what happens when computer

So, in my previous post on the topic of music theory, I described the basic system of building a tuning by repeatedly applying a musical interval to a root note: when this process is performed with a perfect fifth, we get twelve distinct notes before producing a note which is almost identical to the original root note, which is the original genesis of our twelve-note system. I also described how we could perform a different transformation—dividing the space of the octave into equally-spaced chunks—and get a similar but non-identical tuning that nonetheless suffices for most modern music.
At one point, I mentioned and glossed over a stage in musical history, which I want to go into more detail about now. In particular, I want to describe the family of tunings that were popular after Pythagorean tuning fell out of favor in the 16th century until Equal Temperament became popular in the 19th century: the meantone temperaments.

Quarter-Comma Meantone

One big problem with Pythagorean tuning, aside from the wolf that I demonstrated in the last post, is that there are several other pleasing ratios which are very poorly approximated by the notes available. One of those ratios is the one we call a major third: that is, the interval that corresponds to a 5:4 ratio. In Pythagorean tuning, the closest approximation we have is 81:64, which is out-of-tune enough that many musicians who composed for Pythagorean tuning would avoid using it entirely!
Beginning in the sixteenth century, musicians began to adopt alternate tuning systems where the 5:4 ratio—also called the major third—was better approximated and more usable. One way of doing this was to build a system of tuning using the same techniques as in Pythagorean tuning—that is, generating a set of tones by repeatedly increasing frequencies by the 3:2 ratio, also called a fifth—except unlike in Pythagorean tuning, they decided to choose a ratio that differed slightly from the exact 3:2 ratio we used before. Systems that follow the same process as Pythagorean tuning but with a slightly different approximation of the fifth are called meantone temperaments.
But what ratio do we choose? We could just keep trying slightly different approximations of 3:2, seeing what works and what doesn't work, but that's not a very principled way of designing a system of tuning. Instead, let's start with our desired goals and then figure out what ratio we want by working backwards. Let's stipulate right up front that we want a system where major thirds are related to the root by exactly a 5:4 ratio, and then find a frequency that's a close approximation of 3:2—let's call this unknown ratio \(\rho\)—that nevertheless accomodates our initial stipulations.
The variable \(\rho\) is not a standard variable name, so don't use it around music theoreticians and expect them to know what you mean. I just wanted to have a shorter name than "the approximation of the perfect fifth ratio".
So let's start by asking: how did we get the value of a major third in Pythagorean tuning? If we run the process of generating Pythagorean tuning again, we can see that the frequency which most closely approximates the 5:4 ratio was generated after four applications of the 3:2 ratio. To demonstrate this, I'm going to reproduce the relevant parts of Pythagorean tuning in this chart—but in this chart, I'm not going to wrap the frequencies back around into the same octave:
Frequency Formula Ratio w/r/t root
400 Hz \(400 \times (\frac{3}{2})^0\)
600 Hz \(400 \times (\frac{3}{2})^1\) 3:2
900 Hz \(400 \times (\frac{3}{2})^2\) 9:8
1350 Hz \(400 \times (\frac{3}{2})^3\) 27:16
2025 Hz \(400 \times (\frac{3}{2})^4\) 81:64 (which is \(\approx\) 5:4)
In the previous post, I had adjusted the frequencies back down an octave whenever they left the 400–800 Hz range so that they'd be easier to visualize together, but here I didn't: consequently, the frequencies 900 Hz and 1350 Hz are one octave higher than the ones we reported in the last post, and the frequency 2025 Hz is two octaves higher than the one we originally reported. That is to say, 2025 Hz is two octaves plus a major third above 400 Hz in Pythagorean tuning.
But wait! We know the octave ratio is 2:1, and we're currently trying to build a system where the major third is exactly 5:4. Two octaves plus a major third in the system we're attempting to create should be exactly the same as the root multiplied by \(2 \times 2 \times \frac{5}{4} = 5\) (that is, 2000 Hz), but as we already know, Pythagorean tuning gives us the approximation \((\frac{3}{2})^4 = 5.0625\) (or 2025 Hz). In the new system we're building, we still expect to get to a major third by four applications of our approximated fifth, so let's replace the exact fraction \(\frac{3}{2}\) by our variable \(\rho\), and apply some basic algebra to the the resulting equation \( \rho^4 = 2 \times 2 \times \frac{5}{4} \): our mystery frequency turns out to be \(\rho = \sqrt[4]{5}\).
Now that we have that frequency, let's use it just like we used 3:2 in Pythagorean tuning and look at the resulting scale. This time, I'm restricting the resulting notes to the 400 to 800 Hz range and then sorting the resulting frequencies. For comparison, I'm also going to include the values for Pythagorean tuning:
Freq. (Meantone) Cents (Meantone) Freq. (Pythagorean) Cents (Pythagorean)
400.00 Hz 0.00 400.00 Hz 0.00 cents
417.96 Hz 76.04 427.14 Hz 113.68 cents
447.21 Hz 193.16 450.00 Hz 203.91 cents
467.29 Hz 269.20 480.54 Hz 317.59 cents
500.00 Hz 386.31 506.25 Hz 407.82 cents
522.45 Hz 462.35 540.60 Hz 521.50 cents
559.02 Hz 579.47 569.53 Hz 611.73 cents
598.14 Hz 696.58 600.00 Hz 701.95 cents
625.00 Hz 772.62 640.72 Hz 815.64 cents
668.74 Hz 889.73 675.00 Hz 905.86 cents
698.77 Hz 965.78 720.81 Hz 1019.55 cents
747.67 Hz 1082.89 759.37 Hz 1109.77 cents
Well, our fifths are still pretty good approximations: they're within about five cents of a perfect fifth, which is 701.95 cents. And now, our major thirds aren't just approximated: they're exactly perfect! The rest of the scale is a generally reasonable approximation:
The system I've demonstrated here is called quarter-comma meantone. Why? Well, remember that, in this field of music theory, the word comma refers to a 'gap' between two frequencies: in this case, the gap being referenced is the syntonic comma, which is the difference between a perfect third and a third as represented by Pythagorean tuning—or to describe it in the same terms as above, the difference between \((\frac{3}{2})^4\) and \(\rho^4\), which ends up being about 21.51 cents. Consequently, in our system, our fifth ends up differing from a perfect fifth by exactly one-quarter of the syntonic comma, or about 5.38 cents, as demonstrated in the following diagram:
There are a lot of different kinds of meantones temperaments, though, and all of them begin by choosing a different approximation of the fifth. Let's look at a different one!

One-Third-Comma Meantone

Quarter-comma meantone quickly became the most common exemplar of meantone temperament after it was first proposed in the 16th century, but it was also far from the only one proposed or used. One of the first rigorous mathematical descriptions of quarter-comma meantone comes from the 16th-century music theorist and organist Francisco de Salinas. In his writing, he also described a different meantone system with a similar motivation: instead of trying to build a meantone system that accomodates perfect major thirds, what if we built a meantone system that accomodates perfect minor thirds?
A perfect minor third corresponds to a ratio of 6:5, which has a slightly more dissonant, haunting sound than most of the ratios we've been working with. A perfect minor third sounds like this:
I'll elide the math behind this tuning, but it proceeds very much like the process above, except that the Pythagorean approximation of a minor third is derived after nine applications of the 3:2 ratio, which is also several octaves higher, so we end up having to solve a rather different equation for our new \(\rho\) value, but the end result is that our fifth ends up being 694.786 cents: it ends up differing from a perfect major third by exactly one-third of the syntonic comma. Let's start doing the same generative process as before and build our scale:
Well, we've generated our twelve notes, but this looks… well, wrong. For one, there are a handful of surprising empty spaces in the scale: for example, we have the two notes 114.79 and 184.35, which differ by 69.56 cents, but the next note up is 368.70, which differs by 184.35 cents—a gap more than twice as large! For another: recall how when we first generated our Pythagorean scale, we decided to stop generating notes when we got to a note that was comparatively close to the root. However, in the chart above, the last frequency we generated before stopping was 806.96 at cents, which means the next note we were going to generate was going to be 550.55 cents: not close to the root at all! Let's try to keep applying our approximated fifth a few more times and see how long it takes to get reach a frequency that's comparatively close to the root:
Now we've gotten close back to the root note: the red tone which I put in the bottom-right is the last tone we generated, and it's close enough to the root for our purposes (within twenty-five cents, anyway, which is closer than we got before.) But something here is different from before: by using a slightly different approximation of a fifth, we've ended up developed a scale that has not twelve but nineteen notes!
When Salinas proposed this system, he described it as sounding "languid" but not "offensive to the ear". Building an octave from nineteen notes has a different fun property, as well: recall that when using a twelve-note octave, if we keep moving up a fifth (that is, seven steps up in the scale), we'll eventually get back to the root, but only after having stepped through every single note in the scale. This isn't true for other intervals: if we go up two steps, for example, then we'll only hit every other note in the scale before returning to the root. But in a nineteen-notes scale, then every possible choice of step size (except nineteen, of course) will have this same property: moving up any consistent amount will cycle through the scale, hitting every possible note before returning to the root. That means we don't just have a circle of fifths, we can produce, say, a circle of major thirds
or even a circle of minor thirds:
Additionally, with a nineteen-note octave come a number of extra notes which don't have straightforward equivalents in a traditional twelve-note system. For example, in addition to having a major and a minor third, a nineteen-note scale also gives us an interval slightly larger than a major third, which is sometimes called a supermajor third. This clip features a major third and then a supermajor third in sequence:
Similarly, there's a slightly smaller interval than a minor third which is sometimes called a subminor third, played here after a minor third:
Now that we know we can reasonably split an octave into nineteen more-or-less equal chunks, what if we did the same thing we did in the last post, and just split the octave into nineteen equal chunks?
If we do that, it turns out to be incredibly similar to the previous tuning we developed, so close it's difficult to hear the difference. Here's a table of the same notes derived using the meantone approach and the equal temperament approach, as well as the difference in cents between the notes:
Freq. (Meantone) Cents (Meantone) Freq. (ET) Cents (ET) Difference
300.00 Hz 0.00 300.00 0.00 0.00
307.90 Hz 45.02 311.15 63.16 18.14
320.54 Hz 114.65 322.71 126.32 11.66
333.70 Hz 184.29 334.70 189.47 5.18
342.49 Hz 229.31 347.13 252.63 23.33
356.54 Hz 298.94 360.03 315.79 16.85
371.18 Hz 368.58 373.41 378.95 10.37
386.41 Hz 438.22 387.28 442.11 3.89
396.59 Hz 483.23 401.67 505.26 22.03
412.87 Hz 552.87 416.60 568.42 15.55
429.82 Hz 622.51 432.07 631.58 9.07
447.46 Hz 692.15 448.13 694.74 2.59
459.24 Hz 737.16 464.78 757.89 20.73
478.09 Hz 806.80 482.05 821.05 14.25
497.72 Hz 876.44 499.96 884.21 7.78
518.15 Hz 946.07 518.53 947.37 1.30
531.79 Hz 991.09 537.80 1010.53 19.44
553.62 Hz 1060.73 557.78 1073.68 12.96
576.34 Hz 1130.36 578.51 1136.84 6.48
In both cases, our approximation of the perfect fifth is, especially compared to many of the systems we've generated before, not as faithful an approximation. The following clip will play a perfect fifth, followed by the fifth as approximated by a 19-tone tuning:
But our approximation of the major and minor thirds are very close, as we can see from a perfect third followed by the third in our 19-tone tuning:
All this is interesting! We built a scale using a very similar approach as other scales, but it turns out to have some radically different properties that allow for a very different kind of music to be created. And indeed, some composers were eager to work with alternate systems like this. Around the same time that Salinas first proposed this system, a French composer—Guillaume Costeley—independently invented a nearly-identical system Costeley's system was built up from a different mathematical basis, and consequently isn't exactly the same as the one-third-comma meantone that Salinas was describing, but it's remarkably close. I should also mention that, while I presented them in reverse order, Costeley's composition came first, in 1558, while Salinas' book De musica libri septem wouldn't be published until 1577. and wrote a chanson called Seigneur Dieu ta pitié with this system, designed to be played on a newly-invented instrument called the Archicembalo:
While never achieving major popularity among musicians at large, various nineteen-tone tunings have been revisited regularly throughout musical history, and have achieved a modest popularity among some composers, in part because nineteen is still a "workable" number of tones for an octave. Recall in the previous post that, when generating the frequencies for a Pythagorean scale, I decided to stop after having generated a frequency that was was "close enough" to the root, because we wouldn't generate a tone closer to the root until after another 53 steps. That was a motivated choice, but still a choice: I certainly could have kept generating new tones and adding them to the scale until I had generated all 53 notes. After all, we're just concerned with building up a set of frequencies we can use together, so why not compose music on a 53-note scale?
The problem is, of course, that to write music for such a scale, I would need instruments that could play the notes on this scale. The nineteenth-century English music theorist Robert Holford Macdowall Bosanquet actually did invent a device capable of playing music using this 53-note extended Pythagorean tuning! It looks like this: This image was borrowed from this page of photographs of enharmonic harmoniums, and is copyrighted by the Science and Society Picture Library.
Actually playing this instrument is very difficult, and certainly more difficult than mastering a twelve-key piano keyboard! And that's just for a keyboard: attempting to build, say, a guitar fretboard that supports 53 tones per octave would be incredibly difficult! This is a major advantage of a twelve-note scale: it's easy to build physical instruments to accomodate it. Because nineteen is still pretty small, we can still build reasonable physical devices to accomodate it: for example, piano keyboards that include a handful of extra black keys:
Of course, we now have a lot of digital instruments, so for modern music, we can be far more adventurous, and indeed composers like electronic music pioneer Wendy Carlos Wendy Carlos is a personal hero of mine! She's composed music you've certainly heard, including spectacular movie soundtracks for A Clockwork Orange, The Shining, and Tron, and she also pioneered a lot of early work in electronic music. Her 1968 album of Bach pieces played on a Moog—called Switched-On Back—was a major factor in bringing electronic instruments to a wider audience. Look up her music! have taken full advantage of the current obviated need for physical instruments. But still: it's possible to build a nineteen-tone flute in a way that's it's not for a fifty-three-tone flute, which is why nineteen-tone scales are often a first exploration for budding microtonal musicians.

And Even More Scales

So far, I've demonstrated scales that include twelve, nineteen, and fifty-three tones: are there more? Why, of course!
In addition to choosing equal-tempered scales by approximating just intonations, as I've done so far, there are numerous other ways we could construct scales: if we wanted to, we could even just choose equal-tempered scales by running through numbers and discovering whether those choices result in pleasing intervals or not.
Several friends I talked to assumed that twelve-tone equal temperament had been discovered more or less this way, rather than by successive modifications of just intonation!
For example, if we want to keep the same set of intuitions we've built up for twelve-tone scales, but want a greater degree of tonal richness, we could just subdivide each 12-TET (that is, 12-Tone Equal Temperament) interval into two, and we'd get a 24-TET scale. The 24-TET scale also has a precedent in Middle Eastern music theory: while Middle Eastern music doesn't use a 24-tone scale directly (instead opting for a larger numbers of unusual scales built around certain small intervals), the Lebanese historican and music theorist Mikhail Mishaqa described how many of the scales used in Arabic and Persian music can be described and analyzed as being (in effect) selected subsets of a 24-TET tuning. Other popular choices have included 29-TET (which includes closer-to-perfect fifths than 12-TET), 31-TET (which includes reasonably approximated thirds) or even 72-TET, championed by microtonal musicians like Joe Maneri.
We also can go lower than twelve tones per octave and still produce workable scales: in particular, 5-TET and 7-TET are both viable options. Both 5-TET and 7-TET in fact show up in various kinds of non-western music theory: for example, Thai musical instruments are regularly tuned to a close approximation of 7-TET, and some scholars contend that there's a 5-TET basis to one of the two major methods of tuning Javanese gamelan ensembles.
There's yet another wrinkle we can add to our tuning systems: we've been treating the octave as an omnipresent fact of life in all these systems, but we can throw that out the window, too! For example, the well-known Bohlen-Pierce scale is a scale that deliberately ignores the 2:1 octave in its structure, and instead uses a 3:1 ratio, sometimes called a tritave, as a basic element in its tuning, which ends up giving it a very different auditory flavor, as well as making the system very different to compose for. Yet another example: the aforementioned Wendy Carlos has an absolutely wonderful album called Beauty In The Beast where she experiments both with various non-Western tunings as well as some non-octave repeating scales of her own creation, the so-called alpha and beta scales. These scales are created by splitting a smaller interval, such as the perfect fifth or minor third, into equal parts that can then be repeated: these scales don't end up repeating on an octave level, but rather repeat only on the smaller intervals on which they are modeled.

Well, How Do They Sound?

Kind of weird!
…well, they sound weird to most Western listeners, anyway, and I suspect there are multiple factors in play here. Most prominently, many Westerners rarely hear anything but music written in 12-TET, so music written for other tuning sytems sounds a bit… wrong, like it's somehow subtly out-of-tune. For example, many Western listeners aren't used to listening to Indonesian gamelan ensembles, so even a piece like this Gamelan piece by Western composer Lou Harrison will sound unusual and strange:
Yet another one of my heroes! You should also listen to his Concerto for Pipa with String Quartet, too, which is a favorite of mine.
Of course, there's the other factor that many musicians who embrace microtonal music are also musicians who love unusual music in general: for example, a given microtonal enthusiast is probably more likely to be a fan of Schoenberg-style atonal compositions Vi Hart has a wonderful explanatory video about twelve-tone atonal composition, which I'd highly recommend! or experimental aleatoric music, which means the kinds of compositions they write would be unusual to an average listener even if they stuck to 12 notes. That doesn't mean their compositions are bad—far from it!—but it can be hard to have a real apples-to-apples comparison when your points of comparisons are traditional 12-note melodic compositions and experimental, atonal compositions in a 19-note octave.
However, that's by no means universal: there are plenty of more traditional melodic compositions in microtonal scales! For example, here is a more traditionally melodic song composed in the aforementioned Bohlen-Pierce scale by the musician Elaine Walker:
Or here's a short prelude written for a 19-TET guitar by microtonal proponent Ivor Darreg:
This so far has been a very cursory introduction to microtonal music: there's a lot more there than I talked about in this post! It's a field with some lots of interesting corners and a fascinating history, but at the same time, it's also a field full of unexplored directions and corners just waiting for someone to uncover them. And, of course, plenty of fascinating music to listen to, as well!