All-Interval Series Explorer
Interactive guide to twelve-tone AIS structure, operations & combinatoriality
Based on Robert Morris & Daniel Starr, "The Structure of All-Interval Series," Journal of Music Theory 18/2 (Autumn 1974), pp. 364–389
Show pitch classes as
?Pitch-class notationThe 12 notes are numbered 0–11. A=10 (B♭), B=11 (B♮). Toggle to see note names instead (C=0, C♯=1, D=2, E♭=3…).
① Validator
② Operations
③ Analysis
④ SAIS Browser
⑤ Glossary
What is an All-Interval Series (AIS)? A twelve-tone row whose 11 successive intervals are all different — containing each of the intervals 1–11 (mod 12) exactly once. The last note must also be a tritone (interval 6) away from the first. There are exactly 3,856 AISs in normal form.
Enter a 12-tone row
?Input formatsComma-separated: 0,1,4,9
Space-separated: 0 1 4 9
Unseparated: 01493 (A=10, B=11)
All 12 pitch classes must appear exactly once.
Space-separated: 0 1 4 9
Unseparated: 01493 (A=10, B=11)
All 12 pitch classes must appear exactly once.
Pitch Classes ?Colour codingEach colour identifies a tritone pair — two notes 6 semitones apart. Matching colours reveal the row's tritone-nest structure.
Successive Intervals ?Successive intervalsEach number is the directed interval (mod 12) from one PC to the next. In a valid AIS, all 11 are distinct (1–11). The purple 6 marks the tritone — it should only appear as the outer (wraparound) interval.
All-Interval Series Status
Rotation parameter w ?What is w?w is the 1-based index of the note immediately after interval 6 in the succession. Rotating the series by w positions produces another valid AIS — this is called operation Q.
The interval 6 (tritone) splits the series into two halves. w marks where the Q-rotation begins.
Closed operations on AISs. The set of all AISs is closed under four basic operations — P (prime), R (retrograde), I (inversion), and M (×5 mod 12) — and their composite Q (rotation by w). These yield up to 16 distinct forms per series, called a constellation. Click any form to inspect its intervals.
Enter a row
16 constellation forms — click any card to inspect ?ConstellationAll 16 forms are related by operations and share structural properties like tritone-nest. When a series has invariance, two forms coincide, giving only 8 distinct forms.
Selected form
Intervals
The Karnaugh graph arranges all 16 forms so adjacent cells differ by exactly one operation. The grid wraps toroidally: top row adjoins bottom, left column adjoins right.
Deep row analysis. Visualise the tritone-nest, detect invariance type, and check which hexachordal combinatorialities are satisfied.
Enter a row
Tritone-nest (TN). Each of the 6 tritone pairs (e.g. 0↔6, 1↔7…) is shown with a coloured bracket connecting its two positions in the row. AISs related by I, M, or MI always share the same TN. Only 204 of 10,395 possible TNs can occur in a valid AIS.
Tritone-Nest ?Reading the TNEach bracket connects the two positions of a tritone-related pair. The outermost bracket always spans positions 0 and 11. Brackets at the same height can't share adjacent feet — that would duplicate an interval.
Invariant forms. Some constellations have only 8 distinct forms because the series maps to itself under one composite operation. Morris & Starr identify three types: R (wedge row), QI, and QRMI. Famous examples: Berg's Lyric Suite (R-invariant), Nono's Il canto sospeso (R-invariant).
Invariance Type ?Invariance typesR-invariant: the row is its own retrograde (transposed); tritone at centre, w=6. 22 constellations.
QI-invariant: invariant under Q∘I; w=6. 15 constellations.
QRMI-invariant: w even; inner tritone between PCs 3 & 9. 15 constellations.
QI-invariant: invariant under Q∘I; w=6. 15 constellations.
QRMI-invariant: w even; inner tritone between PCs 3 & 9. 15 constellations.
Hexachordal combinatoriality (HC) means two related row-forms can be aligned so that their first hexachords together — and their second hexachords together — each contain all 12 PCs. The table tests each classical and M-type condition.
HC Check ?Reading the tableP = [Hex A | Hex B]. Each row tests if some transposition Tₙ of an operation on Hex B (or A) equals Hex A. ✓ entries give the transposition level n that works.
| F | Condition | Satisfied? | n |
|---|
Source All-Interval Series (SAISs). The 3,856 AISs are grouped into 267 constellations. Each is represented here by its lowest-numbered member — the source AIS. Click any row to preview it and send it to the Operations or Analysis tabs. Filter by invariance type using the buttons below.
Intervals
Key terms from Morris & Starr (1974) explained in plain language for students new to twelve-tone theory.
Morris, Robert & Starr, Daniel. "The Structure of All-Interval Series." Journal of Music Theory 18/2 (Autumn 1974), pp. 364–389. Duke University Press / Yale University Department of Music.