The AllInterval Tetrachord A Musical Application of Almost Difference Sets

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The All-Interval TetrachordA Musical Application
ofAlmost Difference Sets

• Paul Hertz
• University of Wyoming

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(No Transcript)
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Intervals and half-steps

• Intervals are ratios of pitch frequencies
• Also called the distance between pitches
• Upward intervals are 1, downward
• 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

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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, Fs, 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

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Numbering of pitch-classes
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The pitch-class circle
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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!)

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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

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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 Ss interval class
  multiset
• Interval vectors have six entries

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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

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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

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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

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The all-interval tetrachord 0,1,4,6
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The all-interval tetrachord 0,1,3,7
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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

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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 Gs operation

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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

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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
• ?1 times and every element of G not in N
• either ?2 ?1 1 or ?2 ?1 1 times
• If ?1 ?2, D is a difference set (DS)

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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

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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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Questions?