User:Lhearne/Extra-Diatonic Intervals: Difference between revisions
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Regarding the tuning of the intervals however, those corresponding to simple ratios of vibration are, as in Smith, referred to as perfect, however hey are also described as 'justly-intoned', or by Ellis as 'just'. The perfect tuning for the semitone is listed as 16/15, or 182c. The perfect tunings are compared to the Pythagorean tunings, where the Pythagorean tuning of the major Third and sixth are described as 81/80 above the perfect tunings, and of the minor Third, minor Sixth and semitone to be 81/80 below the perfect tunings. Helmholtz notes that the Pythagorean tunings are closer to the equal tempered tunings than the perfect tunings. Helmholtz also describes the Pythagorean Tritone as of 612c. | Regarding the tuning of the intervals however, those corresponding to simple ratios of vibration are, as in Smith, referred to as perfect, however hey are also described as 'justly-intoned', or by Ellis as 'just'. The perfect tuning for the semitone is listed as 16/15, or 182c. The perfect tunings are compared to the Pythagorean tunings, where the Pythagorean tuning of the major Third and sixth are described as 81/80 above the perfect tunings, and of the minor Third, minor Sixth and semitone to be 81/80 below the perfect tunings. Helmholtz notes that the Pythagorean tunings are closer to the equal tempered tunings than the perfect tunings. Helmholtz also describes the Pythagorean Tritone as of 612c. | ||
Ellis includes an [[Ellis' interval table|additional table]] providing names for many different just and tempered intervals, perfect and imperfect. The interval names do not appear to follow any sort of consistent naming system, rather intervals seem to be named case-by-case. They also largely do not correspond to the interval names used by Helmholtz. | |||
The interval names | |||
In ''Notes on the Observations of Musical Beats'', Proc. R. Soc. Lond. 1880, Ellis named many just intervals of the 7-limit (including 3 and 5-limit intervals):<blockquote>‘Fifth 3:2, Fourth 4:3, Major Third 5:4, Minor Third 6:5, Major Sixth 5:3, Sub-Fifth 7:5, Super-Fourth 10:7, Super-major Third 9:7, Sub-minor Sixth 14:9, Sub-minor Third 7:6, Super-major Sixth 12:7, Sub-minor or Harmonic Seventh 7:4, Super-major Second 8:7, Major Tone 9:8, Minor Tone 10:9, Small Major Seventh 9:5 and Diatomic (sic.) Semitone 16:15’</blockquote>He later lists ‘The Major Sevenths 16:9 and 15:8’. The labeling of 16:9 as a Major Seventh and 9:5 as a Small Major Seventh is interesting and at odds with Smith's interval names. Given that 9:5 is larger than 16:9, and no Minor Seventh is mentioned, we can assume 16:9 was mislabeled as a Major Seventh and was understood to be a Minor Seventh, as referred to by Smith. | In ''Notes on the Observations of Musical Beats'', Proc. R. Soc. Lond. 1880, Ellis named many just intervals of the 7-limit (including 3 and 5-limit intervals):<blockquote>‘Fifth 3:2, Fourth 4:3, Major Third 5:4, Minor Third 6:5, Major Sixth 5:3, Sub-Fifth 7:5, Super-Fourth 10:7, Super-major Third 9:7, Sub-minor Sixth 14:9, Sub-minor Third 7:6, Super-major Sixth 12:7, Sub-minor or Harmonic Seventh 7:4, Super-major Second 8:7, Major Tone 9:8, Minor Tone 10:9, Small Major Seventh 9:5 and Diatomic (sic.) Semitone 16:15’</blockquote>He later lists ‘The Major Sevenths 16:9 and 15:8’. The labeling of 16:9 as a Major Seventh and 9:5 as a Small Major Seventh is interesting and at odds with Smith's interval names. Given that 9:5 is larger than 16:9, and no Minor Seventh is mentioned, we can assume 16:9 was mislabeled as a Major Seventh and was understood to be a Minor Seventh, as referred to by Smith. | ||
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Super and sub were further generalised by some theorists and musicians such that an interval a bit smaller than a major is referred to as a ''subminor third'', and an interval a bit larger than a minor third as a ''supraminor third''. | Super and sub were further generalised by some theorists and musicians such that an interval a bit smaller than a major is referred to as a ''subminor third'', and an interval a bit larger than a minor third as a ''supraminor third''. | ||
From this, Keenan defines a consistent interval naming system, meaning one which obeys diatonic interval arithmetic (In each column, the parenthesised prefix is the one that is implied when there is no prefix) | From this, Keenan defines a consistent interval naming system, meaning one which obeys diatonic interval arithmetic (In each column, the parenthesised prefix is the one that is implied when there is no prefix). When adding intervals the indexes are added together to give the index of the resulting interval. Keenan also adds corrections for each interval class to the indexes in order to account for inconsistencies that occur within diatonic interval arithmetic when concerning intervals greater than an octave, so that his system, unlike regular diatonic interval names, may be completely consistent. | ||
As can be seen above, sub, super, augmented and diminished have also carried inconsistent meaning historically, where in Keenan's system they always alter intervals by the same amount. | |||
{| class="wikitable" | {| class="wikitable" | ||
|+ | |+ | ||
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|double augmented | |double augmented | ||
|superaugmented | |superaugmented | ||
|} | |||
The index values correspond most directly to degrees of 31edo, whose interval names by this method are given in the following table: | |||
{| class="wikitable" | |||
|+Extended-diatonic interval names in 31-tET | |||
!31-tET degree | |||
!Ratios | |||
!Names | |||
! | |||
|- | |||
|0 | |||
| | |||
| | |||
| | |||
|- | |||
|1 | |||
| | |||
| | |||
| | |||
|- | |||
|2 | |||
| | |||
| | |||
| | |||
|} | |} | ||