24edo/Interval names and harmonies
24 EDO or 24 ET divides the octave into 24 equal parts and is also a multiple of twelve, therefore, 24 EDO contains all of the original harmonies found in 12 EDO. This page seeks to explore the new harmonies available in a 24 Tone system.
Tone Sizes
While 12 EDO contains only two tone sizes: the whole tone at 200 cents, and the semitone at 100 cents, 24 EDO contains five being that it has three additional new tone sizes. Generally, as it divides the octave into 24 parts, it's a good idea to approach intervals and tones with this mindset of there being a new wider or narrower version of the previous intervals. These tone sizes are mainly used in context of scale steps and sometimes modulation but not usually in context of a chord or scale degrees.
Quarter Tone
The quarter tone is the smallest tone size in 24 EDO. At only 50 cents, It is a highly dissonant interval and has a characteristic washy, beating sound to it and is reminiscent of tuning instrument. Melodically it can function similar to the way a semitone does in 12 EDO but it tends to sound really different as it's such a small interval. Due to the high dissonance, this interval is challenging to make it sound good in a chord within the context of tonal music but can work quite well for composers who wish to explore the dissonance of 24 EDO. Through chord changes, the quarter tone is very effective in creating a sound of a record player going in and out of pitch. It can be a nice effect in smooth jazz progressions or post-modal music to simply move a diatonic chord from 12 EDO up a quarter tone as quarter tone root movement is quite novel in sound. Within scale context, the quarter tone is represented by a lowercase q. An example of quarter tone is C to Ct or enharmonically C to Ddb.
Whole Tones
24 EDO there are not one, but two distinct sizes of larger tones. The wide whole tone usually called "wide tone" at 250 cents, and the natural whole tone, usually called "whole tone" at 200 cents, therefore the natural whole tone is exactly the same interval that appears in 12 EDO as a whole tone. The wide whole tone generally has a more metallic sound than the narrow tone as well as a more moody character compared to the brightness of the natural whole tone. In context of a major chord, the wide tone brings a much colder flavor to the major chord than the whole tone which enhances the brightness of the major chord. The wide tone from the root clashes heavily with minor chords as the minor third and the wide tone are only a quarter tone apart. Diatonic chords tend to move naturally by wide tone movement such as moving an Am chord to a Gd major chord, and other chords can do the same, such as moving a D semi-augmented chord to a Et diminished chord- the latter type of motion in particular being useful in modulations. The wide tone is fairly unique to 24 EDO as in it's too small to be considered a good 7/6 and fits more as being described as a 15/13. The major whole tone is represented by a lowercase w while to wide whole tone is represented by an uppercase W. An example of a whole tone is C to D and a wide tone is C to Dt or enharmonically C to Edb.
Semitones
Like whole tones, there are two distinct sizes of semitones in 24 EDO: The narrow semitone at 100 cents and the wider semitone at 150 cents called "neutral". While the narrow semitone is exactly the same as the 12 EDO semitone, the neutral tone is unique. The neutral tone is called so because it represents both a narrow whole tone and a wide semitone depending on how it's used. It is heavily used in persian, turkish, and other forms of eastern music as well as some east asian scales though normally is slightly sharp or flat from 24 ET. The character of the neutral tone resembles the sound of bells, a car horn, and other sounds that are normally considered "non-musical" which can be a valuable asset to those trying to impressionistically compose music to mimic sounds such as trains and car horns. In fact, the dialing tone in the US is fairly close to a 150 cent neutral tone so one could potentially (and humorously) call it a "dial tone". In addition to all this, the neutral tone is surprisingly useful in chord progressions. An example of a narrow semitone is C to C# or enharmonically, C to Db. An example of a wide semitone is C to Dd.
Seconds
Within a theory context, the above tone sizes will normally be referred to as seconds of the following names from great to small: Infra Second 50c - Minor Second 100c - Neutral Second 150c - Major Second 200c - Ultra Second 250c. Therefore an Infra second is the same as a Quarter Tone, a Minor second is the same as a semitone, a Neutral second is the same enharmonically as a neutral tone/wide semitone/narrow whole tone, a Major second is the same as a whole tone, and an Ultra second is the same as a Wide tone.
Thirds
Like seconds, 24 EDO contains five sizes of thirds which are in order: Infra Third 250c - Minor Third 300c - Neutral third 350c - Major Third 400c - Ultra Third 450c. Obviously, the Infra Third is enharmonically the same as the ultra second but appear differently on the staff and function differently. In a chord such as C E G Dt or 000 - 400 - 700 - 250, 250c is functioning as an ultra ninth but in a chord such as 000 - 250 - 700 - 950, we could say that it's probably functioning as an infra third. The infra third is thought to represent 15/13, the neutral third is 11/9 and the ultra third is 13/10.
Fourths
24 EDO contains five distinct sizes of fourths: Diminished fourth 400c - Narrow fourth 450c - Perfect fourth 500c - Wide fourth 550c - Augmented fourth 600c. The wide fourth is a great representation of the eleventh harmonic 11/8, while the narrow fourth is enharmonically the same as an ultra third. The 11/8 is a fantastic addition to major triads, while the 13/10 can sound good with minor triads.
Fifths
24 EDO contains five distinct sizes of fifths: Diminished Fifth 600 - Narrow fifth 650c - Perfect fifth 700c - Wide fifth 750c - Augmented fifth 800c. The sound of the narrow and wide fifths are very cool and extremely different from 12 ET sounds. The narrow fifth, which is considered to be the most dissonant interval in 24 ET next to the ultra seventh, is a fantastic representation of the 11th subharmonic 16/11. The narrow fifth, sometimes called a "wolf fifth", can be used to create dynamic texture and voice leading but has a rough character that can be challenging to incorporate into the music well, though of course there's the option using it as the fifth of a falling triad. The wide fifth is a good representation of 17/11 and is easily used as the fifth of a rising triad.
Sixths
24 EDO contains five distinct sizes of sixths: Infra Sixth 750c - Minor Sixth 800c - Neutral Sixth 850c - Major Sixth 900c - Ultra Sixth - 950c. The Infra sixth is enharmonically the same as the wide fifth. The neutral sixth is a decent approximation of 13/8, the thirteenth harmonic, however not quite as good as how well 24 approximates 11/8; The interval may more closely approximate 18/11. The Neutral sixth is though to be the sweetest sounding neutral interval in the tuning. The Ultra sixth approximates the interval 26/15 very closely but also can be considered a rather poor 7/4, yet can functionally harmonize in the same manner.
Sevenths
24 EDO contains five distinct sizes of sevenths: Infra Seventh 950c - Minor Seventh 1000c - Neutral Seventh 1050c - Major Seventh 1100c - Ultra Seventh - 1150c. The Infra seventh is enharmonically the same as the Ultra Sixth as it represents 26/15 or sometimes 7/4 but rather poorly. The Neutral seventh is a fantastic 11/6 and the Ultra Seventh represents a good 35/18 and has a very rough character as it is the inversion of the infra second.
Special Enharmonics
24 EDO contains certain enharmonics that are good to keep in mind, the list is as follows:
- An ultra second and infra third are both 250 cents.
- An ultra third and narrow fourth are both 450 cents.
- A wide fifth and infra sixth are both 750 cents.
- An ultra sixth and infra seventh are both 950 cents.
Interval Class Categories
Neutral
Neutral intervals basically are right between the major and minor version of an interval in 12 EDO. For example, the neutral third is between the major and minor third. The name also suggests that the interval can function as either depending on how it's used. In addition, neutral intervals contain very special color to them that makes them unique.
Ultra/Infra
Ultra and Infra are used to describe intervals that are borderline between two classes. For example, an Ultra third borderlines a perfect fourth and can also be called a narrow fourth. Only seconds, thirds, sixths, and sevenths have ultra and infra classes, and the names are never used to describe fourths, fifths, or octaves.
Narrow/Wide
Narrow and wide are used to describe intervals that are between perfect and diminished/augmented. While they aren't normally used to describe anything but fourths, fifths, and sometimes octaves, they can be used to describe extreme altered intervals such as 350 cents being a narrow diminished fourth.
Types of Chords
As mentioned above, 24edo contains all the types of chords contained in 12edo, so, for the sake of ease, only the most basic of those will be listed here. On the other hand, there are at least several new varieties of chord that are added.
Ultra
This type of triad consists of the root, the 450c Ultra third on top of that, and the 250c Infra third on top of that. This triad, which is 24edo's version of the Tendo triad, has a rather bright but strange sound, and there are multiple types of tetrads which can be built on top of this chord.
Major
This type of triad is the same as the major triad found in 12edo. While there's nothing new about this triad structure in as of itself aside from the fact that it no longer the best substitute for the septimal supermajor chord, there are new potential set-ups and follow ups, as well as options for tetrads built on this type of triad. For instance, one can easily stack Infra third on top of the Major triad to make a triad that more closely approximates a harmonic seventh. Alternatively one can stack an Ultra second on top of the Major triad to make a tetrad with functionalities reminiscent of the more traditional German Sixth chord.
Neutral
This type of triad is halfway between the traditional Major and Minor chords, as the third is located at an even 350c above the Tonic. It is more dissonant in sound than either the Major or Minor triads due to the third being located relatively far from harmonic entropy minima, but it is still viable in the right hands.
Minor
This type of triad is the same as the minor triad found in 12edo. As with the Major triad, this triad is no longer the best substitute for the septimal subminor chord, however, there are new potential set-ups and follow ups, as well as options for tetrads built on this type of triad.
Infra
This type of triad consists of the root, the 250c Infra third on top of that, and the 450c Ultra third on top of that. This triad, which is 24edo's version of the Arto triad, has a rather dark and bluesy sound, and there are multiple types of tetrads which can be built on top of this chord.
Rising Ultra
The Rising Ultra triad is a triad which consists of the Root, the 450c Ultra third on top of that, and the 300c Minor third on top of that. When this triad is built on the note located 650 cents above the Tonic, it can act as a set-up for either a Falling Major triad or Falling Neutral triad built on the root located a semitone below it. In addition, it can also be followed up by a Falling Major or Falling Minor triad built on the Tonic, which in turn can lead to modulations.
Rising Major
One of four chords named for a similar type of chord found in Oneirotonic scales, the Rising Major triad is a bright but tense triad which consists of the Root, the 400c Major third on top of that, and the 350c Neutral third on top of that. When built on the Tonic, it can lead into modulations.
Rising Neutral
The rising Neutral triad is a triad which consists of the Root, the 350c Neutral third on top of that, and the 400c Major third on top of that. It can also set up either a Major triad or a Falling Major triad built on the root located a major third below it, and can also lead into either a Major or Minor triad on the same root.
Rising Minor
One of four chords named for a similar type of chord found in Oneirotonic scales, the Rising Minor triad is a triad which consists of the Root, the 300c Major third on top of that, and the 450c Ultra third on top of that, though it can also be conceived as being the result of stacking two 750c Fifths. It is likely to arise in counterpoint under certain circumstances. Like a Rising Neutral triad, it can set up either a Major triad or a Falling Major triad built on the root located a major third below it. However, in addition to this, has some of the same set-ups and follow-ups as the Rising Ultra triad.
Falling Major
One of four chords named for a similar type of chord found in Oneirotonic scales, the Falling Major triad is a bright but tense triad which consists of the Root, the 400c Major third on top of that, and the 250c Infra third on top of that. When this triad is built on the note located 550 cents above the Tonic, especially with octave reduplication of the root, it can easily be followed up with with the same Tonic's traditional V7 chord. In modulation, this same triad can also be followed up with a Dominant seventh chord built on the same root, leading to the tonicization of the neutral seventh. Regardless, the use of this chord requires the proper set-ups in terms of both melody and harmony.
Falling Neutral
The Falling Neutral triad is a triad which consists of the Root, the 350c Neutral third on top of that, and the 300c Minor third on top of that. It is one of two triads that can be considered to sound like a Xenharmonic version of a Diminished triad. Like with a Falling Major Triad, it can easily be followed up with with the same Tonic's traditional V7 chord.
Falling Minor
One of four chords named for a similar type of chord found in Oneirotonic scales, the falling minor triad is a triad which consists of the Root, the 300c Minor third on top of that, and the 350c Neutral third on top of that. It is also one of two triads that can be considered to sound like a Xenharmonic version of a Diminished triad.
Falling Infra
The Falling Infra triad is a triad which consists of the Root, the 250c Infra third on top of that, and the 400c Major third on top of that. Like the Falling Major triad, when this triad is built on the note located 550 cents above the Tonic, especially with octave reduplication of the root, it can easily be followed up with with the same Tonic's traditional V7 chord, and the use of this chord also requires the proper set-ups in terms of both melody and harmony.
Augmented
The Augmented chord in 24edo is differs from its 12edo counterpart mainly in terms of having a wider variety of set-ups and follow-ups.
Diminished
Like with the Augmented chord, the Diminished chord in 24edo has a wider variety of set-ups and follow-ups than in 12edo.