[This post belongs to a series about Dylan’s idea of “mathematical music” in Chronicles]
“I’m not a numerologist”, Dylan says (Chronicles, p. 159). But before and after this statement, he builds up such a metaphysical web around the force of numbers, that the only definition of a numerologist that he does not fit into, is the kind who calculate a lucky number from the letters of their name. Alright, this is after all not a Rod Stewart blog.
In the Rolling Stone interview from November 2001, where he first mentioned the Lonnie Jonhson method explicitly, he says:
Lonnie Johnson, the blues-jazz player, showed me a technique on the guitar in maybe 1964. I hadn’t really understood it when he first showed it to me. It had to do with the mathematical order of the scale on a guitar, and how to make things happen, where it gets under somebody’s skin and there’s really nothing they can do about it, because it’s mathematical.
In Chronicles, he continues:
I had the idea that he was showing me something secretive, though it didn’t make sense to me at the time…
So we have an esoteric system, communicated to him in the secrecy of the back room or wherever he was taken aside, which works, regardless of what the player or listener know, understands, or thinks of it, solely on the force of the mathematical structure of the system — “because it’s mathematical.”
The Pythagorean Tradition of numbers
The belief that something can work simply “because it’s mathematical”, depends in some way or another on the idea that numbers have certain metaphysical qualities with a real influence on things in reality.
This is the foundation of the Pythagorean theory of numbers, which I’ve alluded to before in this series. Most people know the Pythagorean Theorem, about the relations between the sides in a right-angled triangle: a2 + b2 = c2 (Dylan knows it too, even though he got the formula wrong in the Rome interview, where he presented it as “a square equals b square equals c square”, which may reveal a truth on a more profound level, but which would do you no good in your calculus 101 class).
But the classic didactical myth, handed down in numerous treatises throughout Antiquity and the Middle Ages, tells of how Pythagoras walked by a blacksmith who was pounding away on his anvils, and Pythagoras discovered that some of the anvils produced harmonious sounds together, while others did not. He investigated this closer, and found that the mass of the harmonious anvils were in simple proportions to each other — 1:2, 2:3, or 3:4 — while those in more complex relations produced unpleasing sounds. An anvil twice as big as another, would sound an octave lower, whereas one 1.3658 times the size, would sound like… dunno, the Shaggs or something.
The physical facts of this legend have been proven wrong, but what matters is the belief (1) that harmoniousness depends on proportions that can be expressed in simple ratios, (2) that these proportions, which can be described in a purely mathematical form, not only govern harmony in music, but also in the universe as a whole, and (3) that there is some kind of connection between the different kinds and areas of harmony. Thus, playing a tune in a mode which emphasises certain intervals, will influence the balance between the body fluids, and can thus alter the mood of the listeners.
This discovery and the theoretical/religious system that was built around it, became essential to all ideas of harmony and beauty from Antiquity up until the eighteenth century. Plato considered this kind of mathematical harmony to be the fundamental property of the world. In his creation myth Timaios, the creator-god shapes the world beginning with unity, then extending it with ‘the other’ and ‘the intermediary’, and along the corresponding number series 1, 2, 4, 8 and 1, 3, 9, 27, the whole world is created.
In the Middle Ages, this idea was adapted to the Christian frame of thought. In the apocryphical Wisdom of Solomon in the Bible, it says, “You have ordered all things in number, measure, and weight” (Wisdom of Solomon 11: 21), and this verse was quoted time and again in medieval treatises on music.
Thus, what at first sight may look like a dry and slightly tedious exercise in simple arithmetics, is of vast importance because behind the dry façade lies the notion that numbers and numerical relations are reflections of the divine principles governing the universe; that we find the same relations in the universe as a whole, in human beings, in musical sounds, and in visible beauty, and that by knowing the numbers, we can affect humans and glimpse God.
This is why the slight irregularities in the purely mathematical definition of the scale became such a heated topic. The theorists spent gallons of ink on discussing the problem with the division of a tone in two equal halves, which according to the Pythagorean system is impossible, because it is founded on ratios between natural numbers (the equal division of a tone requires the square root of 2, which was unknown to ancient and medieval thinkers).
The Christian heritage from antiquity was largely Platonic. One of the consequences of the humanistic re-appraisal of the classical traditions during the Renaissance, was that other voices from antiquity were added to the stew. Aristotle, with his less mystical and more rationalistic approach, was revived from the twelfth century, and in the field of music theory, Aristoxenos, whose theories were based on geometrical rather than arithmetical considerations, was more palatable to the practically oriented writers of the Renaissance, who were more concerned with actual sound and preferred the pure harmonies of just intonation to the theoretically “correct” but ugly-sounding harmonies.
Approaching Dylan again
If you object that this doesn’t seem to have much to do with Dylan and Lonnie, you’re absolutely right. I’m partly exerting my right to write whatever I want to do — this is my blog — but partly I’m also trying to demonstrate how important the concept of mathematical music has been, way back in history, and how widely the implications it carries reach.
In order to gradually work our way back to Dylan again, one might point to yet another element that entered the picture in the Florentine academies in the fifteenth century: an extension of the notion of the special mystical character of certain numbers. The mainstream medieval tradition had mainly been concerned with twos and threes, but — partly owing to influence from the cabbalistic tradition — a more extended array of meaningful numbers was established and systematized. The Fibonacci sequences and other similar number sequences, and all the sacred numbers of the Bible — just about every number seemed to have a secret meaning, a value beyond the numerical one.
This is the background for Dylan’s perception of the system he learned back in ’64. In the following quotation from Chronicles (p. 158), I have emphasised some words which highlights the strong dichotomy that Dylan sees between the world of 2 and the world of 3:
The system works in a cyclical way. Because you’re thinking in odd numbers instead of even numbers, you’re playing with a different value system. Popular music is usually based on the number 2 and then filled with fabrics, colors, effects and technical wizardry to make a point. But the total effect is usually depressing and oppressive and a dead end which at the most can only last in a nostalgic way. If you’re using an odd numerical system, things that strengthen a performance automatically begin to happen and make it memorable for the ages. You don’t have to plan or think ahead.
What is most striking, I think (apart from the description of popular music as based on the number 2, which quite bluntly disregards the blues/jazz tradition, where a triple feel is predominant), is the statement that these are different worlds, different value systems, which have an automatic effect on the performance: it is not something the performer does, but something that is done through the performer.
Does Dylan believe all this? Yes, I would think so. He is after all a poet, a sponge, a mystic, a sage; he takes what he can gather from coincidence, mixes it all together, and out comes… well, sometimes Knocked out Loaded, but we can forgive him that, since he also produces Blood on the Tracks and Chronicles, which is a fascinating read, even though what he writes is less clear than what an academic might have wanted.
More to come…
(Those of you who have access to Judas! may want to look up my article “Beauty may only turn to Rust” in the 8th issue, where I go into these things in more detail, and relate them to Dylan’s liner notes to Joan Baez in Concert, vol. 2, his aesthetical manifesto.)
Quite fascinating to see the old Pythagorean world — even if in a fairly remote ways — still have some presence in an artist like Dylan. Regardless of the correctness of the physics behind the Pythagorean ideology (the sizes of the anvils etc), the harmoniousness of the simple number relations (1:2, 2:3 3:4 etc) really have governed all music theory and practice up to our own time through modality and tonality, except for modernism (and thereby creating a link to your blog in more or less defense of that revolution against aesthetical beauty — in favour of ‘truth’ — which almost in itself confirms the ideology of harmony and beauty — it’s only that the truth-part no longer was considered to be identical with beauty as in Antiquity and the Middle Ages (and even longer…). Although concessions were made to practicalities in the well-tempered scale, these concessions — as far as I know — were thought to be so small as not to affect the ear (and seemingly didn’t create any large scale problems either), but the other way round actually solved the real ‘practical’ Pythagorean problem, that of the incongruence between the 2s and the 3s: The problem of the circle of fifths which almost but not quite brings you back to a tone some octaves higher than the point of departure. A fifth is (in the Pythagorean system) 2:3, and if you start on for instance a C on the piano and go up 12 fifths from there, you come back to a C seven octaves higher. This works because of the tempered scale (where the fifths are not pure fifths (i.e. 2:3); in a real pythagorean scale it would be almost correct, but only almost. 12 fifths would amount to (2:3) to the twelfth, which is almost 130, whereas the seven octaves would be (1:2) to the seventh, i.e. 128. Thus the circle of fiths (according to Pythagorean scales) would just miss the seventh octave on the way up. But only by very very little…
Looking at — or listening to — the natural overtones gives exactly the same almost Pythagorean structure (which may be why only the first of them really sound if you try to produce them, for instance on a violin or a piano (hold down one key on a piano silently and strike hard on the key one octave below, or an octave and a fifth, or two octaves, or… and you’ll hear the resonance, but if you go much further you won’t hear much (because of the lack of collaboration between 2s and 3s? or…?)
In any case, it’s difficult to determine whether the numbers and the world do match or not. The seem to almost match which I suppose for a ‘real’ Pythagorean would have to be terribly dissatisfying… I wonder if that would make any difference to Dylan?
Even if there are no “natural” tonal laws in nature, and even though scales have no basis outside of human rationalization, the Ideas are now so deeply imbedded in western society (of which dylan is an obvious product) that they have become learned laws. In that, besides the ratios in octaves, which almost every known civilization has discovered, there is no absolute “math” involved in music.Its all leaned, so I dont see how we can have a scale of 12 and the chinese have a scale of 5, and say that the math is concrete, or universal, or that its so relevent that it relates to the metaphysical.
“The theorists spent gallons of ink on discussing the problem with the division of a tone in two equal halves, which according to the Pythagorean system is impossible, because it is founded on ratios between natural numbers (the equal division of a tone requires the square root of 2, which was unknown to ancient and medieval thinkers).”
This part is wrong. Ancients did know the “square root of 2”. In fact’s that’s exactly one of the things Pythagoras proved – and the discovery that it’s an irrational number. But as a practical math thing, it was known way before Pythagoras:
https://en.wikipedia.org/wiki/Square_root_of_2#History
And ancients absolutely had semitones (splitting a tone in two halves):
https://en.wikipedia.org/wiki/Musical_system_of_ancient_Greece
Thanks for bringing this up. You point out exactly the problem: the ancients knew that there had to be a square root of two, but they did not know how to calculate it – or as it says in the article you link to: that the diagonal of a square is incommensurable with its sides – since their mathematics were based on ratios – common measures. So what Mr Pyth discovered was that there are numbers that cannot be expressed as ratios (hence: irrational), but the way to express them was not discovered until the modern era.
As for semitones in ancient music: again: ancient music theory is full of it, because the music used semitones, and according to a geometric argument, of course there had to be some point that was exactly half ways between one tone and another, but it was impossible to calculate it – again because it requires mathematics that is not based on ratios between whole numbers. That’s what they spent those gallons of ink on.
again, off topic and im sure this gets annoying but i cant find it anywhere, can you please tab the howlin wolf song of the folksingers choice show ‘tell me baby’?
thanks,
gary
p.s. i do belive it is the only song from the folksingers choice show that is not on dylanchords, so this could round out that album as well.
It’s already there, under the title Smokestack Lightning
whhaaaaat?!?
thanks
My suspicion is that this is a lot of stuff about not much of anything at all. If you look at early Dylan (and even later Dylan), you can see that one of the things that he starts doing is adding in minors and doing those endless descending bass runs from the 8,7,6,5. I think Johnson simply showed him a variety of the substitutions (1 can go with 3minor; 2 with 4; 3minor with 5 and 7, and taught him that this enables you to walk down from 8 using the tonic, 3 minor, 2 minor, V and IV chords. Dylan does this a million times especially in his open tunings — something I learned from this site years ago!!
yours,
Peter T.
Great site! Keep it running!
Wonderful blog.
The notion that mathematics and music are inseperable is far from dead. Nature provides for us the blueprint for this in the Natural Harmonic Overtone Series, which is literally present in every note eminating from every instrument (Except for synthesizers which can generate pure sine waves). The overtone series propagates in the ratios you refferenced: 2:1 for the octave/first overtone, 3:2 for the fifth/second overtone, 4:3 for the fourth/third overtone, 5:4 for the major third/fourth overtone, 6:5 for the monor third/fifth overtone, &c. These are the pure, beatless intervals you find in so-caled Just Intonation (Among other higher ratios for the seconds, and non-superparticulars for the sixths, &c.).
The ratios for these intervals define the rules of counterpoint; the fact that the fundamental and the first six overtones spell out a dominant seventh chord defines the fundamentals of harmony and harmonic progression; the interference patterns these ratios produce define the basics of rhythm; and the higher ratios and the overall shape of the overtone series defines the elements of melodic motion and trajectory.
If you are really interested in this subject, I have a super-wonky music theory blog that spells all of this out: http://hucbald.blogspot.com .
BTW: I’m collecting links today for a Blogroll that I’m putting on my site. You’ll be on it.
Cheers.
Peter T., what the? One can go with three minor, 2 with four etc.
Would you be a bit less Dylanesque and explain that? Sounds like something I might be able to actually put to practical use. An example of a Dylan progression using the technique would be helpful.
Love this site. Love the blog.
1 can go with 3 minor = when in a major key, you can substitute the iii (third scale step is root of chord, minor quality) for the I (root).
so if you’re playing in C, you can substitute Em for C.
2 for 4 would be substituting Dm for F (if you’re still in the key of C). basically you’re just changing one note of the triad, so the sound is similar, but the quality is different (you’re making it a minor chord, and thus darker).
You guys got it all wrong. Dylan’s style is simple arithmetic…
Devil + Dusty Crossroads – Soul = Music.
-elpollodiablo
I am still trying to figure this out. Wormholes, cones, Pythegorian in 3/4 time can someone explain it to me like I’m a bright 5 year old with perfect pitch? I thought it was more about the beat and the spaces between the beats and how you lay down the contrast, than tones and scales. Seems to me Bob is talking about two very different theories of music and then falls down the rabbit hole into the golden spiral or the Fibonacci sequence in music. Maybe he tapped into the original source.