Chain-of-fifths notation: Difference between revisions

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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 (~~~~♯, ♭~~~~ 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.

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 the [[Pythagorean tuning]], the [[meantone]] tunings, and [[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|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.

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.

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 D♭ 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 using [[subset notation]]. For example, 13edo can be notated as a subset of 26edo.

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.

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.

== Accidental symbols ==

Unicode provides symbols for the standard accidentals of circle-of-fifths notation<ref>https://www.w3.org/2021/03/smufl14/tables/standard-accidentals-12-edo.html</ref> and for the Stein-Zimmermann accidentals of neutral circle-of-fifths notation<ref>https://www.w3.org/2021/03/smufl14/tables/stein-zimmermann-accidentals-24-edo.html</ref>. Some fonts may not include all symbols, so fonts designed for musical notation, such as Bravura or Leland<ref>https://www.smufl.org/fonts/</ref>, are recommended.

Note that there are other accidental sets for neutral circle-of-fifths notation, such as Gould arrow quartertone accidentals<ref>https://www.w3.org/2021/03/smufl14/tables/gould-arrow-quartertone-accidentals-24-edo.html</ref>and Persian accidentals<ref>https://www.w3.org/2021/03/smufl14/tables/persian-accidentals.html</ref>, but Stein-Zimmermann accidentals appear to be the most widespread.

In certain circumstances, such as when typing quickly in an online discussion, substitute symbols are often used instead of the regular symbols. In addition, the Xenharmonic Wiki provides [[:Category:character templates|character templates]] to enter these symbols easily in wiki pages. The following table includes the usual equivalences.

{| class="wikitable center-all left-1"

|+ Accidental symbols in (neutral) circle-of-fifths notation

|-

! Style \ Offset

! -2

! -1½

! -1

! -½

! ±0

! +½

! +1

! +1½

! +2

|- style="vertical-align: top;"

| style="vertical-align: middle;" | Standard

| 𝄫 (U+1D12B)

|

| ♭ (U+266D)

|

| ♮ (U+266E)

|

| ♯ (U+266F)

|

| 𝄪 (U+1D12A)

|- style="vertical-align: top;"

| style="vertical-align: middle;" | Standard + Stein-Zimmermann

| {{flat2|150%}} (U+E264)

| {{sesquiflat|150%}} (U+E281)

| {{flat|150%}} (U+E260)

| {{demiflat|150%}} (U+E280)

| {{natural|150%}} (U+E261)

| {{demisharp|150%}} (U+E282)

| {{sharp|150%}} (U+E262)

| {{sesquisharp|150%}} (U+E283)

| {{sharp2|150%}} (U+E263)

|-

| Substitute

| bb

| db

| b

| d

| h

| t

| #

| t#

| x

|-

| Xen Wiki [[:Category:character templates|character templates]]

| <nowiki>{{</nowiki>[[Template:flat2|flat2]]}}

| <nowiki>{{</nowiki>[[Template:sesquiflat|sesquiflat]]}}

| <nowiki>{{</nowiki>[[Template:flat|flat]]}}

| <nowiki>{{</nowiki>[[Template:demiflat|demiflat]]}}

| <nowiki>{{</nowiki>[[Template:natural|natural]]}}

| <nowiki>{{</nowiki>[[Template:demisharp|demisharp]]}}

| <nowiki>{{</nowiki>[[Template:sharp|sharp]]}}

| <nowiki>{{</nowiki>[[Template:sesquisharp|sesquisharp]]}}

| <nowiki>{{</nowiki>[[Template:sharp2|sharp2]]}}

|-

|}

== Edos up to 100 ==

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* [[Fifthspan]]

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

== References ==

[[Category:Notation]]

[[Category:Method]]

[[Category:Fifth]]

@@ Line 1: / Line 1: @@
-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.
+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 the [[Pythagorean tuning]], the [[meantone]] tunings, and [[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|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.
-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.
+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 D♭ 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 using [[subset notation]]. For example, 13edo can be notated as a subset of 26edo.
 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.
 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.
+== Accidental symbols ==
+Unicode provides symbols for the standard accidentals of circle-of-fifths notation<ref>https://www.w3.org/2021/03/smufl14/tables/standard-accidentals-12-edo.html</ref> and for the Stein-Zimmermann accidentals of neutral circle-of-fifths notation<ref>https://www.w3.org/2021/03/smufl14/tables/stein-zimmermann-accidentals-24-edo.html</ref>. Some fonts may not include all symbols, so fonts designed for musical notation, such as Bravura or Leland<ref>https://www.smufl.org/fonts/</ref>, are recommended.
+Note that there are other accidental sets for neutral circle-of-fifths notation, such as Gould arrow quartertone accidentals<ref>https://www.w3.org/2021/03/smufl14/tables/gould-arrow-quartertone-accidentals-24-edo.html</ref>and Persian accidentals<ref>https://www.w3.org/2021/03/smufl14/tables/persian-accidentals.html</ref>, but Stein-Zimmermann accidentals appear to be the most widespread.
+In certain circumstances, such as when typing quickly in an online discussion, substitute symbols are often used instead of the regular symbols. In addition, the Xenharmonic Wiki provides [[:Category:character templates|character templates]] to enter these symbols easily in wiki pages. The following table includes the usual equivalences.
+{| class="wikitable center-all left-1"
+|+ Accidental symbols in (neutral) circle-of-fifths notation
+|-
+! Style \ Offset
+! -2
+! -1½
+! -1
+! -½
+! ±0
+! +½
+! +1
+! +1½
+! +2
+|- style="vertical-align: top;"
+| style="vertical-align: middle;" | Standard
+| 𝄫<br>(U+1D12B)
+|
+| ♭<br>(U+266D)
+|
+| ♮<br>(U+266E)
+|
+| ♯<br>(U+266F)
+|
+| 𝄪<br>(U+1D12A)
+|- style="vertical-align: top;"
+| style="vertical-align: middle;" | Standard + Stein-Zimmermann
+| {{flat2|150%}}<br>(U+E264)
+| {{sesquiflat|150%}}<br>(U+E281)
+| {{flat|150%}}<br>(U+E260)
+| {{demiflat|150%}}<br>(U+E280)
+| {{natural|150%}}<br>(U+E261)
+| {{demisharp|150%}}<br>(U+E282)
+| {{sharp|150%}}<br>(U+E262)
+| {{sesquisharp|150%}}<br>(U+E283)
+| {{sharp2|150%}}<br>(U+E263)
+|-
+| Substitute
+| bb
+| db
+| b
+| d
+| h
+| t
+| #
+| t#
+| x
+|-
+| Xen Wiki [[:Category:character templates|character templates]]
+| <nowiki>{{</nowiki>[[Template:flat2|flat2]]}}
+| <nowiki>{{</nowiki>[[Template:sesquiflat|sesquiflat]]}}
+| <nowiki>{{</nowiki>[[Template:flat|flat]]}}
+| <nowiki>{{</nowiki>[[Template:demiflat|demiflat]]}}
+| <nowiki>{{</nowiki>[[Template:natural|natural]]}}
+| <nowiki>{{</nowiki>[[Template:demisharp|demisharp]]}}
+| <nowiki>{{</nowiki>[[Template:sharp|sharp]]}}
+| <nowiki>{{</nowiki>[[Template:sesquisharp|sesquisharp]]}}
+| <nowiki>{{</nowiki>[[Template:sharp2|sharp2]]}}
+|-
+|}
 == Edos up to 100 ==
@@ Line 267: / Line 334: @@
 * [[Fifthspan]]
 * [[User:Xenwolf/cofn]] – sortable table with more intervals (all fifths within the interval [4\7, 3\5], the "[[diatonic range]]")
+== References ==
+<references/>
 [[Category:Notation]]
 [[Category:Method]]
 [[Category:Fifth]]