6. Twelfth fright

by anyarchitectanyarchitect on 1220346740|%e %B %Y
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In this article, I shall attempt to tie together some disparate concepts: architecture, mathematics, lovers, trains, music, stories and well, Shakespeare. None of them in any particular order. I hope it will make some sense.

I remember quizzing an architect at her thesis presentation. I could see she had slaved through weeks of toil, and she was justifiably apprehensive at meeting me — after all I have this propensity to go off laterally into areas that are so ill-defined. Even though I did empathize with her; the moment she started explaining her design to me, I had to sadly ask the question that I was notorious for: I asked her to explain the objective aspects of the design to me. Give them to me logically, I said. She was bewildered. I could see her frustration building up. If she was a bit older, maybe she would have loved to shake me by my shoulder and shout "Derrida" or "Foucault" or the names of any of those other European philosophers who are discussed lovingly behind my back, and about whom the students collectively thought I knew nothing about. (Well, I actually don't know much about them. I know a bit about of their thoughts and I do suspect that they must be knowing their mathematics well enough when it comes to getting the change after buying bread for their respective houses)

I remember then telling myself, "Oh no. Twelfth fright, once again" and I had to digress into this explanation:

Here's a scene from some movie: A man waiting anxiously for his beloved to turn up at a railway station. The atmosphere is all charged up. There is a train driven by a steam engine, with all the various hissing and steam all around. The emotions are palpable. There are people milling about, attending to last minute things just before the train left. The man's head is craning above the crowd, trying to catch a glimpse of her somewhere. The peddlers are busy: yelling and screaming to get their monies; oblivious to the ongoing human drama. The clock is ticking fast, and there is a sense of urgency in the hiss of the steam. So many people so close together, but the real close person is so far away…..and then…a silhouette of a female form appears in the entranceway. The audience can easily make out that it must be the same woman the man is looking for but he has yet to see her…. and then the rest of the story unfolds.

What would make the above romantic scene terribly boring? Well, there can be several: How about getting into the details of steam generation? Or the mathematics used in the micro-economics of peddling on a railway station. What about the description of the neck muscles that come into play when a person's head is craning upward. …. and so on. I may end up destroying the whole dramatic impact of the movie scene. But then, I am certain that there would be people who would still find beauty and mystery //even // in those seemingly boring descriptions. That is what Richard Feynman was attempting to explain in his famous BBC interview (the relevant excerpt from that interview is in the PDF file available at www.anyarchitect.org).

Life will always continue in its multi-hued, multi-layered fashion. We have no choice but to accept that statement to be true. In this topic, I am attempting to highlight such an obvious aspect. However, we should not get completely waylaid by the complexity of the processes either. Instead, let us advance objectively wherever we can. Objectivity can sit alongside subjectivity. Creativity can not be bridled because of such a pairing. In fact, I can dare say that can that scene remain romantic if the steam engine was not generating steam, and the peddlers were not into those micro-economic issues, and so on?

This issue is best explained when we examine the mathematics of equally tempered scale in music.

Long time back, musicians were using the naturally tempered scale. That scale was based on rational intervals (once again attributed to the Greeks). Here is the problem description (as quoted from a website, see references )


The problem with a major scale, minor scale, or any combination of scales which have unequal intervals is that musical melodies cannot be transposed to a different tonic. For instance, since a major scale is defined to have exact ratios of frequencies 9:8, 5:4, 4:3, 3:2, 5:3, 2:1, etc., changing the tonic from a C to a D (using a C with a frequency of 264 Hz) in a major scale would result in a piquant difficulty explained here

Here, while D, G, B, and D' have the same frequency in both keys (and E and A are close), F and C' are off by quite a bit. Having the capability to play a given piece of music in either the key of C or the key of D would thus require separate keyboard keys or frets on a fretted instrument for the "same" note. As a result, it would not be possible to play a tune a fixed number of steps lower, because this transposition would require frequencies which the instrument was incapable of playing.


To explain the above problem in another words. Say, there was a piano that was tuned using the natural scale starting from "C" (i.e. 264 Hz) and a singer had practiced to sing a song using that key. One day, she turns up with a problem with her voice. She realizes that she cannot sing using the C key. She can however sing at a higher pitch (i.e. on the D key). Now using the ratios as specified in the natural scale (9:8, 5:4, etc) , the frequencies of the subsequent notes would have to be as given in the above table. But unfortunately, the piano is already tuned to the key of C and it cannot play the required frequencies now on some of the notes. (In fact, only B, D' and D would be at the correct frequency).

This is not a unique situation with the piano. The problem would have happened even with fretted instruments such as a guitar because the frets would have been hammered in at the precise locations of the C scale and not the D scale. Shifting the frets, like the retuning of the piano, would be impossible if the singer comes with a voice problem. In fact, this need to shift the key can only be handled by non-fretted instruments and wind instruments, where the musician has total control over which frequency the instrument produces.

The need for shifting the scale is a common enough occurrence nowadays in music. And it has been solved without reworking the instrument. The shifting of a scale is called "transposing". For example; nowadays, on a guitar, it is achieved by tightening a vice type of device called a capo around the neck of a guitar. How did this happen? Simple: Around the Renaissance, a few musicians who were also mathematically inclined took a pen and paper and worked out a different method for determining the frequencies to be used in music. They called this the equally tempered scale.

This is how it works: First of all here is a definition: An octave is nothing but twice the frequency of a particular note. Within an octave, there are 12 notes (the total number of black and white keys inside an octave on a piano). As per most studies , when an octave is played alongside its parent note, it is quite pleasing (next to playing the same note twice). So the equally tempered scale promoters decided to stick to the Grecian rule of making an octave twice the frequency of the parent note.

Take any one note (f), and 12 notes away should be its octave (2f). Now, each note after f (say f1, f2, f3 … etc.) should also have their respective octaves ( 2f1, 2f2, 2f3, etc.) spaced 12 notes away from each one of them. If you now do some simple mathematical introspection, you'll realize that the only way that can happen accurately is if you get the frequencies of f1, f2, f3, etc. to fall on a geometric scale where the geometric ratio is the twelfth root of two.

The frequencies of the notes in an equally tempered scale no longer adheres to the Grecian ratios (or the natural scale, as it is usually called. Note once again the use of the word natural). The frequency of each note is actually slightly different from those proposed in the natural scale. There are many purists who insist that the older natural scale was …ummm… more natural than what is seen in the equally tempered scale. They get the Twelfth fright :-). The differences can be noticed by expert listeners, but most of us do not. All of western music now use the equally tempered scale, for the simple reason that keys can be transposed if required.

Curiously enough, much of classical Indian music approached the same problem in a different manner: Our fretted instruments used a simple device of movable frets to handle the issue of transposition. Therefore you'll see a sitarist carrying a small wooden mallet which is used to finely adjust the frets on the fretboard of the sitar to yield the correct natural frequencies. Mercifully, classical Indian music did not use any piano type of musical instrument. (No, the harmonium is not considered as a classical Indian musical instrument. It came into use much later.)

So, what is this long winded explanation in aid of? I could not think of a better symbol of creativity than music, and I wanted to show that even in that august subject there are some intricate mathematical issues. Next time, you attend a rock concert or listen to some soothing old Kishore Kumar songs, or whatever else you may fancy; I hope that occasionally you would remember that parallely some maths is also at work. In one of my earlier articles (regarding Godel), I had stated that it is impossible to completely make a fully deterministic model representation of architecture. That is sometimes used as an excuse for lack of objectivity. Not so. One needs to advance objectively in as many aspects of a problem description as one can. Only then can one achieve a multi-layered solution. Hence I have no direct quarrel with any European philosophers, as long as the model gives me some objective answers too.

I often ask my students this question: What distinguishes pulp-fiction from real great works of literature? The answer is that great literary works always have multiple layers of meanings.

Now that I have used a reference of music in this article; let me clarify another common conception. I have heard of references of music theory in architecture. Unless I can see some real strong mathematical principles used in music that are actually valid in architecture also, I would state that such references are poetic at best.

Now a bit of Shakespeare to end this article. In the comedy Twelfth Night, Shakespeare weaves a tale of gender transformation and complex misunderstandings. The girl dressed up as a male, falling in love with a male. All kinds of confusions ensue. Fortunately, in the end; the twin brother lands up and the girl can reveal who she really is, and gets married to the one whom she loves. (I hope I have not offended any Shakespeare expert in the audience for I am no expert on Shakespeare by giving such a short summary of a wonderful play.)

I therefore shall now leave you to establish the multi-layered connections to this article. (subjectively! :-) )


  1. http://www.ericweisstein.com/encyclopedias/music/EqualTemperament.html

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