24edo: Difference between revisions

{{EDO intro|24}}  

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 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. Additionally, like [[22edo]], 24edo tempers out the [[quartisma]], linking the otherwise sub-par [[7-limit]] harmonies with those of the [[11-limit]].  

Its step, at 50 cents, is notable for having some of the highest [[harmonic entropy]] possible, making it, in theory, one of the most dissonant intervals possible (using the relatively common values of {{nowrap|''a'' {{=}} 2}} and {{nowrap|''s'' {{=}} 1%}}, the peak occurs at around 46.4 cents). 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.

== Notation ==

 

=== Ups and down notation ===

{| class="wikitable center-all"

! Approximate Ratios<ref group="note">{{sg|limit=2.3.5.11.13.17.19&nbsp;[[subgroup]]}}</ref>

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

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

![[24edo solfege|Solfege]]

=== Interval qualities in color notation ===

! [[Color name|Color Name]]

! Monzo Format

| style="white-space: nowrap;" | (a, b, 0, 1)

| rowspan="2" |minor

| style="white-space: nowrap;" | (a, b), b &lt; −1

| style="white-space: nowrap;" | (a, b, −1)

| rowspan="2" |mid

| style="white-space: nowrap;" | (a, b, 0, 0, 1)

| style="white-space: nowrap;" | (a, b, 0, 0, −1)

| rowspan="2" |major

| style="white-space: nowrap;" | (a, b, 1)

| style="white-space: nowrap;" | (a, b), b &gt; 1

| style="white-space: nowrap;" | (a, b, 0, −1)

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 ===

| Ultra sixth , infra seventh

=== Interval alterations ===

=== 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.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

*[[24edo Chord Names]]

*[[Ups and Downs Notation#Chords and Chord Progressions]].

{{Uniform map|13|23.5|24.5}}

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

|{{monzo| -19 12 }}

|{{monzo| 3 4 -4 }}

|{{monzo| 18 -4 -5 }}

|{{monzo| 7 0 -3 }}

|{{monzo| -4 4 -1 }}

|{{monzo| 11 -4 -2 }}

|{{monzo| 26 -12 -3 }}

|{{monzo| -15 8 1 }}

|{{monzo| 161 -84 -12 }}

|{{monzo| -8 3 -1 2 }}

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

|{{monzo| 0 -5 1 2 }}

|{{monzo| -4 9 -2 -2 }}

|{{monzo| 11 1 -3 -2 }}

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

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

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

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

|{{monzo| 7 -4 0 1 -1 }}

|{{monzo| -1 5 0 0 -2 }}

|{{monzo| 15 8 0 0 -8 }}

|{{monzo| -7 -1 1 1 1 }}

|{{monzo| 24 -6 0 1 -5 }}

|{{monzo| -3 4 -2 -2 2 }}

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

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

|{{monzo| 9 -1 0 0 0 -2 }}

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

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

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

|{{monzo| 12 -2 -1 -1 0 -1 }}

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

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

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

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

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

|{{monzo| -5 -2 0 0 0 0 2 }}

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

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

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

|{{monzo| 5 1 -1 0 0 0 0 -1 }}

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

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

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

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

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

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

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

| [[Hemiripple]] / [[cohemiripple]]

| [[Godzilla]] (24)<br>[[Bridgetown]]

| [[Barton]]

|{{dash|0, 5, 14|hair}}

|{{dash|C, E{{sesquiflat2}}, G|hair}}

| C subminor<br>C minor semiflat-three

|{{dash|0, 6, 14|hair}}

|{{dash|C, E♭, G|hair}}

|{{dash|C, E{{demiflat2}}, G|hair}}

|{{dash|0, 8, 14|hair}}

|{{dash|C, E, G|hair}}

|{{dash|0, 9, 14|hair}}

|{{dash|C, E{{demisharp2}}, G|hair}}

| C supermajor<br>C major semisharp-three

24edo 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.

|{{dash|0, 5, 14, 19|hair}}

|{{dash|1, vb3, 5, vb7|hair}}

|{{dash|C, E{{sesquiflat2}}, G, B{{sesquiflat2}}|hair}}

| Subminor seven<br>Minor seven semiflat-three semiflat-seven

| |-

|{{dash|0, 6, 14, 20|hair}}

|{{dash|1, b3, 5, b7|hair}}

|{{dash|C, E♭, G, B♭|hair}}

| |-

|{{dash|0, 7, 14, 21|hair}}

|{{dash|1, v3, 5, v7|hair}}

|{{dash|C, E{{demiflat2}}, G, B{{demiflat2}}|hair}}

|{{dash|0, 8, 14, 22|hair}}

|{{dash|1, b3, 5, b7|hair}}

|{{dash|C, E, G, B|hair}}

| |-

|{{dash|0, 8, 14, 22|hair}}

|{{dash|1, b3, 5, b7|hair}}

|{{dash|C, E{{demisharp2}}, G, B{{demisharp2}}|hair}}

| Supermajor seven<br>Major seven semisharp-three semisharp-seven

| |-

|{{dash|0, 8, 14, 20|hair}}

|{{dash|1, 3, 5, b7|hair}}

|{{dash|C, E, G, B♭|hair}}

| |-

|{{dash|0, 8, 14, 19|hair}}

|{{dash|1, 3, 5, vb7|hair}}

|{{dash|C, E, G, B{{sesquiflat2}}|hair}}

| |-

|{{dash|0, 5, 14, 20|hair}}

|{{dash|1, vb3, 5, b7|hair}}

|{{dash|C, E{{sesquiflat2}}, G, B♭|hair}}

|{{dash|0, 9, 14, 19|hair}}

|{{dash|1, ^3, 5, vb7|hair}}

|{{dash|C, E{{demisharp2}}, G, B{{sesquiflat2}}|hair}}

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.

| [[File:Strict-contrary-motion-24edo.png|left|frame|Every sequence of 12edo intervals are reachable by strict contrary motion in 24edo. [[File:24-EDO_Contrary_Motion.flac]]]]

|}

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.

However, while these are playable, the extra frets can make playing chords and navigating the fretboard significantly more challenging for 12edo chords and scales.

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]].

;Quarter-tone grand piano, Czech Museum of Music (this piano is essentially two stacked grand pianos, and as such is massive, in order to avoid sacrificing strings per note)

*[https://www.youtube.com/shorts/Ieqi54XE2lI Demonstration short video by Nahre Sol] (2024)

;Quarter-tone upright piano, Academy of Music in Prague (Czech Republic) (this piano apparently sacrificed number of strings per note in order to be able to fit into a reasonable amount of space)

*[https://www.youtube.com/watch?v=PdP4epQIUrU Demonstration video by Steve Cohn] (2011)

;Quarter-tone flute, made by Eva Kingma

*[https://www.youtube.com/watch?v=F3GD0Omr4Z0 Visit to the workshop of Eva Kingma, followed by test by Manuel Luis Cochofel] (2010) (demonstration of fingering starts at 06:56)

 

{{Main|Music in 24edo}}