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

Mesopotamian interval names, from http://www.historyofmusictheory.com/?page_id=130, accessed October 7, 2018.

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 string length ratios, after Pythagoras, or as positions in a tetrachord.

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, resulting in 81/64, the Pythagorean major third. (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.

Zarlino and Meantone

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Intervals were referred to by the Ancient Greek names through the the 18th century, as Latin names. By the Renaissance it had been discovered that a Pythagorean diminished fourth sounded sweet, and approximated the string length ratio 5/4. This just tuning for the major third was sought after, along with the complementary 6/5 tuning for the minor third, and octave complements to both - 8/5 for the minor sixth and 5/3 for the major sixth. Influential Italian music theorist and composer Gioseffo Zarlino put forth that choirs tuned the diatonic scale to the tuning built from this tetrachord, the intense diatonic scale, also known as the syntonic or syntonus diatonic scale or the Ptolemaic sequence:

1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1

Included in this scale, however, were wolf intervals: imperfect consonances that occurred as tunings of the same interval as perfect consonances. For example, between 1/1 and 3/2, 4/3 and 1/1, 5/3 and 5/4; and 5/4 and 15/8 occurs the perfect fifth, 3/2, whereas between 9/8 and 5/3 occurs the wolf fifth, 40/27, flat of 3/2 by 81/80. This was also the interval by which four 3/2 fifths missed 5/1 (the interval two octaves above 5/4). It was named the syntonic comma after Ptolemy's syntonus or intense diatonic tetrachord which consists of the intervals 9/8, 10/9 and 16/15, where 9/8 and 10/9 differ by this interval. By making the syntonic comma a unison the wolf fifth could be made a perfect fifth. It was discovered that this could be achieved by flattening (tempering) the perfect fifth by some fraction of this comma such that four of these fifths less two octaves gave an approximation of 5/4. Where two fifths less an octave give 9/8, the next two add another 10/9 to result in the 5/4. 9/8 and 10/9 were referred to as the major tone (tunono maggiore) and minor tone (tunono minore), respectively, and where this tuning led to them being equated, it was referred to as Meantone temperament, which is said to 'temper out' the syntonic comma. Zarlino advocated the flattening of the fifth by 2/7 of a comma, leading to 2/7-comma Meantone, but also described 1/3-comma and 1/4-comma Meantone as usable (Zarlino, 1558).

The diagram on the right, from Zarlino's 1558 treatise Le istitutioni harmoniche associates many intervals with their tuning as perfect consonances. The perfect tuning for the ditone was considered then to be 5/4, rather than 81/64. The interval for which 6/5 is considered a perfect tuning was referred to as a semiditone (labelled also in Le istitutioni harmoniche by as Trihemituono). This may seem odd to us now, but in Latin 'semi' referred not to 'half', but to 'smaller', so 'semiditone' translated to something like 'smaller ditone'. Additionally 'semitone' referred to the interval smaller than the 'tone'. Like the tone, this interval possessed two alternative perfect tunings: 16/15, the difference between 15/8 and 2/1, or 5/4 and 4/3, and 25/24, the difference between 6/5 and 5/4. 16/15 was referred to as the major semitone (semituono maggiore) and 25/24 as the minor semitone (semituono maggiore).

In addition to the Latin interval names, derived from the Ancient Greek interval names, we see on the diagram a single interval name in Italian: Essachordo maggiore, referring to the ratio 5/3, which we are tempted to translate to 'major sixth'. Chapter 16, Quel che sia Consonanze semplice, e Composta; & che nel Senario si ritouano le sorme di tutte le somplici consonanze; & onde habbia origine l'Essachordo minore, puts forward that the Essachordo minore, or perhaps 'minor sixth' be tuned to 8/5.

In the 1691 Lettre de Monsieur Huygens à l'Auteur [Henri Basnage de Beauval] touchant le Cycle Harmonique, theorist Christiaan Huygens gave names and ratios to common intervals and mapped them to 31-tET, which very closely approximates 1/4-comma Meantone. Translated from French, 3/2 was labelled a Fifth, 4/3 a Fourth, 5/4 a major Third, 6/5 and minor Third, 5/3 a major Sixth and 8/5 a minor Sixth. Here we really begin to see today's interval names.

English interval names in the Baroque

Harmonics, or The Philosophy of Musical Sounds, Edition 2, Smith, 1759, Section 2: On the Names and Notation of consonance and their intervals, Fig. 2 & 3 , pg. 10

After English superseded Latin as the the main language of scholarship, the Latin interval names were rejected and the convention we saw in Zarlino's Italian for naming the smaller of a pair of sizes of an interval 'minor' and the larger 'major' was further applied.

Music theorist and mathematician Robert Smith provides the diagram and table on the right in his 1749 Harmonics, or, The Philosophy of Musical Sounds, with the description:

'Fig. 2. If a musical string CO and it's parts DO, EO, FO, GO, AO, BO, cO, be in proportion to one another as the numbers 1, 8/9, 4/5, 3/4, 2/3, 3/5, 8/15, 1/2, their vibrations will exhibit the system of 8 sounds which musicians donate by the letters C, D, E, F, G, A, B, c.

Fig. 3. And supposing those strings to be ranged like ordinates to a right line Cc, and their distances CD, DE, EF, FG, GA, AB, BC, not to be the differences of their lengths, as in fig. 2. but to be the magnitudes proportional to the intervals of their sounds, the received Names of these intervals are shewn in the following Table; and are taken from the numbers of the strings or sounds in each interval inclusively; as a Second, Third, Fourth, Fifth, &c, with the epithet of major or minor, according as the name or number belongs to a greater of smaller total interval; the difference of which results chiefly from the different magnitudes of the major and minor second, called the Tone and Hemitone.'

We note that Smith uses the tuning of the diatonic scale that Zarlino put forward: the Ptolemaic Sequence. We may be surprised to see 4:3 here labelled as a minor Fourth, and 3/2 as a major Fifth, but it is obvious that this naming is more consistent than today's. Smith adds that

'Any one of the ratios in the third column of the foregoing Table, except 80 to 81, or any one of them compounded once of oftener with the ratio 2 to 1 or 1 to 2, is called a Perfect ratio when reduced to it's least terms. And when the times of the single vibrations of any two sounds have a perfect ratio, the consonance and it's interval too after called Perfect; and is called Imperfect or Tempered when that perfect ratio and interval is a little increased or decreased.'

'Any small increment of decrement of a perfect interval is called respectively the Sharp or Flat Temperament of the imperfect consonance, and is measured most conveniently by the proportion it bears to the comma'

Therefore in this system 3/2 is the Perfect major Fifth and 5/4 the Perfect major Third. 81/64 might be labelled a comma sharp major Third, 32/27 a comma flat minor Third, and the 1/4-comma Meantone fifth a 1/4-comma flat major Fifth. The interval naming scheme Smith describes may be immediately applied to 5-limit microtonal systems. There is an inconsistency, however, where it seems that 9/8 should be called a Perfect major Second, but that, while 9/5 be named a comma sharp minor Seventh, it's inverse, 10/9, is a Perfect minor Tone.

In An Elementary Treatise on Musical Intervals and Temperament, published in 1876, R. H. M Bosanquet refers to 5/4 as the perfect third, and 81/64 as the Pythagorean third. Bosanquet also labels other intervals of the Pythagorean diatonic scale similarly, i.e. 256/243, the limma, is labelled the Pythagorean semitone, and 27/16 the Pythagorean sixth. 81/80 is labelled the ordinary comma, or simple the comma, and the Pythagorean comma is defined as the difference between twelve fifths and seven octaves. The apotome of 2187/2048 is referred to as Apatomè Pythagoria. The following relationships are then described: (the image is too inappropriate to show, see pg 28)

Table describing the influence of the different consonances on one another, up to the 9th partial, from On the Sensations of Tone as a Phycological Basis for Theory of Music, Helmholtz, 1863, Translation by Ellis, 1875, pg. 187

Helmholtz and Ellis

Through the investigations of Galileo (1638), Newton, Euler (1729), and Bernouilli (1771), theorist Hermann von Helmholtz was aware that ratios governing the lengths of strings existed also for the vibrations of the tones they produced. His investigation of the harmonic series associated with these ratios of vibration led him to the consideration of ratios above the 5-limit. In his seminal On the Sensations of Tone as a Physiological Basis for the Theory of Music, published in German in 1863 and translated into English in 1875 by Alexander Ellis, he listed just intervals as show in the table to the right. It is interesting to note that 8:9 is labelled a 'Second' rather than a 'major Second'. The minor Seventh is shown as 5:9 rather than as 9:16 seemingly because of the 9 partial limit imposed on the table. It is also worth noting that 5:7 is labelled a subminor Fifth. 'Super', indicated in notation with a '+', raises an interval by 35:36, the septimal quarter tone, and 'sub', indicated with a '-' lowers by the same interval with the exception of the Supersecond 7:8, which lies 63:64, the septimal comma above the Second. The subminor fifth is not included in this as no minor Fifth is shown. If we assume that 'sub' lowers an interval 35:36, then the minor Fifth would be 18:25, 80:81 above Smith's 45:64 minor Fifth, however in table 2 below, Ellis labels 18:25 a superfluous Fourth, and it's inverse, 25:32, an acute diminished Fifth, whilst 64:45 is labelled a diminished Fifth and its inverse 32:45 a false Fourth or Tritone. If we label 9:10 as a 'major Second', and 7:8 as a 'supermajor Second' then they differ by 35:36, the major Second is the inverse of the minor Seventh, and the supermajor Second is the octave inverse of the subminor Seventh. Perhaps 'major' has been left off name of the major Fifth, and minor off the name of the minor Fourth since the time of Smith. We can add to this table the remaining octave inversions as well as the super Fourth and sub Fifth.

Table 1. Additional Intervals

Intervals	Notation	Ratio	Cents in the interval
Major Second	D	9:10	182
Sub Fourth	F-	16:21	471
Supermajor Fourth	F#+	7:10	617
Super Fifth	G+	21:32	729
Subminor Sixth	A♭-	9:14	765
Supermajor Sixth	A+	7:12	933

Helmholtz defined the perfect consonances as the Octave, Twelfth and Double Octave as well as Fourth and Fifth. The major Sixth and major Third are next called medial consonances, considered to in the era of Pythagorean tuning to be imperfect consonances, which Helmholtz defined instead to be the minor Third and the minor Sixth.

Regarding the tuning of the intervals however, those corresponding to simple ratios of vibration are, as in Smith, referred to as perfect, however hey are also described as 'justly-intoned', or by Ellis as 'just'. The perfect tuning for the semitone is listed as 16/15, or 182c. The perfect tunings are compared to the Pythagorean tunings, where the Pythagorean tuning of the major Third and sixth are described as 81/80 above the perfect tunings, and of the minor Third, minor Sixth and semitone to be 81/80 below the perfect tunings. Helmholtz notes that the Pythagorean tunings are closer to the equal tempered tunings than the perfect tunings. Helmholtz also describes the Pythagorean Tritone as of 612c.

Ellis includes an additional table providing names for many different just and tempered intervals, perfect and imperfect. The interval names do not appear to follow any sort of consistent naming system, rather intervals seem to be named case-by-case. They also largely do not correspond to the interval names used by Helmholtz.

In Notes on the Observations of Musical Beats, Proc. R. Soc. Lond. 1880, Ellis named many just intervals of the 7-limit (including 3 and 5-limit intervals):

‘Fifth 3:2, Fourth 4:3, Major Third 5:4, Minor Third 6:5, Major Sixth 5:3, Sub-Fifth 7:5, Super-Fourth 10:7, Super-major Third 9:7, Sub-minor Sixth 14:9, Sub-minor Third 7:6, Super-major Sixth 12:7, Sub-minor or Harmonic Seventh 7:4, Super-major Second 8:7, Major Tone 9:8, Minor Tone 10:9, Small Major Seventh 9:5 and Diatomic (sic.) Semitone 16:15’

He later lists ‘The Major Sevenths 16:9 and 15:8’. The labeling of 16:9 as a Major Seventh and 9:5 as a Small Major Seventh is interesting and at odds with Smith's interval names. Given that 9:5 is larger than 16:9, and no Minor Seventh is mentioned, we can assume 16:9 was mislabeled as a Major Seventh and was understood to be a Minor Seventh, as referred to by Smith.

There is an inconsistency associated with the labeling of 9:5 as a Small Major Seventh also, as it lies a 3:2 Fifth above the 6:5 Minor Third, and we know a fifth and a minor third when added together to give a minor, rather than major seventh.

Ellis here uses the name for 8:7 we suggested above, Super-major Second, and includes our suggested Sub-minor Sixth and Super-major Sixth, however rather than Subminor Fifth and Supermajor Fourth, 7:5 and 10:7 are labelled Sub-Fifth and Super-Fourth, where in this instance sub and super are seen to raise and lower by 21:20 instead of by 36:35.

Ellis, in a footnote to his translation of Helmholtz,'s treatise also provides names for a single 11-limit interval. The interval 22:27, of 355c, introduced by Zalzal, says Ellis was termed a neutral Third by Herr J. P. N. Land originally in Over de Toonladders der Arabische Musiek (On the Scales of Arabic Music) in 1880. An interval a fourth higher than this is mentioned, but a ratio is not given, and it is not named. We can ourselves however find it's ratio as 11:18, and guess it's name to be a neutral Sixth, given that it lies a perfect Fourth above the neutral Third. Following a similar process as in our completion of Helmholtz table above, and assuming that the octave inverse of a neutral Third should be a neutral Sixth we may introduce the following 11-limit intervals that see common use among music theorists and microtonal musicians through to today:

Table 3. 11-limit intervals

Intervals	Notation	Ratio	Cents in the interval
Neutral Second	D♭^	81:88	143
Neutral Second	Dv	11:12	151
Neutral Third	E♭^	9:11	347
Neutral Third	Ev	22:27	355
Neutral Sixth	A♭^	27:44	845
Neutral Sixth	Av	11:18	853
Neutral Seventh	B♭^	6:11	1049
Neutral Seventh	Bv	44:81	1057

Each interval name has two sizes that differ by the comma 242:243. The notation included in the table is from HEWM notation, developed as an extension to the Helmholtz-Ellis use of '+' and '-' by Joe Monzo (http://www.tonalsoft.com/enc/h/hewm.aspx).'^' indicates raising 'v' a lowered of 33/32. In HEWM notation '+' and '-' are refined to mean raising and lowering of 81/80 respectively and '>' and '<' are added instead to indicate raising and lowering of 64/63. Letter names correspond instead of the the Ptolemaic sequence, as in Smith's and Helmholtz' descriptions, but to a Pythagorean tuning of the diatonic scale, where '#' and '♭' and respectively raise and lower the apotome, 2187/2048. HEWM notation is not accompanied by an interval naming system.

Common interval names today

These interval names are used by theorists and microtonal musicians today, though 7:5 and 10:7 are given many different names, today also considered to be an augmented fourth and diminished fifth, lesser septimal tritone and greater septimal tritone, or simply as tritones. The fourth and fifth are today called perfect fourth and perfect fifth, and Smith's major Fourth and minor Fifth referred to as augmented fourth and diminished fifth respectively. As can be seen in Tchaikovsky's A Guide to the Practical Study of Harmony, by the beginning of end of the 19th century the familiar conventions for the naming of intervals were set, wherein

• Seconds, thirds, sixths and sevenths appear in the diatonic in two sizes, the larger labelled 'major' and the smaller, 'minor'.
• Major, when raised by a semitone, becomes 'augmented', and minor, lowered by a semitone, 'diminished'.
• The smaller of the two sizes of fourth and the larger of the two sizes of fifth are labelled 'perfect', along with the unison and octave.
• A perfect interval, when raised a semitone is labelled 'augmented', and when lowered a semitone, 'diminished'.

Fokker/Keenan Extended-diatonic interval-names

Considering the 11-limit otonal chord 4:5:6:7:9:11 a chain of thirds, in addition to the familiar major, minor, subminor, supermajor and neutral thirds, Dave Kennan labelled 5:7 a sub-diminished fifth and 7:11 an augmented fifth. 7:10, the inversion of 5:7, is labelled a diminished. 5:7, therefore, is also an augmented fourth. In terms of sevenths, 4:7 is subminor, 5:9 is minor and 11:6 is neutral.

Super and sub were 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.

From this, Keenan defines a consistent interval naming system, meaning one which obeys diatonic interval arithmetic (In each column, the parenthesised prefix is the one that is implied when there is no prefix). When adding intervals the indexes are added together to give the index of the resulting interval. Keenan also adds corrections for each interval class to the indexes in order to account for inconsistencies that occur within diatonic interval arithmetic when concerning intervals greater than an octave, so that his system, unlike regular diatonic interval names, may be completely consistent.

As can be seen above, sub, super, augmented and diminished have also carried inconsistent meaning historically, where in Keenan's system they always alter intervals by the same amount.

Index	Prefix for unisons, fourths, fifths, octaves	Prefix for seconds, thirds, sixths, sevenths, ninths
-4	double diminished	subdiminished
-3	subdiminished	diminished
-2	diminished	subminor
-1	sub	minor
0	(perfect)	neutral
1	super	(major)
2	augmented	supermajor
3	superaugmented	augmented
4	double augmented	superaugmented

The index values correspond most directly to degrees of 31edo, whose interval names by this method are given in the following table:

Extended-diatonic interval names in 31-tET

31-tET degree	Ratios	Names
0	1:1		unison
1	48:49 44:45 36:35 33:32	(diminished second)	super unison
2	27:28 24:25 20:21	subminor second	(augmented unison)
3	15:16 14:15	minor second
4	11:12 10:11	neutral second
5	9:10 8:9	major second
6	7:8	supermajor second	(diminished third)
7	6:7	(augmented second)	subminor third
8	5:6		minor third
9	9:11		neutral third
10	4:5	(subdiminished fourth)	major third
11	11:14 7:9	(diminished fourth)	supermajor third
12	16:21	subfourth	(augmented third)
13	3:4	perfect fourth
14	8:11	super fourth
15	5:7	augmented fourth	(subdiminished fifth)
16	7:10	(superaugmented fourth)	diminished fifth
17	11:16		sub fifth
18	2:3		perfect fifth
19	21:32	(diminished sixth)	super fifth
20	9:14 7:11	subminor sixth	(augmented fifth)
21	5:8	minor sixth	(superaugmented fifth)
22	11:18	neutral sixth
23	3:5	major sixth
24	7:12	supermajor sixth	(diminished seventh)
25	4:7	(augmented sixth)	subminor seventh
26	9:16 5:9		minor seventh
27	11:20 6:11		neutral seventh
28	8:15		major seventh
29	14:27	(diminished octave)	supermajor seventh
30	18:35	sub octave	(augmented seventh)
31	1:2	octave

The interval names shown in brackets could be said to be 'secondary', the others, 'primary'.

After releasing his system Keenan was informed that is was identical to the extended-diatonic interval-naming scheme of Adriaan Fokker but for the acknowledgment of more 11-limit ratios.This system depends on the tempering out of 81/80, where the diatonic major third, from four stacked fifths, approximates the just major third, 5/4. It also depends on the existence of neutral intervals, i.e., that the perfect fifth or equivalently, the chromatic semitone, subtends an even number of degrees of the ET. To simply to our familiar naming scheme for 12-tET, we observe that it applies to 24edo equally as directly as in 31-tET, where the prefixes correspond to degrees of the edo. Exactly the same is also true for 38-tET, twice 19-tET, a meantone which very closely approximates 1/3-comma meantone. Meantone temperament wherein the fifth is divided into two equally sized neutral thirds is referred to as neutral temperament. Whereas meantone temperament is generated by the fifth, iin neutral temperament the generator is half this interval, the neutral third. Where it was seen above that there are two neutral thirds, 9:11 and 22:27 that differ by 243/242, neutral temperament is at its most simple the temperament defined by this equivalence: the tempering out of 243/242, as meantone is defined by the tempering out of 81/80. The temperament that tempers out both 81/80 and 243/242 is called Mohajira, upon which Keenan's scheme can be said to be based.

The primary interval names resulting in this system's application to 24-tET and 38-tET is now show, along with 31-tET again for easy comparison, where 'M', 'm', 'P', 'N', 'A', 'd', 'S' and 's' are shorthand for major, minor, perfect, neutral, augmented, diminished, super and sub, respectively:

24edo: P1 S1/sm2 m2 N2 M2 SM2/sm3 m3 N3 M3 SM3/s4 P4 S4 A4/d5 s5 P5 S5/sm6 m6 N6 M6 SM6/sm7 m7 N7 M7 SM7/s8 P8

31edo: P1 S1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 S4 A4 d5 s5 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 s8 P8

38edo: P1 S1 A1 sm2 m2 N2 M2 SM2 A2/d3 sm3 m3 N3 M3 SM3 d4 s4 P4 S4 A4 SA4/sd5 d5 s5 P5 S5 A5 sm6 m6 N6 M6 SM6 A6/d7 sm7 m7 N7 M7 SM7 d8 s8 P8

to Meantone tunings that are not Mohajira tunings, the regular diatonic interval names can be applied, but with the addition of double augmented and double diminished:

19edo: P1 A1 m2 M2 A2/d3 m3 M3 d4 P4 A4 d5 P5 A5 m6 M6 A6/d7 m7 d8 P8

26edo: P1 A2 d2 m2 M2 A2 d3 m3 M3 A3 d4 P4 A4 AA4/AA5 d5 P5 A5 d6 m6 M6 A6 d7 m7 M7 A7 d8 P8

Keenan adds further that if it is desired to distinguish between ratios that are in 31-tET approximated by the same number of steps, an addition prefix be added to describe the prime limit of the approximated interval. For 3-limit intervals, the obvious choice is 'Pythagorean', for 5-limit Keenan chooses 'classic', for 7, 'septimal, 11, 'undecimal' and 13, 'tridecimal'. When the highest prime is the same, Keenan suggests adding 'small' and 'large' as final prefixes for this purpose.

In non-Meantone tunings, the two definitions of major third - 4:5 and 64:81, the just (or classic) and Pythagorean major thirds no-longer correspond. If intervals are to receive unique names then to one or both of these major thirds must be added a prefix. Keenan has been involved with the development of both types of systems. Only when the major is defined by it's mapping as fourth fifths, i.e. 81/64, can conserve interval arithmetic.

Sagittal - sagispeak

One system in which 81/64 is the major third, 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. As in HEWM notation, Pythagorean intonation is assumed as a basis. Then the prefixes depart from Pythaogrean intonation, altering by commas and introducing other primes. In place of the prefixes 'sub' and 'super', generally signifying an alteration of 36/35 from 5-limit intervals or 64/63 for 3-limit, Sagittal features an accidental of 64/63, which may be used to take a Pythagorean major interval to a supermajor, minor to subminor, or perfect to super or sub. The prefix 'tao' indicates a decrease of 64/63 and and the prefix 'tai' an increase. Whereas in previous interval naming schemes 'major' and 'minor' were synonymous with the 5-limit tunings, in sagispeak they map instead to Pythagorean. A prefix is needed then to take a Pythagorean intoned interval to a 5-limit tuning. Where 5/4 is 81/80 below the the Pythagorean third, the prefixes 'pai' and 'pao' (where 'p' is for 'pental', as in, involving prime 5), which raise or lower a note by 81/80respectively. Similarly, 'vai' and 'vao', which raise or lower a note by 33/32 respectively, leading to ratios of 11.

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. As in Fokker/Keenan Extended-diatonic Interval-names, 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, a very useful property for a microtonal interval naming system to possess. Another helpful property of sagispeak is its 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. In meantone systems, those we are used to, they correspond, but in most edos they do not. 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.

For comparison, 31edo is shown below in sagispeak:

31edo: P1 tai-1/vai-1 tao-m2 m2 vai-2m/vao-M2 M2 tai-M2 tao-m3 m3 vai-m3/vao-M3 M3 tai-M3 tao-4 P4 vai-4 A4 d5 vao-5 P5 tai-5 tao-m6 m6 vai-m6/vao-M6 M6 tai-M6 tao-m7 m7 vai-m7.vao-M7 M7 tai-M7 vao-8/tao-8 P8

Dave Keenan's most recent system

In 2016 Dave Keenan proposed an alternative generalised microtonal interval naming system for edos. In what might be understood as a generalisation of his extended-diatonic interval-naming system described above onto any equal tuning. Employing as prefixes the familiar 'sub', 'super', 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. Note that the interval just wide of a minor third is labelled a 'supraminor third' rather than a 'superminor third'. This reflect recent tendencies among microtonal musicians and theorists.

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. User:PiotrGrochowski/Extra-Diatonic Intervals — 50edo does this with 50edo. It has a very amazingly excellent solution to the 5/4 and 81/64 problem: The 5/4 is named major third and this notation is split in half, while 81/64 is named high major third. 10/9 and 9/8 are both named major second. PiotrGrochowski (info, talk, contribs).

Size-based systems are completely generalisable, but do not conserve interval arithmetic.

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.

1. Perfect
2. Neutral
3. Major, minor, A4 and d5 and, if the chroma is subtended by a single (positive) step of the edo, other augmented and diminished intervals
4. Super, sub, supra and small prefixes to major, minor, perfect interval and to A4 and d5
5. Intermediates
6. Augmented and diminished intervals (for when the chroma is subtended by more than a single (positive) step of the edo)
7. Super, sub, supra and small prefixes to augmented and diminished intervals
8. 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:

1. Label P1 = 1, P8 = n
2. 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
3. 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)
4. Determine the size of the apotome: A = M-m (for any of degree 2, 3, 6 or 7).
5. The initial chromatic intervals may be labelled:
   • A1 = A
   • d8 = P8 - A
   • A4 = P4 + A
   • d5 = P5 - A
6. If A is even, then intervals half way between M and m within a degree may be labelled N.
7. 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.
8. If sub/super < 0, alterations of 64/63 should not be applied.
9. If small/supra < 0, alterations of 81/80 my be applied only if A = 0, wherein all diatonic intervals are given the label 'N'.
10. 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
11. 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
12. 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'.
13. Remaining augmented (M + A) and diminished (m - A) intervals may be labelled 'A' and 'd'.
14. If desired, augmentation and diminution may be iterated through the continued adding or subtracting of A, labelled by additional 'A' or 'd' prefixes respectively.
15. 'S' and 's' may be further applied as if 'A' were 'M' or 'd' were 'm'.
16. If desired and if n is even, the interval subtended by n/2 steps may be labelled 4-5.
17. 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/P5 P8

3edo: P1 P4 P5 P8

4edo: P1 SM2/sm3 P4/P5 SM6/sm7 P8

5edo: P1/m2 M2/m3 M3/P4 P5/m6 M6/m7 M7/P8

6edo: P1 SM2 sm3/s4 P4/P5 S5/sM6 sm7 P8

7edo: P1 N2 N3 P4 P5 N6 N7 P8

8edo: P1 sM2 M2 P4 S4/s5 P5 m7 Sm7 P8

9edo: P1 M2 M3 m3 P4 P5 M6 m6 m7 P8

10edo: P1/m2 N2 M2/m3 N3 M3/P4 S4/s5 P5/m6 N6 M6/m7 N7 M7/P8

11edo: P1 Sm2 Sm3 N3 sM3 P4 P5 Sm6 N6 sM6 sM7 P8

12edo: P1 m2 M2 m3 M3 P4 A4/d5 P5 m6 M6 m7 M7 P8

13edo: P1 N2 m3 M2 N3 P4 N5 N4 P5 N6 m7 M6 N7 P8

14edo: P1 S1/sm2 N2 SM2/sm3 N3 SM3/s4 P4 SA4/sd5 P5 S5/sm6 N6 SM6/sm7 N7 SM7/s8 P8

15edo: P1/m2 S1/Sm2 sM2 M2/m3 Sm3 sM3 M3/P4 S4 s5 P5/m6 Sm6 sM6 M6/m7 Sm7 sM7/s8 M7/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 S1/sm2 m2 M2 SM2/sm3 m3 M3 SM3/s4 P4 A4 d5 P5 S5/sm6 m6 M6 SM6/sm7 m7 M7/s8 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 S1/sm2 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 SM7/s8 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 S1 m2 Sm2 sM2 M2 SM2/sm3 m3 Sm3 sM3 M3 SM3/s4 P4 S4 d5 A4 s5 P5 S5/sm6 m6 Sm6 sM6 M6 SM6/sm7 m7 Sm7 sM7 M7 s8 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 S1 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 s8 P8

38edo: P1 S1 A1 sm2 m2 N2 M2 SM2 A2/d3 sm3 m3 N3 M3 SM3 d4 s4 P4 N4 A4 SA4/sd5 d5 N5 P5 S5 A5 sm6 m6 N6 M6 SM6 A6/d7 sm7 m7 N7 M7 SM7 d8 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.