1. Paul Hertz University of Wyoming
The All-Interval Tetrachord A Musical Application of Almost 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?