Coherence and sameness in well-formed and pairwise well-formed scales

@article{Carey2007CoherenceAS,
  title={Coherence and sameness in well-formed and pairwise well-formed scales},
  author={Norman Carey},
  journal={Journal of Mathematics and Music},
  year={2007},
  volume={1},
  pages={79 - 98}
}
  • Norman Carey
  • Published 1 July 2007
  • Mathematics
  • Journal of Mathematics and Music
Abstract A common theme running through many of the scale studies in recent years is a concern for the distribution of intervals and pitch classes. The question of good distribution becomes increasingly complex with the increase in parameters. Complexity increases when the cardinality, N, increases, and when the number of step sizes increases relative to cardinality. Complexity is also shown to be dependent upon the relative sizes of the step intervals. Two measures of scalar complexity are the… Expand
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References

SHOWING 1-10 OF 28 REFERENCES
Aspects of Well-Formed Scales
Pentatonic, diatonic, and chromatic scales share the same underlying structure, that of the well-formed scale. Well-formedness is defined in terms of a relationship between the order in which aExpand
Scales, Sets, and Interval Cycles: A Taxonomy
Recent studies in the theory of scales by Agmon, Balzano, Carey and Clampitt, Clough and Douthett, Clough and Myerson, and Gamer have in common the central role of the interval cycle. Based on scaleExpand
On Coherence and Sameness, and the Evaluation of Scale Candidacy Claims
We begin with a view of the universe of pitch classes in 12-tone equal temperament, shown in (a) in Example 1. Pitch class n is represented by the fraction n/12, for n = 0, 1, ... 11. Next, we expandExpand
The group-theoretic description of 12-fold and microtonal pitch systems
TLDR
This paper argues for another way of assessing the resources of a pitch system, one that is independent of ratio concerns and that considers the individual intervals as transformations forming a mathematical group. Expand
A model for pattern perception with musical applications part I: Pitch structures as order-preserving maps
TLDR
This first paper discusses the relation between the ability to perceive relative sizes of musical intervals and the choice of reference frame from a given musical context. Expand
Variety and Multiplicity in Diatonic Systems
ample, {B, D, F} is an instance of the class "diminished triad" and also an instance of the larger class "triad." We shall call the former designation specific and the latter generic. Thus the classExpand
Coordination of Interval Sizes in Seven-Tone Collections
In the Appendix to his book on The Structure of Atonal Music, Allen Forte lists 38 seven-pc subsets of the twelve-semitone set (1973 and cf. 1964). These 38 seven-tone sets are listed in Figure 1. InExpand
A Mathematical Model of the Diatonic System
In this article a mathematical model of the diatonic system is presented, that is, the notion "diatonic system" is defined in mathematical terms. We believe the proposed model to be of considerableExpand
Coherent Tone-Systems: A Study in the Theory of Diatonicism
Looking back at the second half of the present century, a future historian of music theory might cite 1962 as the year modem diatonic theory came into being: the newly founded Perspectives ofNewExpand
A model for pattern perception with musical applications part III: The graph embedding of pitch structures
TLDR
The treatment is extended to systems of different scales (as exist in many musical cultures) where a listener's recognition of any one scale in the system interacts with his ability to recognize the others. Expand
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