There is a simple (ahem) reason why the standard (Western) music scale divides an octave into twelve (roughly) equal steps or semitones. This is that it allows a particularly close approximation of both the second and third harmonics of a fundamental tone. This is based on the properties of a vibrating string. Other sound elements (such as surfaces like a drum membrane) or resonating volumes of various shapes have more complicated sequences of harmonics.

For a vibrating string of given density and tension, the frequency of sound of its fundamental tone is inversely proportional to its length. The first harmonic corresponds to half the length and twice the frequency. The second harmonic corresponds to one third of the length and three times the frequency.

The interval corresponding to doubling the frequency is called an octave. Successive halvings of a string’s length correspond in frequencies of f, 2f, 4f, 8f, etc, separated by octaves. The multiplicative nature of this process suggests the use of the logarithm. The logarithms of a sequence of frequencies octaves apart are ln f, ln f + ln 2, ln f + 2 ln 2, ln f + 3 ln f, etc. We can see that we can move by octaves along the scale by adding or subtracting ln 2 to the logarithm of the frequency.

Similarly, we can jump by intervals of the second harmonic (trebling of frequency) by adding ln 3 to the logarithm of the frequency.

Question: is if possible that n ln 2 = m ln 3 ? That is, is it possible that a sufficiently high harmonic can be reached by both a sequence of octaves and a sequence of treblings? The answer is no, as 2n is always even and 3m is always odd. However, it is possible to find some good approximations:

21 = 2 ~ 3 = 31
31 = 3 ~ 4 = 22
23 = 8 ~ 9 = 32

The question of finding really good approximations, even better than these, is solved by the technique of continued fractions, which is based on the Euclidean division algorithm.

$\frac{ln 2}{ln 3} = 0.631\ldots = 1 + \frac{1}{-3 + \frac{1}{3 + \frac{1}{2 + \frac{1}{3.846\ldots}}}}$

For easier visualisation, this is written

$\frac{ln 2}{ln 3} = 0.631\ldots = 1 + \frac{1}{-3 +} \frac{1}{3 +} \frac{1}{2 +} \frac{1}{3.846\ldots}$

Truncating the continued fraction at each step one finds

$\frac{5}{8} < \frac{53}{84} < \cdots < \frac{ln 2}{ln 3} < \cdots < \frac{12}{19} < \frac{2}{3}$

This corresponds to

23 = 8 ~ 9 = 32
28 = 256 ~ 243 = 35
219 = 524288 ~ 531441 = 312

The important observation is that 12:19 ~ ln 2:ln 3 can be interpreted as 12 steps for a factor of 2, and 7 steps for a factor of approximately 3/2. This corresponds to an octave divided into 12 semitones with a major fifth represented by 7 semitones.

The case of 5:8 ~ ln 2:ln 3 corresponds to the pentatonic scale: an octave is divided into the five black keys of a piano, and a major fifth corresponds to three steps, for instance between F# (the first black key of a group of 3) and C# (the first black key of the next group of 2).

WordPress supports LaTeX. Who knew?

$e^{i\pi}+1=0$

They’ve created a monster…

Esta tarde a las 18h en la plaza de Callao hay una concentración conmemorativa del 60 aniversario del Tratado de Roma. Según la Comisión Europea, las asociaciones convocantes son (en orden alfabético):

Estudiantes por Europa, Eurocitizens, Europeístas, Europa en Suma, Jóvenes Federalistas Europeos, Valor Europa.

¿Quiénes son estas asociaciones?

Cross-posted on European Tribune.

In his satire Candide, published in 1759, Voltaire pokes fun at the way the Portuguese Inquisition persecuted jews who had falsely converted to Catholicism:

After the earthquake had destroyed three-fourths of Lisbon, the sages of that country could think of no means more effectual to prevent utter ruin than to give the people a beautiful auto-da-fe; for it had been decided by the University of Coimbra, that the burning of a few people alive by a slow fire, and with great ceremony, is an infallible secret to hinder the earth from quaking.

In consequence hereof, they had seized on a Biscayner, convicted of having married his godmother, and on two Portuguese, for rejecting the bacon which larded a chicken they were eating[7]; after dinner, they came and secured Dr. Pangloss, and his disciple Candide, the one for speaking his mind, the other for having listened with an air of approbation.

Fast-forward to 2016, and Sarkozy’s extremism is indistinguishable from Voltaire’s satire.

Se ha vertido mucha tinta sobre los pactos en la constitución de la mesa del primer parlamento cuasi-post-bipartidista desde la Transición. Por primera vez se constituyen las Cortes españolas sin que esté nada claro el color del futuro gobierno, ni siquiera si habrá futuro gobierno sin tener que repetir las elecciones. Por tanto los posibles pactos para constituir la mesa del Congreso no están necesariamente ligados a un pacto de gobierno.

Cross-posted from European Tribune.

Prominent heterodox economist James Galbraith, who enjoyed an inside view of the last five months of Greek negotiations as an advisor to Yanis Varoufakis, writes the following for a mainstream American audience: Greece, Europe, and the United States (Harper’s, July 16, 2015)

What will become of Europe? Clearly the hopes of the pro-European, reformist left are now over. That will leave the future in the hands of the anti-European parties, including UKIP, the National Front in France, and Golden Dawn in Greece. These are ugly, racist, xenophobic groups; Golden Dawn has proposed concentration camps for immigrants in its platform. The only counter, now, is for progressive and democratic forces to regroup behind the banner of national democratic restoration. Which means that the left in Europe will also now swing against the euro.

The parallel between the Greek crisis and the Prague Spring, with a ruthless mainstream left crushing the hopes of an idealist left in defence of a system, is illustrated with poetic irony by the following tweet by a Social-Democrat finance minister from the former Czechoslovakia:

Meanwhile, in an interview with Jacobin Magazine which we have already been discussing in previous threads on European Tribune, Left Platform Syriza MP Stathis Kouvelakis says the following about the ideology of “left-Europeanism”: Greece: The Struggle Continues (Jacobin, July 14, 2015)

I think that in this case we can clearly see what the ideology at work here is. Although you don’t positively sign up to the project and you have serious doubts about the neoliberal orientation and top-down structure of European institutions, nevertheless you move within its coordinates and can’t imagine anything better outside of its framework.

I imagine that you could have written the same of Communist parties in the 1960s and their support for the Soviet Union. Out of the disappointment of the Prague Spring (on top of the invasion of Hungary a decade earlier) was born the Eurocommunist strand of the 1970s.