Metrics for Scales and Tunings (original) (raw)
Related papers
In search of universal properties of musical scales
Musical scales have both general and culture-specific properties. While most common scales use octave equivalence and discrete pitch relationships, there seem to be no other universal properties. This paper presents an additional property across the world’s musical scales that may qualify for universality. When the intervals of 998 (just intonation) scales from the Scala Archive are represented on an Euler lattice, 96.7% of them form star-convex structures. For the subset of traditional scales this percentage is even 100%. We present an attempted explanation for the star-convexity feature, suggesting that the mathematical search for universal musical properties has not yet reached its limits.
Perfect balance: A novel principle for the construction of musical scales and meters
In T. Collins, D. Meredith, and A. Volk (Eds.), Mathematics and Computation in Music—MCM 2015, volume 9110 of LNAI, pages 97–108, Heidelberg. Springer.
We identify a class of periodic patterns in musical scales or meters that are perfectly balanced. Such patterns have elements that are distributed around the periodic circle such that their 'centre of gravity' is precisely at the circle's centre. Perfect balance is implied by the well established concept of perfect evenness (e.g., equal step scales or isochronous meters). However, we identify a less trivial class of perfectly balanced patterns that have no repetitions within the period. Such patterns can be distinctly uneven. We explore some heuristics for generating and parameterizing these patterns. We also introduce a theorem that any perfectly balanced pattern in a discrete universe can be expressed as a combination of regular polygons. We hope this framework may be useful for understanding our perception and production of aesthetically interesting and novel (microtonal) scales and meters, and help to dis-ambiguate between balance and evenness; two properties that are easily confused. Perfect balance: A novel principle for the construction of musical scales and meters. Available from: https://www.researchgate.net/publication/278683682\_Perfect\_balance\_A\_novel\_principle\_for\_the\_construction\_of\_musical\_scales\_and\_meters [accessed Jun 19, 2015].
Isomorphic controllers and Dynamic Tuning: Invariant fingering over a tuning continuum
Computer Music Journal, 2007
This article introduces the idea of tuning invariance, by which relationships among the intervals of a given scale remain the “same” over a range of tunings. This requires that the frequency differences between intervals that are considered the “same” are “glossed over” to expose underlying similarities. This article shows how tuning invariance can be a musically useful property by enabling (among other things) dynamic tuning, that is, real-time changes to the tuning of all sounded notes as a tuning variable changes along a smooth continuum. On a keyboard that is (1) tuning invariant and (2) equipped with a device capable of controlling one or more continuous parameters (such as a slider or joystick), one can perform novel real-time polyphonic musical effects such as tuning bends and temperament modulations—and even new chord progressions—all within the time-honored framework of tonality. Such novel musical effects are discussed briefly in the section on dynamic tuning, but the bulk of this article deals with the mathematical and perceptual abstractions that are their prerequisite.
Tuning of Musical Notes through Mathematics
2012
Mathematics, an enigma in numbers and calculations, often accompanied by feelings of rejection and disinterest while Music, a flow with emotions, feelings and life. Motivation for investigating the connections between these two apparent opposites’ poles is attempted in this paper. A correlation between Mathematics and Music is shown.. Music theorists sometimes use mathematics to understand music. Mathematics is "the basis of sound" and sounds itself "in its musical aspects... exhibits a remarkable array of number properties", simply because nature itself "is amazingly mathematical". In today’s technology, without mathematics it is difficult to imagine anything feasible. In this paper we have discussed the relation between music and mathematics. How piano keys are interrelated with mathematics, frequencies are correlated and discussed. With the aid of mathematical tools, regression, geometric progression, tuning frequency can be calculated and further re...
A general pitch interval representation: Theory and applications
Journal of New Music Research, 1996
Pitch and pitch-intervals are most often represented-in the western traditioneither by the traditional pitch naming system and the relating pitch-interval names, or as pitch-classes and pitch-class intervals. In this paper we discuss the properties, relationships and limits of these two representations and propose a General Pitch Interval Representation (GPIR) in which the above two constitute specific instances. GPIR can be effectively used in systems that attempt to represent pitch structures of a wide variety of musical styles (from traditional tonal to contemporary atonal) and can easily be extended to other microtonal environments. Special emphasis will be given to the categorisation of intervals according to their frequency of occurrence within a scale. Two applications of the GPIR will be presented: a) in a system that transcribes melodies from an absolute pitch number notation to the traditional staff notation, and b) in a pattern-matching process that attempts to discover repetitions within a melody.
An algorithm for spelling the pitches of any musical scale
Information Sciences, 2019
In this paper, we propose a method for the fundamental task of optimally spelling the pitches of any given musical scale. The input, given as a sequence of pitch-class numbers, can be any randomly compiled subset of the chromatic scale, resulting in either a traditional/known scale or a novel/unknown one. The method consists of generating all potential solutions containing all possible spellings for the pitch classes in a given input sequence, and subjecting them to five filtering stages to find the correct solution. We present an algorithm to accomplish this task, and demonstrate some exemplary outputs. Constructing also a modified version of the algorithm to retrieve and execute all possible input sequences, we also present distributions of various outcomes of the procedure over the input universe to exhibit an overall view of results to be produced by the algorithm, along with some findings obtained by this process.
Models of the perceived distance between pairs of pitch collections are a core component of broader models of the perception of tonality as a whole. Numerous different distance measures have been proposed, including voice-leading, psychoacoustic, and pitch and interval class distances; but, so far, there has been no attempt to bind these different measures into a single mathematical framework, nor to incorporate the uncertain or probabilistic nature of pitch perception (whereby tones with similar frequencies may, or may not, be heard as having the same pitch). To achieve these aims, we embed pitch collections in novel multi-way expectation arrays, and show how metrics between such arrays can model the perceived dissimilarity of the pitch collections they embed. By modeling the uncertainties of human pitch perception, expectation arrays indicate the expected number of tones, ordered pairs of tones, ordered triples of tones and so forth, that are heard as having any given pitch, dyad of pitches, triad of pitches, and so forth. The pitches can be either absolute or relative (in which case the arrays are invariant with respect to transposition). We provide a number of examples that show how the metrics accord well with musical intuition, and suggest some ways in which this work may be developed.
Perceptually Based Theory for World Music Tunings
The larger research program, of which the present report is a part, involves approaching the analysis of world music from the vantage point of perception. With regard to tuning and scales, there have been three main approaches: via abstract numbers, via empirical measurements, and via perceptual responses.