Chain-of-fifths notation: Difference between revisions

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#The '''chain-of-fifths notation''', also known as '''extended Pythagorean notation''', is a [[musical notation]] system that supports a variety of [[tuning system]]s which are [[octave]]-repeating and generated by the [[3/2|fifth]] ([[just]] or [[tempered]]). A good number of [[edo]]s and [[regular temperament]]s can be notated this way, as it generalizes the classical notation system for [[Pythagorean tuning]] and [[meantone]] tunings (including [[12edo]]). It uses the seven natural notes of the [[diatonic]] scale (A to G) and accidentals (♯, ♭, and their multiples) to sharpen and flatten these seven notes by the [[chromatic semitone|chromatic semitone]]. Any regular rank-2 temperament generated by the octave and fifth (i.e. one with the unsplit [[pergen]]) can be notated this way. For [[equal divisions of the octave]] in particular, this becomes the familiar ''circle of fifths''.

The '''chain-of-fifths notation''', also known as '''extended Pythagorean notation''', is a [[musical notation]] system that supports a variety of [[tuning system]]s which are [[octave]]-repeating and generated by the [[3/2|fifth]] ([[just]] or [[tempered]]). A good number of [[edo]]s and [[regular temperament]]s can be notated this way, as it generalizes the classical notation system for [[Pythagorean tuning]] and [[meantone]] tunings (including [[12edo]]). It uses the seven natural notes of the [[diatonic]] scale (A to G) and accidentals (♯, ♭, and their multiples) to sharpen and flatten these seven notes by the [[chromatic semitone|chromatic semitone]]. Any regular rank-2 temperament generated by the octave and fifth (i.e. one with the unsplit [[pergen]]) can be notated this way. For [[equal divisions of the octave]] in particular, this becomes the familiar ''circle of fifths''.

Chain-of-fifths notation can cover all notes only in [[Ring number|single-ring]] edos. Some tunings have multiple mutually-exclusive circles of fifths, such as [[24edo]] which has two, and [[36edo]] which has three. This notation works best for edos of [[sharpness]] 1, and for 7edo, where accidentals have no effects. In tunings where sharps raise by multiple steps, notes in the chromatic scale will run out of order. For example, 17edo's chromatic scale would be {{dash|C, D♭, C♯, D, E♭, D♯, E, F, G♭, F♯, G, A♭, G♯, A, B♭, A♯, B, C|s=hair|d=med}}. If the fifth is flatter than 685.714{{cent}}, the order of the sharps and flats will be inverted. One can avoid these by using [[ups and downs notation]], or for certain edos by using half-sharps (see below). Edos whose fifth has a high relative error makes more sense considered as [[dual-fifth]], and notated using [[subset notation]], such as in the case of 13edo, which can be notated as a subset of 26edo. Nonetheless, such tunings may also be notated without resorting to subset notation, and the direct application of the chain-of-fifths notation to a dual-fifth tuning is generally called the '''native fifth notation'''.

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	#The '''chain-of-fifths notation''', also known as '''extended Pythagorean notation''', is a [[musical notation]] system that supports a variety of [[tuning system]]s which are [[octave]]-repeating and generated by the [[3/2\|fifth]] ([[just]] or [[tempered]]). A good number of [[edo]]s and [[regular temperament]]s can be notated this way, as it generalizes the classical notation system for [[Pythagorean tuning]] and [[meantone]] tunings (including [[12edo]]). It uses the seven natural notes of the [[diatonic]] scale (A to G) and accidentals (♯, ♭, and their multiples) to sharpen and flatten these seven notes by the [[chromatic semitone\|chromatic semitone]]. Any regular rank-2 temperament generated by the octave and fifth (i.e. one with the unsplit [[pergen]]) can be notated this way. For [[equal divisions of the octave]] in particular, this becomes the familiar ''circle of fifths''.		The '''chain-of-fifths notation''', also known as '''extended Pythagorean notation''', is a [[musical notation]] system that supports a variety of [[tuning system]]s which are [[octave]]-repeating and generated by the [[3/2\|fifth]] ([[just]] or [[tempered]]). A good number of [[edo]]s and [[regular temperament]]s can be notated this way, as it generalizes the classical notation system for [[Pythagorean tuning]] and [[meantone]] tunings (including [[12edo]]). It uses the seven natural notes of the [[diatonic]] scale (A to G) and accidentals (♯, ♭, and their multiples) to sharpen and flatten these seven notes by the [[chromatic semitone\|chromatic semitone]]. Any regular rank-2 temperament generated by the octave and fifth (i.e. one with the unsplit [[pergen]]) can be notated this way. For [[equal divisions of the octave]] in particular, this becomes the familiar ''circle of fifths''.

	Chain-of-fifths notation can cover all notes only in [[Ring number\|single-ring]] edos. Some tunings have multiple mutually-exclusive circles of fifths, such as [[24edo]] which has two, and [[36edo]] which has three. This notation works best for edos of [[sharpness]] 1, and for 7edo, where accidentals have no effects. In tunings where sharps raise by multiple steps, notes in the chromatic scale will run out of order. For example, 17edo's chromatic scale would be {{dash\|C, D♭, C♯, D, E♭, D♯, E, F, G♭, F♯, G, A♭, G♯, A, B♭, A♯, B, C\|s=hair\|d=med}}. If the fifth is flatter than 685.714{{cent}}, the order of the sharps and flats will be inverted. One can avoid these by using [[ups and downs notation]], or for certain edos by using half-sharps (see below). Edos whose fifth has a high relative error makes more sense considered as [[dual-fifth]], and notated using [[subset notation]], such as in the case of 13edo, which can be notated as a subset of 26edo. Nonetheless, such tunings may also be notated without resorting to subset notation, and the direct application of the chain-of-fifths notation to a dual-fifth tuning is generally called the '''native fifth notation'''.		Chain-of-fifths notation can cover all notes only in [[Ring number\|single-ring]] edos. Some tunings have multiple mutually-exclusive circles of fifths, such as [[24edo]] which has two, and [[36edo]] which has three. This notation works best for edos of [[sharpness]] 1, and for 7edo, where accidentals have no effects. In tunings where sharps raise by multiple steps, notes in the chromatic scale will run out of order. For example, 17edo's chromatic scale would be {{dash\|C, D♭, C♯, D, E♭, D♯, E, F, G♭, F♯, G, A♭, G♯, A, B♭, A♯, B, C\|s=hair\|d=med}}. If the fifth is flatter than 685.714{{cent}}, the order of the sharps and flats will be inverted. One can avoid these by using [[ups and downs notation]], or for certain edos by using half-sharps (see below). Edos whose fifth has a high relative error makes more sense considered as [[dual-fifth]], and notated using [[subset notation]], such as in the case of 13edo, which can be notated as a subset of 26edo. Nonetheless, such tunings may also be notated without resorting to subset notation, and the direct application of the chain-of-fifths notation to a dual-fifth tuning is generally called the '''native fifth notation'''.