User:CompactStar/Ed10/3: Difference between revisions

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=== Division of a thirteenth (e. g. 10/3) into n equal parts ===

Division of e. g. the 10:3 into equal parts can be conceived of as to directly use this interval as an equivalence, or not. The question of [[equivalence]] has not even been posed yet. The utility of 10:3 or another thirteenth as a base though, is apparent by being the top of the upper structure of jazz voicings. Most importantly, a minor thirteenth is the quadruple of a fourth while a major thirteenth is the triple of a fifth, so diatonic scales will not generate prime edXIIIs though these have 1-3-5-7-10 pentads rather than the tone clusters of an equal division of a (perfect) fourth or fifth. Although they no longer count as equivalent, 2-3 and 4-3 are still as valid suspensions of normal root position pentads as 9-10 and 11-10. Many, though not all, of these scales have a pseudo (false) octave, with various degrees of accuracy.

Division of e. g. the 10:3 into equal parts can be conceived of as to directly use this interval as an equivalence, or not. The question of [[equivalence]] has not even been posed yet. The utility of 10:3 or another thirteenth as a base though, is apparent by being the top of the upper structure of jazz voicings and the complete ambitus of three, later five, of the church modes (Dorian G below low D to E above high D, Phrygian A below low E to F above high E and Mixolydian C below low G to A above above high G and later Aeolian D below low A to B(b) above high A and Ionian F below low C to D above high C; it is unknown whether a scale on Bb was within the question before the Baroque period). Most importantly, a minor thirteenth is the quadruple of a fourth while a major thirteenth is the triple of a fifth, so diatonic scales will not generate prime edXIIIs though these have 1-3-5-7-10 pentads rather than the tone clusters of an equal division of a (perfect) fourth or fifth. Although they no longer count as equivalent, 2-3 and 4-3 are still as valid suspensions of normal root position pentads as 9-10 and 11-10. Many, though not all, of these scales have a pseudo (false) octave, with various degrees of accuracy.

Incidentally, one way to treat 10/3 as an equivalence is the use of the 4:5:6:7:10 chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone. Just as in meantone it takes four 3/2 to get to 5/1, tempering out the comma 81/80. So, doing this yields 9, 12, 21 and 33 tone 3MOS. While the notes are rather farther apart, the scheme is uncannily similar to [[Augmented family|augmented]] temperament. "Macro-augmented" might be a practically perfect term for it if it hasn't been named yet.

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 === Division of a thirteenth (e. g. 10/3) into n equal parts ===
-Division of e. g. the 10:3 into equal parts can be conceived of as to directly use this interval as an equivalence, or not. The question of [[equivalence]] has not even been posed yet. The utility of 10:3 or another thirteenth as a base though, is apparent by being the top of the upper structure of jazz voicings. Most importantly, a minor thirteenth is the quadruple of a fourth while a major thirteenth is the triple of a fifth, so diatonic scales will not generate prime edXIIIs though these have 1-3-5-7-10 pentads rather than the tone clusters of an equal division of a (perfect) fourth or fifth. Although they no longer count as equivalent, 2-3 and 4-3 are still as valid suspensions of normal root position pentads as 9-10 and 11-10. Many, though not all, of these scales have a pseudo (false) octave, with various degrees of accuracy.
+Division of e. g. the 10:3 into equal parts can be conceived of as to directly use this interval as an equivalence, or not. The question of [[equivalence]] has not even been posed yet. The utility of 10:3 or another thirteenth as a base though, is apparent by being the top of the upper structure of jazz voicings and the complete ambitus of three, later five, of the church modes (Dorian G below low D to E above high D, Phrygian A below low E to F above high E and Mixolydian C below low G to A above above high G and later Aeolian D below low A to B(b) above high A and Ionian F below low C to D above high C; it is unknown whether a scale on Bb was within the question before the Baroque period). Most importantly, a minor thirteenth is the quadruple of a fourth while a major thirteenth is the triple of a fifth, so diatonic scales will not generate prime edXIIIs though these have 1-3-5-7-10 pentads rather than the tone clusters of an equal division of a (perfect) fourth or fifth. Although they no longer count as equivalent, 2-3 and 4-3 are still as valid suspensions of normal root position pentads as 9-10 and 11-10. Many, though not all, of these scales have a pseudo (false) octave, with various degrees of accuracy.
 Incidentally, one way to treat 10/3 as an equivalence is the use of the 4:5:6:7:10 chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone. Just as in meantone it takes four 3/2 to get to 5/1,  tempering out the comma 81/80. So, doing this yields 9, 12, 21 and 33 tone 3MOS. While the notes are rather farther apart, the scheme is uncannily similar to [[Augmented family|augmented]] temperament. "Macro-augmented" might be a practically perfect term for it if it hasn't been named yet.