Chain-of-fifths notation: Difference between revisions

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'''.

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|s=hair|d=med}}.

The [https://w3c.github.io/smufl/latest/ Standard Music Font Layout (SMuFL)] specification provides Unicode codepoints for the standard accidentals of chain-of-fifths notation and for the Stein-Zimmermann accidentals of neutral circle-of-fifths notation. Some fonts may not include all symbols, so fonts designed for musical notation, such as Bravura or Leland<ref>[https://www.smufl.org/fonts/ SMuFL | Introducing SMuFL]</ref>, are recommended.

In circumstances where the fonts or codepoints are not quickly accessible, ASCII substitute symbols are 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 these equivalences.

|+ style="font-size: 105%;" | Accidentals in (neutral) circle-of-fifths notation

! Style \ Offset

! &minus;2

! &minus;1&#189;

! &minus;1

! &minus;&#189;

! +&#189;

! +1&#189;

| Half-flat<br />Demiflat<br />Semiflat

| Half-sharp<br />Demisharp<br />Semisharp

| style="vertical-align: middle;" | Standard accidentals<ref>[https://www.w3.org/2021/03/smufl14/tables/standard-accidentals-12-edo.html Standard Music Font Layout | Standard accidentals (12-EDO)]</ref>

| 𝄫<br />(U+1D12B)

| ♭<br />(U+266D)

| ♮<br />(U+266E)

| ♯<br />(U+266F)

| 𝄪<br />(U+1D12A)

| style="vertical-align: middle;" | Standard accidentals<br />+ Stein-Zimmermann accidentals<ref>[https://www.w3.org/2021/03/smufl14/tables/stein-zimmermann-accidentals-24-edo.html Standard Music Font Layout | Stein-Zimmermann accidentals (24-EDO)]</ref>

| {{flat2|200%}}<br />(U+E264)

| {{sesquiflat|200%}}<br />(U+E281)

| {{flat|200%}}<br />(U+E260)

| {{demiflat|200%}}<br />(U+E280)

| {{natural|200%}}<br />(U+E261)

| {{demisharp|200%}}<br />(U+E282)

| {{sharp|200%}}<br />(U+E262)

| {{sesquisharp|200%}}<br />(U+E283)

| {{sharp2|200%}}<br />(U+E263)

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

| {{tl|flat2}}

| {{tl|sesquiflat}}<br />{{tl|sesquiflat2}}

| {{tl|flat}}

| {{tl|demiflat}}<br />{{tl|demiflat2}}

| {{tl|natural}}

| {{tl|demisharp}}<br />{{tl|demisharp2}}

| {{tl|sharp}}

| {{tl|sesquisharp}}<br />{{tl|sesquisharp2}}

| {{tl|sharp2}}

While the Stein-Zimmermann accidentals appear to be the most widespread for neutral circle-of-fifths notation nowadays, and are most likely to be understood by professional musicians, other accidental sets have been developed and used by various musicians.

Note that certain symbols may be very similar or identical to standard or Stein-Zimmermann accidentals despite having different Unicode codepoints.

|+ style="font-size: 105%;" | Alternative accidentals in (neutral) circle-of-fifths notation

! Style \ Offset

! &minus;2

! &minus;1&#189;

! &minus;1

! &minus;&#189;

! +&#189;

! +1&#189;

| style="vertical-align: middle;" | Gould arrow quartertone accidentals<ref>[https://www.w3.org/2021/03/smufl14/tables/gould-arrow-quartertone-accidentals-24-edo.html Standard Music Font Layout | Gould arrow quartertone accidentals (24-EDO)]</ref><ref group="note">Symbols for five-quarter-tones accidentals are also available.</ref>

| {{flat2|200%}}

| {{Bravura||200%}}<br />(U+E271)<br />(U+1D12D)<br />{{Bravura||200%}}<br />(U+E278)

| {{flat|200%}}

| {{Bravura||200%}}<br />(U+E270)<br />(U+1D12C)<br />{{Bravura||200%}}<br />(U+E273)<br />(U+1D12F)

| {{natural|200%}}

| {{Bravura||200%}}<br />(U+E275)<br />(U+1D131)<br />{{Bravura||200%}}<br />(U+E272)<br />(U+1D12E)

| {{sharp|200%}}

| {{Bravura||200%}}<br />(U+E274)<br />(U+1D130)<br />{{Bravura||200%}}<br />(U+E277)

| {{sharp2|200%}}

| style="vertical-align: middle;" | Persian accidentals<ref>[https://www.w3.org/2021/03/smufl14/tables/persian-accidentals.html Standard Music Font Layout | Persian accidentals]</ref>

| {{flat|200%}}

| {{Bravura||200%}}<br />Koron<br />(U+E460)

| {{natural|200%}}

| {{Bravura||200%}}<br />Sori<br />(U+E461)

| {{sharp|200%}}

| style="vertical-align: middle;" | [[Sagittal]] accidentals<ref>[https://www.w3.org/2021/03/smufl14/tables/spartan-sagittal-single-shaft-accidentals.html Standard Music Font Layout | Spartan Sagittal single-shaft accidentals]</ref><ref>[https://www.w3.org/2021/03/smufl14/tables/spartan-sagittal-multi-shaft-accidentals.html Standard Music Font Layout | Spartan Sagittal multi-shaft accidentals]</ref><ref group="note">In mixed Sagittal notation, standard sharps and flats may be used instead of sagittal sharps and flats, and sagittal accidentals may be used to the left of those to alter them. Also, Sagittal notation includes many more accidentals besides those included in the table.</ref>

| {{Bravura||200%}}<br />(U+E335)

| {{Bravura||200%}}<br />(U+E327)

| {{Bravura||200%}}<br />(U+E319)

| {{Bravura||200%}}<br />(U+E30B)

| {{natural|200%}}

| {{Bravura||200%}}<br />(U+E30A)

| {{Bravura||200%}}<br />(U+E318)

| {{Bravura||200%}}<br />(U+E326)

| {{Bravura||200%}}<br />(U+E334)

| style="vertical-align: middle;" | [[Wyschnegradsky]] accidentals<ref>[https://www.w3.org/2021/03/smufl14/tables/wyschnegradsky-accidentals-72-edo.html Standard Music Font Layout | Wyschnegradsky accidentals (72-EDO)]</ref><ref group="note">Wyschnegradsky accidentals also include twelfth-tone ([[72edo]]) accidentals.</ref>

| {{flat2|200%}}

| {{Bravura||200%}}<br />(U+E433)

| {{Bravura||200%}}<br />(U+E430)

| {{Bravura||200%}}<br />(U+E42D)

| {{natural|200%}}

| {{Bravura||200%}}<br />(U+E422)

| {{Bravura||200%}}<br />(U+E425)

| {{Bravura||200%}}<br />(U+E428)

| {{sharp2|200%}}

! Fifth-detuning<br />abs (&#162;), rel (%)

! Major<br />2nd

! Minor<br />2nd

! Augmented<br />1sn

! [[12edo| 12]]

| 7 || &minus;2.0 ( &minus;2.0%) || 2 || 1 || 1

! [[17edo| 17]]

! [[19edo| 19]]

| 11 || &minus;7.2 (&minus;11.4%) || 3 || 2 || 1

! [[22edo| 22]]

! [[26edo| 26]]

| 15 || &minus;9.6 (&minus;20.9%) || 4 || 3 || 1

! [[27edo| 27]]

! [[29edo| 29]]

! [[31edo| 31]]

| 18 || &minus;5.2 (&minus;13.4%) || 5 || 3 || 2

! [[32edo| 32]]

! [[33edo| 33]]

| 19 || &minus;11.0 (&minus;30.4%) || 5 || 4 || 1

! [[37edo| 37]]

! [[39edo| 39]]

! [[40edo| 40]]

| 23 || &minus;12.0 (&minus;39.9%) || 6 || 5 || 1

! [[41edo| 41]]

! [[42edo| 42]]

! [[43edo| 43]]

| 25 || &minus;4.3 (&minus;15.3%) || 7 || 4 || 3

! [[45edo| 45]]

| 26 || &minus;8.6 (&minus;32.3%) || 7 || 5 || 2

! [[46edo| 46]]

! [[47edo| 47]]

| 27 || &minus;12.6 (&minus;49.3%) || 7 || 6 || 1

! [[49edo| 49]]

! [[50edo| 50]]

| 29 || &minus;6.0 (&minus;24.8%) || 8 || 5 || 3

! [[53edo| 53]]

| 31 || &minus;0.1 ( -0.3%) || 9 || 4 || 5

! [[55edo| 55]]

| 32 || &minus;3.8 (&minus;17.3%) || 9 || 5 || 4

! [[56edo| 56]]

! [[59edo| 59]]

! [[61edo| 61]]

! [[63edo| 63]]

! [[64edo| 64]]

| 37 || &minus;8.2 (&minus;43.8%) || 10 || 7 || 3

! [[65edo| 65]]

| 38 || &minus;0.4 ( -2.3%) || 11 || 5 || 6

! [[67edo| 67]]

| 39 || &minus;3.4 (&minus;19.2%) || 11 || 6 || 5

! [[69edo| 69]]

| 40 || &minus;6.3 (&minus;36.2%) || 11 || 7 || 4

! [[70edo| 70]]

! [[71edo| 71]]

! [[73edo| 73]]

! [[74edo| 74]]

| 43 || &minus;4.7 (&minus;28.7%) || 12 || 7 || 5

! [[75edo| 75]]

! [[77edo| 77]]

| 45 || &minus;0.7 ( &minus;4.2%) || 13 || 6 || 7

! [[79edo| 79]]

| 46 || &minus;3.2 (&minus;21.2%) || 13 || 7 || 6

! [[80edo| 80]]

! [[81edo| 81]]

| 47 || &minus;5.7 (&minus;38.2%) || 13 || 8 || 5

! [[83edo| 83]]

! [[88edo| 88]]

| 51 || &minus;6.5 (&minus;47.7%) || 14 || 9 || 5

! [[89edo| 89]]

| 52 || &minus;0.8 ( -6.2%) || 15 || 7 || 8

! [[90edo| 90]]

! [[91edo| 91]]

| 53 || &minus;3.1 (&minus;23.2%) || 15 || 8 || 7

! [[94edo| 94]]

! [[95edo| 95]]

! [[97edo| 97]]

! [[98edo| 98]]

| 57 || &minus;4.0 (&minus;32.6%) || 16 || 9 || 7

! [[99edo| 99]]

! Edo !! Fifth !! Fifth-detuning<br />abs (&#162;), rel (%) !! Major<br />2nd !! Minor<br />2nd !! Augmented<br />1sn

! [[17edo| 17]]

! [[24edo| 24]]

| 14 || &minus;4.0 (&minus;4.0%) || 4 || 2 || 2

! [[27edo| 27]]

! [[31edo| 31]]

| 18 || &minus;5.2 (&minus;13.4%) || 5 || 3 || 2

! [[37edo| 37]]

! [[38edo| 38]]

| 22 || &minus;7.2 (&minus;22.9%) || 6 || 4 || 2

! [[41edo| 41]]

! [[44edo| 44]]

! [[45edo| 45]]

| 26 || &minus;8.6 (&minus;32.3%) || 7 || 5 || 2

! [[52edo| 52]]

| 30 || &minus;9.6 (&minus;41.8%) || 8 || 6 || 2

! [[55edo| 55]]

| 32 || &minus;3.8 (&minus;17.3%) || 9 || 5 || 4

! [[58edo| 58]]

! [[61edo| 61]]

! [[65edo| 65]]

| 38 || &minus;0.4 (&minus;2.3%) || 11 || 5 || 6

! [[69edo| 69]]

| 40 || &minus;6.3 (&minus;36.2%) || 11 || 7 || 4

! [[71edo| 71]]

! [[75edo| 75]]

! [[78edo| 78]]

! [[79edo| 79]]

| 46 || &minus;3.2 (&minus;21.2%) || 13 || 7 || 6

! [[86edo| 86]]

| 50 || &minus;4.3 (&minus;30.7%) || 14 || 8 || 6

! [[89edo| 89]]

| 52 || &minus;0.8 (&minus;6.2%) || 15 || 7 || 8

! [[92edo| 92]]

! [[95edo| 95]]

! [[99edo| 99]]

* [[Syntonic-rastmic subchroma notation]] &ndash; Built on neutral chain-of-fifths notation

* [[Ups and downs notation]] &ndash; Built on chain-of-fifths notation

* [[Diamond-mos notation]] &ndash; Built on neutral chain-of-fifths notation

* [[Sagittal notation]] (''evo flavor'') &ndash; Built on chain-of-fifths notation or neutral chain-of-fifths notation

* [[Extended meantone notation]] &ndash; Built on chain-of-fifths notation or neutral chain-of-fifths notation

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