vignettes/KeysAndChord.md

In Computational-Cognitive-Musicology-Lab/humdrumR: humdrumR

title: Diatonic and Tertian Sets in humdrumR author: "Nathaniel Condit-Schultz" date: "2021-03-03" output: rmarkdown::html_vignette vignette: > %\VignetteIndexEntry{Tonality and Meter in humdrumR} %\VignetteEngine{knitr::rmarkdown} %\VignetteEncoding{UTF-8}

Diatonic Sets

As mentioned in the Pitch and Tonality vignette, a normative diatonic key consists of a set of seven consetutive pitch chroma on the Line of Fifths.

A diatonic set can be orderd either by line-of-fifths position:

LoF -1 0 1 2 3 4 5

Note F C G D A E B

or in "scale-order," which corresponds to steps of $+2$ (or $-5$) modulo 7.

LoF 0 2 4 -1 1 3 5

Note C D E F G A B Step 1 2 3 4 5 6 7

Tertian Sets

The set of seven notes in a diatonic key can be reimagined as a chord---a set of notes played at the same time. Specifically, a full seven-note diatonic chord is refered to as a 13th chord. However, most chords used in tonal music are subsets of the full diatonic set, in particular three-note triads.

When viewing a diatonic set as a chord, we traditionally order the set as a sequence of ascending thirds, corresponding to intervals of $+4$ on the line-of-fifths, modulo 7. These tertian steps are usually not wrapped to the octave, resulting in steps 9, 11, and 13, instead of 2, 4, and 6.

LoF 0 4 1 5 2 -1 3

Note C E G B D F A Step 1 3 5 7 9 11 13

There are $2^7=$ 128 possible subsets that can be formed from the full diatonic set. Of these, the seven possiblities that are built from consecutive tertian steps are theoritically priveledged : i.e., ${{1}$, {1,3}, {1,3,5}, {1,3,5,7}, {1,3,5,7,9}, {1,3,5,7,9,11}, {1,3,5,7,9,11,13}}$.

A few other possible sets are fairly commonplace in Western theory as well: ${1,5, 11}$ ("sus4"), ${1,3,5,9}$ ("add9"), ${1,3,5,13}$ ("add6"), etc.

Computational-Cognitive-Musicology-Lab/humdrumR documentation built on Oct. 22, 2024, 9:28 a.m.