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.  
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]]. 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 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{{flat}} C{{sharp}} D E{{flat}} D{{sharp}} E… For negative sharpness edos the order of the accidentals 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]]. For example, 13edo 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 C, D{{sesquiflat2}}, C{{demisharp2}}, D♭, C♯, D{{demiflat2}}, C{{sesquisharp2}}, D.  
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{{sesquiflat}}, C{{demisharp}}, D{{flat}}, C{{sharp}}, D{{demiflat}}, C{{sesquisharp}}, D|hair|med}}.


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.
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.


== Accidentals ==
== Accidentals ==
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.
The [https://w3c-cg.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 chain-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.
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.


{| class="wikitable center-all left-1"
{| class="wikitable center-all left-1"
|+ Accidentals in (neutral) circle-of-fifths notation
|+ style="font-size: 105%;" | Accidentals in (neutral) chain-of-fifths notation
|-
|-
! Style \ Offset
! Style \ offset
! &minus;2
! −2
! &minus;1½
! −1½
! &minus;1
! −1
! &minus;½
! −½
! ±0
! 0
! +½
! +½
! +1
! +1
Line 37: Line 37:
| Double sharp
| Double sharp
|- style="vertical-align: top;"  
|- style="vertical-align: top;"  
| 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>
| style="vertical-align: middle;" | Standard accidentals<ref>[https://w3c-cg.github.io/smufl/latest/tables/standard-accidentals-12-edo.html Standard Music Font Layout | Standard accidentals (12-EDO)]</ref>
| 𝄫<br>(U+1D12B)
| 𝄫<br>(U+1D12B)
|  
|  
Line 48: Line 48:
| 𝄪<br>(U+1D12A)
| 𝄪<br>(U+1D12A)
|- style="vertical-align: top;"  
|- style="vertical-align: top;"  
| 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>
| style="vertical-align: middle;" | Standard accidentals<br>+ Stein–Zimmermann accidentals<ref>[https://w3c-cg.github.io/smufl/latest/tables/stein-zimmermann-accidentals-24-edo.html Standard Music Font Layout | Stein-Zimmermann accidentals (24-EDO)]</ref>
| {{flat2|200%}}<br>(U+E264)
| {{flat2}}<br>(U+E264)
| {{sesquiflat|200%}}<br>(U+E281)
| {{sesquiflat}}<br>(U+E281)
| {{flat|200%}}<br>(U+E260)
| {{flat}}<br>(U+E260)
| {{demiflat|200%}}<br>(U+E280)
| {{demiflat}}<br>(U+E280)
| {{natural|200%}}<br>(U+E261)
| {{natural}}<br>(U+E261)
| {{demisharp|200%}}<br>(U+E282)
| {{demisharp}}<br>(U+E282)
| {{sharp|200%}}<br>(U+E262)
| {{sharp}}<br>(U+E262)
| {{sesquisharp|200%}}<br>(U+E283)
| {{sesquisharp|200%}}<br>(U+E283)
| {{sharp2|200%}}<br>(U+E263)
| {{sharp2}}<br>(U+E263)
|-
|-
| Substitute symbols
| Substitute symbols
Line 70: Line 70:
| x
| x
|-
|-
| Xen Wiki [[:Category:character templates|character templates]]
| Xen Wiki [[:Category: Character templates|character templates]]
| <nowiki>{{</nowiki>[[Template:flat2|flat2]]}}
| <small>{{tlx|flat2|plaincode}}</small>
| <nowiki>{{</nowiki>[[Template:sesquiflat|sesquiflat]]}}<br><nowiki>{{</nowiki>[[Template:sesquiflat2|sesquiflat2]]}}
| <small>{{tlx|sesquiflat|plaincode}}<br>{{tlx|sesquiflat2|plaincode}}</small>
| <nowiki>{{</nowiki>[[Template:flat|flat]]}}
| <small>{{tlx|flat|plaincode}}</small>
| <nowiki>{{</nowiki>[[Template:demiflat|demiflat]]}}<br><nowiki>{{</nowiki>[[Template:demiflat2|demiflat2]]}}
| <small>{{tlx|demiflat|plaincode}}<br>{{tlx|demiflat2|plaincode}}</small>
| <nowiki>{{</nowiki>[[Template:natural|natural]]}}
| <small>{{tlx|natural|plaincode}}</small>
| <nowiki>{{</nowiki>[[Template:demisharp|demisharp]]}}<br><nowiki>{{</nowiki>[[Template:demisharp2|demisharp2]]}}
| <small>{{tlx|demisharp|plaincode}}<br>{{tlx|demisharp2|plaincode}}</small>
| <nowiki>{{</nowiki>[[Template:sharp|sharp]]}}
| <small>{{tlx|sharp|plaincode}}</small>
| <nowiki>{{</nowiki>[[Template:sesquisharp|sesquisharp]]}}<br><nowiki>{{</nowiki>[[Template:sesquisharp2|sesquisharp2]]}}
| <small>{{tlx|sesquisharp|plaincode}}<br>{{tlx|sesquisharp2|plaincode}}</small>
| <nowiki>{{</nowiki>[[Template:sharp2|sharp2]]}}
| <small>{{tlx|sharp2|plaincode}}</small>
|}
|}


=== Alternative accidentals ===
=== Alternative accidentals ===
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.
While the Stein–Zimmermann accidentals appear to be the most widespread for neutral chain-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.
Note that certain symbols may be very similar or identical to standard or Stein–Zimmermann accidentals despite having different Unicode codepoints.


A particular case is [[ups and downs notation]], which uses [[arrow]]s placed to the left of accidentals (e.g. ^#) or note names (e.g. ^C#). Since different tuning systems associate a different number arrows to different offsets, they are not included below, but the most basic notation can be found at [[24edo #Notation]].
A particular case is [[ups and downs notation]], which uses [[arrow]]s placed to the left of accidentals (e.g. ^#) or note names (e.g. ^C#). Since different tuning systems associate a different number arrows to different offsets, they are not included below, but the most basic notation can be found at [[24edo #Notation]].


{| class="wikitable center-all left-1"
{| class="wikitable center-all left-1"
|+ Alternative accidentals in (neutral) circle-of-fifths notation
|+ style="font-size: 105%;" | Alternative accidentals in (neutral) chain-of-fifths notation
|-
|-
! Style \ Offset
! Style \ offset
! &minus;2
! −2
! &minus;1½
! −1½
! &minus;1
! −1
! &minus;½
! −½
! ±0
! 0
! +½
! +½
! +1
! +1
Line 103: Line 103:
! +2
! +2
|- style="vertical-align: top;"  
|- style="vertical-align: top;"  
| 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>
| style="vertical-align: middle;" | Gould arrow quartertone accidentals<ref>[https://w3c-cg.github.io/smufl/latest/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%}}
| {{Flat2}}
| {{Bravura||200%}}<br>(U+E271)<br>(U+1D12D)<br>{{Bravura||200%}}<br>(U+E278)
| {{Bravura|&#xE271;}}<br>(U+E271)<br>(U+1D12D)<br>{{Bravura|&#xE278;}}<br>(U+E278)
| {{flat|200%}}
| {{Flat}}
| {{Bravura||200%}}<br>(U+E270)<br>(U+1D12C)<br>{{Bravura||200%}}<br>(U+E273)<br>(U+1D12F)
| {{Bravura|&#xE270;}}<br>(U+E270)<br>(U+1D12C)<br>{{Bravura|&#xE273;}}<br>(U+E273)<br>(U+1D12F)
| {{natural|200%}}
| {{Natural}}
| {{Bravura||200%}}<br>(U+E275)<br>(U+1D131)<br>{{Bravura||200%}}<br>(U+E272)<br>(U+1D12E)
| {{Bravura|&#xE275;}}<br>(U+E275)<br>(U+1D131)<br>{{Bravura|&#xE272;}}<br>(U+E272)<br>(U+1D12E)
| {{sharp|200%}}
| {{Sharp}}
| {{Bravura||200%}}<br>(U+E274)<br>(U+1D130)<br>{{Bravura||200%}}<br>(U+E277)
| {{Bravura|&#xE274;}}<br>(U+E274)<br>(U+1D130)<br>{{Bravura|&#xE277;}}<br>(U+E277)
| {{sharp2|200%}}
| {{Sharp2}}
|- style="vertical-align: top;"
|- style="vertical-align: top;"
| 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>
| style="vertical-align: middle;" | Persian accidentals<ref>[https://w3c-cg.github.io/smufl/latest/tables/persian-accidentals.html Standard Music Font Layout | Persian accidentals]</ref>
|  
|  
|  
|  
| {{flat|200%}}
| {{Flat}}
| {{Bravura||200%}}<br>Koron<br>(U+E460)
| {{Bravura|&#xE460;}}<br>Koron<br>(U+E460)
| {{natural|200%}}
| {{Natural}}
| {{Bravura||200%}}<br>Sori<br>(U+E461)
| {{Bravura|&#xE461;}}<br>Sori<br>(U+E461)
| {{sharp|200%}}
| {{Sharp}}
|  
|  
|  
|  
|- style="vertical-align: top;"  
|- style="vertical-align: top;"  
| 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>
| style="vertical-align: middle;" | [[Sagittal]] accidentals<ref>[https://w3c-cg.github.io/smufl/latest/tables/spartan-sagittal-single-shaft-accidentals.html Standard Music Font Layout | Spartan Sagittal single-shaft accidentals]</ref><ref>[https://w3c-cg.github.io/smufl/latest/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|&#xE335;}}<br>(U+E335)
| {{Bravura||200%}}<br>(U+E327)
| {{Bravura|&#xE327;}}<br>(U+E327)
| {{Bravura||200%}}<br>(U+E319)
| {{Bravura|&#xE319;}}<br>(U+E319)
| {{Bravura||200%}}<br>(U+E30B)
| {{Bravura|&#xE30B;}}<br>(U+E30B)
| {{natural|200%}}
| {{natural}}
| {{Bravura||200%}}<br>(U+E30A)
| {{Bravura|&#xE30A;}}<br>(U+E30A)
| {{Bravura||200%}}<br>(U+E318)
| {{Bravura|&#xE318;}}<br>(U+E318)
| {{Bravura||200%}}<br>(U+E326)
| {{Bravura|&#xE326;}}<br>(U+E326)
| {{Bravura||200%}}<br>(U+E334)
| {{Bravura|&#xE334;}}<br>(U+E334)
|- style="vertical-align: top;"  
|- style="vertical-align: top;"  
| 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>
| style="vertical-align: middle;" | [[Wyschnegradsky]] accidentals<ref>[https://w3c-cg.github.io/smufl/latest/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%}}
| {{flat2}}
| {{Bravura||200%}}<br>(U+E433)
| {{Bravura|&#xE433;}}<br>(U+E433)
| {{Bravura||200%}}<br>(U+E430)
| {{Bravura|&#xE430;}}<br>(U+E430)
| {{Bravura||200%}}<br>(U+E42D)
| {{Bravura|&#xE42D;}}<br>(U+E42D)
| {{natural|200%}}
| {{natural}}
| {{Bravura||200%}}<br>(U+E422)
| {{Bravura|&#xE422;}}<br>(U+E422)
| {{Bravura||200%}}<br>(U+E425)
| {{Bravura|&#xE425;}}<br>(U+E425)
| {{Bravura||200%}}<br>(U+E428)
| {{Bravura|&#xE428;}}<br>(U+E428)
| {{sharp2|200%}}
| {{sharp2}}
|}
|}


Line 151: Line 151:
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.
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.


<div><div style="display: inline-grid; margin-right: 25px;">
{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|+ style="font-size: 105%;" | Diatonic edos fit for chain-of-fifths notation
|+ Diatonic edos fit for chain-of-fifths notation
|-
|-
! Edo
! Edo
! Fifth
! Fifth
! Fifth-detuning <br> abs (¢), rel (%)
! Fifth-detuning<br>abs (¢), rel (%)
! Major <br> 2nd
! Major<br>2nd
! Minor <br> 2nd
! Minor<br>2nd
! Augmented <br> 1sn
! Augmented<br>1sn
|-
|-
! [[12edo| 12]]
! [[12edo|12]]
| 7 || &minus;2.0 ( -2.0%) || 2 || 1 || 1
| 7 || −2.0 ( −2.0%) || 2 || 1 || 1
|-
|-
! [[17edo| 17]]
! [[17edo|17]]
| 10 || +3.9 ( +5.6%) || 3 || 1 || 2
| 10 || +3.9 ( +5.6%) || 3 || 1 || 2
|-
|-
! [[19edo| 19]]
! [[19edo|19]]
| 11 || &minus;7.2 (&minus;11.4%) || 3 || 2 || 1
| 11 || −7.2 (−11.4%) || 3 || 2 || 1
|-
|-
! [[22edo| 22]]
! [[22edo|22]]
| 13 || +7.1 (+13.1%) || 4 || 1 || 3
| 13 || +7.1 (+13.1%) || 4 || 1 || 3
|-
|-
! [[26edo| 26]]
! [[26edo|26]]
| 15 || &minus;9.6 (&minus;20.9%) || 4 || 3 || 1
| 15 || −9.6 (−20.9%) || 4 || 3 || 1
|-
|-
! [[27edo| 27]]
! [[27edo|27]]
| 16 || +9.2 (+20.6%) || 5 || 1 || 4
| 16 || +9.2 (+20.6%) || 5 || 1 || 4
|-
|-
! [[29edo| 29]]
! [[29edo|29]]
| 17 || +1.5 ( +3.6%) || 5 || 2 || 3
| 17 || +1.5 ( +3.6%) || 5 || 2 || 3
|-
|-
! [[31edo| 31]]
! [[31edo|31]]
| 18 || &minus;5.2 (&minus;13.4%) || 5 || 3 || 2
| 18 || −5.2 (−13.4%) || 5 || 3 || 2
|-
|-
! [[32edo| 32]]
! [[32edo|32]]
| 19 || +10.5 (+28.1%) || 6 || 1 || 5
| 19 || +10.5 (+28.1%) || 6 || 1 || 5
|-
|-
! [[33edo| 33]]
! [[33edo|33]]
| 19 || &minus;11.0 (&minus;30.4%) || 5 || 4 || 1
| 19 || −11.0 (−30.4%) || 5 || 4 || 1
|-
|-
! [[37edo| 37]]
! [[37edo|37]]
| 22 || +11.6 (+35.6%) || 7 || 1 || 6
| 22 || +11.6 (+35.6%) || 7 || 1 || 6
|-
|-
! [[39edo| 39]]
! [[39edo|39]]
| 23 || +5.7 (+18.6%) || 7 || 2 || 5
| 23 || +5.7 (+18.6%) || 7 || 2 || 5
|-
|-
! [[40edo| 40]]
! [[40edo|40]]
| 23 || &minus;12.0 (&minus;39.9%) || 6 || 5 || 1
| 23 || −12.0 (−39.9%) || 6 || 5 || 1
|-
|-
! [[41edo| 41]]
! [[41edo|41]]
| 24 || +0.5 ( +1.7%) || 7 || 3 || 4
| 24 || +0.5 ( +1.7%) || 7 || 3 || 4
|-
|-
! [[42edo| 42]]
! [[42edo|42]]
| 25 || +12.3 (+43.2%) || 8 || 1 || 7
| 25 || +12.3 (+43.2%) || 8 || 1 || 7
|-
|-
! [[43edo| 43]]
! [[43edo|43]]
| 25 || &minus;4.3 (&minus;15.3%) || 7 || 4 || 3
| 25 || −4.3 (−15.3%) || 7 || 4 || 3
|-
|-
! [[45edo| 45]]
! [[45edo|45]]
| 26 || &minus;8.6 (&minus;32.3%) || 7 || 5 || 2
| 26 || −8.6 (−32.3%) || 7 || 5 || 2
|-
|-
! [[46edo| 46]]
! [[46edo|46]]
| 27 || +2.4 ( +9.2%) || 8 || 3 || 5
| 27 || +2.4 ( +9.2%) || 8 || 3 || 5
|-
|-
! [[47edo| 47]]
! [[47edo|47]]
| 27 || &minus;12.6 (&minus;49.3%) || 7 || 6 || 1
| 27 || −12.6 (−49.3%) || 7 || 6 || 1
|-
|-
! [[49edo| 49]]
! [[49edo|49]]
| 29 || +8.2 (+33.7%) || 9 || 2 || 7
| 29 || +8.2 (+33.7%) || 9 || 2 || 7
|-
|-
! [[50edo| 50]]
! [[50edo|50]]
| 29 || &minus;6.0 (&minus;24.8%) || 8 || 5 || 3
| 29 || −6.0 (−24.8%) || 8 || 5 || 3
|-
|-
! [[53edo| 53]]
! [[53edo|53]]
| 31 || &minus;0.1 ( -0.3%) || 9 || 4 || 5
| 31 || −0.1 ( -0.3%) || 9 || 4 || 5
|-
|-
! [[55edo| 55]]
! [[55edo|55]]
| 32 || &minus;3.8 (&minus;17.3%) || 9 || 5 || 4
| 32 || −3.8 (−17.3%) || 9 || 5 || 4
|-
|-
! [[56edo| 56]]
! [[56edo|56]]
| 33 || +5.2 (+24.2%) || 10 || 3 || 7
| 33 || +5.2 (+24.2%) || 10 || 3 || 7
|-
|-
! [[59edo| 59]]
! [[59edo|59]]
| 35 || +9.9 (+48.7%) || 11 || 2 || 9
| 35 || +9.9 (+48.7%) || 11 || 2 || 9
|-
|-
! [[61edo| 61]]
! [[61edo|61]]
| 36 || +6.2 (+31.7%) || 11 || 3 || 8
| 36 || +6.2 (+31.7%) || 11 || 3 || 8
|-
|-
! [[63edo| 63]]
! [[63edo|63]]
| 37 || +2.8 (+14.7%) || 11 || 4 || 7
| 37 || +2.8 (+14.7%) || 11 || 4 || 7
|-
|-
! [[64edo| 64]]
! [[64edo|64]]
| 37 || &minus;8.2 (&minus;43.8%) || 10 || 7 || 3
| 37 || −8.2 (−43.8%) || 10 || 7 || 3
|-
|-
! [[65edo| 65]]
! [[65edo|65]]
| 38 || &minus;0.4 ( -2.3%) || 11 || 5 || 6
| 38 || −0.4 ( -2.3%) || 11 || 5 || 6
|-
|-
! [[67edo| 67]]
! [[67edo|67]]
| 39 || &minus;3.4 (&minus;19.2%) || 11 || 6 || 5
| 39 || −3.4 (−19.2%) || 11 || 6 || 5
|-
|-
! [[69edo| 69]]
! [[69edo|69]]
| 40 || &minus;6.3 (&minus;36.2%) || 11 || 7 || 4
| 40 || −6.3 (−36.2%) || 11 || 7 || 4
|-
|-
! [[70edo| 70]]
! [[70edo|70]]
| 41 || +0.9 ( +5.3%) || 12 || 5 || 7
| 41 || +0.9 ( +5.3%) || 12 || 5 || 7
|-
|-
! [[71edo| 71]]
! [[71edo|71]]
| 42 || +7.9 (+46.8%) || 13 || 3 || 10
| 42 || +7.9 (+46.8%) || 13 || 3 || 10
|-
|-
! [[73edo| 73]]
! [[73edo|73]]
| 43 || +4.9 (+29.8%) || 13 || 4 || 9
| 43 || +4.9 (+29.8%) || 13 || 4 || 9
|-
|-
! [[74edo| 74]]
! [[74edo|74]]
| 43 || &minus;4.7 (&minus;28.7%) || 12 || 7 || 5
| 43 || −4.7 (−28.7%) || 12 || 7 || 5
|-
|-
! [[75edo| 75]]
! [[75edo|75]]
| 44 || +2.0 (+12.8%) || 13 || 5 || 8
| 44 || +2.0 (+12.8%) || 13 || 5 || 8
|-
|-
! [[77edo| 77]]
! [[77edo|77]]
| 45 || &minus;0.7 ( -4.2%) || 13 || 6 || 7
| 45 || −0.7 ( −4.2%) || 13 || 6 || 7
|-
|-
! [[79edo| 79]]
! [[79edo|79]]
| 46 || &minus;3.2 (&minus;21.2%) || 13 || 7 || 6
| 46 || −3.2 (−21.2%) || 13 || 7 || 6
|-
|-
! [[80edo| 80]]
! [[80edo|80]]
| 47 || +3.0 (+20.3%) || 14 || 5 || 9
| 47 || +3.0 (+20.3%) || 14 || 5 || 9
|-
|-
! [[81edo| 81]]
! [[81edo|81]]
| 47 || &minus;5.7 (&minus;38.2%) || 13 || 8 || 5
| 47 || −5.7 (−38.2%) || 13 || 8 || 5
|-
|-
! [[83edo| 83]]
! [[83edo|83]]
| 49 || +6.5 (+44.8%) || 15 || 4 || 11
| 49 || +6.5 (+44.8%) || 15 || 4 || 11
|-
|-
! [[88edo| 88]]
! [[88edo|88]]
| 51 || &minus;6.5 (&minus;47.7%) || 14 || 9 || 5
| 51 || −6.5 (−47.7%) || 14 || 9 || 5
|-
|-
! [[89edo| 89]]
! [[89edo|89]]
| 52 || &minus;0.8 ( -6.2%) || 15 || 7 || 8
| 52 || −0.8 ( -6.2%) || 15 || 7 || 8
|-
|-
! [[90edo| 90]]
! [[90edo|90]]
| 53 || +4.7 (+35.3%) || 16 || 5 || 11
| 53 || +4.7 (+35.3%) || 16 || 5 || 11
|-
|-
! [[91edo| 91]]
! [[91edo|91]]
| 53 || &minus;3.1 (&minus;23.2%) || 15 || 8 || 7
| 53 || −3.1 (−23.2%) || 15 || 8 || 7
|-
|-
! [[94edo| 94]]
! [[94edo|94]]
| 55 || +0.2 ( +1.4%) || 16 || 7 || 9
| 55 || +0.2 ( +1.4%) || 16 || 7 || 9
|-
|-
! [[95edo| 95]]
! [[95edo|95]]
| 56 || +5.4 (+42.9%) || 17 || 5 || 12
| 56 || +5.4 (+42.9%) || 17 || 5 || 12
|-
|-
! [[97edo| 97]]
! [[97edo|97]]
| 57 || +3.2 (+25.9%) || 17 || 6 || 11
| 57 || +3.2 (+25.9%) || 17 || 6 || 11
|-
|-
! [[98edo| 98]]
! [[98edo|98]]
| 57 || &minus;4.0 (&minus;32.6%) || 16 || 9 || 7
| 57 || −4.0 (−32.6%) || 16 || 9 || 7
|-
|-
! [[99edo| 99]]
! [[99edo|99]]
| 58 || +1.1 ( +8.9%) || 17 || 7 || 10
| 58 || +1.1 ( +8.9%) || 17 || 7 || 10
|}
|}</div>


<div style="display: inline-grid; margin-right: 25px;">
{| class="wikitable center-all"
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Diatonic edos fit for neutral chain-of-fifths notation
|-
|-
|+ Diatonic edos fit for neutral chain-of-fifths notation
! Edo !! Fifth !! Fifth-detuning<br>abs (¢), rel (%) !! Major<br>2nd !! Minor<br>2nd !! Augmented<br>1sn
|-
|-
! Edo
! [[17edo|17]]
! Fifth
! Fifth-detuning <br> abs (¢), rel (%)
! Major <br> 2nd
! Minor <br> 2nd
! Augmented <br> 1sn
|-
! [[17edo| 17]]
| 10 || +3.9 ( +5.6%) || 3 || 1 || 2
| 10 || +3.9 ( +5.6%) || 3 || 1 || 2
|-
|-
! [[24edo| 24]]
! [[24edo|24]]
| 14 || &minus;4.0 (&minus;4.0%) || 4 || 2 || 2
| 14 || −4.0 (−4.0%) || 4 || 2 || 2
|-
|-
! [[27edo| 27]]
! [[27edo|27]]
| 16 || +9.2 (+20.6%) || 5 || 1 || 4
| 16 || +9.2 (+20.6%) || 5 || 1 || 4
|-
|-
! [[31edo| 31]]
! [[31edo|31]]
| 18 || &minus;5.2 (&minus;13.4%) || 5 || 3 || 2
| 18 || −5.2 (−13.4%) || 5 || 3 || 2
|-
|-
! [[37edo| 37]]
! [[37edo|37]]
| 22 || +11.6 (+35.6%) || 7 || 1 || 6
| 22 || +11.6 (+35.6%) || 7 || 1 || 6
|-
|-
! [[38edo| 38]]
! [[38edo|38]]
| 22 || &minus;7.2 (&minus;22.9%) || 6 || 4 || 2
| 22 || −7.2 (−22.9%) || 6 || 4 || 2
|-
|-
! [[41edo| 41]]
! [[41edo|41]]
| 24 || +0.5 ( +1.7%) || 7 || 3 || 4
| 24 || +0.5 ( +1.7%) || 7 || 3 || 4
|-
|-
! [[44edo| 44]]
! [[44edo|44]]
| 26 || +7.1 (+26.2%) || 8 || 2 || 6
| 26 || +7.1 (+26.2%) || 8 || 2 || 6
|-
|-
! [[45edo| 45]]
! [[45edo|45]]
| 26 || &minus;8.6 (&minus;32.3%) || 7 || 5 || 2
| 26 || −8.6 (−32.3%) || 7 || 5 || 2
|-
|-
! [[52edo| 52]]
! [[52edo|52]]
| 30 || &minus;9.6 (&minus;41.8%) || 8 || 6 || 2
| 30 || −9.6 (−41.8%) || 8 || 6 || 2
|-
|-
! [[55edo| 55]]
! [[55edo|55]]
| 32 || &minus;3.8 (&minus;17.3%) || 9 || 5 || 4
| 32 || −3.8 (−17.3%) || 9 || 5 || 4
|-
|-
! [[58edo| 58]]
! [[58edo|58]]
| 34 || +1.5 ( +3.6%) || 10 || 4 || 6
| 34 || +1.5 ( +3.6%) || 10 || 4 || 6
|-
|-
! [[61edo| 61]]
! [[61edo|61]]
| 36 || +6.2 (+31.7%) || 11 || 3 || 8
| 36 || +6.2 (+31.7%) || 11 || 3 || 8
|-
|-
! [[65edo| 65]]
! [[65edo|65]]
| 38 || &minus;0.4 ( -2.3%) || 11 || 5 || 6
| 38 || −0.4 (−2.3%) || 11 || 5 || 6
|-
|-
! [[69edo| 69]]
! [[69edo|69]]
| 40 || &minus;6.3 (&minus;36.2%) || 11 || 7 || 4
| 40 || −6.3 (−36.2%) || 11 || 7 || 4
|-
|-
! [[71edo| 71]]
! [[71edo|71]]
| 42 || +7.9 (+46.8%) || 13 || 3 || 10
| 42 || +7.9 (+46.8%) || 13 || 3 || 10
|-
|-
! [[75edo| 75]]
! [[75edo|75]]
| 44 || +2.0 (+12.8%) || 13 || 5 || 8
| 44 || +2.0 (+12.8%) || 13 || 5 || 8
|-
|-
! [[78edo| 78]]
! [[78edo|78]]
| 46 || +5.7 (+37.3%) || 14 || 4 || 10
| 46 || +5.7 (+37.3%) || 14 || 4 || 10
|-
|-
! [[79edo| 79]]
! [[79edo|79]]
| 46 || &minus;3.2 (&minus;21.2%) || 13 || 7 || 6
| 46 || −3.2 (−21.2%) || 13 || 7 || 6
|-
|-
! [[86edo| 86]]
! [[86edo|86]]
| 50 || &minus;4.3 (&minus;30.7%) || 14 || 8 || 6
| 50 || −4.3 (−30.7%) || 14 || 8 || 6
|-
|-
! [[89edo| 89]]
! [[89edo|89]]
| 52 || &minus;0.8 ( -6.2%) || 15 || 7 || 8
| 52 || −0.8 (−6.2%) || 15 || 7 || 8
|-
|-
! [[92edo| 92]]
! [[92edo|92]]
| 54 || +2.4 ( +18.3%) || 16 || 6 || 10
| 54 || +2.4 (+18.3%) || 16 || 6 || 10
|-
|-
! [[95edo| 95]]
! [[95edo|95]]
| 56 || +5.4 (+42.9%) || 17 || 5 || 12
| 56 || +5.4 (+42.9%) || 17 || 5 || 12
|-
|-
! [[99edo| 99]]
! [[99edo|99]]
| 58 || +1.1 ( +8.9%) || 17 || 7 || 10
| 58 || +1.1 (+8.9%) || 17 || 7 || 10
|}
|}
</div></div>


== Expansions ==
== Expansions ==
* [[Syntonic-rastmic subchroma notation]] – built on neutral chain-of-fifths notation
* [[Kite's ups and downs notation]] – built on chain-of-fifths notation
* [[Ups and downs notation]] – built on chain-of-fifths notation
* [[Stein–Zimmermann–Gould notation]] – built on neutral chain-of-fifths notation
** Neutral ups and downs notation (&rarr; [[Alternative symbols for ups and downs notation]])
* [[Syntonic–rastmic subchroma notation]] – built on neutral chain-of-fifths notation
* [[Diamond-mos notation]] – built on neutral chain-of-fifths notation
* [[Sagittal notation]] (''evo flavor'') – built on chain-of-fifths notation or neutral chain-of-fifths notation
* [[Sagittal notation]] (''evo flavor'') – built on chain-of-fifths notation or neutral chain-of-fifths notation
* [[Extended meantone notation]] – built on chain-of-fifths notation or neutral chain-of-fifths notation
* [[Extended meantone notation]] – built on chain-of-fifths notation or neutral chain-of-fifths notation
* [[Fractional sharp notation]] – generalizes neutral chain-of-fifths notation to other fractions
* [[Diamond-mos notation]] – generalizes neutral chain-of-fifths notation to non-diatonic scales


== See also ==
== See also ==
Line 410: Line 407:
* [[Fifthspan]]
* [[Fifthspan]]
* [[Pythagorean tuning]]
* [[Pythagorean tuning]]
* [[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,&nbsp;3\5], the "[[diatonic range]]")


== Notes ==
== Notes ==
Line 417: Line 414:
== References ==
== References ==
<references />
<references />
{{Navbox notation}}


[[Category:Notation]]
[[Category:Notation]]
[[Category:Method]]
[[Category:Method]]
[[Category:Fifth]]
[[Category:Fifth]]

Latest revision as of 11:07, 12 May 2026

The chain-of-fifths notation, also known as extended Pythagorean notation, is a musical notation system that supports a variety of tuning systems which are octave-repeating and generated by the fifth (just or tempered). A good number of edos and regular temperaments 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. 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 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 C – D♭ – C♯ – D – E♭ – D♯ – E – F – G♭ – F♯ – G – A♭ – G♯ – A – B♭ – A♯ – B – C. If the fifth is flatter than 685.714 ¢, 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 (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 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 C – D – C – D – C – D – C – D.

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.

Accidentals

The 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 chain-of-fifths notation. Some fonts may not include all symbols, so fonts designed for musical notation, such as Bravura or Leland[1], 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 character templates to enter these symbols easily in wiki pages. The following table includes these equivalences.

Accidentals in (neutral) chain-of-fifths notation
Style \ offset −2 −1½ −1 −½ 0 +½ +1 +1½ +2
Name Double flat Sesquiflat Flat Half-flat
Demiflat
Semiflat 		Natural Half-sharp
Demisharp
Semisharp 		Sharp Sesquisharp Double sharp
Standard accidentals[2] 𝄫
(U+1D12B) 		♭
(U+266D) 		♮
(U+266E) 		♯
(U+266F) 		𝄪
(U+1D12A)
Standard accidentals
+ Stein–Zimmermann accidentals[3]
(U+E264)
(U+E281)
(U+E260)
(U+E280)
(U+E261)
(U+E282)
(U+E262)
(U+E283)
(U+E263)
Substitute symbols bb db b d h t # t# x
Xen Wiki character templates {{flat2}} {{sesquiflat}}
{{sesquiflat2}} 		{{flat}} {{demiflat}}
{{demiflat2}} 		{{natural}} {{demisharp}}
{{demisharp2}} 		{{sharp}} {{sesquisharp}}
{{sesquisharp2}} 		{{sharp2}}

Alternative accidentals

While the Stein–Zimmermann accidentals appear to be the most widespread for neutral chain-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.

A particular case is ups and downs notation, which uses arrows placed to the left of accidentals (e.g. ^#) or note names (e.g. ^C#). Since different tuning systems associate a different number arrows to different offsets, they are not included below, but the most basic notation can be found at 24edo #Notation.

Alternative accidentals in (neutral) chain-of-fifths notation
Style \ offset −2 −1½ −1 −½ 0 +½ +1 +1½ +2
Gould arrow quartertone accidentals[4][note 1]
(U+E271)
(U+1D12D)

(U+E278)
(U+E270)
(U+1D12C)

(U+E273)
(U+1D12F)
(U+E275)
(U+1D131)

(U+E272)
(U+1D12E)
(U+E274)
(U+1D130)

(U+E277)
Persian accidentals[5]
Koron
(U+E460)
Sori
(U+E461)
Sagittal accidentals[6][7][note 2]
(U+E335)
(U+E327)
(U+E319)
(U+E30B)
(U+E30A)
(U+E318)
(U+E326)
(U+E334)
Wyschnegradsky accidentals[8][note 3]
(U+E433)
(U+E430)
(U+E42D)
(U+E422)
(U+E425)
(U+E428)

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 pentasharpness and sharpness respectively.

Diatonic edos fit for chain-of-fifths notation
Edo Fifth Fifth-detuning
abs (¢), rel (%) 		Major
2nd 		Minor
2nd 		Augmented
1sn
12 7 −2.0 ( −2.0%) 2 1 1
17 10 +3.9 ( +5.6%) 3 1 2
19 11 −7.2 (−11.4%) 3 2 1
22 13 +7.1 (+13.1%) 4 1 3
26 15 −9.6 (−20.9%) 4 3 1
27 16 +9.2 (+20.6%) 5 1 4
29 17 +1.5 ( +3.6%) 5 2 3
31 18 −5.2 (−13.4%) 5 3 2
32 19 +10.5 (+28.1%) 6 1 5
33 19 −11.0 (−30.4%) 5 4 1
37 22 +11.6 (+35.6%) 7 1 6
39 23 +5.7 (+18.6%) 7 2 5
40 23 −12.0 (−39.9%) 6 5 1
41 24 +0.5 ( +1.7%) 7 3 4
42 25 +12.3 (+43.2%) 8 1 7
43 25 −4.3 (−15.3%) 7 4 3
45 26 −8.6 (−32.3%) 7 5 2
46 27 +2.4 ( +9.2%) 8 3 5
47 27 −12.6 (−49.3%) 7 6 1
49 29 +8.2 (+33.7%) 9 2 7
50 29 −6.0 (−24.8%) 8 5 3
53 31 −0.1 ( -0.3%) 9 4 5
55 32 −3.8 (−17.3%) 9 5 4
56 33 +5.2 (+24.2%) 10 3 7
59 35 +9.9 (+48.7%) 11 2 9
61 36 +6.2 (+31.7%) 11 3 8
63 37 +2.8 (+14.7%) 11 4 7
64 37 −8.2 (−43.8%) 10 7 3
65 38 −0.4 ( -2.3%) 11 5 6
67 39 −3.4 (−19.2%) 11 6 5
69 40 −6.3 (−36.2%) 11 7 4
70 41 +0.9 ( +5.3%) 12 5 7
71 42 +7.9 (+46.8%) 13 3 10
73 43 +4.9 (+29.8%) 13 4 9
74 43 −4.7 (−28.7%) 12 7 5
75 44 +2.0 (+12.8%) 13 5 8
77 45 −0.7 ( −4.2%) 13 6 7
79 46 −3.2 (−21.2%) 13 7 6
80 47 +3.0 (+20.3%) 14 5 9
81 47 −5.7 (−38.2%) 13 8 5
83 49 +6.5 (+44.8%) 15 4 11
88 51 −6.5 (−47.7%) 14 9 5
89 52 −0.8 ( -6.2%) 15 7 8
90 53 +4.7 (+35.3%) 16 5 11
91 53 −3.1 (−23.2%) 15 8 7
94 55 +0.2 ( +1.4%) 16 7 9
95 56 +5.4 (+42.9%) 17 5 12
97 57 +3.2 (+25.9%) 17 6 11
98 57 −4.0 (−32.6%) 16 9 7
99 58 +1.1 ( +8.9%) 17 7 10
Diatonic edos fit for neutral chain-of-fifths notation
Edo Fifth Fifth-detuning
abs (¢), rel (%)		 Major
2nd		 Minor
2nd		 Augmented
1sn
17 10 +3.9 ( +5.6%) 3 1 2
24 14 −4.0 (−4.0%) 4 2 2
27 16 +9.2 (+20.6%) 5 1 4
31 18 −5.2 (−13.4%) 5 3 2
37 22 +11.6 (+35.6%) 7 1 6
38 22 −7.2 (−22.9%) 6 4 2
41 24 +0.5 ( +1.7%) 7 3 4
44 26 +7.1 (+26.2%) 8 2 6
45 26 −8.6 (−32.3%) 7 5 2
52 30 −9.6 (−41.8%) 8 6 2
55 32 −3.8 (−17.3%) 9 5 4
58 34 +1.5 ( +3.6%) 10 4 6
61 36 +6.2 (+31.7%) 11 3 8
65 38 −0.4 (−2.3%) 11 5 6
69 40 −6.3 (−36.2%) 11 7 4
71 42 +7.9 (+46.8%) 13 3 10
75 44 +2.0 (+12.8%) 13 5 8
78 46 +5.7 (+37.3%) 14 4 10
79 46 −3.2 (−21.2%) 13 7 6
86 50 −4.3 (−30.7%) 14 8 6
89 52 −0.8 (−6.2%) 15 7 8
92 54 +2.4 (+18.3%) 16 6 10
95 56 +5.4 (+42.9%) 17 5 12
99 58 +1.1 (+8.9%) 17 7 10

Expansions

See also

Notes

  1. ↑ Symbols for five-quarter-tones accidentals are also available.
  2. ↑ 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.
  3. ↑ Wyschnegradsky accidentals also include twelfth-tone (72edo) accidentals.

References

  1. ↑ SMuFL | Introducing SMuFL
  2. ↑ Standard Music Font Layout | Standard accidentals (12-EDO)
  3. ↑ Standard Music Font Layout | Stein-Zimmermann accidentals (24-EDO)
  4. ↑ Standard Music Font Layout | Gould arrow quartertone accidentals (24-EDO)
  5. ↑ Standard Music Font Layout | Persian accidentals
  6. ↑ Standard Music Font Layout | Spartan Sagittal single-shaft accidentals
  7. ↑ Standard Music Font Layout | Spartan Sagittal multi-shaft accidentals
  8. ↑ Standard Music Font Layout | Wyschnegradsky accidentals (72-EDO)


View • Talk • EditMusical notation
Universal Sagittal notation
Just intonation Functional Just System • Ben Johnston's notation (Johnston–Copper notation) • Helmholtz–Ellis notation • Color notation
MOS scales Diamond-mos notation • KISS notation (Quasi-diatonic MOS notation)
Temperaments Chain-of-fifths notation • Stein–Zimmermann–Gould notation • Ups and downs notation • Syntonic–rastmic subchroma notation • Extended meantone notation • Fractional sharp notation
See musical notation for a longer list of systems by category. See Category:Notation for the most complete, comprehensive list, but not sorted by category.
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