Chain-of-fifths notation: Difference between revisions

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The '''circle-of-fifths notation''' (aka '''extended Pythagorean notation''') is suitable to open up the variety of ~~tones of a selection~~ of [[edo]]s and [[regular temperament]]s ~~of fifth generator. The principle is based on one of the intervals taking over the role of the fifth of~~ the traditional classical notation system ~~(in~~ [[~~12edo~~]] or the [[meantone]] ~~tuning)~~. ~~The classical notation system~~ uses seven root notes and accidentals (<span style="font-size:larger">♯, ♭</span> and their multiples) to sharpen and flatten these root notes by the ~~same amount~~ (which is an octave-reduced stack of 7 fifths).

The '''circle-of-fifths notation''' (aka '''extended Pythagorean notation''') is suitable to open up the variety of [[tuning system]]s which are octave repeating and generated by the fifth. A good number of [[edo]]s and [[regular temperament]]s can be notated this way, as it generalizes the traditional classical notation system for the [[Pythagorean tuning]], the [[meantone]] tunings, and later [[12edo]]. It uses seven root notes of the [[diatonic]] scale and accidentals (<span style="font-size:larger">♯, ♭</span> and their multiples) to sharpen and flatten these root notes by the [[chromatic semitone]] (which is an octave-reduced stack of 7 fifths).

Edos that are best supported by this system are those whose fifth does not deviate too much from the pure fifth [[3/2]] (702{{cent}}) and that can be represented by only one ring of fifths. [[24edo]], as a counter-example to this, contains two rings. If we as well demand that whole tones (2 × P5 - P8), diatonic semitones (3 × P8 - 5 × P5), and chromatic semitones (shifts caused by one accidental, 7 × P5 - 4 × P8), use a positive number of steps, we exclude all edos below 12 and also {{EDOs| 13, 16, 18, and 23 }}. They make more sense notated as subsets. For example, 13edo can be notated as a subset of [[26edo]].

To notate edos, one of the intervals must be selected as the fifth. Edos that are best supported by this system are those whose fifth does not deviate too much from the pure fifth [[3/2]] (702{{cent}}) and that can be represented by only one ring of fifths. [[24edo]], as a counter-example to this, contains two rings. If we as well demand that whole tones (2 × P5 - P8), diatonic semitones (3 × P8 - 5 × P5), and chromatic semitones (shifts caused by one accidental, 7 × P5 - 4 × P8), use a positive number of steps, we exclude all edos below 12 and also {{EDOs| 13, 16, 18, and 23 }}. They make more sense notated as subsets. For example, 13edo can be notated as a subset of [[26edo]].

The '''neutral circle-of-fifths notation''' (aka '''quartertone notation''') uses an extended accidental set including '''demisharps''' and '''demiflats'''. It works for any tuning system generated by a neutral third. The [[mohaha]] temperament and its typical edo tunings ([[17edo]], 24edo, [[31edo]], [[38edo]], [[45edo]]) are well represented by this system.

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-The '''circle-of-fifths notation''' (aka '''extended Pythagorean notation''') is suitable to open up the variety of tones of a selection of [[edo]]s and [[regular temperament]]s of fifth generator. The principle is based on one of the intervals taking over the role of the fifth of the traditional classical notation system (in [[12edo]] or the [[meantone]] tuning). The classical notation system uses seven root notes and accidentals (<span style="font-size:larger">♯, ♭</span> and their multiples) to sharpen and flatten these root notes by the same amount (which is an octave-reduced stack of 7 fifths).
+The '''circle-of-fifths notation''' (aka '''extended Pythagorean notation''') is suitable to open up the variety of [[tuning system]]s which are octave repeating and generated by the fifth. A good number of [[edo]]s and [[regular temperament]]s can be notated this way, as it generalizes the traditional classical notation system for the [[Pythagorean tuning]], the [[meantone]] tunings, and later [[12edo]]. It uses seven root notes of the [[diatonic]] scale and accidentals (<span style="font-size:larger">♯, ♭</span> and their multiples) to sharpen and flatten these root notes by the [[chromatic semitone]] (which is an octave-reduced stack of 7 fifths).
-Edos that are best supported by this system are those whose fifth does not deviate too much from the pure fifth [[3/2]] (702{{cent}}) and that can be represented by only one ring of fifths. [[24edo]], as a counter-example to this, contains two rings. If we as well demand that whole tones (2 × P5 - P8), diatonic semitones (3 × P8 - 5 × P5), and chromatic semitones (shifts caused by one accidental, 7 × P5 - 4 × P8), use a positive number of steps, we exclude all edos below 12 and also {{EDOs| 13, 16, 18, and 23 }}. They make more sense notated as subsets. For example, 13edo can be notated as a subset of [[26edo]].
+To notate edos, one of the intervals must be selected as the fifth. Edos that are best supported by this system are those whose fifth does not deviate too much from the pure fifth [[3/2]] (702{{cent}}) and that can be represented by only one ring of fifths. [[24edo]], as a counter-example to this, contains two rings. If we as well demand that whole tones (2 × P5 - P8), diatonic semitones (3 × P8 - 5 × P5), and chromatic semitones (shifts caused by one accidental, 7 × P5 - 4 × P8), use a positive number of steps, we exclude all edos below 12 and also {{EDOs| 13, 16, 18, and 23 }}. They make more sense notated as subsets. For example, 13edo can be notated as a subset of [[26edo]].
 The '''neutral circle-of-fifths notation''' (aka '''quartertone notation''') uses an extended accidental set including '''demisharps''' and '''demiflats'''. It works for any tuning system generated by a neutral third. The [[mohaha]] temperament and its typical edo tunings ([[17edo]], 24edo, [[31edo]], [[38edo]], [[45edo]]) are well represented by this system.