How is pitch defined mathematically by frequency?
Given pitch p and frequency f, what is the mathematic relation that defines pitch. In “Mathematics and Music” by David Wright (2009) we are told “pitch is frequency” implying that p = f, but in “A Geometry of Music” we have the equation p = c1 + c2log 2 (f/440). More commonly, p= log 2 f is cited, which is empirical based on the octave doubling of frequency exponentially.
In “The Topos of Music” by Mazzola pitch is never defined but apparently taken to be real and therefore a topology of real numbers. The author states that pitch has always been problematic because of consonance and dissonance intervals, which are represented by frequency ratios. “Here, we simply want to recall that a unified mathematic foundation of music thinking in the paradigm of simple consonant frequency ratios could not survive the differential development of theories in the contrapuntal setting, the psychological foundation of musical relations as introduced by Rene Descartes, and the discovery of physical partials by Marin Mersenne.” While we are not sure if pitch is real or rational, we apparently are quite sure music is a Grothendieck topology. This requires that pitch is a continuous function, but we know that notes are discrete.
In ”Music A Mathematical Offering,” David Benson says that pitch “should at first be thought of as corresponding to frequency of vibration.” He goes on to say that the notion of pitch needs to be modified for a number of reasons. “The first is that most vibrations do not consist of a single frequency, and naming a ‘defining’ frequency can be difficult.” The perceived pitch can represent at frequency not actually present in the wave form, we are told. Yet there seems to be no problem defining the frequency of the fundamental vibration when tuning a guitar string, and we also note that Benson has no problem treating frequency as a continuous function, in spite of the problem that frequency cannot be resolved to a point and is not in fact continuous.
In “The Math Behind the Music,” Leon Harkleroad says “For simplicity, I shall treat the pitch of a note as being determined by the frequency of air-pressure variations. The true state of affairs is more complex. By definition pitch is a subjective sensation in response to a note. Factors other than frequency can influence the pitch we hear. For instance, when the loudness of a note changes, some people may sense a change in pitch as well, even if the frequency remains constant.” Harkleroad goes on to scoff at the idea of a lattice formed by union and intersection operations on the 12-tone set.
In “Music and Mathematics From Pythagoras to Fractals,” by Fauvel, Flood, and Wilson we learn that “Pitch is a more complicated matter. To us today it seems obvious that high-pitch notes are ‘high’ and low-pitch ones are ‘low’. … So it comes as a shock to learn that in classical Greece high-pitch notes weren’t heard as ‘high.’” So apparently now we cannot even tell whether or not pitch is continuously rising like frequency.
Sometimes it seems that almost every mathematician has written a thesis on the mathematic nature of music so it seems that since music is squarely mathematics, it must be true that all mathematics apply to music.
No one seems to notice that the point here is that both p = f and p = log 2 f are true at the same time, which makes a curving-lifting map.
The issue here is not the union of music and mathematics, but rather the point at which music and mathematics intersect. What are the special mathematics of music? It does not seem that anyone in mathematics can follow the path that connects mathematics to music. It cannot be that pitch is the theory of music notes any more than frequency explains how light is a particle.
In fact, the idea that "pitch is frequency" trivializes music, and if there is one thing most would agree on, it is that music is not trivial. What is remarkable is that these mathematic authors show absolutely unlimited restraint in invoking every possible mathematic theorem to use in music. Music, they seem to think, is the same thing as mathematics rather that a particular subset of math.
There is no way to define pitch in a modern way without using Zermelo-Fraenkel theory. I can make this assertion confidentially because set theory is pre-emptive in both music and mathematics. If you cannot define musical sets, then you cannot define music operations. The operations of tonal movement that determine tonality are constructed using basic descriptive set theory (union, intersection, and compliment). There is no way to understand tonality if pitch is frequency, because frequency is one-dimension. Music spaces cannot be frequency spaces, as Euler suggested in his pitch value space.
If I am not correct about this assertion, I should think a mathematician would immediately correct my error.
The American Standard Association defines pitch as follows: "Pitch is that attribute of auditory sensation in terms of which sounds may be ordered on a scale extending from low to high. Pitch depends primarily upon the frequency of the sound stimulus, but it also depends upon the sound pressure and waveform of the stimulus."
Pitch indeed is an attribute of auditory sensation. It belongs to the domain of cognitive science, not of mathematics. Would-be music theorists introduce confusions with related concepts such as "standard pitch", "pitch standard", "note", "pitch class" and the like, often without providing a clear definition of any of these.
The word "pitch", in English and in the musical sense, is said to appear in the 1670's, perhaps ultimately deriving from various words, such as prikke, prik, prick, in various Germanic languages, denoting a point.
As to the terms "low" and "high", they are fully language-dependent. They appeared rather late in English (around 1300, and "low" before "high"), first to denote the loudness of sound. These metaphors were unknown in Latin, for instance, where a high pitched sound was said acutus (acute, i.e. piercing), a low one gravis (i.e. heavy); similarly in ancient Greek (oxys, acute, and barys, heavy, respectively). Such terminologies evidence the fact that no clear difference was made initially between the sensation of loudness and that of higness.
First, I want to say that I really appreciate discussing this with you! Thank you.
You are using archaic concepts! Pitch, it seems is an occult object, as they used to say of gravity. In fact, almost every thing in Wikipedia and on the web uses this idea that music is a theory of pitch; and then don't even bother to define pitch. That is how you get nonsense. Do you think the color of light is merely a sensation and there is no difference between color and brightness?
I think that since the ZF axioms are not intuitive to you, maybe it is possible that the older Peano axioms for numbers make more sense to you. I think no harm comes from the simplification.
See if you agree with this Peano statement:
Each notes has a unique successor but there is one note that is the 0 note that is not the successor of any note. Two distinct notes cannot have the same successor. If music is a set of notes such that 0 is in music and for every note in music, the successor is also in music, then every note is in music.
Makes sense, right? Then music is defined by sets! We have not said anything about tempering or even if there are 12-tones. There are no variable except the fundamental so higher overtones on the string are not allow. The octave of course is the first multiple, but the other overtones simply are not defined but the frets are simple multiples. If you can't find fault with the statement, you old system is dead. Your in check. The Peano axioms are pre-emptive to your old way of reasoning about numbers in music. Look, we can count the piano keys 0, 1, 2, 3, ..... right? And the pitch values like C, C#, D .... are mapped one to one with the piano key numbers. So the piano keys and the notes they play are mathematically the same type of set. That makes the whole system of numbers into an alphabet, and the actual pitch of each note as a real number for the frequency does not matter.
What is important is that only multiples of the fundamental are allowed. The definition of multiple here is 1. Anything times 1 is 1. The octave unequivocally is 1. That means how ever the octave is divided into pieces, the pieces are also multiples. But in music multiplication and addition are the same, so we always say that C plus 1 is C#.
We have the octave as a unit ruler, and we mark the ruler with 12 smaller units. The units can be added up so that 12 + 1 = 1. The number automatically make a circle.
You know pitch is not important because we can represent every thing in the key with those roman numerals that are the same for all pitch. You need to see that guitar tunings are also all the same when it comes to pitch, but each guitar key is different. Open G and Open D are as different as English and French, and clearly distinct tonally but you cannot tell when the tuning changes.
You really think that recognizing higher frequency as qualitatively high is learned?
Terence, you start from the premise that music is an affair of discrete notes, or pitches, or call them what you want. This is true of many musics in the world, among others of piano or guitar music. But it is not true of all musics (a) because some musics, even although based on notes, allow for fluctuating notes and even of notes "crossing" each other -- I mean, where one note conceptually higher than another actually at times is lower; and (b) some recent Western musicians developed a "music of noise", where any idea of notes or pitches is merely absent.
For the music of which you are thinking, one may indeed consider that notes follow each other at equal intervals, and may be represented as a harmonic series (i.e. the series of integers). For this however to be true in the acoustic domain, you have to presuppose equal temperament, i.e. that each interval in the octave is equal to the nth root of 2, where n is the number of intervals in the octave; and in addition you have to consider that successive notes do not form an arithmetic (or harmonic) series, but a geometric one. In other words, you can neglect temperament only by considering the notes merely as distinctive entities, not as mathematical ones: so doing, you would turn from mathematics to semiotics.
You seem to consider a "pitch" to be a "note" with some defined degree or highness or lowness and, more specifically, a note of which the degree of highness or lowness is determined by a pitch standard. I fully agree with you that this kind of pitch is unimportant and that the only thing that really matters is the relative arrangement of notes between themselves. This, in music theory, is usually called "relative pitch" as opposed to "absolute pitch".
I am absolutely certain that recognizing notes as qualitatively higher or lower than others is a cultural matter. Some languages, such as Latin, do not know this terminology. Would you think that, say, 2 (the integer) is "higher" than 1, or merely "larger"? This is but a question of words and of language: there is nothing that makes 2 inherently "higher" than 1. Note that on a keyboard, what is "high" actually is at the same level, but to the right (unless you play on a reversed keyboard; they do exist; there are even pianos with two reversed keyboards, one on which the right is to the left...).
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