DOI: 10.1126/science.1134163, 330 (2007);315 Science

Dmitri TymoczkoResponse to Comment on "The Geometry of Musical Chords"

This copy is for your personal, non-commercial use only.

clicking here.colleagues, clients, or customers by , you can order high-quality copies for yourIf you wish to distribute this article to others

  here.following the guidelines

can be obtained byPermission to republish or repurpose articles or portions of articles

  ): February 11, 2014 www.sciencemag.org (this information is current as of

The following resources related to this article are available online at

http://www.sciencemag.org/content/315/5810/330.3.full.htmlversion of this article at:

including high-resolution figures, can be found in the onlineUpdated information and services,

http://www.sciencemag.org/content/315/5810/330.3.full.html#relatedfound at:

can berelated to this article A list of selected additional articles on the Science Web sites

http://www.sciencemag.org/content/315/5810/330.3.full.html#ref-list-1, 1 of which can be accessed free:cites 1 articlesThis article

http://www.sciencemag.org/cgi/collection/tech_commentTechnical Comments

http://www.sciencemag.org/cgi/collection/comp_mathComputers, Mathematics

subject collections:This article appears in the following

registered trademark of AAAS. is aScience2007 by the American Association for the Advancement of Science; all rights reserved. The title

CopyrightAmerican Association for the Advancement of Science, 1200 New York Avenue NW, Washington, DC 20005. (print ISSN 0036-8075; online ISSN 1095-9203) is published weekly, except the last week in December, by theScience

on

Feb

ruar

y 11

, 201

4w

ww

.sci

ence

mag

.org

Dow

nloa

ded

from

o

n F

ebru

ary

11, 2

014

ww

w.s

cien

cem

ag.o

rgD

ownl

oade

d fr

om

on

Feb

ruar

y 11

, 201

4w

ww

.sci

ence

mag

.org

Dow

nloa

ded

from

Response to Comment on“The Geometry of Musical Chords”Dmitri Tymoczko

The basic sonorities of traditional Western tonality divide the octave nearly evenly and are foundnear the center of the orbifolds T3/S3 and T4/S4. Many common musical patterns exploit this fact,which permits efficient voice leading between structurally similar chords. In actual music, thesepatterns sometimes appear incompletely or are accompanied by additional notes. Using orbifoldsin musical analysis therefore requires interpretive skill.

Theorists agree that when analyzing music,there are two distinct phenomena toconsider: the actual notes (the surface)

and the common musical patterns those notesmay imperfectly embody (the background).Orbifolds can be used to represent either phe-nomenon, although they are typically most use-ful when modeling background patterns.

Triads and seventh chords are the conceptualbuilding blocks of Western tonality, the onlycomplete harmonies recognized by traditionaltheory. These sonorities are found near the centerof the orbifolds T3/S3 and T4/S4. Theoristsagree that in actual music these harmonies aresometimes incomplete and are sometimesaccompanied by additional notes (doublingsand “nonharmonic tones”). Thus, we cannotexpect the surface of every tonal piece alwaysto inhabit the center of some orbifold.

Figure 1A, reproduced from (1), is typical-ly accompanied by an additional voice, as inFig. 1B. The full progression lies on thesingular boundary of T4/S4. Directly plottingFig. 1B on this orbifold is not maximally infor-mative, as its lowest voice operates according towhat are generally recognized to be distinctivemusical principles. My report therefore separatesthe progression into two parts: the upper voices,which exhibit efficient voice leading betweenstructurally similar chords, and the bass, which

adds harmonic support by leaping to the rootof each chord (1). The upper voices exploit thegeometry of T3/S3; the bass plays a differentmusical role.

Incomplete chords pose a related challenge.Most theorists would understand Fig. 1C toimply a succession of triads, as in Fig. 1B. Wecan represent the musical surface by plottingthe incomplete chords of Fig. 1C on the or-bifold T2/S2; we can represent the backgroundby plotting Fig. 1A or 1B on the appropriateorbifold. Again, the upper three voices of thebackground pattern (Fig. 1A) make the mostinteresting use of orbifold geometry.

Brown and Headlam (2) observe that tonalphrases sometimes cadence on unisons. We cantypically model these cadences as incompletemanifestations of a prototypical five-voice back-

ground pattern (Fig. 2). The pattern’s top fourvoices use maximally efficient voice leading toconnect nearly transpositionally related multi-sets, interestingly exploiting the geometry ofT4/S4. If, alternatively, we are interested in themusical surface, we can again use orbifolds. AsHeadlam and Brown note, when a tonal piececadences on a unison it moves from the centerof an orbifold to its singular boundary. Thus,these cadences do possess a distinctive geomet-rical signature, even if we restrict our attentionto the musical surface.

Given these two analytical possibilities, itis unclear why Headlam and Brown claimthat orbifolds cannot effectively model suchprogressions. Perhaps they have not clearlydistinguished the musical surface from thebackground or have misinterpreted my remarksabout the background (that the basic tonalsonorities inhabit the center of their respectiveorbifolds) as remarks about the surface. It is alsopossible that they have misinterpreted myReport (1) as an attempt to model all the style-specific norms of 18th- and 19-century compo-sition (3). Instead, it was an effort to model afew general musical principles common to awider range of Western musical styles.

Orbifolds are tools that can be used for avariety of musical purposes. Like any tools,they have limitations, but these limitations areconsiderably less constraining when the toolsare skillfully deployed.

References and Notes1. D. Tymoczko, Science 313, 72 (2006).2. D. Headlam, M. Brown, Science 315, 330 (2007);

www.sciencemag.org/cgi/content/full/315/5810/330b.3. Headlam and Brown’s fourth footnote suggests that this

may be so. My report asked when structurally similarchords can be linked by efficient voice leading. Thisquestion admits a trivial answer, because one can alwaysmove all of a chord’s notes in the same direction by thesame small amount. My remarks about voice independence,beyond reflecting a general Western musical value, weremeant to exclude this trivial solution from the discussion.(I did not suggest that any composer used independentvoice leadings exclusively.) Headlam and Brown, however,interpret me as attempting to model the avoidance ofparallel perfect intervals, a style-specific convention thatwas not very important in either medieval or modern music.

29 August 2006; accepted 20 December 200610.1126/science.1134163

TECHNICALCOMMENT

Department of Music, Princeton University, Princeton, NJ08544, USA. E-mail: dmitri@princeton.edu

A B C

Fig. 1. (A) A common classical upper-voicepattern that exploits the geometry of T3/S3. (B)In actual music, the three-voice pattern is oftenaccompanied by an additional bass voice, whosefunction is to provide harmonic support by sound-ing the root of each chord. (C) A two-voice pas-sage evoking (B).

19 JANUARY 2007 VOL 315 SCIENCE www.sciencemag.org330c

Fig. 2. Many tonal cadences, including all of the examples in (2), can be understood as variants of aprototypical five-voice pattern (A). The pattern’s prototypicality is illustrated by the fact that it concludesseveral of the first pieces in Bach’s Well-Tempered Clavier (WTC) (C to F), as well as several of Beethoven’s firstpiano-sonata movements (G to K). By contrast, Headlam and Brown’s cadence (B) concludes only one of thefirst 20 Beethoven piano-sonata movements and none of the first 20 pieces in the WTC. (A) The prototypicalfive-voice pattern. (B) The cadence from figure 1 in (2). (C) Bach, WTC Prelude 3. (D) Bach, WTC Fugue 3. (E)Bach, WTC Prelude 6. (F) Bach, WTC Prelude 10. (G to H) Beethoven, Piano Sonata No. 1, movements 1 and 2.(I to K) Beethoven, Piano Sonata No. 2, movements 1, 3, and 4.

www.sciencemag.org SCIENCE VOL 315 19 JANUARY 2007 330c

TECHNICAL COMMENT