Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
A Structural Theory of Rhythm Notation Based on Tree Representations and Term Rewriting Read chapter Read first chapter

2015 | OriginalPaper | Chapter

Can a Musical Scale Have 14 Generators?

Author : Emmanuel Amiot

Published in: Mathematics and Computation in Music

Publisher: Springer International Publishing

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

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.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter ¿El Caballo Viejo? Latin Genre Recognition with Deep Learning and Spectral Periodicity
next chapter On the Step-Patterns of Generated Scales that are Not Well-Formed
Footnotes
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
This was mentioned to Norman Carey by Mark Wooldridge, see [4], Chap. 3.
 
3
This case was suggested by David Clampitt in a private communication; it also appears in [13].
 
4
Sloane’s integer sequence A000010.
 
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
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
Sloane’s sequence A005277 in his online encyclopedia of integer sequences [15]. For the whole sequence including odd numbers, see A007617.
 
8
Some degree of generalization is possible, see [6] for instance, but results merely in translations of the set.
 
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
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
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
This could also be proved directly from \(\varphi (D)=D\).
 
13
It is well known that affine transformations permute interval vectors.
 
14
Because \(\varphi \) is one to one.
 
15
This is the group generated by \(A-A\), in all generality, cf. [12], 7.26.
 
16
The different generators are the k / b where \(0<k<b\) is coprime with b.
 
17
Such geometric sequences occur in Auto-Similar Melodies [2], like the famous initial motive in Beethoven’s Fifth Symphony, autosimilar under ratio 3.
 
Literature
1.
2.
3.
Callender, C.: Sturmian canons. In: Yust, J., Wild, J., Burgoyne, J.A. (eds.) MCM 2013. LNCS, vol. 7937, pp. 64–75. Springer, Heidelberg (2013) CrossRef
4.
Carey, N.: Distribution Modulo 1 and musical scales. Ph.D. thesis, University of Rochester (1998). Available online
5.
Carey, N., Clampitt, D.: Aspects of well formed scales. Music Theory Spectr. 11(2), 187–206 (1989)CrossRef
6.
Clough, J., Douthett, J.: Maximally even sets. J. Music Theory 35, 93–173 (1991)CrossRef
7.
Clough, J., Myerson, G.: Variety and multiplicity in diatonic systems. J. Music Theory 29, 249–270 (1985)CrossRef
8.
9.
Carey, N.: Lambda words: a class of rich words defined over an infinite alphabet. J. Integer Seq. 16(3), 13.3.4 (2013)MathSciNetMATH
10.
Lewin, D.: Re: intervalic relations between two collections of notes. J. Music Theory 3, 298–301 (1959)
11.
Lewin, D.: Generalized Musical Intervals and Transformations. Yale University Press, New Haven (1987)
12.
13.
Quinn, I.: A unified theory of chord quality in chromatic universes. Ph.D. Dissertation, Eastman School of Music (2004)
14.
Rahn, D.: Basic Atonal Theory. Longman, New York (1980)
15.
Metadata
Title
Can a Musical Scale Have 14 Generators?
Author
Emmanuel Amiot
Copyright Year
2015
Book
Mathematics and Computation in Music

Print ISBN: 978-3-319-20602-8

Electronic ISBN: 978-3-319-20603-5

Copyright Year: 2015

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

Premium Partner

    Image Credits