# Pitch Class Calculator

## Introduction

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).

## Bibliography

- 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.

## Acknowledgements

- Ricardo Mazzini Bordini
- Alexandre Mascarenhas Espinheira