Advanced Mathematics 202:
Decimal Music

The French Revolution brought humanity many things, among which are the values of liberty, equality, and secularism. However, in their hot-blooded revolt, they also brought the widespread use of the decimal system, taking the place of the old-fashioned Carolingian system, which resembled somewhat the Imperial system of today’s USA. They even thought of bringing up decimal time, and a decimal calendar, where the twelve months of the year each had three “weeks” of ten days. At the moment of writing these lines, we are Octidi the 28th of Messidor (Monday, July 16th, 2018). Each of these days would then be divided into ten hours, which themselves were divided into ten, and so on and so forth until as short as you need. This was in official use about two years, from 1793 to 1795, but never really caught on. One of the only things that the French didn’t try to decimalize is music. That’s why I’ve started to write this admittedly exploratory article, where I will try to conjure, to the best of my ability, a decimal music system. This is purely experimental, but it should be quite entertaining.


Décaphonie — The Decimal Harmony

Today’s Western musical system is based on an octave, divided equally in twelve notes. This octave is created by doubling the frequency of a given pitch. Say we start with 440 Hz, the current agreed upon concert pitch for A. That means that its octave, still called A, but higher pitched, will be 880 Hz. This is very basic stuff, if you’ve ever read about music in your life.

Something slightly more advanced is the notion of twelve equally-spaced notes. Since, when you keep doubling a pitch it increases exponentially, to divide an octave equally we need to apply a logarithmic transformation. When we do that, we obtain what we call cents (¢). You’ll notice that if you play a tone at 440 Hz, then one at 540 Hz, 640 Hz, and so on, they will not appear equally spaced. The higher you go, the smaller their distance will seem to be. However, when you play a note, then another note 100 ¢ higher, then 200 ¢, 300 ¢, and so on, they will sound equal from one another. That’s how a regular piano or guitar is designed.

We’ll need two equations. First, to find the cents value between two notes in Hertz:

$$n = 1200 \cdot log_{2} \left(\frac{b}{a}\right) \tag{1}$$

where \(a\) is the smallest note, \(b\) is the biggest, and \(n\) is the value in cents between the two notes.

The second one is the inverse, where we know the value in Hertz of the first note, and the difference in cents we need to apply, but we want to know the pitch of a second note:

$$b = a \cdot 2^{\frac{n}{1200}} \tag{2}$$

Now, we have to throw all of that knowledge through the window, burn all the books, and guillotine Boethius, in true revolutionary fashion! We will keep referring to cents throughout this article.


Starting the Cycle

First of all, we need to announce what is the basis of our musical system is. You will agree that the doubling of a given pitch is rather arbitrary, and rather non-decimal. Therefore, I hereby declare that a tenfold increase in pitch value will be our unit of musical height. I will name it la décime, from Latin decimus to French, as an analogy with the octave. “The decime”, pronounced like the first syllables of “December” or of “decimate”—whichever you prefer—could be a potential English translation. So, the difference between a note at 100 Hz and 1,000 Hz is one decime.

We also need a concert pitch, if only to make convenient examples without always having to change or decide on a new frequency. In pure decimalist fashion, let’s settle on 100 Hz. In our current musical system, this would be like a really sharp G2, which is originally at 98 Hz (in our 440 Hz concert pitch octavian dodecaphonic system); precisely, it would be 34.98 ¢ sharp (calculated using equation 1).

So, our first decime’s range is from 100 Hz to 1,000 Hz. In cents, this range covers approximately 3,986.3 ¢. Once again, when we draw the parallel to our Western musical system, this would mean from +34.98G2 to +21.30B5—a range of three octaves and a 5/4 (a just intonation major third). This is where things get interesting. That means that decimal equivalence will happen at a just intonation interval!

If you didn’t follow me, there, I use “decimal equivalence” as an analogy to “octave equivalence”. Octave equivalence is why we use the same letters for notes regardless of the octave they’re in. Whether you play a C on bass, or a C on soprano, we view the two notes as equivalent and don’t treat them differently. These notes are what we call dyadic–equivalent notes will always be a power of two of one another, in terms of Hertz. In our decimal system, what we treat as equivalent will sound three octaves and a just major third higher than the original pitch.

Musical systems without octave equivalence are not a new thing, although they sound quite foreign to our ear, used very strongly to hearing octaves. Indeed, non-octave repeating (NOR) scales have existed for a while, and perhaps the most popular of these scales is the Bohlen-Pierce scale. If you want to read more on NOR scales, click on this link to read my article on xenharmonic music on Heavy Blog Is Heavy. You can hear an example of Bohlen-Pierce on Sevish’s song “Orbital”, or Zia’s “Love Song”.


Carving Steps

Now that we’ve got our decimal octave—the decime—, we can start carving steps into it to make good use for music. In boring decimal fashion, let’s divide the decime by ten. I feel like these steps should be called centimes, because they’re one order of magnitude smaller than decimes. Each centime has a difference of about 398.63 ¢ with one another.

This value has nothing of much interest. It doesn’t relate to a just interval. It’s just a slightly flat major third. Let’s nevertheless superpose it to our Western dodecaphonic music, so we have some reference. Here’s our decime, starting on +34.98G2, the ten centimes, and the first decime equivalence:

+34.98G2+33.64B2+32.30D#3+30.90G3+29.54B3+28.15D#4+26.78G4+25.44B4+24.07D#5+22.68G5+21.30B5

The bold notes are the starting notes of a decime, they are judged equivalent in our decimal music system. Here’s the same decime, but in a more visual presentation:

Figure 1. The 10 centimes of a decime, based on the starting note at 100 Hz. The text represents the deviation, in cents, from the written note. The first and last notes are decimal-equivalent.

The keen eyed among you might have noticed one thing: three centimes is almost equal to an octave. In fact, it’s so close that it would be almost indistinguishable from a dodecaphonic octave, if they were to be played back to back. Of course, their 4.08 ¢ difference would be noticeable if they were played at the same time, however, because of the interference pattern. Three centimes is therefore almost equal to a 3-EDO (equal divisions of the octave) scale; stacked major thirds.

But we’re not done yet!


Inception

With the resolution of a major third, our centime is hardly a convenient tool for composing music. The only way to get the feel of a minor third would be to go to higher octaves and wait for a good approximation. Fortunately, the decimal system is infinitely recursive. That means that we can always take one step and divide it further into ten sub-steps. Indeed, with only one more iteration of this process, we achieve a much finer definition. Imagine the major third being divided into 10 equally-spaced notes. That would mean around thirty notes per octave, at this point! Of course, we don’t care about octaves. Octaves are overrated. So let’s continue with centimes, and the smaller millimes.

One millime is one hundredth of a decime, and one tenth of a centime. With an easy logical step, we find that the difference between two millimes is 39.863 ¢. That’s smaller than a quarter-tone, so that five millimes is almost equal to a whole tone (199.3 ¢ vs. 200 ¢).

Let’s talk about equivalence before going further. We’ve talked about octave equivalence, or our analogous decimal equivalence. That’s the only equivalence in regular, dyadic, dodecaphonic, Western music, but our decimal system, with its recursive behaviour, allows for different kinds of equivalence, not unlike a fractal. When we divide a step—like a decime—into ten subdivisions, we are basically saying that the steps into it are the same as those in the decimes above and below. Think of it like a ruler of one metre long. When you look at it, it’s divided into ten decimetres, each divided into ten centimetres. The “1 cm” mark is no different from the “11 cm” mark: it’s the first one of its subdivision. The only difference is that one is farther away than the other, just like one note will be higher than the other, but they are both equivalent in every respect.

That means that we can have decimal equivalence, but also centimal equivalence, and further millimal equivalence, and so on. How to treat them is perhaps beyond my grasp, at the moment; I’m merely pointing out the fact that arises from the rules of the decimal system. Perhaps multiple degrees of equivalence is not a worthwhile idea, but I dare to think it can be an interesting property of decimal music. Say, you feel that a perfect 24th for equivalence is a bit too broad for you, you could decide to instead focus on the centimal equivalence, which is close to a major third. Or, why not?, you could also decide to completely disregard equivalence and go with what feels right!

Back to our topic, here’s the centimes in a range of one decime, starting on that 100 Hz note we previously established as our concert pitch. Here, since the notes are closer together, I will start using quarter-tonal notation (reverse flat sign for half-flat, and half sharp sign) in addition to the cent correction.

Figure 2. The 10 millimes of a centime, based on the starting note at 100 Hz. The text represents the deviation, in cents, from the written note. The first and last notes are centimal-equivalent.

I think I need not illustrate this concept further. If you’ve ever been acquainted with the decimal system of measurement, you’ll quickly understand that it’s easily applicable to music via the cents measurement. More precisely, the subdivision of a starting pitch and its tenfold multiplication.


A Short Interlude: Why Not Something Else?

Of course, one of the critiques of this thought experiment will question this basic premise. Why multiply a pitch by ten, to then divide it in ten equally-spaced notes? Why not start with 1,000 ¢ and then subdivide everything by ten from here on? Well, the answer is very simple: it’s not out of the ordinary enough. Obviously, if you’re not interested in the “out-there”, you’re free to take your 1,000 ¢ and divide it by ten: it’ll give you steps of 100 ¢, which you can play on a regular instrument and easily notate using a wide array of notation softwares. The only thing that will behave differently from Occidental music theory is octave equivalence. Whereas your newfound “octave” will not cycle back after twelve steps, but rather ten, making it sound like a flat 7th. There’s no reason why you shouldn’t explore this path, if it interests you, but I chose another path; completely arbitrarily.

Why not take the octave, and divide it into ten parts? That would be decimal, right? Well, yeah… That temperament is called 10-EDO, and many pieces have already been written using that system. Once again, it’s a perfectly valid path to explore, but it’s not the one I chose. I wanted a more… revolutionary perspective. (Although I don’t claim it will have an effect nearly as important as the French Revolution had.) For an example of 10-EDO, listen to Zia’s “Who Loves You… Me!”.

The point here is that the way I decided to do things for this article is purely arbitrary. It’s maybe not the best way, and definitely not the only way we could come up with a decimal musical system. It’s just a fun thought experiment, so enjoy it! Now, let’s go back in line.


Decimal Rhythm

I’m fairly content with millimes and the way we uncovered the decimal harmonic system. Some might wish to further subdivide—into micromes, nanomes, or even picomes!—, but 39.863 ¢ is perfectly fine for me and for this article to continue. However, music is the conjunction of harmony and rhythm, therefore an entire half of decimal music is still untouched!

I’m not fooling anyone, however. Anyone with even a moderate knowledge of rhythm will already have guessed that this part will be much shallower. While there are many interesting and complex ways to perceive harmony, and organize notes and pitches, it seems that moving forwards through time is inherently less complex. Although, if you’ve already read this column in the past, you’ve come across some pretty challenging and incredible rhythmic ideas! Rejoice, for they all can be adapted and applied to decimal rhythm as well!


Readjusting the Measures

Let’s start with the basics. In modern Western music theory, \(\frac{4}{4}\) is known as common time. The vast majority of what you probably hear on the daily is in that time signature. Everything is divided into four, and then further subdivided into four, and so on. In order to make everything in base 10 from now on, we’ll have to change the numbers here.

If you’ve read my post on non-dyadic time signatures, you have already that notion in you. If you haven’t, please take some time to read the chapter dedicated to that.

All we need to do, to be rhythmically decimal, is to turn \(\frac{4}{4}\) into \(\frac{10}{10}\). Musically, this is no different than writing \(\frac{10}{4}\) at a different tempo, but the point is to change the standard itself, so that it becomes base 10. From now on, this is the standard measure, this is the common decimal time:

Figure 3. An example of a common decimal time measure, including 10 crotchets.

This common ten-time measure doesn’t implicitly dictate how the notes should be grouped together and felt. It’s not obvious to the system to splitting the bar into two groups of five notes, and then into groups of three and two beats. If you wish to feel this rhythm, you’re free to do so! And, since this is only one of the infinite possibilities of time signatures, you could also quite simply create bars of \(\frac{4}{10}\) if you wish to play that silly Western pop music in our advanced decimal music system. The audience will never notice the difference!

Nota bene: I will continue using the British appellation of note length, since its names don’t imply a dyadic subdivision of the measure as much as “half”, “quarter”, “eighth”, and “sixteenth” notes do. Just get used to it, for now.


The Decimal Nature of Time

If you’re, like me, somewhat used to seeing and understanding music notations, prepare for a (slight) shock. Because, yes—maybe you saw it coming—, each of these crotchets will be subdivided into ten quavers; and, alternatively, these ten crotchets add up to a minim. That also has the repercussion that a “whole note”, a semibreve, lasts for ten common decimal measures. Strange, but not unworkable.

Figure 4. You guessed it: 100 quavers fit into the common decimal time measure. 10 quavers fit into a crotchet. (Click to enlarge)

Here we go, five scores of quavers and you know naught about what to do with them, do you? This may look unsettling, at first, but, when you think that you can use any tempo you like, then you realize that crotchets and quavers can have any length of time desired. I have to roll back on what I’ve previously said about playing a song in \(\frac{4}{10}\). It would not be nowhere near equal to a \(\frac{4}{4}\) composition, when you take into account that notes are not divisible by two, but by ten.

While this may seem like an insurmountable obstacle, I believe this is only due to the shackles of Western music theory. Of course, we’ve grown up and learned to use the tools that the theory gave us, and honed our craft over many years, decades, even! And now, to come across such a wildly different vision of music might be daunting. You might ridicule it, or at the very least brush it with the back of the hand and not think twice about using it. However, I think that its limitations—or, rather, its idiosyncrasies—can enhance creativity by forcing you into a new, unexplored area.


Conclusion

This was a fun thought experiment: trying to come up with a new musical system. Of course, it’s far from being a complete system just yet. Fortunately, the world is full of musical nerds, and, especially, xenharmonic nerds. Hopefully, this proof of concept proves to be interesting enough to be taken by someone with more knowledge and talent than me so that it can be fully fleshed out and theorized upon. Maybe this was a bad idea all along! In that case, well, it will at least have been fun to conjure.

As for me, I’ll try to come up with an actual decimal composition when time allows it. However, I’m not the most familiar with microtonal softwares, so if you’re more seasoned than me and feel like it’s a challenge for you: go ahead and create the first ever decimal music composition! I would be glad to hear it for certain!

Hopefully this has entertained your brains as much as it has mine!


Read More

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

201: Sequences

On July 17 2018, this entry was posted and tagged: ,
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