← Home   ·   Lindberg-scale tool   ·   Spectral Analyzer   ·   Instrument ranges   ·   Messiaen modes

All-Interval Chords

Chords that contain every interval exactly once — the two tetrachords, chains built from them, the twelve-note rows, and the microtonal systems where larger ones live.

The four notes of the chord

The six intervals

Four notes make six pairs. In an all-interval tetrachord each pair lands on a different interval class, so all six appear once and none repeats.

Sequence

Chords you can add next

Each chord is voiced in its own register, stacked upward from the base note, so the chain reads left to right as well as bottom to top. Notes shared with the previous chord are outlined.

Export

“Retrograde is a transposition” keeps only the rows that read the same backwards as forwards once moved up by a tritone — the rows Wikipedia writes as P0 = R6. Eighty-eight of the 1,928 inversion classes have this property.

Fix intervals at chosen positions

Leave a slot on “any” to let the search fill it. Each of the eleven intervals 1–11 is used exactly once, so fixing a few forces the rest into a small number of possibilities.

Preloaded rows

No search run yet.

Selected row

The twelve notes, each interval taken upward

All-interval chords outside 12-tone space

Twelve-tone space runs out at four notes, because four notes need six interval classes and twelve-tone space has exactly six. Bigger all-interval chords need bigger interval systems. Peck (2016) catalogues 11,438 of them across twenty systems. Nine of those systems are cyclic — plain equal divisions of the octave, written Zv for v equal steps — and this tool computes those in full.

The eleven systems this tool does not compute

The other eleven systems in Peck's catalogue are not equal divisions of the octave. Their intervals are elements of groups such as the dihedral group of order 8, the semidihedral group of order 16, or the direct product of the symmetric group on three letters with a cycle of three. A chord in one of those systems is not a set of pitches at all — Peck's example for S3×Z3 is a nine-note sonority split into three registers, where the "interval" is a permutation of which triad sits where. Adding them would mean building a small group-theory engine and inventing a display for each system, which is a different tool. They are listed here so nothing is silently dropped.

What an all-interval chord is

Take any two notes. Count the semitones between them, and take the smaller of the two ways round — C up to A is nine semitones, A up to C is three, so the answer is three. That number, from 1 to 6, is the interval class. Twelve-tone music has exactly six of them: 1 (semitone), 2 (whole tone), 3 (minor third), 4 (major third), 5 (fourth), 6 (tritone).

An all-interval chord uses each interval class exactly once, and no interval class twice. That makes it the most intervallically economical chord available: every kind of interval is present, nothing is wasted on repetition.

The interval-class vector (ICV) is the tally. Write down how many times each of the six interval classes appears, in order. An all-interval chord has the ICV [1, 1, 1, 1, 1, 1]. The directed-interval vector (DIV) is the stricter tally used by David Lewin: it counts each interval in both directions separately, so it has twelve slots, one for each distance 0 to 11.

Why twelve-tone space stops at four notes

A chord of k notes has k(k−1)/2 pairs of notes. If every pair is to have a different interval class, the system must contain exactly that many interval classes. Four notes make six pairs, and twelve-tone space has six interval classes — so four notes fit exactly. Three notes make only three pairs, which leaves three interval classes unused. Five notes make ten pairs, and there are only six interval classes to go round, so something must repeat. Four is the only size that works.

At that size there are exactly 48 all-interval tetrachords, in two set classes: [0,1,4,6], called 4-Z15, and [0,1,3,7], called 4-Z29. Twenty-four chords each. The "Z" records that the two have the same interval content without being transpositions or inversions of one another — they are Z-related. This tool recomputes all 48 from scratch every time the page loads.

A famous instance: Schoenberg's Das Buch der hängenden Gärten Op. 15 ends its first song on {3, 5, 8, 9}, which is a member of 4-Z15. Peck uses it as his opening example. Alban Berg's Vier Gesänge Op. 2 has a passage (bars 20–22) built entirely of all-interval tetrachords, alternating the two set classes.

The other reading: all-interval rows

There is a second, different sense of "all-interval". A twelve-tone row is an ordering of all twelve pitch classes. It is an all-interval row when the eleven steps from one note to the next use each distance 1 to 11 exactly once. Here direction matters — a step of 8 upward is not the same as a step of 4 upward — so eleven intervals can all be different even though there are only six interval classes.

Of the 479,001,600 possible twelve-tone rows, 46,272 are all-interval. Fixing the first note to remove transposition leaves 3,856. Treating a row and its inversion as the same halves it to 1,928. Eighty-eight of those read the same backwards as forwards-transposed, which gives 1,008 row types in Schoenberg's sense. This tool enumerates them by backtracking and reproduces all four counts.

Because the eleven intervals sum to 1+2+…+11 = 66, and 66 is 6 more than five full octaves, the first and last notes of an all-interval row are always a tritone apart, and stacking the row upward always spans exactly 66 semitones — five and a half octaves. The tool checks both properties on all 3,856 rows.

The first one found was Fritz Heinrich Klein's Mutterakkord of 1921 — the Mother chord — which Berg later used in the Lyric Suite. Nicolas Slonimsky's Grandmother chord of 1938 adds a pattern: the intervals alternate odd and even, the odd ones shrinking by a whole tone each time and the even ones growing.

What Peck generalises

Robert Peck's 2016 paper asks the question in a wider setting. An "interval" need not be a distance in semitones; following David Lewin, it can be any element of a mathematical group acting on a space of musical objects. Peck counts the all-interval chords of 2 to 8 notes across every such system, and finds 11,438 chords in twenty interval systems.

He also identifies what kind of combinatorial object these chords are. A difference set is a set of numbers in which every non-zero difference appears the same number of times. When every difference appears exactly once it is a planar difference set, and the chord is all-interval in the strictest sense — each directed interval appears once too. The 12-tone tetrachords are not that strict: their DIV is (4,1,1,1,1,1,2,1,1,1,1,1), with the tritone appearing twice because a tritone is its own inverse. They are almost difference sets — every difference appears once except one, which appears twice. Peck writes this as (12, 4, 1, 10).

Sources

Every count on this page is computed in your browser when the page loads or when you press a button. The figures above were checked against these four sources before the tool was written; the checks are in test.js next to this page.