Chain-of-fifths notation: Difference between revisions

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The '''~~circle~~-of-fifths notation''' (aka '''extended Pythagorean notation''') is suitable for a 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 [[12edo]]. It uses the seven natural notes of the [[diatonic]] scale and accidentals (<span style="font-size:larger">♯, ♭</span> and their multiples) to sharpen and flatten these seven notes by the [[chromatic semitone|augmented unison aka the chromatic semitone]].

The '''chain-of-fifths notation''' (aka '''extended Pythagorean notation''') is suitable for a 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 [[12edo]]. It uses the seven natural notes of the [[diatonic]] scale and accidentals (<span style="font-size:larger">♯, ♭</span> and their multiples) to sharpen and flatten these seven notes by the [[chromatic semitone|augmented unison aka the chromatic semitone]]. Any regular rank-2 temperament generated by the 8ve and the 5th (i.e. one with the unsplit [[pergen]]) can be notated this way.

~~Circle~~-of-fifths notation only works for [[Ring number|single-ring]] edos. A counter-example is [[24edo]], which is double-ring. This notation works best for edos of [[sharpness]] 1, and for 7edo, where accidentals have no effects. For any multi-sharpness edos, this notation causes the notes to run out of order. For example, 17edo would run C Db C# D Eb D# E… For negative sharpness edos the accidentals will be inverse. 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 as subsets. For example, 13edo can be notated as a subset of 26edo.

Chain-of-fifths notation only works for [[Ring number|single-ring]] edos. A counter-example is [[24edo]], which is double-ring. This notation works best for edos of [[sharpness]] 1, and for 7edo, where accidentals have no effects. For any multi-sharpness edos, this notation causes the notes to run out of order. For example, 17edo would run C Db C# D Eb D# E… For negative sharpness edos the accidentals will be inverse. 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 as subsets. For example, 13edo can be notated as a subset of 26edo.

~~Any regular~~ rank-2 temperament generated by ~~the 8ve~~ and ~~the 5th (~~i.e. ~~one~~ with the ~~unsplit~~ [[~~pergen~~]]) ~~can be notated~~ this ~~way~~. ~~Because it's rank~~-2, ~~the circle~~ of fifths is ~~actually a~~ (~~theoretically infinite~~) ~~chain of fifths~~.

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, 41edo (sharp-4) has C Ddb Ct Db C# Dd C#t D.

The '''neutral circle-of-fifths notation''' (aka '''circle-of-half-fifths notation''', '''circle-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, 41edo (sharp-4) has C Ddb Ct Db C# Dd C#t D.

Chain-of-third-fifths notation, chain-of-quarter-fifths notation, etc., are theoretical possibilities. In practice, ups and downs are usually used for third-sharps or quarter-sharps.

~~Circle~~-of-third-fifths notation, ~~circle~~-of-quarter-fifths notation, etc., are theoretical possibilities. In practice, ups and downs are usually used for third-sharps or quarter-sharps.

== Edos up to 100 ==

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{| class="wikitable center-all"

|-

|+Diatonic edos fit for ~~circle~~-of-fifths notation

|+Diatonic edos fit for chain-of-fifths notation

|-

! Edo

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{| class="wikitable center-all"

|-

|+Diatonic edos fit for neutral ~~circle~~-of-fifths notation

|+Diatonic edos fit for neutral chain-of-fifths notation

|-

! Edo

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== Expansions ==

* [[Syntonic-rastmic subchroma notation]] – built on neutral ~~circle~~-of-fifths notation

* [[Syntonic-rastmic subchroma notation]] – built on neutral chain-of-fifths notation

* [[Ups and downs notation]] – built on ~~circle~~-of-fifths notation

* [[Ups and downs notation]] – built on chain-of-fifths notation

** Neutral ups and downs notation (→ [[Alternative symbols for ups and downs notation]])

* [[Sagittal notation]] (''evo flavor'') – built on ~~circle~~-of-fifths notation or neutral ~~circle~~-of-fifths notation

* [[Sagittal notation]] (''evo flavor'') – built on chain-of-fifths notation or neutral chain-of-fifths notation

== See also ==

* [[Nominal-accidental chain]]

* [[~~Circle~~ of fifths]]

* [[Chain of fifths]]

* [[Fifthspan]]

* [[User:Xenwolf/cofn]] – sortable table with more intervals (all fifths within the interval [4\7, 3\5], the "[[diatonic range]]")

@@ Line 1: / Line 1: @@
-The '''circle-of-fifths notation''' (aka '''extended Pythagorean notation''') is suitable for a 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 [[12edo]]. It uses the seven natural notes of the [[diatonic]] scale and accidentals (<span style="font-size:larger">♯, ♭</span> and their multiples) to sharpen and flatten these seven notes by the [[chromatic semitone|augmented unison aka the chromatic semitone]].
+The '''chain-of-fifths notation''' (aka '''extended Pythagorean notation''') is suitable for a 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 [[12edo]]. It uses the seven natural notes of the [[diatonic]] scale and accidentals (<span style="font-size:larger">♯, ♭</span> and their multiples) to sharpen and flatten these seven notes by the [[chromatic semitone|augmented unison aka the chromatic semitone]]. Any regular rank-2 temperament generated by the 8ve and the 5th (i.e. one with the unsplit [[pergen]]) can be notated this way.
-Circle-of-fifths notation only works for [[Ring number|single-ring]] edos. A counter-example is [[24edo]], which is double-ring. This notation works best for edos of [[sharpness]] 1, and for 7edo, where accidentals have no effects. For any multi-sharpness edos, this notation causes the notes to run out of order. For example, 17edo would run C Db C# D Eb D# E… For negative sharpness edos the accidentals will be inverse. 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 as subsets. For example, 13edo can be notated as a subset of 26edo.
+Chain-of-fifths notation only works for [[Ring number|single-ring]] edos. A counter-example is [[24edo]], which is double-ring. This notation works best for edos of [[sharpness]] 1, and for 7edo, where accidentals have no effects. For any multi-sharpness edos, this notation causes the notes to run out of order. For example, 17edo would run C Db C# D Eb D# E… For negative sharpness edos the accidentals will be inverse. 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 as subsets. For example, 13edo can be notated as a subset of 26edo.
-Any regular rank-2 temperament generated by the 8ve and the 5th (i.e. one with the unsplit [[pergen]]) can be notated this way. Because it's rank-2, the circle of fifths is actually a (theoretically infinite) chain of fifths.
+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, 41edo (sharp-4) has C Ddb Ct Db C# Dd C#t D.
-The '''neutral circle-of-fifths notation''' (aka '''circle-of-half-fifths notation''', '''circle-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, 41edo (sharp-4) has C Ddb Ct Db C# Dd C#t D.
+Chain-of-third-fifths notation, chain-of-quarter-fifths notation, etc., are theoretical possibilities. In practice, ups and downs are usually used for third-sharps or quarter-sharps.
-Circle-of-third-fifths notation, circle-of-quarter-fifths notation, etc., are theoretical possibilities. In practice, ups and downs are usually used for third-sharps or quarter-sharps.
 == Edos up to 100 ==
@@ Line 14: / Line 12: @@
 {| class="wikitable center-all"
 |-
-|+Diatonic edos fit for circle-of-fifths notation
+|+Diatonic edos fit for chain-of-fifths notation
 |-
 ! Edo
@@ Line 176: / Line 174: @@
 {| class="wikitable center-all"
 |-
-|+Diatonic edos fit for neutral circle-of-fifths notation
+|+Diatonic edos fit for neutral chain-of-fifths notation
 |-
 ! Edo
@@ Line 259: / Line 257: @@
 == Expansions ==
-* [[Syntonic-rastmic subchroma notation]] – built on neutral circle-of-fifths notation
+* [[Syntonic-rastmic subchroma notation]] – built on neutral chain-of-fifths notation
-* [[Ups and downs notation]] – built on circle-of-fifths notation
+* [[Ups and downs notation]] – built on chain-of-fifths notation
 ** Neutral ups and downs notation (→ [[Alternative symbols for ups and downs notation]])
-* [[Sagittal notation]] (''evo flavor'') – built on circle-of-fifths notation or neutral circle-of-fifths notation
+* [[Sagittal notation]] (''evo flavor'') – built on chain-of-fifths notation or neutral chain-of-fifths notation
 == See also ==
 * [[Nominal-accidental chain]]
-* [[Circle of fifths]]
+* [[Chain of fifths]]
 * [[Fifthspan]]
 * [[User:Xenwolf/cofn]] – sortable table with more intervals (all fifths within the interval [4\7, 3\5], the "[[diatonic range]]")