Xenharmonic Wiki

24edo: Difference between revisions

← Older editNewer edit →
VisualWikitext
Sintel (talk | contribs)
autopatrolled, emailconfirmed
1,548 edits
Cleanup; linking; +about 1mu
Line 9: Line 9:
{{Wikipedia|Quarter tone}}
{{Wikipedia|Quarter tone}}


24edo is also known as '''quarter-tone tuning''', since it evenly divides the 12-tone equal tempered semitone in two. Quarter-tones are the most commonly used microtonal tuning due to its retention of the familiar 12 tones, since it is the smallest microtonal equal temperament that contains all the 12 notes, and also because of its use in theory and occasionally in practice in [[Arabic,_Turkish,_Persian|Arabic]] music.  
24edo is also known as '''quarter-tone tuning''', since it evenly divides the 12-tone equal tempered semitone in two. Quarter-tones are the most commonly used microtonal tuning due to its retention of the familiar 12 tones, since it is the smallest microtonal equal temperament that contains all the 12 notes, and also because of its use in theory and occasionally in practice in [[Arabic, Turkish, Persian music|Arabic music]].  


It is easy to jump into this tuning and make microtonal music right away using common 12 equal software and even instruments as illustrated in ''[[DIY Quartertone Composition with 12 equal tools]]''.
It is easy to jump into this tuning and make microtonal music right away using common 12 equal software and even instruments as illustrated in ''[[DIY Quartertone Composition with 12 equal tools]]''.
Line 16: Line 16:
The [[5-limit]] approximations in 24edo are the same as those in 12edo, therefore 24edo offers nothing new as far as approximating the 5-limit is concerned.   
The [[5-limit]] approximations in 24edo are the same as those in 12edo, therefore 24edo offers nothing new as far as approximating the 5-limit is concerned.   


The 7th harmonic-based intervals ([[7/4]], [[7/5]] and [[7/6]]) are almost as bad in 24edo as in 12edo. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12edo requires high-degree tunings like [[36edo|36et]], [[72edo|72et]], [[84edo|84et]] or [[156edo|156et]]. However, it should be noted that 24edo, like [[22edo]], ''does'' temper out the [[quartisma]], linking the otherwise sub-par [[7-limit]] harmonies with those of the [[11-limit]].  
The 7th harmonic-based intervals ([[7/4]], [[7/5]] and [[7/6]]) are almost as bad in 24edo as in 12edo. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12edo requires high-degree tunings like [[36edo|36et]], [[72edo|72et]], [[84edo|84et]] or [[156edo|156et]]. However, it should be noted that 24edo, like [[22edo]], ''does'' temper out the [[quartisma]], linking the otherwise sub-par [[7-limit]] harmonies with those of the [[11-limit]].  


Speaking of 11-limit representation in 24edo, the 11th harmonic, and most intervals derived from it, (11/10, 11/9, 11/8, 11/6, 12/11, 15/11, 16/11, 18/11, 20/11) are very well approximated in this EDO. The 24-tone interval of 550 cents is 1.3 cents flatter than 11:8 and is almost indistinguishable from it. In addition, the interval approximating 11:9 is 7 steps which is exactly half the perfect fifth.  
Speaking of 11-limit representation in 24edo, the 11th harmonic, and most intervals derived from it, ([[11/10]], [[11/9]], [[11/8]], [[11/6]], [[12/11]], [[15/11]], [[16/11]], [[18/11]], [[20/11]]) are very well approximated in this edo. The 24-tone interval of 550 cents is 1.3 cents flatter than 11/8 and is almost indistinguishable from it. In addition, the interval approximating 11/9 is 7 steps which is exactly half the perfect fifth.  


The tunings supplied by [[72edo]] cannot be used for all low-limit just intervals, but they can be used on the 17-limit [[k*N_subgroups|3*24 subgroup]] 2.3.125.35.11.325.17 [[just intonation subgroup]], making some of the excellent approximations of 72 available in 24edo. Chords based on this subgroup afford considerable scope for harmony, including in particular intervals and chords using only 2, 3, 11 and 17. Another approach would be to treat 24-EDO as a 2.3.11.17.19 [[Just intonation subgroup|subgroup]] temperament, on which it is quite accurate.  
The tunings supplied by [[72edo]] cannot be used for all low-limit just intervals, but they can be used on the 17-limit [[k*N subgroups|3*24 subgroup]] 2.3.125.35.11.325.17 [[just intonation subgroup]], making some of the excellent approximations of 72 available in 24edo. Chords based on this subgroup afford considerable scope for harmony, including in particular intervals and chords using only 2, 3, 11 and 17. Another approach would be to treat 24edo as a 2.3.11.17.19 [[subgroup]] temperament, on which it is quite accurate.  


=== Prime harmonics ===
=== Prime harmonics ===
{{harmonics in equal|24}}
{{Harmonics in equal|24}}


=== Subsets and supersets ===
=== Subsets and supersets ===
24edo is the 6th [[highly composite EDO]]. Its divisors are {{EDOs|1, 2, 3, 4, 6, 8, 12}}.
24edo is the 6th [[highly composite edo]]. Its nontrivial divisors are {{EDOs| 2, 3, 4, 6, 8, and 12 }}.


== Notation ==
== Notation ==
Line 37: Line 37:
! Degree
! Degree
! Cents
! Cents
! Approximate Ratios<ref>based on treating 24-EDO as a 2.3.11.17.19 [[subgroup]]; other approaches are possible.</ref>
! Approximate Ratios<ref>based on treating 24edo as a 2.3.11.17.19 [[subgroup]]; other approaches are possible.</ref>
! colspan="3" | [[ups and downs notation]]
! colspan="3" | [[ups and downs notation]]
![[24edo solfege|Solfege]]
! [[24edo solfege|Solfege]]
|-
|-
| 0
| 0
Line 47: Line 47:
| unison
| unison
| C
| C
|Do
| Do
|-
|-
| 1
| 1
Line 55: Line 55:
| up-unison, downminor 2nd
| up-unison, downminor 2nd
| ^C, vDb
| ^C, vDb
|Da/Ru
| Da/Ru
|-
|-
| 2
| 2
Line 63: Line 63:
| aug unison, minor 2nd
| aug unison, minor 2nd
| C#, Db
| C#, Db
|Ro
| Ro
|-
|-
| 3
| 3
Line 71: Line 71:
| mid 2nd
| mid 2nd
| vD
| vD
|Ra
| Ra
|-
|-
| 4
| 4
Line 79: Line 79:
| major 2nd
| major 2nd
| D
| D
|Re
| Re
|-
|-
| 5
| 5
Line 87: Line 87:
| upmajor 2nd, downminor 3rd
| upmajor 2nd, downminor 3rd
| ^D, vEb
| ^D, vEb
|Ri/Mu
| Ri/Mu
|-
|-
| 6
| 6
Line 95: Line 95:
| minor 3rd
| minor 3rd
| Eb
| Eb
|Mo
| Mo
|-
|-
| 7
| 7
Line 103: Line 103:
| mid 3rd
| mid 3rd
| vE
| vE
|Ma
| Ma
|-
|-
| 8
| 8
Line 111: Line 111:
| major 3rd
| major 3rd
| E
| E
|Me
| Me
|-
|-
| 9
| 9
Line 119: Line 119:
| upmajor 3rd, down-4th
| upmajor 3rd, down-4th
| ^E, vF
| ^E, vF
|Mi/Fu
| Mi/Fu
|-
|-
| 10
| 10
Line 127: Line 127:
| fourth
| fourth
| F
| F
|Fo
| Fo
|-
|-
| 11
| 11
Line 135: Line 135:
| up-4th, mid-4th
| up-4th, mid-4th
| ^F
| ^F
|Fa/Su
| Fa/Su
|-
|-
| 12
| 12
Line 143: Line 143:
| aug 4th, dim 5th
| aug 4th, dim 5th
| F#, Gb
| F#, Gb
|Fe/So
| Fe/So
|-
|-
| 13
| 13
Line 151: Line 151:
| down-5th, mid-5th
| down-5th, mid-5th
| vG
| vG
|Fi/Sa
| Fi/Sa
|-
|-
| 14
| 14
Line 159: Line 159:
| fifth
| fifth
| G
| G
|Se
| Se
|-
|-
| 15
| 15
Line 167: Line 167:
| up-fifth, downminor 6th
| up-fifth, downminor 6th
| ^G, vAb
| ^G, vAb
|Si/Lu
| Si/Lu
|-
|-
| 16
| 16
Line 175: Line 175:
| minor 6th
| minor 6th
| Ab
| Ab
|Lo
| Lo
|-
|-
| 17
| 17
Line 183: Line 183:
| mid 6th
| mid 6th
| vA
| vA
|La
| La
|-
|-
| 18
| 18
Line 191: Line 191:
| major 6th
| major 6th
| A
| A
|Le
| Le
|-
|-
| 19
| 19
Line 199: Line 199:
| upmajor 6th, downminor 7th
| upmajor 6th, downminor 7th
| ^A, vBb
| ^A, vBb
|Li/Tu
| Li/Tu
|-
|-
| 20
| 20
Line 207: Line 207:
| minor 7th
| minor 7th
| Bb
| Bb
|To
| To
|-
|-
| 21
| 21
Line 215: Line 215:
| mid 7th
| mid 7th
| vB
| vB
|Ta
| Ta
|-
|-
| 22
| 22
Line 223: Line 223:
| major 7th
| major 7th
| B
| B
|Te
| Te
|-
|-
| 23
| 23
Line 231: Line 231:
| upmajor 7th, down-8ve
| upmajor 7th, down-8ve
| ^B, vC
| ^B, vC
|Ti/Du
| Ti/Du
|-
|-
| 24
| 24
Line 239: Line 239:
| perfect 8ve
| perfect 8ve
| C
| C
|Do
| Do
|}
|}


Line 246: Line 246:
In many other edos, 5/4 is downmajor and 11/9 is mid. To agree with this, the term mid is generally preferred over down or downmajor.
In many other edos, 5/4 is downmajor and 11/9 is mid. To agree with this, the term mid is generally preferred over down or downmajor.


=== Interval qualities in color notation ===
Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:
Combining ups and downs notation with [[color notation]], qualities can be loosely associated with colors:


{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
! quality
! Quality
![[color name]]
! [[Color name|Color Name]]
! monzo format
! Monzo Format
! examples
! Examples
|-
|-
| downminor
| downminor
Line 293: Line 294:
|}
|}


Ups and downs notation can be used to name chords. See [[24edo Chord Names]] and [[Ups and Downs Notation#Chords and Chord Progressions]].  
Ups and downs notation can be used to name chords. See [[24edo Chord Names]] and [[Ups and Downs Notation #Chords and Chord Progressions]].  


=== William Lynch's notation ===
=== William Lynch's notation ===
Line 851: Line 852:
* Ellis, Don. ''[https://archive.org/details/don-ellis-quarter-tones/ Quarter Tones: A Text with Musical Examples, Exercises and Etudes]''. 1975.
* Ellis, Don. ''[https://archive.org/details/don-ellis-quarter-tones/ Quarter Tones: A Text with Musical Examples, Exercises and Etudes]''. 1975.
* [[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Icosikaitetraphonic Scales for Guitar: Theory and Scales for Twenty-four Equal Divisions of the Octave]''. 2009. (Features a practical approach to understanding the tuning, and over 550 scale examples on nine-string finger board charts, which allows for both symmetrical tuning visualization and standard guitar tuning- helpful for bassists and large range guitarists as well. Includes MOS, DE, and *all* the scales/modes from the list above.)
* [[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Icosikaitetraphonic Scales for Guitar: Theory and Scales for Twenty-four Equal Divisions of the Octave]''. 2009. (Features a practical approach to understanding the tuning, and over 550 scale examples on nine-string finger board charts, which allows for both symmetrical tuning visualization and standard guitar tuning- helpful for bassists and large range guitarists as well. Includes MOS, DE, and *all* the scales/modes from the list above.)
== See also ==
* [[Equal-step tuning|Equal multiplications]] of MIDI-resolution units
** [[48edo]] (2mu tuning)
** [[96edo]] (3mu tuning)
** [[192edo]] (4mu tuning)
** [[384edo]] (5mu tuning)
** [[768edo]] (6mu tuning)
** [[1536edo]] (7mu tuning)
** [[3072edo]] (8mu tuning)
** [[6144edo]] (9mu tuning)
** [[12288edo]] (10mu tuning)
** [[24576edo]] (11mu tuning)
** [[49152edo]] (12mu tuning)
** [[98304edo]] (13mu tuning)
** [[196608edo]] (14mu tuning)


== External links ==
== External links ==
* [http://tonalsoft.com/enc/q/quarter-tone.aspx quarter-tone / 24-edo - Encyclopedia of Microtonal Music Theory] [https://www.webcitation.org/5xeFMH6cd Permalink]
* [http://tonalsoft.com/enc/q/quarter-tone.aspx Tonalsoft Encyclopedia | ''quarter-tone / 24-edo / 24-ed2''] [https://www.webcitation.org/5xeFMH6cd Permalink]
* [http://www.96edo.com/24_EDO.html About 24-EDO] by Shaahin Mohajeri [https://www.webcitation.org/5xeFBNdQW Permalink]
* [http://www.96edo.com/24_EDO.html About 24-EDO] by Shaahin Mohajeri [https://www.webcitation.org/5xeFBNdQW Permalink]
* [https://docs.google.com/file/d/0Bzrl-iLY6DeEVkl1VjBGdEJlOTg/edit Notation and Chord Names for 24-EDO] by William Lynch
* [https://docs.google.com/file/d/0Bzrl-iLY6DeEVkl1VjBGdEJlOTg/edit Notation and Chord Names for 24-EDO] by William Lynch
* [http://www.tonalsoft.com/sonic-arts/darreg/dar8.htm The place of QUARTERTONES in Today's Xenharmonics] by [[Ivor Darreg]]     
* [http://www.tonalsoft.com/sonic-arts/darreg/dar8.htm The place of QUARTERTONES in Today's Xenharmonics] by [[Ivor Darreg]]     


[[Category:24edo| ]] <!-- main article -->
[[Category:Equal divisions of the octave|##]] <!-- 2-digit number -->
[[Category:Listen]]
[[Category:Listen]]
[[Category:Quartertone]]
[[Category:Quartertone]]
[[Category:Quartismic]]
[[Category:Quartismic]]
[[Category:Subgroup]]
[[Category:Subgroup temperaments]]


[[Category:Twentuning]]
[[Category:Twentuning]]
Retrieved from "https://en.xen.wiki/w/24edo"