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''.

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|hair|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 will run out of order. For example, 17edo's notes would be {{dash|C, D♭, C♯, D, E♭, D♯, E, F, G♭, F♯, G, A♭, G♯, A, B♭, A♯, B, C|hair|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'''.

The '''neutral chain-of-fifths notation''' (~~aka~~ '''chain-of-half-fifths notation''', '''chain-of-neutral-thirds notation''', or less accurately, '''quartertone notation''') uses an extended accidental set including '''half-sharps''' and '''half-flats'''. It works for any rank-2 temperament generated by an octave and a neutral third, i.e. those with a [[pergen]] of (P8, P5/2), such as the [[mohaha]] temperament. It also works for certain edos of even sharpness (except sharp-0 edos, in which sharps and flats have no effects). Not all even-sharpness edos allow this notation. For example, 34edo (sharp-4) does not, because its half-fifth is 10\34, and 10 and 34 are not coprime. The GCD is 2, thus there are two rings of half-fifths. In other words, the edo must be [[Ring number #Generalizations|single-ring]] with respect to the half-fifth. All edos with sharpness 2 or −2 qualify. If a qualifying edo's sharpness is not ±2, the notes will run out of order. For example, in 41edo, which is sharp-4, the notes within a (major) whole tone are {{dash|C, D{{sesquiflat2}}, C{{demisharp2}}, D♭, C♯, D{{demiflat2}}, C{{sesquisharp2}}, D|hair|med}}.

The '''neutral chain-of-fifths notation''' (a.k.a. '''chain-of-half-fifths notation''', '''chain-of-neutral-thirds notation''', or less accurately, '''quartertone notation''') uses an extended accidental set including '''half-sharps''' and '''half-flats'''. It works for any rank-2 temperament generated by an octave and a neutral third, i.e. those with a [[pergen]] of (P8, P5/2), such as the [[mohaha]] temperament. It also works for certain edos of even sharpness (except sharp-0 edos, in which sharps and flats have no effects). Not all even-sharpness edos allow this notation. For example, 34edo (sharp-4) does not, because its half-fifth is 10\34, and 10 and 34 are not coprime. The GCD is 2, thus there are two rings of half-fifths. In other words, the edo must be [[Ring number #Generalizations|single-ring]] with respect to the half-fifth. All edos with sharpness 2 or −2 qualify. If a qualifying edo's sharpness is not ±2, the notes will run out of order. For example, in 41edo, which is sharp-4, the notes within a (major) whole tone are {{dash|C, D{{sesquiflat2}}, C{{demisharp2}}, D♭, C♯, D{{demiflat2}}, C{{sesquisharp2}}, D|hair|med}}.

Finer divisions (chain-of-third-fifths, chain-of-quarter-fifths, and beyond) are also theoretical possibilities. In practice, ups and downs are usually used when sharps raise by three or more steps.

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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''.
-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|hair|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 will run out of order. For example, 17edo's notes would be {{dash|C, D♭, C♯, D, E♭, D♯, E, F, G♭, F♯, G, A♭, G♯, A, B♭, A♯, B, C|hair|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'''.
-The '''neutral chain-of-fifths notation''' (aka '''chain-of-half-fifths notation''', '''chain-of-neutral-thirds notation''', or less accurately, '''quartertone notation''') uses an extended accidental set including '''half-sharps''' and '''half-flats'''. It works for any rank-2 temperament generated by an octave and a neutral third, i.e. those with a [[pergen]] of (P8, P5/2), such as the [[mohaha]] temperament. It also works for certain edos of even sharpness (except sharp-0 edos, in which sharps and flats have no effects). Not all even-sharpness edos allow this notation. For example, 34edo (sharp-4) does not, because its half-fifth is 10\34, and 10 and 34 are not coprime. The GCD is 2, thus there are two rings of half-fifths. In other words, the edo must be [[Ring number #Generalizations|single-ring]] with respect to the half-fifth. All edos with sharpness 2 or −2 qualify. If a qualifying edo's sharpness is not ±2, the notes will run out of order. For example, in 41edo, which is sharp-4, the notes within a (major) whole tone are {{dash|C, D{{sesquiflat2}}, C{{demisharp2}}, D♭, C♯, D{{demiflat2}}, C{{sesquisharp2}}, D|hair|med}}.
+The '''neutral chain-of-fifths notation''' (a.k.a. '''chain-of-half-fifths notation''', '''chain-of-neutral-thirds notation''', or less accurately, '''quartertone notation''') uses an extended accidental set including '''half-sharps''' and '''half-flats'''. It works for any rank-2 temperament generated by an octave and a neutral third, i.e. those with a [[pergen]] of (P8, P5/2), such as the [[mohaha]] temperament. It also works for certain edos of even sharpness (except sharp-0 edos, in which sharps and flats have no effects). Not all even-sharpness edos allow this notation. For example, 34edo (sharp-4) does not, because its half-fifth is 10\34, and 10 and 34 are not coprime. The GCD is 2, thus there are two rings of half-fifths. In other words, the edo must be [[Ring number #Generalizations|single-ring]] with respect to the half-fifth. All edos with sharpness 2 or −2 qualify. If a qualifying edo's sharpness is not ±2, the notes will run out of order. For example, in 41edo, which is sharp-4, the notes within a (major) whole tone are {{dash|C, D{{sesquiflat2}}, C{{demisharp2}}, D♭, C♯, D{{demiflat2}}, C{{sesquisharp2}}, D|hair|med}}.
 Finer divisions (chain-of-third-fifths, chain-of-quarter-fifths, and beyond) are also theoretical possibilities. In practice, ups and downs are usually used when sharps raise by three or more steps.