24edo interval names and harmonies
24edo or 24et divides the octave into 24 equal parts and is also a multiple of twelve, therefore, 24edo contains all of the original harmonies found in 12edo. This page seeks to explore the new harmonies available in a 24-tone system.
While 12edo contains only two tone sizes: the whole tone at 200 cents, and the semitone at 100 cents, 24edo contains five being that it has three additional new tone sizes. Generally, as it divides the octave into 24 parts, it is 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.
The quarter tone is the smallest tone size in 24edo. 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 12edo but it tends to sound really different as it is 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 24edo. 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 12edo 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.
24edo has 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 12edo 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 24edo as in it is 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.
Like whole tones, there are two distinct sizes of semitones in 24edo: 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 12edo 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 is 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 24et. 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 dialling 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.
Within a theory context, the above tone sizes will normally be referred to as seconds of the following names from great to small: inframinor second 50¢ – minor second 100¢ – neutral second 150¢ – major second 200¢ – ultramajor second 250¢. Therefore an inframinor 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 ultramajor second is the same as a wide tone.
Like seconds, 24edo contains five sizes of thirds which are in order: inframinor third 250¢ – minor third 300¢ – neutral third 350¢ – major third 400¢ – ultramajor third 450¢. Obviously, the inframinor third is enharmonically the same as the ultramajor second but appear differently on the staff and function differently. In a chord such as C E G Dt or 0-400-700-250, 250¢ is functioning as an ultramajor ninth but in a chord such as 0-250-700-950, we could say that it is probably functioning as an inframinor third. The inframinor third is thought to represent 15/13, the neutral third is 11/9 and the ultra third is 13/10.
24edo contains five distinct sizes of fourths: diminished fourth 400¢ – paraminor fourth 450¢ – perfect fourth 500¢ – paramajor fourth 550¢ – augmented fourth 600¢. The paramajor fourth is a great representation of the eleventh harmonic 11/8, while the paraminor fourth is enharmonically the same as an ultramajor third. The 11/8 is a fantastic addition to major triads, while the 13/10 can sound good with minor triads.
24edo contains five distinct sizes of fifths: diminished fifth 600¢ – paraminor fifth 650¢ – perfect fifth 700¢ – paramajor fifth 750¢ – augmented fifth 800¢. The sound of the paraminor and paramajor fifths are very cool and extremely different from 12et sounds. The paraminor fifth, which is considered to be the most dissonant interval in 24et next to the ultramajor seventh, is a fantastic representation of the 11th subharmonic (16/11). The paraminor fifth, sometimes mistakenly 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 is the option using it as the fifth of a lowered triad. The paramajor fifth is a good representation of 17/11 and is easily used as the fifth of a raised triad.
24edo contains five distinct sizes of sixths: inframinor sixth 750¢ – minor sixth 800¢ – neutral sixth 850¢ – major sixth 900¢ – ultramajor sixth 950¢. The inframinor sixth is enharmonically the same as the paramajor 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 ultramajor 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.
24edo contains five distinct sizes of sevenths: inframinor seventh 950¢ – minor seventh 1000¢ – neutral seventh 1050¢ – major seventh 1100¢ – ultramajor seventh – 1150¢. The inframinor seventh is enharmonically the same as the ultramajor sixth as it represents 26/15 or sometimes 7/4 but rather poorly. The neutral seventh is a fantastic 11/6 and the ultramajor seventh represents a good 35/18 and has a very rough character as it is the inversion of the inframinor second.
24edo contains certain enharmonics that are good to keep in mind, the list is as follows:
- An ultramajor second and inframinor third are both 250 cents.
- An ultramajor third and paraminor fourth are both 450 cents.
- A paramajor fifth and inframinor sixth are both 750 cents.
- An ultramajor sixth and inframinor seventh are both 950 cents.
Interval class categories
Neutral intervals basically are right between the major and minor version of an interval in 12edo. 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 is used. In addition, neutral intervals contain very special color to them that makes them unique.
Ultramajor and Inframinor are used to describe major and minor intervals which have been modified by a quartertone away from the neutral position. Only seconds, thirds, sixths, and sevenths have ultramajor and inframinor classes
Paramajor and Paraminor are used to describe the quartertone intervals that are on either side of a perfect fourth or perfect fifth, with paramajor intervals being sharper than their perfect counterparts by a quartertone and paraminor intervals being sharper than their perfect counterparts by a quartertone. The terms "paramajor" and "paraminor" have their roots in the terms "major fourth" and "minor fifth" as used by Ivan Wyschnegradsky. The reason the "para-" prefix (meaning "resembling" or "alongside") has been added to the words "major" and "minor" as seen in Wyschnegradsky's original terms for these intervals is because the quartertone intervals surrounding the perfect fourth and perfect fifth relate to each other in a manner resembling the relationship between conventional major and minor for other scale degrees, except that this relationship occurs in a context where the note halfway between them is actually part of the base scale rather than the two notes in question.
The terms "Ultra" and "Infra" themselves occur in prefix form when modifying Primes and Octaves, as well as when modifying Augmented and Diminished intervals, thus leading to the terms "Ultraprime", "Ultraoctave", "Infraoctave", "Infra-Augmented", "Ultra-Augmented", "Infra-Diminished" and "Ultra-Diminished"; however, the term "Infraprime" is not used since the perfect unison is already the smallest form of prime. Note, for example, that an Infra-Augmented Fourth is the enharmonically equivalent to a Paramajor Fourth in 24edo because the rastma tempered out.
Types of basic 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.
- Main article: Arto and Tendo Theory
This type of triad consists of the root, the 450¢ ultramajor third on top of that, and the 250¢ inframinor 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.
This type of triad is the same as the major triad found in 12edo. While there is 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 inframinor third on top of the major triad to make a triad that more closely approximates a harmonic seventh. Alternatively one can stack an ultramajor second on top of the Major triad to make a tetrad with functionalities reminiscent of the more traditional German sixth chord.
This type of triad is halfway between the traditional major and minor chords, as the third is located at an even 350¢ 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.
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.
- Main article: Arto and Tendo Theory
This type of triad consists of the root, the 250¢ inframinor third on top of that, and the 450¢ ultramajor 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.
The cocytic ultramajor triad is a triad which consists of the root, the 450¢ ultramajor third on top of that, and the 300¢ 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 lowered major triad or lowered neutral triad built on the root located a semitone below it. In addition, it can also be followed up by a lowered major or lowered minor triad built on the tonic, which in turn can lead to modulations.
The cocytic 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.
The cocytic neutral triad is a triad which consists of the root, the 350¢ neutral third on top of that, and the 400¢ major third on top of that. It can also set up either a major triad or a lowered 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.
The cocytic minor triad is a triad which consists of the root, the 300¢ minor third on top of that, and the 450¢ ultramajor third on top of that, though it can also be conceived as being the result of stacking two 750¢ fifths. It is likely to arise in counterpoint under certain circumstances. Like a raised 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 raised ultramajor triad.
The dusthumic major triad is a bright but tense triad which consists of the root, the 400¢ major third on top of that, and the 250¢ inframinor 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 harmonic and melodic set-ups, with one of the most reliable set-ups early on being the 0¢-550¢-700¢ suspension chord, though the IIImin and Imaj triads also work later on.
The dusthumic neutral triad is a triad which consists of the root, the 350¢ neutral third on top of that, and the 300¢ 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 lowered major triad, it can easily be followed up with the same tonic's traditional V7 chord.
The dusthumic minor triad is a triad which consists of the root, the 300¢ minor third on top of that, and the 350¢ 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.
The dusthumic inframinor triad is a triad which consists of the root, the 250¢ inframinor third on top of that, and the 400¢ major third on top of that. Like the lowered 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 the same tonic's traditional V7 chord. However, the use of this chord also requires the proper set-ups in terms of both melody and harmony, especially the 0¢-550¢-700¢ suspension chord early on, though the bVImaj triad and the Imin triad also work later on.
The augmented chord in 24edo differs from its 12edo counterpart mainly in terms of having a wider variety of set-ups and follow-ups.
Like with the augmented chord, the diminished chord in 24edo has a wider variety of set-ups and follow-ups than in 12edo.
Rules for finding other usable chord types
- See also: Dinner party rules #24edo
In addition to the types of basic chord, one can derive other types of usable chords through the Dinner Party Rules, which were first compiled by a YouTuber going by the username Quartertone Harmony.