This is a commentary accompanying Rabinovitch (2019), situating the research within some previous... more This is a commentary accompanying Rabinovitch (2019), situating the research within some previous work on scale theory.
... Formal and general investigations into precisely which set classes may support such cycles ar... more ... Formal and general investigations into precisely which set classes may support such cycles are found in Lewin 1996 and Clampitt 1997. ... According to Hyer, for Riemann "the parallel, relative, and leitton-wechsel are always applied to tonic, dominant, or subdominant triads, and ...
One of the most significant attitudinal shifts in the history of music occurred in the Renaissanc... more One of the most significant attitudinal shifts in the history of music occurred in the Renaissance, when an emerging triadic consciousness moved musicians towards a new scalar formation that placed major thirds on a par with perfect fifths. In this paper we revisit the confrontation between the two idealized scalar and modal conceptions, that of the ancient and medieval world and that of the early modern world, associated especially with Zarlino. We do this at an abstract level, in the language of algebraic combinatorics on words. In scale theory the juxtaposition is between well-formed and pairwise well-formed scales and modes, expressed in terms of Christoffel words or standard words and their conjugates, and the special Sturmian morphisms that generate them. Pairwise well-formed scales are encoded by words over a threeletter alphabet, and in our generalization we introduce special positive automorphisms of F 3, the free group over three letters.
This paper studies the mathematical basis for a new study of modes of well-formed (WF) scales, an... more This paper studies the mathematical basis for a new study of modes of well-formed (WF) scales, and presents a new characterization of special standard Sturmian morphisms. We introduce WF words, which coincide with the step-interval patterns of modes of well-formed scales. WF words can be represented as conjugates of some Christoffel word (generalized Lydian mode). To every WF word we may assign a pair of affine automorphisms fw and gw. These assignments induce a pair of involutions over the set of WF words: the plain adjoint and the twisted adjoint. We study the properties of these adjoints; in particular we show how the plain adjoint coincides with duality over the set of Christoffel words and also that the twisted adjoint extends Sturmian involution to the set of WF words. Thomas Noll's divider incidence result holds, inter alia, that w is special standard if and only if fw(1) = 1.
This book surveys, at a very elementary level, selected relationships between mathematics and mus... more This book surveys, at a very elementary level, selected relationships between mathematics and music, and it is true that there is a dearth of wide-ranging treatments that presume very little background in both mathematics and music. There are some people who will benefit from the accessible presentation of musical acoustics, tuning and temperament, or of serial operations; who will enjoy the introduction to permutation groups through campanology (change-ringing), or who will learn something about probability and/or music history from the survey of chance methods of composition. But I fear that a student or layperson is likely to be repelled rather than drawn in: the author seems peculiarly incurious about his subject. Harkleroad is satisfied to expose some musical aspects that may be quantified, and to set forth mathematical methods of composition that mostly produce musically trivial results (dice games) or that by their reductionism trivialize the music in question (the treatment of serial composition). The bibliography, which lists ‘materials [that] were mentioned in the text, were used as reference sources by me, or discuss relevant topics’, includes no music-theoretical work from the past 20 years, a period of intense activity in mathematical music theory. The only article cited from a music theory journal is Milton Babbitt’s 1961 ‘Set structure as a compositional determinant’ [1]. George Perle’s Serial Composition and Atonality [2], Allen Forte’s The Structure of Atonal Music [3], John Rahn’s Basic Atonal Theory [4], and Easley Blackwood’s The Structure of Recognizable Diatonic Tunings [5] are the only other sources from the music theory side, strictly speaking. That fewer than one in eight of the citations are to the music theory literature, none from the past two decades, suggests a complacent attitude that is confirmed in the writing. The problem is not the introductory level of the discussions. It is possible to avoid technical language and complicated formulae and still convey some of the depth of the issues at hand. The best science writers for lay audiences, whether journalists such as James Gleick [6], or scientists such as Roger Penrose [7], Melvin Konner [8], or Richard Dawkins [9], manage to bring their readers to the frontiers of their fields, where both the known and unknown are conveyed, reach exceeds grasp, and the mystery and excitement of the search are communicated. Already on the first page there are the sorts of statements that are not so much untrue as unexamined: ‘Math and music do have much in common.At heart, abstract patterns form the stockin-trade of both. To express these patterns, each field has developed its own symbolic language, used the world over regardless of nationality’ (p. 1). We may pass over the first statement: ‘abstract patterns’ is a good starting point in a discussion of commonalities, even if ‘stock-in-trade’ is an oddly diminishing expression to follow. But the ethnocentrism of the next sentence should not be
The short paper below presents the definition of the pairwise well-formed scale concept, and a fe... more The short paper below presents the definition of the pairwise well-formed scale concept, and a few of the significant mathematical and musical features entailed by that definition. The verifications that are easily available are supplied here; for the more difficult proofs which are here omitted, the reader is directed to my dissertation (Clampitt 1997). While the definition itself is quite abstract, the body of implications and equivalences that constitute the theory include several musically attractive properties. With the significant exception of one structural subcategory, all other pairwise well-formed scales participate in modulating cycles that generalize the maximally smooth cycles defined in Cohn 1996 and intersect with the Cohn functions defined in Lewin 1996.
The ethnic names associated with the diatonic modes (e.g., Dorian, Lydian, et al.) were assigned ... more The ethnic names associated with the diatonic modes (e.g., Dorian, Lydian, et al.) were assigned in one way by the ancient Greeks and in a different way in the anonymous medieval treatise Alia musica. Music historians usually say that this renaming was the result of a confusion, and leave it at that, but Edward Gollin showed that there was a logic here that could be captured in transformational terms. In this paper we add/uncover another layer to the palimpsest with the observation that the respective Greek and medieval nomenclatures are correlated with one of the fundamental distinctions in mathematical scale theory/word theory, the distinction between plain and twisted adjoint folding patterns of modes, as represented by conjugacy classes of Christoffel words.
The conjugation class of a special Sturmian morphism carries a natural linear order by virtue of ... more The conjugation class of a special Sturmian morphism carries a natural linear order by virtue of the two elementary conjugations \(conj_a\) and \(conj_b\) with the single letters a and b, with the standard morphism of the class as the smallest element in this order. We show that a lexicographic order on the morphisms of the given conjugation class can be defined that matches the conjugation order.
The description of the Major and Minor modes as fillings of a triadic division of the octave offe... more The description of the Major and Minor modes as fillings of a triadic division of the octave offers the possibility to study them as Pairwise Well-Formed Modes. As a consequence one obtains two projections: the diatonic projection yields the well-known Ionian and Aeolian modes and provides a link between the triadic modes and the pseudo-classical modes. The syntonic projection looks unfamiliar at first sight, but closer inspection shows that these modes provide a common ground for the natural, harmonic, and melodic manifestations of both the Major and the Minor modes.
The context of this paper is the theory of modes of non-degenerate well-formed scales (generalize... more The context of this paper is the theory of modes of non-degenerate well-formed scales (generalized diatonic or pentatonic scales), within the framework of algebraic combinatorics of words, specifically musical modes encoded as members of the monoid of words in \(A^{*}\) over a two-letter alphabet A, and the monoid of Sturmian morphisms that act on \(A^{*}\). The paper relates lexicographic orderings of words modes of (non-degenerate) well-formed scales (especially the canonical examples, the diatonic modes) and lexicographic orderings of the special Sturmian morphisms associated with the modes, to the musical scale and circle-of-fifths orderings. These lexicographic orderings are related to Zarlino’s 1571 re-ordering of Glarean’s 1547 listing of six authentic diatonic modes.
Pairwise well-formed modes are investigated as substitutions on words over a three-letter alphabe... more Pairwise well-formed modes are investigated as substitutions on words over a three-letter alphabet. Motivating examples are the major and minor modes, here described in terms of the substitutions a → ac, b → ba, c → cab and a → ab, b → ca, c → bac. Pairwise well-formedness is investigated through the inspection of the two-letter-mergings of a substiution, namely concatenations with letter insertions from the right and letter projections from the left. The appropriate generalizations of the two mentioned examples are the plain and twisted kaleidoscope substitutions. Four of their mergings turn out to be Sturmian morphisms with interesting interdependencies. The diatonic mergings are musically related to the medieval modes, which are fifth-generated. The syntonic mergings are third-generated and provide a common neutral anchor for the major and minor modes. The paper thus provides a new proposal in order to integrate the diatonic side of harmonic tonality with its triadic aspects, which we characterize as its syntonic side.
... In the past two decades Norman Carey and David Clampitt developed mathematical music theory f... more ... In the past two decades Norman Carey and David Clampitt developed mathematical music theory for the study of scales, regions and related concepts. At the same time mathematicians such as Aldo de Luca, Jean Berstel, Valérie Berthé, Christian Kassel, and Christophe ...
One of the most telling features of the application of combinatorics on words in mathematical mus... more One of the most telling features of the application of combinatorics on words in mathematical music theory is the extent to which aspects of the history of theory are captured by the word theory model. Well-formed modes, including the usual diatonic modes, are modeled by Christoffel words and their conjugates. A class of Sturmian morphisms produces words representing the Glarean modes, and gives rise to standard factorizations, which in turn model the authentic and plagal divisions of the diatonic modes. The principal discussions are the renaming of the ancient Greek modes in the tenth-century Alia musica, and the reordering of the authentic modes of Glarean's Dodecachordon by Zarlino in 1571. The latter exemplifies a recently demonstrated property in combinatorics on words, the matching of the conjugation order with the lexicographic order for the conjugation class of a given special Sturmian morphism. The two main topics are unified in this presentation.
This is a commentary accompanying Rabinovitch (2019), situating the research within some previous... more This is a commentary accompanying Rabinovitch (2019), situating the research within some previous work on scale theory.
... Formal and general investigations into precisely which set classes may support such cycles ar... more ... Formal and general investigations into precisely which set classes may support such cycles are found in Lewin 1996 and Clampitt 1997. ... According to Hyer, for Riemann "the parallel, relative, and leitton-wechsel are always applied to tonic, dominant, or subdominant triads, and ...
One of the most significant attitudinal shifts in the history of music occurred in the Renaissanc... more One of the most significant attitudinal shifts in the history of music occurred in the Renaissance, when an emerging triadic consciousness moved musicians towards a new scalar formation that placed major thirds on a par with perfect fifths. In this paper we revisit the confrontation between the two idealized scalar and modal conceptions, that of the ancient and medieval world and that of the early modern world, associated especially with Zarlino. We do this at an abstract level, in the language of algebraic combinatorics on words. In scale theory the juxtaposition is between well-formed and pairwise well-formed scales and modes, expressed in terms of Christoffel words or standard words and their conjugates, and the special Sturmian morphisms that generate them. Pairwise well-formed scales are encoded by words over a threeletter alphabet, and in our generalization we introduce special positive automorphisms of F 3, the free group over three letters.
This paper studies the mathematical basis for a new study of modes of well-formed (WF) scales, an... more This paper studies the mathematical basis for a new study of modes of well-formed (WF) scales, and presents a new characterization of special standard Sturmian morphisms. We introduce WF words, which coincide with the step-interval patterns of modes of well-formed scales. WF words can be represented as conjugates of some Christoffel word (generalized Lydian mode). To every WF word we may assign a pair of affine automorphisms fw and gw. These assignments induce a pair of involutions over the set of WF words: the plain adjoint and the twisted adjoint. We study the properties of these adjoints; in particular we show how the plain adjoint coincides with duality over the set of Christoffel words and also that the twisted adjoint extends Sturmian involution to the set of WF words. Thomas Noll's divider incidence result holds, inter alia, that w is special standard if and only if fw(1) = 1.
This book surveys, at a very elementary level, selected relationships between mathematics and mus... more This book surveys, at a very elementary level, selected relationships between mathematics and music, and it is true that there is a dearth of wide-ranging treatments that presume very little background in both mathematics and music. There are some people who will benefit from the accessible presentation of musical acoustics, tuning and temperament, or of serial operations; who will enjoy the introduction to permutation groups through campanology (change-ringing), or who will learn something about probability and/or music history from the survey of chance methods of composition. But I fear that a student or layperson is likely to be repelled rather than drawn in: the author seems peculiarly incurious about his subject. Harkleroad is satisfied to expose some musical aspects that may be quantified, and to set forth mathematical methods of composition that mostly produce musically trivial results (dice games) or that by their reductionism trivialize the music in question (the treatment of serial composition). The bibliography, which lists ‘materials [that] were mentioned in the text, were used as reference sources by me, or discuss relevant topics’, includes no music-theoretical work from the past 20 years, a period of intense activity in mathematical music theory. The only article cited from a music theory journal is Milton Babbitt’s 1961 ‘Set structure as a compositional determinant’ [1]. George Perle’s Serial Composition and Atonality [2], Allen Forte’s The Structure of Atonal Music [3], John Rahn’s Basic Atonal Theory [4], and Easley Blackwood’s The Structure of Recognizable Diatonic Tunings [5] are the only other sources from the music theory side, strictly speaking. That fewer than one in eight of the citations are to the music theory literature, none from the past two decades, suggests a complacent attitude that is confirmed in the writing. The problem is not the introductory level of the discussions. It is possible to avoid technical language and complicated formulae and still convey some of the depth of the issues at hand. The best science writers for lay audiences, whether journalists such as James Gleick [6], or scientists such as Roger Penrose [7], Melvin Konner [8], or Richard Dawkins [9], manage to bring their readers to the frontiers of their fields, where both the known and unknown are conveyed, reach exceeds grasp, and the mystery and excitement of the search are communicated. Already on the first page there are the sorts of statements that are not so much untrue as unexamined: ‘Math and music do have much in common.At heart, abstract patterns form the stockin-trade of both. To express these patterns, each field has developed its own symbolic language, used the world over regardless of nationality’ (p. 1). We may pass over the first statement: ‘abstract patterns’ is a good starting point in a discussion of commonalities, even if ‘stock-in-trade’ is an oddly diminishing expression to follow. But the ethnocentrism of the next sentence should not be
The short paper below presents the definition of the pairwise well-formed scale concept, and a fe... more The short paper below presents the definition of the pairwise well-formed scale concept, and a few of the significant mathematical and musical features entailed by that definition. The verifications that are easily available are supplied here; for the more difficult proofs which are here omitted, the reader is directed to my dissertation (Clampitt 1997). While the definition itself is quite abstract, the body of implications and equivalences that constitute the theory include several musically attractive properties. With the significant exception of one structural subcategory, all other pairwise well-formed scales participate in modulating cycles that generalize the maximally smooth cycles defined in Cohn 1996 and intersect with the Cohn functions defined in Lewin 1996.
The ethnic names associated with the diatonic modes (e.g., Dorian, Lydian, et al.) were assigned ... more The ethnic names associated with the diatonic modes (e.g., Dorian, Lydian, et al.) were assigned in one way by the ancient Greeks and in a different way in the anonymous medieval treatise Alia musica. Music historians usually say that this renaming was the result of a confusion, and leave it at that, but Edward Gollin showed that there was a logic here that could be captured in transformational terms. In this paper we add/uncover another layer to the palimpsest with the observation that the respective Greek and medieval nomenclatures are correlated with one of the fundamental distinctions in mathematical scale theory/word theory, the distinction between plain and twisted adjoint folding patterns of modes, as represented by conjugacy classes of Christoffel words.
The conjugation class of a special Sturmian morphism carries a natural linear order by virtue of ... more The conjugation class of a special Sturmian morphism carries a natural linear order by virtue of the two elementary conjugations \(conj_a\) and \(conj_b\) with the single letters a and b, with the standard morphism of the class as the smallest element in this order. We show that a lexicographic order on the morphisms of the given conjugation class can be defined that matches the conjugation order.
The description of the Major and Minor modes as fillings of a triadic division of the octave offe... more The description of the Major and Minor modes as fillings of a triadic division of the octave offers the possibility to study them as Pairwise Well-Formed Modes. As a consequence one obtains two projections: the diatonic projection yields the well-known Ionian and Aeolian modes and provides a link between the triadic modes and the pseudo-classical modes. The syntonic projection looks unfamiliar at first sight, but closer inspection shows that these modes provide a common ground for the natural, harmonic, and melodic manifestations of both the Major and the Minor modes.
The context of this paper is the theory of modes of non-degenerate well-formed scales (generalize... more The context of this paper is the theory of modes of non-degenerate well-formed scales (generalized diatonic or pentatonic scales), within the framework of algebraic combinatorics of words, specifically musical modes encoded as members of the monoid of words in \(A^{*}\) over a two-letter alphabet A, and the monoid of Sturmian morphisms that act on \(A^{*}\). The paper relates lexicographic orderings of words modes of (non-degenerate) well-formed scales (especially the canonical examples, the diatonic modes) and lexicographic orderings of the special Sturmian morphisms associated with the modes, to the musical scale and circle-of-fifths orderings. These lexicographic orderings are related to Zarlino’s 1571 re-ordering of Glarean’s 1547 listing of six authentic diatonic modes.
Pairwise well-formed modes are investigated as substitutions on words over a three-letter alphabe... more Pairwise well-formed modes are investigated as substitutions on words over a three-letter alphabet. Motivating examples are the major and minor modes, here described in terms of the substitutions a → ac, b → ba, c → cab and a → ab, b → ca, c → bac. Pairwise well-formedness is investigated through the inspection of the two-letter-mergings of a substiution, namely concatenations with letter insertions from the right and letter projections from the left. The appropriate generalizations of the two mentioned examples are the plain and twisted kaleidoscope substitutions. Four of their mergings turn out to be Sturmian morphisms with interesting interdependencies. The diatonic mergings are musically related to the medieval modes, which are fifth-generated. The syntonic mergings are third-generated and provide a common neutral anchor for the major and minor modes. The paper thus provides a new proposal in order to integrate the diatonic side of harmonic tonality with its triadic aspects, which we characterize as its syntonic side.
... In the past two decades Norman Carey and David Clampitt developed mathematical music theory f... more ... In the past two decades Norman Carey and David Clampitt developed mathematical music theory for the study of scales, regions and related concepts. At the same time mathematicians such as Aldo de Luca, Jean Berstel, Valérie Berthé, Christian Kassel, and Christophe ...
One of the most telling features of the application of combinatorics on words in mathematical mus... more One of the most telling features of the application of combinatorics on words in mathematical music theory is the extent to which aspects of the history of theory are captured by the word theory model. Well-formed modes, including the usual diatonic modes, are modeled by Christoffel words and their conjugates. A class of Sturmian morphisms produces words representing the Glarean modes, and gives rise to standard factorizations, which in turn model the authentic and plagal divisions of the diatonic modes. The principal discussions are the renaming of the ancient Greek modes in the tenth-century Alia musica, and the reordering of the authentic modes of Glarean's Dodecachordon by Zarlino in 1571. The latter exemplifies a recently demonstrated property in combinatorics on words, the matching of the conjugation order with the lexicographic order for the conjugation class of a given special Sturmian morphism. The two main topics are unified in this presentation.
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