The All-Interval Tetrachord A Musical Application of Almost Difference Sets

1. The All-Interval TetrachordA Musical Application ofAlmost Difference Sets Paul Hertz University of Wyoming

3. Intervals and half-steps • Intervals are ratios of pitch frequencies • Also called the distance between pitches • Upward intervals are > 1, downward < 1 • The octave has a ratio of 2 : 1 (or 1 : 2) • The half-step (or semitone) is the smallest interval commonly used • In 12-tone equal temperament (the chromatic scale), there are twelve half-steps per octave

4. Pitch-classes and integer notation • Pitches separated by one or more octaves (in a 2n : 1 ratio for some integer n) are members of the same pitch-class, or pc • The set of all Cs (Ds, Es, F#s, Abs etc.) is a pc • Pitch-classes are equivalent to note names • There are 12 pitch-classes, which we can write as the integers 0, 1, 2, … ,11. • We usually let C (i.e. the set of all Cs) be 0

5. Numbering of pitch-classes

6. The pitch-class circle

7. Intervals between pitch-classes • If we consider intervals separated by one or more octaves to be equivalent, there are twelve intervals from one pitch-class to another • We can then write intervals as 0, 1, 2, … ,11. • The interval denoted by n is the result of going up n half-steps (or down 12 – n half-steps) • The interval from a pc p to a pc q is q – p (order matters!)

8. Interval classes • The distance between two pitch-classes depends on which pc you choose to start measuring from • To eliminate this problem, the interval class ic(p,q) between two pcs p and q is defined as ic(p,q) = min(p – q, q – p) • There are seven possible interval classes • Interval classes are the shortest distance from one pitch-class to another on the pitch-class circle

9. The interval vector • Sets of pitch-classes are called pc sets • The interval vector of a pc set tells us about its intervallic content • For any pc set S, consider the multiset of all interval classes {ic(p,q) : p,q in S} • The kth entry of the interval vector is the number of times k appears in S’s interval class multiset • Interval vectors have six entries

10. All-interval tetrachords • AITs have an interval vector of [1,1,1,1,1,1] • Each nonzero interval class is represented exactly once • Composers first used AITs around 1910, but they first based entire compositions on them in the 1950s or 60s • Music theorists named them in the 1960s • Two examples are {0,1,4,6} and {0,1,3,7}

11. Transposition and inversion • The transposition of the pc set {p1, … , pk} by the interval n is the pc set {p1 + n, … , pk + n} • The inversion of the pc set {p1, … , pk} is the pc set {12 – p1, … , 12 – pk} • Transposition and inversion do not change the interval vector • Any combination of transpositions and inversions of an AIT will produce another AIT

12. AITs and the pitch-class circle • Transposition by the interval n is clockwise rotation by n pitch-classes • Inversion is reflection across the vertical line from C (0) to F#/Gb (6) • On the pitch-class circle, each of the edges and diagonals of an AIT has a different length • Rotations and reflections of a pc set polygon remain congruent to the original polygon • Any two congruent pc sets have the same interval vector

13. The all-interval tetrachord {0,1,4,6}

14. The all-interval tetrachord {0,1,3,7}

15. The Z-relation and multiplication • {0,1,4,6} and {0,1,3,7} cannot be transposed or inverted into each other • Their polygons are not congruent • Music theorists call pc sets with identical interval vectors that are unrelated through transposition or inversion Z-related • All Z-related pc sets are related under multiplication by 5: {0, 1, 4, 6} * 5 = {0, 5, 20, 30} = {0, 5, 8, 6} = {5, 6, 8, 0} = 5 + {0, 1, 3, 7}

16. Groups • A set which is closed under a binary operation • The operation is associative • There is an identity element • Every element has an inverse • A subgroup of a group G is a subset of G which is a group under G’s operation

17. Pitch-class and interval groups • The operation is addition modulo 12 • The identity is 0 • The inverse of x is 12 – x • The inverse of a pitch-class p is the inversion of p • The interval from p to q is the inverse of the interval from q to p • This is Z12, the cyclic group of order 12

18. Almost difference sets • A subset D of a group G • Let N be a subgroup of G • The difference multiset {d1 – d2 : d1, d2 in D and d1 ≠ d2} contains every nonidentity element of N λ1 times and every element of G not in N either λ2 = λ1 – 1 or λ2 = λ1 + 1 times • If λ1 = λ2, D is a difference set (DS)

19. AITs are almost difference sets • AITs contain every interval class once • 0 and 6 are the only intervals that are their own inverses • {0, 6} is a subgroup of Z12 • Intervals are differences • An AIT, therefore, is an ADS of Z12 where 6 occurs twice in the difference multiset and all other nonzero intervals occur once

20. All-interval sets in microtonal scales • The definition of AIT can be extended to include any number of pitches per octave • These are known as microtonal scales • If the number of pitches per octave is even, an all-interval set is an ADS; if odd, a DS • ADS are generally harder to find than DS • The Prime Power Conjecture states that DS only exist in groups with order pn, p prime • Therefore all-interval sets for 15, 21, 33, 35, 39, 45, 51 etc. pitches per octave do not exist

21. Questions?