4 N4 d5 A4 N5 sM5 P5 S5 sm6 m6 Sm6 N6 sM6 M6 SM6 sm7 m7 Sm7 N7 sM7 M7 SM7 N8 P8
43edo: P1 S1 1-2 A1/sm2 m2 dd3 AA1 M2 SM2/d3 2-3 A2/sm3 m3 dd4 AA2 M3 SM3/d4 3-4 A3/s4 P4 dd5 AA3 A4 d5 dd6 AA4 P5 S5/A6 5-6 A5/sm6 m6 dd7 AA5 M6 SM6/d7 6-7 A6/sm7 m7 dd8 AA6 M7 SM7/d8 7-8 s8 P8
46edo: P1 S1 sm2 m2 Sm2 A1 d3 sM2 M2 SM2 sm3 m3 Sm3 A2 d4 sM3 M3 SM3 s4 P4 S4 sd5 d5 sA4/Sd5 A4 SA4 s5 P5 S5 sm6 m6 Sm6 A4 d7 sM6 M6 SM6 sm7 m7 Sm7 A6 d8 sM7 M7 SM7 s8 P8
In 43edo we encounter the first time we have to use double Augmented and diminished intervals. 43edo marks the first instance in which Jones' alternative ups and downs interval names do not match those from this system. In his system, for example, AA2 would simply be sM3, but in this system since sM3 implies an approximation to 5/4 and the M3 already represents 5/4, and therefore is equivalent to sM3, we cannot do this. This tells us however that no simple ratio is approximated by the interval, and perhaps it is better understood as an AA2. Larger edos contain unlabeled intervals (without resorting to extended diatonic interval names). The association of 'super' and 'sub' with 64/63 and with 'supra' and 'small' with 81/80 may effect the assignment of primary interval names, but for all of these edos, as well as all those mentioned before, when 'S' and 's' are used, they still signify a raising or lowering by a single step of the edo, and thus appear equivalent to the Ups and Downs version. The comma associations add that, though use of enharmonic equivalences and secondary interval names may be necessary, intervals from MOS scales may be spelled in a consistent way across tuning to different edos.
Other rank-2 temperaments' MOS scales
On top of those discussed thus far, other temperaments generated by the P5 or a fraction of it are also supported to some extent, where their MOS scales may be represented, including Semaphore, Augmented, Porcupine, Diminished, Negri, Tetracot and Slendric.
Semaphore[5] 2|2: P1 SM2/sm3 P4 P5 SM6/sm7 P8
Semaphore[9] 4|4: P1 M2 SM2/sm3 SM3/s4 P4 P5 S5/sm6 SM6/sm7 m7 P8
Augmented[6] 1|0 (3): P1 Sm3 sM3 P5 Sm6 sM7 P8
Augmented[9] 1|1 (3): P1 Sm2 Sm3 sM3 P4 P5 Sm6 SM6 sM7 P8
Augmented[12] 2|1 (3): P1 Sm2 M2 Sm3 sM3 P4 sA4 P5 Sm6 Sm7 SM6 sM7 P8
Augmented[15] 2|2 (3): P1 Sm2 sM2 M2 Sm3 sM3 P4 Sd5 sA4 P5 Sm6 Sm7 SM6 m7 sM7 P8
Porcupine[7] 3|3: P1 sM2 Sm3 P4 P5 sM6 Sm7 P8
Porcupine[8] 4|3: P1 sM2 Sm3 P4 s5 P5 sM6 Sm7 P8
Porcupine[15] 7|7: P1 S1 sM2 M2 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 m7 Sm7 s8 P8
Negri[9] 4|4: P1 Sm2 SM2/sm3 sM3 P4 P5 Sm6 SM6/sm7 sM7 P8
Negri[10] 4|4: P1 Sm2 SM2/sm3 sM3 P4 A4 P5 Sm6 SM6/sm7 sM7 P8
Tetracot[6] 3|2: P1 sM2 N3 S4 N6 Sm7 P8
Tetracot[7] 4|2: P1 sM2 N3 S4 P5 N6 Sm7 P8
Tetracot[13] 6|6: P1 N2 sM2 Sm3 N3 P4 S4 s5 P5 N6 sM6 Sm7 N7 P8
Slendric[5] 2|3: P1 SM2 s4 S5 sm7 P8
Slendric[6] 3|2: P1 SM2 s4 P5 S5 sm7 P8
Slendric[11] 5|5: P1 S1 SM2 sm3 s4 P4 P5 S5 SM6 sm7 s8 P8
We may also write temperaments with a 9/8 but no 3/2. The most well known of these is Machine:
Machine[5] 2|2: P1 M2 M3 m6 m7 P8
Machine[6] 3|2: P1 M2 M3 A4 m6 m7 P8
Machine[11] 5|5: P1 d3 M2 d4 M3 d5 A4 m6 A5 m7 A6 P8
Formal summary
The resultant system may be formally summarised as follows:
Definition 1a. M and m label the two sizes of 2nd, 3rd, 6th and 7th in the Pythagorean diatonic scale.
Definition 1b. The smaller 4th and larger 5th are labelled P.
Definition 1c. The single size of 1 and 8 is labelled P.
Definition 2. A chroma above M or P is A and below m or P is d.
Definition 3a. Within generic interval classes 2, 3, 6 and 7, half way between M and m is N.
Definition 3b. Within generic interval classes 1 and 4, half way between P and A is N.
Definition 3c. Within generic interval classes 5 and 8, half way between P and d is N.
Corollary: 7edo can be written P1 N2 N3 P4 P5 N6 N7 N8.
Definition 4. Half way between adjacent generic interval classes lie the intermediates.
Corollary: 5edo can be written 1-2 2-3 3-4 5-6 6-7 7-8.
Definition 5a. 'S' when applied before M, and 's' when applied before m, raise or lower by 64/63 respectively.
Definition 5b. 'S' when applied before m, and 's' when applied before M, raise or lower by 81/80 respectively.
Definition 5c. 'S' when applied to P5 (leading to S5) and 's' when applied to P4 (leading to s4) raise and lower by 64/63 respectively.
Definition 5d. 'S' when applied to P4 (leading to S4) and 's' when applied to P5 (leading to s5) raise and lower by 81/80 respectively.
Definition 5e. 'S' when applied to P1 and 's' when applied to P8 may raise and lower by 64/63, or by 81/80.
Definition 6a. When 'S' and 's' imply alterations of 64/63, they have long-form 'super' and 'sub' respectively.
Definition 6b. When 'S' and 's' imply alterations of 81/80, they have long-form 'supra' and 'small' respectively.
Definition 7. If alterations of 81/80 and of 64/63 need to be distinguished from one another in short-form, alterations of 81/80 can be written 'SR' and 'sl'.
edos with extreme fifths
Thus far the best fifth of all edos described lies in the 'diatonic range', between 4 steps of 7edo and 3 steps of 5edo. The best fifths of some edos lies outside this range, in either directly. Whereas in edos where the best fifth lies between 7 steps of 12edo and 4 steps of 7edo, 81/80, the meantone comma is tempered out, wherein four fifths approximate the fifth harmonic (a sM3 raised two octave), in edos with fifths flatter than this, 135/128, a meantone chromatic semitone is instead tempered out, resulting in the four fifths instead approximate the Sm3, 6/5 (raised two ocaves). This system is called Mavila temperament. In the other direction, whereas the best fifth lies between 7 steps of 12edo and 3 steps of 5edo, 64/63, the septimal or Archytas comma is tempered out, wherein four fifths approximate a SM3, 9/7, raised two octaves, in edos who's best fifth is sharper than this, 9/7, as well as 5/4 are approximated by the perfect fourth, tempering out 16/15 and 28/27. This system is called Father temperament. The application of this system to edos of both of these fifth sizes is addressed below.
Mavila
In Mavila, the perfect 5th is flatter than in 7edo, so major intervals are below minor and augmented below major. In Mavila the major third approximates 6/5 and the minor third 5/4, tempering out 135/128. This presents no problem to the scheme however, and the rules are applied just the same. The small major third, 81/80 below 6/5 or 81/64 comes to 32/27, the minor 3rd, and the sub minor 3rd remains 7/6.
Mavila[7] 3|3 can be written
P1 M2 m3 P4 P5 M6 m7 P8,
the same as the diatonic scale.
Mavila[9] 4|4 can be written
P1 M2 M3 m3 P4 P5 M6 m6 m7 P8,
which is also the primary interval name set for 9edo.
If we don't want to have major being below minor, we can hide it with some secondary interval names:
P1 M2 Sm3 sM3 P4 P5 Sm6 sM6 m7 P8, arriving at Augmented[9].
We say that our first Mavila[9] interval names are not well-ordered, where for an interval name set to be well-ordered, for each degree major must be above minor. By extension we defined a well-ordered interval names set as one in which ... ≤ dd ≤ d ≤ m ≤ M ≤ A ≤ AA ≤ ... or ... ≤ dd ≤ d ≤ P ≤ A ≤ AA ≤ ..., and where s_ ≤ _ ≤ S_ (where '_' represents any of ... dd, d, m, (P), M, A, AA ...).
Using the fact that 64/63 is represented by a single step in 9edo, it is left as an exercise for the reader to prove that 9edo supports Negri temperament (by replacing some names with sub and super prefixed names to arrive at Negri[9] 4|4).
The primary interval names for Augmented[9] 1|1 (3) are well-ordered.
Mavila[16] 8|7 can be writtten
P1 d1 M2 m2 M3 m3 A4 P4 A5 P5 d5 M6 m6 M7 m7 A8 P8.
The enharmonic equivalences for 16edo can be generated by: M=Sm=sm or m=sM=SM
This leads us to our primary well-ordered interval name set for 16edo:
P1 S1 Sm2 sM2 Sm3 sM3 s4 P4 4-5 P5 S5 Sm6 sM6 Sm7 sM7 s8 P8,
wherein we can see that it supports Diminished temperament.
Mavila[23]11|11 can be written as:
P1 d1 A2 M2 m2 A3 M3 m3 d3 A4 P4 d4 A5 P5 d5 A6 M6 m6 d6 M7 m7 d7 A8 P8
And the primary well-ordered interval names for 23edo are appear as:
P1 S1 1-2 M2 m2 2-3 M3 m3 3-4 A4 P4 d4 A5 P5 d5 5-6 M6 m6 6-7 M7 m7 7-8 d8 P8
Father
In the other direction, the fifths of Father temperament are sharper than those of 5edo, leading to minor second going backwards. In Father temperament, 5/4 and 4/3 are tempered to a unison, along with 9/7. As 64/63 is tempered out, alterations of 64/63 act as identity alterations.The M2, larger than m3 is also a Sm3. sM2, then, returns to m3.
The primary interval names for Father[5] 2|2,
P1 M2 P4 P5 m7 P8,
present no problems.
In the primary interval names for Father[8] 4|3, however:
P1 m3 M2 P4 M3 P5 m7 M6 P8,
which are the same as the primary intervals for 8edo, but with M3 rather than 4-5, we see our diatonic interval names begin to cross over. We will add to our definition of well-ordered interval names that no interval names from interval-class n-1 may be subtended by a larger number of steps that any interval names from interval-class n. As above, we can may use some secondary interval names to address it, leading to
P1 sM2 M2 P4 S4 P5 m7 Sm7 P8,
as the primary well-ordered interval name set.
If we also re-write M2 and m7 as Sm3 and sM6, we get Porcupine[8] 4|3.
Using some secondary interval names to 'fix' the order leads us to
P1 sM2 M2 P4 S4 P5 m7 M6 P8
The primary interval names for Father[13] 6|6:
P1 M7 m3 M2 d5 P4 M3 m6 P5 A4 m7 M6 m2 P8,
look very unruly.
We will fix up the ordering again with secondary interval names:
P1 S1 sM2 M2 Sm3 P4 S4 s5 P5 sM6 m7 Sm7 s8 P8,.
The primary interval names for 13edo are similar:
P1 N2 m3 M2 N3 P4 N5 N4 P5 N6 m7 M6 N7 P8.
Minimally fixing the order leads us to
P1 N2 sM2 M2 N3 P4 S4 s5 P5 N6 m7 Sm7 N7 P8,
which we have seen before as Tetracot[13] 6|6.
Miracle, 11edo and 21edo
All the scales discusses to this point use either smalls and supras or subs and supers, so in the rare instance that we see a S1 or s8, we can infer whether or not it's small/supra or sub/super, probably without even thinking too much about it. Rarely do MOS scales in this scheme require alterations of both 81/80 and 64/63. One important temperament that includes such scales is Miracle. We do not encounter either S1 in the 10 and 11-note MOS:
Miracle[10] 5|4: P1 Sm2 SM2 N3 s4 sA4 S5 N6 sm7 sM7 P8
Miracle[11] 5|5: P1 Sm2 SM2 N3 s4 Sd5 sA4 S5 N6 sm7 sM7 P8
We first see S1 and s8 in the 21-note MOS:
Miracle[21] 10|10: P1 S1 Sm2 N2 SM2 sm3 N3 sM3 s4 P4 Sd5 sA4 P5 S5 Sm6 N6 SM6 sm7 N7 sM7 s8 P8,
the largest MOS scale we have attempted yet to write. S1 in this scale is 64/63. We could maybe guess this from the presence of S5, but it is not obvious. If we need to make it clear when we are referring to small/supra, we can write their short-hand instead as 'sl' for 'small' and 'Sa' for supra. Miracle[21] would then be re-written:
P1 S1 Sam2 N2 SM2 sm3 N3 slM3 s4 P4 Sad5 slA4 P5 S5 Sam6 N6 SM6 sm7 N7 slM7 s8 P8, if complete clarity is needed, otherwise if long-form names are provided there is no need. It looks unwieldy to me and so I would avoid it, but it is there as a possibility, if the intervals of Mavila[21] need be written with only 4 letters allowed for each interval name.
Having encountered 11 and 21-note scales now, and haven't not described 11 and 21edo, I will add these here.
The fifth in 11edo is too flat even for it to be considered to support Mavila. Let's see what happens:
P1 M2 M3 m2 m3 P4 P5 M6 M7 m6 m7 P8.
Adding neutrals gives us our primary interval names for 11edo:
P1 M2 M3 N3 m3 P4 P5 M6 N6 m6 m7 P8.
This is clearly not Mavila, so we don't know what's tempered out, such that we might add our alterations to arrive at a well-ordered interval name set. Let's review the 11-note scales we have encountered above:
Slendric[11] 5|5: P1 S1 SM2 sm3 s4 P4 P5 S5 SM6 sm7 s8 P8
Miracle[11] 5|5: P1 Sm2 SM2 N3 s4 Sd5 sA4 S5 N6 sm7 sM7 P8
From these union of these scales we can see from P4=Sd5 that 135/128, the Mavila comma is tempered out. We apply our Mavila re-spellings to arrive at:
P1 Sm2 Sm3 N3 sM3 P4 P5 Sm6 N6 sM6 sM7 P8 as a well-ordered interval name set.
We should not expect our Machine[11] scale to be represented in this spelling of 11edo: A spelling of 11edo that shows that it supports Machine uses a different mapping, using the 9/8 from two 22edo P5s. We could spell 11edo as every other note of 22edo if we wish to see how it supports Machine.
21edo can be written as three 7edos as it's best fifth is that of 7edo. Since 81/80 is -1 steps in 21edo, we use 64/63 alterations:
P1 S1 sm2 N2 SM2 sm3 N3 SM3 s4 P4 SA4 sd5 P5 S5 sm6 N6 SM6 sm7 N7 SM7 d8 P8.
Using the available conversion sm = Sm (and therefor SM = sM), we can confirm that 21edo supports Miracle and Whitewood temperaments. This is left as an exercise for the the inspired reader.
6edo
Though 6edo is normally only ever considered as a subset of 12edo, given that we have encountered 6-note MOS I'll give it a red hot go.
Our P5 and P4 in 6edo is our half-octave, 4-5, so 9/8 is tempered out and our chromatic scale only covers 2edo: P1/M2/M3/A4 m2/m3/P4/4-5/P5/M6/M7 m6/m7/P8. If we want to write 6edo is a well-ordered way, we might choose:
P1 SM2 sm3 P4/4-5/P5 SM6 sm7 P8.
Writing s4 and S5 instead of sm3 and SM6 would give us Slendric[6] 3|2.
What of Augmented[6] 1|0 (3): P1 Sm3 sM3 P5 Sm6 sM7 P8?
This tells us that in 6edo 81/80 is mapped to -2 steps of 6edo. This is not a problem, as we can use alterations of 64/63, mapped to 1 step, though I don't see why anyone would want to think of 6edo in this way.
Similarly to Machine in 11edo, Machine in 6edo uses a different (much better) mapping of 9/8: That of 12edo. 6edo is much better spelled as a subset of 12edo, where we can see if supports Machine.
The primary interval names for the remaining trivial edos are trivially derived and are given along with all those described so far in the section below.
28edo
We encounter a new problem with 28edo. 28edo's best fifth is that of 7edo. It's primary intervals names are as such:
P1 sA1 1-2 Sm2 N2 sM2 2-3 Sm3 N3 sM3 3-4 S4 P4 sA4 4-5 Sd5 P5 s5 5-6 Sm6 N6 sM6 6-7 Sm7 N7 sM7 7-8 Sd8 P8
in 28edo 81/80 is represented by -1 steps. We en-devour to maintain as best we can in our primary interval names the original premise behind the notation - that the prefix 's' takes an interval down a single step of the edo and 'S' a single step up. In our primary interval names for 28edo we have S4 below P4 and d5 above P5. We can avoid this confusion however by using the secondary interval names for P4 and P5 in 28edo - N4 and N5. In our list of edos below this change is made.
How to
Though we have derived our interval names through many examples thus far, the process has not been explained as yet such that the reader may immediately apply them. This will be addressed here, with a step-by-step derivation guide. This guide gives all possible labels to each interval.
For n-edo:
- Label P1 = 1, P8 = n
- Find the best approximations of 3/2, 5/4 and 7/4, which are to be labelled P5, sM3 and sm7 (This is equivalent to finding the 7-limit patent val).
- P5 = round(ln(3/2)/ln(n)) steps
- sM3 = round(ln(5/4)/ln(n)) steps
- sm7 = round(ln(7/4)/ln(n)) steps
- Using the best fifth, label the diatonic intervals. i.e.
- M2 = (2*P5) mod n
- M6 = (3*P5) mod n
- M3 = (4*P5) mod n
- M7 = (5*P5) mod n
- (P4, m7, m3, m6, m2) = n - (P4, M2, M6, M3, M7)
- Determine the size of the apotome: A = M-m (for any of degree 2, 3, 6 or 7).
- The initial chromatic intervals may be labelled:
- A1 = A
- d8 = P8 - A
- A4 = P4 + A
- d5 = P5 - A
- If A is even, then intervals half way between M and m within a degree may be labelled N.
- Determine the size of meantone and septimal comma alterations (steps 7-12 may be skipped if A = 1)
- meantone comma = M3 - sM3. If equal to 0 then 81/80 is tempered out, the edo is said to support Meantone temperament and alterations of 81/80 are not needed.
- septimal comma = m7 - sm7. If equal to 0 then 64/63 is tempered out, the edo is said to support Superpyth temperament, and alterations of 64/63 are not needed.
- If sub/super < 0, alterations of 64/63 should not be applied.
- If small/supra < 0, alterations of 81/80 my be applied only if A = 0, wherein all diatonic intervals are given the label 'N'.
- Acknowledging points 7, 8 and 9, label 's' and 'S' alterations to diatonic intervals, where:
- sM (small major) = M - meantone comma
- Sm (supra minor) = m + meantone comma
- SM (super major) = M + septimal comma
- sm (sub minor) = m - septimal comma
- Acknowledging points 7, 8 and 9, label 's' and 'S' alterations to P4 and P5 where:
- S4 (supra 4th) = P4 + meantone comma
- s5 (small 5th) = P5 - meantone comma
- s4 (sub 4th) = P4 - septimal comma
- S5 (super 5th) = P5 + septimal comma
- If meantone comma alterations are used, S1 (supra unison) may be labelled at P1 + meantone comma and d8 (small octave) at P8 - meantone comma. If septimal comma alterations are used, S1 (super unison) may be labelled at P1 + septimal comma and s8 (sub octave) may be labelled at P8 - septimal comma. If both are needed and not equal, and after steps 13-16 are completed as desired the interval corresponding to a super or supra unison (equivalently, sub or small octave) is yet otherwise unlabeled then to ascertain whether S1 is a supra or super unison (or equivalently s8 a small or sub octave), for all degrees, the short hand for 'small' is to be 'sl' rather than 's', and for 'supra' is to be 'SR' rather than 'S'.
- Remaining augmented (M + A) and diminished (m - A) intervals may be labelled 'A' and 'd'.
- If desired, augmentation and diminution may be iterated through the continued adding or subtracting of A, labelled by additional 'A' or 'd' prefixes respectively.
- 'S' and 's' may be further applied as if 'A' were 'M' or 'd' were 'm'.
- If desired and if n is even, the interval subtended by n/2 steps may be labelled 4-5.
- If desired and if m2 is subtended by an even number of steps, the remaining intermediates may be labelled: (1-2, 2-3, 3-4, 5-6, 6-7, 7-8) = (P1, M2, M3, P5, M6, M7) + m2/2.
Lists of edos and MOS scales
Primary well-ordered (unless otherwise noted) interval names for edos
2edo: P1 P4/4-5/P5 P8
3edo: P1 3-4/P4 P5/5-6 P8
4edo: P1 2-3 P4/4-5/P5 6-7 P8
5edo: P1/1-2 2-3 3-4/P4 P5/5-6 6-7 7-8/P8
6edo: P1 SM2 sm3/s4 P4/4-5/P5 S5/sM6 sm7 P8
7edo: P1 N2 N3 P4 P5 N6 N7 P8
8edo: P1 sM2 M2 P4 4-5 P5 m7 Sm7 P8
9edo: P1 1-2 2-3 3-4 P4 P5 5-6 6-7 7-8 P8
10edo: P1/1-2 N2 2-3 N3 3-4/P4 4-5 P5/5-6 N6 6-7 N7 7-8/P8
11edo: P1 Sm2 Sm3 N3 sM3 P4 P5 Sm6 N6 sM6 sM7 P8
12edo: P1 m2 M2 m3 M3 P4 4-5 P5 m6 M6 m7 M7 P8
13edo: P1 N2 m3 M2 N3 P4 N5 N4 P5 N6 m7 M6 N7 P8
14edo: P1 1-2 N2 2-3 N3 3-4 P4 4-5 P5 5-6 N6 6-7 N7 7-8 P8
15edo: P1/1-2 S1/Sm2 sM2 2-3 Sm3 sM3 3-4/P4 S4 s5 P5/5-6 Sm6 sM6 6-7 Sm7 sM7/s8 7-8/P8
16edo: P1 1-2 Sm2 sM2 Sm3 sM3 3-4 P4 4-5 P5 5-6 Sm6 sM6 Sm7 sM7 7-8 P8
17edo: P1 N1 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 N8 P8
19edo: P1 1-2 m2 M2 2-3 m3 M3 3-4 P4 A4 d5 P5 5-6 m6 M6 6-7 m7 M7 7-8 P8
21edo: P1 S1 sm2 N2 SM2 sm3 N3 SM3 s4 P4 SA4 sd5 P5 S5 sm6 N6 SM6 sm7 N7 SM7 d8 P8
22edo: P1 m2 Sm2 sM2 M2 m3 Sm3 sM3 M3 P4 S4 4-5 s5 P5 m6 Sm6 sM6 M6 m7 Sm7 sM7 M7 P8
23edo: P1 S1 1-2 Sm2 sM2 2-3 Sm3 sM3 3-4 A4 P4 S4 s5 P5 d5 5-6 Sm6 sM6 6-7 Sm7 sM7 7-8 s8 P8 (can't quite get it well-ordered)
24edo: P1 1-2 m2 N2 M2 2-3 m3 N3 M3 3-4 P4 N4 4-5 N5 P5 5-6 m6 N6 M6 6-7 m7 N7 M7 7-8 P8
26edo: P1 A2 d2 m2 M2 A2 d3 m3 M3 A3 d4 P4 A4 4-5 d5 P5 A5 d6 m6 M6 A6 d7 m7 M7 A7 d8 P8
27edo: P1 m2 Sm2 N2 sM2 M2 m3 Sm3 N3 sM3 M3 P4 N4 d6 A3 N5 P5 m6 Sm6 N6 sM6 M6 m7 Sm7 N7 sM7 M7 P8
28edo: P1 sA1 1-2 Sm2 N2 sM2 2-3 Sm3 N3 sM3 3-4 S4 N4 sA4 4-5 Sd5 N5 s5 5-6 Sm6 N6 sM6 6-7 Sm7 N7 sM7 7-8 Sd8 P8
29edo: P1 1-2 m2 Sm2 sM2 M2 2-3 m3 Sm3 sM3 M3 3-4 P4 S4 d5 A4 s5 P5 5-6 m6 Sm6 sM6 M6 6-7 m7 Sm7 sM7 M7 7-8 P8
31edo: P1 N1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 N4 A4 d5 N4 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 N8 P8
34edo: P1 1-2 m2 Sm2 N2 sM2 M2 2-3 m3 Sm3 N3 sM3 M3 3-4 P4 S4 N4 4-5 N5 s5 P5 5-6 m6 Sm6 N6 sM6 M6 6-7 m7 Sm7 N7 sM7 M7 7-8 P8
38edo: P1 S1 1-2 sm2 m2 N2 M2 SM2 2-3 sm3 m3 N3 M3 SM3 3-4 s4 P4 N4 A4 4-5 d5 N5 P5 S5 5-6 sm6 m6 N6 M6 SM6 6-7 sm7 m7 N7 M7 SM7 7-8 s8 P8
41edo: P1 N1 sm2 m2 Sm2 N2 sM2 M2 SM2 sm3 m3 Sm3 N3 sM3 M3 SM3 s4 P4 S4 N4 sA4 Sd5 N5 sM5 P5 S5 sm6 m6 Sm6 N6 sM6 M6 SM6 sm7 m7 Sm7 N7 sM7 M7 SM7 N8 P8
43edo: P1 S1 1-2 A1/sm2 m2 dd3 AA1 M2 SM2/d3 2-3 A2/sm3 m3 dd4 AA2 M3 SM3/d4 3-4 A3/s4 P4 dd5 AA3 A4 d5 dd6 AA4 P5 S5/A6 5-6 A5/sm6 m6 dd7 AA5 M6 SM6/d7 6-7 A6/sm7 m7 dd8 AA6 M7 SM7/d8 7-8 s8 P8 (not possible to find a well-ordered interval name set)
46edo: P1 S1 sm2 m2 Sm2 A1 d3 sM2 M2 SM2 sm3 m3 Sm3 A2 d4 sM3 M3 SM3 s4 P4 S4 sd5 d5 4-5 A4 SA4 s5 P5 S5 sm6 m6 Sm6 A4 d7 sM6 M6 SM6 sm7 m7 Sm7 A6 d8 sM7 M7 SM7 s8 P8 (not possible to find a well-ordered interval name set)
Interval names for MOS scales
Augmented[6] 1|0 (3): P1 Sm3 sM3 P5 Sm6 sM7 P8
Augmented[9] 1|1 (3): P1 Sm2 Sm3 sM3 P4 P5 Sm6 SM6 sM7 P8
Augmented[12] 2|1 (3): P1 Sm2 M2 Sm3 sM3 P4 sA4 P5 Sm6 Sm7 SM6 sM7 P8
Augmented[15] 2|2 (3): P1 Sm2 sM2 M2 Sm3 sM3 P4 Sd5 sA4 P5 Sm6 Sm7 SM6 m7 sM7 P8
Barbados[5] 2|2: P1 2-3 P4 P5 6-7 P8
Barbados[9] 4|4: P1 M2 2-3 3-4 P4 P5 5-6 6-7 m7 P8
Blackwood[10] 1|0 (5): P1 sM2 M2/m3 sM3 P4 s5 P5 sM6 M6/m7 sM7/s8 P8
Blackwood[15] 1|1 (5): P1 S1/Sm2 sM2 M2/m3 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 M6/m7 Sm7 sM7/s8 P8
Diminished[8] 1|0 (4): P1 sM2 Sm3 P4 sA4/Sd5 Sm6 sM6 sM7 P8
Diminished[12] 1|1 (4): P1 Sm2 sM2 Sm3 sM3 P4 sA4/Sd5 P5 Sm6 sM6 Sm7 sM7 P8
Injera[12] 3|2 (2): P1 sm2 M2 sm3 SM3 P4 SA4/sd5 P5 sm6 M6 m7 SM7 P8
Injera[14] 3|3 (2): P1 sm2 M2 sm3 m3 SM3 P4 SA4/sd5 P5 sm6 M6 SM6 m7 SM7 P8
Machine[5] 2|2: P1 M2 M3 m6 m7 P8
Machine[6] 3|2: P1 M2 M3 A4 m6 m7 P8
Machine[11] 5|5: P1 d3 M2 d4 M3 d5 A4 m6 A5 m7 A6 P8
Mavila[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8
Mavila[9] 4|4: P1 M2 M3 m3 P4 P5 M6 m6 m7 P8
Mavila[16] 8|7: P1 d1 M2 m2 M3 m3 A4 P4 A5 P5 d5 M6 m6 M7 m7 A8 P8
Meantone[5] 2|2: P1 M2 P4 P5 m7 P8
Meantone[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8
Meantone[12] 6|5: P1 m2 M2 m3 M3 P4 A4 P5 m6 M6 m7 M7 P8
Meantone[19] 9|9: P1 A1 m2 M2 A2 m3 M3 d4 P4 A4 d5 P5 A5 m6 M6 d7 m7 M7 d8 P8
Miracle[10] 5|4: P1 Sm2 SM2 N3 s4 sA4 S5 N6 sm7 sM7 P8
Miracle[11] 5|5: P1 Sm2 SM2 N3 s4 Sd5 sA4 S5 N6 sm7 sM7 P8
Negri[9] 4|4: P1 Sm2 SM2/sm3 sM3 P4 P5 Sm6 SM6/sm7 sM7 P8
Negri[10] 4|4: P1 Sm2 SM2/sm3 sM3 P4 A4 P5 Sm6 SM6/sm7 sM7 P8
Neutral[7] 3|3: P1 N2 N3 P4 P5 N6 N7 P8
Neutral[10] 5|4: P1 N2 M2 N3 P4 N4 P5 N6 m7 N7 P8
Neutral[17] 8|8: P1 N1 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 N8 P8
Pajara[10] 2|2 (2): P1 Sm2 M2 sM3 P4 sA4/Sd5 P5 Sm6 m7 sM7 P8
Pajara[12] 3|2 (2): P1 Sm2 M2 Sm3 sM3 P4 sA4/Sd5 P5 Sm6 sM6 m7 sM7 P8
Porcupine[7] 3|3: P1 sM2 Sm3 P4 P5 sM6 Sm7 P8
Porcupine[8] 4|3: P1 sM2 Sm3 P4 s5 P5 sM6 Sm7 P8
Porcupine[15] 7|7: P1 S1 sM2 M2 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 m7 Sm7 s8 P8
Semaphore[5] 2|2: P1 SM2/sm3 P4 P5 SM6/sm7 P8
Semaphore[9] 4|4: P1 M2 SM2/sm3 SM3/s4 P4 P5 S5/sm6 SM6/sm7 m7 P8
Slendric[5] 2|3: P1 SM2 s4 S5 sm7 P8
Slendric[6] 3|2: P1 SM2 s4 P5 S5 sm7 P8
Slendric[11] 5|5: P1 S1 SM2 sm3 s4 P4 P5 S5 SM6 sm7 s8 P8
Superpyth[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8
Superpyth[12] 6|5: P1 m2 M2 m3 M3 P4 A4 P5 m6 M6 m7 M7 P8
Superpyth[17] 8|8: P1 m2 A1 M2 m3 d4 M3 P4 d5 A4 P5 m6 A5 M6 m7 d8 M7 P8
Tetracot[6] 3|2: P1 sM2 N3 S4 N6 Sm7 P8
Tetracot[7] 4|2: P1 sM2 N3 S4 P5 N6 Sm7 P8
Tetracot[13] 6|6: P1 N2 sM2 Sm3 N3 P4 S4 s5 P5 N6 sM6 Sm7 N7 P8
Conclusion
Using only 'S' and 's' as qualifiers to M, m, P, A and d, along with N and intermediate degrees, a system is developed wherein for most edos below 50 intervals can be systematically named such that for primary interval names, 'S' and 's' raise and lower, respectively, by 1 step of an edo , whilst the intervals of many MOS scales may be consistently named across different tunings, taking the best of both Igliashon Jones' extra diatonic interval names and Sagispeak. One does not need to understand the comma associations to make use of the interval names. While the intervals of some MOS scales may hold consistent names in edos in an Ups and Downs based scheme, there are many common scales that do not in such a system, that do in this one, such as scales of Diminished and Augmented temperament. What's more, the use of neutrals and intermediates leads to quicker recognition of MOS scales that may be supported in edos.