When I was asked by Thomas Christensen to participate in this plenary session, my first inclinati... more When I was asked by Thomas Christensen to participate in this plenary session, my first inclination was to review and celebrate the published work of the last decade in the fields of music theory to which I have contributed. But as I thought through the last ten years, I reminded myself that half of my contributions to music have been in composition and that my interests and attitudes, both musical and academic, have broadened in ways I could not have predicted in 1987. So, for a while, I was at a loss to know how to respond. After further reflection, I realized that I should compose some music for this session rather than present a paper. Actually, my decision does reflect an important and recent direction in musical scholarship: the emphasis upon musical experience, in all its guises and disguises, either in opposition to, or in cooperation with, the ongoing study of musical structure. Besides, music theory scholarship and pedagogy has been a major influence on my compositional poetics, and I have been worried that the present-day separation of composition from theory and analysis in many academic institutions may have gone too far. [2] I found I wanted to express in music the sense of wonder at the rich, deep, provocative, and often beautiful insights, thoughts, and conceptions that I had encountered in music-theory writings of the last ten years. So, after reviewing the literature, I selected 35 quotations, one per author, to be read over a specially designed musical fabric of slowly evolving harmony. The order of the quotations was chosen for aesthetic reasons and does not consciously reflect any bias or prominence of interest on my part. The limitation to culling relatively short quotations that suggest issues rather than documenting results or reviewing detailed and subtle reasoning eliminated more than many important contributions. But perhaps the other speakers will discuss such texts. [3] With some trepidation, I have provided a handout to accompany your listening to my piece, if you so desire. Let me assure you the handout is not a necessary part of this presentation; nevertheless, let us look at it for a moment. The list under the heading in boldface: "Order of Authors" lists the authors of the quotations in order of appearance in my piece; a series of asterisks indicates a short section of music without text. When listening to the piece, those of you who are immediately curious about who wrote a text you hear can find out by glancing at the list but without losing your concentration on the music of words and sounds. Of course, others of you may enjoy not knowing the identity of the authors until the piece is over. In any case, the reverse side of the handout provides a bibliography for the quotes. [4] The composition is in 50 sections, 35 of which have texts. The numbers 35 and 50 might suggest to some of you
posits the notion of balance and contrasts it with evenness, which may be associated with the pat... more posits the notion of balance and contrasts it with evenness, which may be associated with the patterns of step intervals in either maximally even sets or well-formed scales. Scales defined as "perfectly balanced" are optimal in this regard. In this paper, we propose that perfectly balanced scales that display circular palindromic "richness" and also exhibit relatively few step "differences" may prove to be advantageous from a cognitive and musical perspective. For context, all circularly rich sets among 5-, 6-, and 7-note sets in 12-TET are identified. Finally, the minimal perfectly balanced sets in divisions of 30, 42, and 70 identified in Milne, Bulger, and Herff (2017) above are examined for the two constraints proposed here.
Lambda words are sequences obtained by encoding the differences between ordered elements of the f... more Lambda words are sequences obtained by encoding the differences between ordered elements of the form i + jθ, where i and j are non-negative integers and 1 < θ < 2. Lambda words are right-infinite words defined over an infinite alphabet that have connections with Sturmian words, Christoffel words, and interspersion arrays. We show that Lambda words are infinite rich words. Furthermore, any Lambda word may be mapped onto a right-infinite word over a three-letter alphabet. Although the mapping preserves palindromes and non-palindromes of the Lambda word, the resulting Gamma word is not rich.
The fundamental results in mathematical music theory due to Clough and Myerson, Cardinality equal... more The fundamental results in mathematical music theory due to Clough and Myerson, Cardinality equals Variety (CV) and Structure yields Multiplicity (SM), correspond to statements about Christoffel words. These results may be stated and proved in a purely word-theoretic environment by means of the concept of a partitioned factor. In this setting, the generalized circle of fifths of Clough and Myerson is replaced by the lexicographic order of word theory, and their notion of the generic description of a musical line is replaced by that of a partitioned factor of a circular Christoffel word. Sturmian words are the infinite counterparts to Christoffel words, characterized as aperiodic but of minimal complexity, i.e. words such that for all there are n+1 factors of length n. Berthé showed that the factors of a given length have at most three frequencies (probabilities). Drawing on previous work, we extend to results on factors under a fixed partitioning (decompositions of factors of length n into concatenations of words whose lengths are given by a composition of n into k parts). Any factor of a Sturmian word thus partitioned into k components belongs to one of k+1 types (varieties). We show how to compute the frequencies of the varieties. These results recapture CV and SM in Sturmian words, and we suggest musical interpretations, parallel to those in the finitary case.
Http Dx Doi Org 10 1080 17459730701376743, Jul 1, 2007
... 10.3. H θ the set (or hierarchy) of well-formed scales generated by θ. For example, contains ... more ... 10.3. H θ the set (or hierarchy) of well-formed scales generated by θ. For example, contains the black-key pentatonic scale and the white-key diatonic scale, as embedded in the twelve-tone chromatic scale. ... Proof: Carey 1313. ...
Reviewed by Norman Carey The resurgence of interest over the past thirty years in spirituality an... more Reviewed by Norman Carey The resurgence of interest over the past thirty years in spirituality and mysticism has brought in its wake a host of studies in music, as music claims an illustrious position in esoteric studies. The scholarly presses have, wittingly or not, eased access into the depths of musical arcana with new translations of ancient treatises with speculative components, including those of Ptolemy, Boethius, and Aristides Quintilianus. Other presses, such as Inner Traditions, Shambhala, Phanes Press, and Samuel Weiser, have swelled the tide with reprints of many other old sources (mainly neo-Platonic and Pythagorean treatises) and by presenting new authors. 1 These studies largely focus on the effect that music has upon the psyche and, correspondingly, on the musical ("harmonious") nature of the cosmos. Studies involving the spiritual effect of music often prescribe a strict regimen of "pure" intervals, i.e., those derived from the Qrst several members of the overtone series. How to mold pure intervals into the ideal scale is still very much an ongoing quest. Indeed, many composers, performers, and theorists are passionately concerned with matters of tuning and temperament, as journals such as David Doty's 1/1 and John Chalmers's Xenharmonikon, as well as at least one very lively online discussion group attest. 2 Aside from Chalmers and Doty, other champions of the Qeld include
Lambda words are sequences obtained by encoding the differences between ordered elements of the f... more Lambda words are sequences obtained by encoding the differences between ordered elements of the form i+j\theta, where i and j are non-negative integers and 1 < \theta <2. Lambda words are right-infinite words defined over an infinite alphabet that have connections with Sturmian words, Christoffel words, and interspersion arrays. We show that Lambda words are infinite rich words. Furthermore, any Lambda word may be mapped onto a right-infinite word over a three-letter alphabet. Although the mapping preserves palindromes and non-palindromes of the Lambda word, the resulting Gamma word is not rich.
The Nicomachus Triangle, a two-dimensional representation of powers of 2 and 3, provides a starti... more The Nicomachus Triangle, a two-dimensional representation of powers of 2 and 3, provides a starting point for the development of an infinite class of right infinite Lambda words, Λθ . The word is formed by encoding differences in the sequence $\{\mathcal{M}_{\theta}\}_{i} = \{a+b\theta\}_{i}$ , a, b ∈ ℕ. Although the word is on an infinite alphabet, it is traversable via environments containing no more than three letters. When θ = ϑ = log23, the word encodes all well-formed scales and regions generated by the intervals octave and perfect twelfth. The study sheds additional light on the role of palindromes in musical tone structures. Regions are palindromes on two letters, and form the largest palindromes in the Lambda word, as it develops. The regions have a significant dual representation, connecting them to the palindromic prefixes of a characteristic Sturmian word. The Lambda word is rich in palindromes beyond regions. In particular, a palindrome is formed between any two successive appearances of the same letter. Although Λϑ is of particular importance musically, Lambda words are interesting in their own right as word theoretic objects. The paper ends with a brief look at the Fibonacci Lambda word, Λφ .
Diatonic set theory, as established by John Clough and others (see Clough 1979, 1980, Carey and C... more Diatonic set theory, as established by John Clough and others (see Clough 1979, 1980, Carey and Clampitt 1989, Carey 1998), applies the tools of standard set theory of 12-tone ET to the heptatonic set of seven tones of the diatonic scale. The two universes differ from each other in a number of ways other than simple cardinality. Although 12 is nearly twice as big as 7, the fact that 7 is prime and 12 composite contributes to a number of subtle differences between them. Every positive integer less than 7 is a unit mod 7, thus every diatonic interval generates the entire set. In the set of 12 tones, there are only four units, 1, 5, 7, and 11. Further, because of tuning, the geographies, if you will, of the sets also differ. In the equal-tempered 12-tone landscape, every place looks like everyplace else. In the diatonic scale, each generic span is inhabited by several different specific intervals. Because of this, the terrain is everywhere distinct, contributing to the phenomenon of gravitational asymmetry and of tonality.
The two theorems of the title constitute the mathematical results underlying well-formed scale th... more The two theorems of the title constitute the mathematical results underlying well-formed scale theory. This paper includes the purely mathematical portion of a manuscript from 1988, which the authors cited the following year in N. Carey and D. Clampitt [Aspects of well-formed scales, Music Theory Spectrum 11 (1989), pp. 187–206]. Both theorems concern finite sets of the fractional parts of multiples of a positive real number θ. Such sets may be taken to represent musical scales generated by a single interval, and they have nice properties when their cardinalities are the denominators of convergents or semi-convergents (intermediate convergents) in the continued fraction representation of θ. Here, the results and their proofs are brought together in a notationally consistent framework.
When I was asked by Thomas Christensen to participate in this plenary session, my first inclinati... more When I was asked by Thomas Christensen to participate in this plenary session, my first inclination was to review and celebrate the published work of the last decade in the fields of music theory to which I have contributed. But as I thought through the last ten years, I reminded myself that half of my contributions to music have been in composition and that my interests and attitudes, both musical and academic, have broadened in ways I could not have predicted in 1987. So, for a while, I was at a loss to know how to respond. After further reflection, I realized that I should compose some music for this session rather than present a paper. Actually, my decision does reflect an important and recent direction in musical scholarship: the emphasis upon musical experience, in all its guises and disguises, either in opposition to, or in cooperation with, the ongoing study of musical structure. Besides, music theory scholarship and pedagogy has been a major influence on my compositional poetics, and I have been worried that the present-day separation of composition from theory and analysis in many academic institutions may have gone too far. [2] I found I wanted to express in music the sense of wonder at the rich, deep, provocative, and often beautiful insights, thoughts, and conceptions that I had encountered in music-theory writings of the last ten years. So, after reviewing the literature, I selected 35 quotations, one per author, to be read over a specially designed musical fabric of slowly evolving harmony. The order of the quotations was chosen for aesthetic reasons and does not consciously reflect any bias or prominence of interest on my part. The limitation to culling relatively short quotations that suggest issues rather than documenting results or reviewing detailed and subtle reasoning eliminated more than many important contributions. But perhaps the other speakers will discuss such texts. [3] With some trepidation, I have provided a handout to accompany your listening to my piece, if you so desire. Let me assure you the handout is not a necessary part of this presentation; nevertheless, let us look at it for a moment. The list under the heading in boldface: "Order of Authors" lists the authors of the quotations in order of appearance in my piece; a series of asterisks indicates a short section of music without text. When listening to the piece, those of you who are immediately curious about who wrote a text you hear can find out by glancing at the list but without losing your concentration on the music of words and sounds. Of course, others of you may enjoy not knowing the identity of the authors until the piece is over. In any case, the reverse side of the handout provides a bibliography for the quotes. [4] The composition is in 50 sections, 35 of which have texts. The numbers 35 and 50 might suggest to some of you
posits the notion of balance and contrasts it with evenness, which may be associated with the pat... more posits the notion of balance and contrasts it with evenness, which may be associated with the patterns of step intervals in either maximally even sets or well-formed scales. Scales defined as "perfectly balanced" are optimal in this regard. In this paper, we propose that perfectly balanced scales that display circular palindromic "richness" and also exhibit relatively few step "differences" may prove to be advantageous from a cognitive and musical perspective. For context, all circularly rich sets among 5-, 6-, and 7-note sets in 12-TET are identified. Finally, the minimal perfectly balanced sets in divisions of 30, 42, and 70 identified in Milne, Bulger, and Herff (2017) above are examined for the two constraints proposed here.
Lambda words are sequences obtained by encoding the differences between ordered elements of the f... more Lambda words are sequences obtained by encoding the differences between ordered elements of the form i + jθ, where i and j are non-negative integers and 1 < θ < 2. Lambda words are right-infinite words defined over an infinite alphabet that have connections with Sturmian words, Christoffel words, and interspersion arrays. We show that Lambda words are infinite rich words. Furthermore, any Lambda word may be mapped onto a right-infinite word over a three-letter alphabet. Although the mapping preserves palindromes and non-palindromes of the Lambda word, the resulting Gamma word is not rich.
The fundamental results in mathematical music theory due to Clough and Myerson, Cardinality equal... more The fundamental results in mathematical music theory due to Clough and Myerson, Cardinality equals Variety (CV) and Structure yields Multiplicity (SM), correspond to statements about Christoffel words. These results may be stated and proved in a purely word-theoretic environment by means of the concept of a partitioned factor. In this setting, the generalized circle of fifths of Clough and Myerson is replaced by the lexicographic order of word theory, and their notion of the generic description of a musical line is replaced by that of a partitioned factor of a circular Christoffel word. Sturmian words are the infinite counterparts to Christoffel words, characterized as aperiodic but of minimal complexity, i.e. words such that for all there are n+1 factors of length n. Berthé showed that the factors of a given length have at most three frequencies (probabilities). Drawing on previous work, we extend to results on factors under a fixed partitioning (decompositions of factors of length n into concatenations of words whose lengths are given by a composition of n into k parts). Any factor of a Sturmian word thus partitioned into k components belongs to one of k+1 types (varieties). We show how to compute the frequencies of the varieties. These results recapture CV and SM in Sturmian words, and we suggest musical interpretations, parallel to those in the finitary case.
Http Dx Doi Org 10 1080 17459730701376743, Jul 1, 2007
... 10.3. H θ the set (or hierarchy) of well-formed scales generated by θ. For example, contains ... more ... 10.3. H θ the set (or hierarchy) of well-formed scales generated by θ. For example, contains the black-key pentatonic scale and the white-key diatonic scale, as embedded in the twelve-tone chromatic scale. ... Proof: Carey 1313. ...
Reviewed by Norman Carey The resurgence of interest over the past thirty years in spirituality an... more Reviewed by Norman Carey The resurgence of interest over the past thirty years in spirituality and mysticism has brought in its wake a host of studies in music, as music claims an illustrious position in esoteric studies. The scholarly presses have, wittingly or not, eased access into the depths of musical arcana with new translations of ancient treatises with speculative components, including those of Ptolemy, Boethius, and Aristides Quintilianus. Other presses, such as Inner Traditions, Shambhala, Phanes Press, and Samuel Weiser, have swelled the tide with reprints of many other old sources (mainly neo-Platonic and Pythagorean treatises) and by presenting new authors. 1 These studies largely focus on the effect that music has upon the psyche and, correspondingly, on the musical ("harmonious") nature of the cosmos. Studies involving the spiritual effect of music often prescribe a strict regimen of "pure" intervals, i.e., those derived from the Qrst several members of the overtone series. How to mold pure intervals into the ideal scale is still very much an ongoing quest. Indeed, many composers, performers, and theorists are passionately concerned with matters of tuning and temperament, as journals such as David Doty's 1/1 and John Chalmers's Xenharmonikon, as well as at least one very lively online discussion group attest. 2 Aside from Chalmers and Doty, other champions of the Qeld include
Lambda words are sequences obtained by encoding the differences between ordered elements of the f... more Lambda words are sequences obtained by encoding the differences between ordered elements of the form i+j\theta, where i and j are non-negative integers and 1 < \theta <2. Lambda words are right-infinite words defined over an infinite alphabet that have connections with Sturmian words, Christoffel words, and interspersion arrays. We show that Lambda words are infinite rich words. Furthermore, any Lambda word may be mapped onto a right-infinite word over a three-letter alphabet. Although the mapping preserves palindromes and non-palindromes of the Lambda word, the resulting Gamma word is not rich.
The Nicomachus Triangle, a two-dimensional representation of powers of 2 and 3, provides a starti... more The Nicomachus Triangle, a two-dimensional representation of powers of 2 and 3, provides a starting point for the development of an infinite class of right infinite Lambda words, Λθ . The word is formed by encoding differences in the sequence $\{\mathcal{M}_{\theta}\}_{i} = \{a+b\theta\}_{i}$ , a, b ∈ ℕ. Although the word is on an infinite alphabet, it is traversable via environments containing no more than three letters. When θ = ϑ = log23, the word encodes all well-formed scales and regions generated by the intervals octave and perfect twelfth. The study sheds additional light on the role of palindromes in musical tone structures. Regions are palindromes on two letters, and form the largest palindromes in the Lambda word, as it develops. The regions have a significant dual representation, connecting them to the palindromic prefixes of a characteristic Sturmian word. The Lambda word is rich in palindromes beyond regions. In particular, a palindrome is formed between any two successive appearances of the same letter. Although Λϑ is of particular importance musically, Lambda words are interesting in their own right as word theoretic objects. The paper ends with a brief look at the Fibonacci Lambda word, Λφ .
Diatonic set theory, as established by John Clough and others (see Clough 1979, 1980, Carey and C... more Diatonic set theory, as established by John Clough and others (see Clough 1979, 1980, Carey and Clampitt 1989, Carey 1998), applies the tools of standard set theory of 12-tone ET to the heptatonic set of seven tones of the diatonic scale. The two universes differ from each other in a number of ways other than simple cardinality. Although 12 is nearly twice as big as 7, the fact that 7 is prime and 12 composite contributes to a number of subtle differences between them. Every positive integer less than 7 is a unit mod 7, thus every diatonic interval generates the entire set. In the set of 12 tones, there are only four units, 1, 5, 7, and 11. Further, because of tuning, the geographies, if you will, of the sets also differ. In the equal-tempered 12-tone landscape, every place looks like everyplace else. In the diatonic scale, each generic span is inhabited by several different specific intervals. Because of this, the terrain is everywhere distinct, contributing to the phenomenon of gravitational asymmetry and of tonality.
The two theorems of the title constitute the mathematical results underlying well-formed scale th... more The two theorems of the title constitute the mathematical results underlying well-formed scale theory. This paper includes the purely mathematical portion of a manuscript from 1988, which the authors cited the following year in N. Carey and D. Clampitt [Aspects of well-formed scales, Music Theory Spectrum 11 (1989), pp. 187–206]. Both theorems concern finite sets of the fractional parts of multiples of a positive real number θ. Such sets may be taken to represent musical scales generated by a single interval, and they have nice properties when their cardinalities are the denominators of convergents or semi-convergents (intermediate convergents) in the continued fraction representation of θ. Here, the results and their proofs are brought together in a notationally consistent framework.
The diatonic theories reviewed in the first chapter bring into relief the importance of five fact... more The diatonic theories reviewed in the first chapter bring into relief the importance of five factors. First, there is general, if not universal, agreement that the diatonic scale is a generated set. Second, there is a consensus that the diatonic scale is embedded within what is often defined as a 12-tone, equal-tempered universe. Third, the diatonic scale exhibits unique multiplicity, allowing for a hierarchization based upon intervallic rarity. Fourth, there is considerable concern shown for the question of coherence, meaning either the absence of contradiction and ambiguity (Balzano), or the absence of contradiction only (Agmon). Finally, some authors have recognized the importance of a small generalized comma.
This dissertation presents the theory of well-formed scales in the context of the mathematics of ... more This dissertation presents the theory of well-formed scales in the context of the mathematics of distribution modulo 1. Deleted: volume descriptions. Chapter III presents proofs for the assertions of the theory. Three entirely new aspects of the theory are asserted and proven in Chapter III (Theorems 1, 4, and 5). These concern a definition of well-formedness, the nature of the step sequence of the well-formed scale, and the conditions that govern coherence, or generic ordering in well-formed scales and constitute and significant new contribution to the theory. This is also true of the nine corollaries to Theorem 2. Applications and extensions of the theory are the subject of Chapter IV. The theory of well-formed scales is applied to compositional issues including microtonalism, and an extension of the theory into the realm of rhythmic analysis, in which I show that hierarchies of well-formed scales are an appropriate model for the study of rhythm of Bartók. The first two chapters review recent diatonic theory and reconstruct the theory of well-formed scales. Appendix A presents the mathematical concepts and notation that are used throughout the dissertation, particularly in the third chapter. Appendix B illustrates mappings between mathematical and musical realms. Appendix C presents the code for " ScaleDesigner, " a computer program I wrote in order to explore the microtonal possibilities of well-formed scales. ScaleDesigner is another new contribution in this dissertation. Appendix D is the score and analysis of a song I composed, " Amarantha, " which exploits the contrapuntal advantages of a non-triadic diatonic [2] harmony. Appendix E is a glossary. 2
Uploads
Papers by Norman Carey