View source for 53 equal temperament

{{Short description|Musical tuning system of 53 pitches}}

[[File:Syntonic tuning continuum.svg|right|thumb|Figure&nbsp;1: 53&nbsp;TET on the syntonic temperament's tuning continuum at 701.89&nbsp;cents, from {{harvp|Milne|Sethares|Plamondon|2007}}<ref>
{{cite journal
 |last1 = Milne     |first1 = Andrew
 |last2 = Sethares  |first2 = William
 |last3 = Plamondon |first3 = James
 |year = 2007
 |title = Isomorphic controllers and dynamic tuning: Invariant fingering over a tuning continuum
 |journal = Computer Music Journal
 |volume = 31 |issue = 4 |pages = 15–32
 |s2cid = 27906745
 |doi = 10.1162/comj.2007.31.4.15 |doi-access = free
 |url=http://www.mitpressjournals.org/doi/pdf/10.1162/comj.2007.31.4.15
 |via=mitpressjournals.org
}}
</ref>]]

In music, '''53&nbsp;equal temperament''', called '''53&nbsp;TET''', '''53&nbsp;[[equal division of the octave|EDO]]''', or '''53&nbsp;ET''', is the [[musical temperament|tempered]] scale derived by dividing the octave into 53&nbsp;equal steps (equal frequency ratios) ({{audio|53-tet scale on C.mid|Play}}). Each step represents a frequency ratio of {{math|  {{big|2}}{{sup|1&hairsp;∕&hairsp;53}} ,}} or 22.6415&nbsp;[[cent (music)|cents]] ({{Audio|Holdrian comma on C.mid|Play}}), an interval sometimes called the '''[[#Holdrian comma|Holdrian comma]]'''.

53&nbsp;TET is a tuning of [[equal temperament]] in which the tempered perfect fifth is 701.89&nbsp;cents wide, as shown in Figure&nbsp;1, and sequential pitches are separated by 22.642&nbsp;cents.

The 53-TET tuning equates to the unison, or ''tempers out'', the intervals {{math|{{sfrac| 32 805 | 32 768 }},}} known as the [[schisma]], and {{math|{{sfrac| 15 625 | 15 552 }},}} known as the [[kleisma]]. These are both 5&nbsp;limit intervals, involving only the primes 2, 3, and 5 in their factorization, and the fact that 53&nbsp;TET tempers out both characterizes it completely as a 5&nbsp;limit temperament: It is the only [[regular temperament]] tempering out both of these intervals, or [[comma (music)|comma]]s, a fact which seems to have first been recognized by Japanese music theorist [[Shohé Tanaka]]. Because it tempers these out, 53&nbsp;TET can be used for both [[schismatic temperament]], tempering out the schisma, and [[Hanson temperament]] (also called kleismic), tempering out the kleisma.

The interval of {{math|{{sfrac| 7 | 4 }}}} is closest to the 43rd note (counting from 0) and {{math|  {{big|2}}{{sup|43&hairsp;∕&hairsp;53}} {{=}} 1.7548  }} is only 4.8&nbsp;cents sharp from the [[harmonic seventh|harmonic&nbsp;7th]] {{math|  {{=}} {{sfrac| 7 | 4 }}}} in 53&nbsp;TET, and using it for [[7-limit tuning|7-limit]] harmony means that the [[septimal kleisma]], the interval {{sfrac| 225 | 224 }}, is also tempered out.

== History and use ==
<!-- This section is linked from [[Pythagorean comma]] -->
Theoretical interest in this division goes back to antiquity. [[Jing Fang]] (78–37&nbsp;BCE), a Chinese music theorist, observed that a series of 53&nbsp;[[just fifth]]s {{math|{{big|( [}}{{sfrac| 3 | 2 }}{{big|]}}{{sup|53}}}} {{big|)}} is very nearly equal to 31&nbsp;octaves ({{math|{{big|2}}{{sup|31}}}}). He calculated this difference with six-digit accuracy to be {{sfrac| 177 147 | 176 776 }}.<ref>{{cite journal |author1-link=Ernest G. McClain |last1=McClain |first1=Ernest |first2=Ming Shui |last2=Hung |year=1979 |title=Chinese cyclic tunings in late antiquity |journal=[[Ethnomusicology (journal)|Ethnomusicology]] |volume=23 |issue=2 |pages=205–224 }}</ref><ref>{{cite web |title=後漢書/卷91 - 维基文库，自由的图书馆 |lang=zh |trans-title=Book of the Later Han Dynasty / Volume 91 - Wikisource, the free library |url=https://zh.wikisource.org/wiki/%E5%BE%8C%E6%BC%A2%E6%9B%B8/%E5%8D%B791 |access-date=2022-06-23 |website=zh.wikisource.org }}</ref> Later the same observation was made by the mathematician and music theorist [[Nicholas Mercator]] ({{circa|1620–1687}}), who calculated this value precisely as {{math|{{sfrac| 3{{sup|53}} | 2{{sup|84}} }} {{=}} {{sfrac| 19 383 245 667 680 019 896 796 723 | 19 342 813 113 834 066 795 298 816 }},}}{{verification needed|date=November 2019}} which is known as [[Mercator's comma]].<ref>{{cite web |last=Monzo |first=Joe |year=2005 |url=http://www.tonalsoft.com/enc/m/mercator-comma.aspx |title=Mercator's comma |website=Tonalsoft }}</ref> Mercator's comma is of such small value to begin with {{nobr|({{math| ≈ 3.615}} cents),}} but 53&nbsp;equal temperament flattens each fifth by only {{math|{{sfrac| 1 | 53 }}}} of that comma ({{nobr| ≈ {{math|0.0682}} cent}} {{nobr| ≈ {{math|{{sfrac| 1 | 315 }}}} [[syntonic comma]]}} {{nobr| ≈ {{math|{{sfrac| 1 | 344 }}}} [[pythagorean comma]]).}} Thus, 53&nbsp;tone equal temperament is for all practical purposes equivalent to an extended [[Pythagorean tuning]].

After Mercator, [[William Holder]] published a treatise in 1694 which pointed out that 53&nbsp;equal temperament also very closely approximates the [[just major third]] (to within 1.4&nbsp;cents), and consequently 53&nbsp;equal temperament accommodates the intervals of [[five-limit tuning|5&nbsp;limit]] [[just intonation]] very well.<ref>{{harvp|Holder|1967}}</ref><ref>{{cite book |last=Stanley |first=Jerome |year=2002 |title=William Holder and His Position in Seventeenth-Century Philosophy and Music Theory |publisher=The Edwin Mellen Press }} — see also {{harvp|Holder|1967}} </ref> This property of 53&nbsp;TET may have been known earlier; [[Isaac Newton]]'s unpublished manuscripts suggest that he had been aware of it as early as 1664–1665.<ref>{{cite book |last=Barbieri |first=Patrizio |year=2008 |title=Enharmonic Instruments and Music, 1470–1900 |publisher=Latina, Il Levante Libreria Editrice |page=350 |url=http://www.patriziobarbieri.it/1.htm |archive-url=https://web.archive.org/web/20090215045859/http://www.patriziobarbieri.it/1.htm |archive-date=2009-02-15 }}</ref>

=== Music ===
In the 19th&nbsp;century, people began devising instruments in 53&nbsp;TET, with an eye to their use in playing near-just [[Limit (music)|5-limit]] music. Such instruments were devised by [[Robert Holford Macdowall Bosanquet|R.H.M. Bosanquet]]<ref name=Helmholtz-Ellis-1954>{{cite book |last1=von&nbsp;Helmholtz |first1=H.L.F. |author-link=Hermann von Helmholtz |editor-last=Ellis |editor-first=Alexander |year=1954 |title=On the Sensations of Tone |edition=2nd English |lang=en |publisher=Dover Publications |pages=328–329 }}</ref>{{rp|style=ama|p= 328–329}} and the American tuner [[James Paul White|J.P. White]].<ref name=Helmholtz-Ellis-1954/>{{rp|style=ama|p= 329}} Subsequently, the temperament has seen occasional use by composers in the west, and by the early 20th&nbsp;century, 53&nbsp;TET had become the most common form of tuning in [[Ottoman music|Ottoman classical music]], replacing its older, unequal tuning. [[Arabic music]], which for the most part bases its theory on [[quartertone]]s, has also made some use of it; the Syrian violinist and music theorist [[Twfiq Al-Sabagh]] proposed that instead of an equal division of the octave into 24&nbsp;parts a 24&nbsp;note scale in 53&nbsp;TET should be used as the master scale for Arabic music.{{Citation needed|date=September 2021}}

Croatian composer [[Josip Štolcer-Slavenski]] wrote one piece, which has never been published, which uses [[Robert Holford Macdowall Bosanquet|Bosanquet's Enharmonium]] during its first movement, entitled ''Music for Natur-ton-system''.<ref name=Slavenski-2017-06-21a>
{{cite web
 |first=Josip |last=Slavencki |author-link=Josip Štolcer-Slavenski
 |date=21 June 2007a
 |title=Preface
 |department=53&nbsp;EDO piece
 |type=manuscript
 |publisher=The Faculty of Music
 |place=Belgrade, Serbia
 |url=https://commons.wikimedia.org/wiki/File:J_Slavenski_Preface_to_53EDO_piece.jpg
 |via=Wikimedia Commons
}}
</ref><ref name=Slavenski-2017-06-21b>
{{cite web
 |first=Josip |last=Slavencki |author-link=Josip Štolcer-Slavenski
 |date=21 June 2007b
 |title=Title
 |department=53&nbsp;EDO movement
 |type=manuscript
 |publisher=The Faculty of Music
 |place=Belgrade, Serbia
 |via=Wikimedia Commons
 |url=https://commons.wikimedia.org/wiki/File:J_Slavenski_Title_with_53EDO.jpg
}}
</ref><ref name=Khramov-2018-02-Slavenski>
{{cite web
 |first=Josip |last=Slavenski |author-link=Josip Štolcer-Slavenski
 |date=February 2018
 |title=Music Natural 53e6v
 |department=53&nbsp;EDO movement
 |editor-last=Khramov |editor-first=Mykhaylo
 |website=soundcloud.com
 |url=https://soundcloud.com/mykhaylo-khramov/slavenski-musicnatural53e6v |via=soundcloud.com
}}<br/>
{{cite web
 |editor-last=Khramov |editor-first=Mykhaylo
 |date=February 2018
 |title=Link to ZIP with materials
 |department=53&nbsp;EDO movement
 |url=https://drive.google.com/file/d/1KcQU0bvkElYJG54qesu43XLf5_kJzjgH/view
 |via=Google Drive
}}
</ref>{{efn|
"Croatian composer Josip Štolcer-Slavenski wrote one piece,<ref name=Slavenski-2017-06-21a/><ref name=Slavenski-2017-06-21b/> which has never been published, which uses Bosanquet's Enharmonium during its first movement, entitled Music for Natur-ton-system".<ref name=Khramov-2018-02-Slavenski/>
}}


Furthermore, General Thompson worked in league with the London-based guitar maker [[Louis Panormo]] to produce the Enharmonic Guitar.<ref>{{cite magazine |first=James |last=Westbrook |year=2012 |title=General Thompson’s enharmonic guitar |magazine=[[Soundboard (magazine)|Soundboard]] |volume=38 |issue=4 |pages=45–52 |url=https://talks.cam.ac.uk/talk/index/54328 }}</ref>

==Notation==
[[File:Koma_gosterim.png|thumb|Notation used in Ottoman classical music, where the whole notes are divided into 9 commas.|440x440px]]
Attempting to use standard notation, seven-letter notes plus sharps or flats, can quickly become confusing. This is unlike the case with [[19 equal temperament|19&nbsp;TET]] and [[31 equal temperament|31&nbsp;TET]] where there is little ambiguity. By not being meantone, it adds some problems that require more attention. Specifically, the Pythagorean major third ([[ditone]]) and just major third are distinguished, as are the Pythagorean minor third (semiditone) and just minor third. The fact that the [[syntonic comma]] is not tempered out means that notes and intervals need to be defined more precisely. [[Ottoman classical music]] uses a notation of flats and sharps for the 9&nbsp;comma tone.

Furthermore, since 53 is not a multiple of 12, notes such as G{{music|sharp}} and A{{music|flat}} are not enharmonically equivalent, nor are the corresponding [[key signature]]s. As a result, many key signatures will require the use of double sharps (such as G{{music|sharp}} major / E{{music|sharp}} minor), double flats (such as F{{music|flat}} major / D{{music|flat}} minor), or microtonal alterations.

'''Extended pythagorean notation''', using only sharps and flats, gives the following chromatic scale:

* C, B{{music|sharp}}, A{{music|sharp}}{{music|doublesharp}}, E{{music|tripleflat}}, D{{music|flat}}, C{{music|sharp}}, B{{music|doublesharp}}, F{{music|tripleflat}}, E{{music|doubleflat}},
* D, C{{music|doublesharp}}, B{{music|sharp}}{{music|doublesharp}}, F{{music|doubleflat}}, E{{music|flat}}, D{{music|sharp}}, C{{music|sharp}}{{music|doublesharp}}, G{{music|tripleflat}}, F{{music|flat}},
* E, D{{music|doublesharp}}, C{{music|doublesharp}}{{music|doublesharp}}/A{{music|doubleflat}}{{music|doubleflat}}, G{{music|doubleflat}},
* F, E{{music|sharp}}, D{{music|sharp}}{{music|doublesharp}}, A{{music|tripleflat}}, G{{music|flat}}, F{{music|sharp}}, E{{music|doublesharp}}, D{{music|doublesharp}}{{music|doublesharp}}/B{{music|doubleflat}}{{music|doubleflat}}, A{{music|doubleflat}},
* G, F{{music|doublesharp}}, E{{music|sharp}}{{music|doublesharp}}, B{{music|tripleflat}}, A{{music|flat}}, G{{music|sharp}}, F{{music|sharp}}{{music|doublesharp}}, C{{music|tripleflat}}, B{{music|doubleflat}},
* A, G{{music|doublesharp}}, F{{music|doublesharp}}{{music|doublesharp}}/D{{music|doubleflat}}{{music|doubleflat}}, C{{music|doubleflat}}, B{{music|flat}}, A{{music|sharp}}, G{{music|sharp}}{{music|doublesharp}}, D{{music|tripleflat}}, C{{music|flat}},
* B, A{{music|doublesharp}}, G{{music|doublesharp}}{{music|doublesharp}}/E{{music|doubleflat}}{{music|doubleflat}}, D{{music|doubleflat}}, C

Unfortunately, the notes run out of letter-order, and up to quadruple sharps and flats are required. As a result, a just major 3rd must be spelled as a diminished 4th. 

'''Ups and downs notation'''<ref>{{cite web |title=Ups and downs notation |website=Xenharmonic Wiki (en.xen.wiki) |url=https://en.xen.wiki/w/Ups_and_downs_notation |access-date=2024-08-19 |lang=en |df=dmy-all }}</ref> keeps the notes in order and also preserves the traditional meaning of sharp and flat. It uses up and down arrows, written as a caret or a lower-case "v", usually in a sans-serif font. One arrow equals one step of 53-TET. In note names, the arrows come first, to facilitate chord naming. The many enharmonic equivalences allow great freedom of spelling.

* C, ^C, ^^C, vvC{{music|sharp}}/vD{{music|flat}}, vC{{music|sharp}}/D{{music|flat}}, C{{music|sharp}}/^D{{music|flat}}, ^C{{music|sharp}}/^^D{{music|flat}}, vvD, vD,
* D, ^D, ^^D, vvD{{music|sharp}}/vE{{music|flat}}, vD{{music|sharp}}/E{{music|flat}}, D{{music|sharp}}/^E{{music|flat}}, ^D{{music|sharp}}/^^E{{music|flat}}, vvE, vE,
* E, ^E, ^^E/vvF, vF,
* F, ^F, ^^F, vvF{{music|sharp}}/vG{{music|flat}}, vF{{music|sharp}}/G{{music|flat}}, F{{music|sharp}}/^G{{music|flat}}, ^F{{music|sharp}}/^^G{{music|flat}}, vvG, vG,
* G, ^G, ^^G, vvG{{music|sharp}}/vA{{music|flat}}, vG{{music|sharp}}/A{{music|flat}}, G{{music|sharp}}/^A{{music|flat}}, ^G{{music|sharp}}/^^A{{music|flat}}, vvA, vA,
* A, ^A, ^^A, vvA{{music|sharp}}/vB{{music|flat}}, vA{{music|sharp}}/B{{music|flat}}, A{{music|sharp}}/^B{{music|flat}}, ^A{{music|sharp}}/^^B{{music|flat}}, vvB, vB,
* B, ^B, ^^B/vvC, vC, C

=== Chords of 53 equal temperament ===
Since 53-TET is a Pythagorean system, with nearly pure fifths, justly-intonated major and minor triads cannot be spelled in the same manner as in a [[meantone temperament|meantone]] tuning. Instead, the major triads are chords like C-F{{music|b}}-G (using the Pythagorean-based notation), where the major third is a diminished fourth; this is the defining characteristic of [[schismatic temperament]]. Likewise, the minor triads are chords like C-D{{music|#}}-G. In 53-TET, the [[dominant seventh chord]] would be spelled C-F{{music|b}}-G-B{{music|b}}, but the [[otonal]] tetrad is C-F{{music|b}}-G-C{{music|bb}}, and C-F{{music|b}}-G-A{{music|#}} is still another seventh chord. The [[utonal]] tetrad, the inversion of the otonal tetrad, is spelled C-D{{music|#}}-G-G{{music|x}}.

Further septimal chords are the diminished triad, having the two forms C-D{{music|#}}-G{{music|b}} and C-F{{music|bb}}-G{{music|b}}, the subminor triad, C-F{{music|bb}}-G, the supermajor triad C-D{{music|x}}-G, and corresponding tetrads C-F{{music|bb}}-G-B{{music|bb}} and C-D{{music|x}}-G-A{{music|#}}. Since 53-TET tempers out the [[septimal kleisma]], the septimal kleisma augmented triad C-F{{music|b}}-B{{music|bbb}} in its various inversions is also a chord of the system. So is the Orwell tetrad, C-F{{music|b}}-D{{music|x}}{{music|x}}-G{{music|x}} in its various inversions.

Ups and downs notation permits more conventional spellings. Since it also names the intervals of 53&nbsp;TET,<ref>{{cite web |title=53edo intervals |website=Xenharmonic Wiki (en.xen.wiki) |url=https://en.xen.wiki/w/53edo#Intervals |access-date=2024-08-19 |lang=en |df=dmy-all }}</ref> it provides precise chord names too. The pythagorean minor chord with a {{sfrac| 32 | 27 }} third is still named Cm and still spelled C–E{{music|b}}–G. But the 5-limit ''up''minor chord uses the upminor 3rd 6/5 and is spelled C–^E{{music|b}}–G. This chord is named C^m. Compare with ^Cm (^C–^E{{music|b}}–^G). 

* Major triad: C-vE-G (downmajor)
* Minor triad: C-^E{{music|b}}-G (upminor)
* Dominant 7th: C-vE-G-B{{music|b}} (down add-7)
* Otonal tetrad: C-vE-G-vB{{music|b}} (down7)
* Utonal tetrad: C-^E{{music|b}}-G-^A (upminor6)
* Diminished triad: C-^E{{music|b}}-G{{music|b}} (updim)
* Diminished triad: C-vE{{music|b}}-G{{music|b}} (downdim)
* Subminor triad: C-vE{{music|b}}-G (downminor)
* Supermajor triad: C-^E-G (upmajor)
* Subminor tetrad: C-vE{{music|b}}-G-vA (downminor6)
* Supermajor tetrad: C-^E-G-^B{{music|b}} (up7)
* Augmented triad: C-vE-vvG{{music|#}} (downaug dud-5)
* Orwell triad: C-vE-vvG-^A (downmajor dud-5 up6)

== Interval size ==
[[Image:53ed2.svg|thumb|right|250px|7-Limit just intonation intervals approximated in 53&nbsp;TET]]
Because a distance of 31&nbsp;[[Holdrian comma|steps]] in this scale is almost precisely equal to a [[Just intonation|just]] [[perfect fifth]], in theory this scale can be considered a slightly tempered form of [[Pythagorean tuning]] that has been extended to 53&nbsp;tones. As such the intervals available can have the same properties as any Pythagorean tuning, such as fifths that are (practically) pure, major thirds that are wide from just (about {{math|{{sfrac| 81 | 64 }}}} opposed to the purer {{math|{{sfrac| 5 | 4 }},}} and minor thirds that are conversely narrow ({{math|{{sfrac| 32 | 27 }}}} compared to {{math|{{sfrac| 6 | 5 }}}}).

However, 53&nbsp;TET contains additional intervals that are very close to just intonation. For instance, the interval of 17&nbsp;steps is also a major third, but only 1.4&nbsp;cents narrower than the very pure just interval {{math|{{sfrac| 5 | 4 }}.}} 53&nbsp;TET is very good as an approximation to any interval in 5&nbsp;limit just intonation. Similarly, the pure just interval {{math|{{sfrac| 6 | 5 }}}} is only 1.3&nbsp;cents wider than 14&nbsp;steps in 53&nbsp;TET.

The matches to the just intervals involving the 7th&nbsp;harmonic are slightly less close (43&nbsp;steps are 4.8&nbsp;cents sharp for [[harmonic seventh|{{math|{{sfrac| 7 | 4 }}}}]]), but all such intervals are still quite closely matched with the highest deviation being the {{math|{{sfrac| 7 | 5 }}}}&nbsp;tritone. The 11th&nbsp;harmonic and intervals involving it are less closely matched, as illustrated by the undecimal neutral seconds and thirds in the table below. 7-limit ratios are colored light gray, and 11- and 13-limit ratios are colored dark gray.

{| class="wikitable sortable" style="text-align: center;"
! bgcolor="#ffffb4" | Size<br/>([[Holdrian comma|steps]])
! bgcolor="#ffffb4" | Size<br/>([[musical cents|cents]])
! bgcolor="#ffffb4" | Interval name
! bgcolor="#ffffb4" | {{small|Nearest<br/>Just ratio}}
! bgcolor="#ffffb4" | Just<br/>([[musical cents|cents]])
! bgcolor="#ffffb4" | Error<br/>([[musical cents|cents]])
! bgcolor="#ffffb4" | Limit
|-
| 53
| 1200 
| [[perfect octave]]
| {{math| {{sfrac| 2 | 1 }} }}
| 1200 
| 0
| 2
|-
| 52
| 1177.36
| grave [[perfect octave|octave]]
| {{math| {{sfrac| 160 | 81 }} }}
| 1178.49
| &minus;1.14
| 5
|-
| 51
| 1154.72
| just [[augmented seventh]]
| {{math| {{sfrac| 125 | 64 }} }}
| 1158.94
| &minus;4.22
| 5
|-
| 50
| 1132.08
| diminished [[perfect octave|octave]]
| {{math| {{sfrac| 48 | 25 }} }}
| 1129.33
| +2.75
| 5
|-
| 48
| 1086.79
| just [[major seventh]]
| {{math| {{sfrac| 15 | 8 }} }}
| 1088.27
| &minus;1.48
| 5
|-
| 45
| 1018.87
| just [[minor seventh]]
| {{math| {{sfrac| 9 | 5 }} }}
| 1017.60
|  +1.27
| 5
|-
| 44
| 996.23
| Pythagorean [[minor seventh]]
| {{math| {{sfrac| 16 | 9 }} }}
| 996.09
|  +0.14
| 3
|-
| 43
| 973.59
| [[harmonic seventh|accute augmented sixth]]
| {{math| {{sfrac| 225 | 128 }} }}
| 976.54
| &minus;2.95
| 5
|-
| bgcolor="#EEEEEE" | 43
| bgcolor="#EEEEEE" | 973.59
| bgcolor="#EEEEEE" | [[harmonic seventh]]
| bgcolor="#EEEEEE" | {{math| {{sfrac| 7 | 4 }} }}
| bgcolor="#EEEEEE" | 968.83
| bgcolor="#EEEEEE" | +4.76
| bgcolor="#EEEEEE" | 7
|-
| 43
| 973.59
| [[harmonic seventh|accute diminished seventh]]
| {{math| {{sfrac| 17 496 | 10 000 }} }}
| 968.43
| +5.15
| 5
|-
| 42
| 950.94
| just [[augmented sixth]]
| {{math| {{sfrac| 125 | 72 }} }}
| 955.03
| &minus;4.09
| 5
|-
| 42
| 950.94
| just [[diminished seventh]]
| {{math| {{sfrac| 216 | 125 }} }}
| 946.92
| +4.02
| 5
|-
| 39
| 883.02
| [[major sixth]]
| {{math| {{sfrac| 5 | 3 }} }}
| 884.36
| &minus;1.34
| 5
|-
| bgcolor="#D4D4D4" | 37
| bgcolor="#D4D4D4" | 837.73
| bgcolor="#D4D4D4" | tridecimal neutral sixth
| bgcolor="#D4D4D4" | {{math| {{sfrac| 13 | 8 }} }}
| bgcolor="#D4D4D4" | 840.53
| bgcolor="#D4D4D4" | &minus;2.8
| bgcolor="#D4D4D4" | 13
|-
| 36
| 815.09
| [[minor sixth]]
| {{math| {{sfrac| 8 | 5 }} }}
| 813.69
|  +1.40
| 5
|-
| 31
| 701.89
| [[perfect fifth]]
| {{math| {{sfrac| 3 | 2 }} }}
| 701.96
| &minus;0.07
| 3
|-
| 30
| 679.25
| grave fifth
| {{math| {{sfrac| 40 | 27 }} }}
| 680.45
| &minus;1.21
| 5
|-
| 28
| 633.96
| just [[diminished fifth]]<br/>{{small|(greater [[tritone]])}}
| {{math| {{sfrac| 36 | 25 }} }}
| 631.28
| +2.68
| 5
|-
| 27
| 611.32
| Pythagorean augmented fourth
| {{math| {{sfrac| 729 | 512 }} }}
| 611.73
| &minus;0.41
| 3
|-
| 27
| 611.32
| greater ‘classic’ [[tritone]]
| {{math| {{sfrac| 64 | 45 }} }}
| 609.78
| +1.54
| 5
|-
| 26
| 588.68
| lesser ‘classic’ [[tritone]]
| {{math| {{sfrac| 45 | 32 }} }}
| 590.22
| &minus;1.54
| 5
|-
| bgcolor="#EEEEEE" | 26
| bgcolor="#EEEEEE" | 588.68
| bgcolor="#EEEEEE" | [[septimal tritone]]
| bgcolor="#EEEEEE" | {{math| {{sfrac| 7 | 5 }} }}
| bgcolor="#EEEEEE" | 582.51
| bgcolor="#EEEEEE" | +6.17
| bgcolor="#EEEEEE" | 7
|-
| 25
| 566.04
| just [[augmented fourth]]<br/>{{small|(lesser [[tritone]])}}
| {{math| {{sfrac| 25 | 18 }} }}
| 568.72
| &minus;2.68
| 5
|-
| bgcolor="#D4D4D4" | 24
| bgcolor="#D4D4D4" | 543.40
| bgcolor="#D4D4D4" | undecimal [[major fourth]]
| bgcolor="#D4D4D4" | {{math| {{sfrac| 11 | 8 }} }}
| bgcolor="#D4D4D4" | 551.32
| bgcolor="#D4D4D4" | &minus;7.92
| bgcolor="#D4D4D4" | 11
|-
| 24
| 543.40
| double diminished fifth
| {{math| {{sfrac| 512 | 375 }} }}
| 539.10
|  +4.30
| 5

|-
| bgcolor="#D4D4D4" | 24
| bgcolor="#D4D4D4" | 543.40
| bgcolor="#D4D4D4" | undecimal augmented fourth
| bgcolor="#D4D4D4" | {{math| {{sfrac| 15 | 11 }} }}
| bgcolor="#D4D4D4" | 536.95
| bgcolor="#D4D4D4" | +6.45
| bgcolor="#D4D4D4" | 11
|-
| 23
| 520.76
| acute fourth
| {{math| {{sfrac| 27 | 20 }} }}
| 519.55
|  +1.21
| 5
|-
| 22
| 498.11
| [[perfect fourth]]
| {{math| {{sfrac| 4 | 3 }} }}
| 498.04
|  +0.07
| 3
|-
| 21
| 475.47
| grave fourth
| {{math| {{sfrac| 320 | 243 }} }}
| 476.54
| &minus;1.07
| 5
|-
| bgcolor="#EEEEEE" | 21
| bgcolor="#EEEEEE" | 475.47
| bgcolor="#EEEEEE" | septimal narrow fourth
| bgcolor="#EEEEEE" | {{math| {{sfrac| 21 | 16 }} }}
| bgcolor="#EEEEEE" | 470.78
| bgcolor="#EEEEEE" | +4.69
| bgcolor="#EEEEEE" | 7
|-
| 20
| 452.83
| just augmented third
| {{math| {{sfrac| 125 | 96 }} }}
| 456.99
| &minus;4.16
| 5
|-
| bgcolor="#D4D4D4" | 20
| bgcolor="#D4D4D4" | 452.83
| bgcolor="#D4D4D4" | tridecimal augmented third
| bgcolor="#D4D4D4" | {{math| {{sfrac| 13 | 10 }} }}
| bgcolor="#D4D4D4" | 454.21
| bgcolor="#D4D4D4" | &minus;1.38
| bgcolor="#D4D4D4" | 13
|-
| bgcolor="#EEEEEE" | 19
| bgcolor="#EEEEEE" | 430.19
| bgcolor="#EEEEEE" | [[septimal major third]]
| bgcolor="#EEEEEE" | {{math| {{sfrac| 9 | 7 }} }}
| bgcolor="#EEEEEE" | 435.08
| bgcolor="#EEEEEE" | &minus;4.90
| bgcolor="#EEEEEE" | 7
|-
| 19
| 430.19
| just diminished fourth
| {{math| {{sfrac| 32 | 25 }} }}
| 427.37
|  +2.82
| 5
|-
| 18
| 407.54
| [[ditone|Pythagorean ditone]]
| {{math| {{sfrac| 81 | 64 }} }}
| 407.82
| &minus;0.28
| 3
|-
| 17
| 384.91
| [[Just intonation|just]] [[major third]]
| {{math| {{sfrac| 5 | 4 }} }}
| 386.31
| &minus;1.40
| 5
|-
| 16
| 362.26
| grave major third
| {{math| {{sfrac| 100 | 81 }} }}
| 364.80
| &minus;2.54
| 5
|-
| bgcolor="#D4D4D4" | 16
| bgcolor="#D4D4D4" | 362.26
| bgcolor="#D4D4D4" | [[neutral third]], tridecimal
| bgcolor="#D4D4D4" | {{math| {{sfrac| 16 | 13 }} }}
| bgcolor="#D4D4D4" | 359.47
| bgcolor="#D4D4D4" |  +2.79
| bgcolor="#D4D4D4" | 13
|-
| bgcolor="#D4D4D4" | 15
| bgcolor="#D4D4D4" | 339.62
| bgcolor="#D4D4D4" | [[neutral third]], undecimal
| bgcolor="#D4D4D4" | {{math| {{sfrac| 11 | 9 }} }}
| bgcolor="#D4D4D4" | 347.41
| bgcolor="#D4D4D4" | &minus;7.79
| bgcolor="#D4D4D4" | 11
|-
| 15
| 339.62
| acute minor third
| {{math| {{sfrac| 243 | 200 }} }}
| 337.15
|  +2.47
| 5
|-
| 14
| 316.98
| just [[minor third]]
| {{math| {{sfrac| 6 | 5 }} }}
| 315.64
|  +1.34
| 5
|-
| 13
| 294.34
| [[semiditone|Pythagorean semiditone]]
| {{math| {{sfrac| 32 | 27 }} }}
| 294.13
|  +0.21
| 3
|-
| 12
| 271.70
| just augmented second
| {{math| {{sfrac| 75 | 64 }} }}
| 274.58
| &minus;2.88
| 5
|-
| bgcolor="#EEEEEE" | 12
| bgcolor="#EEEEEE" | 271.70
| bgcolor="#EEEEEE" | [[septimal minor third]]
| bgcolor="#EEEEEE" | {{math| {{sfrac| 7 | 6 }} }}
| bgcolor="#EEEEEE" | 266.87
| bgcolor="#EEEEEE" | +4.83
| bgcolor="#EEEEEE" | 7
|-
| 11
| 249.06
| just diminished third
| {{math| {{sfrac| 144 | 125 }} }}
| 244.97
|  +4.09
| 5
|-
| bgcolor="#EEEEEE" | 10
| bgcolor="#EEEEEE" | 226.41
| bgcolor="#EEEEEE" | [[septimal whole tone]]
| bgcolor="#EEEEEE" | {{math| {{sfrac| 8 | 7 }} }}
| bgcolor="#EEEEEE" | 231.17
| bgcolor="#EEEEEE" | &minus;4.76
| bgcolor="#EEEEEE" | 7
|-
| 10
| 226.41
| diminished third
| {{math| {{sfrac| 256 | 225 }} }}
| 223.46
|  +2.95
| 5
|-
| 9
| 203.77
| [[whole tone]], [[major tone]],<br/> greater tone, [[major second|just&nbsp;second]]
| {{math| {{sfrac| 9 | 8 }} }}
| 203.91
| &minus;0.14
| 3
|-
| 8
| 181.13
| grave [[whole tone]], [[minor tone]],<br/> lesser tone, {{nobr|grave [[major second|second]]}}
| {{math| {{sfrac| 10 | 9 }} }}
| 182.40
| &minus;1.27
| 5
|-
| bgcolor="#D4D4D4" | 7
| bgcolor="#D4D4D4" | 158.49
| bgcolor="#D4D4D4" | [[neutral second]], greater undecimal
| bgcolor="#D4D4D4" | {{math| {{sfrac| 11 | 10 }} }}
| bgcolor="#D4D4D4" | 165.00
| bgcolor="#D4D4D4" | &minus;6.51
| bgcolor="#D4D4D4" | 11
|-
| 7
| 158.49
| doubly grave whole tone
| {{math| {{sfrac| 800 | 729 }} }}
| 160.90
| &minus;2.41
| 5
|-
| bgcolor="#D4D4D4" | 7
| bgcolor="#D4D4D4" | 158.49
| bgcolor="#D4D4D4" | [[neutral second]], lesser undecimal
| bgcolor="#D4D4D4" | {{math| {{sfrac| 12 | 11 }} }}
| bgcolor="#D4D4D4" | 150.64
| bgcolor="#D4D4D4" | +7.85
| bgcolor="#D4D4D4" | 11
|-
| 6
| 135.85
| accute [[semitone|diatonic semitone]]
| {{math| {{sfrac| 27 | 25 }} }}
| 133.24
|  +2.61
| 5
|-
| 5
| 113.21
| [[semitone|greater Pythagorean semitone]]
| {{math| {{sfrac| 2 187 | 2 048 }} }}
| 113.69
| &minus;0.48
| 3
|-
| 5
| 113.21
| just [[diatonic semitone]],<br/> [[semitone|just minor second]]
| {{math| {{sfrac| 16 | 15 }} }}
| 111.73
|  +1.48
| 5
|-
| 4
| 90.57
| [[major limma]]
| {{math| {{sfrac| 135 | 128 }} }}
| 92.18
| &minus;1.61
| 5
|-
| 4
| 90.57
| [[semitone|lesser Pythagorean semitone]]
| {{math| {{sfrac| 256 | 243 }} }}
| 90.22
|  +0.34
| 3
|-
| 3
| 67.92
| just [[chromatic semitone]]
| {{math| {{sfrac| 25 | 24 }} }}
| 70.67
| &minus;2.75
| 5
|-
| 3
| 67.92
| greater diesis
| {{math| {{sfrac| 648 | 625 }} }}
| 62.57
| +5.35
| 5
|-
| 2
| 45.28
| just [[diesis]]
| {{math| {{sfrac| 128 | 125 }} }}
| 41.06
|  +4.22
| 5
|-
| 1
| 22.64
| [[syntonic comma]]
| {{math| {{sfrac| 81 | 80 }} }}
| 21.51
|  +1.14
| 5
|-
| 0
| 0
| [[perfect unison]]
| {{math| {{sfrac| 1 | 1 }} }}
| 0
| 0
| 1
|}

== Scale diagram ==
<!--Correct using standard ratios:C=1/1,C#=25/24,Db=16/15,D=9/8,D#=75/64,Eb=6/5,E=5/4,Fb=32/25,E#=125/96,F=4/3,F#=45/32,Gb=64/45,G=3/2,G#=25/16,Ab=8/5,A=5/3,A#=225/128,Bb=16/9,B=15/8,Cb=48/25,B#=125/64,C=2/1-->
The following are 21 of the 53 notes in the chromatic scale. The rest can easily be added.
{| class="wikitable" style="text-align: center;"
|- style="background-color: #ffeeee;"
| '''Interval (steps)'''
| 
| colspan="2" | 3
| colspan="2" | 2
| colspan="2" | 4
| colspan="2" | 3
| colspan="2" | 2
| colspan="2" | 3
| colspan="2" | 2
| colspan="2" | 1
| colspan="2" | 2
| colspan="2" | 4
| colspan="2" | 1
| colspan="2" | 4
| colspan="2" | 3
| colspan="2" | 2
| colspan="2" | 4
| colspan="2" | 3
| colspan="2" | 2
| colspan="2" | 3
| colspan="2" | 2
| colspan="2" | 1
| colspan="2" | 2
| 
|- style="background-color: #ffeeee;"
| | '''Interval (cents)'''
| 
| colspan="2" | 68
| colspan="2" | 45
| colspan="2" | 91
| colspan="2" | 68
| colspan="2" | 45
| colspan="2" | 68
| colspan="2" | 45
| colspan="2" | 23
| colspan="2" | 45
| colspan="2" | 91
| colspan="2" | 23
| colspan="2" | 91
| colspan="2" | 68
| colspan="2" | 45
| colspan="2" | 91
| colspan="2" | 68
| colspan="2" | 45
| colspan="2" | 68
| colspan="2" | 45
| colspan="2" | 23
| colspan="2" | 45
| 
|- style="background-color: #fffbee;"
| | '''Note name (Pythagorean notation)'''
| colspan="2" | C
| colspan="2" | E{{music| tripleflat}}
| colspan="2" | C{{music| sharp}}
| colspan="2" | D
| colspan="2" | F{{music| doubleflat}}
| colspan="2" | D{{music| sharp}}
| colspan="2" | F{{music| flat}}
| colspan="2" | D{{music| doublesharp}}
| colspan="2" | C{{music| doublesharp}}{{music| doublesharp}}/A{{music| doubleflat}}{{music| doubleflat}}
| colspan="2" | F
| colspan="2" | G{{music| flat}}
| colspan="2" | F{{music| sharp}}
| colspan="2" | G
| colspan="2" | B{{music| tripleflat}}
| colspan="2" | G{{sharp}}
| colspan="2" | B{{music| doubleflat}}
| colspan="2" | C{{music| doubleflat}}
| colspan="2" | A{{sharp}}
| colspan="2" | C{{music| flat}}
| colspan="2" | A{{music| doublesharp}}
| colspan="2" | G{{music| doublesharp}}{{music| doublesharp}}/E{{music| doubleflat}}{{music| doubleflat}}
| colspan="2" | C
|- style="background-color: #fffbee;"
| | '''Note name (ups and downs notation)'''
| colspan="2" | C
| colspan="2" | vvC{{sharp}}/vD{{flat}}
| colspan="2" | C{{sharp}}/^D{{flat}}
| colspan="2" | D
| colspan="2" | vvD{{sharp}}/vE{{flat}}
| colspan="2" | D{{sharp}}/^E{{flat}}
| colspan="2" | vE
| colspan="2" | ^E
| colspan="2" | ^^E/vvF
| colspan="2" | F
| colspan="2" | vF{{sharp}}/G{{flat}}
| colspan="2" | F{{sharp}}/^G{{flat}}
| colspan="2" | G
| colspan="2" | vvG{{sharp}}/vA{{flat}}
| colspan="2" | G{{sharp}}/^A{{flat}}
| colspan="2" | vA
| colspan="2" | vvA{{sharp}}/vB{{flat}}
| colspan="2" | A{{sharp}}/^B{{flat}}
| colspan="2" | vB
| colspan="2" | ^B
| colspan="2" | ^^B/vvC
| colspan="2" | C
|- style="background-color: #eeeeff;"
| | '''Note (cents)'''
| colspan="2" | &nbsp;&nbsp;0&nbsp;&nbsp;
| colspan="2" | &nbsp;68&nbsp;
| colspan="2" | 113
| colspan="2" | 204
| colspan="2" | 272
| colspan="2" | 317
| colspan="2" | 385
| colspan="2" | 430
| colspan="2" | 453
| colspan="2" | 498
| colspan="2" | 589
| colspan="2" | 611
| colspan="2" | 702
| colspan="2" | 770
| colspan="2" | 815
| colspan="2" | 883
| colspan="2" | 974
| colspan="2" | <small>1018</small>
| colspan="2" | <small>1087</small>
| colspan="2" | <small>1132</small>
| colspan="2" | <small>1155</small>
| colspan="2" | <small>1200</small>
|- style="background-color: #eeeeff;"
| | '''Note (steps)'''
| colspan="2" | 0
| colspan="2" | 3
| colspan="2" | 5
| colspan="2" | 9
| colspan="2" | 12
| colspan="2" | 14
| colspan="2" | 17
| colspan="2" | 19
| colspan="2" | 20
| colspan="2" | 22
| colspan="2" | 26
| colspan="2" | 27
| colspan="2" | 31
| colspan="2" | 34
| colspan="2" | 36
| colspan="2" | 39
| colspan="2" | 43
| colspan="2" | 45
| colspan="2" | 48
| colspan="2" | 50
| colspan="2" | 51
| colspan="2" | 53
|}

== Holdrian comma ==
In [[music theory]] and [[musical tuning]] the '''Holdrian comma''', also called '''Holder's comma''', and rarely the '''Arabian comma''',<ref name=Touma>{{cite book |first=H.H. |last=Touma |author-link=Habib Hassan Touma |year=1996 |title=The Music of the Arabs |page=23 |translator-first=Laurie |translator-last=Schwartz |place=Portland, OR |publisher=Amadeus Press |ISBN=0-931340-88-8 }}</ref> is a small [[interval (music)|musical interval]] of approximately 22.6415&nbsp;[[cent (music)|cents]],<ref name=Touma/> equal to one step of 53&nbsp;equal temperament, or <math>\ \sqrt[53]{2\;}\ </math> ({{Audio|Holdrian comma on C.mid|play}}). The name [[comma (music)|"comma"]], however, is technically misleading, since this interval is an irrational number and it does not describe a compromise between intervals of any tuning system. The interval gets the name "comma" because it is a close approximation of several [[comma (music)|commas]], most notably the [[syntonic comma]] (21.51&nbsp;cents) ({{Audio|Syntonic comma on C.mid|play}}), which was widely used as a unit of tonal measurement during [[William Holder|Holder's]] time.

The origin of Holder's comma resides in the fact that the [[Ancient Greece|Ancient Greeks]] (or at least to the Roman [[Boethius]]{{efn|
According to [[Boethius]], [[Pythagoras]]' disciple [[Philolaus|Philolaus of Croton]] would have said that the tone consisted in two [[diatonic semitone]]s and a [[comma (music)|comma]]; the diatonic semitone consisted in two [[diaschisma]]ta, each formed of two commas.<ref>
{{cite book
 |first=Anicius Manlius Severinus |last=Boethius |author-link=Boethius
 |title=De institutione musica
 |at=Book&nbsp;3, Chapter&nbsp;8
}}
</ref><ref>
{{cite book
 |first=J.M. |last=Barbour |author-link=J. Murray Barbour
 |year=1951
 |title=Tuning and Temperament: A historical survey
 |page=123
}}</ref>
}})
believed that in the [[Pythagorean tuning]] the tone could be divided in nine commas, four of which forming the diatonic semitone and five the chromatic semitone. If all these commas are exactly of the same size, there results an octave of {{nobr|5 tones + 2 diatonic}} semitones, {{nobr|&hairsp; 5 × 9 + 2 × 4 {{=}} 53 equal}} commas. Holder<ref name=Holder-1731>
{{cite book |first=W. |last=Holder |author-link=William Holder |year=1731 |title=A Treatise of the Natural Grounds, and Principles of Harmony |edition=3rd |place=London, UK |page=79 }}</ref> attributes the division of the octave in 53&nbsp;equal parts to [[Nicholas Mercator]],{{efn|
"The late ''[[Nicholas Mercator]]'', a Modest Person, and a Learned and Judicious Mathematician, in a Manuscript of his, of which I have had a Sight."<ref name=Holder-1731/>
}}
who himself had proposed that {{nobr|{{small|{{sfrac| 1 | 53 }}}} part}} of the octave be named the "artificial comma".

=== Mercator's comma and the Holdrian comma === <!--[[Mercator's comma]] redirects here-->
'''Mercator's comma''' is a name often used for a closely related interval because of its association with [[Nicholas Mercator]].{{efn|
{{harvp|Holder|1731}} writes that [[Marin Mersenne]] had calculated {{nobr| 58{{small| {{sfrac| 1 | 4 }}}} commas}} in the octave; [[Nicholas Mercator|Mercator]] "working by the [[logarithm]]s, finds out but 55, and a little more."<ref name=Holder-1731/>
}}
One of these intervals was first described by [[Jing Fang]] in {{nobr|45 {{sc|BCE}}.}}<ref name=Touma/> Mercator applied logarithms to determine that <math>\ \sqrt[55]{2\;}\ </math> (≈ 21.8182&nbsp;cents) was nearly equivalent to a syntonic comma of ≈ 21.5063&nbsp;cents (a feature of the prevalent [[meantone temperament]] of the time). He also considered that an "artificial comma" of <math>\ \sqrt[53]{2\;}\ </math> might be useful, because 31&nbsp;octaves could be practically approximated by a cycle of 53&nbsp;[[just fifth]]s. [[William Holder|Holder]], for whom the ''Holdrian'' comma is named, favored this latter unit because the intervals of 53&nbsp;equal temperament are closer to [[just intonation]] than to [[55 equal temperament|55&nbsp;TET]]. Thus Mercator's comma and the Holdrian comma are two distinct but nearly equal intervals.

== Use in Turkish makam theory ==
The Holdrian comma has been employed mainly in Ottoman/Turkish music theory by [[Kemal Ilerici]], and by the Turkish composer [[Erol Sayan]]. The name of this comma is {{Lang|tr|Holder koması}} in Turkish.
{| class="wikitable"
! Name of interval
! Commas
! Cents
! Symbol
|-
| Koma
| 1
| 22.64
| F
|-
| Bakiye
| 4
| 90.57
| B
|-
| Küçük Mücennep
| 5
| 113.21
| S
|-
| Büyük Mücennep
| 8
| 181.13
| K
|-
| Tanini
| 9
| 203.77
| T
|-
| Artık Aralık (12)
| 12
| 271.70
| A (12)
|-
| Artık Aralık (13)
| 13
| 294.34
| A (13)
|}

For instance, the [[Turkish makam#Rast makam|Rast makam]] (similar to the Western [[major scale]], or more precisely to the [[justly-tuned major scale]]) may be considered in terms of Holdrian commas:

:[[Image:makam_aeu_rast_wp.png|500px]]
where {{music|d}} denotes a Holdrian comma flat,{{efn|
In common Arabic and Turkish practice, the third note '''e'''{{music|d}} and the seventh note '''b'''{{music|d}} in Rast are even lower than in this theory, almost exactly halfway between western major and minor thirds above '''c''' and '''g''', i.e. closer to 6.5&nbsp;commas (three-quarter tone) above '''d''' or '''a''' and 6.5 below '''f''' or '''c''', the thirds '''c–e'''{{music|d}} and '''g–b'''{{music|d}} often referred to as a "[[neutral third]]s" by musicologists.
}}
while in contrast, the Nihavend makam (similar to the Western [[minor scale]]):

:[[Image:makam_aeu_nihavent_wp.png|500px]]
where {{music|flat}} denotes a five-comma flat,
has [[medium second]]s between '''d–e'''{{music|flat}}, '''e–f''', '''g–a'''{{music|flat}}, '''a'''{{music|flat}}–'''b'''{{music|flat}}, and '''b'''{{music|flat}}–'''c&prime;''', a medium second being somewhere in between 8 and 9&nbsp;commas.<ref name=Touma/>

===Notes===
{{notelist}}

== References ==
{{reflist|25em}}

{{refbegin|small=y}}
* {{cite book
 |last=Holder |first=William
 |year=1967 |orig-year=1694
 |title=A Treatise on the Natural Grounds, and Principles of Harmony
 |edition=facsimile
 |publisher=Broude Brothers
 |place=New York, NY
 |pages=103–106}}
{{refend}}

==External links==
* {{cite web
 | first=Prent | last=Rodgers
 | date=May 2007
 | title=Whisper song in 53&nbsp;EDO
 | edition=slower  | type=podcast
 | website=Bumper Music
 | url=http://bumpermusic.blogspot.com/2007/05/whisper-song-in-53-edo-now-526-slower.html 
}}
* {{cite journal
 | first=Larry | last=Hanson
 | year=1989
 | title=Development of a 53&nbsp;tone keyboard layout
 | journal=Xenharmonicon
 | volume=XII  | pages=68–85
 | publisher=Frog Peak Music
 | place=Hanover, NH
 | url=http://www.anaphoria.com/hanson.PDF 
 | via=Anaphoria.com | access-date=2021-01-04  |df=dmy-all
}}
* {{cite web
 | title=Algebra of Tonal Functions
 | date=2007-05-01   |df=dmy-all
 | website=Sonantometry |type=blog
 | url=http://sonantometry.blogspot.com/2007_05_01_archive.html
}} — Tonal functions as 53&nbsp;TET grades.
* {{cite web
 | last=Barbieri | first=Patrizio
 | year=2008
 | title=Enharmonic instruments and music, 1470–1900
 | website=Latina, Il Levante Libreria Editrice
 | place=Italy
 | url=http://www.patriziobarbieri.it/1.htm
 | archive-url=https://web.archive.org/web/20090215045859/http://www.patriziobarbieri.it/1.htm
 | archive-date=2009-02-15
}} 
* {{cite web
 |first=Jim |last=Kukula
 |date=August 2005
 |series=Fractal microtonal music
 |title=Equal temperament with 53&nbsp;pitches per octave
 |website=Interdependent Science
 |url=http://www.interdependentscience.com/music/calliopist.html
 |access-date=2021-01-04   |df=dmy-all
}}

{{Microtonal music}}
{{Musical tuning}}

{{DEFAULTSORT:53 Equal Temperament}}
[[Category:Equal temperaments]]
[[Category:Microtonality]]
[[Category:Commas (music)]]

[[fr:Tempérament par division multiple]]