24edo/Interval names and harmonies
24edo 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.
Tone sizes
While 12edo contains only two sizes of seconds, thirds, sixths, and sevenths, 24edo has three additional new tone sizes, giving it five different kinds. 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.
Quarter tone
The quarter tone is the smallest tone size in 24edo. At only 50 ¢ in size, it is a highly dissonant interval and has a characteristic washy, beating sound to it and is reminiscent of tuning an instrument. Melodically, it can function similar to the way a semitone does in 12edo but it tends to sound very 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–C 
, or enharmonically, C–D 
.
Neutral second
While the minor second and major second, at 100 and 200 ¢, respectively, are the same as that of 12edo, 24edo also has another kind of second halfway between the two, called a "neutral" second. This can serve as both a narrow whole tone and a wide semitone depending on how it is used. It is heavily used in Arabic, Turkish, Persian, 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 ¢ 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 neutral second is C–D 
.
Seconds
In the 5-limit, 24edo has one whole tone (major second) representing both 9/8 and 10/9, like other meantone temperaments; at 200 ¢, it is exactly the same interval that appears in 12edo as a whole tone. However, it also has an "ultramajor" second, which is 250 ¢ in size; this generally has a more metallic sound than the whole 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 ultramajor second is fairly unique to 24edo in that it is too small to be considered an accurate 7/6 and too large to be considerd an accurate 8/7; as such, it 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–D and a wide tone is C–D , or enharmonically, C–E .
Within a theory context, the 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 ¢), and 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 halfway in betwen a tone and semitone, a major second is the same as a whole tone, and an ultramajor second is a quarter tone wider than a whole tone.
Thirds
Like seconds, 24edo contains five sizes of thirds which are in order: the inframinor third (250 ¢), minor third (300 ¢), neutral third (350 ¢), major third (400 ¢), andultramajor third (450 ¢). The inframinor third is enharmonically the same as the ultramajor second, but it appears differently on the staff and functions differently. In a chord such as C–E–G–D (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 represents 15/13, the neutral third is 11/9, and the ultramajor third is 13/10.
Fourths
24edo contains five distinct sizes of fourths: the diminished fourth (400 ¢), paraminor fourth (450 ¢), perfect fourth (500 ¢), paramajor fourth (550 ¢), and augmented fourth (600 ¢). The paramajor fourth is an extremely accurate representation of the eleventh harmonic (11/8), while the paraminor fourth is enharmonically the same as an ultramajor third. 11/8 is often considerd to be a fantastic addition to major triads, while the 13/10 can sound good with minor triads.
Fifths
24edo contains five distinct sizes of fifths: the diminished fifth (600 ¢), paraminor fifth (650 ¢), perfect fifth (700 ¢), paramajor fifth (750 ¢), and augmented fifth 800¢. The sound of the paraminor and paramajor fifths are often considerd to be 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.
Sixths
24edo contains five distinct sizes of sixths: the inframinor sixth (750 ¢), minor sixth (800 ¢), neutral sixth (850 ¢), major sixth (900 ¢), and ultramajor sixth (950 ¢). The inframinor sixth is enharmonically the same as the paramajor fifth. The neutral sixth is a good approximation of 13/8, but 850 ¢ is not as accurate as a fit to 13/8 as 550 ¢ is to 11/8, and in fact it more closely approximates 18/11 instead. 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. It also approximates 7/4 or 12/7, yet can functionally harmonize in the same manner, despite being too small to be an accurate 7/4 and too large to be an accurate 12/7.
Sevenths
24edo contains five distinct sizes of sevenths: the inframinor seventh (950 ¢), minor seventh (1000 ¢), neutral seventh (1050 ¢), major seventh (1100 ¢), and 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.
Special enharmonics
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 ¢.
- An ultramajor third and paraminor fourth are both 450 ¢.
- A paramajor fifth and inframinor sixth are both 750 ¢.
- An ultramajor sixth and inframinor seventh are both 950 ¢.
Interval class categories
Neutral
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/Inframinor
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/Paraminor
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"[1]) has been added to the words "major" and "minor" that were 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.
Ultra-/Infra-
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.
Ultramajor
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.
Major
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.
Neutral
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.
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.
Inframinor
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.
Cocytic ultramajor
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 ¢ 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.
Cocytic major
The cocytic major triad is a bright but tense triad which consists of the root, the 400 ¢ major third on top of that, and the 350 ¢ neutral third on top of that. When built on the tonic, it can lead into modulations.
Cocytic neutral
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.
Cocytic minor
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.
Dusthumic major
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 ¢ 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.
Dusthumic neutral
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.
Dusthumic minor
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.
Dusthumic inframinor
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 ¢ 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.
Augmented
The augmented chord in 24edo 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.
Rules for finding other usable chord types
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.