Extended-diatonic interval names: Difference between revisions

Line 443:

This review was both motivated by, and has been integral to, the development of the author's own interval naming scheme. SHEFKHED Interval names, or ''Smith/Helmholtz/Ellis/Fokker/Keenan/Hearne Extended-diatonic interval names'', are essentially an extension of Fokker/Keenan Extended-diatonic interval-names, with a nod to Smith, Helmholtz and Ellis, redesigned with Pythagorean intonation at the core, from Sagittal/Sagispeak, where prefixes correspond to alteration by specific commas, into non-meantone edos, keeping interval arithmetic conserved. 'S' and 's', for 'super' and 'sub' suggest alteration by 64/63, and for seconds, thirds, sixths and sevenths, sub and super intervals remain the same as they have been since Helmholtz/Ellis. Similarly, 'C' and 'c' suggest alteration by 81/80. For perfect intervals 'C' and 'c' are short for 'comma-wide' and 'comma-narrow' respectively, derivative of part of Smith's interval naming scheme, and for all other intervals they are short for 'classic', after Keenan's use of the word. In this way the 5/4 major third in non-meantone system receives a label that is still suggestive of it being a familiar major third.

As in Keenan/Fokker Extended-diatonic interval-names, N lies exactly between M and n, splitting the apotome. 'hA' and 'hd' for 'hemi-augmented' and 'hemi-diminished' are added for similar use from perfect intervals as N from major and minor, e.g. in 31-tET 11/8 is labelled hA4 rather than S4, and 33/32 hA1 rather than S1. 'Intermediates' are also included when the limma, is split, rather than the apotome. Where neutrals are associated with the tempering out of 243/~~243~~, intermediates are associated with the tempering out of 676/675, where 13/15 is equated with half of 4/3, labeled '2-3'. Finally, 'wide' and 'narrow', with short-form 'W' and 'n' fill the role of ups and downs from Ups and Downs, and a similar role to their namesake in Fokker/Keenan Extended-interval names. This scheme conserves interval arithmetic wherever it may be applied, which is to all edos and many, but not all MOS and JI scales, where names for the intervals of scales are conserved across different tunings.

As in Keenan/Fokker Extended-diatonic interval-names, N lies exactly between M and m, splitting the apotome. 'hA' and 'hd' for 'hemi-augmented' and 'hemi-diminished' are added for similar use from perfect intervals as N from major and minor, e.g. in 31-tET 11/8 is labelled hA4 rather than S4, and 33/32 hA1 rather than S1. 'Intermediates' are also included when the limma, is split, rather than the apotome. Where neutrals are associated with the tempering out of 243/242, intermediates are associated with the tempering out of 676/675, where 13/15 is equated with half of 4/3, labeled '2-3'. Finally, 'wide' and 'narrow', with short-form 'W' and 'n' fill the role of ups and downs from Ups and Downs, and a similar role to their namesake in Fokker/Keenan Extended-interval names. This scheme conserves interval arithmetic wherever it may be applied, which is to all edos and many, but not all MOS and JI scales, where names for the intervals of scales are conserved across different tunings.

In addition to the 'primary names' seen below, secondary names can be included which may reveal commas tempered out, and in turn what the diatonic intervals 'sound like', as well as the correspondence between chromatic and enharmonic movement and commatic movement. For example, in 22-tet, listed below, the secondary names for the diatonic intervals show 'S' and 's' subscripts, suggesting the tempering out of 64/63 and providing the information that in 22-tET the 'structural' major third is also, or 'sound like' a super major third, for example. The classic major third is secondarily named as an augmented second, so we know how to chromatically 'find it' / how it derives from the diatonic scale. These equivalences define 7-limit Superpyth temperament.

@@ Line 443: / Line 443: @@
 This review was both motivated by, and has been integral to, the development of the author's own interval naming scheme. SHEFKHED Interval names, or ''Smith/Helmholtz/Ellis/Fokker/Keenan/Hearne Extended-diatonic interval names'', are essentially an extension of Fokker/Keenan Extended-diatonic interval-names, with a nod to Smith, Helmholtz and Ellis, redesigned with Pythagorean intonation at the core, from Sagittal/Sagispeak, where prefixes correspond to alteration by specific commas, into non-meantone edos, keeping interval arithmetic conserved. 'S' and 's', for 'super' and 'sub' suggest alteration by 64/63, and for seconds, thirds, sixths and sevenths, sub and super intervals remain the same as they have been since Helmholtz/Ellis. Similarly, 'C' and 'c' suggest alteration by 81/80. For perfect intervals 'C' and 'c' are short for 'comma-wide' and 'comma-narrow' respectively, derivative of part of Smith's interval naming scheme, and for all other intervals they are short for 'classic', after Keenan's use of the word. In this way the 5/4 major third in non-meantone system receives a label that is still suggestive of it being a familiar major third.
-As in Keenan/Fokker Extended-diatonic interval-names, N lies exactly between M and n, splitting the apotome. 'hA' and 'hd' for 'hemi-augmented' and 'hemi-diminished' are added for similar use from perfect intervals as N from major and minor, e.g. in 31-tET 11/8 is labelled hA4 rather than S4, and 33/32 hA1 rather than S1. 'Intermediates' are also included when the limma, is split, rather than the apotome. Where neutrals are associated with the tempering out of 243/243, intermediates are associated with the tempering out of 676/675, where 13/15 is equated with half of 4/3, labeled '2-3'. Finally, 'wide' and 'narrow', with short-form 'W' and 'n' fill the role of ups and downs from Ups and Downs, and a similar role to their namesake in Fokker/Keenan Extended-interval names. This scheme conserves interval arithmetic wherever it may be applied, which is to all edos and many, but not all MOS and JI scales, where names for the intervals of scales are conserved across different tunings.
+As in Keenan/Fokker Extended-diatonic interval-names, N lies exactly between M and m, splitting the apotome. 'hA' and 'hd' for 'hemi-augmented' and 'hemi-diminished' are added for similar use from perfect intervals as N from major and minor, e.g. in 31-tET 11/8 is labelled hA4 rather than S4, and 33/32 hA1 rather than S1. 'Intermediates' are also included when the limma, is split, rather than the apotome. Where neutrals are associated with the tempering out of 243/242, intermediates are associated with the tempering out of 676/675, where 13/15 is equated with half of 4/3, labeled '2-3'. Finally, 'wide' and 'narrow', with short-form 'W' and 'n' fill the role of ups and downs from Ups and Downs, and a similar role to their namesake in Fokker/Keenan Extended-interval names. This scheme conserves interval arithmetic wherever it may be applied, which is to all edos and many, but not all MOS and JI scales, where names for the intervals of scales are conserved across different tunings.
 In addition to the 'primary names' seen below, secondary names can be included which may reveal commas tempered out, and in turn what the diatonic intervals 'sound like', as well as the correspondence between chromatic and enharmonic movement and commatic movement. For example, in 22-tet, listed below, the secondary names for the diatonic intervals show 'S' and 's' subscripts, suggesting the tempering out of 64/63 and providing the information that in 22-tET the 'structural' major third is also, or 'sound like' a super major third, for example. The classic major third is secondarily named as an augmented second, so we know how to chromatically 'find it' / how it derives from the diatonic scale. These equivalences define 7-limit Superpyth temperament.