User:Lhearne/Extra-Diatonic Intervals: Difference between revisions

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Kilmer also writes that 'the ancient Mesopotamian musicians/“musicologists” knew what we call today the Pythagorean series of fifths, and that the series could be accomplished within a single octave by means of “inversion.” '. The Mesopotamian's music and theory was passed down through the Babylonians and the Assyrians to the Ancient Greeks, as well as their mathematics, particularly concerning musical and acoustical sound ratios (Ibid, [http://math-cs.aut.ac.ir/~shamsi/HoM/Hodgkin%20-%20A%20History%20of%20Mathematics%20From%20Mesopotamia%20to%20Modernity.pdf Hodgekin, 2005]).

Such mathematical and musical ideas are attributed to Pythagoras, who undoubtedly made them popular., although many scholars suggest he may have learned these ideas from his Babylonian and Egyptian mentors. None the less, Pythagoras' idea that that by dividing the length of a string into ratios of halves, thirds, quarters and fifths created the musical intervals of an octave, a perfect fifth, an octave again, and a major third form the basis of Ancient Greek music theory (http://www.historyofmusictheory.com/?page_id=20). His tuning of the diatonic scale by only octaves and perfect fifths (Pythagorean intonation) is influential through to today.

Intervals in Ancient Greek music were written either as frequency ratios, after Pythagoras, or as positions in a tetrachord. Some ratios/intervals were also given names:

2/1, the octave, was named ''diapason'' meaning ''<nowiki/>'''through all [strings]'

3/2, the perfect fifth was labelled ''diapente,'' meaning 'through 5 [strings]'

4/3, the perfect fourth, was labelled ''diatessaron'', meaning 'through 4 [strings]'

''dieses,'' 'sending through', refers to any interval smaller than about 1/3 of a perfect fourth

''tonos'' referred both to the interval of a whole tone, and something more akin to mode or key in the modern sense (Chalmers)

''ditone'' referred to the interval made by stacking two 9/8 whole tones, was referred to as ''tonos'', resulting in 81/64, the Pythagorean major third, as a ''ditone''.

256/243 - the ''limma'', which is the ratio between left over after subtracting two 9/8 tones (together making a ditone) a perfect fourth, the ''diatonic semitone'' of the Pythagorean diatonic scale

2187/2048 - the ''apotome'', which is the ratio between the tone and the limma, the ''chromatic semitone'' of the Pythagorean diatonic scale

The Ancient greek term ''diatonon'', meaning 'through tones', refers to the genus with two whole tones and a semitone, or any genus in which no interval is greater than one half of the fourth (Chlamers). The Pythaogrean diatonic scale is the scale that may be built from one two Pythagorean tetrachords, and the left over interval of 9/8.

=== Our current diatonic interval names ===

2/1, 3/2 and 4/3 were labelled 'perfect' because they were seen to be the perfect consonances. The two sizes of second, third, sixth and seventh in the diatonic scale are labelled ''major'' or ''minor'', the larger sizes of each labelled major. Any perfect or major interval raised by a chromatic semitone (the difference, for e.g. between C and C#) is referred to as 'Augmented' and any minor or perfect interval lowered by a chromatic semitone is referred to as 'diminished'.

=== Common micro-tonal interval names ===

Well-renowned just intonation composer Lou Harrison was fascinated with the interval 7/6, which he called a ''subminor third''. This name caught on and a subminor third is considered amoungst most microtonal musicians and theorists today to be represented by 7/6. in a generalisation of this idea, 9/7 is most commonly reffered to as a ''supermajor third,'' 12/7 a ''supermajor sixth'', 14/9 a ''subminor sixths,'' 8/7 a ''supermajor second,'' 7/4 a ''subminor seventh'', 27/14 a ''supermajor seventh'' and 28/27 a ''subminor second.''

This system was 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''. Notice that 'supra' is used instead of 'super', but 'sub' is still used. Similarly defined are submajor and supraminor seconds, sixths and sevenths.

== Premise: ==

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 Kilmer also writes that 'the ancient Mesopotamian musicians/“musicologists” knew what we call today the Pythagorean series of fifths, and that the series could be accomplished within a single octave by means of “inversion.” '. The Mesopotamian's music and theory was passed down through the Babylonians and the Assyrians to the Ancient Greeks, as well as their mathematics, particularly concerning musical and acoustical sound ratios (Ibid, [http://math-cs.aut.ac.ir/~shamsi/HoM/Hodgkin%20-%20A%20History%20of%20Mathematics%20From%20Mesopotamia%20to%20Modernity.pdf Hodgekin, 2005]).
 Such mathematical and musical ideas are attributed to Pythagoras, who undoubtedly made them popular., although many scholars suggest he may have learned these ideas from his Babylonian and Egyptian mentors. None the less, Pythagoras' idea that that by dividing the length of a string into ratios of halves, thirds, quarters and fifths created the musical intervals of an octave, a perfect fifth, an octave again, and a major third form the basis of Ancient Greek music theory (http://www.historyofmusictheory.com/?page_id=20). His tuning of the diatonic scale by only octaves and perfect fifths (Pythagorean intonation) is influential through to today.
+Intervals in Ancient Greek music were written either as frequency ratios, after Pythagoras, or as positions in a tetrachord. Some ratios/intervals were also given names:
+/1, the octave, was named ''diapason'' meaning ''<nowiki/>'''through all [strings]'
+/2, the perfect fifth was labelled ''diapente,'' meaning 'through 5 [strings]'
+/3, the perfect fourth, was labelled ''diatessaron'', meaning 'through 4 [strings]'
+''dieses,'' 'sending through', refers to any interval smaller than about 1/3 of a perfect fourth
+''tonos'' referred both to the interval of a whole tone, and something more akin to mode or key in the modern sense (Chalmers)
+''ditone'' referred to the interval made by stacking two 9/8 whole tones, was referred to as ''tonos'', resulting in 81/64, the Pythagorean major third, as a ''ditone''.
+/243 - the ''limma'', which is the ratio between left over after subtracting two 9/8 tones (together making a ditone) a perfect fourth, the ''diatonic semitone'' of the Pythagorean diatonic scale
+/2048 - the ''apotome'', which is the ratio between the tone and the limma, the ''chromatic semitone'' of the Pythagorean diatonic scale
+The Ancient greek term ''diatonon'', meaning 'through tones', refers to the genus with two whole tones and a semitone, or any genus in which no interval is greater than one half of the fourth (Chlamers). The Pythaogrean diatonic scale is the scale that may be built from one two Pythagorean tetrachords, and the left over interval of 9/8.
+=== Our current diatonic interval names ===
+/1, 3/2 and 4/3 were labelled 'perfect' because they were seen to be the perfect consonances. The two sizes of second, third, sixth and seventh in the diatonic scale are labelled ''major'' or ''minor'', the larger sizes of each labelled major. Any perfect or major interval raised by a chromatic semitone (the difference, for e.g. between C and C#) is referred to as 'Augmented' and any minor or perfect interval lowered by a chromatic semitone is referred to as 'diminished'.
+=== Common micro-tonal interval names ===
+Well-renowned just intonation composer Lou Harrison was fascinated with the interval 7/6, which he called a ''subminor third''. This name caught on and a subminor third is considered amoungst most microtonal musicians and theorists today to be represented by 7/6. in a generalisation of this idea, 9/7 is most commonly reffered to as a ''supermajor third,'' 12/7 a ''supermajor sixth'', 14/9 a ''subminor sixths,'' 8/7 a ''supermajor second,'' 7/4 a ''subminor seventh'', 27/14 a ''supermajor seventh'' and 28/27 a ''subminor second.''
+This system was 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''. Notice that 'supra' is used instead of 'super', but 'sub' is still used. Similarly defined are submajor and supraminor seconds, sixths and sevenths.
 == Premise: ==