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 (♯, ♭ and their multiples) to sharpen and flatten these seven notes by the [[chromatic semitone|augmented unison aka the chromatic semitone]].

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 ~~7-edo and~~ edos of [[sharpness]] 1 ~~or -1~~. For ~~all other~~ edos, this notation causes the notes to run out of order. For example, ~~17-edo~~ would run C Db C# D Eb D# ~~E..~~. One can avoid ~~this~~ by using [[ups and downs notation]], or for certain edos by using half-sharps (see below).

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.

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 '''circle-of-half-fifths notation''' ~~(aka~~ '''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 ~~are useless~~). Not all even-sharpness edos allow this notation. For example, ~~34-edo~~ (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, ~~41-edo~~ (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.

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==

== Edos up to 100 ==

Edos up to 100 are listed in the following tables. The unit (if not stated otherwise) is ''edosteps'' of the corresponding edo which is given in the first column of each row. The tables contain only diatonic edos (i.e. A1 and m2 have edostepspans > 0). The last two columns are the edo's [[Sharpness|pentasharpness and sharpness]] respectively.

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

|-

!Edo

! Edo

!Fifth

! Fifth

!Fifth-detuning abs(¢), rel(%)

! Fifth-detuning abs (¢), rel (%)

!Major

! Major 2nd

2nd

! Minor 2nd

!Minor

! Augmented 1sn

2nd

!Augmented

~~unison~~

|-

![[12edo|12]]

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{| class="wikitable center-all"

|-

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

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

|-

!Edo

! Edo

! Fifth

!Fifth-detuning abs(¢), rel(%)

! Fifth-detuning abs (¢), rel (%)

!Major

! Major 2nd

2nd

! Minor 2nd

! Minor

! Augmented 1sn

2nd

!Augmented

~~unison~~

|-

![[17edo|17]]

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|}

==Expansions ==

== Expansions ==

*[[Syntonic-rastmic subchroma notation]] – built on quartertone notation

* [[Syntonic-rastmic subchroma notation]] – built on quartertone notation

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

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

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

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

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

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

==See also==

== See also ==

*[[Nominal-accidental chain]]

* [[Nominal-accidental chain]]

*[[Circle of fifths]]

* [[Circle of fifths]]

*[[Fifthspan]]

* [[Fifthspan]]

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

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

[[Category:Notation]]

[[Category:Method]]

[[Category:Fifth]]

@@ 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]].
-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 7-edo and edos of [[sharpness]] 1 or -1. For all other edos, this notation causes the notes to run out of order. For example, 17-edo would run C Db C# D Eb D# E... One can avoid this by using [[ups and downs notation]], or for certain edos by using half-sharps (see below).
+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.
 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 '''circle-of-half-fifths notation''' (aka '''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 are useless). Not all even-sharpness edos allow this notation. For example, 34-edo (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, 41-edo (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.
 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==
+== Edos up to 100 ==
 Edos up to 100 are listed in the following tables. The unit (if not stated otherwise) is ''edosteps'' of the corresponding edo which is given in the first column of each row. The tables contain only diatonic edos (i.e. A1 and m2 have edostepspans > 0). The last two columns are the edo's [[Sharpness|pentasharpness and sharpness]] respectively.
@@ Line 16: / Line 16: @@
 |+Diatonic edos fit for circle-of-fifths notation
 |-
-!Edo
+! Edo
-!Fifth
+! Fifth
-!Fifth-detuning <br> abs(¢), rel(%)
+! Fifth-detuning <br> abs (¢), rel (%)
-!Major
+! Major <br> 2nd
-nd
+! Minor <br> 2nd
-!Minor
+! Augmented <br> 1sn
-nd
-!Augmented
-unison
 |-
 ![[12edo|12]]
@@ Line 179: / Line 176: @@
 {| class="wikitable center-all"
 |-
-|+Diatonic edos fit for circle-of-half-fifths notation
+|+Diatonic edos fit for neutral circle-of-fifths notation
 |-
-!Edo
+! Edo
 ! Fifth
-!Fifth-detuning <br> abs(¢), rel(%)
+! Fifth-detuning <br> abs (¢), rel (%)
-!Major
+! Major <br> 2nd
-nd
+! Minor <br> 2nd
-! Minor
+! Augmented <br> 1sn
-nd
-!Augmented
-unison
 |-
 ![[17edo|17]]
@@ Line 264: / Line 258: @@
 |}
-==Expansions ==
+== Expansions ==
-*[[Syntonic-rastmic subchroma notation]] – built on quartertone notation
+* [[Syntonic-rastmic subchroma notation]] – built on quartertone notation
-*[[Ups and downs notation]] – built on circle-of-fifths notation
+* [[Ups and downs notation]] – built on circle-of-fifths notation
-**Neutral ups and downs notation (→ [[Alternative symbols for ups and downs notation]])
+** Neutral ups and downs notation (→ [[Alternative symbols for ups and downs notation]])
-*[[Sagittal notation]] (''evo flavor'') – built on circle-of-fifths notation or quartertone notation
+* [[Sagittal notation]] (''evo flavor'') – built on circle-of-fifths notation or quartertone notation
-==See also==
+== See also ==
-*[[Nominal-accidental chain]]
+* [[Nominal-accidental chain]]
-*[[Circle of fifths]]
+* [[Circle of fifths]]
-*[[Fifthspan]]
+* [[Fifthspan]]
-*[[User:Xenwolf/cofn]] – sortable table with more intervals (all fifths within the interval [4\7, 3\5], the "[[diatonic range]]")
+* [[User:Xenwolf/cofn]] – sortable table with more intervals (all fifths within the interval [4\7, 3\5], the "[[diatonic range]]")
 [[Category:Notation]]
 [[Category:Method]]
 [[Category:Fifth]]