User:Lhearne/Extra-Diatonic Intervals: Difference between revisions
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''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 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 neutral and intermediate interval names; the second includes | 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 === | === Neutrals === | ||
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‘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’. | ‘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. | 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: | 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 | Semaphore[5] 2|2 may be described as | ||
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=== 10edo, Pajara and a problem === | === 10edo, Pajara and a problem === | ||
The primary interval names for 10edo consist of all the neutrals and all the intermediates: | 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: | 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 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: == | == 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 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. | 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. | ||