Music & Mathematics (01)

The direct transposition of mathematics into music usually produces quite trivial results. However, it is some exceptions. One is this.

Compute the first N (we put 100) digits of a transcendent number, one of those numbers that cannot be reduced to a fraction that have infinite digits after the comma, such as π (Pi-Greco), φ (Phi, the golden section) or e (the base of natural logarithms).
In this case, we use the golden section φ (Phi). The result is
1.61803398874989484848204586834365638111772030917980576
28621354486227052604628181890244970720720418939113748
Take the figures as they are, without paying attention to the decimal point, that is
1,6,1,8,0,3,3,9,8,8,7,4,4,9,8,8,8,4,8,2,2,4,5,8,6,8,3,3,3,6.5, … etc.
This will be our basis to produce heights and durations. The idea is that the generation of decimal digits in these numbers is not entirely random. In fact, the figures do not have the same distribution, but above all the series is full of repetitions, repeated configurations, etc.
To get the heights, we transform our digits into notes with a coding. We put 1 = the bass of the plane and go up for semitones. So the lowest do will be 4 and then, for octave, the others do will be 16, 28, 40, 52, 64, 76, 88.
Now, we heat the entire series, which goes from 0 to 9, so that the minimum (0) corresponds to 40 (C3) and the maximum (9) to 64 (C5). We get the following series of notes:
42.56,42,61,40,48,64,61,61,58,50,50,61,64,50,61,61,61,45.40, … etc
In general, it appears that
- 0 = 40 = C3
- 1 = 42 = D3
- 2 = 45 = F3
- 3 = 48 = G#3
- 4 = 50 = A#3
- 5 = 53 = C#4
- 6 = 56 = E4
- 7 = 58 = F#4
- 8 = 61 = A4
- 9 = 64 = C5
Of course we could also have used another interval, more or less wide than 2 octaves obtaining different results.
Now we place the duration. We decide that
- 0 = semicroma
- 1 = croma
- 2 = semiminima
- 3 = minima
And we heat the numerical series as above, but narrowing it between 0 and 3 without decimals. It follows that
- 0, 1, 2 = 0 = semicroma
- 3, 4, 5 = 1 = croma
- 6, 7, 8 = 2 = semiminima
- 9 = 3 = minima
obtaining the following series: 0.2,2,2,0,1,1,3,2,2,2,1,3,3,3,1,2,1,2, … etc.
In this example we always use canonical duration (not irregular) to have no difficulties in writing. However, nothing prevents you from using irregular durations, facing some writing problems. P.es, also using the duration of a terrated chroma, you could find a succession such as: semiminima – Croma Terzinata – semiminima and I want to see how you write it. Oh my God, in many contemporary songs it is even worse, but in this example we are on the simple.
well. At this point we have a series of heights and one of duration of equal length. We decide a metronome and play. here is the final result . Nice, nervous, a bit ’ to the Xenakis even if less complex.

At the careful reader a peculiarity will not have escaped. Using the same starting series for heights and durations, the duration increases as the heights rise. To avoid it, just retrieve one of the two resulting series. In this example we retrieve the durations.

By changing the extension, then the results are different. here the heights are redeemed between 4 and 64 using a large part of the floor extension and making it practically unsuitable from a human at this speed.
Finally, here an overlap of this this fragment (1-64) and the previous one (40-64 with retrograde duration)

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