User:Lhearne/Extra-Diatonic Intervals: Difference between revisions

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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.

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.

== Premise: ==

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’ .

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: ==

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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.

== 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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-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.
+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.
 == Premise: ==
-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’ .
+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: ==
@@ Line 248: / Line 248: @@
 edo: 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.