24edo: Difference between revisions

| en = 24edo

{{Wikipedia|Quarter tone}}

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 ==

The [[5-limit]] approximations in 24edo are the same as those in 12edo, so 24edo offers nothing new as far as approximating the 5-limit is concerned.

The 7th harmonic and its intervals ([[7/4]], [[7/5]], [[7/6]], and [[9/7]]) are almost as inaccurate in 24edo as in 12edo. To achieve a satisfactory level of approximation to intervals of 7 while maintaining the 12 notes of 12edo requires high-degree tunings like [[36edo|36et]], [[72edo|72et]], [[84edo|84et]], [[156edo|156et]], or [[192edo|192et]]. However, 24edo excels at the 11th harmonic and most intervals involving 11 ([[11/10]], [[11/9]], [[11/8]], [[11/6]], [[12/11]], [[15/11]], [[16/11]], [[18/11]], [[20/11]]). 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. 24edo is also good at the 13th harmonic, which makes it a good 2.3.5.11.13 system. Specifically, intervals of 13/5 are particularly well approximated. And of course, 24edo shares its 17 and 19 tunings with 12edo, meaning that 7 and to an extent 5 are the only low primes 24edo tunes particularly poorly.  

While the 7th harmonic is poorly tuned, the intervals 24edo has do serve as reasonable substitutes to 7-limit intervals melodically: a supermajor chord is available at [0 9 14] and a subminor chord at [0 5 14], though they're more ultramajor and inframinor.

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, 17, and 19. Expanding this, one will find that 24edo is consistent in the no-7s 19-odd-limit, though the 2.3.11.17.19 [[subgroup]] is where it is the most accurate.

Its step, at 50 cents, is notable for being generally seen as one of the most dissonant intervals possible (in fact, typical harmonic entropy models show a peak around this point). Intervals less than 40 cents tend to be perceived as being closer to a unison, and thus, more consonant as a result, while intervals larger than approximately 60 cents are often perceived as having less "tension", and thus are also considered to be more consonant.

{{Harmonics in equal|24|prec=2}}

=== Subsets and supersets ===

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

== Intervals ==

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.

Ups and downs are spoken as up, sharp, upsharp, etc. Note that up can be respelled as downsharp.

=== Stein–Zimmermann accidentals ===

{{Sharpness-sharp2|24}}

| style="width: 40px;" | [[File:HeQu1.svg|21px|center]]

| A "semisharp" or "half-sharp" accidental comprising one half of a regular musical sharp symbol.

| style="width: 40px;" | [[File:HeQu3.svg|32px|center]]

| A "sharp and a half" or "sesquisharp" accidental, comprising the above half-sharp symbol connected to the right side of a normal sharp.

| A "flat and a half" or "sesquiflat" accidental, comprising a half-flat symbol and a regular flat symbol placed back to back.

'''Pros:''' familiar, intuitive, and fairly easy to learn.  

'''Cons:''' can clutter a score easily (especially when used in microtonal key signatures), can get confusing when sight read at faster paces.

=== Persian quartertone accidentals ===

| width="40px" | [[File:Koron_sign.svg|39px|center]]

| width="40px" | [[File:Sori_sign.svg|39px|center]]

| '''Sori''' = quarter-tone sharp

'''Pros:''' easy to read.

'''Cons:''' hard to write on a computer, does not fit with standard notation well.

This notation uses the same sagittal sequence as edos [[17edo #Sagittal notation|17]], [[31edo #Sagittal notation|31]], and [[38edo #Sagittal notation|38]], is a subset of the notations for edos [[48edo #Sagittal notation|48]] and [[72edo #Sagittal notation|72]], and is a superset of the notations for edos [[12edo #Sagittal notation|12]], [[8edo #Sagittal notation|8]], and [[6edo #Sagittal notation|6]].

==== Evo flavor ====

rect 447 0 607 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

default [[File:24-EDO_Evo_Sagittal.svg]]

File:24-EDO_Revo_Sagittal.svg

rect 80 0 300 50 [[Sagittal_notation]]

rect 463 0 623 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

default [[File:24-EDO_Revo_Sagittal.svg]]

</imagemap>

 

 

 

 

 

 

 

 

 

 

 

 

=== William Lynch's interval and chord names ===

 

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 24edo that do not exist in 12edo:

{| class="wikitable center-all right-6"

|-

|-

| 50

| Quarter tone, infra second, wide unison

|-

| 150

| Neutral second

|-

| 250

| Ultra second, infra third

|-

| 350

| Neutral third

|-

| 450

| Minor fourth, ultra third, narrow fourth

|-

| 550

| Wide fourth

|-

| 650

| Narrow fifth

|-

| 750

| Wide fifth, infra sixth

|-

| 850

| Neutral sixth

|-

| 950

| Ultra sixth, infra seventh

|-

| 1050

| Neutral seventh

|-

| 1150

| Ultra seventh, narrow octave

|}

* 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.

 

* 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

=== Further discussion of interval and chord naming ===

* [[24edo Chord Names]]

* [[Ups and downs notation#Chords and Chord Progressions]].

== Approximation to JI ==

[[File:24ed2.svg|250px|thumb|right|none|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 24edo]]

=== Interval mappings ===

{{Q-odd-limit intervals|24}}

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal<br>8ve stretch (¢)

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

|-

| 2.3.5.11

| 81/80, 121/120, 128/125

| {{mapping| 24 38 56 83 }}

| −1.08

| 2.82

| 5.63

|-

| 2.3.5.11.13

| 66/65, 81/80, 128/125, 144/143

| {{mapping| 24 38 56 83 89 }}

| −1.37

| 2.59

| 5.19

|-

| 2.3.5.11.13.17

| 51/50, 66/65, 81/80, 128/125, 144/143

| {{mapping| 24 38 56 83 89 98 }}

| −0.94

| 2.55

| 5.11

|-

| 2.3.5.11.13.17.19

| 51/50, 66/65, 76/75, 81/80, 128/125, 144/143

| {{mapping| 24 38 56 83 89 98 102 }}

| −0.89

| 2.37

| 4.74

|}

=== Uniform maps ===

{{Uniform map|edo=24}}

=== Commas ===

This is a partial list of the [[commas]] that 24edo [[tempering out|tempers out]] with its patent [[val]], {{val| 24 38 56 67 83 89 }}.

! [[Harmonic limit|Prime<br>limit]]

|-

| 3

| <abbr title="531441/524288">(12 digits)</abbr>

| <abbr title="262144/253125">(12 digits)</abbr>

| <abbr title="2923003274661805836407369665432566039311865085952/2922977339492680612451840826835216578535400390625">(98 digits)</abbr>

| <abbr title="214990848/214358881">(18 digits)</abbr>

| <abbr title="117440512/117406179">(18 digits)</abbr>

| <abbr title="1771561/1771470">(14 digits)</abbr>

|-

| 13

| [[144/143]]

| {{monzo| 4 2 0 0 -1 -1 }}

| {{monzo| 8 -1 -1 0 0 0 -1 }}

| {{monzo| -3 -2 2 2 0 0 -1 }}

| {{monzo| 2 -1 -2 0 0 0 0 1 }}

| {{monzo| -3 -2 -1 0 0 0 0 2 }}

| Go comma

|-

| 19

| [[5776/5775]]

=== Rank-2 temperaments ===

* [[List of 24et rank two temperaments by badness]]

* [[List of edo-distinct 24et rank two temperaments]]