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| Extra-diatonic interval names are an attempt to formalise an interval naming system that is seeing occasion informal and undefined use in the description of Xenharmonic music, in an attempt to improve pedagogy and communication. They have thus far been applied to equal temperaments and rank-two temperaments, but should allow further application.This exposition begins with the original premise, after which the original scheme is put forward, before an alternative second scheme built from the first is described. Examples are given along the way.
| | <nowiki/>''<nowiki/>''This is now here [[Extended-diatonic interval names]] |
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| == Premise: ==
| | and here [[SHEFKHED interval names]] |
| 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.
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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 almost equivalent to the primary interval names of 17edo,
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| P1 m2 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 M7 P8,
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| consisting of the neutrals, perfects, majors and minors.
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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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| 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 ===
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| 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 ===
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| 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 ===
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| 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 ===
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| 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 m2 N2 M2 m3 N3 M3 P4 N4 N5 P5 m6 N6 M6 m7 N7 M7 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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| 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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| == 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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