# Pythagorean Scale and Markets, from Alex Castaldo and Victor Niederhoffer

April 24, 2007 |

The musical scale that Pythagoras invented and that forms the foundation for all Western music is based on relationships between small integers. Both the frequencies of musical notes and the intervals between them are ratios of certain integers, as follows:

Scale of Just Intonation

Note            C    D    E    F    G    A     B    C'

Frequency  1/1  9/8  5/4  4/3  3/2  5/3  15/8  2/1

Interval      9/8 10/9 16/15 9/8 10/9  9/8  16/15

How could similar relationships be uncovered for the S&P? As a first step we might force all S&P daily moves into bins one point wide, that is we could call all moves of 1.0 to 1.9 points "one point moves," and so forth.

We calculated the number of occurrences of each integer move, and also the number that would be expected if we assumed a normal distribution having the same mean and standard deviation. The result may be familiar to some readers but new to others, so we show it here:

S&P Daily moves in increments of 1 point

1999 to 2007/2/28
NormDist

LO HI Cases ExpCases
-500 -40.1 10 2.46
-40 -39.1 1 0.63
-39 -38.1 0 0.79
-38 -37.1 1 0.98
-37 -36.1 2 1.21
-36 -35.1 3 1.49
-35 -34.1 3 1.83
-34 -33.1 1 2.22
-33 -32.1 4 2.68
-32 -31.1 7 3.22
-31 -30.1 3 3.85
-30 -29.1 9 4.58
-29 -28.1 2 5.41
-28 -27.1 9 6.35
-27 -26.1 8 7.42
-26 -25.1 6 8.62
-25 -24.1 5 9.95
-24 -23.1 13 11.42
-23 -22.1 13 13.03
-22 -21.1 20 14.79
-21 -20.1 16 16.69
-20 -19.1 13 18.73
-19 -18.1 15 20.89
-18 -17.1 16 23.17
-17 -16.1 17 25.55
-16 -15.1 19 28.01
-15 -14.1 24 30.54
-14 -13.1 28 33.10
-13 -12.1 29 35.67
-12 -11.1 39 38.23
-11 -10.1 43 40.73
-10 -9.1 53 43.15
-9 -8.1 46 45.45
-8 -7.1 55 47.59
-7 -6.1 52 49.56
-6 -5.1 52 51.31
-5 -4.1 54 52.82
-4 -3.1 60 54.06
-3 -2.1 70 55.01
-2 -1.1 79 55.66
-1 -0.1 76 56.00
0 0.9 94 56.02
1 1.9 85 55.71
2 2.9 91 55.09
3 3.9 89 54.17
4 4.9 74 52.95
5 5.9 69 51.47
6 6.9 54 49.74
7 7.9 52 47.80
8 8.9 59 45.67
9 9.9 53 43.38
10 10.9 35 40.98
11 11.9 43 38.48
12 12.9 38 35.93
13 13.9 28 33.36
14 14.9 26 30.79
15 15.9 25 28.26
16 16.9 17 25.79
17 17.9 18 23.40
18 18.9 14 21.11
19 19.9 14 18.94
20 20.9 10 16.89
21 21.9 7 14.98
22 22.9 6 13.20
23 23.9 5 11.57
24 24.9 6 10.09
25 25.9 15 8.74
26 26.9 4 7.53
27 27.9 8 6.45
28 28.9 7 5.50
29 29.9 2 4.66
30 30.9 2 3.92
31 31.9 3 3.28
32 32.9 2 2.73
33 33.9 2 2.26
34 34.9 1 1.86
35 35.9 2 1.52
36 36.9 3 1.24
37 37.9 0 1.00
38 38.9 2 0.81
39 39.9 4 0.65
40 500 16 2.52

The main features of the distribution as compared to the normal are: an excess of small changes (say from -6 to +6 points), a deficit of medium sized moves (of about +-16 points) and a modest excess of very large moves (+-35 points or more). A description in terms of these three features is a better one, in my opinion, than simply focusing on the size of the negative tail (as too many people do).

## From Laurel Kenner:

In mathematics, Dr. Castaldo is on Parnassus and I am on the "Gradus ad." I do know a little about music and writing, and so I will nevertheless venture to observe that Pythagoras did not actually invent the musical scale. As Stuart Isacoff puts it in his masterful book "Temperament":

"Pythagoras's discovery was that the most 'agreeable' harmonies [e.g., the octave, fifth and fourth] are formed by the simplest kind of mathematical relationships. If the vibrations of one tone are twice as fast as the vibrations of another's, for example, the two will blend so smoothly the result will sound almost like a single entity [he is referring to an octave]. The separate constituents of this musical marriage are oscillating in the proportion 2:1."

Closer to our day, Helmholz wrote that consonant tonal relationships are embedded in the essential physical structure of notes — the sound waves. Look at a graph, and you'll see greater spikes in intensity around the octave, fifth, fourth, etc. These spikes are known as harmonics. Their intensity levels depend on the shape of the instrument that produces the sound, and the resulting mixture contributes to the distinctive sound of each instrument.

(Hold a bass note on the piano down and strike the same note, stacatto, a few octaves higher and listen closely — you'll hear sonorities of tones still higher. Or experiment with harmonics on a vibrating guitar string.)

The market's ever-changing significant levels might be viewed (heard) as harmonics of past turning points. The Chair's insights on the continuing psychological impact of catastrophic events would seem to be in this genre.

The mathematical relations found in music are tempting to apply to the physical structure of the universe, and people have done that at least since Pythagoras. But I will leave that to the physicists.

We might be socialized to market intervals in much the same way as to musical intervals. I found this interesting.

Evolutionary Effects by Robert Fink, 2004

General human evolution has provided us with voices that are acoustically musical, and with ear receptors that are appreciative of, or attracted to, acoustically-musical sounds (i.e., not noisy). Why this evolution?

Without these physiological capacities, then:

* Mothers would not coo to their babies; nor would the babies love the sound of it;

* Nor would evolution of language and the socializing sounds of the voice have been as possible;

* Nor would the noisy (i.e., not-acoustically musical) sounds from any nearby destructive event or attack, or of the sounds of breakage, screams or cries of pain, have served as a noisy warning [unattractive or repelling] to alarm or alert us — Some sounds make us come, others make us run….

And, as a result, our collectivized survival might not have been as efficient, and we could have gone the way of the extinct Dodo.

All those same capacities [regarding being able to distinguish noise from "musical" sound] also served to allow the development of musical systems to arise and evolve wherever there were curious people with time to play or experiment with the stimuli around them.

## Tom Ryan extends:

I find this Pythagorean scale discussion fascinating and have thought about various applications of musical scales and notes to market prices often over the years. Pythagoras believed the universe was an immense monochord and many of the Pythagorean teachings at Crotona are remarkable insights, especially with respect to what we now call string theory.

Pythagoras believed that by studying music mathematically, one could develop an understanding of structures in nature. One of the more interesting aspects that I have found in this is the lambdoma table, which is a 16×16 dimensional matrix of ratios starting with 1/1 and ending with 16/16,

1/1 1/2 1/3 1/4 1/5 1/6 1/7…….1/16
2/1 2/2 2/3 2/4 2/5 2/6 2/7…….2/16
3/1 3/2 ………..
.
.
.
.
16/1 16/2 16/3……………….16/16

The lambdoma is composed of two series. The first represents the divisions of a string which represent frequencies or tones. The second level represents harmonics. For example the first 16 harmonics of C are 256 hz (C), 512 hz (C), 768 hz(G) 1024 hz(C) 1280 (E) 1536 (G) 1792(B-) 2048(C) 2304(D) 2560(E) 2816(F#) 3072(G) 3328(A-) 3584(Bb-) 3840(B) and finally 4096 hz (C again since 4096/256 =16/1). This series represents the overtones or partial and whole harmonics of C and are represented by ratios in the table

Through the ratios one can study other types of natural structures as well. Botanists in particular have studied geometrical structure and many plant structures (leaves, flowers) have been noted to be geometrically developed in consistent ways where the key geometric ratios fall into a contiguous grouping or overtones of a fundamental, i.e. a pattern of squares within the lambdoma matrix. Leaves for instance often have simultaneous ratios of thirds (5:4) and fifths (3:2). The Renaissance studies including Da Vinci's notebooks note that the human body develops along particular lines as well, namely an abundance of major sixths (3:5) and minor sixths (5:8).

These whole number ratios or pattern of harmonics, as Laurel noted, form the basis for musical scales,

Octave 1:2
Fifth  2:3
Fourth  3:4
Major sixth     3:5
Major third     4:5
Minor sixth     5:8
Minor seventh   5:9
Major second    8:9
Major seventh   8:15
Minor second    15:16
Tritone 32:45

The question is whether specific harmonic patterns occur at times in the market and whether these can be quantified in some way. Victor discusses market prices as music in his book EdSpec.

One way would be to classify movements as ratios, and see the patterns of ratios as they unfold, classifying movements in price as patterns on a scale or within the lambdoma. From that one might be able to find a few meager predictable patterns. But there is a problem with this: to calculate ratios in a contiguous strip of real time prices one must arbitrarily choose reference points in order to calculate the ratios.

This means that there is a substantial level of subjectivity in developing the ratios from which the patterns can be studied. But one could arbitrarily divide the day into segments and look at ranges or deltas within those segments (say 30 minutes) and then calculate ratios of adjacent periods. There are probably an infinite variety of ways to segment and calculate ratios and therein lies the dilemma.

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