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=Basics=
=Theory=
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 - [[DIY_Quartertone_Composition_with_12_equal_tools|see this page]].
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 - [[DIY_Quartertone_Composition_with_12_equal_tools|see this page]].


=24edo as a temperament=
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]].
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]].
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 24-tone equal temperament. 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|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.


= Notation =
== Ups and down notation ==
{| class="wikitable"
{| class="wikitable"
|-
|-
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| style="text-align:center;" | ^P1, vm2
| style="text-align:center;" | ^P1, vm2
| style="text-align:center;" | up-unison, downminor 2nd
| style="text-align:center;" | up-unison, downminor 2nd
| style="text-align:center;" | C^, Dbv
| style="text-align:center;" | ^C, vDb
|-
|-
| style="text-align:center;" | 2
| style="text-align:center;" | 2
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| style="text-align:center;" | ~2
| style="text-align:center;" | ~2
| style="text-align:center;" | mid 2nd
| style="text-align:center;" | mid 2nd
| style="text-align:center;" | Dv
| style="text-align:center;" | vD
|-
|-
| style="text-align:center;" | 4
| style="text-align:center;" | 4
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| style="text-align:center;" | ^M2, vm3
| style="text-align:center;" | ^M2, vm3
| style="text-align:center;" | upmajor 2nd, downminor 3rd
| style="text-align:center;" | upmajor 2nd, downminor 3rd
| style="text-align:center;" | D^, Ebv
| style="text-align:center;" | ^D, vEb
|-
|-
| style="text-align:center;" | 6
| style="text-align:center;" | 6
Line 91: Line 95:
| style="text-align:center;" | ~3
| style="text-align:center;" | ~3
| style="text-align:center;" | mid 3rd
| style="text-align:center;" | mid 3rd
| style="text-align:center;" | Ev
| style="text-align:center;" | vE
|-
|-
| style="text-align:center;" | 8
| style="text-align:center;" | 8
Line 107: Line 111:
|576 (240<sub>16</sub>)
|576 (240<sub>16</sub>)
| style="text-align:center;" | 22/17
| style="text-align:center;" | 22/17
| style="text-align:center;" | ^M3, vP4
| style="text-align:center;" | ^M3, v4
| style="text-align:center;" | upmajor 3rd, down-fourth
| style="text-align:center;" | upmajor 3rd, down-4th
| style="text-align:center;" | E^, Fv
| style="text-align:center;" | ^E, vF
|-
|-
| style="text-align:center;" | 10
| style="text-align:center;" | 10
Line 125: Line 129:
|704 (2C0<sub>16</sub>)
|704 (2C0<sub>16</sub>)
| style="text-align:center;" | 11/8
| style="text-align:center;" | 11/8
| style="text-align:center;" | ^P4
| style="text-align:center;" | ^4, ~4
| style="text-align:center;" | up-fourth
| style="text-align:center;" | up-4th, mid-4th
| style="text-align:center;" | F^
| style="text-align:center;" | ^F
|-
|-
| style="text-align:center;" | 12
| style="text-align:center;" | 12
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|832 (340<sub>16</sub>)
|832 (340<sub>16</sub>)
| style="text-align:center;" | 16/11
| style="text-align:center;" | 16/11
| style="text-align:center;" | vP5
| style="text-align:center;" | v5, ~5
| style="text-align:center;" | down-fifth
| style="text-align:center;" | down-5th, mid-5th
| style="text-align:center;" | Gv
| style="text-align:center;" | vG
|-
|-
| style="text-align:center;" | 14
| style="text-align:center;" | 14
Line 161: Line 165:
|960 (3C0<sub>16</sub>)
|960 (3C0<sub>16</sub>)
| style="text-align:center;" | 17/11
| style="text-align:center;" | 17/11
| style="text-align:center;" | ^P5, vm6
| style="text-align:center;" | ^5, vm6
| style="text-align:center;" | up-fifth, downminor 6th
| style="text-align:center;" | up-fifth, downminor 6th
| style="text-align:center;" | G^, Abv
| style="text-align:center;" | ^G, vAb
|-
|-
| style="text-align:center;" | 16
| style="text-align:center;" | 16
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| style="text-align:center;" | ~6
| style="text-align:center;" | ~6
| style="text-align:center;" | mid 6th
| style="text-align:center;" | mid 6th
| style="text-align:center;" | Av
| style="text-align:center;" | vA
|-
|-
| style="text-align:center;" | 18
| style="text-align:center;" | 18
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| style="text-align:center;" | ^M6, vm7
| style="text-align:center;" | ^M6, vm7
| style="text-align:center;" | upmajor 6th, downminor 7th
| style="text-align:center;" | upmajor 6th, downminor 7th
| style="text-align:center;" | A^, Bbv
| style="text-align:center;" | ^A, vBb
|-
|-
| style="text-align:center;" | 20
| style="text-align:center;" | 20
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| style="text-align:center;" | ~7
| style="text-align:center;" | ~7
| style="text-align:center;" | mid 7th
| style="text-align:center;" | mid 7th
| style="text-align:center;" | Bv
| style="text-align:center;" | vB
|-
|-
| style="text-align:center;" | 22
| style="text-align:center;" | 22
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| style="text-align:center;" | ^M7, vP8
| style="text-align:center;" | ^M7, vP8
| style="text-align:center;" | upmajor 7th, down-8ve
| style="text-align:center;" | upmajor 7th, down-8ve
| style="text-align:center;" | B^, Cv
| style="text-align:center;" | ^B, vC
|-
|-
| style="text-align:center;" | 24
| style="text-align:center;" | 24
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|}
|}


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 24-tone equal temperament. 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. Some good chords in 24-tET are (the numbers are edosteps, e.g. 4 is a major second, 8 is a major third):
Ups and downs notation can be used to name chords. Alterations are always enclosed in parentheses, additions never are. An up, down or mid after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).  
 
0-4-8-11-14 ("major" chord with a 9:8 and a 11:8 above the root)
 
Its inversion, 0-3-6-10-14 ("minor")
 
0-7-14 ("neutral")
 
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.)


===Commas===
In many other edos, especially larger ones, 5/4 is downmajor and 11/9 is mid. To agree with this, the term mid is generally preferred over down or downmajor.
24 EDO tempers out the following commas. (Note: This assumes val &lt; 24 38 56 67 83 89 |.)


{| class="wikitable"
{| class="wikitable"
|-
|-
! | Comma
!edosteps
! | Monzo
!notes
! | Value (Cents)
!written name
! | Name 1
!spoken name
! | Name 2
!JI chord, if any
! | Name 3
|-
|-
| style="text-align:center;" | 531441/524288
|0-5-14
| style="text-align:center;" |<nowiki> | -19 12 </nowiki>&gt;
|C vEb G
| style="text-align:center;" | 23.46
|Cvm
| style="text-align:center;" | Pythagorean Comma
|C downminor
| style="text-align:center;" |
|6:7:9
| style="text-align:center;" |
|-
| style="text-align:center;" | 648/625
| style="text-align:center;" |<nowiki> | 3 4 -4 </nowiki>&gt;
| style="text-align:center;" | 62.57
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" |
|-
| style="text-align:center;" | 128/125
| style="text-align:center;" |<nowiki> | 7 0 -3 </nowiki>&gt;
| style="text-align:center;" | 41.06
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" |
|-
| style="text-align:center;" | 81/80
| style="text-align:center;" |<nowiki> | -4 4 -1 </nowiki>&gt;
| style="text-align:center;" | 21.51
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 2048/2025
| style="text-align:center;" |<nowiki> | 11 -4 -2 </nowiki>&gt;
| style="text-align:center;" | 19.55
| style="text-align:center;" | Diaschisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 5201701/5149091
|0-6-14
| style="text-align:center;" |<nowiki> | 26 -12 -3 </nowiki>&gt;
|C Eb G
| style="text-align:center;" | 17.60
|Cm
| style="text-align:center;" | Misty Comma
|C minor
| style="text-align:center;" |
|10:12:15 <u>or</u> 6/(6:5:4)
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 32805/32768
|0-7-14
| style="text-align:center;" |<nowiki> | -15 8 1 </nowiki>&gt;
|C vE G
| style="text-align:center;" | 1.95
|C~
| style="text-align:center;" | Schisma
|C mid
| style="text-align:center;" |
|18:22:27
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 1465155/1465142
|0-8-14
| style="text-align:center;" |<nowiki> | 161 -84 -12 </nowiki>&gt;
|C E G
| style="text-align:center;" | 0.02
|C
| style="text-align:center;" | Atom
|C major or C
| style="text-align:center;" |
|4:5:6
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 49/48
|0-9-14
| style="text-align:center;" |<nowiki> | -4 -1 0 2 </nowiki>&gt;
|C ^E G
| style="text-align:center;" | 35.70
|C^
| style="text-align:center;" | Slendro Diesis
|C upmajor or C up
| style="text-align:center;" |
|14:18:21 <u>or</u> 9/(9:7:6)
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 245/243
|0-8-13
| style="text-align:center;" |<nowiki> | 0 -5 1 2 </nowiki>&gt;
|C E vG
| style="text-align:center;" | 14.19
|C(v5)
| style="text-align:center;" | Sensamagic
|C down-five
| style="text-align:center;" |
|
| style="text-align:center;" |  
|-
|-
| style="text-align:center;" | 19683/19600
|0-7-13
| style="text-align:center;" |<nowiki> | -4 9 -2 -2 </nowiki>&gt;
|C vE vG
| style="text-align:center;" | 7.32
|C~(v5)
| style="text-align:center;" | Cataharry
|C mid down-five
| style="text-align:center;" |
|
| style="text-align:center;" |  
|-
|-
| style="text-align:center;" | 6144/6125
|0-4-11-14
| style="text-align:center;" |<nowiki> | 11 1 -3 -2 </nowiki>&gt;
|C D ^F G
| style="text-align:center;" | 5.36
|C^4,9
| style="text-align:center;" | Porwell
|C up-four add-nine
| style="text-align:center;" |
|8:9:11:12
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 121/120
|0-9-14-19
| style="text-align:center;" |<nowiki> | -3 -1 -1 0 2 </nowiki>&gt;
|C ^E G ^A
| style="text-align:center;" | 14.37
|C^6
| style="text-align:center;" | Biyatisma
|C up-six
| style="text-align:center;" |
|14:18:21:24
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 176/175
|0-5-14-20
| style="text-align:center;" |<nowiki> | 4 0 -2 -1 1 </nowiki>&gt;
|C vEb G Bb
| style="text-align:center;" | 9.86
|Cm7(v3)
| style="text-align:center;" | Valinorsma
|C minor-seven down-three
| style="text-align:center;" |
|
| style="text-align:center;" |  
|-
|-
| style="text-align:center;" | 896/891
|0-5-14-19
| style="text-align:center;" |<nowiki> | 7 -4 0 1 -1 </nowiki>&gt;
|C vEb G vBb
| style="text-align:center;" | 9.69
|Cvm7
| style="text-align:center;" | Pentacircle
|C downminor-seven
| style="text-align:center;" |
|12:14:18:21
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 243/242
|0-7-14-21
| style="text-align:center;" |<nowiki> | -1 5 0 0 -2 </nowiki>&gt;
|C vE G vB
| style="text-align:center;" | 7.14
|C~7
| style="text-align:center;" | Rastma
|C mid-seven
| style="text-align:center;" |
|18:22:27:33
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | 385/384
|0-7-14-21-28
| style="text-align:center;" |<nowiki> | -7 -1 1 1 1 </nowiki>&gt;
|C vE G vB D
| style="text-align:center;" | 4.50
|C~9
| style="text-align:center;" | Keenanisma
|C mid-nine
| style="text-align:center;" |
|
| style="text-align:center;" |  
|-
|-
| style="text-align:center;" | 9801/9800
|0-7-14-21-28-35
| style="text-align:center;" |<nowiki> | -3 4 -2 -2 2 </nowiki>&gt;
|C vE G vB D ^F
| style="text-align:center;" | 0.18
|C~11
| style="text-align:center;" | Kalisma
|C mid-eleven
| style="text-align:center;" | Gauss' Comma
|
| style="text-align:center;" |  
|-
|-
| style="text-align:center;" | 91/90
|0-7-14-21-28-35-42
| style="text-align:center;" |<nowiki> | -1 -2 -1 1 0 1 </nowiki>&gt;
|C vE G vB D ^F A
| style="text-align:center;" | 19.13
|C~13
| style="text-align:center;" | Superleap
|C mid-thirteen
| style="text-align:center;" |
|
| style="text-align:center;" |  
|-
|-
| style="text-align:center;" | 676/675
|0-8-14-28-35-42
| style="text-align:center;" |<nowiki> | 2 -3 -2 0 0 2 </nowiki>&gt;
|C E G D ^F vA
| style="text-align:center;" | 2.56
|C9,~11~13no7
| style="text-align:center;" | Parizeksma
|C nine mid-eleven mid-thirteen no-seven
| style="text-align:center;" |
|4:5:6:9:11:13
| style="text-align:center;" |
|}
|}
For a more complete list of chords, see [[Ups and Downs Notation#Chords and Chord Progressions|Ups and Downs Notation - Chords and Chord Progressions]].


=Intervals=
==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.
 
These are the intervals of 24 EDO that do not exist in 12 EDO: 2


[[24_EDO_Interval_names_and_Harmonies|See full article on 24 Edo intervals.]]
These are the intervals of 24 EDO that do not exist in 12 EDO:


{| class="wikitable"
{| class="wikitable"
Line 685: Line 633:
|}
|}


=Notation=
===Interval Alterations===
There have been disputes about disadvantages of various systems for notating quarter tones. Here are some of the few systems along with pros and cons.
The special alterations of the intervals and chords of 12 equal can be notated like this:
 
Supermajor or "'''Tendo'''" is a major interval raised a quarter tone
 
Subminor or "'''Arto'''" is a minor interval lowered a quarter tone
 
Neutral are intervals that exist between the major and minor version of an interval
 
The prefix '''under''' indicates a perfect interval lowered by one quarter tone
 
The prefix '''over''' indicates a perfect interval raised by a quarter tone
 
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===
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.
 
They are:
 
Super + perfect interval such as "perfect fifth" means to raise it by a quarter tone
 
Sub + perfect interval means to lower a quarter tone
 
Sharp is to raise by one half tone
 
Flat is to raise by a half tone
 
Neutral, arto and tendo refer to triads or tetrads
 
Neutral, arto, or tendo + interval name of 2nd, 3rd, 6th, or 7th is to alter respectively
 
Examples:
 
Neutral Super Eleventh or neut^11 = C neutral 7th chord with a super 11th thrown on top
 
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=
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==
Line 803: Line 789:
Cons: Not practical, tends to clutter a score
Cons: Not practical, tends to clutter a score


==In Persian music==
==Persian Accidentals==


{| class="wikitable"
{| class="wikitable"
Line 832: Line 818:
[[File:sagittal_24.PNG|alt=sagittal 24.PNG|sagittal 24.PNG]]
[[File:sagittal_24.PNG|alt=sagittal 24.PNG|sagittal 24.PNG]]


=Interval Alterations=
=Chord Types=
The special alterations of the intervals and chords of 12 equal can be notated like this:
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.


Supermajor or "'''Tendo'''" is a major interval raised a quarter tone
As for entirely new chords, the most obvious is the neutral or mid triad 0-7-14. However there are other options such as


Subminor or "'''Arto'''" is a minor interval lowered a quarter tone
0-9-14 (Ultra Triad or upmajor triad) and 0-5-14 (Infra Triad or downminor triad), the chord names being based on what kind of third is in the chord.


Neutral are intervals that exist between the major and minor version of an interval
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.


The prefix '''under''' indicates a perfect interval lowered by one quarter tone
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.


The prefix '''over''' indicates a perfect interval raised by a quarter tone
More good chords in 24-tET:


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.
0-4-8-11-14 ("major" chord with a 9:8 and a 11:8 above the root)


=Chord Structures=
Its inversion, 0-3-6-10-14 ("minor")
24edo features a rich variety on 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.


As for entirely new chords, 24edo features many possibilities for chords. The most obvious is the neutral or mid triad 0-7-14 however there are other options such as
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.)
 
0-9-14 (Ultra Triad or upmajor triad) and 0-5-14 (Infra Triad or downminor triad), the chord names being based on what kind of third is in the chord.
 
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.


William Lynch considers these as some possible good tetrads:
William Lynch considers these as some possible good tetrads:
Line 876: Line 855:
| | arto
| | arto
| | 0 5 14 20
| | 0 5 14 20
| | 1 bv3 5 b7
| | 1 vb3 5 b7
| | [[File:arto_tetrad_on_C.mp3]]
| | [[File:arto_tetrad_on_C.mp3]]
|-
|-
| | tendo
| | tendo
| | 0 9 14 19
| | 0 9 14 19
| | 1 ^3 5 bv7
| | 1 ^3 5 vb7
| | ...
| | ...
|}
|}


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.


=Naming Chords in 24edo=
=Commas=
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.
24 EDO tempers out the following [[commas]]. (Note: This assumes val &lt; 24 38 56 67 83 89 |.)


They are:
{| class="wikitable" style="text-align:center;"
 
|-
Super + perfect interval such as "perfect fifth" means to raise it by a quarter tone
! | [[Ratio]]
 
! | [[Monzo]]
Sub + perfect interval means to lower a quarter tone
! | [[Cents]]
 
![[Color notation/Temperament Names|Color Name]]
Sharp is to raise by one half tone
! | Name 1
 
! | Name 2
Flat is to raise by a half tone
! | Name 3
 
|-
Neutral, arto and tendo refer to triads or tetrads
| style="text-align:center;" | 531441/524288
 
| style="text-align:center;" |<nowiki> | -19 12 </nowiki>&gt;
Neutral, arto, or tendo + interval name of 2nd, 3rd, 6th, or 7th is to alter respectively
| style="text-align:center;" | 23.46
 
|Lalawa
Examples:
| style="text-align:center;" | Pythagorean Comma
 
| style="text-align:center;" |
Neutral Super Eleventh or neut^11 = C neutral 7th chord with a super 11th thrown on top
| style="text-align:center;" |
 
|-
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
| style="text-align:center;" | 648/625
 
| style="text-align:center;" |<nowiki> | 3 4 -4 </nowiki>&gt;
Alternatively, [[Ups_and_Downs_Notation|ups and downs notation]] can be used. Here are the zo, gu, ilo, yo and ru triads:
| style="text-align:center;" | 62.57
 
|Quadgu
{| class="wikitable"
| style="text-align:center;" | Major Diesis
| style="text-align:center;" | Diminished Comma
| style="text-align:center;" |
|-
| style="text-align:center;" | 128/125
| style="text-align:center;" |<nowiki> | 7 0 -3 </nowiki>&gt;
| style="text-align:center;" | 41.06
|Trigu
| style="text-align:center;" | Diesis
| style="text-align:center;" | Augmented Comma
| style="text-align:center;" |
|-
| style="text-align:center;" | 81/80
| style="text-align:center;" |<nowiki> | -4 4 -1 </nowiki>&gt;
| style="text-align:center;" | 21.51
|Gu
| style="text-align:center;" | Syntonic Comma
| style="text-align:center;" | Didymos Comma
| style="text-align:center;" | Meantone Comma
|-
| style="text-align:center;" | 2048/2025
| style="text-align:center;" |<nowiki> | 11 -4 -2 </nowiki>&gt;
| style="text-align:center;" | 19.55
|Sagugu
| style="text-align:center;" | Diaschisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 5201701/5149091
| style="text-align:center;" |<nowiki> | 26 -12 -3 </nowiki>&gt;
| style="text-align:center;" | 17.60
|Sasa-trigu
| style="text-align:center;" | Misty Comma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 32805/32768
| style="text-align:center;" |<nowiki> | -15 8 1 </nowiki>&gt;
| style="text-align:center;" | 1.95
|Layo
| style="text-align:center;" | Schisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 1465155/1465142
| style="text-align:center;" |<nowiki> | 161 -84 -12 </nowiki>&gt;
| style="text-align:center;" | 0.02
|Sepbisa-quadbigu
| style="text-align:center;" | Atom
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 49/48
| style="text-align:center;" |<nowiki> | -4 -1 0 2 </nowiki>&gt;
| style="text-align:center;" | 35.70
|Zozo
| style="text-align:center;" | Slendro Diesis
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 245/243
| style="text-align:center;" |<nowiki> | 0 -5 1 2 </nowiki>&gt;
| style="text-align:center;" | 14.19
|Zozoyo
| style="text-align:center;" | Sensamagic
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 19683/19600
| style="text-align:center;" |<nowiki> | -4 9 -2 -2 </nowiki>&gt;
| style="text-align:center;" | 7.32
|Labirugu
| style="text-align:center;" | Cataharry
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 6144/6125
| style="text-align:center;" |<nowiki> | 11 1 -3 -2 </nowiki>&gt;
| style="text-align:center;" | 5.36
|Sarurutrigu
| style="text-align:center;" | Porwell
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 121/120
| style="text-align:center;" |<nowiki> | -3 -1 -1 0 2 </nowiki>&gt;
| style="text-align:center;" | 14.37
|Lologu
| style="text-align:center;" | Biyatisma
| style="text-align:center;" |
| style="text-align:center;" |
|-
| style="text-align:center;" | 176/175
| style="text-align:center;" |<nowiki> | 4 0 -2 -1 1 </nowiki>&gt;
| style="text-align:center;" | 9.86
|Lorugugu
| style="text-align:center;" | Valinorsma
| style="text-align:center;" |
| style="text-align:center;" |
|-
|-
! | [[Kite's color notation|color of the 3rd]]
| style="text-align:center;" | 896/891
! | JI chord
| style="text-align:center;" |<nowiki> | 7 -4 0 1 -1 </nowiki>&gt;
! | notes as edosteps
| style="text-align:center;" | 9.69
! | notes of C chord
|Saluzo
! | written name
| style="text-align:center;" | Pentacircle
! | spoken name
| style="text-align:center;" |
| style="text-align:center;" |
|-
|-
| style="text-align:center;" | zo
| style="text-align:center;" | 243/242
| style="text-align:center;" | 6:7:9
| style="text-align:center;" |<nowiki> | -1 5 0 0 -2 </nowiki>&gt;
| style="text-align:center;" | 0-5-14
| style="text-align:center;" | 7.14
| style="text-align:center;" | C Ebv G
|Lulu
| style="text-align:center;" | C.vm
| style="text-align:center;" | Rastma
| style="text-align:center;" | C downminor
| style="text-align:center;" |  
| style="text-align:center;" |  
|-
|-
| style="text-align:center;" | gu
| style="text-align:center;" | 385/384
| style="text-align:center;" | 10:12:15
| style="text-align:center;" |<nowiki> | -7 -1 1 1 1 </nowiki>&gt;
| style="text-align:center;" | 0-6-14
| style="text-align:center;" | 4.50
| style="text-align:center;" | C Eb G
|Lozoyo
| style="text-align:center;" | Cm
| style="text-align:center;" | Keenanisma
| style="text-align:center;" | C minor
| style="text-align:center;" |  
| style="text-align:center;" |  
|-
|-
| style="text-align:center;" | ilo
| style="text-align:center;" | 9801/9800
| style="text-align:center;" | 18:22:27
| style="text-align:center;" |<nowiki> | -3 4 -2 -2 2 </nowiki>&gt;
| style="text-align:center;" | 0-7-14
| style="text-align:center;" | 0.18
| style="text-align:center;" | C Ev G
|Bilorugu
| style="text-align:center;" | C~
| style="text-align:center;" | Kalisma
| style="text-align:center;" | C mid
| style="text-align:center;" | Gauss' Comma
| style="text-align:center;" |  
|-
|-
| style="text-align:center;" | yo
| style="text-align:center;" | 91/90
| style="text-align:center;" | 4:5:6
| style="text-align:center;" |<nowiki> | -1 -2 -1 1 0 1 </nowiki>&gt;
| style="text-align:center;" | 0-8-14
| style="text-align:center;" | 19.13
| style="text-align:center;" | C E G
|Thozogu
| style="text-align:center;" | C
| style="text-align:center;" | Superleap
| style="text-align:center;" | C major or C
| style="text-align:center;" |  
| style="text-align:center;" |  
|-
|-
| style="text-align:center;" | ru
| style="text-align:center;" | 676/675
| style="text-align:center;" | 14:18:27
| style="text-align:center;" |<nowiki> | 2 -3 -2 0 0 2 </nowiki>&gt;
| style="text-align:center;" | 0-9-14
| style="text-align:center;" | 2.56
| style="text-align:center;" | C E^ G
|Bithogu
| style="text-align:center;" | C.^
| style="text-align:center;" | Parizeksma
| style="text-align:center;" | C upmajor or C dot up
| style="text-align:center;" |  
| style="text-align:center;" |  
|}
|}
For a more complete list, see [[Ups_and_Downs_Notation#Chord names in other EDOs|Ups and Downs Notation - Chord names in other EDOs]].


=Rank two temperaments=
=Rank two temperaments=
Line 1,413: Line 1,495:
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|Iceface tuning]].


=Compositions=
=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]


Line 1,434: Line 1,514:
[https://www.youtube.com/watch?v=yzvXEMYgHCY Autumn Girl], by Mason Green
[https://www.youtube.com/watch?v=yzvXEMYgHCY Autumn Girl], by Mason Green


==About==
"Prométhée enchaîné" by [http://en.wikipedia.org/wiki/Fromental_Hal%C3%A9vy Fromental Halévy] (considered the first mainstream western orchestral composition to use quarter tones.)
"Prométhée enchaîné" by [http://en.wikipedia.org/wiki/Fromental_Hal%C3%A9vy Fromental Halévy] (considered the first mainstream western orchestral composition to use quarter tones.)


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