24edo: Difference between revisions

| de = 24-EDO

{{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 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]]''.

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.

=== Prime harmonics ===

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

=== Subsets and supersets ===

== Intervals ==

{| class="wikitable center-all left-3"

|-

! Approximate ratios<ref group="note">{{sg|limit=2.3.5.11.13.17.19-[[subgroup]] (no-sevens 19-limit)}}</ref>

! colspan="3" | [[Ups and downs notation]] ([[Enharmonic unisons in ups and downs notation|EUs]]: vvA1 and d2)

! colspan="3" | [[SKULO interval names|SKULO notation]] {{nowrap|(U or S {{=}} 1)}}

! [[24edo solfege|Solfege]]

|-

| 0

| 0

| 1/1

| unison

| D

| unison

| P1

| D

| Do

|-

| 1

| 50

| 33/32, 34/33

| ^P1, vm2

| up-unison, downminor 2nd

| ^D, vEb

| super unison, uber unison

| S1, U1

| SD, UD

| Da/Ru

|-

| 2

| 100

| 16/15, 17/16, 18/17, 19/18

| A1, m2

| aug unison, minor 2nd

| D#, Eb

| aug unison, minor 2nd

| A1, m2

| D#, Eb

| Ro

|-

| 3

| 150

| 13/12, 12/11, 11/10

| ~2

| mid 2nd

| vE

| neutral 2nd

| N2

| UEb, uE

| 4

| 200

| 9/8, 10/9

| M2

| major 2nd

| E

| major 2nd

| M2

| E

| Re

|-

| 5

| 250

| 15/13, 22/19

| ^M2, vm3

| upmajor 2nd, downminor 3rd

| ^E, vF

| supermajor 2nd, subminor 3rd

| SE, sF

| Ri/Mu

|-

| 6

| 300

| 6/5, 13/11, 19/16

| minor 3rd

| F

| minor 3rd

| m3

| F

| Mo

|-

| 7

| 350

| 11/9, 16/13, 27/22, 39/32

| ~3

| mid 3rd

| vF#

| neutral 3rd

| N3

| UF, uF#

|-

| 8

| 400

| 5/4, 24/19

| M3

| major 3rd

| F#

| major 3rd

| M3

| F#

| Me

|-

| 9

| 450

| 13/10, 17/13, 22/17

| ^M3, v4

| upmajor 3rd, down-4th

| ^F#, vG

| supermajor 3rd, sub 4th

| Mi/Fu

|-

| 10

| 4/3

| P4

| G

| P4

| G

| Fo

|-

| 11

| 550

| ^4, ~4

| up-4th, mid-4th

| ^G

| uber 4th/neutral 4th

| U4/N4

| UG

| Fa/Su

|-

| 12

| 600

| 17/12, 24/17, 45/32, 64/45

| A4, d5

| aug 4th, dim 5th

| G#, Ab

| aug 4th, dim 5th

| A4, d5

| G#/Ab

| Fe/So

|-

| 13

| 650

| 16/11, 22/15

| v5, ~5

| down-5th, mid-5th

| vA

| unter 5th/neutral 5th

| u5/N5

| uA

| Fi/Sa

| 14

| 700

| 3/2

| P5

| fifth

| A

| Se

|-

| 15

| 750

| 17/11, 20/13

| ^5, vm6

| ^A, vBb

| super 5th, subminor 6th

| S5, sm6

| SA, sBb

| Si/Lu

|-

| 800

| 8/5, 19/12

| Bb

| minor 6th

| m6

| Bb

| Lo

|-

| 17

| 850

| 13/8, 18/11, 44/27, 64/39

| ~6

| N6

| UBb, uB

| La

|-

| 18

| 900

| 5/3, 22/13, 32/19

| B

| major 6th

| M6

| B

| Le

|-

| 950

| 19/11, 26/15

| ^M6, vm7

| upmajor 6th, downminor 7th

| ^B, vC

| supermajor 6th, subminor 7th

| SB, sC

| Li/Tu

|-

| 20

| 1000

| 9/5, 16/9

| m7

| minor 7th

| C

| minor 7th

| m7

| C

| To

|-

| 11/6, 20/11

| ~7

| mid 7th

| vC#

| neutral 7th

| N7

| Ta

|-

| 22

| 1100

| 15/8, 17/9, 32/17

| M7

| major 7th

| C#

| major 7th

| M7

| C#

| Te

|-

| 23

| 1150

| 33/17, 64/33

| ^M7, vP8

| upmajor 7th, down-8ve

| ^C#, vD

| sub 8ve, unter 8ve

| s8, u8

| C#, uD

| Ti/Du

|-

| 24

| 1200

| 2/1

| P8

| perfect 8ve

| D

| perfect 8ve

| P8

| D

| Do

|}

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.

== Notation ==

=== Ups and downs notation ===

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

{{sharpness-sharp2a|24}}

=== Stein–Zimmermann accidentals ===

{{Sharpness-sharp2|24}}

{| class="wikitable center-1"

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

'''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.  

{| class="wikitable"

|-

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

| '''Koron''' = quarter-tone flat

|-

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

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

|}

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

=== Sagittal notation ===

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 ====

File:24-EDO_Evo_Sagittal.svg

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

rect 20 80 130 106 [[33/32]]

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

</imagemap>

==== Evo-SZ flavor ====

File:24-EDO_Evo-SZ_Sagittal.svg

rect 407 0 567 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]

rect 20 80 130 106 [[33/32]]

default [[File:24-EDO_Evo-SZ_Sagittal.svg]]

</imagemap>