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Why the chromatic scale has 12 tones

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 2, 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 perfect 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 perfect 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).

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¿Quién es quién en la #MarchForEurope de Madrid?

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?

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Sarkozy’s auto-da-fe

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.
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De pactos parlamentarios

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.

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The Prague moment of the European Left

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.
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Ptochotrapezocracy, or rule by bankrupt banks, is a term used (and, as far as I can tell, coined) by Yanis Varoufakis to describe the system of capitalism post-crisis [1]. While the European council applies itself to the task of apportioning blame for the clusterfuck that is the Greek debt crisis (though now it seems the Council President Donald Tusk wants nothing to do with the mess) it seems to me that policy in and around Greece will in the next days and weeks be determined by the state of the Greek banks, with all concern for “Europe” thrown to the wind in the face of more pressing demands.


A summary of the argument below the fold:

  • Lifting trade credit and import restrictions should be a priority for the Greek government, more than lifting restrictions on cash withdrawals
  • The greek banks are still solvent, but the ECB won’t allow ELA to be relaxed for the purpose of financing imports unless the ongoing bank run is stopped
  • The bank run can only be stopped by freezing deposits until the economy is stabilised, as done in Cyprus two years ago
  • If the ECB pulls ELA entirely, in addition to a deposit freeze there will have to be a deposit haircut, of maybe 35% of the €65bn of domestic time deposits (€60bn of domestic overnight deposits would be preserved, but frozen)
  • In order to avoid a bail-in of deposits even in the event of a recapitalization, the ECB’s supervisory arm SSM should have restructured the banks as early as February 4, when in fact the ECB shifted half of its liquidity provision to ELA
  • This is because €31bn of liquidity flowed out of the Greek banks in the month of January, the worst month of the 9-month bank run

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Varoufakis on ESM bank recapitalization

At today’s press conference (see my full transcript below the fold) between the Eurogroup and his one-on-one meeting with Mario Draghi, Yanis Varoufakis made a very significant observation on bank recapitalization:

The second reason concerned the financing. The proposal for financing the 5-month extension period of the current agreement – the funding proposal that was on the table – by the Institutions was technically inadequate: the numbers simply didn’t add up. And when we asked questions about, in particular, the utilisation of parts of the HFSF 10.9 billion – which was there in order to recapitalise banks and strengthen them in view of the problem with the NPLs – we received an answer that involved the ESM and that, as we know, effectively baked into this short-term arrangement a new program. Both technically and legally that did not make much sense and I’m sure that other member states would have a problem with it, Germany in particular.

What Yanis Varoufakis is referring to is the following. In the framework of the European Bank Recovery and Resolution Directive, the European Stability Mechanism has established a direct bank recapitalisation instrument. From the FAQ [PDF]:

Until the creation of the direct recapitalisation instrument, the ESM could only recapitalise financial institutions indirectly. In this case, the ESM provides a loan to the government of a euro area Member State. With these funds the government then recapitalises the financial institutions, which is how the ESM provided assistance to Spain. However, such assistance adds to the beneficiary country’s public debt, which could have a negative impact on market sentiment. …

Until the creation of the direct recapitalisation instrument, the ESM could only recapitalise financial institutions indirectly. In this case, the ESM provides a loan to the government of a euro area Member State. With these funds the government then recapitalises the financial institutions, which is how the ESM provided assistance to Spain. However, such assistance adds to the beneficiary country’s public debt, which could have a negative impact on market sentiment.

So far, so good. But crucially (my emphasis)

There will be conditions applying to the recapitalised institution, established under EU state aid rules. In addition, the ESM, in liaison with the Commission and the ECB, can add additional institution-specific conditions. These can include rules on the governance of the institution, remuneration of management and bonuses. Other policy conditions may be related to the general economic policies of the ESM Member concerned. They will be included in the Memorandum of Understanding (MoU) attached to the financial assistance.

This is something that the Greek government will apparently not stand for.

But this has more immediate implications. It’s looking likely that the ECB Governing Council may withdraw emergency liquidity assistance (ELA) from the Greek banks no later than Wednesday (the current “second programme” expires at the end of Tuesday), if not earlier (though I think it could also freeze ELA now and withdraw it on Wednesday). At that point, the Single Supervisory Mechanism could declare the four major Greek banks insolvent and set into motion the procedures foreseen by the Bank Recovery and Resolution Directive. Given the insolvency of the Greek state, “recovery” would involve ESM recapitalisation as outlined in the FAQ above. But, as Varoufakis hints, the creditors would attempt to impose policy conditionality on the Greek state, at which point the Greek government would refuse recapitalisation. This would leave resolution (i.e., liquidation) as the only solution for the Greek banks. And it is quite likely that resolution would involve a bail-in of Greek depositors as Greek banks have very little by way of long-term funding other than time deposits. If the ECB withdraws emergency liquidity from the Greek banks on Wednesday, a decision to wind down the Greek banks could be taken as early as Friday, two days before the referendum.

Note that this diagram by Sven Giegold MEP refers to an early version of the legislation and does not reflect later amendments.

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Merkel cuts Greece loose

With all eyes on the Eurogroup today, and the antics of the creditors speculating on a bank run in Greece, Angela Merkel’s declaration to the Bundestag on the upcoming European Council (press release with video) went completely under the radar, not that such speeches get widely reported anyway. And to judge by the media reporting, you would think that Merkel expressed support for Greece in the Euro. I mean, look at the BBC:

In her statement to the German parliament, Mrs Merkel said Germany was working hard to keep Greece in the euro, but said Athens had to follow through on reform commitments. “I’m still convinced – where there’s a will, there’s a way,” she said. “If those in charge in Greece can muster the will, an agreement with the three institutions is still possible.”

But, in reality, in a self-righteous tone Merkel blames the feckless Greeks for their failure to allow the EU to help them help themselves to their feet, and so sets the stage for cutting Greece loose from the Euro. I expect this will be clear from the transcript below (with my translation). Read more…

The double-dip recession as a Eurozone policy failure

The Spanish Economic Association (, from it Spanish acronym AEE) has introduced its own version of the business cycle dating pioneered by the US National Bureau of Economic Research. Here I look at the business cycle dating by NBER and AEE, as well as the Eurozone business cycle dating by the CEPR. My conclusions are:

  • the 2008 recession in Europe is indeed a knock-on effect of the US subprime crisis
  • Zapatero was right when he said on the campaign trail for his March 2008 re-election that there was no recession in Spain
  • bank failures in the Eurozone are a consequence of recession and not the other way around
  • automatic stabilizers and stimulus worked in 2009
  • the sovereign debt crisis was entirely artificial and manufactured
  • the second dip of the Spanish recession was caused by Zapatero bowing to Eurozone pressure and instituting austerity
  • the second Eurozone recession was caused by Trichet’s ‘impeccable’ rate raises in 2011
  • the end of the Spanish recession was helped by the European bank rescue of 2012
  • Draghi’s extraordinary monetary policy measures have not been enough to pull the Eurozone out of recession but have prevented it from getting worse

There is a timeline at the bottom of this post so you can draw your own.

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