after Pythagoras

One of the most famous discoveries of Pythagoras, or of the Pythagorean School (it is often difficult to tell the difference), is, according to G.S. Kirk & J.E. Raven, "**that the chief musical intervals are expressible in simple mathematical ratios between the first four integers**" [*The Presocratic Philosophers*, Cambridge University Press, 1964, p.229]. Thus, the "**Octave=2:1, fifth=3:2, fourth=4:3**" [p.230]. These ratios *harmonize*, not only *mathematically* but *musically* -- they are pleasing both to the mind and to the ear. This impressed the hell out of the Pythagoreans, who also honored the "first four integers" because those add up to **ten**, the perfect number, and can be displayed in a triangle (like all "triangular" numbers), the "Tetractys of the Decade": . The Pythagoreans are supposed to have sworn their oaths by this device. In music, adding a fifth to a four, which requires multiplying the ratios, results in the octave: **3/2 x 4/3 = 12/6 = 2**. Unfortunately, as with some other Pythagorean mathematical inquiries, the simplicity, or even the truth, of this result disappears on further investigation.

Calling intervals the "fourth," "fifth," or "octave" (i.e. "eighth"), when they are part of a system of seven tones, is a little confusing. Adding four to five doesn't even equal eight, much less seven, but nine. What is going on, however, is the device of the "inclusive" counting of ordinal numbers, where we start a new cycle of numbering (the first) with the end of the previous cycle (*octava*, the eighth). This form of counting is discussed elsewhere in relation to calendars. The fourth and fifth thus reduce to three and four as numbers, which then do add up to seven.

The seven note scale of the Greeks is the **diatonic** or **heptatonic** scale, to which have been assigned the letters **A** through **G**,

C | D | E | F | G | A | B | C |
---|---|---|---|---|---|---|---|

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

first | second | third | fourth | fifth | sixth | seventh | octave |

One response might be that not every interval is equal. We hear that it is only a "half step" from **E** to **F** and from **B** to **C**, while it is a "whole step" between the other notes. What this is supposed to mean we can see on a piano, where there are black keys between the white keys, but no black keys between **E** and **F** or between **B** and **C**. This may muddle the mathematics of the ratios. However, we can check.

C | D | E | F | G | A | B | C | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

24/24 | 27/24 | 30/24 | 32/24 | 36/24 | 40/24 | 45/24 | 48/24 | ||||||||

3/24 | 3/24 | 2/24 | 4/24 | 4/24 | 5/24 | 3/24 |

I might ask then what the scale would look like if we *wanted* the scale to *evenly divide* the octave, with equal intervals between the notes. Since the problem of the musical scale is, as John Stillwell says, "*multiplication perceived as addition*" [*Yearning for the Impossible, The Surprising Truths of Mathematics*, A.K. Peters, Ltd., 2006, p.4], what we need to do is reduce multiples to sums. This can simply be done with logarithms, which by addition give us the products of multiplication (through the "law of exponents"). The logarithm of **2** is **0.301029996**. If we divide this by **7** we get **0.043004285**. Adding this in successive sums through six, taking the anti-log (i.e. raised to the power of ten),

C | D | E | F | G | A | B | C |
---|---|---|---|---|---|---|---|

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

24/24 | 27/24 | 30/24 | 32/24 | 36/24 | 40/24 | 45/24 | 48/24 |

24/24 | 26.5/24 | 29.3/24 | 32.3/24 | 35.7/24 | 39.4/24 | 43.5/24 | 48/24 |

As it happens, most of the traditional ratios are not used. Why this is so we can see from the following table, where we take the interval of the fifth (3/2) and begin adding it successively -- where this is now done on the *twelve note scale*, the "chromatic" scale, where the black piano keys are added to the white ones (distinguished as "sharps," **#**, or "flats," , of the "natural," , notes). We can compare the result with a baseline principle of the octave, that the ratio of *any interval* will be *doubled* in the following octave.

C | C# D | D | D# E | E | F | F# G | G | G# A | A | A# B | B | C |
---|---|---|---|---|---|---|---|---|---|---|---|---|

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

2/1 | 9/4 | 5/2 | 8/3 | 3 | 10/3 | 15/4 | 4/1 | |||||

27/8 |

The following table works out this process. The blue ratios are our reference values for the octaves (the firsts and the eighths, "inclusively" counted). In red we follow the additions (by multiplication) of the fifths. Once we get a red value for any key, then we divide it down octave by octave to the first one. This gives us values for all the intervals and all the notes, although most are now very far from being "simple mathematical ratios." But, unfortunately, the problem of inconsitency found in the previous table occurs again.

C | C# D | D | D# E | E | F | F# G | G | G# A | A | A# B | B | C |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1/1 | 21872048 | 9/8 | 1968316384 | 81/64 | 177147131072 | 729512 | 3/2 | 65614096 | 27/16 | 5904932768 | 243/128 | 2/1 |

2/1 | 21871024 | 9/4 | 196838192 | 81/32 | 17714765536 | 729256 | 3 | 65612048 | 27/8 | 5904916384 | 243/64 | 4/1 |

4/1 | 2187512 | 196834098 | 81/16 | 17714732768 | 729128 | 65611024 | 590498192 | 243/32 | 8/1 | |||

8/1 | 2187256 | 196832048 | 17714716384 | 72964 | 6561512 | 590494096 | 16/1 | |||||

16/1 | 2187128 | 196831024 | 1771478192 | 6561256 | 590492048 | 32/1 | ||||||

32/1 | 19683512 | 1771474096 | 590491024 | 64/1 | ||||||||

64/1 | 1771472048 | 128/1 | ||||||||||

5314414096 |

After seven octaves, adding twelve fifths brings us to a value for **C**, **531441/4096**, which is different from the value we get, **128/1**, derived directly from our original value of the eighth, at **2/1**. The ratio between these two values is **1.013643265** (**531441/524288**, or **3 ^{12}/2^{19}**), which gets called the "

This means that the whole Pythagorean probject is now in shambles -- although, as Stillwell says, "the Pythagoreans may never have noticed" [p.20]. These scales *cannot* be constructed with "simple mathematical ratios." Schopenhauer, one of the surpreme philosophers of music, was aware of this, as we find him saying:

...thus, a perfectly pure harmonious system of tones is impossible not only physically, but even arithmetically. The numbers themselves, by which the tones can be expressed, have insoluable irrationalities. [The World as Will and Representation, Volume I, §52, E.F.J. Payne translation, Dover Publications, 1966, p.266]

This creates a dilemma for real musicians, which is how the scales are to be constructed at all. To be sure, music can be played using the intervals derived from adding fifths, or even using the original ratios, and the ear may not object -- despite using notes created by systems that are ultimately inconsistent. The differences are, after all, rather small, even for the original and traditional ratios. But it is annoying. There is a sort of Pythagorean itch that keeps us thinking that there should be a proper mathematical solution to the matter. This is not going to be as simple as what Pythagoras expected, but the belief continues that the fundamental ratio, the octave, **2:1**, can be reconciled with the division of the scale into other intervals.

Although Schopenhauer, with his characteristic pessimism, does not seem to have appreciated it, something can be done about the problem. The solution, or at least one solution, is to adjust the ratio of the fifth so that *it is commensurable* with seven octaves. Seven octaves is

There is some solace for Pythagoras here. The intervals are mostly no longer simple ratios of integers. The fifth is not **3:2**. But now it *is* **2 ^{7/12}**. That is a notation unfamiliar to the Greeks, who didn't get all that far thinking about powers and roots, but we can be sure that Pythagoras would have eaten it up. The red ratios in the table above thus can be reconstructed with successive powers of

C | C# D | D | D# E | E | F | F# G | G | G# A | A | A# B | B | C |
---|---|---|---|---|---|---|---|---|---|---|---|---|

2^{0} | 2^{1/12} | 2^{1/6} | 2^{1/4} | 2^{1/3} | 2^{5/12} | 2^{1/2} | 2^{7/12} | 2^{2/3} | 2^{3/4} | 2^{5/6} | 2^{11/12} | 2^{1} |

2^{1} | 2^{13/12} | 2^{7/6} | 2^{5/4} | 2^{4/3} | 2^{17/12} | 2^{3/2} | 2^{5/3} | 2^{7/4} | 2^{11/6} | 2^{23/12} | 2^{2} | |

2^{2} | 2^{25/12} | 2^{9/4} | 2^{7/3} | 2^{29/12} | 2^{5/2} | 2^{8/3} | 2^{17/6} | 2^{35/12} | 2^{3} | |||

2^{3} | 2^{37/12} | 2^{13/4} | 2^{41/12} | 2^{7/2} | 2^{11/3} | 2^{23/6} | 2^{4} | |||||

2^{4} | 2^{49/12} | 2^{17/4} | 2^{53/12} | 2^{14/3} | 2^{29/6} | 2^{5} | ||||||

2^{5} | 2^{21/4} | 2^{65/12} | 2^{35/6} | 2^{6} | ||||||||

2^{6} | 2^{77/12} | 2^{7} |

It is now evident from this table that the interval between successive values is always **2 ^{1/12}**. This is the perfect marriage of addition and multiplication. By adding the exponent

Finally, we can calculate the actual frequencies of sound for the octave above Middle **C**. This is based on the standard of exactly 440.0 Hz for **A** above Middle **C**. We multiply that by the reciprocal of **2 ^{3/4}** to get the frequency of Middle

C | C# D | D | D# E | E | F | F# G | G | G# A | A | A# B | B | C |
---|---|---|---|---|---|---|---|---|---|---|---|---|

2^{0} | 2^{1/12} | 2^{1/6} | 2^{1/4} | 2^{1/3} | 2^{5/12} | 2^{1/2} | 2^{7/12} | 2^{2/3} | 2^{3/4} | 2^{5/6} | 2^{11/12} | 2^{1} |

261.6 | 277.2 | 293.7 | 311.1 | 329.6 | 349.2 | 370.0 | 392.0 | 415.3 | 440.0 | 466.2 | 493.9 | 523.3 |

These values sometimes differ by one or two tenths of a Hertz in comparison to the values I originally copied from Isaac Asimov and that I cite in the section on the Electromagnetic Spectrum. These small differences are insignificant but I am not aware why they would occur --Asimov's values may have been rounded off in a different way.

Philosophy of Science, Mathematics

One of the most intriguing areas of mathematics that drew the interest of the Pythagoreans was in **number series** (where *series* in Latin is both the singular and the plural). Our practice today of speaking of "squares" and "cubes" for numbers that are multiplied by themselves two or three times is an artifact of the way that the Pythagoreans liked to *visualize* their numbers as patterns of little dots. Beginning with **4**, **square** numbers can always be represented by little squares: The Pythagoreans then discovered that the addition of successive odd numbers produces successive squares.

integers | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

odds | 1 | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | 25 | 27 | 29 |

squares | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 | 169 | 196 | 225 |

The favorite series of the Pythagoreans, however, were the **triangular** () numbers. As we might imagine, these are the numbers with which we can construct little triangles of dots, beginning with **3**: The rule for the construction of this series is very simple: add all the successive integers up to the desired point. **Three** is **1+2**, **15 = 1+2+3+4+5**, etc. A formula to calculate any triangular in the series directly, without adding all the previous numbers, was discovered by **Carl Friedrich Gauss** (1777–1855): ** _{n} = (n(n+1))/2**. Here

Although I never heard of them in any high school math class, triangulars draw some small attention in modern mathematics, for instance in the proposition of **Pierre Fermat** (1601-1665) that *every number* "is either triangular or the sum of two or three triangular numbers" [cf. Constance Reid, *From Zero to Infinity*, 1955, A.K. Peters, Ltd., 2006, p.76]. However, a series with a *very similar rule* is encountered constantly in calculus and elsewhere. "Factorials" are the series produced by the successive *multiplication* of integers, rather than addition as with the triangulars. Factorials quickly become very large. In the table below, I have resorted to the notation for factorials, with an exclamation mark (!), because the actual numbers would make the table too big. The factorial of **13** (13!) is already **6,227,020,800**. One of my calculators cannot go over **69!** without displaying an error message for a number that is too large to be calculated -- **69!** itself is already **1.711 x 10 ^{98}**. The great virtue of factorials is that their reciprocals rapidly become exceedingly small. One use of factorials may be seen in the continued series for the constant

integers | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

triangulars | 1 | 3 | 6 | 10 | 15 | 21 | 28 | 36 | 45 | 55 | 66 | 78 | 91 | 105 | 120 |

factorials | 1 | 2 | 6 | 24 | 120 | 720 | 7! | 8! | 9! | 10! | 11! | 12! | 13! | 14! | 15! |

The Pythagoreans particuarly honored the triangulars because one of them is the number **10**, and the Pythagoreans honored **10** as the perfect number. They are supposed to have sworn their oaths by the "Tetractys of the Decade," i.e. the little triangular image, Their reasons for taking **10** as the perfect number now seem a little silly -- that **10** is perfect because we have ten fingers and toes -- or redundant -- that most peoples count in tens (which is probably because we have ten fingers and toes -- although the Babylonians counted in *sixties* and the Maya in *twenties*). Nevertheless, their reverence for **10** had various interesting consequences, such as in their cosmology, where they thought there needed to be ten planets. It also gave to the triangulars a prominance that they have not enjoyed since. One Pythagorean discovery about triangulars is that every **square** number is the sum to two successive triangulars, e.g. **10+15 = 25 = 5 ^{2}**.

integers | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

triangulars | 1 | 3 | 6 | 10 | 15 | 21 | 28 | 36 | 45 | 55 | 66 | 78 | 91 | 105 | 120 |

squares | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 121 | 144 | 169 | 196 | 225 |

The Pythagorean proof for this discovery gives us some important insight into how the very idea of proof developed in Greek mathematics. The Pythagoreans liked to *visualize* their proofs, like their numbers. Thus, they reasoned that, as any square can be divided by a diagonal into two triangles, any *square number* should be divisible into two *triangular numbers*: . The truth of the proposition can be read off the diagram. The earliest proofs for the Pythagorean Theorem were evidently of this kind, with the proofs using more abstract argument coming later. The triangulars thus occupy a critical position in the earliest days of the history of mathematics.

We can do an algebraic proof of the Pythagorean discovery using Gauss's formula for triangulars. We use the formula for , add to it the formula where we substitute **n+1** for **n**, and the rest is basic algebra. The nice thing about this proof is that for each **n**, we can see which triangulars we are talking about, and have a separate expression at the and for the square that is in question. For **n=20**, our first triangular is **210**, our second is **231**, and the square is **(n+1) ^{2} = 441**.

A proposition much like the Pythagorean discovery relative to squares and triangulars is presently one of the most famous unsolved problems in mathematics. This is "Goldbach's Conjecture," named after **Christian Goldbach** (1690–1764), which is that *every even number greater than two can be expressed as the sum to two prime numbers*. Actually, in 1742 Goldbach himself wrote a letter to the great mathematician **Leonhard Euler** (1707–1783) in which he proposed that every integer greater than two can be expressed as the sum of three primes. Since **1** is no longer regarded as a prime number, this is currently restated as *every integer greater than five can be expressed as the sum of three primes*. Euler replied to Goldbach that this proposition would actually be a *corollary* of one about even numbers as the sum of two primes. Thus "Goldbach's Conjecture" about even numbers (the "strong" conjecture) is directly due to Euler, but Golbach gets the credit for suggesting this *kind* of proposition. Thus, like the Pythagorean work, modern mathematics is still looking at relationships between number series, in this case between evens and primes. But, unlike the Pythagoreans with their diagram, Goldbach's Conjecture remains unproven -- although by 1995 the French mathematician Olivier Ramaré proved that every even number equal to or greater than **4** is the sum of no more than six primes. So we get closer.

Mathematics & Music, after Pythagoras

Philosophy of Science, Mathematics