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The '''24edo''' system divides the octave into 24 equal parts of exactly 50 cents each. It 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 and 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. 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]]''.
The '''24edo''' system divides the octave into 24 equal parts of exactly 50 cents each. It 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 and 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. 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]]''.


=Theory=
== Theory ==
{{Odd harmonics in edo|edo=24}}
{{Odd harmonics in edo|edo=24}}


The [[Harmonic_Limit|5-limit]] approximations in 24-tone equal temperament are the same as those in 12-tone equal temperament, therefore 24-tone equal temperament offers nothing new as far as approximating the 5-limit is concerned. The 7th harmonic-based intervals ([[7/4|7:4]], [[7/5|7:5]] and [[7/6|7:6]]) are almost as bad in 24-tET as in 12-tET. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12-tET requires high-degree tunings like [[36edo|36-tET]], [[72edo|72-tET]], [[84edo|84-tET]] or [[156edo|156-tET]].  However, it should be noted that 24edo, like [[22edo]], ''does'' temper out [[Quartisma|117440512/117406179]], linking the otherwise sub-par 7-limit harmonies with those of the 11-limit, and 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 [[5-limit]] approximations in 24-tone equal temperament are the same as those in 12-tone equal temperament, therefore 24-tone equal temperament 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 24-tET as in 12-tET. To achieve a satisfactory level of approximation while maintaining the 12 notes of 12-tET 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, and 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 72 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|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 subgroup temperament, on which it is quite accurate.
The tunings supplied by 72 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 subgroup temperament, on which it is quite accurate.  
===Differences between distributionally-even scales and smaller edos===
 
=== Differences between distributionally-even scales and smaller edos ===
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= Notation =
== Notation ==


== Ups and down notation ==
=== Ups and down notation ===


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For a more complete list of chords, see [[24edo Chord Names]] and [[Ups and Downs Notation #Chords and Chord Progressions]].
For a more complete list of chords, see [[24edo Chord Names]] and [[Ups and Downs Notation #Chords and Chord Progressions]].


== William Lynch's notation ==
=== William Lynch's notation ===


24 EDO breaks intervals into two sets of five cartegories. Infra - Minor - Neutral - Major - Ultra for seconds, thirds, sixths, and sevenths; and diminished - narrow - perfect - wide - augmented for fourths, fifths, unison, and octave. For other strange enharmonics, wide and narrow can be used in conjunction with augmented and diminished intervals such as 550 cents being called a narrow diminished fifth and 850 cents being called a wide augmented fifth. See the [[24_EDO_Interval_names_and_Harmonies|full article on 24 Edo intervals.]]
24 EDO breaks intervals into two sets of five cartegories. Infra - Minor - Neutral - Major - Ultra for seconds, thirds, sixths, and sevenths; and diminished - narrow - perfect - wide - augmented for fourths, fifths, unison, and octave. For other strange enharmonics, wide and narrow can be used in conjunction with augmented and diminished intervals such as 550 cents being called a narrow diminished fifth and 850 cents being called a wide augmented fifth. See the [[24_EDO_Interval_names_and_Harmonies|full article on 24 Edo intervals.]]
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=== Interval Alterations ===
=== Interval alterations ===


The special alterations of the intervals and chords of 12 equal can be notated like this:
The special alterations of the intervals and chords of 12 equal can be notated like this:
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The Latin words "tendo" (meaning "expand") and "arto" (meaning "contract") can be used to replace the words "supermajor" and "subminor" in order to shorten the names of the intervals.
The Latin words "tendo" (meaning "expand") and "arto" (meaning "contract") can be used to replace the words "supermajor" and "subminor" in order to shorten the names of the intervals.


=== Chord Names ===
==== Chord names ====


Naming chords in 24edo can be achieved by adding a few things to the already [[#|existing]] set of terms that are used to name 12edo chords.
Naming chords in 24edo can be achieved by adding a few things to the already [[#|existing]] set of terms that are used to name 12edo chords.
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Arto Sub Seventh Tendo Thirteenth or artsub7^13 = Arto tetrad with an arto seventh and a tendo thirteenth on top Minor Seventh Flat Five Arto Ninth Super Eleventh or m7b5^9^11
Arto Sub Seventh Tendo Thirteenth or artsub7^13 = Arto tetrad with an arto seventh and a tendo thirteenth on top Minor Seventh Flat Five Arto Ninth Super Eleventh or m7b5^9^11


=Quartertone Accidentals=
== Quartertone accidentals ==
Besides ups and downs, there are various systems for notating quarter tones. Here are some of them, along with their pros and cons.
Besides ups and downs, there are various systems for notating quarter tones. Here are some of them, along with their pros and cons.


==Mainstream Quartertone Notation==
=== Mainstream quartertone notation ===


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Cons: Clutters a score easily, can get confusing when sight read at faster paces
Cons: Clutters a score easily, can get confusing when sight read at faster paces


==Alternate Quartertone Accidentals==
=== Alternate quartertone accidentals ===


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Cons: Not practical, tends to clutter a score
Cons: Not practical, tends to clutter a score


==Persian Accidentals==
=== Persian accidentals ===


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Cons: Hard to write on a computer, doesn't fit with standard notation well
Cons: Hard to write on a computer, doesn't fit with standard notation well


==Sagittal Notation==
=== Sagittal notation ===
[[Sagittal_notation|Sagittal notation]] works extremely well for 24 Edo notation as well as other systems.
[[Sagittal notation]] works extremely well for 24edo notation as well as other systems.


It's easy on the eyes, easy to recognize the various symbols and keeps a score looking tidy and neat.
It is easy on the eyes, easy to recognize the various symbols and keeps a score looking tidy and neat.


A possibility for the best approach would be to not use traditional sharps and flats altogether and replace them
A possibility for the best approach would be to not use traditional sharps and flats altogether and replace them
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Cons: Not as familiar as traditional notation, and thus not immediately accessible to many traditional musicians who are just starting out with microtonality
Cons: Not as familiar as traditional notation, and thus not immediately accessible to many traditional musicians who are just starting out with microtonality


We also have, from the appendix to [[The Sagittal Songbook]] by [[JacobBarton|Jacob A. Barton]], a diagram of how to notate 24-EDO in the Revo flavor of Sagittal:
We also have, from the appendix to [[The Sagittal Songbook]] by [[Jacob Barton|Jacob A. Barton]], a diagram of how to notate 24-EDO in the Revo flavor of Sagittal:


[[File:24edo Sagittal.png|800px]]
[[File:24edo Sagittal.png|800px]]


=Chord Types=
== Chord types ==
24edo features a rich variety of not only new chords, but also alterations that can be used with regular 12 Edo chords. For example, an approximation of the ninth, eleventh, and thirteenth harmonic can be added to a major triad to create a sort of super-extended chord structure of a major chord: 4:5:6:9:11:13.
24edo features a rich variety of not only new chords, but also alterations that can be used with regular 12 Edo chords. For example, an approximation of the ninth, eleventh, and thirteenth harmonic can be added to a major triad to create a sort of super-extended chord structure of a major chord: 4:5:6:9:11:13.


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These chords though tend to lack the forcefulness to sound like resolved, tonal sonorities, but can be resolved of that issue by using tetrads in place of triads.
These chords though tend to lack the forcefulness to sound like resolved, tonal sonorities, but can be resolved of that issue by using tetrads in place of triads.


For example, the neutral triad can have the neutral 7th added to it to make a full neutral tetrad: 0-7-14-21. However, another option is to replace the neutral third with an 11/8 to produce a sort of 11 limit neutral tetrad. 0-14-21-35 [[William_Lynch|William Lynch]] considers this chord to be the most consonant tetrad in 24edo involving a neutral tonality. 24 edo also is very good at 15 limit and does 13 quite well allowing barbodos 10:13:15 and barbodos minor triad 26:30:39 to be used as an entirely new harmonic system.
For example, the neutral triad can have the neutral 7th added to it to make a full neutral tetrad: 0-7-14-21. However, another option is to replace the neutral third with an 11/8 to produce a sort of 11 limit neutral tetrad. 0-14-21-35 [[William Lynch]] considers this chord to be the most consonant tetrad in 24edo involving a neutral tonality. 24 edo also is very good at 15 limit and does 13 quite well allowing barbodos 10:13:15 and barbodos minor triad 26:30:39 to be used as an entirely new harmonic system.


More good chords in 24-tET:
More good chords in 24-tET:
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Its inversion, 0-3-6-10-14 ("minor")
Its inversion, 0-3-6-10-14 ("minor")


0-5-10 (another kind of "neutral", splitting the fourth in two. The 0-5-10 can be extended into a pentatonic scale, 0-5-10-14-19-24 ([[Godzilla|godzilla]]), that is close to equi-pentatonic and also close to several Indonesian slêndros. In a similar way 0-7-14 extends to 0-4-7-11-14-18-21-24 ([[Mohajira|mohajira]]), a heptatonic scale close to several Arabic scales.)
0-5-10 (another kind of "neutral", splitting the fourth in two. The 0-5-10 can be extended into a pentatonic scale, 0-5-10-14-19-24 ([[Godzilla|godzilla]]), that is close to equi-pentatonic and also close to several Indonesian slêndros. In a similar way 0-7-14 extends to 0-4-7-11-14-18-21-24 ([[mohajira]]), a heptatonic scale close to several Arabic scales.)


William Lynch considers these as some possible good tetrads:
William Lynch considers these as some possible good tetrads:
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The tendo chord can also be spelled 1 ^3 5 ^6. Due to convenience, the names Arto and tendo have been changed to Ultra and Infra.
The tendo chord can also be spelled 1 ^3 5 ^6. Due to convenience, the names Arto and tendo have been changed to Ultra and Infra.


= Commas =
== Commas ==


24 EDO [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 24 38 56 67 83 89 }}.  
24edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 24 38 56 67 83 89 }}.  


{| class="commatable wikitable center-1 center-2 right-4 center-5"
{| class="commatable wikitable center-1 center-2 right-4 center-5"
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<references/>
<references/>


=Rank two temperaments=
== Rank-2 temperaments ==
[[List_of_24et_rank_two_temperaments_by_badness|List of 24et rank two temperaments by badness]]
* [[List of 24et rank two temperaments by badness]]
 
* [[List of edo-distinct 24et rank two temperaments]]
[[List_of_edo-distinct_24et_rank_two_temperaments|List of edo-distinct 24et rank two temperaments]]


Important MOSes include:  
Important MOSes include:  
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=Scales / Modes=
== Scales / modes ==


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=Instruments=
== Instruments ==
The ever-arising question in microtonal music, how to play it on instruments designed for 12edo, has a relatively simple answer in the case of 24edo: use two standard instruments tuned a quartertone apart. This [[Microtonal_Keyboards#twelvenoteoctavescales|"12 note octave scales"]] approach is used in a wide part of the existing literature - see below.
The ever-arising question in microtonal music, how to play it on instruments designed for 12edo, has a relatively simple answer in the case of 24edo: use two standard instruments tuned a quartertone apart. This [[Microtonal_Keyboards#twelvenoteoctavescales|"12 note octave scales"]] approach is used in a wide part of the existing literature - see below.


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24-tone "1/4-tone" Guitar by Ron Sword / Sword guitars
24-tone "1/4-tone" Guitar by Ron Sword / Sword guitars


Hidekazu Wakabayashi tuned a piano and harp to where the normal sharps and flats are tuned 50 cents higher in which he called [[Iceface_Tuning|Iceface tuning]].
Hidekazu Wakabayashi tuned a piano and harp to where the normal sharps and flats are tuned 50 cents higher in which he called [[Iceface tuning]].


=Music=
== Music ==
<span style="">[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Mohajeri/Microhex3.mp3 Microhex3]</span> <span style="">[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Mohajeri/Microhex4.mp3 Microhex4]</span> and <span style="">[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Mohajeri/Microhex5.mp3 Microhex5]</span> by [http://www.96edo.com/About_me.html Shaahin Mohajeri]
<span style="">[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Mohajeri/Microhex3.mp3 Microhex3]</span> <span style="">[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Mohajeri/Microhex4.mp3 Microhex4]</span> and <span style="">[http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Mohajeri/Microhex5.mp3 Microhex5]</span> by [http://www.96edo.com/About_me.html Shaahin Mohajeri]


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[http://en.wikipedia.org/wiki/List_of_quarter_tone_pieces List of quartertone pieces on Wikipedia]
[http://en.wikipedia.org/wiki/List_of_quarter_tone_pieces List of quartertone pieces on Wikipedia]


=Practical Theory / Books=
== Practical theory / books ==


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'''"Icosakaiteraphonic Scales for Guitar"''' - A Book for Twenty-Four Equal Divisions of the Octave on guitar, or 'Quarter-tones'. 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.
'''"Icosakaiteraphonic Scales for Guitar"''' - A Book for Twenty-Four Equal Divisions of the Octave on guitar, or 'Quarter-tones'. 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.


=External links=
== External links ==
[http://tonalsoft.com/enc/q/quarter-tone.aspx quarter-tone / 24-edo - Encyclopedia of Microtonal Music Theory] [http://www.webcitation.org/5xeFMH6cd Permalink]
* [http://tonalsoft.com/enc/q/quarter-tone.aspx quarter-tone / 24-edo - Encyclopedia of Microtonal Music Theory] [http://www.webcitation.org/5xeFMH6cd Permalink]
 
* [http://www.96edo.com/24_EDO.html About 24-EDO] by Shaahin Mohajeri [http://www.webcitation.org/5xeFBNdQW Permalink]
[http://www.96edo.com/24_EDO.html About 24-EDO] by Shaahin Mohajeri [http://www.webcitation.org/5xeFBNdQW Permalink]
* [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]]     
[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|Ivor Darreg]]     


[[Category:24edo| ]] <!-- main article -->
[[Category:24edo| ]] <!-- main article -->
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[[Category:Twentuning]]
[[Category:Twentuning]]


[[Category:Todo:cleanup]]
{{Todo| cleanup }}
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