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International Conference on Mathematics and Computation in Music

MCM 2015: Mathematics and Computation in Music pp 349–360Cite as

Can a Musical Scale Have 14 Generators?

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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9110))

Abstract

A finite arithmetic sequence of real numbers has exactly two generators: the sets \(\{a, a+f, a+2f,\dots a+ (n-1)f = b\}\) and \(\{b, b-f, b-2f, \dots , b-(n-1)f = a\}\) are identical. A different situation exists when dealing with arithmetic sequences modulo some integer c. The question arises in music theory, where a substantial part of scale theory is devoted to generated scales, i.e. arithmetic sequences modulo the octave. It is easy to construct scales with an arbitrary large number of generators. We prove in this paper that this number must be a totient number, and a complete classification is given. In other words, starting from musical scale theory, we answer the mathematical question of how many different arithmetic sequences in a cyclic group share the same support set. Extensions and generalizations to arithmetic sequences of real numbers modulo 1, with rational or irrational generators and infinite sequences (like Pythagorean scales), are also provided.

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Notes

  1. 1.

    I am indebted to a reviewer for reminding me of Mazzola’s ‘circle chords’ which provide ‘a generative fundament for basic chords in harmony’, cf. [12] pp. 514 for a reference in English.

  2. 2.

    This was mentioned to Norman Carey by Mark Wooldridge, see [4], Chap. 3.

  3. 3.

    This case was suggested by David Clampitt in a private communication; it also appears in [13].

  4. 4.

    Sloane’s integer sequence A000010.

  5. 5.

    These sets were introduced by Clough and Myerson [7], they can be seen as scales where the elements are as evenly spaced as possible on a number of given sites and comprise the diatonic, whole-tone, pentatonic and octatonic scales among others. See also [1, 6, 8].

  6. 6.

    A reviewer sums up nicely the two ‘plethoric’ cases by identifying them with multiple orbits of affine endomorphisms, i.e. different affine maps generating the same orbit-sets. See also [2] about orbits of affine maps modulo n.

  7. 7.

    Sloane’s sequence A005277 in his online encyclopedia of integer sequences [15]. For the whole sequence including odd numbers, see A007617.

  8. 8.

    Some degree of generalization is possible, see [6] for instance, but results merely in translations of the set.

  9. 9.

    I have to stress the musical interest of such bizarre objects, actively researched both in the domain of word/scale theory [4, 5, 9] and aperiodic rhythms [3] and providing compositional material.

  10. 10.

    A famous concept in music theory, see for instance [14]. I had initially found an alternative proof based on majorizations of the Fourier Transform, omitted here in favour of a shorter one.

  11. 11.

    The minimum value of \({{\varvec{IV}}}\) and the number of its repeated occurrences could be computed – it is 0 for \(d<c/2\) – but are irrelevant to the discussion.

  12. 12.

    This could also be proved directly from \(\varphi (D)=D\).

  13. 13.

    It is well known that affine transformations permute interval vectors.

  14. 14.

    Because \(\varphi \) is one to one.

  15. 15.

    This is the group generated by \(A-A\), in all generality, cf. [12], 7.26.

  16. 16.

    The different generators are the k / b where \(0<k<b\) is coprime with b.

  17. 17.

    Such geometric sequences occur in Auto-Similar Melodies [2], like the famous initial motive in Beethoven’s Fifth Symphony, autosimilar under ratio 3.

References

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Acknowledgments

I thank David Clampitt for fruitful discussions on the subject, and Ian Quinn whose ground-breaking work edged me on to explore the subject in depth, and my astute and very helpful anonymous reviewers.

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Authors and Affiliations

  1. Classes Préparatoire aux Grandes Ecoles, Perpignan, France

    Emmanuel Amiot

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  1. Emmanuel Amiot
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Correspondence to Emmanuel Amiot .

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Editors and Affiliations

  1. De Montfort University, Leicester, United Kingdom

    Tom Collins

  2. Aalborg University, Aalborg, Denmark

    David Meredith

  3. Utrecht University, Utrecht, The Netherlands

    Anja Volk

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Amiot, E. (2015). Can a Musical Scale Have 14 Generators?. In: Collins, T., Meredith, D., Volk, A. (eds) Mathematics and Computation in Music. MCM 2015. Lecture Notes in Computer Science(), vol 9110. Springer, Cham. https://doi.org/10.1007/978-3-319-20603-5_35

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  • DOI: https://doi.org/10.1007/978-3-319-20603-5_35

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-20602-8

  • Online ISBN: 978-3-319-20603-5

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