72 equal temperament
{{redirect|Third tone|the Chinese third tone|Standard Chinese phonology#Tones}}
In music, 72 equal temperament, called twelfth-tone, 72 TET, 72 EDO, or 72 ET, is the tempered scale derived by dividing the octave into twelfth-tones, or in other words 72 equal steps (equal frequency ratios). {{audio|72-tet scale on C.mid|Play}} Each step represents a frequency ratio of {{radic|2|72}}, or {{nobr|{{sfrac| 16 | 2 | 3 }} cents,}} which divides the 100 cent 12 EDO "halftone" into 6 equal parts (100 cents ÷ {{nobr|{{sfrac| 16 | 2 | 3 }} }} {{=}} 6 steps, exactly) and is thus a "twelfth-tone" ({{Audio|1 step in 72-et on C.mid|Play}}). Since 72 is divisible by 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72, 72 EDO includes all those equal temperaments. Since it contains so many temperaments, 72 EDO contains at the same time tempered semitones, third-tones, quartertones and sixth-tones, which makes it a very versatile temperament.
This division of the octave has attracted much attention from tuning theorists, since on the one hand it subdivides the standard 12 equal temperament and on the other hand it accurately represents overtones up to the twelfth partial tone, and hence can be used for 11 limit music. It was theoreticized in the form of twelfth-tones by Alois Hába
{{cite book
|first=A. |last=Hába |author-link=Alois Hába
|orig-year=1927 |year=1978
|title=Harmonické základy ctvrttónové soustavy |lang=cs, de
|trans-title= German translation Neue Harmonielehre des diatonischen, chromatischen Viertel-, Drittel-, Sechstel- und Zwölftel-tonsystems English translation Harmonic Fundamentals of the Quarter-Tone System
|translator-first=Fr. |translator-last=Kistner
|publisher=C.F.W. Siegel (1927) / Universal (1978)
|place=Leipzig (1927) / Wien, 1978
}}
: Revised German edition:
{{cite book
|first=A. |last=Hába |author-link=Alois Hába
|editor-first=Erich |editor-last=Steinhard
|orig-year=1927, 1978 |year=2001
|title=Grundfragen der mikrotonalen Musik |lang=de
|trans-title=Foundations of Microtonal Music |edition=rev.
|publisher=Musikedition Nymphenburg Filmkunst-Musikverlag
|place=München, DE
|volume=3
|others=Kistner, Fr. (original translation)
}}
and Ivan Wyschnegradsky,
{{cite journal
|first=I. |last=Wyschnegradsky |author-link=Ivan Wyschnegradsky
|year= 1972
|title=L'ultrachromatisme et les espaces non octaviants
|journal=La Revue Musicale
|issue=290–291 |pages=71–141
}}
{{cite book
|editor-first=Franck |editor-last=Jedrzejewski
|orig-year=1953 |publication-date=1996
|title=La Loi de la Pansonorité |lang=fr
|trans-title=The Laws of Multitonal Music
|type=manuscript
|publisher=Ed. Contrechamps
|place=Geneva, CH
|others=Criton, Pascale (preface)
|ISBN=978-2-940068-09-8
}}
{{cite book
|editor-first=Franck |editor-last=Jedrzejewski
|orig-year=1936 |publication-date=2005
|title=Une philosophie dialectique de l'art musical |lang=fr
|trans-title=A Dialectical Philosophy of Musical Art
|type=manuscript
|publisher=Ed. L'Harmattan
|place=Paris, FR
|ISBN=978-2-7475-8578-1
}}
who considered it as a good approach to the continuum of sound. 72 EDO is also cited among the divisions of the tone by Julián Carrillo, who preferred the sixteenth-tone (96 EDO) as an approximation to continuous sound in discontinuous scales.
History and use
=Byzantine music=
The 72 equal temperament is used in Byzantine music theory,{{cite conference
|url=http://smc07.uoa.gr/SMC07%20Proceedings/SMC07%20Paper%2020.pdf
|first1=G.
|last1=Chryssochoidis
|first2=D.
|last2=Delviniotis
|first3=G.
|last3=Kouroupetroglou
|title=A semi-automated tagging methodology for Orthodox ecclesiastic chant acoustic corpora
|book-title=Proceedings SMC'07
|conference=4th Sound and Music Computing Conference
|place=Lefkada, Greece
|date=11–13 July 2007
|access-date=24 April 2008
|archive-date=15 August 2007
|archive-url=https://web.archive.org/web/20070815212401/http://smc07.uoa.gr/SMC07%20Proceedings/SMC07%20Paper%2020.pdf
|url-status=live
}}
dividing the octave into 72 equal moria, which itself derives from interpretations of the theories of Aristoxenos, who used something similar. Although the 72 equal temperament is based on irrational intervals (see above), as is the 12 tone equal temperament (12 EDO) mostly commonly used in Western music (and which is contained as a subset within 72 equal temperament), 72 equal temperament, as a much finer division of the octave, is an excellent tuning for both representing the division of the octave according to the ancient Greek diatonic and the chromatic genera in which intervals are based on ratios between notes, and for representing with great accuracy many rational intervals as well as irrational intervals.
=Other history and use=
A number of composers have made use of it, and these represent widely different points of view and types of musical practice. These include Alois Hába, Julián Carrillo, Ivan Wyschnegradsky, and Iannis Xenakis.{{citation needed|date=May 2014}}
Many other composers use it freely and intuitively, such as jazz musician Joe Maneri, and classically oriented composers such as Julia Werntz and others associated with the Boston Microtonal Society. Others, such as New York composer Joseph Pehrson are interested in it because it supports the use of miracle temperament, and still others simply because it approximates higher-limit just intonation, such as Ezra Sims and James Tenney. There was also an active Soviet school of 72 EDO composers, with less familiar names: Evgeny Alexandrovich Murzin, Andrei Volkonsky, Nikolai Nikolsky, Eduard Artemiev, Alexander Nemtin, Andrei Eshpai, Gennady Gladkov, Pyotr Meshchianinov, and Stanislav Kreichi.{{citation needed|date=May 2014}}
The ANS synthesizer uses 72 equal temperament.
Notation
The Maneri-Sims notation system designed for 72 EDO uses the accidentals {{music|down}} and {{music|up}} for {{nobr|{{small|{{sfrac| 1 | 12 }}}} tone}} down and up (1 step {{nobr| {{=}} {{sfrac| 16 | 2 | 3 }} cents),}} {{music|half check down}} and {{music|half check up}} for {{nobr|{{small|{{sfrac| 1 | 6 }}}} down}} and up (2 steps {{nobr| {{=}} {{sfrac| 33 | 1 | 3 }} cents),}} and {{music|check}} and {{music|check up}} for septimal quarter tone up and down (3 steps {{nobr|{{=}} 50 cents}} {{nobr| {{=}} half a 12 EDO sharp).}}
They may be combined with the traditional sharp and flat symbols (6 steps = 100 cents) by being placed before them, for example: {{music|half check down}}{{music|flat}} or {{music|check up}}{{music|b}}, but without the intervening space. A {{small|{{sfrac| 1 | 3 }}}} tone may be one of the following {{music|up}}{{music|check up}}, {{music|down}}{{music|check down}}, {{music|half check down}}{{music|#}}, or {{music|half check up}}{{music|b}} (4 steps = {{sfrac| 66 | 2 | 3 }}) while 5 steps may be {{music|half check up}}{{music|check up}}, {{music|down}}{{music|sharp}}, or {{music|up}}{{music|b}} ({{sfrac| 83 | 1 | 3 }} cents).
Interval size
Below are the sizes of some intervals (common and esoteric) in this tuning. For reference, differences of less than 5 cents are melodically imperceptible to most people, and approaching the limits of feasible tuning accuracy for acoustic instruments. Note that it is not possible for any pitch to be further than {{nobr|{{sfrac| 8 | 1 | 3 }} cents}} from its nearest 72 EDO note, since the step size between them is {{nobr|{{sfrac| 16 | 2 | 3 }} cents.}} Hence for the sake of comparison, pitch errors of about 8 cents are (for this fine a tuning) poorly matched, whereas the practical limit for tuning any acoustical instrument is at best about 2 cents, which would be very good match in the table – this even applies to electronic instruments if they produce notes that show any audible trace of vibrato.{{citation needed|date=November 2024}}
| class="wikitable sortable"
!align=center| Interval name !align=center| Size !align=center| Size !align=center| MIDI audio !align=center| Just !align=center| Just !align=center| MIDI audio !align=center| Error |
| align=center| octave
|align=center| 72 |align=center| 1200 |align=center| |align=center| 2:1 |align=center| 1200 |align=center| |align=center| 0 |
| align=center| harmonic seventh
|align=center| 58 |align=center| 966.67 |align=center| |align=center| 7:4 |align=center| 968.83 |align=center| |align=center| −2.16 |
| align=center| perfect fifth
|align=center| 42 |align=center| 700 |align=center| {{Audio|Perfect fifth on C.mid|play}} |align=center| 3:2 |align=center| 701.96 |align=center| {{Audio|Just perfect fifth on C.mid|play}} |align=center| −1.96 |
| align=center| septendecimal tritone
|align=center| 36 |align=center| 600 |align=center| {{audio|Tritone on C.mid|play}} |align=center| 17:12 |align=center| 603.00 |align=center| |align=center| −3.00 |
| align=center| septimal tritone
|align=center| 35 |align=center| 583.33 |align=center| {{Audio|35 steps in 72-et on C.mid|play}} |align=center| 7:5 |align=center| 582.51 |align=center| {{Audio|Lesser septimal tritone on C.mid|play}} |align=center| +0.82 |
| align=center| tridecimal tritone
|align=center| 34 |align=center| 566.67 |align=center| {{Audio|34 steps in 72-et on C.mid|play}} |align=center| 18:13 |align=center| 563.38 |align=center| |align=center| +3.28 |
| align=center| 11th harmonic
|align=center| 33 |align=center| 550 |align=center| {{Audio|Eleven quarter tones on C.mid|play}} |align=center| 11:8 |align=center| 551.32 |align=center| {{Audio|Eleventh harmonic on C.mid|play}} |align=center| −1.32 |
| align=center| (15:11) augmented fourth
|align=center| 32 |align=center| 533.33 |align=center| {{Audio|32 steps in 72-et on C.mid|play}} |align=center| 15:11 |align=center| 536.95 |align=center| {{Audio|Undecimal augmented fourth on C.mid|play}} |align=center| −3.62 |
| align=center| perfect fourth
|align=center| 30 |align=center| 500 |align=center| {{Audio|Perfect fourth on C.mid|play}} |align=center| 4:3 |align=center| 498.04 |align=center| {{Audio|Just perfect fourth on C.mid|play}} |align=center| +1.96 |
| align=center| septimal narrow fourth
|rowspan=2 align=center| 28 |rowspan=2 align=center| 466.66 |rowspan=2 align=center| {{Audio|28 steps in 72-et on C.mid|play}} |align=center| 21:16 |align=center| 470.78 |align=center| {{Audio|Twenty-first harmonic on C.mid|play}} |align=center| −4.11 |
| align=center| 17:13 narrow fourth
|align=center| 17:13 |align=center| 464.43 |align=center| |align=center| +2.24 |
| align=center| tridecimal major third
|rowspan=2 align=center| 27 |rowspan=2 align=center| 450 |rowspan=2 align=center| {{Audio|Nine quarter tones on C.mid|play}} |align=center| 13:10 |align=center| 454.21 |align=center| {{Audio|Tridecimal major third on C.mid|play}} |align=center| −4.21 |
| align=center| septendecimal supermajor third
|align=center| 22:17 |align=center| 446.36 |align=center| |align=center| +3.64 |
| align=center| septimal major third
|align=center| 26 |align=center| 433.33 |align=center| {{Audio|26 steps in 72-et on C.mid|play}} |align=center| 9:7 |align=center| 435.08 |align=center| {{Audio|Septimal major third on C.mid|play}} |align=center| −1.75 |
| align=center| undecimal major third
|align=center| 25 |align=center| 416.67 |align=center| {{Audio|25 steps in 72-et on C.mid|play}} |align=center| 14:11 |align=center| 417.51 |align=center| {{Audio|Undecimal major third on C.mid|play}} |align=center| −0.84 |
| align=center| quasi-tempered major third
|align=center| 24 |align=center| 400 |align=center| {{Audio|Major third on C.mid|play}} |align=center| 5:4 |align=center| 386.31 |align=center| {{Audio|Just major third on C.mid|play}} |align=center| 13.69 |
| align=center| major third
|align=center| 23 |align=center| 383.33 |align=center| {{Audio|23 steps in 72-et on C.mid|play}} |align=center| 5:4 |align=center| 386.31 |align=center| {{Audio|Just major third on C.mid|play}} |align=center| −2.98 |
| align=center bgcolor="#B4B4B4"| tridecimal neutral third
|align=center bgcolor="#B4B4B4"| 22 |align=center bgcolor="#B4B4B4"| 366.67 |align=center bgcolor="#B4B4B4"| {{Audio|22 steps in 72-et on C.mid|play}} |align=center bgcolor="#B4B4B4"| 16:13 |align=center bgcolor="#B4B4B4"| 359.47 |align=center bgcolor="#B4B4B4"| |align=center bgcolor="#B4B4B4"| +7.19 |
| align=center| neutral third
|align=center| 21 |align=center| 350 |align=center| {{Audio|Neutral third on C.mid|play}} |align=center| 11:9 |align=center| 347.41 |align=center| {{Audio|Neutral third on C.mid|play}} |align=center| +2.59 |
| align=center| septendecimal supraminor third
|align=center| 20 |align=center| 333.33 |align=center| {{Audio|20 steps in 72-et on C.mid|play}} |align=center| 17:14 |align=center| 336.13 |align=center| |align=center| −2.80 |
| align=center| minor third
|align=center| 19 |align=center| 316.67 |align=center| {{Audio|19 steps in 72-et on C.mid|play}} |align=center| 6:5 |align=center| 315.64 |align=center| {{Audio|Just minor third on C.mid|play}} |align=center| +1.03 |
| align=center| quasi-tempered minor third
|align=center| 18 |align=center| 300 |align=center| {{Audio|Minor third on C.mid|play}} |align=center| 25:21 |align=center| 301.85 |align=center| |align=center| −1.85 |
| align=center bgcolor="#B4B4B4"| tridecimal minor third
|align=center bgcolor="#B4B4B4"| 17 |align=center bgcolor="#B4B4B4"| 283.33 |align=center bgcolor="#B4B4B4"| {{Audio|17 steps in 72-et on C.mid|play}} |align=center bgcolor="#B4B4B4"| 13:11 |align=center bgcolor="#B4B4B4"| 289.21 |align=center bgcolor="#B4B4B4"| {{Audio|Tridecimal minor third on C.mid|play}} |align=center bgcolor="#B4B4B4"| −5.88 |
| align=center| septimal minor third
|align=center| 16 |align=center| 266.67 |align=center| {{Audio|16 steps in 72-et on C.mid|play}} |align=center| 7:6 |align=center| 266.87 |align=center| {{Audio|Septimal minor third on C.mid|play}} |align=center| −0.20 |
| align=center| tridecimal {{nobr|{{small|{{sfrac| 5 | 4 }}}} tone}}
|align=center| 15 |align=center| 250 |align=center| {{Audio|Five quarter tones on C.mid|play}} |align=center| 15:13 |align=center| 247.74 |align=center| |align=center| +2.26 |
| align=center| septimal whole tone
|align=center| 14 |align=center| 233.33 |align=center| {{Audio|14 steps in 72-et on C.mid|play}} |align=center| 8:7 |align=center| 231.17 |align=center| {{Audio|Septimal major second on C.mid|play}} |align=center| +2.16 |
| align=center| septendecimal whole tone
|align=center| 13 |align=center| 216.67 |align=center| {{audio|13 steps in 72-et on C.mid|play}} |align=center| 17:15 |align=center| 216.69 |align=center| |align=center| −0.02 |
| align=center| whole tone, major tone
|align=center| 12 |align=center| 200 |align=center| {{Audio|Major second on C.mid|play}} |align=center| 9:8 |align=center| 203.91 |align=center| {{Audio|Major tone on C.mid|play}} |align=center| −3.91 |
| align=center| whole tone, minor tone
|align=center| 11 |align=center| 183.33 |align=center| {{Audio|11 steps in 72-et on C.mid|play}} |align=center| 10:9 |align=center| 182.40 |align=center| {{Audio|Minor tone on C.mid|play}} |align=center| +0.93 |
| align=center| greater undecimal neutral second
|align=center| 10 |align=center| 166.67 |align=center| {{Audio|10 steps in 72-et on C.mid|play}} |align=center| 11:10 |align=center| 165.00 |align=center| {{Audio|Greater undecimal neutral second on C.mid|play}} |align=center| +1.66 |
| align=center| lesser undecimal neutral second
|align=center| 9 |align=center| 150 |align=center| {{Audio|Neutral second on C.mid|play}} |align=center| 12:11 |align=center| 150.64 |align=center| {{Audio|Neutral second on C.mid|play}} |align=center| −0.64 |
| align=center bgcolor="#B4B4B4"| greater tridecimal {{nobr|{{small|{{sfrac| 2 | 3 }}}} tone}}
|rowspan=3 align=center bgcolor="#B4B4B4"| 8 |rowspan=3 align=center bgcolor="#B4B4B4"| 133.33 |rowspan=3 align=center bgcolor="#B4B4B4"| {{Audio|8 steps in 72-et on C.mid|play}} |align=center bgcolor="#B4B4B4"| 13:12 |align=center bgcolor="#B4B4B4"| 138.57 |align=center bgcolor="#B4B4B4"| {{audio|Greater tridecimal two-third tone on C.mid|play}} |align=center bgcolor="#B4B4B4"| −5.24 |
| align=center| great limma
|align=center| 27:25 |align=center| 133.24 |align=center| {{audio|Semitone Maximus on C.mid|play}} |align=center| +0.09 |
| align=center bgcolor="#B4B4B4"| lesser tridecimal {{nobr|{{small|{{sfrac|2|3}}}} tone}}
|align=center bgcolor="#B4B4B4"| 14:13 |align=center bgcolor="#B4B4B4"| 128.30 |align=center bgcolor="#B4B4B4"| {{audio|Lesser tridecimal two-third tone on C.mid|play}} |align=center bgcolor="#B4B4B4"| +5.04 |
| align=center| septimal diatonic semitone
|rowspan=2 align=center| 7 |rowspan=2 align=center| 116.67 |rowspan=2 align=center| {{Audio|7 steps in 72-et on C.mid|play}} |align=center| 15:14 |align=center| 119.44 |align=center| {{Audio|Septimal diatonic semitone on C.mid|play}} |align=center| −2.78 |
| align=center | diatonic semitone
|align=center| 16:15 |align=center| 111.73 |align=center| {{Audio|Just diatonic semitone on C.mid|play}} |align=center| +4.94 |
| align=center| greater septendecimal semitone
|rowspan=2 align=center| 6 |rowspan=2 align=center| 100 |rowspan=2 align=center| {{audio|Minor second on C.mid|play}} |align=center| 17:16 |align=center| 104.95 |align=center| {{audio|Just major semitone on C.mid|play}} |align=center| −4.95 |
| align=center| lesser septendecimal semitone
|align=center| 18:17 |align=center| 98.95 |align=center| {{audio|Just minor semitone on C.mid|play}} |align=center| +1.05 |
| align=center| septimal chromatic semitone
|align=center| 5 |align=center| 83.33 |align=center| {{Audio|5 steps in 72-et on C.mid|play}} |align=center| 21:20 |align=center| 84.47 |align=center| {{Audio|Septimal chromatic semitone on C.mid|play}} |align=center| −1.13 |
| align=center| chromatic semitone
|rowspan=2 align=center| 4 |rowspan=2 align=center| 66.67 |rowspan=2 align=center| {{Audio|4 steps in 72-et on C.mid|play}} |align=center| 25:24 |align=center| 70.67 |align=center| {{Audio|Just chromatic semitone on C.mid|play}} |align=center| −4.01 |
| align=center| septimal third-tone
|align=center| 28:27 |align=center| 62.96 |align=center| {{audio|Septimal minor second on C.mid|play}} |align=center| +3.71 |
| align=center| septimal quarter tone
|align=center| 3 |align=center| 50 |align=center| {{Audio|Quarter tone on C.mid|play}} |align=center| 36:35 |align=center| 48.77 |align=center| {{Audio|Septimal quarter tone on C.mid|play}} |align=center| +1.23 |
| align=center| septimal diesis
|align=center| 2 |align=center| 33.33 |align=center| {{Audio|1 step in 36-et on C.mid|play}} |align=center| 49:48 |align=center| 35.70 |align=center| {{Audio|Septimal diesis on C.mid|play}} |align=center| −2.36 |
| align=center| undecimal comma
|align=center| 1 |align=center| 16.67 |align=center| {{Audio|1 step in 72-et on C.mid|play}} |align=center| 100:99 |align=center| 17.40 |align=center| |align=center| −0.73 |
- {{Audio|72-et diatonic scale on C.mid|play diatonic scale in 72 EDO}}
- {{Audio|Just diatonic scale on C.mid|contrast with just diatonic scale}}
- {{Audio|Diatonic scale on C.mid|contrast with diatonic scale in 12 EDO}}
Although 12 EDO can be viewed as a subset of 72 EDO, the closest matches to most commonly used intervals under 72 EDO are distinct from the closest matches under 12 EDO. For example, the major third of 12 EDO, which is sharp, exists as the 24 step interval within 72 EDO, but the 23 step interval is a much closer match to the 5:4 ratio of the just major third.
12 EDO has a very good approximation for the perfect fifth (third harmonic), especially for such a small number of steps per octave, but compared to the equally-tempered versions in 12 EDO, the just major third (fifth harmonic) is off by about a sixth of a step, the seventh harmonic is off by about a third of a step, and the eleventh harmonic is off by about half of a step. This suggests that if each step of 12 EDO were divided in six, the fifth, seventh, and eleventh harmonics would now be well-approximated, while 12 EDO‑s excellent approximation of the third harmonic would be retained. Indeed, all intervals involving harmonics up through the 11th are matched very closely in 72 EDO; no intervals formed as the difference of any two of these intervals are tempered out by this tuning system. Thus, 72 EDO can be seen as offering an almost perfect approximation to 7-, 9-, and 11 limit music. When it comes to the higher harmonics, a number of intervals are still matched quite well, but some are tempered out. For instance, the comma 169:168 is tempered out, but other intervals involving the 13th harmonic are distinguished.
Unlike tunings such as 31 EDO and 41 EDO, 72 EDO contains many intervals which do not closely match any small-number (< 16) harmonics in the harmonic series.
Scale diagram
File:Regular diatonic tunings 72-tone versus 12-tone.pngs notated with the Maneri-Sims system]]
Because 72 EDO contains 12 EDO, the scale of 12 EDO is in 72 EDO. However, the true scale can be approximated better by other intervals.
See also
References
{{reflist|25em}}
External links
{{refbegin}}
- {{cite web
|title=The Boston Microtonal Society
|type=official site
|url=http://bostonmicrotonalsociety.org/
|via=bostonmicrotonalsociety.org
|access-date=2005-12-05
|archive-date=2011-02-09
|archive-url=https://web.archive.org/web/20110209143213/http://bostonmicrotonalsociety.org/
|url-status=usurped
}}
- {{cite web
|title=Wyschnegradsky notation for twelfth-tone
|url=https://sagittal.org/gift/Image6.gif
|via=sagittal.org
|access-date=24 June 2022
|archive-date=4 April 2023
|archive-url=https://web.archive.org/web/20230404235843/https://sagittal.org/gift/Image6.gif
|url-status=live
}}
- {{cite web
|title=Sagittal
|type=website main page
|url=https://www.sagittal.org/
|via=sagittal.org
}}
- {{cite web
|title=Sagittal notation
|website=The Xenharmonic wiki
|via=en.xen.wiki
|url=https://en.xen.wiki/w/Sagittal_notation
|access-date=2022-06-24
|archive-date=2022-10-22
|archive-url=https://web.archive.org/web/20221022094010/https://en.xen.wiki/w/Sagittal_notation
|url-status=live
}}
- {{cite web
|title = Alterations
|date = 27 September 2017
|publisher = Ekmelic Music
|url = http://www.ekmelic-music.org/en/extra/alter.htm
|url-status = dead
|via = ekmelic-music.org
|access-date = 16 October 2017
|archive-url = https://web.archive.org/web/20171004131828/http://www.ekmelic-music.org/en/extra/alter.htm
|archive-date = 4 October 2017
}} — symbols for Maneri-Sims notation and others
- {{cite web |first=Ioannis |last=Spyrakis |title={{math|Η Ελληνική Σελίδα για τη Βυζαντινή Μουσική}} |lang=el, en |trans-title=Byzantine music, electroacoustic music, musicology, education |url=http://www.g-culture.org/ioannis/ |via=g-culture.org |url-status=live |access-date=19 November 2024 |archive-date=5 April 2023 |archive-url=https://web.archive.org/web/20230405082932/https://www.g-culture.org/ioannis/ }}
{{refend}}
{{Microtonal music}}
{{Musical tuning}}
{{Byzantine music}}