24edo: Difference between revisions

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 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 [[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.  

There are multiple ways of notating 24edo. While an arguably common form can be seen on [[Wikipedia: Quarter tone #Interval size in equal temperament|Wikipedia's page on quartertones]], there are other forms, and it is these other forms that will be considered here. For the full list, along with some chord progression information, see [[24edo interval names and harmonies|full article on 24 Edo intervals]], and also [[24edo Chord Names]] and [[Ups and Downs Notation#Chords and Chord Progressions]].

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.

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.

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

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

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.

As for entirely new chords, the most obvious is the neutral or mid triad 0-7-14. However there are other options such as

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]] 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.

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")

 

0-5-10 (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]]), 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.)

24edo is the first edo to have both a [[sqrt(25/24)]] distinct from [[25/24]] and a correct [[5-odd-limit]].

 

It is thus the first edo which allows to lead the two voices of a major third to a minor third by [[strict contrary motion]]. And vice versa.

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