Paul Hertz University of Wyoming The All-Interval Tetrachord A Musical Application of Almost Difference Sets Paul Hertz University of Wyoming

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

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

Numbering of pitch-classes

The pitch-class circle

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!)

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

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

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}

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

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

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

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

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}

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

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

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)

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

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

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