Pitch Class Calculator


Pitch Class Calculator is an auxiliary tool for Post-Tonal music analysis. Jamary Oliveira's PCN has inspired this tool.

Available operations

  • Transposition
  • Inversion
  • Transposition to zero
  • Repeats cleaning
  • Sort
  • Normal form
  • Prime form
  • Forte class number
  • Interval class vector
  • Index vector
  • Multiply (by 5 and 7)
  • Class members
  • Subsets
  • Complement
  • IcVSIM (Interval class vector similarity)
  • Row inversion
  • Rotation
  • Retrograde
  • Row matrix
  • Symmetry axis
  • Voice-leading space

Voice leading space

by Ricardo Bordini

Triadic transformations involve two types of operations: contextual inversions that preserve notes in common and parsimonious voice-leading. Contextual inversion inverts a triad around a note or two, and parsimonious voice-leading connects the triads as smoothly as possible. The most parsimonious voice-leading is one that preserves two common notes and another moves by semitone (relationship of parallels and leading-tones). Slightly less parsimonious voice-leading is one that preserves two notes and another moves by two semitones (relative) or that preserve one note and the other two move by semitone (obverses of homonyms, leading-tone and relative). (For more details see Straus, 2016, p. 188-89.) The idea of a space of notes in a tonal range appeared for the first time in Lerdahl in 1988. This idea was later expanded to a space for chains in an atonal scope by Morris in 1998. For more about spaces of chains for trichords and tetrachords, see Straus (2016, p. 179-183).


  • Bordini, Ricardo. 2018. Expanded Atonal Voice-Leading Space for Trichords: an auxiliary model for generating pre-compositional material. MUSICA THEORICA. Salvador: TeMA, 201805, p. 108-127.
  • Isaacson, Eric J. 1990. Similarity of Interval-Class Content between Pitch-Class Sets: The IcVSIM Relation. Journal of Music Theory, Vol. 34, No. 1. (Spring, 1990), pp. 1-28.
  • Oliveira, Jamary. 1995. Informática em Música: o parâmetro altura. Reísa 1: 148p.
  • Straus, Joseph Nathan. 2016. Introduction to Post-Tonal Theory. 4. ed. New York: W. W. Norton.


  • Ricardo Mazzini Bordini
  • Alexandre Mascarenhas Espinheira