SHEFKHED interval names: Difference between revisions

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Today a small number of competing diatonic-based interval naming schemes exist for the description of microtonal music. After a review of the historical development of Western interval names, and of current proposed schemes, a scheme is developed, taking the best and leaving alone the worst aspects of the existing standards. In addition to the standard diatonic interval name qualifiers - 'M', 'm', 'P', 'A' and 'd', the three most commonly used microtonal qualifies, 'N', 'S' and 's' are used, along with interval-class degrees and the additional qualifiers 'c' and 'C'. Using these ''SHEFKHED interval names'' or ''Smith/Helmholtz/Ellis/Fokker/Keenan/Hearne Extended-diatonic interval names'', almost all small to medium sized [[Equal Temperaments|equal temperaments]] (ETs) can be named such that 'S' and 's' and/or 'C' and 'c' correspond to a displacement of an interval up or down a single degree of the ET, respectively. Many commonly used [[MOS scale|MOS scales]] may also be described using this scheme such that these scales' interval names are expressed consistently in in any tuning that supports them. The scheme, which can also be easily mapped to many of the current interval naming standards, facilitating translation between them, should improve pedagogy and communication in microtonal music.

Today a small number of competing diatonic-based interval naming schemes exist for the description of microtonal music. After a review of the historical development of Western interval names, and of current proposed schemes, a scheme is developed, taking the best and leaving alone the worst aspects of the existing standards. In addition to the standard diatonic interval name qualifiers - 'M', 'm', 'P', 'A' and 'd', the three most commonly used microtonal qualifies, 'N', 'S' and 's' are used, along with interval-class degrees and the additional qualifiers 'c' and 'C'. Using these ''SHEFKHED interval names'' or ''Smith/Helmholtz/Ellis/Fokker/Keenan/Hearne Extended-diatonic interval names'', almost all small to medium sized [[Equal Temperaments|equal temperaments]] (ETs) can be named such that 'S' and 's' and/or 'C' and 'c' correspond to a displacement of an interval up or down a single degree of the ET, respectively. Many commonly used [[MOS scale|MOS scales]] may also be described using this scheme such that these scales' interval names are expressed consistently in in any tuning that supports them. The scheme, which can also be easily mapped to many of the current interval naming standards, facilitating translation between them, should improve pedagogy and communication in microtonal music

==Background: Interval names from antiquity to today==

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15edo: P1 C1/Cm2 cM2 M2/m3 Cm3 cM3 P4 C4 c5 P5 Cm6 cM6 M6/m7 Cm7 cM7/c8 P8

~~20edo, however, is difficult, as in the 13~~-~~limit, it~~'~~s patent val maps only to notes of 10edo~~, ~~so only half the notes are available~~. ~~One option would be to use ups and downs:~~

The remaining 5''n-''edos are difficult, however.

20edo: P1 ^P1/^m2 N2 vM2 M2/m3 ^m3 N3 vM3/v4 P4 ^4 ~~N4/N5~~ v5 P5 ^5/~~vN6 N6~~ vM6 M6/m7 ^m7 N7 vM7/v8 P8

In the 13-limit, 20edo's patent val maps only to notes of 10edo, so only half the notes are available, while the fifths of 10edo are very sharp and 5/4 rather flat, we might wonder if using the less well approximated sharp third might be better. We can test this most simply by finding the 7-odd limit interval (interval consisting of no odd number greater than 7) with the highest error for either mapping. For the patent mapping of 5 and the second best mapping of five, the error associated with the intervals of the 7-odd limit are as follows: (only the intervals in the first half of the octave are included, as the intervals in the top half of a purely tuned octave contain exactly the same error as their octave-inverses.

{| class="wikitable"

|+20edo 7-limit Error

!Interval

!Error patent (degrees)

!Error alternative (degrees)

|-

|4/3

|0.30

|-

|5/4

|0.44

|0.66

|-

|6/5

|0.74

|0.36

|-

|7/5

|0.29

|0.71

|-

|7/6

|0.45

|-

|8/7

|0.15

|}

The second best mapping of 5 is better by this measure, so we may notate 20edo using this mapping: 20c ('c' here is called a wart, indicating the use of the second best approximations of the third prime, 5).

20edo (20c): P1 C1/Cm2 N2 cM2 M2/m3 Cm3 N3 cM3 P4 C4 N4/N5 c5 P5 Cm6 N6 cM6 M6/m7 Cm7 N7 cM7/c8 P8.

In 25edo 5/4 is two degrees below the M3, so the interval in-between does not have a separate function using the patent val in the 7-limit. In 25edo the approximation of 5 is excellent, so we check the second best approximations of 7 and 3.

{| class="wikitable"

|+25edo 7-limit Error

!Interval

!Error 25p (degrees)

!Error 25b (degrees)

!Error 25d (degrees)

|-

|4/3

|0.38

|0.62

|0.38

|-

|5/4

|0.05

|-

|6/5

|0.42

|0.58

|0.42

|-

|7/5

|0.14

|0.76

|-

|7/6

|0.56

|0.44

|-

|8/7

|0.18

|0.82

|}

The patent val, 25p performs best here. We may still use either 25b or 25d if we desire, however if we want to use 25p, we may employ ups and downs to make the intervals that do not carry a seperate function under this mapping:

25edo: P1 ^P1/^m2 Cm2 cM2 vM2 M2/m3 ^m3 Cm3 cM3 vM3/v4 P4 ^4 C4 c5 v5 P5 ^5/^m6 Cm6 cM6 vM6 M6/m7 ^m7 Cm7 cM7 vM7/vP8 P8

In 25edo 81/80 is represented by 2 degrees rather than by a single degree, so our scheme doesn't completely work for 25edo, but our scheme is based on the diatonic scale, which in 25edo has pretty much completely broken down.

~~Another would be to use a different mapping for primes in 20edo; a non-patent val. No-limit TOP error, arguably the simplest way to find the~~ '~~best~~' ~~val for an edo gives~~

=== 7''n''-edos ===

@@ Line 1: / Line 1: @@
-Today a small number of competing diatonic-based interval naming schemes exist for the description of microtonal music. After a review of the historical development of Western interval names, and of current proposed schemes, a scheme is developed, taking the best and leaving alone the worst aspects of the existing standards. In addition to the standard diatonic interval name qualifiers - 'M', 'm', 'P', 'A' and 'd', the three most commonly used microtonal qualifies, 'N', 'S' and 's' are used, along with interval-class degrees and the additional qualifiers 'c' and 'C'. Using these ''SHEFKHED interval names'' or ''Smith/Helmholtz/Ellis/Fokker/Keenan/Hearne Extended-diatonic interval names'', almost all small to medium sized [[Equal Temperaments|equal temperaments]] (ETs) can be named such that 'S' and 's' and/or 'C' and 'c' correspond to a displacement of an interval up or down a single degree of the ET, respectively. Many commonly used [[MOS scale|MOS scales]] may also be described using this scheme such that these scales' interval names are expressed consistently in in any tuning that supports them. The scheme, which can also be easily mapped to many of the current interval naming standards, facilitating translation between them, should improve pedagogy and communication in microtonal music.
+Today a small number of competing diatonic-based interval naming schemes exist for the description of microtonal music. After a review of the historical development of Western interval names, and of current proposed schemes, a scheme is developed, taking the best and leaving alone the worst aspects of the existing standards. In addition to the standard diatonic interval name qualifiers - 'M', 'm', 'P', 'A' and 'd', the three most commonly used microtonal qualifies, 'N', 'S' and 's' are used, along with interval-class degrees and the additional qualifiers 'c' and 'C'. Using these ''SHEFKHED interval names'' or ''Smith/Helmholtz/Ellis/Fokker/Keenan/Hearne Extended-diatonic interval names'', almost all small to medium sized [[Equal Temperaments|equal temperaments]] (ETs) can be named such that 'S' and 's' and/or 'C' and 'c' correspond to a displacement of an interval up or down a single degree of the ET, respectively. Many commonly used [[MOS scale|MOS scales]] may also be described using this scheme such that these scales' interval names are expressed consistently in in any tuning that supports them. The scheme, which can also be easily mapped to many of the current interval naming standards, facilitating translation between them, should improve pedagogy and communication in microtonal music
 ==Background: Interval names from antiquity to today==
@@ Line 1,457: / Line 1,457: @@
 edo: P1 C1/Cm2 cM2 M2/m3 Cm3 cM3 P4 C4 c5 P5 Cm6 cM6 M6/m7 Cm7 cM7/c8 P8
-edo, however, is difficult, as in the 13-limit, it's patent val maps only to notes of 10edo, so only half the notes are available. One option would be to use ups and downs:
+The remaining 5''n-''edos are difficult, however.
-edo: P1 ^P1/^m2 N2 vM2 M2/m3 ^m3 N3 vM3/v4 P4 ^4 N4/N5 v5 P5 ^5/vN6 N6 vM6 M6/m7 ^m7 N7 vM7/v8 P8
+In the 13-limit, 20edo's patent val maps only to notes of 10edo, so only half the notes are available, while the fifths of 10edo are very sharp and 5/4 rather flat, we might wonder if using the less well approximated sharp third might be better. We can test this most simply by finding the 7-odd limit interval (interval consisting of no odd number greater than 7) with the highest error for either mapping. For the patent mapping of 5 and the second best mapping of five, the error associated with the intervals of the 7-odd limit are as follows: (only the intervals in the first half of the octave are included, as the intervals in the top half of a purely tuned octave contain exactly the same error as their octave-inverses.
+{| class="wikitable"
+|+20edo 7-limit Error
+!Interval
+!Error patent (degrees)
+!Error alternative (degrees)
+|-
+|4/3
+|0.30
+|0.30
+|-
+|5/4
+|0.44
+|0.66
+|-
+|6/5
+|0.74
+|0.36
+|-
+|7/5
+|0.29
+|0.71
+|-
+|7/6
+|0.45
+|0.45
+|-
+|8/7
+|0.15
+|0.15
+|}
+The second best mapping of 5 is better by this measure, so we may notate 20edo using this mapping: 20c ('c' here is called a wart, indicating the use of the second best approximations of the third prime, 5).
+edo (20c): P1 C1/Cm2 N2 cM2 M2/m3 Cm3 N3 cM3 P4 C4 N4/N5 c5 P5 Cm6 N6 cM6 M6/m7 Cm7 N7 cM7/c8 P8.
+In 25edo 5/4 is two degrees below the M3, so the interval in-between does not have a separate function using the patent val in the 7-limit. In 25edo the approximation of 5 is excellent, so we check the second best approximations of 7 and 3.
+{| class="wikitable"
+|+25edo 7-limit Error
+!Interval
+!Error 25p (degrees)
+!Error 25b (degrees)
+!Error 25d (degrees)
+|-
+|4/3
+|0.38
+|0.62
+|0.38
+|-
+|5/4
+|0.05
+|0.05
+|0.05
+|-
+|6/5
+|0.42
+|0.58
+|0.42
+|-
+|7/5
+|0.14
+|0.14
+|0.76
+|-
+|7/6
+|0.56
+|0.44
+|0.44
+|-
+|8/7
+|0.18
+|0.18
+|0.82
+|}
+The patent val, 25p performs best here. We may still use either 25b or 25d if we desire, however if we want to use 25p, we may employ ups and downs to make the intervals that do not carry a seperate function under this mapping:
+edo: P1 ^P1/^m2 Cm2 cM2 vM2 M2/m3 ^m3 Cm3 cM3 vM3/v4 P4 ^4 C4 c5 v5 P5 ^5/^m6 Cm6 cM6 vM6 M6/m7 ^m7 Cm7 cM7 vM7/vP8 P8
+In 25edo 81/80 is represented by 2 degrees rather than by a single degree, so our scheme doesn't completely work for 25edo, but our scheme is based on the diatonic scale, which in 25edo has pretty much completely broken down.
-Another would be to use a different mapping for primes in 20edo; a non-patent val. No-limit TOP error, arguably the simplest way to find the 'best' val for an edo gives
+=== 7''n''-edos ===