Advanced Mathematics 201: Sequences

The first semester of Advanced Mathematics has now ended. I’ve exhausted the rhythm topics I wanted to address, and I wanted to talk more about verticality. Today is sort of bridge between the two semesters, as what we’ll see can be applied to rhythm as well as harmony and melody: number sequences.

Divergent Series

Divergent series are sequences of numbers in which a term diverges from its starting point. This means that, with infinite iterations of the series, said term will reach ∞ or -∞. They are opposite to convergent series, the terms of which get closer and closer to a certain number. Let’s see a few divergent series, and then we’ll discuss about a few convergent examples.

The Harmonic Series, or the Overtone Series

One of the most well-known divergent series in music is the Harmonic series, where the denominator gets bigger with each step:

$$\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + … \sum_{n=1}^\infty \frac{1}{n}$$

This is an interesting series, especially for musicians, since it shows the overtone series.

The Harmonic Series, by Wikipedia user MusicMaker5376 (CC BY-SA 3.0). The number above the note represents the difference in cents to the displayed notes in 12-tone equal temperament.

The first and most obvious application of this series to music playing and composition is Just Intonation (JI), which we’ll cover more in a subsequent class. Briefly, however, JI is a method of creating a tonal system different than equal temperament. In equal temperament, you find a number (let’s say 12), and divide an arbitrary selection (let’s say an octave) with it, equally, using cents. In JI, there are multiple ways of proceeding, but one is to select a pitch (let’s say 440 Hz), and derive all other notes based on simple fractions.

Harmonics of a string. The fraction represents the length of the string at which a node is forced.

That’s where the harmonic series comes in. Since we’re talking about string harmonics, the ratios shown above represent the string length vibrating. When you’re playing a natural harmonic on a string, you’re forcing a node along the length of the string, so that it vibrates between smaller intervals, thus producing a higher pitched sound. If you follow it step by step you have:

FractionHarmonic (fret)Pitch (Hz)Approximate NoteDifference to Note (¢)
\(\frac{1}{1}\)0440A0
\(\frac{1}{2}\)12880A0
\(\frac{1}{3}\)7, 191320E+1.955
\(\frac{1}{4}\)5, 241760A0
\(\frac{1}{5}\)3.9, 8.8, 15.9, 27.92200C♯-13.686
\(\frac{1}{6}\)3.2, 312640E+1.955
\(\frac{1}{7}\)2.7, 5.8, 9.7, 14.7, 21.7, 33.73080G-31.174
\(\frac{1}{8}\)2.3, 8.1, 17.0, 36.03520A0
\(\frac{1}{9}\)2.0, 4.4, 10.2, 14.0, 26.0, 38.03960B+3.910
\(\frac{1}{10}\)1.8, 6.2, 20.8, 39.94400C♯-13.686

JI systems will take notes from the harmonic series, up to a certain arbitrary point, and bring them all back to the same octave, by divisions of 2. However, there are other ways to use this series creatively. For example, it could be a way to develop a rhythmic theme in Karnatic music, first using \(\frac{1}{1}\) (quarter notes), \(\frac{1}{2}\) (eighth notes), \(\frac{1}{3}\) (eighth note triplets), \(\frac{1}{4}\) (sixteenth notes), \(\frac{1}{5}\) (sixteenth note quintuplets), and so on. It’s also an essential tool for crafting polyrhythms, polymetric, and polytempo parts related to each other with simple ratios. Using swing feels, the series gives off harder and harder reverse swings, with 1:2 (iambic triplet feel), 1:3 (iambic dotted eighth feel), 1:4 (iambic hard quintuplet feel), and so on. There are multiple ways to use the numbers given by this mathematical sequence, and those are but a few ideas of mine, but feel encouraged to come up with different relations with musical concepts.

The Subharmonic Series, or the Undertone Series

Simply put, this is the mirror image of the harmonic series. In order to be consistent with the way I described the harmonic series, I will also refer to the fractions of the undertone series as the length of a string vibration. While you could achieve overtone frequencies on a regular instrument, say a guitar, using natural harmonics, you would have to extend the length of the string to achieve the subharmonic series’ frequencies. Here it is, in a table similar to the previous one:

FractionPitch (Hz)Approximate NoteDifference to Note (¢)
\(\frac{1}{1}\)440A0
\(\frac{2}{1}\)220A0
\(\frac{3}{1}\)146.667D-1.955
\(\frac{4}{1}\)110A0
\(\frac{5}{1}\)88F+13.686
\(\frac{6}{1}\)73.333D+1.955
\(\frac{7}{1}\)62.857B-31.174
\(\frac{8}{1}\)55A0
\(\frac{9}{1}\)48.889G+3.910
\(\frac{10}{1}\)44F-13.686

While it is impossible to achieve naturally these undertones on regular stringed instruments, Samuel Gaudet and Claude Gauthier, of the Moncton University, developed a three-spoked guitar that could naturally play such complex resonant frequencies. It is called the tritare in French, but I guess an appropriate translation for English would be the “tritar”. It is a modified electric guitar where, instead of the bridge where the string vibration ends on a regular guitar, the string separated into two and goes to two new nuts at the end of supernumerary necks, which are not fretted but are amplified. You can listen to it in action in a Facebook video by Samuel Gaudet, showcasing one of his compositions for the instrument:

I think it’s a shame that this instrument has spent almost a score of years in nigh oblivion, with next to no acknowledgement or use. I would love to get my hand on one of these tritars myself, and encourage every one of you do the same!

Another method for achieving subharmonic frequencies is with a technique called the “third bridge method”, where you add a node–a new bridge–under the string to achieve different resonating frequencies. More precisely, it should be placed at different string lengths than those found in the harmonic series, which would defeat the purpose because the strings can already resonate at these points. Many experimental music players–Glenn Branca included–used this technique, and some companies build instruments with the third bridge concept in mind. For example, this Australian guitar company, called New Complexity, offers an extended bridge that can be tuned and that is amplified with another pickup:

Please note, also that those are not microtonal instruments. This only alters the resonating frequencies that accompany the vibrating string. However, once could tune an instrument to the undertone series, and therefore create microtonal music. Still, the strings, tuned to undertones, would themselves resonate with regular string overtones (from the harmonic series), unless they are played on instruments allowing for undertones to resonate naturally, such as the tritar and extended bridge instruments.

The Fibonacci Sequence

Famous is the Fibonacci sequence! You probably already heard of it: start with 0 and 1, and add together the last number and the previous one to create a new number, then repeat. In equation form, it would be \(F_n = F_{n-1} + F_{n-2}\). This leads to the well-known series of numbers

$$0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …$$

This sequence is found in many places in nature, but that’s not due to some divine property, just that nature likes simplicity and efficiency, of which that sequence shows a lot. But what is its application in music?

It is many. I think that one recurring theme in this article is to be creative. You know the numbers, you know the sequence, therefore you only need to find ways to apply them. One example can be found in one of my old compositions. I’ll bring this example back later, because I included more than one number series to create this passage.

Here, I have taken the Fibonacci sequence to dictate which notes of the chromatic scale I shall play, A being 0. If the number becomes higher than 12, I just subtracted 12 until the note arrived within the first octave. Then I arbitrarily chose where to play the note and at what octave to play it, depending on playing convenience, so long as the note itself didn’t change. I took the first twenty-four iterations of the sequence, omitting the 0 at the beginning for no reason other than that I chose to do so. Here is what the notes look like when juxtaposed with the Fibonacci sequence:

StepFibonacci NumberNote
10A
21A♯
31A♯
42B
53C
65D
78F
813A♯
921F♯
1034G
1155E
1289D
13144A
14233D
15377C
16610G
17987C
181,597A♯
192,584C♯
204,181D
216,765F♯
2210,946B
2317,711G♯
2428,657A♯
2546,368A

Interestingly enough, even after 25 steps, not all twelve notes of the chromatic scale were played; missing is D♯. This still leads to a quite atonal piece, but it would be possible to use it with the degrees of a heptatonic scale, for more consonance: only subtract 7 instead of 12 to the resulting Fibonacci number until you have a figure that’s between 1 and 7.

Another way to use that sequence using the chromatic scale is to craft a scale with it using steps. “1-1-2-3-5” indeed provides a pentatonic scale, although a rather unpleasing one (subjective comment). Based on C, this scale would include C, C♯, D, E, and G. However, if you mix these numbers up a little and don’t use them in this particular order, you can find the Raga Devaranji (5-2-1-3-1), although these steps are poor approximation for the Indian Raga.

Similar to the Fibonacci numbers are the Lucas numbers, which, instead of starting with 0 and 1 are based on 2 and 1. While the rules are the same, the resulting numbers are quite different:

$$L_n = L_{n-1} + L_{n-2}$$

$$2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, …$$

Moreover, you can generalize the Fibonacci procedure and start with any number pair you can think of; this is called a G series:

$$G_n = G_{n-1} + G_{n-2}$$

Here is an example, with the starting pair 7 and 5:

$$7, 5, 12, 17, 29, 46, 75, 121, 196, 317, 513, 830, 1343, …$$


Convergent Series and Mathematical Constants

Contrary to divergent series, convergent ones get closer and closer to a certain number with every step. These numbers often end up being mathematical constants, so I will also use this chapter to talk about them, whether or not they derive from convergent series.

The Alternating Harmonic Series, and \(ln(2)\)

With a slight modification to how we iterate the aforementioned harmonic series, we get a convergent series that approximates the natural logarithm of 2, \(ln(2)\):

$$\frac{1}{1} – \frac{1}{2} + \frac{1}{3} – \frac{1}{4} + \frac{1}{5} – …$$

Here, instead of only adding the next number, we alternate between subtraction and addition. The series itself can hardly translate into useful musical material–prove me wrong, I only wish that–, but its mathematical constant–\(ln(2)\)–can serve as a selection of numbers from which to choose in order to create music. The only thing is that we’re using a decimal counting system and a duodecimal musical system: the 12 chromatic notes of the octave. Fortunately, a variety of online base converters can help you to switch from base-10 to base-12 and apply this string of numbers more conveniently to music. Usually, the symbols for the number 10 and 11 in base-12 are ↊ and ↋, or X and E, or A and B (there is no real consensus yet); I will use X for 10 and E for 11 out of convenience.

$$ln(2) \approx 0.6931471805599455_{(10)} = 0,839912483369XE59_{(12)}$$

For this example, let’s say our zeroth note is A again, we find this series of notes:

$$A – F – C – F♯ – F ♯ – A♯ – B – C♯ – F – C – C – D♯ – F♯ – G – G♯ – D – F♯$$

Other Mathematical Constants

This is but one example of a constant. Each constant will be different, here is a short selection:

ConstantBase 10 ApproximationBase 12 Approximation
Archimedes’ (\(\pi\))3.14159263.1848093
Euler’s number (\(e\))2.71828182.8752351
Pythagoras’ (\(\sqrt{2}\))1.41421351.4E7916E
Feigenbaum’s \(\alpha\)2.50290782.6050368
Feigenbaum’s \(\delta\)4.66920164.8044693
Apéry’s (\(\zeta(3)\))1.20205691.2511X28
Golden Ratio (\(\varphi\))1.61803391.74EE674
Euler-Mascheroni’s (\(\gamma\))0.57721560.6E15187
Conway’s (\(\lambda\))1.30357721.3786E88
Khinchin’s (\(K\))2.68545202.8285648
Glaisher-Kinkelin’s (\(A\))1.28242711.34804XX

From these constants, you can derive musical notes, rhythms, and many other interesting things for music! Moreover, you can pick any number, it doesn’t always have to start before the decimal point.


Complex Sequences

Some number patterns don’t converge or diverge towards infinity. Some behave seemingly randomly, which proves even more interesting for music, because it sounds less predictable! Let’s discuss some of them.

The Collatz Conjecture

This conjecture states that any number you choose will eventually reach 1 if you apply these two simple rules: if the number is even, divide it by 2; if it is odd, multiply it by 3 and add 1. It’s a fun problem to play with and try for ourselves, but don’t have any idea of finding a number that never reaches 1: scientists have looked to numbers in the trillions, and even they only have little more than a thousand steps to go through before eventually reaching 1.

Let’s bring the composition I introduced earlier, but here I’ll show you the complete passage:

This passage is a theme of 24 notes based off of the Fibonacci sequence for its melody, but its rhythmic pattern is based off of the Collatz conjecture. The time signatures used are rather arbitrary and do not reflect any underlying number sequence. I started with \(\frac{4}{4}\) out of convenience, and made the necessary adjustments so that the Fibonacci sequence always started back on a 1 beat. I decided to start with the number 24, and each step in the Collatz formula is separated by a short rest. As for the resulting number of the equation, it dictates how many notes are in an uninterrupted grouping:

$$24 \div 2 = 12 \tag{1}$$
$$12 \div 2 = 6 \tag{2}$$
$$6 \div 2 = 3 \tag{3}$$
$$3 \times 3 + 1 = 10 \tag{4}$$
$$10 \div 2 = 5 \tag{5}$$
$$5 \times 3 + 1 = 16 \tag{6}$$
$$16 \div 2 = 8 \tag{7}$$
$$8 \div 2 = 4 \tag{8}$$
$$4 \div 2 = 2 \tag{9}$$
$$2 \div 2 = 1 \tag{10}$$

I then cycled once again through the 4 – 2 – 1 series, because why not? I thought it sounded better this way. But once you hit any of those three numbers, you’re caught inside an endless loop of 4 – 2 – 1. Quite fascinating! Feel free to make your own and include it in your music in some way!

The Racamán Sequence

This is one that I only very recently heard of. It can be summed up by “subtract if you can, otherwise add”. The important part is that you need to subtract or add the amount equal to the step you’re in: at step 1, you add 1; step 2, you add 2; step 3, you add 3; step 4, you can finally subtract 4! If the number has already been used, you can’t go there a second time. This leads to a mesmerizing behaviour, which is very well explained and visualized in this Numberphile video. There also is the audio version of this sequence included:


Conclusion

I think that will be all for today. The topic is vast and can look quite overwhelming, but if you just start by looking at just one series, one constant, and take it apart and try to find ways in which you could turn it into music, you will have a lot of fun. That doesn’t mean, obviously, that the music will be good–it probably won’t be!–, but it will be more interesting than a lot of “good” music.


Advanced Mathematics

101: Morse Code
102: Microrhythms
103: Polyamory
104: Notes inégales
105: Introduction to Karnatic Rhythms

201: Sequences

On June 17 2018, this entry was posted.
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