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

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== Premise: ==

Extra-diatonic names should be simple, generalisable, widely applicable, backwards compatible with standard diatonic notation and reflecting current informal practice as closely as possible. Extra-diatonic interval names are fifth based; extended from the familiar major, minor and perfect interval names so that diatonic interval arithmetic is conserved. ‘M’, ‘m’, and ‘P’ remain the short-hand for major, minor and perfect. ‘A’ and ‘d’ for Augmented and diminished may also be used in the familiar way. In cases where the chroma (the chromatic semitone, or augmented unison) is represented by multiple steps in the tuning the prefix ‘super’ raises major and perfect intervals by a single step while ‘sub’ lowers minor and perfect intervals, with short-hand ‘S’ and ‘s’. ‘S’ and ‘s’ may also be used to raise minor and lower major intervals respectively, reflecting occasion practice. In this case ‘S’ is short-hand for ‘supra’, ~~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’, 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' .

== 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.

''Neutrals'' and ''intermediates'' are also included, where neutrals occur between opposing sizes of a single generic interval the intermediates between each generic interval and the next.

Interval names for equal tunings are ranked in ~~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.

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.

=== Neutrals ===

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The secondary interval names show that the chroma is equivalent to a unison in 7edo.

~~Neutral[10] 5|4 may be written as~~

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.

P1 N2 M2 N3 P4 s5 P5 N6 m7 N7 P8

Neutral[10] 5|4 may then be written as

P1 N2 M2 N3 P4 N4 P5 N6 m7 N7 P8

Neutral[17] 8|8 may be written as

P1 S1 N2 M2 m3 N3 P4 ~~S4 s5~~ P5 m6 N6 M6 m7 N7 s8 P8,

P1 N1 N2 M2 m3 N3 P4 N4 N5 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,

P1 m2 N2 M2 m3 N3 P4 N4 N5 P5 m6 N6 M6 m7 N7 M7 P8,

consisting of the neutrals, perfects, majors and minors~~, as well as S4 and s5~~.

consisting of the neutrals, perfects, majors and minors.

=== Intermediates ===

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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:

~~P11~~-2 N2 2-3 N3 3-4/P4 4-5 P5/5-6 N6 6-7 N7 7-8/P8

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 10edo are as follows:

m2 Sm2/sM2 M2/m3 Sm3/sM3 M3 S4/s5 m6 Sm6/sM6 M6/m7 Sm7/sM7 M7.

m2 Sm2/sM2 M2/m3 Sm3/sM3 M3 N4/N5 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.

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== 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’.

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’.

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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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~~ ===

=== Blacksmith and further extension ===

10edo also support Blacksmith temperament, and we may think to write Blacksmith[10] 1|0 (5) as:

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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.

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.

~~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~~

Blacksmith[15] 1|1 (5) can be written as all the intermediates, supras minors and smalls:

P1/m2 Sm2 sM2 ~~M2/m3~~ Sm3 sM3 M3/P4 ~~Sd5 sA4~~ P5/m6 Sm6 sM6 ~~M6/m7~~ Sm7 sM7 M7/P1

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,

~~The~~ primary interval names ~~for~~ 15edo ~~include S4 and s5 rather than Sd5 and sA4:~~

These being the primary interval names of 15edo.

~~1-2 Sm2 sM2 2-3 Sm3 sM3 3-4 S4 s5 5-6 Sm6 sM6 6-7 Sm7 sM7 7-8~~

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.

=== Other rank-2 temperaments' MOS scales ===

@@ Line 2: / Line 2: @@
 == Premise: ==
-Extra-diatonic names should be simple, generalisable, widely applicable, backwards compatible with standard diatonic notation and reflecting current informal practice as closely as possible. Extra-diatonic interval names are fifth based; extended from the familiar major, minor and perfect interval names so that diatonic interval arithmetic is conserved. ‘M’, ‘m’, and ‘P’ remain the short-hand for major, minor and perfect. ‘A’ and ‘d’ for Augmented and diminished may also be used in the familiar way. In cases where the chroma (the chromatic semitone, or augmented unison) is represented by multiple steps in the tuning the prefix ‘super’ raises major and perfect intervals by a single step while ‘sub’ lowers minor and perfect intervals, with short-hand ‘S’ and ‘s’. ‘S’ and ‘s’ may also be used to raise minor and lower major intervals respectively, reflecting occasion practice. In this case ‘S’ is short-hand for ‘supra’, 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’, 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' .
 == 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.
+''Neutrals'' and ''intermediates'' are also included, where neutrals occur between opposing sizes of a single generic interval the intermediates between each generic interval and the next.
-Interval names for equal tunings are ranked in 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.
+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.
 === Neutrals ===
@@ Line 22: / Line 22: @@
 The secondary interval names show that the chroma is equivalent to a unison in 7edo.
-Neutral[10] 5|4 may be written as
+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.
-P1 N2 M2 N3 P4 s5 P5 N6 m7 N7 P8
+Neutral[10] 5|4 may then be written as
+P1 N2 M2 N3 P4 N4 P5 N6 m7 N7 P8
 Neutral[17] 8|8 may be written as
-P1 S1 N2 M2 m3 N3 P4 S4 s5 P5 m6 N6 M6 m7 N7 s8 P8,
+P1 N1 N2 M2 m3 N3 P4 N4 N5 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,
+P1 m2 N2 M2 m3 N3 P4 N4 N5 P5 m6 N6 M6 m7 N7 M7 P8,
-consisting of the neutrals, perfects, majors and minors, as well as S4 and s5.
+consisting of the neutrals, perfects, majors and minors.
 === Intermediates ===
@@ Line 74: / Line 76: @@
 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
+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 10edo are as follows:
-m2 Sm2/sM2 M2/m3 Sm3/sM3 M3 S4/s5 m6 Sm6/sM6 M6/m7 Sm7/sM7 M7.
+m2 Sm2/sM2 M2/m3 Sm3/sM3 M3 N4/N5 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.
@@ Line 95: / Line 97: @@
 == 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’.
+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’.
 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.
@@ Line 120: / Line 122: @@
 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 ===
+=== Blacksmith and further extension ===
 edo also support Blacksmith temperament, and we may think to write Blacksmith[10] 1|0 (5) as:
@@ Line 127: / Line 129: @@
 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.
+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.
-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
--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:
--2 Sm2 sM2 2-3 Sm3 sM3 3-4 S4 Sd5 sA4 5-6 Sm6 sM6 6-7 Sm7 sM7 7-8, or as
+Blacksmith[15] 1|1 (5) can be written as all the intermediates, supras minors and smalls:
-P1/m2 Sm2 sM2 M2/m3 Sm3 sM3 M3/P4 Sd5 sA4 P5/m6 Sm6 sM6 M6/m7 Sm7 sM7 M7/P1
+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,
-The primary interval names for 15edo include S4 and s5 rather than Sd5 and sA4:
+These being the primary interval names of 15edo.
--2 Sm2 sM2 2-3 Sm3 sM3 3-4 S4 s5 5-6 Sm6 sM6 6-7 Sm7 sM7 7-8
+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.
 === Other rank-2 temperaments' MOS scales ===