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Chain-of-fifths notation: Difference between revisions

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Major rewrite. Removed the opinion stated as fact that perfect, pentatonic and superflat edos are better notated as subsets of larger edos (counter-example: 7edo). Clarified which edos can use quartertone notation, and which can best use it. Other changes as well.
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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''', 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''.


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).  
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'''.


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''' (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,&nbsp;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}}.


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.  
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.


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.  
== Accidentals ==
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.


==Edos up to 100==
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"
|+ style="font-size: 105%;" | 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<br>Demiflat<br>Semiflat
| Natural
| Half-sharp<br>Demisharp<br>Semisharp
| Sharp
| Sesquisharp
| Double sharp
|- style="vertical-align: top;"
| 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+266D)
|
| ♮<br>(U+266E)
|
| ♯<br>(U+266F)
|
| 𝄪<br>(U+1D12A)
|- style="vertical-align: top;"
| 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}}<br>(U+E264)
| {{sesquiflat}}<br>(U+E281)
| {{flat}}<br>(U+E260)
| {{demiflat}}<br>(U+E280)
| {{natural}}<br>(U+E261)
| {{demisharp}}<br>(U+E282)
| {{sharp}}<br>(U+E262)
| {{sesquisharp|200%}}<br>(U+E283)
| {{sharp2}}<br>(U+E263)
|-
| Substitute symbols
| bb
| db
| b
| d
| h
| t
| #
| t#
| x
|-
| Xen Wiki [[:Category: Character templates|character templates]]
| <small>{{tlx|flat2|plaincode}}</small>
| <small>{{tlx|sesquiflat|plaincode}}<br>{{tlx|sesquiflat2|plaincode}}</small>
| <small>{{tlx|flat|plaincode}}</small>
| <small>{{tlx|demiflat|plaincode}}<br>{{tlx|demiflat2|plaincode}}</small>
| <small>{{tlx|natural|plaincode}}</small>
| <small>{{tlx|demisharp|plaincode}}<br>{{tlx|demisharp2|plaincode}}</small>
| <small>{{tlx|sharp|plaincode}}</small>
| <small>{{tlx|sesquisharp|plaincode}}<br>{{tlx|sesquisharp2|plaincode}}</small>
| <small>{{tlx|sharp2|plaincode}}</small>
|}
 
=== 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 [[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"
|+ style="font-size: 105%;" | Alternative accidentals in (neutral) chain-of-fifths notation
|-
! Style \ offset
! −2
! −1½
! −1
! −½
! 0
! +½
! +1
! +1½
! +2
|- style="vertical-align: top;"
| 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}}
| {{Bravura|&#xE271;}}<br>(U+E271)<br>(U+1D12D)<br>{{Bravura|&#xE278;}}<br>(U+E278)
| {{Flat}}
| {{Bravura|&#xE270;}}<br>(U+E270)<br>(U+1D12C)<br>{{Bravura|&#xE273;}}<br>(U+E273)<br>(U+1D12F)
| {{Natural}}
| {{Bravura|&#xE275;}}<br>(U+E275)<br>(U+1D131)<br>{{Bravura|&#xE272;}}<br>(U+E272)<br>(U+1D12E)
| {{Sharp}}
| {{Bravura|&#xE274;}}<br>(U+E274)<br>(U+1D130)<br>{{Bravura|&#xE277;}}<br>(U+E277)
| {{Sharp2}}
|- style="vertical-align: top;"
| 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}}
| {{Bravura|&#xE460;}}<br>Koron<br>(U+E460)
| {{Natural}}
| {{Bravura|&#xE461;}}<br>Sori<br>(U+E461)
| {{Sharp}}
|
|
|- style="vertical-align: top;"
| 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|&#xE335;}}<br>(U+E335)
| {{Bravura|&#xE327;}}<br>(U+E327)
| {{Bravura|&#xE319;}}<br>(U+E319)
| {{Bravura|&#xE30B;}}<br>(U+E30B)
| {{natural}}
| {{Bravura|&#xE30A;}}<br>(U+E30A)
| {{Bravura|&#xE318;}}<br>(U+E318)
| {{Bravura|&#xE326;}}<br>(U+E326)
| {{Bravura|&#xE334;}}<br>(U+E334)
|- style="vertical-align: top;"
| 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}}
| {{Bravura|&#xE433;}}<br>(U+E433)
| {{Bravura|&#xE430;}}<br>(U+E430)
| {{Bravura|&#xE42D;}}<br>(U+E42D)
| {{natural}}
| {{Bravura|&#xE422;}}<br>(U+E422)
| {{Bravura|&#xE425;}}<br>(U+E425)
| {{Bravura|&#xE428;}}<br>(U+E428)
| {{sharp2}}
|}
 
== 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.
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 circle-of-fifths notation
! Edo
! Fifth
! Fifth-detuning<br>abs (¢), rel (%)
! Major<br>2nd
! Minor<br>2nd
! Augmented<br>1sn
|-
|-
!Edo
! [[12edo|12]]
!Fifth
| 7 || −2.0 ( −2.0%) || 2 || 1 || 1
!Fifth-detuning <br> abs(¢), rel(%)
!Major
2nd
!Minor
2nd
!Augmented
unison
|-
|-
![[12edo|12]]
! [[17edo|17]]
|7||-2.0 ( -2.0%) ||2||1||1
| 10 || +3.9 ( +5.6%) || 3 || 1 || 2
|-
|-
![[17edo|17]]
! [[19edo|19]]
|10||+3.9 ( +5.6%) ||3||1||2
| 11 || −7.2 (−11.4%) || 3 || 2 || 1
|-
|-
![[19edo|19]]
! [[22edo|22]]
|11||-7.2 (-11.4%) ||3||2||1
| 13 || +7.1 (+13.1%) || 4 || 1 || 3
|-
|-
![[22edo|22]]
! [[26edo|26]]
|13||+7.1 (+13.1%) ||4||1||3
| 15 || −9.6 (−20.9%) || 4 || 3 || 1
|-
|-
![[26edo|26]]
! [[27edo|27]]
|15||-9.6 (-20.9%) ||4||3||1
| 16 || +9.2 (+20.6%) || 5 || 1 || 4
|-
|-
![[27edo|27]]
! [[29edo|29]]
|16||+9.2 (+20.6%)||5||1||4
| 17 || +1.5 ( +3.6%) || 5 || 2 || 3
|-
|-
![[29edo|29]]
! [[31edo|31]]
|17||+1.5 ( +3.6%)||5||2||3
| 18 || −5.2 (−13.4%) || 5 || 3 || 2
|-
|-
![[31edo|31]]
! [[32edo|32]]
|18||-5.2 (-13.4%)||5||3||2
| 19 || +10.5 (+28.1%) || 6 || 1 || 5
|-
|-
![[32edo|32]]
! [[33edo|33]]
|19||+10.5 (+28.1%) ||6||1||5
| 19 || −11.0 (−30.4%) || 5 || 4 || 1
|-
|-
![[33edo|33]]
! [[37edo|37]]
|19||-11.0 (-30.4%)||5||4||1
| 22 || +11.6 (+35.6%) || 7 || 1 || 6
|-
|-
![[37edo|37]]
! [[39edo|39]]
|22||+11.6 (+35.6%) ||7||1||6
| 23 || +5.7 (+18.6%) || 7 || 2 || 5
|-
|-
![[39edo|39]]
! [[40edo|40]]
|23||+5.7 (+18.6%)||7||2||5
| 23 || −12.0 (−39.9%) || 6 || 5 || 1
|-
|-
![[40edo|40]]
! [[41edo|41]]
|23||-12.0 (-39.9%) ||6||5||1
| 24 || +0.5 ( +1.7%) || 7 || 3 || 4
|-
|-
![[41edo|41]]
! [[42edo|42]]
|24||+0.5 ( +1.7%) ||7||3||4
| 25 || +12.3 (+43.2%) || 8 || 1 || 7
|-
|-
![[42edo|42]]
! [[43edo|43]]
|25||+12.3 (+43.2%) ||8||1||7
| 25 || −4.3 (−15.3%) || 7 || 4 || 3
|-
|-
![[43edo|43]]
! [[45edo|45]]
|25||-4.3 (-15.3%)||7||4||3
| 26 || −8.6 (−32.3%) || 7 || 5 || 2
|-
|-
![[45edo|45]]
! [[46edo|46]]
|26||-8.6 (-32.3%) ||7||5||2
| 27 || +2.4 ( +9.2%) || 8 || 3 || 5
|-
|-
![[46edo|46]]
! [[47edo|47]]
|27||+2.4 ( +9.2%) ||8||3||5
| 27 || −12.6 (−49.3%) || 7 || 6 || 1
|-
|-
![[47edo|47]]
! [[49edo|49]]
|27||-12.6 (-49.3%) ||7||6||1
| 29 || +8.2 (+33.7%) || 9 || 2 || 7
|-
|-
![[49edo|49]]
! [[50edo|50]]
|29||+8.2 (+33.7%) ||9||2||7
| 29 || −6.0 (−24.8%) || 8 || 5 || 3
|-
|-
![[50edo|50]]
! [[53edo|53]]
|29||-6.0 (-24.8%) ||8||5||3
| 31 || −0.1 ( -0.3%) || 9 || 4 || 5
|-
|-
![[53edo|53]]
! [[55edo|55]]
|31||-0.1 ( -0.3%) ||9||4||5
| 32 || −3.8 (−17.3%) || 9 || 5 || 4
|-
|-
![[55edo|55]]
! [[56edo|56]]
|32||-3.8 (-17.3%) ||9||5||4
| 33 || +5.2 (+24.2%) || 10 || 3 || 7
|-
|-
![[56edo|56]]
! [[59edo|59]]
|33||+5.2 (+24.2%) ||10||3||7
| 35 || +9.9 (+48.7%) || 11 || 2 || 9
|-
|-
![[59edo|59]]
! [[61edo|61]]
|35||+9.9 (+48.7%) ||11||2||9
| 36 || +6.2 (+31.7%) || 11 || 3 || 8
|-
|-
![[61edo|61]]
! [[63edo|63]]
|36||+6.2 (+31.7%) ||11||3||8
| 37 || +2.8 (+14.7%) || 11 || 4 || 7
|-
|-
![[63edo|63]]
! [[64edo|64]]
|37||+2.8 (+14.7%) ||11||4||7
| 37 || −8.2 (−43.8%) || 10 || 7 || 3
|-
|-
![[64edo|64]]
! [[65edo|65]]
|37||-8.2 (-43.8%) ||10||7||3
| 38 || −0.4 ( -2.3%) || 11 || 5 || 6
|-
|-
![[65edo|65]]
! [[67edo|67]]
|38||-0.4 ( -2.3%) ||11||5||6
| 39 || −3.4 (−19.2%) || 11 || 6 || 5
|-
|-
![[67edo|67]]
! [[69edo|69]]
|39||-3.4 (-19.2%) ||11||6||5
| 40 || −6.3 (−36.2%) || 11 || 7 || 4
|-
|-
![[69edo|69]]
! [[70edo|70]]
|40||-6.3 (-36.2%) ||11||7||4
| 41 || +0.9 ( +5.3%) || 12 || 5 || 7
|-
|-
![[70edo|70]]
! [[71edo|71]]
|41||+0.9 ( +5.3%) ||12||5||7
| 42 || +7.9 (+46.8%) || 13 || 3 || 10
|-
|-
![[71edo|71]]
! [[73edo|73]]
|42||+7.9 (+46.8%) ||13||3||10
| 43 || +4.9 (+29.8%) || 13 || 4 || 9
|-
|-
![[73edo|73]]
! [[74edo|74]]
|43||+4.9 (+29.8%) ||13||4||9
| 43 || −4.7 (−28.7%) || 12 || 7 || 5
|-
|-
![[74edo|74]]
! [[75edo|75]]
|43||-4.7 (-28.7%) ||12||7||5
| 44 || +2.0 (+12.8%) || 13 || 5 || 8
|-
|-
![[75edo|75]]
! [[77edo|77]]
|44||+2.0 (+12.8%) ||13||5||8
| 45 || −0.7 ( −4.2%) || 13 || 6 || 7
|-
|-
![[77edo|77]]
! [[79edo|79]]
|45||-0.7 ( -4.2%) ||13||6||7
| 46 || −3.2 (−21.2%) || 13 || 7 || 6
|-
|-
![[79edo|79]]
! [[80edo|80]]
|46||-3.2 (-21.2%) ||13||7||6
| 47 || +3.0 (+20.3%) || 14 || 5 || 9
|-
|-
![[80edo|80]]
! [[81edo|81]]
|47||+3.0 (+20.3%) ||14||5||9
| 47 || −5.7 (−38.2%) || 13 || 8 || 5
|-
|-
![[81edo|81]]
! [[83edo|83]]
|47||-5.7 (-38.2%) ||13||8||5
| 49 || +6.5 (+44.8%) || 15 || 4 || 11
|-
|-
![[83edo|83]]
! [[88edo|88]]
|49||+6.5 (+44.8%) ||15||4||11
| 51 || −6.5 (−47.7%) || 14 || 9 || 5
|-
|-
![[88edo|88]]
! [[89edo|89]]
|51||-6.5 (-47.7%) ||14||9||5
| 52 || −0.8 ( -6.2%) || 15 || 7 || 8
|-
|-
![[89edo|89]]
! [[90edo|90]]
|52||-0.8 ( -6.2%) ||15||7||8
| 53 || +4.7 (+35.3%) || 16 || 5 || 11
|-
|-
![[90edo|90]]
! [[91edo|91]]
|53||+4.7 (+35.3%) ||16||5||11
| 53 || −3.1 (−23.2%) || 15 || 8 || 7
|-
|-
![[91edo|91]]
! [[94edo|94]]
|53||-3.1 (-23.2%) ||15||8||7
| 55 || +0.2 ( +1.4%) || 16 || 7 || 9
|-
|-
![[94edo|94]]
! [[95edo|95]]
|55||+0.2 ( +1.4%) ||16||7||9
| 56 || +5.4 (+42.9%) || 17 || 5 || 12
|-
|-
![[95edo|95]]
! [[97edo|97]]
|56||+5.4 (+42.9%) ||17||5||12
| 57 || +3.2 (+25.9%) || 17 || 6 || 11
|-
|-
![[97edo|97]]
! [[98edo|98]]
|57||+3.2 (+25.9%)||17||6|| 11
| 57 || −4.0 (−32.6%) || 16 || 9 || 7
|-
|-
![[98edo|98]]
! [[99edo|99]]
|57||-4.0 (-32.6%)||16||9||7
| 58 || +1.1 ( +8.9%) || 17 || 7 || 10
|-
|}</div>
![[99edo|99]]
|58||+1.1 ( +8.9%)||17||7||10
|}


<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 circle-of-half-fifths notation
! Edo !! Fifth !! Fifth-detuning<br>abs (¢), rel (%) !! Major<br>2nd !! Minor<br>2nd !! Augmented<br>1sn
|-
!Edo
! Fifth
!Fifth-detuning <br> abs(¢), rel(%)
!Major
2nd
! Minor
2nd
!Augmented
unison
|-
|-
![[17edo|17]]
! [[17edo|17]]
|10 ||+3.9 ( +5.6%)||3||1||2
| 10 || +3.9 ( +5.6%) || 3 || 1 || 2
|-
|-
![[24edo|24]]
! [[24edo|24]]
|14 ||-4.0 (-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 ||-5.2 (-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 ||-7.2 (-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 ||-8.6 (-32.3%)||7||5 ||2
| 26 || −8.6 (−32.3%) || 7 || 5 || 2
|-
|-
![[52edo|52]]
! [[52edo|52]]
| 30 ||-9.6 (-41.8%)||8||6 ||2
| 30 || −9.6 (−41.8%) || 8 || 6 || 2
|-
|-
![[55edo|55]]
! [[55edo|55]]
| 32 ||-3.8 (-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 ||-0.4 ( -2.3%)||11||5 ||6
| 38 || −0.4 (−2.3%) || 11 || 5 || 6
|-
|-
![[69edo|69]]
! [[69edo|69]]
| 40 ||-6.3 (-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 ||-3.2 (-21.2%)||13||7 ||6
| 46 || −3.2 (−21.2%) || 13 || 7 || 6
|-
|-
![[86edo|86]]
! [[86edo|86]]
|50||-4.3 (-30.7%)||14 || 8 ||6
| 50 || −4.3 (−30.7%) || 14 || 8 || 6
|-
|-
![[89edo|89]]
! [[89edo|89]]
|52||-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 ==
* [[Kite's ups and downs notation]] – built on chain-of-fifths notation
* [[Stein–Zimmermann–Gould notation]] – built on neutral chain-of-fifths notation
* [[Syntonic–rastmic subchroma notation]] – built on 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
* [[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 ==
* [[Nominal-accidental chain]]
* [[Chain of fifths]]
* [[Fifthspan]]
* [[Pythagorean tuning]]
* [[User:Xenwolf/cofn]] – sortable table with more intervals (all fifths within the interval [4\7,&nbsp;3\5], the "[[diatonic range]]")
== Notes ==
<references group="note" />


==Expansions ==
== References ==
*[[Syntonic-rastmic subchroma notation]] – built on neutral circle-of-fifths notation
<references />
*[[Ups and downs notation]] – built on circle-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


==See also==
{{Navbox notation}}
*[[Nominal-accidental chain]]
*[[Circle of fifths]]
*[[Fifthspan]]
*[[User:Xenwolf/cofn]] – sortable table with more intervals (all fifths within the interval [4\7, 3\5], the "[[diatonic range]]")


[[Category:Notation]]
[[Category:Notation]]
[[Category:Method]]
[[Category:Method]]
[[Category:Fifth]]
[[Category:Fifth]]
Retrieved from "https://en.xen.wiki/w/Chain-of-fifths_notation"