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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]]


== 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’, while ‘s’ remains shorthand for ‘sub’. 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’ .
 
== Additions and examples: ==
''Neutrals'' and ''intermediates'' are also included, where neutrals occur between the major and minor varieties of generic intervals 2, 3, 6 and 7, the intermediates between each generic interval and the next.
 
Interval names for equal tunings are ranked in four tiers. The first tier includes perfect, neutral and intermediate interval names; the second includes major and minor. The third includes super and sub prefixes to major, minor and perfect intervals. Augmented and diminished are included in the second tier when the chroma is a single step of the tuning, otherwise they occur in the fourth 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.
 
=== Neutrals ===
N2, N3, N6 and N7, i.e. neutral 2nds, 3rds, 6ths and 7ths, falling exactly in-between the major and minor intervals of the same interval class, add native support for neutral-thirds and whitewood temperaments, where the N3 divides the P5 in exact halves.
 
Then Neutral[7] 3|3 can then be written:
 
P1 N2 N3 P4 P5 N6 N7 P8.
 
The same names give the primary interval names for 7edo, whose secondary intervals names are:
 
A1 m2/M2 m3/M3 A4 d5 m6/M6 m7/M7 d8.
 
The secondary interval names show that the chroma is equivalent to a unison in 7edo.
 
Neutral[10] 5|4 may be written as
 
P1 N2 M2 N3 P4 s5 P5 N6 m7 N7 P8
 
Neutral[17] 8|8 may be written as
 
P1 S1 N2 M2 m3 N3 P4 S4 s5 P5 m6 N6 M6 m7 N7 s8 P8,
 
which is almost equivalent to the primary interval names of 17edo,
 
P1 m2 N2 M2 m3 N3 P4 S4 s5 P5 m6 N6 M6 m7 N7 M7 P8,
 
consisting of the neutrals, perfects, majors and minors, as well as S4 and s5.
 
=== Intermediates ===
To provide native support for Semaphore, Pajara and Injera, intermediates are also added to the system.
 
‘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’.
 
‘1-2’ lies exactly half-way between P1 and m2, dividing the m2 in half. It may be read ‘unison-second’ or ‘unicond’. Its octave-inverse, ‘7-8’, lies exactly half-way between M7 and P8 and may be read ‘seventh-octave’ or ‘sevtave’.
 
‘3-4’ lies exactly half-way between M3 and P4, dividing the M6 in half. It may be read ‘third-fourth’ or ‘thourth’. It’s octave-inverse, ‘5-6’, lies exactly half-way between P5 and m6 and may be read ‘fifth-sixth’ or ‘fixths’.
 
5edo can be spelt with the list of only these intermediates:
 
1-2 2-3 3-4 5-6 6-7 7-8.
 
The primary interval names for 5edo give the perfects also as equivalent to some of the intermediates:
 
P1/1-2 2-3 3-4/P4 P5/5-6 6-7 7-8/P8.
 
The secondary interval names for 5edo are as follows:
 
m2 M2/m3 M3 P5 M6/m7 M7.
 
Semaphore[5] 2|2 may be described as
 
P1 2-3 P4 P5 6-7 P8,
 
and Semaphore[9] 4|4 as
 
P1 M2 2-3 3-4 P4 P5 5-6 6-7 m7 P8.
 
The final intermediate, ‘4-5’ lies exactly between P4 and P5 and divides the octave in half. It may be read ‘fourth-fifth’ or ‘firth’.
 
The primary interval names for 12edo are then:
 
P1 m2 M2 m3 M3 P4 4-5 P5 m6 M6 m7 M7 P8.
 
=== 10edo, Pajara and a problem ===
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:
 
P11-2 N2 2-3 N3 3-4/P4 4-5 P5/5-6 N6 6-7 N7 7-8/P8
 
The secondary interval names for 10edo are as follows:
 
m2 Sm2/sM2 M2/m3 Sm3/sM3 M3 S4/s5 m6 Sm6/sM6 M6/m7 Sm7/sM7 M7.
 
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.
 
We know 10edo and 12edo both support Pajara temperament. Pajara[10] 2|2 (2) consists of:
 
P1 Sm2 M2 sM3 P4 4-5 P5 Sm6 m7 sM7 P8,
 
and Pajara[12] 3|2 (2) of
 
P1 Sm2 M2 Sm3 sM3 P4 4-5 P5 Sm6 sM6 m7 sM7 P8.
 
We can see Pajara[10] in 10edo, but in 12edo, wouldn’t sM3 be m3?
 
This is where my system diverges from Igliashon Jones’. We have to break our first rule here, or at least add some conditions to it.
 
== Divergent second scheme: ==
To address this problem of consistency, we now state that when 81/80 is tempered out, M=sM and m=Sm, and when 64/63 is tempered out, M=SM and m=sm. In the case of sm and SM, ‘S’ and ‘s’ raise and lower by 64/63, and in the case of Sm and sM, ‘S’ and ‘s’ raise and lower by 81/80. In this way extra-diatonic interval names are equivalent to Sagispeak interval names, where for sm and SM ‘S’ and ‘s’ are equivalent to ‘tai’ and ‘pao’ and for Sm and sM ‘S’ and ‘s’ are equivalent to ‘pai’ and ‘pao’.
 
It is important to note that given this change, 'S' and 's' may alter an interval by a different number of steps in an edo depending on which interval names they prefix. This may seem confusing, but it seems to reflect existing informal practice.
 
In 12edo, which represents the union between the two, where both 64/63 and 81/80 are tempered out, ‘S’ and ‘s’ do not raise or lower intervals at all. We can now easily see that 12edo supports Pajara, where simply removing all the ‘s’s and ‘S’s from Pajara[12] gives us our primary interval names of 12edo.
 
=== 14edo and Injera ===
The primary interval names for 14edo includes all the neutrals, perfects and intermediates:
 
P1 1-2 N2 2-3 N3 3-4 P4 4-5 P5 5-6 N6 6-7 N7 7-8 P8.
 
The secondary interval names for 14edo are as follows:
 
A1 sm2 m2/M2 SM2/sm3 m3/M3 SM3/s4 A4 SA4/sd5 d5 S5/sm6 m6/M6 SM6/sm7 m7/M7 SM7 d8.
 
We know that 12edo and 14edo support Injera, where Injera[12] 3|2 (2) may be labelled
 
P1 sm2 M2 sm3 SM3 P4 4-5 P5 sm6 M6 m7 SM7 P8,
 
and Injera[14] 3|3 (2) labelled
 
P1 sm2 M2 sm3 m3 SM3 P4 4-5 P5 sm6 M6 SM6 m7 SM7 P8.
 
We can see both in 14edo, and to get 12edo from Injera[12], as with Pajara, we remove all the ‘s’ and ‘S’ prefixes.
 
=== Blacksmith and another (solved) problem ===
10edo also support Blacksmith temperament, and we may think to write Blacksmith[10] 1|0 (5) as:
 
1-2 sM2 2-3 sM3 3-4 s5 5-6 sM6 6-7 sM7 7-8, or
 
P1/m2 sM2 M2/m3 sM3 M3/P4 s5 P5/m6 sM6 M6/m7 sM7 M7/P1.
 
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. ‘s4’ has been used to refer to 21/16, and ‘S5’ to 32/21. Accordingly we add that s5 is lower than P5 by 33/32, that S4 is higher than P4 by 33/32 (acting as Sagispeak’s ‘vao’ or ‘pakao’ and ‘vai’ or ‘pakai’ prefixes), that s4 is lower than P4 by 64/63 and that S5 is higher than P5 by 64/63.
 
Given that Neutral[17] and 17edo, listed above use S1 to imply 33/32, we will define that the comma by which ‘S’ raises P1 and ‘s’ lowers P8 is 33/32.
 
We still need to describe one more interval in Blacksmith[10]. That’s no problem, however: Given that it’s a M2 above sM3, we can call it sA4, leading us to
 
1-2 sM2 2-3 sM3 3-4 sA4 5-6 sM6 6-7 sM7 7-8, or
 
P1/m2 sM2 M2/m3 sM3 M3/P4 sA4 P5/m6 sM6 M6/m7 sM7 M7/P1.
 
We need to add then that ‘S’ and ‘s’ may raise diminished, and lower augmented intervals by 81/80 as they do to minor and major respectively.
 
We will add for consistency also that when ‘S’ raised an augmented interval, or ‘s’ lowers it, the change is by 64/63.
 
Blacksmith[15] 1|1 (5) can be written as all the intermediates, supra minors and sub majors, as well as Sd5 and sA4:
 
1-2 Sm2 sM2 2-3 Sm3 sM3 3-4 S4 Sd5 sA4 5-6 Sm6 sM6 6-7 Sm7 sM7 7-8, or as
 
P1/m2 Sm2 sM2 M2/m3 Sm3 sM3 M3/P4 Sd5 sA4 P5/m6 Sm6 sM6 M6/m7 Sm7 sM7 M7/P1
 
The primary interval names for 15edo include S4 and s5 rather than Sd5 and sA4:
 
1-2 Sm2 sM2 2-3 Sm3 sM3 3-4 S4 s5 5-6 Sm6 sM6 6-7 Sm7 sM7 7-8
 
=== Other rank-2 temperaments' MOS scales ===
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 and Diminished.
 
Augmented[6] 1|0 (3): P1 Sm3 sM3 P5 Sm6 sM7 P8
 
Augmented[9] 1|1 (3): P1 Sm2 Sm3 sM3 P4 P5 Sm6 SM6 sM7 P8
 
Augmented[12] 2|1 (3): P1 Sm2 M2 Sm3 sM3 P4 sA4 P5 Sm6 Sm7 SM6 sM7 P8
 
Augmented[15] 2|2 (3): P1 Sm2 sM2 M2 Sm3 sM3 P4 Sd5 sA4 P5 Sm6 Sm7 SM6 m7 sM7 P8
 
Diminished[8] 1|0 (4): P1 sM2 Sm3 P4 4-5 Sm6 sM6 sM7 P8
 
Diminished[12] 1|1 (4): P1 Sm1 sM2 Sm3 sM3 P4 4-5 P5 Sm6 sM6 Sm7 sM7 P8
 
Porcupine[7] 3|3: P1 sM2 Sm3 P4 P5 sM6 Sm7 P8
 
Porcupine[8] 4|3: P1 sM2 Sm3 P4 s5 P5 sM6 Sm7 P8
 
Porcupine[15] 7|7: P1 S1 sM2 M2 Sm3 sM3 P4 S4 s5 P5 Sm6 sM6 m7 Sm7 s8 P8
 
=== Formal summary ===
We will sum up our definitions and corollary's for the divergent system:
 
'''Definition 1a.''' M and m label the two sizes of 2nd, 3rd, 6th and 7th in the pythagorean diatonic scale.
 
'''Definition 1b.''' The smaller 4th and larger 5th are labelled P.
 
'''Definition 1c.''' The single size of 1 and 8 is labelled P.
 
'''Definition 2.''' A chroma above M or P is A, a chroma below m or P is d.
 
'''Definition 3.''' Within generic interval classes, half way between M and m is N.
 
'''Corollary:''' 7edo can be written P1 N2 N3 P4 P5 N6 N7 N8.
 
'''Definition 4.''' Half way between adjacent generic interval classes lie the intermediates.
 
'''Corollary:''' 5edo can be written 1-2 2-3 3-4 5-6 6-7 7-8.
 
'''Definition 5a.''' 'S', when applied before m, and 's' when applied before M, raise or lower by 81/80 respectively.
 
'''Definition 5b.''' 'S' when applied before M, and 's' when applied before m, raise or lower by 36/35 respectively.
 
'''Definition 5c.''' 'S' when applied to P1 and P4 (leading to S1 and S4) and 's' when applied to P8 and P5 (leading to s8 and and s5) raise and lower by 33/32 respectively.
 
'''Definition 5d.''' 'S' when applied to P5 (leading to S5) and 's' when applied to P4 (leading to s4) raise and lower by 64/63 respectively.
 
=== Further application in edos ===
The primary interval names are shown below for some larger edos:
 
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.
 
22edo: P1 m2 Sm2 sM2 M2 m3 Sm3 sM3 P4 S4 4-5 s5 P5 m6 Sm6 sM6 m7 Sm7 sM7 M7 P8.
 
24edo: P1 1-2 m2 N2 M2 2-3 m3 N3 M3 3-4 P4 S4 4-5 s5 P5 5-6 m6 N6 M6 6-7 m7 N7 M7 7-8 P8.
 
26edo: P1 A2 d2 m2 M2 A2 d3 m3 M3 A3 d4 P4 A4 4-5 d5 P5 A5 d6 m6 M6 A6 d7 m7 M7 A7 d8 P8.
 
27edo: P1 m2 Sm2 N2 sM2 M2 m3 Sm3 N3 sM3 M3 P4 S4 d6 A3 s5 P5 m6 Sm6 N6 sM6 M6 m7 Sm7 N7 sM7 M7 P8.
 
29edo: P1 1-2 m2 Sm2 sM2 M2 2-3 m3 Sm3 sM3 M3 3-4 P4 S4 d5 A4 s5 P5 5-6 m6 Sm6 sM6 M6 6-7 m7 Sm7 sM7 M7 7-8 P8.
 
31edo: P1 S1 sm2 m2 N2 M2 SM2 sm3 m3 N3 M3 SM3 s4 P4 S4 A4 d5 s5 P5 S5 sm6 m6 N6 M6 SM6 sm7 m7 N7 M7 SM7 s8 P8.
 
Larger edos contain unlabeled intervals (without resorting to extended diatonic interval names)
 
== Further divergent scheme ==
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.
 
'''Definition 1a.''' M and m imply two sizes of the generic interval in the Pythagorean diatonic scale (M larger, m smaller),
 
'''Definition 1b.''' P implies a single size. Corollary: Only 1 and 8 are P.
 
'''Lemma:''' P intervals are also M as well as m.
 
'''Definition 2.''' A chroma above M is A, a chroma below m is d.
 
'''Definition 3.''' Within generic interval classes, half way between M and m is N.
 
'''Corollary:''' 7edo can be written N1 N2 N3 N4 N5 N6 N7 N8.
 
'''Definition 4.''' Half way between adjacent generic interval classes lie the intermediates.
 
'''Corollary:''' 5edo can be written 1-2 2-3 3-4 5-6 6-7 7-8.
 
'''Definition 5a.''' 'S', when applied before m, and 's' when applied before M, raise or lower by 81/80 respectively.
 
'''Definition 5b.''' 'S' when applied before M, and 's' when applied before m, raise or lower by 64/63 respectively.
 
'''Corollary:''' Sm1 is 81/80 and sM1 is 64/63. sM8 is 81/40 and sm8 is 63/36.
 
Using this system 34edo and 41edo can be labelled (without resorting to extended diatonic interval names):
 
34edo: P1 1-2 m2 Sm2 N2 sM2 M2 2-3 m3 Sm3 N3 sM3 M3 3-4 m4 Sm4 N4 4-5 N5 sM5 M5 5-6 m6 Sm6 N6 sM6 M6 6-7 m7 Sm7 N7 sM7 M7 7-8 P8.
 
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.
 
== Summary and further options ==
So far the goals in the premise have been achieved. Looking especially at the interval names for edos presented here, we can see they are simple and align with informal interval names. One argument that might be made against this system is that (unlike the interval size indicators), 'S' and 's' mean different things in different places. This complicates interval arithmetic, where an S may not be cancelled by an s, or carry over as would be expected after addition to the diatonic intervals (P, M, m, A and d). However, this is only true for scales or edos where more than one class of 'S' and 's' indicators is present and not represented by the same interval / number of steps. This is not true for any of the above examples, which cover most common tunings. It was never necessary to provide unique labels for edos much larger than 41edo anyway, above which ups and downs / the original premise becomes inconsistent in it's mappings anyway, where, for example, in 72edo, a 9/7, a 64/63 above 81/64, is labelled 'SSM3' instead of 'SM3'. In my scheme the inconsistency manifests in that in 72edo 'S' before M raises by 2 steps, but 'S' before m raises by 1. In this way they remain comparable, but my scheme may be better applied to rank-2 temperaments.
 
If one desires that this sort of complication be avoided, one can substitute s for z (small) and S for L (large) when alterations of 81/80 are desired (although obviously small begins with 's', 's' is already being used as 'sub').
 
Alternatively as a middle ground possibility, one could keep 'S' and 's' for both 81/80 and 64/63, but where for 81/80 the long-form for 's' in this case is small, and for 'S' remains 'supra'. Then we can supposedly have it both ways, where the long-form shows us that they are different, but no short-form qualifiers are added, so these differences can be ignored in almost all scales and edos, but are there when needed. If for some reason you need an interval 64/63 above m3, or you need both a 81/80 and a 64/63 above 4/3, then for indicators 'small' and 'supra' could be written 'sl' and 'SR'.
 
For both of these options N4 can be used like it is in the further divergent scheme to indicated an 11/8, 33/32 above 4/3 (and N5 to indicate 33/32 below 3/2). For both options, as in the further divergent scheme, the ability to name the interval 33/32 is lost while the ability to name the intervals 81/80 and 64/63 is gained. This is because N1, between M1 and m1, is P1 rather than 33/32 above P1.
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