The “universe” in which we work can be defined in a multi-step process. Somewhat more complicated, but the ordering that frequency brings is of “low-pitched” sounds have low frequencies, and so with high. Ordering corresponds closely to our intuitive notion of pitch: Sinusoids, and to a large number of sounds, via their fundamental component. We use the measurement of frequency to give an ordering to the set of all For example, although the fundamentalįrequency of the piano’s middle C is 261 Hz, there are also components at higherįrequencies that give “body” to the note. When we refer to the frequency of a sound, we The sinusoidal component with this lowest frequency is called theįundamental of the sound. Related in a particularly simple way: they are all integer multiplies of a baseįrequency. Strings, columns of air, or reeds, the frequencies of the sound’s components are In certain cases of particular interest, such as sounds produced by vibrating For comparison, the notes on a piano range from 27 Hz (low A) to 4.2 It is conventional to describe the range of the human ear asĢ0 Hz - 20,000 Hz (20 kHz), although most people’s hearing falls rapidly after 15 Frequency is measured in cycles per second, or The termįrequency refers to the number of times per second that the oscillation These sine waves are the components of the sound. This function can be broken up into a sum of pure sine waves of differentįrequencies. We will consider a sound to be representedīy a real function of time describing this variation.

TheĪverage ambient air pressure is roughly 100 Newtons of force per square meter ofĪrea, and the variations caused by ordinary conversation are about seven orders Sounds are vibrations in the air: variations in air pressure about a mean. In a case where I would stress or accent (or shout) a word when describing these

Well-defined or circumscribed meaning, it appears (where defined) in bold. Where a term is used in a more restrictive sense than is common, or already has a The symmetry groups of tilings or crystals. Those musical transformations that preserve melodic and harmonic relationsīetween notes, or interchange those relations. Structure-preserving self-mappings of this space, will correspond to precisely Attempting to pull musical patterns into a secondĭimension will reveal that these patterns may be thought of as embedded in aġ2-point space with peculiar connectivity. Musical scale that is closely related to the process of autocorrelation We will see that there is a natural way to describe the internal structure of a Mathematical and musical aspects evolving in parallel. In order toĪddress a readership of wide-ranging backgrounds, I present these ideas with the Of abstract algebra: elementary number theory, groups, and graphs. Intervals, the major scale and the concept of key. It assumes some familiarity with musical ideas: octaves and The purpose of this paper is to examine some unique properties of commonly used “Nothing can be farther from the working musician’s mind thanĬounting, nothing farther from the working mathematician’s mind than singing, and These phenomena areĮxplored for the particularly common case of the twelve-note equally-tempered Structure-preserving transformations of melodies. Ideas from crystallography may be applied to the description of Patterns on string instruments are embedded in a space with an unusual For example, the cyclicĪutocorrelation of a musical scale describes its interval content. Musical patterns may be investigated with the mathematical tools moreĬommonly applied in science and engineering. In addition, I have been working on a MIDI controller that exploits some of the patterns described in this paper.

There is also a Web page with an interactive graphic illustrating the graph H 12 discussed below. I have posted a video lecture covering roughly the first half of this paper. © 1991, Audio Engineering Society and Andrew Duncan. COMBINATORIAL MUSIC THEORY Journal of the Audio Engineering Society, vol.

Classical compositions unusual scales