Defining Modular Transformations

Santa, Matthew

doi:10.1525/mts.1999.21.2.02a00030

Cited by 5 publications

(3 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Weitere Fragestellungen der Set Theory haben zur Untersuchung von ›pitch-class sets‹ mit speziellen Eigenschaften geführt, unter denen das siebentönige diatonische ›set‹ eine zentrale Stelle einnimmt. 14 Unter Anwendung der Set Theory haben verschiedene Autoren Akkordformationen in weiteren Tonsystemen untersucht, wie dem oktatonischen System (van den Toorn 1983, 1986, Santa 1999, dem pentatonischen und ganztönigen System (Santa 1999), den modalen Systemen von Olivier Messiaen (Bernard 1986, Cheong 2002, 2003, Neidhöfer 2005 und im ›double aggregate‹, in dem jeder Ton im chromatischen Total doppelt vertreten ist (Morris 2003).…”

Section: Set Theory Christoph Neidhöferunclassified

Set Theory

Neidhöfer¹

2005

ZGMTH

View full text Add to dashboard Cite

Section: Set Theory Christoph Neidhöferunclassified

Set Theory

Neidhöfer¹

2005

ZGMTH

View full text Add to dashboard Cite

“…The aligned‐cycle model also resonates with ‘pc equivalence classes’ in Morris (), pp. 153–4; ‘MODTRANS’ in Santa (); and ‘dual interval spaces’ in Brown ( and ). More recently, Gollin ( and ) introduced the multi‐aggregate cycle and investigated these cyclic structures in the music of Bartók.…”

mentioning

confidence: 99%

Aligned Cycles in Thomas Adès's Piano Quintet

Stoecker¹

2013

Music Analysis

View full text Add to dashboard Cite

In this article I investigate aligned cycles in Thomas Adès's Piano Quintet, Op. 20. First identified by Berg in a letter to Schoenberg and later discussed by George Perle and Dave Headlam, an aligned cycle is the result of two or more interval cycles unfolding in the same direction in a note‐against‐note alignment – for example, a rising whole‐tone scale (an interval‐2 cycle) simultaneously with a rising chromatic scale (an interval‐1 cycle). I begin by exploring in more detail the theoretical properties of three‐voice aligned cycles and use concentric circles to show how these cyclic structures generate a closed, tight‐knit family of harmonies. I then reveal how Adès uses aligned cycles to provide harmonic, melodic and motivic material throughout his Quintet. These cyclic structures are combined not only with other kinds of cycles (one‐, two‐ and five‐voice, and multi‐aggregate) but also with the non‐cyclic material in the piece. Finally, I discuss how all the transformations of the different cycles function within the large‐scale formal design of the piece.

show abstract

“…56 56 Santa (1999) discusses scale‐to‐scale operations related to interscalar transposition, and Hook (2007) discusses interscalar transpositions in pitch space. …”

mentioning

confidence: 99%

Scale Theory, Serial Theory and Voice Leading

Tymoczko

2008

Music Analysis

View full text Add to dashboard Cite

Article S cale T heory , S erial T heory and V oice L eading dmitri tymoczko dmitri tymoczko S cale T heory , S erial T heory and V oice L eading Musical norms evolve over time. In the eleventh century parallel perfect fifths were tolerated, and perfect fourths were considered more consonant than thirds. In the eighteenth century parallel perfect fifths were not tolerated, and thirds were considered more consonant than perfect fourths. And in the twentieth century all manner of traditional prohibitions collapsed as composers revelled in the sense that everything was permitted. Despite such changes, however, certain musical principles remain relatively constant across styles. One of the most important of these dictates that harmonies should, in general, be connected by efficient voice leading. That is, notes should be distributed among individual musical voices so that no voice moves very far as harmonies change. Ex. 1 supplies a few representative passages. In each case, the music exploits the shortest possible path between successive chords, in a sense to be defined below. 1 Efficient voice leading is not simply a matter of performers' physical comfort, although that is certainly a factor. It also enables listeners to segregate the auditory stimulus into a series of independent musical lines, which is a prerequisite for understanding polyphony. 2 Efficient voice leading is so ubiquitous that we tend to take it for granted. But upon reflection it is rather remarkable that composers manage so consistently to find the shortest route from chord to chord. There are 57,366,997,447 distinct voice leadings between two hexachords. 3 There are 288 bijective voice leadings from a half-diminished seventh chord to the twelve dominant seventh chords. 4 Yet almost any composer, theorist or undergraduate student of Ex. 1 Efficient voice leading in several musical styles: (a) Ad organum faciendum (eleventh century); (b) Landini, Sy dolce non sono; (c) J.S. Bach, Das wohltemperierte Klavier I, Fugue No. 3 in CC major, BWV 869 (final cadence, upper four voices only); (d) Wagner, the 'Tarnhelm' motive (upper three voices only); and (e) a common jazz ii-V-I voicing (upper four voices only)dmitri tymoczko harmony can quickly find the most efficient voice leading between two hexachords, or a maximally efficient bijective voice leading from a particular half-diminished seventh chord to any of the twelve dominant seventh chords. The puzzle lies in the mismatch between the large number of possibilities and our apparent ease in sorting through them.We can complicate our puzzle by noting that students receive little explicit conceptual instruction in identifying efficient voice leadings. Music teachers tend to provide examples augmented with relatively uninformative exhortations: we enjoin our students to obey the 'law of the shortest way' -that is, to minimise the overall voice-leading distance between successive chords 5 -without telling them precisely what this means or how to do it. And students comply, easily and for the most part w...

show abstract

Defining Modular Transformations

Cited by 5 publications

References 0 publications

Set Theory

Set Theory

Aligned Cycles in Thomas Adès's Piano Quintet

Scale Theory, Serial Theory and Voice Leading

Contact Info

Product

Resources

About