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| 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 each equal division of the octave between up to 29, and several larger commonly used equal temperaments 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 scale|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.
| | <nowiki/>''<nowiki/>''This is now here [[Extended-diatonic interval names]] |
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| == Background ==
| | and here [[SHEFKHED interval names]] |
| [[File:Mesopotamian interval names table.jpg|thumb|500x500px|Mesopotamian interval names, from http://www.historyofmusictheory.com/?page_id=130, accessed October 7, 2018.]]
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| === The origin of diatonic interval names ===
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| Music theory describing the use of hepatonic-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 ([http://www.jstor.org/stable/985853. Kilmer, 2016]). The table to the right following table displays the Ancient Mesopotamian interval names accompanied by their modern names.
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| 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, [http://math-cs.aut.ac.ir/~shamsi/HoM/Hodgkin%20-%20A%20History%20of%20Mathematics%20From%20Mesopotamia%20to%20Modernity.pdf Hodgekin, 2005]).
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| 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 intonation) is influential through to today.
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| === Ancient Greek interval names ===
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| 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:
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| 2/1, the octave, was named ''diapason'' meaning ''<nowiki/>'''through all [strings]'
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| 3/2, the perfect fifth was labelled ''diapente,'' meaning 'through 5 [strings]'
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| 4/3, the perfect fourth, was labelled ''diatessaron'', meaning 'through 4 [strings]'
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| ''dieses,'' 'sending through', refers to any interval smaller than about 1/3 of a perfect fourth
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| ''tonos'' referred both to the interval of a whole tone, and something more akin to mode or key in the modern sense (Chalmers)
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| ''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''.
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| [[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
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| [[2187/2048]] - the ''apotome'', which is the ratio between the tone and the limma, the ''chromatic semitone'' of the Pythagorean diatonic scale
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| 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 (Chlamers). The Pythaogrean diatonic scale is the scale that may be built from one two Pythagorean tetrachords, and the left over interval of 9/8.
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| === Our current diatonic interval names ===
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| 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'.
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| === Common microtonal interval names ===
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| 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 sixths,'' 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.
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| === [[Sagittal notation|Sagittal]] - [http://forum.sagittal.org/viewforum.php?f=9 sagispeak] ===
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| 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 [http://tonalsoft.com/enc/h/hewm.aspx 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.
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| 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 benifits 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.
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| 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.
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| === Dave Keenan's system ===
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| In 2016 Dave Keenan proposed an alternative generalised [http://dkeenan.com/Music/EdoIntervalNames.pdf 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 [[Neutral7|Neutral[7]]] 3|3 using [[Modal UDP Notation|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.
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| 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.
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| === Size based systems ===
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| Microtonal theorist [[Aaron Andrew Hunt]] devised [http://musictheory.zentral.zone/huntsystem4.html the Hunt system], which includes interval name assignments for JI 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).
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| Neo-medieval musicians and early music historian and theorist [[Margo Schulter]] described her own [http://www.bestii.com/~mschulter/IntervalSpectrumRegions.txt 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.
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| 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.
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| <ul><li>[[User:PiotrGrochowski/Extra-Diatonic Intervals]] gives each 43edo interval a name, then maps each desired interval to a 43edo interval. [[User:PiotrGrochowski|PiotrGrochowski]] ([[Editor PiotrGrochowski|info]], [[User talk:PiotrGrochowski|talk]], [[Special:Contributions/PiotrGrochowski|contribs]]) 14:59, 7 October 2018 (UTC)</li></ul>
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| === Ups and Downs ===
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| One final interval naming system, associated with the [[Ups and Downs Notation|ups and downs notation]] system, belonging to microtonal theorist and musicians [[KiteGiedraitis|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. 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.
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| 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 Saggital, 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.
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| == Premise: ==
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| 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. 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' .
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| == Additions and examples: ==
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| ''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.
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| Interval names for equal tunings are ranked in five tiers. The first tier includes perfect and intermediate interval names; the second comprises of the neutrals and the third, major and minor. The fourth includes super and sub prefixes to major, minor and perfect intervals. Augmented and diminished are included in the third tier when the chroma is a single step of the tuning, otherwise they occur in the fifth tier, along with their ‘s’ and ‘S’ prefixes. 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, and ‘secondary’ the second.
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| === Neutrals ===
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| 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 and whitewood temperaments, 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'.
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| Then Neutral[7] 3|3 can then be written:
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| P1 N2 N3 P4 P5 N6 N7 P8.
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| The same names give the primary interval names for 7edo, whose secondary intervals names are:
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| A1 m2/M2 m3/M3 A4 d5 m6/M6 m7/M7 d8.
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| The secondary interval names show that the chroma is equivalent to a unison in 7edo.
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| Extended 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.
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| Neutral[10] 5|4 may then be written as
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| P1 N2 M2 N3 P4 N4 P5 N6 m7 N7 P8
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| Neutral[17] 8|8 may be written as
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| P1 N1 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 N8 P8,
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| which is equivalent to the primary interval names of 17edo.
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| === Intermediates ===
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| To provide native support for Semaphore, Pajara and Injera, intermediates are also added to the system.
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| ‘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 ‘sixths-seventh’ or ‘sinth’.
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| ‘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’.
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| ‘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 ‘fixths’.
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| 5edo can be spelt with the list of only these intermediates:
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| 1-2 2-3 3-4 5-6 6-7 7-8.
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| The primary interval names for 5edo give the perfects also as equivalent to some of the intermediates:
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| P1/1-2 2-3 3-4/P4 P5/5-6 6-7 7-8/P8.
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| The secondary interval names for 5edo are as follows:
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| m2 M2/m3 M3 P5 M6/m7 M7.
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| Semaphore[5] 2|2 may be described as
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| P1 2-3 P4 P5 6-7 P8,
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| and Semaphore[9] 4|4 as
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| P1 M2 2-3 3-4 P4 P5 5-6 6-7 m7 P8.
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| 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’.
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| The primary interval names for 12edo are then:
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| P1 m2 M2 m3 M3 P4 4-5 P5 m6 M6 m7 M7 P8.
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| === 10edo, Pajara and a problem ===
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| The primary interval names for 10edo consist of all the neutrals and all the intermediates with all the perfects as alternatives for some of the intermediates:
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| P1/1-2 N2 2-3 N3 3-4/P4 4-5 P5/5-6 N6 6-7 N7 7-8/P8
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| The secondary interval names for 10edo are as follows:
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| m2 Sm2/sM2 M2/m3 Sm3/sM3 M3 N4/N5 m6 Sm6/sM6 M6/m7 Sm7/sM7 M7.
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| We can see that 10edo supports Neutral thirds scales, given that we can make the interval names for Neutral[10] using the primary and secondary interval names for 10edo.
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| We know 10edo and 12edo both support Pajara temperament. Pajara[10] 2|2 (2) consists of:
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| P1 Sm2 M2 sM3 P4 4-5 P5 Sm6 m7 sM7 P8,
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| and Pajara[12] 3|2 (2) of
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| P1 Sm2 M2 Sm3 sM3 P4 4-5 P5 Sm6 sM6 m7 sM7 P8.
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| We can see Pajara[10] in 10edo, but in 12edo, wouldn’t sM3 be m3?
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| 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.
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| == Divergent second scheme: ==
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| 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 [http://forum.sagittal.org/viewforum.php?f=9 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’.
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| 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.
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| 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.
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| === 14edo and Injera ===
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| The primary interval names for 14edo includes all the neutrals, perfects and intermediates:
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| P1 1-2 N2 2-3 N3 3-4 P4 4-5 P5 5-6 N6 6-7 N7 7-8 P8.
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| The secondary interval names for 14edo are as follows:
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| A1 sm2 m2/M2 SM2/sm3 m3/M3 SM3/s4 A4 SA4/sd5 d5 S5/sm6 m6/M6 SM6/sm7 m7/M7 SM7 d8.
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| We know that 12edo and 14edo support Injera, where Injera[12] 3|2 (2) may be labelled
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| P1 sm2 M2 sm3 SM3 P4 4-5 P5 sm6 M6 m7 SM7 P8,
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| and Injera[14] 3|3 (2) labelled
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| P1 sm2 M2 sm3 m3 SM3 P4 4-5 P5 sm6 M6 SM6 m7 SM7 P8.
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| We can see both in 14edo, and to get 12edo from Injera[12], as with Pajara, we remove all the ‘s’ and ‘S’ prefixes.
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| === Blacksmith and further extension ===
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| 10edo also support Blacksmith temperament, and we may think to write Blacksmith[10] 1|0 (5) as:
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| 1-2 sM2 2-3 sM3 3-4 s5 5-6 sM6 6-7 sM7 7-8, or
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| P1/m2 sM2 M2/m3 sM3 M3/P4 s5 P5/m6 sM6 M6/m7 sM7 M7/P1.
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| 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.
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| | |
| Blacksmith[15] 1|1 (5) can be written as all the intermediates, supras and smalls:
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| 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,
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| These being the primary interval names of 15edo.
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| 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.
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| | |
| === Other rank-2 temperaments' MOS scales ===
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| 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 Augmented, Porcupine, Diminished, Negri, Tetracot and Slendric.
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| Augmented[6] 1|0 (3): P1 Sm3 sM3 P5 Sm6 sM7 P8
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| Augmented[9] 1|1 (3): P1 Sm2 Sm3 sM3 P4 P5 Sm6 SM6 sM7 P8
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| Augmented[12] 2|1 (3): P1 Sm2 M2 Sm3 sM3 P4 sA4 P5 Sm6 Sm7 SM6 sM7 P8
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| Augmented[15] 2|2 (3): P1 Sm2 sM2 M2 Sm3 sM3 P4 Sd5 sA4 P5 Sm6 Sm7 SM6 m7 sM7 P8
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| Diminished[8] 1|0 (4): P1 sM2 Sm3 P4 4-5 Sm6 sM6 sM7 P8
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| Diminished[12] 1|1 (4): P1 Sm2 sM2 Sm3 sM3 P4 4-5 P5 Sm6 sM6 Sm7 sM7 P8
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| Porcupine[7] 3|3: P1 sM2 Sm3 P4 P5 sM6 Sm7 P8
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| Porcupine[8] 4|3: P1 sM2 Sm3 P4 s5 P5 sM6 Sm7 P8
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| Porcupine[15] 7|7: P1 S1 sM2 M2 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 m7 Sm7 s8 P8
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| Negri[9] 4|4: P1 Sm2 2-3 sM3 P4 P5 Sm6 6-7 sM7 P8
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| Negri[10] 4|4: P1 Sm2 2-3 sM3 P4 A4 P5 Sm6 6-7 sM7 P8
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| Tetracot[6] 3|2: P1 sM2 N3 S4 N6 Sm7 P8
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| Tetracot[7] 4|2: P1 sM2 N3 S4 P5 N6 Sm7 P8
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| Tetracot[13] 6|6: P1 N2 sM2 Sm3 N3 P4 S4 s5 P5 N6 sM6 Sm7 N7 P8
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| Slendric[5] 2|3: P1 SM2 s4 S5 sm7 P8
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| Slendric[6] 3|2: P1 SM2 s4 P5 S5 sm7 P8
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| Slendric[11] 5|5: P1 S1 SM2 sm3 s4 P4 P5 S5 SM6 sm7 s8 P8
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| We may also write temperaments with a 9/8 but no 3/2. The most well known of these is Machine:
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| Machine[5] 2|2: P1 M2 M3 m6 m7 P8
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| Machine[6] 3|2: P1 M2 M3 A4 m6 m7 P8
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| Machine[11] 5|5: P1 d3 M2 d4 M3 d5 A4 m6 A5 m7 A6 P8
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| | |
| === Further application in edos ===
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| The primary interval names are shown below for some larger edos:
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| 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
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| 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
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| 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
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| 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
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| 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
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| 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
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| | |
| 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
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| 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
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| 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
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| 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
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| 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
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| | |
| Larger edos contain unlabeled intervals (without resorting to extended diatonic interval names)
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| === Formal summary ===
| |
| We will sum up our definitions and corollary's for the divergent system:
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| '''Definition 1a.''' M and m label the two sizes of 2nd, 3rd, 6th and 7th in the pythagorean diatonic scale.
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| '''Definition 1b.''' The smaller 4th and larger 5th are labelled P.
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| '''Definition 1c.''' The single size of 1 and 8 is labelled P.
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| '''Definition 2.''' A chroma above M or P is A and below m or P is d.
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| '''Definition 3a.''' Within generic interval classes 2, 3, 6 and 7, half way between M and m is N.
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| '''Definition 3b'''. Within generic interval classes 1 and 4, half way between P and A is N.
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| '''Definition 3c.''' Within generic interval classes 5 and 8, half way between P and d is N.
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| '''Corollary:''' 7edo can be written P1 N2 N3 P4 P5 N6 N7 N8.
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| '''Definition 4.''' Half way between adjacent generic interval classes lie the intermediates.
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| '''Corollary:''' 5edo can be written 1-2 2-3 3-4 5-6 6-7 7-8.
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| '''Definition 5a.''' 'S' when applied before M, and 's' when applied before m, raise or lower by 64/63 respectively.
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| '''Definition 5b.''' 'S' when applied before m, and 's' when applied before M, raise or lower by 81/80 respectively.
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| '''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.
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| '''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.
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| '''Definition 5e.''' 'S' when applied to P1 and 's' when applied to P8 may raise and lower by 64/63, or by 81/80.
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| '''Definition 6a.''' When 'S' and 's' imply alterations of 64/63, they have long-form 'super' and 'sub' respectively.
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| '''Definition 6b.''' When 'S' and 's' imply alterations of 81/80, they have long-form 'supra' and 'small' respectively.
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| '''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'.
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| | |
| === 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.
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| Mavila[7] 3|3 can be written
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| P1 M2 m3 P4 P5 M6 m7 P8,
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| the same as the diatonic scale.
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| Mavila[9] 4|4 can be written
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| P1 M2 M3 m3 P4 P5 M6 m6 m7 P8,
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| If we don't have major being below minor, we can hide it with some secondary interval names:
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| P1 M2 Sm3 sM3 P4 P5 Sm6 sM6 m7 P8, arriving at Augmented[9].
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| We might think that the primary interval names of 9edo are the same as our first spelling of Mavila[9] 4|4 above, however we note that the third step is half-way between M2 and m3, and so is primarily a serd. The fourth step, half-way between M3 and P4 is a thourth. If the M3 was a serd, it is also a m2, meaning that the second step is half-way between P1 and m2, and is primarily a unicond. Our primary interval names for 9edo are as follows:
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| P1 1-2 2-3 3-4 P4 P5 5-6 6-7 7-8 P8.
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| Rewriting M2 as Sm2 and m7 as sM7 and putting sM3 and Sm6 back gives us Negri[9].
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| Mavila[16] 8|7 can be writtten
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| P1 A2 M2 m2 M3 m3 d3 P4 A5 P5 A6 M6 m6 M7 m7 d7 P8.
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| The primary interval names for 16edo are the same, but for the inclusion of intermediates:
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| P1 1-2 M2 m2 M3 m3 3-4 P4 4-5 P5 5-6 M6 m6 M7 m7 7-8 P8.
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| Following the same path as in 9edo, we could also write 16edo as:
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| P1 1-2 Sm2 sM2 Sm3 sM3 3-4 P4 4-5 P5 5-6 Sm6 sM6 Sm7 sM7 7-8 P8,
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| wherein we can see that it supports Diminished temperament.
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| Mavila[23]11|11 can be written as:
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| P1 d1 A2 M2 m2 A3 M3 m3 d3 A4 P4 d4 A5 P5 d5 A6 M6 m6 d6 M7 m7 d7 A8 P8
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| The primary interval names for 23edo are the same but for the inclusion of intermediates:
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| P1 d1 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 A8 P8
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| === Father ===
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| 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.
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| The primary interval names for Father[5] 2|2,
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| P1 M2 P4 P5 m7 P8,
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| present no problems.
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| In the primary interval names for Father[8] 4|3, however:
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| P1 m3 M2 P4 M3 P5 m7 M6 P8,
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| 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. If this is a problem, we may use some secondary interval names to address it, i.e.
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| P1 sM2 M2 P4 S4 P5 m7 Sm7 P8.
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| If we also re-write M2 and m7 as Sm3 and sM6, we get Porcupine[8] 4|3.
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| Using some secondary interval names to 'fix' the order leads us to
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| P1 sM2 M2 P4 4-5 P5 m7 M6 P8
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| The primary interval names for Father[13] 6|6:
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| P1 M7 m3 M2 d5 P4 M3 m6 P5 A4 m7 M6 m2 P8,
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| look very unruly.
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| We will fix up the ordering again with secondary interval names:
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| P1 S1 sM2 M2 Sm3 P4 S4 s5 P5 sM6 m7 Sm7 s8 P8,.
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| The primary interval names for 13edo are similar:
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| P1 N2 m3 M2 N3 P4 N5 N4 P5 N6 m7 M6 N7 P8.
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| Minimally fixing the order leads us to
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| P1 N2 sM2 M2 N3 P4 S4 s5 P5 N6 m7 Sm7 N7 P8,
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| which we have seen before as Tetracot[13] 6|6.
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| === Miracle, 11edo and 21edo ===
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| 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:
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| Miracle[10] 5|4: P1 Sm2 SM2 N3 s4 sA4 S5 N6 sm7 sM7 P8
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| Miracle[11] 5|5: P1 Sm2 SM2 N3 s4 Sd5 sA4 S5 N6 sm7 sM7 P8
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| We first see S1 and s8 in the 21-note MOS:
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| 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,
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| 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 'SR' for supra. Miracle[21] would then be re-written:
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| P1 S1 SRm2 N2 SM2 sm3 N3 slM3 s4 P4 SRd5 slA4 P5 S5 SRm6 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.
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| Having encountered 11 and 21-note scales now, and haven't not described 11 and 21edo, I will add these here.
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| The fifth in 11edo is too flat even for it to be considered to support Mavila. Let's see what happens:
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| P1 M2 M3 m2 m3 P4 P5 M6 M7 m6 m7 P8.
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| Adding neutrals gives us our primary interval names for 11edo:
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| P1 M2 M3 N3 m3 P4 P5 M6 N6 m6 m7 P8.
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| 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:
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| Slendric[11] 5|5: P1 S1 SM2 sm3 s4 P4 P5 S5 SM6 sm7 s8 P8
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| Miracle[11] 5|5: P1 Sm2 SM2 N3 s4 Sd5 sA4 S5 N6 sm7 sM7 P8
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| 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:
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| P1 Sm2 Sm3 N3 sM3 P4 P5 Sm6 N6 sM6 sM7 P8 as a well-ordered interval name set.
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| 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.
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| 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:
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| P1 S1 sm2 N2 SM2 sm3 N3 SM3 s4 P4 SA4 sd5 P5 S5 sm6 N6 SM6 sm7 N7 SM7 d8 P8.
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| 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.
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| | |
| === 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.
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| | |
| 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:
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| P1 SM2 sm3 P4/4-5/P5 SM6 sm7 P8.
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| Writing s4 and S5 instead of sm3 and SM6 would give us Slendric[6] 3|2.
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| What of Augmented[6] 1|0 (3): P1 Sm3 sM3 P5 Sm6 sM7 P8?
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| 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.
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| 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.
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| | |
| 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.
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| | |
| === 28edo ===
| |
| We encounter a new problem with 28edo. 28edo's best fifth is that of 7edo. It's primary intervals names are as such:
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| | |
| 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
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| | |
| 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. We made realise from this situation that P4 really is a m4, and P5 a M5. An alternative scheme is developed on this premise, which leads to N4 and N5 as the primary interval names for these intervals, and avoids any initial confusion, detailed after the lists immediately below. In the primary interval names for all edos listed our original premise, that 'S' and 's' correspond to alterations of 1 step of the edo up and down respectively.
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| | |
| == Lists of edos and MOS scales ==
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| | |
| === Primary interval names for edos ===
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| 2edo: P1 P4/4-5/P5 P8
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| 3edo: P1 3-4/P4 P5/5-6 P8
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| 4edo: P1 2-3 P4/4-5/P5 6-7 P8
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| 5edo: P1/1-2 2-3 3-4/P4 P5/5-6 6-7 7-8/P8
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| 6edo: P1 SM2 sm3/s4 P4/4-5/P5 S5/sM6 sm7 P8
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| 7edo: P1 N2 N3 P4 P5 N6 N7 P8
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| 8edo: P1 m3 M2 P4 4-5 P5 m7 M6 P8
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| 9edo: P1 1-2 2-3 3-4 P4 P5 5-6 6-7 7-8 P8
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| 10edo: P1/1-2 N2 2-3 N3 3-4/P4 4-5 P5/5-6 N6 6-7 N7 7-8/P8
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| 11edo: P1 M2 M3 N3 m3 P4 P5 M6 N6 m6 m7 P8
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| 12edo: P1 m2 M2 m3 M3 P4 4-5 P5 m6 M6 m7 M7 P8
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| 13edo: P1 N2 m3 M2 N3 P4 N5 N4 P5 N6 m7 M6 N7 P8
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| 14edo: P1 1-2 N2 2-3 N3 3-4 P4 4-5 P5 5-6 N6 6-7 N7 7-8 P8
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| | |
| 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
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| 16edo: P1 1-2 M2 m2 M3 m3 3-4 P4 4-5 P5 5-6 M6 m6 M7 m7 7-8 P8
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| 17edo: P1 N1 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 N8 P8
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| 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
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| | |
| 21edo: P1 S1 sm2 N2 SM2 sm3 N3 SM3 s4 P4 SA4 sd5 P5 S5 sm6 N6 SM6 sm7 N7 SM7 d8 P8
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| | |
| 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
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| | |
| 23edo: P1 d1 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 A8 P8
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| | |
| 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
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| | |
| 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
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| 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
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| 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
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| 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
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| 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
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| 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
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| 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
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| 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
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| 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
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| === Interval names for MOS scales ===
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| Augmented[6] 1|0 (3): P1 Sm3 sM3 P5 Sm6 sM7 P8
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| Augmented[9] 1|1 (3): P1 Sm2 Sm3 sM3 P4 P5 Sm6 SM6 sM7 P8
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| Augmented[12] 2|1 (3): P1 Sm2 M2 Sm3 sM3 P4 sA4 P5 Sm6 Sm7 SM6 sM7 P8
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| Augmented[15] 2|2 (3): P1 Sm2 sM2 M2 Sm3 sM3 P4 Sd5 sA4 P5 Sm6 Sm7 SM6 m7 sM7 P8
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| Blackwood[10] 1|0 (5): P1/1-2 sM2 2-3 sM3 3-4/P4 s5 P5/ 5-6 sM6 6-7 sM7 7-8/P8
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| Blackwood[15] 1|1 (5): 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
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| Diminished[8] 1|0 (4): P1 sM2 Sm3 P4 4-5 Sm6 sM6 sM7 P8
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| Diminished[12] 1|1 (4): P1 Sm2 sM2 Sm3 sM3 P4 4-5 P5 Sm6 sM6 Sm7 sM7 P8
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| Injera[12] 3|2 (2): P1 sm2 M2 sm3 SM3 P4 4-5 P5 sm6 M6 m7 SM7 P8
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| Injera[14] 3|3 (2): P1 sm2 M2 sm3 m3 SM3 P4 4-5 P5 sm6 M6 SM6 m7 SM7 P8
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| Machine[5] 2|2: P1 M2 M3 m6 m7 P8
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| Machine[6] 3|2: P1 M2 M3 A4 m6 m7 P8
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| Machine[11] 5|5: P1 d3 M2 d4 M3 d5 A4 m6 A5 m7 A6 P8
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| Mavila[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8
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| Mavila[9] 4|4: P1 M2 M3 m3 P4 P5 M6 m6 m7 P8
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| Mavila[16] 8|7: P1 A2 M2 m2 M3 m3 d3 P4 A5 P5 A6 M6 m6 M7 m7 d7 P8
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| Meantone[5] 2|2: P1 M2 P4 P5 m7 P8
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| Meantone[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8
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| Meantone[12] 6|5: P1 m2 M2 m3 M3 P4 A4 P5 m6 M6 m7 M7 P8
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| Meantone[19] 9|9: P1 A1 m2 M2 A2 m3 M3 d4 P4 A4 d5 P5 A5 m6 M6 d7 m7 M7 d8 P8
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| Miracle[10] 5|4: P1 Sm2 SM2 N3 s4 sA4 S5 N6 sm7 sM7 P8
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| Miracle[11] 5|5: P1 Sm2 SM2 N3 s4 Sd5 sA4 S5 N6 sm7 sM7 P8
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| Negri[9] 4|4: P1 Sm2 2-3 sM3 P4 P5 Sm6 6-7 sM7 P8
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| Negri[10] 4|4: P1 Sm2 2-3 sM3 P4 A4 P5 Sm6 6-7 sM7 P8
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| Neutral[7] 3|3: P1 N2 N3 P4 P5 N6 N7 P8
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| Neutral[10] 5|4: P1 N2 M2 N3 P4 N4 P5 N6 m7 N7 P8
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| Neutral[17] 8|8: P1 N1 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 N8 P8
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| Pajara[10] 2|2 (2): P1 Sm2 M2 sM3 P4 4-5 P5 Sm6 m7 sM7 P8
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| Pajara[12] 3|2 (2): P1 Sm2 M2 Sm3 sM3 P4 4-5 P5 Sm6 sM6 m7 sM7 P8
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| Porcupine[7] 3|3: P1 sM2 Sm3 P4 P5 sM6 Sm7 P8
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| Porcupine[8] 4|3: P1 sM2 Sm3 P4 s5 P5 sM6 Sm7 P8
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| Porcupine[15] 7|7: P1 S1 sM2 M2 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 m7 Sm7 s8 P8
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| Semaphore[5] 2|2: P1 2-3 P4 P5 6-7 P8
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| Semaphore[9] 4|4: P1 M2 2-3 3-4 P4 P5 5-6 6-7 m7 P8
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| Slendric[5] 2|3: P1 SM2 s4 S5 sm7 P8
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| Slendric[6] 3|2: P1 SM2 s4 P5 S5 sm7 P8
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| Slendric[11] 5|5: P1 S1 SM2 sm3 s4 P4 P5 S5 SM6 sm7 s8 P8
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| Superpyth[7] 3|3: P1 M2 m3 P4 P5 M6 m7 P8
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| Superpyth[12] 6|5: P1 m2 M2 m3 M3 P4 A4 P5 m6 M6 m7 M7 P8
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| Superpyth[17] 8|8: P1 m2 A1 M2 m3 d4 M3 P4 d5 A4 P5 m6 A5 M6 m7 d8 M7 P8
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| Tetracot[6] 3|2: P1 sM2 N3 S4 N6 Sm7 P8
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| Tetracot[7] 4|2: P1 sM2 N3 S4 P5 N6 Sm7 P8
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| Tetracot[13] 6|6: P1 N2 sM2 Sm3 N3 P4 S4 s5 P5 N6 sM6 Sm7 N7 P8
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| == Conclusion ==
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| 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. While the intervals of some MOS scales may hold consistent names in an ups and downs based scheme, there are many common scales that cannot be 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.
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| == Further divergent scheme ==
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| In the diatonic scale, major and minor label the smaller and larger 2nds, 3rds, 6ths and 7ths. 4ths and 5ths also come in 2 different sizes, but they are labelled perfect and augmented for 4ths; and diminished and perfect for fifths. A simpler and more extensive, but kinda crazy extra-diatonic interval naming scheme is devised upon using major and minor labels for 4ths and 5ths.
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| '''Definition 1a.''' M and m imply two sizes of the generic interval in the Pythagorean diatonic scale (M larger, m smaller),
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| '''Definition 1b.''' P implies a single size.
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| '''Corollary:''' Only 1 and 8 are P.
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| '''Lemma:''' P intervals are also M as well as m.
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| '''Definition 2.''' A chroma above M is A, a chroma below m is d.
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| '''Definition 3.''' Within generic interval classes, half way between M and m is N.
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| '''Corollary:''' 7edo can be written N1 N2 N3 N4 N5 N6 N7 N8.
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| '''Definition 4.''' Half way between adjacent generic interval classes lie the intermediates.
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| '''Corollary:''' 5edo can be written 1-2 2-3 3-4 5-6 6-7 7-8.
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| '''Definition 5a.''' 'S' when applied before M, and 's' when applied before m, raise or lower by 64/63 respectively, with long-form 'super' and 'sub' respectively.
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| '''Definition 5b.''' 'S', when applied before m, and 's' when applied before M, raise or lower by 81/80 respectively, with long-form 'supra' and 'small' respectively.
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| '''Corollary:''' Sm1 is 81/80 and sM1 is 64/63. sM8 is 81/40 and sm8 is 63/36.
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| '''Definition 6.''' 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'.
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| Using this scheme 41edo can be written:
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| 41edo: P1 SM1 sm2 m2 Sm2 N2 sM2 M2 SM2 sm3 m3 Sm3 N3 sM3 M3 SM3 sm4 m4 Sm4 N4 m5 M4 N5 sM5 M5 SM5 sm6 m6 Sm6 N6 sM6 M6 SM6 sm7 m7 Sm7 N7 sM7 M7 SM7 sm8 P8.
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