We like to think that music is mathematical but according to Wikipedia there is no axiomatic basis for music. How are musical sets constructed using basic set theoretic tools of union, intersection, and complimentation?
I propose using sequences in tablature music for guitar to study how polyphonic objects are constructed by adding point-wise limits to the continuous function of pitch. Tablature music is a rich algebraic language that has substantial archival, educational, cultural, and economic significance but no mathematic theory of tablature exists.
I have mastered reading and writing tablature. Anyone familiar with my study of multiple parathyroid tumors in the Journal of Theoretic Biology back in 1985 can see I have substantial training in mathematics.
I need a mathematician who knows model theory or algebraic topology to review my work. Please see the attached manuscript if you are interested in making a modern mathematic model for music theory.
The basic problem: Are music sets constructible?
Having both a PhD in engineering and recently retired after about 35 years as a professor of music, in which I am also professionally trained, I do not believe there is any theory of music based on axioms. However, I can make a few comments that you may find help is understanding why.
First of all, there is no general theory of music to "axiomatize". The notion that music is mathematical goes back at least as far as Pythagoras (ca. 2500 years ago), though music is clearly much older (I suspect it may predate speech). The Pythagoreans are usually associated with the idea of the "music of the spheres", meaning that important aspects of music can be described mathematically (such as the harmonic series upon which much "classical", i.e., tonal, music theory is based). Most so-called popular music is based on a combination of musical tonalities and modalities.
But there are many types of music. Schoenberg's system for music composition using 12 notes (so-called "atonality) has an enormous mathematical literature based largely in set theory, but not axioms in the Euclidean sense. Once again, such theory is descriptive rather than generative.
Music tablature is typically designed around the playing characteristics of particular musical instruments, like lutes and drums. Common practice music notation (CPN) is far more general in its descriptive power than tablature, since it notates sounds rather than actions to be taken by a performer on a particular instrument. Bach, for instance, often wrote music for no particular instrument (such as his "Inventions"). Bach's music could be written in tablature for a particular instrument, but CPN allows it to be readily adapted to any instrument, from a lute to a sound synthesizer.
Regarding Moore's answer, we agree there is no current axiomatic definition, but you have not proved that there is no consistent theory of music.
I wonder if we could begin with Euler's tonnetz or tone net described in 1739 and still accepted, because the tone net implies that there is a defined set.
The question here is whether the torus is correct, or if it is actually a sphere.
I don't necessarily agree with F. Richard's answer here (though it is well said). One idea that could possibly be regarded as axiomatic is that of pitch classes or frequencies. The fact that a single note or melody can be played on a variety of musical instruments (strings, pipes etc) shows that it is a class. We could not for eg play a C note "in and for itself" because playing it on a violin, say, does not represent the entire set. (Recall Lockes' "All of them together and none of them at once"). Another concept that most certainly can be regarded as axiomatic, at least as strong as anything else in logic or geometry, is the Ratio. As F. Richard alluded to, when Pythagoras discovered that ratios of stretched strings correspond to the integers, he is said to have exclaimed that :everything is number". But a more accurate translation might be logos, which translates to Latin as ratio. Now this is right at the very beginnings of science but is completely ignored in the philosophy of science, maths and physics. But the fact is that the same laws of wave motions apply to light and even matter. And despite what many have since claimed, the invariance of the ratio is tacitly assumed all the way throughout science. Denying this fact leads to a reductio ad absurdum. So in a sense the "music of the spheres" lives on in another way.
There can be no objection to assuming there is a null set and that sets are constructed in the order null set, {null set}, {null set {null set]?
Unless I am mistaken, this assumption leads to S03 symmetry and disproves Euler's assumption that pitch value and musical key make an ordered pair (p, k). It must be {pitch, {pitch, key]}. The pitch and the key have the same fundamental. The value of the null set is 0 if the fundamental is abstract, and 1 if specific. The pair (0, 1) therefore makes a new direction and an integer unit.
The abstract measure of pitch has nothing to do with measures of frequency on the real line. Intonation fields reduced to pitch appear to be merely points and intervals falling on a line, making the measure of pitch seem trivial as if guitar does not depend on the algebraic source language. Constructible sentences in tablature music for guitar show how the guitar tuning ring signature is invertible and can be recovered up to intonation isomorphism using the guitar as typed alphabet finite state machine. The guitar model recognizes when the algebraic language of tablature satisfies the abstract inner measure of the guitar tuning theory.
Suppose I give you tablature but do not tell you what the guitar tuning is to interpret the numbers. Then all you need to go is try every guitar tuning in order until eventually you come to the tuning that is correct to read the tab. Clearly, the Zermelo-Frankel axioms apply, which can all be represented if we assume the axiom of standard construction.
There is no such thing as THE theory of music, because music is not a universal, and very little can be considered universal in the musics of the world.
Let me first correct some of the statements of previous answers:
1) The Pythagoreans did not really describe music (it the sense in which we understand it) as mathematical, they built a theory of ratios of whole numbers, which they called "music".
2) The Western notion that music could be based on harmonic numbers has more to do with physics (and physiology) than with mathematics: the ear functions as a Fourier analyzer and tells something about physical sounds; the mathematical theory of these only follows.
3) The idea that music could be founded on harmonics (in acoustics or in mathematics) really concerns a music founded on (more or less) periodic sounds, as is the case with Western music, but not with many others.
4) Staff notation is NOT and never was intended to notate sounds -- or at least not to notate them as acoustic entities. It notates distinctive units in what must be considered a semiotic system. These are categories (categories of pitch, for instance), but these are never defined with any level of (quantified) accuracy (neither absolute nor relative pitch, for instance). But not all musics of the world are semiotic systems of this kind, and not all of them consider pitch a distinctive unit.
Specific types of theories about specific kinds of music could be considered based on axioms. Schenkerian theory, for instance, is based on the (quasi) axiom of the tonal triad. [For some basic idea about this, see https://en.wikipedia.org/wiki/Klang_(music)].
Euler's Tonnetz is a possible starting point for a music based on twelve notes in the octave (twelve pitch classes), with fifths and thirds considered the main intervals. It would not serve for musics based on another number of pitch classes, or other main intervals, nor for musics based on concepts of fluctuating pitches or intervals, nor for musics where pitch may not be considered an important characteristic (think of musics of percussion, or of noises).
Guitar tablatures appear IMO based on several axioms that cannot be accepted for music at large:
- Music consists of "pitches" (periodic sounds).
- There are twelve pitches in the octave.
- The intervals (distances between pitches) can be described by numeric ratios. (That this is not true is evidenced by the fact that equal temperament, in which guitars usually are tuned, produce irrational intervals).
Etc.
Nicolas' answer coverts most of the broad picture; To return to your very first query, construction of sets of musical objects using the basic operators was (ab)used by Xenakis and theorised (not very neatly) by Allen Forte, you may look up numerous bibliographical references on these.
Set theory is only a start. Modern analysis and composition makes use of group theroy, starting with MOrris in the 80's and with D. Lewin's Generalized Interval Systems.
More high-level mathematical tools are currently in use: category theory is prominent in Mazzola's huge Topos of Music which purports to give a theoretical framework for all music — a rather tall order, but at least some of the stuff does have explanatory and generative power.
For a bird's eye view on current mathematical research in music look up the Journal of Mathematics and Music. Sometimes Perspectives of New Music and Journal of Music Theory too.
This discussion is all meta-mathematics.
The Zermelo-Frankel axioms are just theorems and they can be proved true or false in music.
For example, the axiom of extension applies because if there are two distinct intervals, then there is a third interval contained in one and not the other.
It is hard to argue that the standard model axiom does not apply. For example, we have the musical key defined as a Ferge abstraction {pitch value set| k is the tone center|. This creates a base-point topology in which transposition is induced as a second operation (since pitch can only rise or fall).
This show the musical key is a subset of the pitch values because it is defined by adding a point-wise limit to the continuous function of pitch. If the musical key is the power set or even the product set of the musical scale, then music is not countable.
But clearly the guitar is finite and the value of every fret and string number is less than the pitch value. This means the Lowenheim-Skolem theorem proves guitar is a Turing machine! It comes under Godel's completeness theorem, not the incompleteness. Music is finite: a continuum minus finitely many points. A guitar is a 6-tupple.
We have then the musical key or the guitar model is an element of the pitch value set, and both pitch value set and the musical key set (or guitar model set) are elements of the music structure. I have not required either tempering or even a 12-tone set as an assumption.
The tonnetz is a 2-dimensional projection. S1 (circle) classifies Z/Z12, not the other way (that is Z/Z12 works but it is just an approximation of the sphere).
I've attached a set of drawing and a legend which go with the original manuscript I posted.
My question is clear and answerable: Are the basic axioms of set theory true or false? (An alternative is the sequences in music are not well founded formulas or contain independent variable not bound or closed. But then music is not mathematical. It does not obey the Heine-Borel Theorem. Then there is no way to learn guitar.)
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Terence B Allen
Here's a comparison between piano music and tablature. There is no mathematic function that relates piano music to tabs but there is a mathematic algorithm (indicated by the vector in the tabs for Double Drop D Tuning).
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Nicolas Meeus
Sorbonne Université
I am a professional musicologist and music theorist, and I must confess that what you people write is Chinese to me. Music can be notated in staff notation, or in various kinds of tablature, or in various types of alphanumeric notations. Passing from the one to the other (which is called "transcription") can now be automated, especially with the help of MusicXML and MEI (Music Encoding Initiative).
BUT this all is valid only provided that the music can be notated at all. And the transcriptions exclusively concern the notated aspects of the music. Notation is utterly unable to notate everything. What it notates is called "notes", which represent a combination of pitch and duration. But, because notations (like music itself) are semiotic systems, the notated pitches and durations are but abstract categories escaping any attempt at quantification. Notes are distinctive elements of a semiotic system. And, in addition, not every music can be notated, which probably means that not every music is semiotic.
The so called pitch classes of musical set theory are abstractions. If they are considered classes properly speaking (which I doubt they are), then the members of these classes must themselves be considered abstractions, and so also are the intervals between them. And none of this can be evaluated in terms of "true/false". Musical set theory is not a set theory in the sense of Zermelo-Frankel.
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Terence B Allen
If music is not set theory then which of the axioms does not a apply? The axioms are just theorems so if they are false, that is provable. To say the axioms do not apply is to claim music is a unique invention of man beyond mathematics. Nonsense!
Computers that make tablature are purely algebraic and do not attach a true or false value to sequences. That is why I say, tablature is a game played with a Turing machine. The tab I showed for the piano music is correct only because I have learned the language of the tuning. If you think, I have not found the best tab, can you find a better one?
Here a statement true in music but not guitar: E2, G#2, and B2 are a true statement of an E major chord. But if you try to play the chord in standard tuning the statement is false because E2 and G#2 cannot be sounded on the first string at the same time. In fact there are 5 notes on the first string that are found only once on guitar so that makes 25 intervals that are impossible to sound on guitar. The interval E2-G2# is impossible in standard tuning. The means the music key on guitar is a subset of the musical key that does not contain every element of the musical key.
That is why Open G Key of C is a different language than the Key of C in Double Drop D, with different tonality. Since the guitar key is internal to guitar, it is foreign if you have not learned the tuning and therefore makes no sense.
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Nicolas Meeus
Sorbonne Université
@Terence B Allen, you write: "To say the axioms do not apply is to claim music is a unique invention of man beyond mathematics.Nonsense!"
But of course, this is exactly what I claim, that music is a unique invention of man!!! If you cannot conceive this, then we are not speaking of the same thing and even more, I am afraid, we don't speak the same language.
You write: "E2, G#2, and B2 are a true statement of an E major chord." If you want, but in what sense do these notes, and the notion of an E major chord, pertain to music at large? You are speaking here of elements that in themselves belong to an elaborate theory, resulting from layers and layers of "axioms" or, better, of mere hypotheses.
The terms "E2", "G#2" and "B2" rest on the hypothesis that music is made of pitches, themselves corresponding to frequencies of periodic vibrations, and implicitly that these pitches differ by a number of unit intervals, semitones, etc. etc. Now this may be true for some particular types of Western common practice music, but certainly not for music, nor even for Western music, at large.
"But if you try to play the chord in standard tuning the statement is false because E2 and G#2 cannot be sounded on the first string at the same time." This is not a theoretical, nor a mathematical limitation, but merely one resulting from the very concrete limitations of the way in which guitars are tuned today. Now if you take a guitar to represent a mathematical truth, then I can but repeat that we may not speak the same language.
"That is why Open G Key of C is a different language than the Key of C in Double Drop D, with different tonality." These are words without any meaning in any music theory that I know. I don't understand "Open G Key of C", nor "the Key of C in Double Drop D", nor "with different tonality". To me, "Key" and "Tonality" are words with specific meanings, specified in many dictionaries, encyclopedias, and the like, and which at first sight do not allow the kind of usage you make of them here. As to "Double Drop D", I can figure out that it may belong to guitarits' slang, and I even think I migth understand what it means; but if my understanding is correct, then the expression certainly does not concern "tonality".
If you are a mathematician, you should know that one cannot invent the meaning of words as one uses them: one can only use them in their shared meaning. I don't think you should ask question about what music theory is if you cannot agree with the common meaning of "music" on one hand, of "theory" on the other.
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Terence B Allen
Let's start over.
Do you agree that the musical scale is a well-ordering. That is, is there a first and last element and a place for every note in between?
Then ZF axioms are true>
Do you accept that pitch values are continous? Then you accept the axiom of choice.
If music is computable then ZF!
Walter Everett wrote 2 volumes about the Beatles and never mentioned the guitar tuning they used once.
I told Everett that Lucy in the Sky is written in Open D and the Key has to be E for the bar cords to fall on the neck of the guitar in the right way.
I say this because I tried the lick in a dozen tunings and Open D Key E is the magic setting that is amazing. Its like you realize what Lennon found when he composed the tune.
Now the nature of proofs in tablature is you never know that you have the best tablature, but only that you have the best known version in a collection.
Maybe there is a better tuning and key for the tune, but I bet you can't find one.
I've sketch the tab here with the intro and showing how the rest is all power chords. If you follow this logic to the chorus you see how the chord are open, fret 5, fret 7 where you sing "Lucy in the Sky with Diamonds." That's the part that proves the Key E Open D and not the more obvious Open E Key of E. Now if you have a better version please let me know!
(By the way, since guitar tunings are well-ordered, it cannot be true that we won't eventually find the guitar tuning that Lennon used. It certainly isn't standard tuning.)
I had some You-tube lectures on guitar tunings used by the Beatles. One of my favorites is "Across the Universe" in Honolulu Blue tuning! Unmistakable tonality!
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Nicolas Meeus
Sorbonne Université
Terence, I think that we should better stop here. You write for instance "That's the part that proves the Key E Open D and not the more obvious Open E Key of E." This is a concatenation of words of which I can make no sense. I merely cannot understand what the words "key" and "open" are supposed to mean in such a phrase.
I agree with what you want about "the musical scale"; but even the concept of scale is not a valid axiom for music theory at large: there are many musics in the world that do not know the concept of scale, nor that of pitch.
Of course, it is possible to deduce the tuning of a guitar from pieces played on them. But what does this have to do with music theory? It somehow concerns the theory of guitar playing, but that's all...
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Terence B Allen
Thank you for for interest in the subject.
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Paul Sampson
Northumbria University
As you will probably know, 17 note scales have been proposed as an optimal tuning for arabic lutes. If the guitar - and presumably also a lute - is a Turing Machine, I'd like to see it used to compute pi to 17 decimal places.
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Rick Ballan
The University of Sydney
Wikipedia is very much mistaken. The concept of ratio is probably the most axiomatic in all science. It goes back to the very beginnings of science with Pythagoras' discovery that the numbers correspond to musical harmonies, which to the Greeks was synonymous with nature. Further, the connection with string lengths might very well have produced the idea that time could be represented by lengths in space. We find a common thread via Keplers' harmony of the world. Indeed, as I've shown the concept of ratio still exists all throughout science but has been concealed or 'forgotten'. The sine wave for eg could not have been written down if it was not assumed a priori that ratios = numbers of cycles and that a wave supplies its own measure of space and time. So the statement is blatantly untrue.
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Terence B Allen
To Paul Sampson: The domain of the augmented frame includes only integers, so calculating a real number is not possible. The Turing machine output is a subset of the model.
We can use R, Q, and Z as approximations of Borel sets. The guitar set is the collection of all intervals, the tuning is the relation, the tablature sequences are the domain, and the map between the guitar position value set and the pitch value set define a modal S4 logic. Real and rational analysis is compelling but not simple and not topologically correct.
To Rick Ballan: Are we talking about guitar as a subset of music?
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Nicolas Meeus
Sorbonne Université
@Paul Sampson: so far as I can tell, the 17-note (Pythagorean) scale has been proposed by Safi al-Din in the 13th or early 14th century as a possible Arabic scale. Safi al-Din believed that Pythagorean enharmonies could stand for the non-diatonic (Zalzalian) degrees of the Arabic scale. Pythagorean enharmonies, however, are but a comma apart, while Zalzalian intervals (such as the neutral third) involve differences closer to a quarter tone. Safi al-Din himself admitted his mistake, but the idea of a 17-note Arabic scale remained among those who do not bother to know better.
@Rick Ballan: Pythagoras never discovered that "the numbers correspond to musical harmonies", he merely discovered the properties of ratios of integers, and called them "harmonic" (in the mathematical sense). The association of this "harmony" with music came only later, as a way of practically illustrating the properties of these ratios. The Greek conception was not that music was synonymous with nature, but that nature was ruled by ratios of integers. Music properly speaking has little to do in all this.
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Paul Sampson
Northumbria University
@NicolasMeeus: I drew attention to the 17 note scale (the particular one proposed by Safi al-Din - or Safi al-Din al-Urmawi, if you wish to be more formal - is only one of the several 17 note models invented to cater for the [non]-problem of the Pythagorean comma) only to provide a 'hook' for the calculation of pi to 17 decimal places by a guitar. I might just as well have namedropped Harry Partch to request a banjo to calculate pi to 43 places.
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Terence B Allen
Is it true that all n-tonal sets fit in an octave? I mean, when you have 17-tones you are still dividing the octaves into equal units according to the 17th root of 2?
Because if that is the case, you have the same topology with a different sub filter.
Since the guitar strings are all the same length, how long the strings are does not affect which musical language is invoked by the tuning. That shows the intonation level is not the determinant of the tonality. The tuning is. Each tuning has its own set of musical keys because the guitar key (musical key on the guitar) is inside the tuning. The guitar keys are constant when the capo is used.
That means Key of C in one tuning does not sound the same as the Key of C in another tuning. The observer can tell that 2 tunings are not the same sound but can't tell when the tuning changes and can't say why the tunings are different. Unless you experiment with tunings, you think that playing the correct pitch values is all that is required. This leads to a fallacy: Write the guitar music by pitch and then proof the music by pitch and the result will always seem correct. The correct way to learn guitar is to write the tab by the tuning rules and then proof by pitch.
This experiment shows what I mean: Suppose you move the capo up the guitar neck 1 fret at a time and at the same time transpose the guitar key down 1 step to compensate. The result is that the timbre gets higher and higher as the strings shorten, and the key remains constant, but at each capo position a different tonality results, as compared to playing the same guitar key and allowing the musical key to rise with capo.
On the piano transposition does not make a new key but on guitar it does. You might think that the guitar is different than piano (and it is) but whatever is true in music must be true for guitar, and vice versa. That is, guitar music and music are similar and equal in every harmonic ratio but there is never a way that guitar music and music can be superimposed on one another and match point for point.
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Paul Sampson
Northumbria University
@Terence B Allen: It is not necessarily true that a p-note scale (e.g. p=17) must divide the octave into equal units according to the p-th root of 2. Had that been the case, I would have been unable to claim that Safi al-Din Urmawi's scale (which most certainly did not employ the 17th root of 2) was/is only one of several. You must surely know that equal-tempered scales are a relatively recent invention and that earlier dodecaphonic (and of course heptatonic) scales were similarly uneven.
There's no particular limit on how many ways you might come up with a p-note scale, for any p, even assuming a frequency doubling at the end of it. Thus obviously a seven note C-Major run from C to B in an ancient Pythagorean tuning will sound different from a run in the modern equal tempered scale. As for what prehistoric scale runs might have sounded like, by definition, we have little idea. That it was still music is, however, not in doubt (the two epistemological deficiencies being entirely different categories of unknown-ness).
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Terence B Allen
I disagree that equal-tempered scales are modern. The guitar is tempered. Any instrument with equal string length using the 12th root of 2 is tempered. You may be thinking of the tempered keyboard, but that is at least 300 years old. The tempered keyboard is what made Euler think of the tonnetz. The method of temperament using small incrementals lead to the nonsense that pitch values are "integerized" when in fact they are integers tempered or not.
If you don't fit the scale into an ocatve unit, then do you agree the the notes are simple multiples of the fundamental frequency?
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Paul Sampson
Northumbria University
300 years is modern, compared to the hundreds of thousands of years that our species has likely been 'making music' in whatever arbitrary sense of the word. Even the thousands of years of history where the concept of a (frequency doubling) 'octave' (itself a misleading historical accident of nomenclature - did users of oriental pentatonic scales call a frequency doubling span a hexave?) has been employed.
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Terence B Allen
The mathematic principle that musical systems are defined completely as simple multiples of the fundamental is Pythagorean.
The music scale is constructed in the Pythagorean method by placing the 12th fret at the exact mid-point of the string.
When the strings are equal in length then there is no need to temper the scale which is the same for each string. (In practice the string diameter requires slight adjustment of effective string length.)
The guitar and older predecessors always sound in tune with every key. The problem of temperament results on the keyboard when strings are different in length. The use of incrementals and fractions approximates music but only integers are allowed in music sequences (by definition).
At any rate, concerning the structure of music theory (all theories are also geometric structures), tempering has no relevance. When we write music, we don't care about tempering, we care about harmony.
In music we are concerned with sequences of integers. That's all there is, an alphabet of notes. Pitch can be eliminated but the alphabet remains. We move music up or down in pitch, the structure is the same. Theory does not depend on pitch. Music is abstracted from pitch.
I doubt there was ever a time in the historic period that octave was not recognized as a unit. The Greeks knew about octave doubling when they defined harmonic numbers as multiples. I mean, Pythagoras stands at the dawn of history; so we can say that in effect "he" discovered the geometric construction of the square root of 2 and we know that the octave is the same as the square root of 2 (as in p = log 2 f). Isn't that right?
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Nicolas Meeus
Sorbonne Université
" we know that the octave is the same as the square root of 2" ???!!!@@@###???
What I know is that the octave corresponds to a doubling or a halvening in frequency -- i.e. to 2, in either direction (as Alfred Jarry would have said).
In the music theory that I practice, the square root of 2 corresponds to the tritone.
But this is a free world... (I hope).
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Paul Sampson
Northumbria University
It's probably pre-Pythagorean, but as that's the earliest written evidence I suppose we have, we'll allow the guy the credit.
But I can't let you have this 'dawn of history' stuff. Pythagoras is well inside the historical period and I'd hardly call that the dawn of history when we know characters - real ones - by name from at least two thousand years earlier who themselves weren't near any such dawn. And it's still not prehistoric (we have texts). Prehistoric civilisations with city sized social constructions go back another few thousand years. That's a long time for fairly sophisticated (and decidedly not well-tempered) musical activity. If you can build cities, it's reasonably certain you can make a flute, a drum, a guitar, a lyre, a horn, of some form. Not to mention singing of course, likely going back hundreds of thousands of years. But none of these people knew anything about the twelfth root of two. Obviously I have no proof of this, but, well, really?
As for the square root of two, I'm not sure - off the top of my head - that Pythagoras himself, in particular, 'knew' of it. I'm pretty certain that the Ancient Greeks knew that the diagonal of a unit square was not rational, because it's quite easy to prove with only the properties of integers known at the time (cf Euclid and factorisations), but square-rootedness as a 'category'? Hmm.
In any case it's still a long way from the concept of a square root to a more general n-th root and specifically a twelfth root. And of course an octave is not the same as the square root of two (which would be a diminished fifth, or augmented fourth, aka trtone) - I think you must have mistyped something there!
The mid-point of the string is indeed coincident with frequency doubling and it's highly probable that every culture, even prehistoric ones, recognised that there was something pretty especially 'samey' about a note's pitch being doubled, quadrupled, halved, quartered etc.
But that's only my opinion. I just can't imagine that a human male and a human female, six thousand years ago, would not regard singing in 'unison' as singing the 'same' note - though they were often as not likely an octave apart. (But that could be just my failure of imagination).
I'd go further and suggest that the perfect fifth (where you divide your string a third of the way along) was probably also perceived, very early on, as something special. Beyond that, I just don't know. But that's all you really need to get good non-trivial scales (i.e. those with more than two pitch classes [dominant/tonic]) off the ground, since the fifth of the fifth (say, starting from F to C to G) gives you a third kind of note, the fifth of the fifth of the fifth another (introducing D) etc. The next 'fifthing' generates an A - and at this point you have all the notes you need for a pentatonic scale, F, G, A, C, D. That's whole musical cultures right there.
Carrying on another two steps, from A to E, then to B, gets us what we need for our standard occidental heptatonic scale C, D, E, F, G, A, B. Continuing still further, we introduce - by further 'fifthing' - F#, C#, G#, D#, A# - until after eleven 'fifthings' in total you end up almost but not quite where you started, at something very near F (a last fifth jump from that A#~Bb up to F-ish).
(3/2)^12 = 129.746337891, which is 128 (which would be exactly seven doublings of the original F pitch) times 1.01364 - a little sharp of F [that small difference being the well-known Pythagorean Comma].
Finally, sometimes we don't even care about harmony. In the west, plainchant may be concerned with - at most - interval, but it does not fret (sorry) about harmony and is yet most definitely music. Harmony is a relatively recent - and largely western - obsession. But your mathematical theory of music cannot restrict itself to such parochial issues of harmoniousness and will have to accommodate everything.
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Terence B Allen
Wait. Tritone is 6 frets and an octave is 12.
So let me prove the octave is the square root of 2.
Let's make a graph of the octave. So we have 12 steps on the x-axis and 12 steps on the y-axis and then we find the octave point (12, 12).
If we move the y-axis to the right we make a triangle. The sides of the triangle are 12 but so is the hypotenuse. Count 'em. A triangle with 12 steps on all 3 sides, right? That is not Pythagorean.
Now we have at least one right angle (between x and y axes) so therefore we have a triangle with 3 equal sides and 3 90 degree angles. It lies on the surface of a sphere and not a plane, as Euler thought. Since the hypotenuse should be the square root of 2, it clear the octave is the equivalent of the geometric construction of the square root of 12. It takes little to see the 12-root of 2 makes 12 equal units. The Greeks did know this.
From "The Universal Book of Mathematics" by David Darling. "Pythagoras left no writings and virtually nothing is known about him as an individual." Darling goes on to assert the bit about ratios in music which you mention, but the integers win over ratios because they are simple and ratios are not. The 5th is not defined by a ratio, it is defined by counting 7 steps up or 5 down. It is intervals between frequencies we know; what does it mean to divide two frequencies? (unless you are talking about a quotient space, which of course is a set).
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Paul Sampson
Northumbria University
"So let me prove the octave is the square root of 2."
I'm afraid that - at this point - I must give up. Good luck with your axioms.
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2 Recommendations
Nicolas Meeus
Sorbonne Université
"If we move the y-axis to the right we make a triangle. The sides of the triangle are 12 but so is the hypotenuse."
I would have thought the hypothenuse now is 24... But your mathematics (12+12=12) are too advanced for me: I am afraid I too must give up.
(A final point, though: calculating the 12th root of 2 hasn't been done, nor could have been done, before Simon Stevin's continuous fractions at the end of the 16th century (CE!) and the invention of logarithms by Lord Neper in the first half of the 17th.)
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Terence B Allen
I think we should summarize our different viewpoints, especially where we are saying the exact opposite.
I say the harmonic numbers in music are completely mathematical but you seem to say music is beyond mathematics.
For example, the square root of 2 geometry and p = log 2 f are the same thing. Are you rejecting the equation p = log 2 f ? What about p = f?
Are you rejecting guitar music is a subset of music?
Are you rejecting the idea that different guitar tunings are different tonal classes defined by different guitar tuning languages? Are all the Beatles songs in standard tuning?
Do you accept the guitar key is inside the tuning?
Are you rejecting the idea that music theory has a geometric structure with a distance formula?
Clearly, you reject the ZF axioms. Are you also rejecting the Peano axioms? Are you rejecting the principle that only simple multiples of the fundamental are allowed in any harmonic model?.
Are you saying that real numbers or fractions explain music better than integers?
Do you accept Euler's tonnetz and the Z/12Z number system?
Do you think music mathematics are trivial or nontrivial? Is the space connected or disconnected?
These questions are substantial because they concern how we view our music universe. Nothing about music theory as we know it changes under axioms, but our understanding of the musical space affects how we understand higher order polyphonic music where we have the pitch specific-device independent music we hear and the device specific-pitch independent music that must be seen to be understood. Look at two tabs for the same music in different tunings: same notes are written completely differently.
I think this has been a fascinating discussion but I am confused about what your central thesis is. Could we focus now on what is your fundamental principle for harmony, in a nutshell?
I also think is interesting that we have not yet heard from a mathematician with expertise in algebraic topology, descriptive set theory, or finite state machines, who can clarify exactly what is the mathematics profession's understanding of music theory as form of mathematics. MIDI shows music is computable but not how to naturally represent music with numbers. That is the problem: How is music represented according to modern representation theory?
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Nicolas Meeus
Sorbonne Université
Terence, I don't know why I answer your questions anymore... I should perhaps begin stating that I am a professional historian of music theory. My specific domains include theories of modality and tonality, Arabic and Western medieval theories, Schenkerian theory, etc. It is from this knowledge that I will try to answer your questions.
1. "I say the harmonic numbers in music are completely mathematical". What do you call "the harmonic numbers in music"? You seem to believe that any music is made of "harmonic numbers", but this merely is not true.
2. "Are you rejecting the equation p = log 2 f ? What about p = f?" I suppose that your p = pitch and your f = frequency. By "log 2", do you mean a log of base 2? The relation between pitch and frequency is context dependent, so that p=f is only approximately true. As to p = log 2 f (which, if true, probably would be true also in any other logarithmic system), it is said that our perceptions are logarithmic, so that if your p denotes our perception of pitch, then indeed p = log f is approximately true.
3. "Are you rejecting guitar music is a subset of music?" Of course not. Note however that the guitar is a rather recent Western instrument, and that fretting the neck of a lute instrument appears to be an even more recent development. Guitar music therefore is a quite limited subset...
4. "Are you rejecting the idea that music theory has a geometric structure with a distance formula?" I don't understand what you may mean. A theory is a linguistic construct about a phenomenon. There is no such thing as "music theory", there are music theories of various kinds (and musics of various kinds).
5. "Are you rejecting the principle that only simple multiples of the fundamental are allowed in any harmonic model?" I think that you should explain what you understand by "harmonic model". Harmony appeared in the West in the 16th century, and certainly without any consideration of "simple multiples of the fundamental". The fundamental of what? A perfect major triad does not consist in simple multiples (unless you admit 5/4 and 3/2 as simple multiples), and these are not "multiples of its fundamental". Each note in a triad has its own fundamental; if the frequencies of these notes are periodic, then their partials are harmonic. The triad also has a fundamental, but of a different kind from that of each of its notes. If you assume that the term "fundamental" has the same meaning in both cases, then you are mistaken.
6. "Are you saying that real numbers or fractions explain music better than integers?" I think that neither "explain music", but if your statement involves a part of the truth, then yes, I am saying that real numbers and fractions explain music better that integers.
7. "Do you accept Euler's tonnetz". Well, that Euler built something that was later dubbed the Tonnetz is a historical fact. I don't see how one could reject it.
8. "and the Z/12Z number system?" I have nothing against it. I presume it can say something of musics based on 12 degrees in the octave; but many musics of the world are not.
9. "Do you think music mathematics are trivial or nontrivial? Is the space connected or disconnected?" I hardly could answer this one before you define your terms.
10. "Could we focus now on what is your fundamental principle for harmony, in a nutshell?" But I thought you were asking questions about music theory at large! May we now understand that your interest is in Western music of the last five centuries or so? I wrote many papers on the theory of harmony, but none is founded on a "fundamental principle".
11. "How is music represented according to modern representation theory?" I am afraid you assume that there exists something univocally called "music". This merely is not true.
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Paul Sampson
Northumbria University
Apropos of p and logn(F) (base irrelevant), but not - I'm afraid - of Mr Allen's pursuit of a musical GUT, @NicolasMeeus reminded me of a quote by Xenakis in a 1986 conversation between him and Feldman (Morton, not Marty):
- For instance, if you’re playing an ascending glissando on the violin, there will be an acceleration if your finger is uniform in its movement. It’s like a geometric progression of the distance. lf you need one uniform ascending movement of the glissando then the musician has to slow down his movement of the finger. They have to learn that but they don’t teach that at the conservatories. So you know, you’ll never obtain this uniform movement of the glissando with an orchestra.
As the violin is fretless, and attempting glissandi on a guitar is affected by speed bumps, perhaps more germane to the current discussion is the immediately succeeding sentence
- In any case western notation is an approximation, an abstraction of the sound. Fortunate, because that gives the performer the possibility to make something out of it.
An earlier post in this thread by @EmmanuelAmiot (although, sadly, he does not appear to be a fan of Xenakis' theorising) and others cite various works on the mathematics of music from authors grounded in music. Perhaps Mr Allen - as he indicates that he is missing input from a 'proper' mathematician, and that he himself has a mathematical background from biology - might care to consider textbooks on music written by actual mathematicians.
I can recommend David Benson's "Music: A Mathematical Offering" - which holds much maths to chew on (discrete maths for musical scale modelling and partial differential equations for the wave equation) and certainly discusses everything mentioned in this thread. Gareth Loy's "Musimathics: The Mathematical Foundations of Music" which covers much of the same ground in two volumes also looks good. Though I confess to not yet having finished those.
Either should adequately explain why the square root of two is not the 'octave' - a word which I will persist in putting inside scare quotes since its frequency doubling association with its etymologically misleading parochial 'eightness' punches way above its actual weight in the vast landscape of music.
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Terence B Allen
I want to try one more time because I think that this is like the optical illusion that looks like a wine goblet, and your about to see it as two faces.
I agree I am not explaining the square root of 2 very well. But remember I didn't make up the math I am just trying to explain it.
We can agree that we have at concert pitch the set A1 = 55 Hz, A2 = 110, A3 = 220, A4 = 440.
We don't say that A4 is exp 440 or log 440. We have p = f but it is also clear that each higher note doubles the frequency of the note before, and each lower note halves the frequency. So we have a tower of octaves with powers of 2 going up and in effect square roots of 2 going down. Log 2 it turns is counted in 1 and 0, like a binary code.
Now it is not true that any log can replace log 2 because in fact there is only 1 point in log 2 f that is actually true to p = f. The log curve and the line have only 1 point in common and that point is the octave where they intersect. This is true because log 0 = 1 and log 1 = 2, which we treat like 0 and 1. That is, we have only two values: 2 to power 0 is 1 and 2 to power 1 is 2. Those are the only point that are defined and the only points we need.
Now suppose we have a chromatic or harmonic circle (5ths). Then there is a spiral above the circle that is formed if we allow the pitch to continue rising. The circle is the closed form of the spiral.
The way that this happens is described in mathematics like a spiral staircase, where you are walking up the steps and there is always another person above you at the same octave position. This makes the spiral continuous and the octave is the winding number that counts how many time the spiral has made one full revolution.
The octave is precise to a point and that makes the pitch values and the pitch value intervals continuous.
The point is very important because in mathematics frequency is not continuous. This means that if we say " A1 = 55 Hz, A2 = 110, A3 = 220, A4 = 440" is the definition of pitch we have a trivial function and a connected space where arithmetic is just real number arithmetic which includes fractions and integers.
It sounds like "trivial" and "simply connected' are good things but they lead to paradox, while in mathematics "nontrivial" and "disconnected" are good things.
You guys are the ones who can recognize that there are different languages on the guitar. That is why I am saying that "Lucy in the Sky with Diamonds" is in the language of Open D. I know that like I know french when I hear it. You cannot play it in Open G.
Now you cannot tell how a guitar is tuned by listening, but if you use the Key of C in Open G it is absolutely clearly different than Open D. This sounds like nonsense to you because its a foreign language you don't understand. I have been writting tablature for 40 years. I have a library of at least 10, 000 tabs and in many cases I have tablature in a dozen different tuning key combinations. I have transcribed every Joplin rag for guitar, every Beatles tune, and almost the entire "Motion Picture Moods" by Rappe.
I tell you what. Why don't you send me some piano music that you think cannot be played on guitar. Obviously I can't match the velocity of anything beyond Andante and need something reasonably cantabile. I send you the tab. You can see if its a language that way.
Regarding the tonnetz, which means a tone net (like a piece of graph paper ruled by pitch values), what is important is that the alternative to the doughnut is a sphere. A sphere makes sense, torus does not. These are the only possibilities according to Mobius. I think you can see the sphere shrinks down to a point (the fundamental) but Euler's tone net can never reduce to a point. The idea music is trivial, connected, simple is a mathematically incorrect idea based on a powerful illusion. There is no pitch that does not have a state of system attached to it. The system defines the tonality and not the pitch.
Thank you so much form your comments!
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Paul Sampson
Northumbria University
Aha! I believe I finally grasp the cause of your misapprehension.
"So we have a tower of octaves with powers of 2 going up and in effect square roots of 2 going down."
No. What you're doing here is mentally along the right lines, in that you're reversing the process, from up to down. But the required inverse operation is negation. Square rooting would be the inverse/reverse of squaring. But you aren't squaring, you're just doubling. Thus ...
We have a tower of octaves with (positive) powers of 2 going up and negative powers of 2 going down.
Doubling is 1, 2, 4, 8, 16 - which is to say 20, 21, 22, 23, 24 ... and halving is 1, 1/2, 1/4. 1/8 ... which is to say 20, 2-1, 2-2, 2-3, ...
Squaring would be 2, 22, (22)2 = 24 = 16, ((22)2)2 = 28 = 256 ... and the inverse operation, corresponding to the other direction, would indeed be 2, 21/2, (21/2)1/2 = 21/4, ((21/2)1/2)1/2 = 21/8 ...
They're two entirely different series. The frequency of a string's vibration doesn't square with each successive 'octave', it doubles. Likewise the frequency halves (not square roots) with each preceding one. I think that you've made an extra leap and applied the (correct) idea of halving (as the inverse operation) to the power of the two and not just to the two itself. Indeed you can most easily see this with the very example you've provided, Just switch the direction and your concert pitch As go from 440 to 220 to 110 etc. Not from 440 to 20.9762 to 4.58 (which is what you'd get by taking square roots of successive values in the other direction).
"Why don't you send me some piano music that you think cannot be played on guitar."
Have a look at almost anything by Charles Ives.
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Nicolas Meeus
Sorbonne Université
Terence,
The pitch standard needs not be at A4=440 (it rarely is exactly). Let's merely admit that for our purpose the highness or lowness of the sounds (their pitch in one specific sense) is their frequency f. From a purely acoustical point of view, climbing octaves, from any starting point, indeed corresponds to f x 2n, and descending corresponds to f x 21/n, where n is the number of octaves.
Now it is true that in our common parlance we do not say that successive intervals are resulting from powers or roots, but from additions or substractions. This is why cognitivists say that our perception is logarithmic. This is not a mathematical truth, only a psychological or perceptual one. It is probably not exactly true over all the audible range, but true enough to have entered our way of speaking. For instance, we say that an octave is a fifth plus a fourth, not a fifth times a fourth.
That is to say that we may represent any succession of identical intervals as a succession of integers. You now suggest that we use the series of integers to represent the number of octaves, but we might as well use the series to represent semitones. In other words, we may associate the integers with powers of 2, or with powers of 21/12 -- and the octaves, in this case, with 12, 24, 36, etc.
All this truly is equivalent, it is but a matter of convention. The use of log2 for music was first described in 1647, in a letter from Caramuel de Lobkowitz to Athanasius Kircher: this was the first musical logarithm. Sauveur, and later Savard, used common logarithms (log10) for the same purpose, dividing the octave in about 301 units (not a very useful division, but an astonishingly successful one). Logs on base 21/2 were proposed by Gaspard de Prony in the end of the 18th century, with the semitone as unit; and on base 21/1200 (dividing the semitone in 100) by Ellis around 1850. Passing from one system to the other is but a matter of multiplying by a constant.
After this, I can no more follow you: we merely do not have the same vocabulary. I don't understand what you mean when you say that "frequency is not continuous". Nor when you speak of "the language of open D" (it probably refers to a tuning, but in a way of which I am not aware). I know perfectly well what a tablature is, I have studied the history of tablature notation, but I begin to wonder what you call a tablature. I know what Tonnetz means (you may want to know that Euler did not call it that name, he termed it a speculum, a mirror), but I cannot see how it could form a sphere (see this image: https://upload.wikimedia.org/wikipedia/commons/6/67/TonnetzTorus.gif).
And above all, I don't understand your statement that "The system defines the tonality and not the pitch." I don't understand what the words "system", "define", "tonality" and "pitch" mean in such a sentence. Really. Perhaps you could explain...
I am not arguing for the pleasure, Terence, I really don't understand your terms, and I'd like you to translate them in common English (which obviously isn't my native language). If you can't translate your terms, I am afraid we really cannot continue the discussion.
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I. E. Kaporin
Russian Academy of Sciences
the correct claim should be: "octaves ... descending corresponds to f x 2-n
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I. E. Kaporin
Russian Academy of Sciences
Surprisingly, we are still not discussing "pink noise" issues (which seem to be a real basis for understanding)
https://scholar.google.ru/scholar?q=pink+noise+music+Zipf&btnG=&hl=en&as_sdt=0%2C5
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Terence B Allen
OK. Now we are talking math. We have a formula we agree on. You do believe music is math. Your formula for octaves is only partly correct, however, because it is also true that we have f + n in which we use the exponents, not the bases. The octaves go 1, 2, 3 and also 2, 4, 8, 16, at the same time. Remember, bases multiply and exponents add. The fact that multiplication and addition are the same in music is possible since the identity of the octave includes both 0 and 1, that is both addition and multiplication. We have only one operation.
What we have then, according to the agreed upon formula is a log space, and not a tonnetz. Log space is a language structure (see Wikipedia). If we follow your formula then you don't have a tonnetz, you have a sphere.
If you get how the chromatic or harmonic circles are cursves lifted in a spiral so the octave unit is constant across the entire frequency domain, then think about transposition. You are walking up the spiral stair case and there is always someone that is say, 5 steps above you. Then the number 5 is all that is required to transpose on spiral to another. The spiral has to close on a sphere.
Now let's say that you have two spirals, one at zero and another at 5. The spiral at zero is always the same so we know the whole system by 5. So one string is a constant system, always the same but two string depend on the interval between the strings and nothing else.
Now let's say that we have 6 spirals and each spiral has its own prime number like 5 that can only be known by the tuning rule. This makes the 6-way transposition to 6 string, each with their own prime number, into something like a combination lock.
The tuning numbers are prime numbers because they are products (sums) ot themselves and the system fundamental, but all the other numbers are known as multiples (sums) of the system fundamental and the string fundamental. The prime numbers of the tuning act as an encryption key that is secret. The Turing test then is to see if we can write the tablature for a Beatles song if we don't know how the guitar is tuned. The fact that all Beatles songs are Published in Standard Tuning does not mean that the language of the composition. In fact, it may be impossilbe to actually recover the tuning, but I don't accept that.
In fact the guitar tuning is a musical key to the 6th degree since in effect each string has its own musical key. That means the musical key on the guitar, which is called the guitar key, is unique. Every guitar key is unique but every musical key is the same. You are confused that the Key of C in Open G is not the Key of C in Open D because foreign languages never make sense to the naive.
We know in mathematics that the musical key cannot be a universal set. The musical key cannot contain every possible musical key including itself.
So we have say standard tuning (0 5 5 5 4 5). This says the lowest string is zero, the next string is tuned to fret 5, and so on. So if 0 is intonated at E2 we have EADGBE.
Or if its G Major tuning we have (0 5 7 5 4 3). At D2 this comes out as DGDGBD. At E2 the tuning is called Open A Major, but the language is the same. In the same way, Open D and Open E are the same tuning. Of course, this makes no sense if you don't know the tuning. But it does make sense. Perfect sense.
When the tuning changes by the smallest increment., the guitarist must completely re-learn the music, but no learning is required if the guitar intonation level changes.
That is, if I use a capo or detune all the string equally, the language does not change but if a single string is altered then every thing is different.
Guitar music is a collection of different languages that seem to be the same. It is difficult to tell (0 5 7 5 4 3) from (0 7 5 4 3 5) which is Open D Major DADF#AD because there are so many musical statements that can be expressed but lute tuning (0 5 5 5 4 5) is almost useless because it is practically impossible to play a dominant 7 chord.
Each guitar tuning has a set of keys that are unique and usually only maybe 6 keys are useful and the other keys are just made by shifting the good keys a fret or two. The fact that some figure are impossible or difficult to play while other figures are present in redundant forms makes the syntax, the order the notes are played, on guitar important.
I don't agree Pink noise is the issue, the issue here is predicate logic. Tablature has predicate logic because if you write the tablature incorrectly, then a person who knows the tuning can correct the fret numbers without knowing what the music says. I don't see how pink noise relates to language. Is pink noise the same for every guitar tuning?
By the way, David Benson says in his book that he doesn't know how to define pitch. Nothing in his book explains anything about music theory. Certainly, nothing he says is relevant to tunings or scoratura.
You say that music has no truth values (true or false) but if we change a note in a composition, the error is immediately recognized when the music is played. In fact, we could change prehaps up to 50% of notes in a composition and still recognize the music. English has this property too. But in a binary code, a single error destroys all meaning. If music is just pitch then we cannot explain how we know when music is correct.
I wonder if it is possible for you to say what is the definition of a language so then we can see if music languages fit the definition. If I say there are foreign language and you say no there is only English, there must be a way to determine who is right.
I think you should consider this: Since no one knows about guitar languages, you can be the first to tell your colleges about a new discovery. The way for you to discover this is obviously to investigate what I am saying rather that dismiss the possibility there is something about music that remains to be learned. Mathematics of music has not changed since Euler's mistake.
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Terence B Allen
Here Charles Ives in tablature.
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Dear Terence,
Axioms. This is exactly what I am considering in my work. You are welcome to visit my profile to view what I have found. You can skip transcriptions of renaissance works, and so on, and view active research documents such as: The Coma a Constant, Geometric Representations R to g11, The Gamut - a quadrangle representation, etc....
I also have uncovered three interlocking sequences which is causing me problems (because I always try to link back to renaissance documents, and can only partly successfully do so in this case), so I am working on a question.. Help from a mathematician, ideally specializing in geometry would be "god sent".
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Terence B Allen
To Anne Allaire
We are interested in the same things but have a different model.
We have a different way of counting the pitch values and the musical keys.
Like most, you are thinking that pitch values are defined by real numbers and therefore you are using real analysis or fractions to define pitch as simply the mathematics of the real numbers. This gives the topology of real numbers with the Pythagorean metric, also called the absolute value topology. That is, you think there is a pair (pitch, key) = (pitch + x, key + y). What I think is F(pitch, key) = 0. We have a triple (pitch, key, fundamental) not a pair.
This means you are trying to "integerize" pitch values that are already integers because they are letters in an alphabetic that have a vanishing point or zero. It isn't tempering that makes pitch values countable. It is induction on the system fundamental. First we have the null set, then we have 0. Then we have 1 thing (a zero), and 0 < 1, so we have 2. We can substitute 0, 1, 2, 3.. for A, A#, B, ... without making any assumptions. We can also if we like say that B is the ratio of 2/26 in the English alphabet and have a consistent system for analysis (but not a simple theory).
What we need is to understand how the alphabet of pitch values becomes arithmetic of diaphanous polynomials. In music, equations always are first-order and always are real valued functions of (x, y, z) in the form a + b = c. We know this as transposition and as tone movement. For the guitar, we have 6 diaphanous equations that make a highly-determined language because of restriction on the musical key that make new operations.
What is strange here is that tones can move in 3 directions but only 2 are algebraic directions. That is only 2 directions are recognized by the audio observer as a change in pitch. Then there is the 3rd logic movement where pitch and position (pitch and key, say) move in composition in opposite directions. For example, the transposing piano (144 keys) and the change of notes on guitar string at constant pitch (so frets go up and pitch down or vice versa). Or if the capo position goes up and the key is transposed down an equal amount: Tonality changes but pitch is constant. When you change guitar tuning there is no change in pitch but a distinct change in tonality is clear.
You must begin by defining sets to understand how transposition comes about as an operation. That is, pitch goes up and down but then where does transposition come from. The answer is it comes when the tonal center of the musical defines a new set of operations.
You are using this idea in your chart of 49 modal keys. (You are demonstrating an Abelian defined by a multiplication table product rule. First, you choose the musical key which imposes a point-wise restriction on pitch, and then you add a second point in designating the mode.
Now here is the interesting part. Why did you only create 7 modes and not 12. Because you restricted the modes to have only scale values. Presumably the reason you did this is common usage. The odd-note modes do not make sense. No one uses them. That is an example of grammar in music, some modes are more likely correct than others. Modes and keys do not have equal probability.
In the same way,no one uses weird guitar tunings that have useless keys that are not useful. If you consider the musical scale it seems that there is no point in continuing to add point-wise limits on the scale once the musical key is identified. But if we have to deal with the power set or even the product set of the scale, we are hopelessly lost because we have no way to count sets.
Instead of using ratios, sequences are by definition composed of integers. It is the finite nature of music systems that means they can be understood directly under Godel's completeness theorem and the Lowenheim-Skoelem Theorem.
The goal here is to understand how to make a continuous model without using real valued-functions. This puts music at the pre-1900 period of mathematic thought, before Baire showed how union and intersection make a metric space.
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Dear Terence,
This is all very interesting and it will take me some time to internalize and will continue this answer later. For now:
Please note that all my work is based on the values provided by Pythagoras, and that because I am deciphering 16th century music I must remain as close as I can to period theory and geometric understanding of music. Therefore I avoid anything that is modern in terms of music analysis and geometry. I try to use period tools to understand period documents which are still beyond modern musicologists.
Tones do move in two directions, however Modes move in three directions, and in our tempered age we don't understand this relationship.
As far as 7 modes instead of 12; Medieval theorists identified 8 quite clearly, so I am still missing one. This is very important, to be true to this period's music I must remain within what was possible at that time.
I just uploaded to my profile a sequence that is still giving me problems. I invite you to look into it and offer some solution.
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Terence B Allen
My system is based on Pythagoras, too. We are not changing music but they way we understand how to count notes. I appreciate your thoughtful comments.
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David Bruce Smith
New York City College of Technology
I am not sure exactly why one would indicate that music theory can be non-axiomatic. Axioms are propositions that are considered true but unprovable ("obvious"), and are the basic givens for a particular formal system. Thus, in Euclid we have such axioms as
"Things that are equal to the same thing are also equal to each other"
IT is thus possible to construct a set of axioms to describe the constraints within a formal system: 12 tone, and set theory are examples of formal systems that can use axiomatic principles to construct their system. Formal Set Theory in Music is actually a Z-12 cyclic group (for pitch classes) (http://groupprops.subwiki.org/wiki/Cyclic_group:Z12) and thus qualifies for all axioms used to define this type of group.
Since one can define arbitrary axioms and then from that generate a set of rules that defines a system, I would rather state that NONFORMAL musical systems may be non-axiomatic. However I do agree that the set of relationships based on natural harmonic series and associated development of tuning systems would qualify as an axiomatic example.
https://proofwiki.org/wiki/Definition:Group_Axioms
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Terence B Allen
The Z/12Z system is the same as Euler's tone net. But in fact. this makes no sense because the torus has 2 independent generators where only a single initial element is allowed. Z/Z12 is a useful approximation just like fractions but question remains how the integers are induced. If we arbitrary make pitch values equal to pitch then the space is connected in a simple way that is the same as real numbers. In real analysis there is no way that music can make a language-level structure. Mathematic functions have no intellectual value and pitch can only rise and fall, so how is tonality grand?
In fact the integers are classified by the S1 circle in projective space. The only axioms that can make sense are the descriptive set theory axioms. Otherwise music theory does is not consistent.
I asked a question about topology and got a straight aswer: Music is not 2-fold, it is 3-fold. The old theory of music is dead, it just won't lie down,
See: If H is a 1-dimensional simplicial complex with 6 vertices, each pair of vertices being a 1-simplex, is it possible that H has no realization in R2?
https://www.researchgate.net/post/If_H_is_a_1-dimensional_simplicial_complex_with_6_vertices_each_pair_of_vertices_being_a_1-simplex_is_it_possible_that_H_has_no_realization_in_R2#view=571911d4cbd5c2a5bc5329bd
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Rick Ballan
The University of Sydney
This has been an interesting thread. I really do believe that there is a terrible bias in our (western) education system that unconsciously treats the 'hard' sciences such as physics as universal while musical harmony is relegated to a subjective 'art'. For example, we believe that the speed of light in vacuo is a universal constant, just as true in the time of the dinosaurs or in the Amazon jungle than in a lab in the twentieth century. But this same universality is not applied to the fact that the harmonic series applies not only to the motion of stretched strings and air pressure in pipes but is a principle of wave superposition itself. If we play, for instance, a sine wave frequency of 60Hz and another of 90Hz without making any further assumptions as to how these waves are produced then the resultant frequency of the wave will be 30Hz, the greatest common divisor. This will be the fundamental and shows strong resemblance to the tonic in musical harmony. it's a feature that I would propose is just as universal as anything else in science. Yet this phenomenon is called the "missing" fundamental and is regarded as a psychological effect. The problem was that if we change one of the frequencies to say 91Hz instead of 90Hz then experiments still confirmed that we perceived a tone close to the original 30Hz but the GCD between 60Hz and 91Hz is now 1Hz. One possibility was that we were perceiving the difference tone or AM envelope - 91 – 60 = 29Hz is after all close to 30Hz. But experiment soon confirmed that this was not the correct pitch. Hence scientists concluded that the mind makes 'best guesses' to the closest harmonics. But I have found that the problem resided in the definition of frequency itself which was too narrow. It is too complicated to spell out here but there is a more general definition of frequency that can be developed that predicts the correct fundamental. In this example the frequency is 30.2Hz. In other words, scientists has unconsciously decided in advance that musical harmony was a psychological effect and this prevented them from exploring any other options. The concept of ratio is as deep as mathematics. It is fundamental to the very idea that the world can be described in number and these ratios are embodied in the representation of the 'time line' itself. There is little doubt in my mind that this representation of time by lengths in space comes from Pythagoras and his discovery that the ratios of the stretched string correspond to the natural numbers.
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Nicolas Meeus
Sorbonne Université
Rick, the acoustical phenomenon that the superposition of sine waves of frequencies f1 and f2 produces a resultant complex wave with a frequency equal to f2-f1 indeed seems universal. But to what extent does it apply to music?
I know of NO music that makes use of sine waves. It is not even likely that music ever makes use of strictly periodic sounds, because these strictly speaking should be of infinite duration.
The case that you mention, two waves of frequency 91 and 60, roughly describes what happens in sounds with inharmonic partials. It can easily be demonstrated that the only possible GCD frequency in this case is 1Hz (as you recognize yourself). This means that there cannot be any strict periodicity around 30Hz or 30.2Hz.
You may claim that you have a "a more general definition of frequency" than the usual one, but that is a dangerous claim in the case of an exact science, because exact sciences require exact definitions.
It is well know that in the case of two periodic vibrations in inharmonic relation (i.e. where the relation is close to harmonic), the resulting complex vibration is of fluctuating amplitude (this is called "beating", as you know). It could also be shown, but that involves messy trigonometry, that the resulting vibration at about the difference frequency is also fluctuating. (And "fluctuating frequency", which may be considered a contradiction in the terms, might be what you mean by your "more general definition".)
Both amplitude and frequency fluctuate at the rate of the GCD frequency, 1Hz in your case. A similar problem has been studied in the case of frequency modulation, the literature about which may provide some answers.
The fact that we perceive a pitch at the approximate (and fluctuating) difference frequency merely shows that our perception of pitch is not exactly linked with frequency perception. Without leaving the domain of exact science, one might argue that this has to do with the excitation of the basilar membrane and of the Corti cells in an area corresponding to the approximate difference frequency.
But, once again: no music makes use of sine frequencies, and none probably of strictly periodic sounds. It is true that our Western music, more than any other, does make use of almost periodic sounds, which are essential in a polyphonic music, based on the difference between consonance and dissonance. But we do not even know how our music was sung before the advent of polyphony in the West, say before the 10th century, how fluctuating vocal emissions were in earlier times, etc.
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2 Recommendations
Terence B Allen
"It is important to maintain balance vis a vis formalized and unformalized language. There is no conflict between the fact that certain methodological principles can be applied only to formalized languages and the fact that scientific statements are normally made in natural languages. Indeed, Tarski's axiomatizations of part of methodlogy involve axioms stated largely in natural language. Moreover, a formalized language has little interest for Tarski unless it is at least adequately translatable into natural language.
(Editor's comment in footnote to Alfred Tarski's Logic, Semantics, Metamathematics).
Music has a metamathemaitc structure because metamathematics is concerned with how a formal system like music has a syntactic structure in which frequency is the witness function. That is, how is music a language expressed in frequency of sound?
This problem has already been solved in math but not yet applied to music.
We need an axiom for music in a natural language anyone can understand.
The basic problem is that frequency cannot be resolved to a point without assuming that frequency domain has a topologic structure. (Spectral Resolution theorem). Usually we assume that frequency is a real number so that defines the mathematics of pitch in a way that makes it seem the system fundamental is defined by its pitch. But the fundamental defines the system at any pitch. The fundamental unit 1is not defined by pitch but by our one and only axiom: The octave is 2 times the fundamental (on which we all agree and for all time).
The problem is the other higher overtones are not in the system, only the octave and its semitones, like a ruler with 12 inches. The inches have the same metric as the foot. The semitones have the same metric as the octave. The higher overtones are not in the octave metric and they are not defined (even thought they may be close in frequency).
The 12-tone scale is purely mathematic and not tempered.
This solves the problem of making a language out of frequency because it tell us the topology of music is not the topology real numbers (a compelling and useful theory nonetheless) but instead pitch values make a log 2 space that is Boolean. That makes music what Tarski called a truth structure - it can all be reduced to 0 and 1. No real numbers. Just a language of 0 and 1 (like tablature).
The space in which musical tone exist is so intuitive that it seems as if real numbers work fine, which they do in electric currents and midi. We add intervals in harmonics but in log space addition and multiplication are the same. The overtones are multiples and sums of the fundamental. The fundamental F = (0, 1) means the fundamental is the identity of both addition and multiplication. But dividing notes and even subtraction is not defined in music (because there are no negative frequency values there is no subtraction).
This means that if C is the fundamental the first overtone multiple/addition is C#.
I really appreciate Rick Ballen's comments. Obviously we have the math to understand the music, but so far we have no mathematicians that consider music to be a form of mathematics. In the long marriage of music and math it seems that music is always to blame when the math doesn't perform as expected.
Topologize, topologize, tolopogize, a famous mathematician said.
There are only two topologies possible: the torus and the sphere.
The sphere is correct but the magic is the sphere can map on the Euler's torus, so the torus seems to work too.
The mathematic structure of music is astonishingly beautiful, and shows how false paradigms can persist in the modern information technology.
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Nicolas Meeus
Sorbonne Université
I have only one answer to this: NO. Or else, but it is only another way to state the same: what you call "music" has nothing in common with what I call "music".
- " In the long marriage of music and math it seems that music is always to blame when the math doesn't perform as expected."
I would rather blame the ignorance -- and I mean ignorance of music.
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1 Recommendation
Tibor Szasz
Hochschule für Musik Freiburg
No building block in music has an absolute identity in and of itself, but only within the relative context of a given piece of music. Implied in this relativity of the text within the context is the notion that the momentary identity gained within any given local context must be "updated" as soon as the local context changes. I have demonstrated this truth repeatedly to my students - most of whom come to my class believing that there is such a palpable "reality" as a C major. My favorite method of demonstrating to the students the uselessness of abstract analyses which are concerned only with "name callings" such as a "C major chord"
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Tibor Szasz
Hochschule für Musik Freiburg
(Continued) is to have one single musical formula such as the "Romanesca" identified in works ranging from 1700 to 2000. After the students do the work of "identifying" this formula, I ask them to find any two identified fragments which are even remotely similar. The answer is usually "none of the Romanesca fragments are musically similar other than in an absolutely abstract manner. In other words, each Romanesca fragment gains an identity all of its own, that is, totally different from any other Romanesca fragment in music history. Good music analyses will always be sensitive to the variations of a few seemingly invariant structures placed within ever varying contexts.
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1 Recommendation
Nicolas Meeus
Sorbonne Université
@Tibor Szasz, this indeed is what Schenker said. He made it his motto: Semper idem, sed non eodem modo, "always the same think, but never in the same manner". There always is some invariant [in tonal music, at least], but what really counts, what makes each piece unique are all the variants that elaborate the invariant. This is the essence of Schenker's theory, too often misinterpreted. Schenker does not mean that all tonal works have the same background structure, he stresses on the contrary that even although they all share the same background, they all nevertheless are unique.
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1 Recommendation
Tibor Szasz
Hochschule für Musik Freiburg
@Nicolaus Meeus: Apropos Schenker, read my most recent article titled "Towards a New Edition of Liszt's Sonata in B minor", Journal of the American Liszt Society, 2017 issue. Let me know if you cannot find the article.
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Tibor Szasz
Hochschule für Musik Freiburg
@Nicolas Meeus: apologies for the typo in your name.
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Willy LJ Van Buggenhout
Vrije Universiteit Brussel
... a concert hal, musical insturments, musicians, an orchestra... creaties expectations.
Listeners are led by these expectations.
They expect: a tonality of a kind, rhythm
- a symphony orchestra creates other expectations that a Balinese gamelan orchestra.
And specially musical instruments create expectations.
When there is a piano on the stage you seldom expect to hear John Cage's prepared piano.
You could name these "expectations" axioms of a kind, departing points, meetings point, semantics, agreements, history, ...
But the again: Music - as any art form - is to change what you expect from it.
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Leon Conrad
Co-Founder: Academy of Oratory; Founder: The Traditional Tutor; Orator in Residence: The Next Society Institute
I recommend you look at Hans Kayser's work on Harmonics - it is based on a Pythagorean understanding of mathematics which integrates length, proportion (with application to architecture), sound and metre. Rhythm and harmony add interesting challenges but both involve variations on listeners' expectations. Applying a 'modern' approach to analysing music mathematically is an interesting exercise, but I suspect it would be just end up being an exercise in mapping one application of a calculus to another application of the same underlying calculus. Do maths. Do music. Either way, why not ask why do maths? why do music?
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Gennady Fedulov
Dear Terence B Allen,
"Obviously we have the math to understand the music, but so far we have no mathematicians that consider music to be a form of mathematics. In the long marriage of music and math it seems that music is always to blame when the math doesn't perform as expected."
I hope that the time will come when the phrase "we will see mathematicians that will consider music to be a form of mathematics" will come true. In any case, I try to present the creation of music as a process of Combinatorial Optimization, where:
1. An objective function must be set, the optimization of which should ensure the receipt of music with specified properties.
2. Limitations must be set, which should take into account all the possibilities and limitations for the extraction of sounds.
For example, I never cease to be amazed at how Beethoven composed the ingenious "Adagio cantabile" from the Pathetic Sonata. How, with the help of a minimal combination of sounds, such ingenious music was achieved. What is striking is that any addition or deletion of any note completely violates the whole integrity of harmony. It seems to me that at first Beethoven heard all the harmony in his brain, and then he accurately recorded this harmony in musical notation. From this point of view, I see the process of creating music as a Schedule Theory of note sequences. However, Schedule Theory is a mathematical theory in which one need to optimize the objective function under given constraints. For example, not all combinations of sounds sound good. A combination of two and more notes sounds different in different octaves. That is, here you can involve the Theory of Counts and consider the musical combination of sounds as subgraphs with specified properties.
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