[John Chalmers, Divisions of the Tetrachord]

A [musical] chord of four distinct tones, none of which is an octave transposition of another, is a tetrad.

. . . . . . . . .

[Joe Monzo]

"Common-practice" (classical) music-theory is based on triads, but jazz and blues use chords which in theory are never smaller than tetrads. Blues is the more straightforward of the two, in which all chords are one of the functions I, IV, or V, and are all of the dominant-7th type. In jazz theory, there are considered to be five main types of tetrads: diminished, half-diminished, minor, dominant, and major. Below is a table showing the interval-string (is) for all 5 types and their inversions, in 12-edo tuning:

12-edo interval strings of the 5 main chord types (tetrads) and their inversions chord-members are listed in ascending pitch order, from left to right (parentheses indicate the last interval which completes the octave) diminished Root 3rd 5th 7th Root 3rd 5th 7th Position: Root 3 3 3 (3) 1st-inv 3 3 3 (3) 2nd-inv 3 3 3 (3) 3rd-inv 3 3 3 (3) half-diminished Root 3rd 5th 7th Root 3rd 5th 7th Position: Root 3 3 4 (2) 1st-inv 3 4 2 (3) 2nd-inv 4 2 3 (3) 3rd-inv 2 3 3 (4) minor Root 3rd 5th 7th Root 3rd 5th 7th Position: Root 3 4 3 (2) 1st-inv 4 3 2 (3) 2nd-inv 3 2 3 (4) 3rd-inv 2 3 4 (3) dominant Root 3rd 5th 7th Root 3rd 5th 7th Position: Root 4 3 3 (2) 1st-inv 3 3 2 (4) 2nd-inv 3 2 4 (3) 3rd-inv 2 4 3 (3) major Root 3rd 5th 7th Root 3rd 5th 7th Position: Root 4 3 4 (1) 1st-inv 3 4 1 (4) 2nd-inv 4 1 4 (3) 3rd-inv 1 4 3 (4)

The two less-common types are found only in the harmonic-minor scale, and are permutations containing the same interval-vector as 'major':

12-edo interval strings of the 2 secondary chord types (tetrads) and their inversions minor/major-7th Root 3rd 5th 7th Root 3rd 5th 7th Position: Root 3 4 4 (1) 1st-inv 4 4 1 (3) 2nd-inv 4 1 3 (4) 3rd-inv 1 3 4 (4) augmented/major-7th Root 3rd 5th 7th Root 3rd 5th 7th Position: Root 4 4 3 (1) 1st-inv 4 3 1 (4) 2nd-inv 3 1 4 (4) 3rd-inv 1 4 4 (3)

Below are links to download .pdf files of 12-edo diagrams of the tetrads in all the major and minor keys of traditional 31-tone musical notation:

- download: 12-edo diagram of tetrads in all major keys (.pdf file)
- download: 12-edo diagram of tetrads in all harmonic-minor keys (.pdf file)

. . . . . . . . .

The tonalsoft.com website is almost entirely the work of one person: me, Joe Monzo. Please reward me for my knowledge and effort by choosing your preferred level of financial support. Thank you.

a&b temperament [a&b are numbers]

55-edo (comma) (Mozart's tuning)

1/1 or 1:1 (ratio)

2/1 or 2:1 (ratio)

a&b temperament [a&b are numbers]

apotome (Greek interval)

aristoxenean (temperament family)

atomic (temperament family)

augmented / diesic (temperament family)

augmented-2nd / aug-2 / #2 (interval)

augmented-4th / aug-4 / #4 (interval)

augmented-5th / aug-5 / #5 (interval)

augmented-6th / aug-6 / #6 (interval)

augmented-9th / aug-9 / #9 (interval)

blackjack (tuning)

cent / ¢ (unit of interval measurement)

centitone / iring (unit of interval measurement)

chromatic-semitone / augmented-prime (interval)

daseian (musical notation)

dekamu / 10mu (MIDI-unit)

diapason (Greek interval)

diapente (Greek interval)

diatessaron (Greek interval)

diatonic semitone (minor-2nd) (interval)

diesic (temperament family)

diezeugmenon (Greek tetrachord)

diminished-5th / dim5 / -5 / b5 (interval)

diminished-7th / dim7 / o7 (interval)

doamu / 2mu (MIDI-unit)

dodekamu / 12mu (MIDI-unit)

dominant-7th (dom-7, x7) (chord)

dorian (mode)

eleventh / 11th (interval)

enamu / 1mu (MIDI-unit)

endekamu / 11mu (MIDI-unit)

enharmonic semitone (interval)

ennealimmal (temperament family)

enneamu / 9mu (MIDI-unit)

farab (unit of interval measurement)

fifth / 5th (interval)

flu (unit of interval measurement)

Gentle introduction to Fokker periodicity-blocks (by Paul Erlich)

fourth / 4th (interval)

Gentle introduction to Fokker periodicity-blocks (by Paul Erlich)

grad (unit of interval measurement)

hexamu / 6mu (MIDI-unit)

Hurrian Hymn (Monzo reconstruction)

hypate (Greek note)

hypaton (Greek tetrachord)

hyperbolaion / hyperboleon (Greek tetrachord)

hypophrygian (Greek mode)

imperfect (interval quality)

iring / centitone (unit of interval measurement)

1/1 or 1:1 (ratio)

2/1 or 2:1 (ratio)

jot (unit of interval measurement)

JustMusic: A New Harmony [Monzo's book]

JustMusic prime-factor notation [Monzo essay]

kwazy (temperament family)

leimma / limma (Greek interval)

lichanos (Greek note)

limma / leimma (Greek interval)

locrian (mode)

lydian (mode)

magic (temperament family)

Mahler 7th/1 [Monzo score and analysis]

marvel (temperament family)

meantone (temperament family)

mem (unit of interval measurement)

meride (unit of interval measurement)

mese (Greek note)

meson (Greek tetrachord)

millioctave / m8ve (unit of interval measurement)

mina (unit of interval measurement)

minerva (temperament family)

miracle (temperament family)

mixolydian (mode)

monzo (prime-exponent vector)

Monzo, Joe (music-theorist)

morion / moria (unit of interval measurement)

mutt (temperament family)

mystery (temperament family)

octamu / oktamu / 8mu (MIDI-unit)

octave (interval)

oktamu / octamu / 8mu (MIDI-unit)

orwell (temperament family)

p4, perfect 4th, perfect fourth (interval)

p5, perfect 5th, perfect fifth (interval)

pantonality of Schoenberg [Monzo essay]

paramese (Greek note)

paranete (Greek note)

parhypate (Greek note)

pentamu / 5mu (MIDI-unit)

prime-factor notation (JustMusic) [Monzo essay]

proslambanomenos (Greek note)

savart (unit of interval measurement)

schismic / skhismic (temperament family)

Schoenberg's pantonality [Monzo essay]

second / 2nd (interval)

semisixths (temperament family)

semitone (unit of interval measurement)

seventh / 7th (interval)

sixth / 6th (interval)

sk (unit of interval measurement)

skhismic / schismic (temperament family)

sruti tuning [Monzo essay]

studloco (tuning)

subminor 3rd (interval)

Sumerian tuning [speculations by Monzo]

synemmenon (Greek tetrachord)

temperament-unit / tu (unit of interval measurement)

tenth / 10th (interval)

tetrachord-theory tutorial [by Monzo]

tetradekamu / 14mu (MIDI-unit)

tetramu / 4mu (MIDI-unit)

third / 3rd (interval)

thirteenth / 13th (interval)

tina (unit of interval measurement)

tone (interval, and other definitions)

tredek (unit of interval measurement)

triamu / 3mu (MIDI-unit)

tridekamu / 13mu (MIDI-unit)

trihemitone (Greek interval)

trite (Greek note)

tu / temperament-unit (unit of interval measurement)

Türk sent (unit of interval measurement)

twelfth / 12th (interval)

whole-tone (interval)

woolhouse-unit (unit of interval measurement)