## In search of universal properties of musical scales

### BibTeX

@MISC{Honingh_insearch,

author = {Aline Honingh and Rens Bod},

title = {In search of universal properties of musical scales},

year = {}

}

### OpenURL

### Abstract

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. 1

### Citations

34 |
The Group-theoretic Description of 12-fold and Microtonal Pitch systems
- Balzano
- 1980
(Show Context)
Citation Context ...t of which have been defined mathematically. Well-known scales like the diatonic scales or the chromatic 12-tone scale are special in the sense that they possess many of these properties (Agmon 1989; =-=Balzano 1980-=-). However, the aforementioned authors also show that none of these properties are required for a set of notes to form a scale, and thus they do not present universal properties and do neither contrib... |

24 |
Maximally Even Sets
- Clough, Douthett
- 1991
(Show Context)
Citation Context ...and other definitions of a scale to determine what then exactly a scale is, and what it is not (Lindley and Turner-Smith 1993). Some properties of scales have been proposed, such as maximal evenness (=-=Clough and Douthett 1991-=-), Myhill’s property (Clough and Myerson 1985), well-formedness (Carey and Clampitt 1989), and cardinality equals variety (Clough and Myerson 1985), most of which have been defined mathematically. Wel... |

21 | Letter to a Musical Friend - Longuet-Higgins, Second |

16 | Reasoning about categories in conceptual spaces
- Gärdenfors, Williams
- 2001
(Show Context)
Citation Context ...ollowing this property. Finally it may be noteworthy that star-convexity is not unique for musical scales, but seems to be a prevalent property in many other areas of human perception, from language (=-=Gärdenfors and Williams 2001-=-) to vision (Jaeger 2009). In this light, the star-convexity of scales may perhaps only be an instantiation of a more general cognitive property for the domain of music. 8 Acknowledgements The authors... |

14 |
Variety and Multiplicity in Diatonic Systems
- Clough, Myerson
- 1985
(Show Context)
Citation Context ...what then exactly a scale is, and what it is not (Lindley and Turner-Smith 1993). Some properties of scales have been proposed, such as maximal evenness (Clough and Douthett 1991), Myhill’s property (=-=Clough and Myerson 1985-=-), well-formedness (Carey and Clampitt 1989), and cardinality equals variety (Clough and Myerson 1985), most of which have been defined mathematically. Well-known scales like the diatonic scales or th... |

9 | R.: Convexity and the Well-formedness of Musical Objects
- Honingh, Bod
- 2005
(Show Context)
Citation Context ...and prehistoric music (Nettl 1956; Sachs 1962), rather common to current musical systems (Burns 1999). In this paper, we will give an in-depth examination of two previously proposed scale properties (=-=Honingh and Bod 2005-=-), convexity and star-convexity, across a wide range of music cultures, and we try to relate these properties to more intuitive features of scales. In contrast to the scale properties listed above, th... |

8 |
The Origin and Well-Formedness of Tonal Pitch Structures
- Honingh
- 2006
(Show Context)
Citation Context ...s (of the set) lying on the line segment from x0 to any point in the set are contained in the set. We have shown elsewhere that under a basis transformation of the space, a convex set remains convex (=-=Honingh 2006-=-). 4 Convex just intonation scales While we previously demonstrated that the major and minor diatonic scales, and a small selection of other 5-limit JI scales, are (star-)convex (Honingh and Bod 2005)... |

8 | Ideen zu einer Lehre von den Tonvorstellungen - Riemann - 1914 |

6 |
Aspects of well-formed scales
- Carey, Clampitt
- 1989
(Show Context)
Citation Context ...s not (Lindley and Turner-Smith 1993). Some properties of scales have been proposed, such as maximal evenness (Clough and Douthett 1991), Myhill’s property (Clough and Myerson 1985), well-formedness (=-=Carey and Clampitt 1989-=-), and cardinality equals variety (Clough and Myerson 1985), most of which have been defined mathematically. Well-known scales like the diatonic scales or the chromatic 12-tone scale are special in th... |

6 |
Tentamen novae theoriae musicae
- Euler
(Show Context)
Citation Context ...aving obtained the octave division of n equal intervals, from each interval the most consonant ratio is chosen, according to Euler’s Gradus function 6 (which calculates the simplicity of an interval)(=-=Euler 1739-=-). percentage of (star−)convex scales 100 80 60 40 20 0 0 10 20 30 40 number of notes in the scale star−convex convex Figure 7: Using rule 1 (see text for details), the percentages of convex and star-... |

3 |
A mathematical model of the diatonic system
- Agmon
- 1989
(Show Context)
Citation Context ...n 1985), most of which have been defined mathematically. Well-known scales like the diatonic scales or the chromatic 12-tone scale are special in the sense that they possess many of these properties (=-=Agmon 1989-=-; Balzano 1980). However, the aforementioned authors also show that none of these properties are required for a set of notes to form a scale, and thus they do not present universal properties and do n... |

3 |
Facing the music
- Ball
- 2008
(Show Context)
Citation Context ...the octave, with some pitch steps bigger than others. The asymmetry offers clues about a melody’s tonal centre, letting a listener quickly figure out where the tune is in relation to the tonic note (=-=Ball 2008-=-; Browne 1981). Furthermore, the use of discrete pitch relationships, as well as the concept of octave equivalence seem, while not universal in early and prehistoric music (Nettl 1956; Sachs 1962), ra... |

3 | Just Intonation. The Hague: Martinus Nijhoff - Fokker - 1949 |

2 |
Unison Vectors and Periodicity Blocks in the Three-Dimensional (3-5-7-) Harmonic Lattice of Notes
- Fokker
- 1969
(Show Context)
Citation Context ...he Euler lattice, unless they are approximations of a just intonation scale. Fokker’s periodicity blocks form a method by which an equal tempered approximation of a just intonation scale can be made (=-=Fokker 1969-=-). In an Euler-lattice as shown in figure 1 or 2, unison vectors can be found which represent very small ratios, which are known 1 This lattice representation and minor variants of it appear in numero... |

1 |
Tonal implications of the diatonic set. Theory Only 5
- Browne
- 1981
(Show Context)
Citation Context ..., with some pitch steps bigger than others. The asymmetry offers clues about a melody’s tonal centre, letting a listener quickly figure out where the tune is in relation to the tonic note (Ball 2008; =-=Browne 1981-=-). Furthermore, the use of discrete pitch relationships, as well as the concept of octave equivalence seem, while not universal in early and prehistoric music (Nettl 1956; Sachs 1962), rather common t... |

1 | Intervals, scales and tuning - unknown authors - 1999 |

1 |
Pre-instrumental scales
- Ellis
- 1965
(Show Context)
Citation Context ...ed on oral tradition. For a scale to be included in the Scala archive, the scale needs to be notated in either ratios or cents. As ethnomusicological measurements of scales face various difficulties (=-=Ellis 1965-=-; Schneider 2001) this can lead to misinterpretations of the scale. The second problem is that, since no complete definition of a scale exists, and since for the Scala Archive scales have been collect... |

1 |
Natural color categories are convex sets. Key note lecture at the Amsterdam Colloquium 2009
- Jaeger
- 2009
(Show Context)
Citation Context ... noteworthy that star-convexity is not unique for musical scales, but seems to be a prevalent property in many other areas of human perception, from language (Gärdenfors and Williams 2001) to vision (=-=Jaeger 2009-=-). In this light, the star-convexity of scales may perhaps only be an instantiation of a more general cognitive property for the domain of music. 8 Acknowledgements The authors wish to thank two anony... |

1 |
Mathematical Models of Musical Scales: A New Approach. Verlag für systematische Musikwissenschaft
- Lindley, Turner-Smith
- 1993
(Show Context)
Citation Context ... define unambiguously a mode, tonality, or some special linear construction”. It is difficult from this and other definitions of a scale to determine what then exactly a scale is, and what it is not (=-=Lindley and Turner-Smith 1993-=-). Some properties of scales have been proposed, such as maximal evenness (Clough and Douthett 1991), Myhill’s property (Clough and Myerson 1985), well-formedness (Carey and Clampitt 1989), and cardin... |

1 |
Music in Primitive Culture, Chapter 4: Scale and Melody
- Nettl
- 1956
(Show Context)
Citation Context ...o the tonic note (Ball 2008; Browne 1981). Furthermore, the use of discrete pitch relationships, as well as the concept of octave equivalence seem, while not universal in early and prehistoric music (=-=Nettl 1956-=-; Sachs 1962), rather common to current musical systems (Burns 1999). In this paper, we will give an in-depth examination of two previously proposed scale properties (Honingh and Bod 2005), convexity ... |

1 |
The wellsprings of music. The Hague: Martinus Nijhoff
- Sachs
- 1962
(Show Context)
Citation Context ...note (Ball 2008; Browne 1981). Furthermore, the use of discrete pitch relationships, as well as the concept of octave equivalence seem, while not universal in early and prehistoric music (Nettl 1956; =-=Sachs 1962-=-), rather common to current musical systems (Burns 1999). In this paper, we will give an in-depth examination of two previously proposed scale properties (Honingh and Bod 2005), convexity and star-con... |

1 | Archive (2010). http://www.huygens-fokker.org/docs/scales.zip. Downloaded July 2010, version 74 - Scala |

1 | scale: From ”tone measurements” to sonological analysis in ethnomusicology - Sound |