# Permutation product set

A **permutation product set** (PPS) is obtained from a chord C = {1,*a*_1,*a*_2,...,*a*_*n*} as follows:

Let *b*_1,...,*b_' n*

**be the intervals between successive notes of the chord:**

*b_i*=*a_i'**/a_*(

*i*-1). These

*n*intervals can be permuted in

*n*! ways, yielding

*n*! different chords:

{1,*b*_s(1),*b*_s(1)**b*_s(2),...} where s is a permutation of {1,2,...,*n*}

The union of these *n* chords is the PPS of C. PPSes may or may not be octave equivalent.

Permutation product sets were introduced by Marcel De Velde in 2009 to explain the diatonic scale.

## Special cases

If C is a harmonic series, {1/1,2/1,...,*n*/1}, then the PPS of C is called the *n*-limit harmonic permutation product set (HPPS). *n* can be even.

The octave equivalent 6-limit HPPS is the union of the major and minor diatonic scales:

1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1

The octave equivalent 8-limit HPPS has 33 notes.

The octave equivalent 16-limit HPPS has 1775 notes.