This article is about the different ways one can think about musical scales and the logic that underlies them. For any given scale you could ask, how were the particular pitches of this scale selected from the continuum? What sort of reasoning underlies the choices? These questions can be asked in analyzing existing scales, and they’re also central for anyone wanting to develop new scales.
Intonation systems in all their variety are the subject of mountains of learned discourse, but this article is not a deep delve; it’s a broad and shallow take. The hope is just that before we’re done we will have glimpsed a few inventive and off-the-beaten-track perspectives on the possibilities for different sorts of scale-logic.
One way to answer the question “why those particular pitches?” is to say, “The choices are simply based on what sounds good. The ear has its intuitions for what pitch relationships work together; no further scrutiny is needed.” This is an excellent answer! But even having said that, there’s still a lot to be learned through analysis. The usual starting point for scale analysis is the assumption that pitch corresponds to frequency*. This means, conveniently, that we can turn a pitch into a number – cycles per second — and we can then subject pitches and pitch relationships to mathematical analysis. In this article we won’t get deep into the math, but we do accept the idea that numbers can be a meaningful language for talking about scale.
*[The notion that perceived pitch corresponds to frequency involves some simplifications, but it is well enough grounded for the current discussion.]
There are two leading scale logics in the music we encounter most often in the world today, so we’ll start our discussion with snapshots of those. This will be familiar territory to many readers, and those who are new to it can pick up a rudimentary education from sources online or off, so I won’t do more here than to state their main underlying ideas. From there I’ll continue with descriptions of several other possible scale-logics, and for these I’ll go into just a bit more depth since they’re not as widely known.
Just intonation
Just intonation is based on the idea that our ears interpret pitch relationships in terms of ratios of rates of vibration, and that the most meaningful pitch relationships correspond to the simplest ratios. Just scales are those in which the intervals that make up the scale can be expressed as a series of simple frequency ratios. This ratio-based thinking does indeed seem to correspond well to the way the ears respond to and interpret musical intervals, and for that reason people often think of just intonation as more “natural” than other approaches to tuning.
There is an important limitation inherent in just scales. The intervals between scale degrees vary in size. This creates a richness that scales with uniform intervals lack, but it also introduces complications when it comes to modulating from one key to another. The scale tones do not align between the original key and the new key, and so the new key seemingly calls for a whole new set of notes, which for most instruments quickly becomes impractical. There are several ways to respond to this situation, but historically, in European music, one response has been most significant: tempering. Tempering involves finding ways to adjust some scale tones a tiny bit up or down, creating compromise pitches that, while not ideal, are close enough to the ideal pitches to work in more than one key. This is a fascinating but demanding topic and I won’t tackle it here. But it does lead us to our next type of scale-logic.
Equal Divisions of the Octave
In EDOs, the octave is divided into some number of equally sized steps or intervals. Equal scales have the advantage that a system having uniform intervals is wonderfully convenient in many practical ways, including the ability to play in all keys. But this comes at a cost: the mathematical logic that underlies equal divisions does not correspond to the “natural” ratio-based logic of just intonation.
Far and away the most common number of divisions per octave is twelve. The main reason that 12-tone EDO has become standard is that it happens, very conveniently, to do a reasonably good job of approximating the most important intervals of simple just scales. In this sense 12 EDO functions as a tempered scale as described above. Form a purist perspective, the intervals of 12-EDO are imperfectly tuned, but for most listeners 12-equal works well enough in providing reasonable facsimiles of important just intervals, and it does so without the challenges that a true just scale would entail.
But now recall that EDOs are not restricted to 12 divisions. The scale-logic of equal divisions can be used to generate any number of very different-sounding scales, from a reductively simple two or three tones per octave to an insanely microtonal 84 or 96 or beyond. These non-twelve tunings (with a few notable exceptions) bear no meaningful relationship to just intonation, nor need they. Each of them presents its own tonal world to explore. Subjectively speaking, we find (to use a word favored by intonation theorist Ivor Darreg) each EDO scale seems to create its own distinctive musical mood. The possibilities are rich.
It’s also possible to generate scales based on equal divisions of intervals other than the octave. This idea takes us into pretty esoteric territory, but it can certainly be interesting and fun for scale theorists and mathematicians. For most listeners this underlying logic doesn’t translate into musically meaningful tonal relationships except insofar as the intervals that’s arise may by chance approximate other more familiar intervals. But I want to mention in brief one particularly interesting instance here. If you start with something familiar like 12 EDO, but base your division not on the octave but on an interval just a bit larger than the octave — a stretched octave, as it were — you can generate a scale which may seem most peculiar when you look at its ratios, but which can make an odd but intriguing kind of sense to the ear. This is especially true when the timbres of the instruments playing the music are similarly stretched — that is, the ostensibly harmonic overtones within the instrumental tone quality are slightly more spaced out in a way that corresponds to the octave-stretching of the scale.*
*[For these ideas we are indebted to Bill Sethares. His is the foundational work on the relationship between the tuning and the timbre. See Tuning Timbre Spectrum Scale for his book-length exposition, or here for an article-length discussion. We’ll be seeing Bill again before we’re through, as his work informs much of this article.]
…And now, having touched (very briefly) on the two most most prominent scale-logics, we can ask, what other sorts of logics might be possible? Here are a few.
Scales Based on the Harmonic Series
This approach fits within the definition of just intonation as discussed above, but at the same time it suggests a slightly different way of thinking about scale building. It is possible to devise scales built upon harmonic series. Scales built on the series are by definition just intonation scales, because the underlying arithmetic is indeed one of ratios. Like other just scales they can be seen as having a sort of “natural” quality, and it’s not hard to convince yourself when you hear them that they do make sort of intuitive sense to the ear.
Here’s an example of a scale based on the harmonic series. The intervals in the lowest part of the harmonic series are too large to feel scale-like (octaves and fifths and such), but the intervals get smaller as you go up, and when you get up around the 8th harmonic the interval sizes are in accordance with what we’re used to as typical scale intervals, like major and minor seconds. So you can simply take harmonics 8-16 and call them a scale. The resulting tones cover an octave with typical scale-sized steps; they can easily be used to create melodies and harmonies, and the ear accepts the results as comprising a coherent tonal world. As with other scales, you can extend the range of this scale by repeating octave duplications of the same eight notes through higher or lower octaves. For a more extensive harmonic series scale you could continue upward in the harmonic series through the 17th harmonic, 18th, 19th, and so forth as far as you wish. As you continue up the series the number of tones per octave progressively increases as the intervals get smaller, and the relationships between the tones become more esoteric and less intuitive to the ear. The result is less obviously scale-like, but it creates room for more adventurous explorations.
In scales based on the harmonic series, the pitches are potentially in agreement with the overtones in the timbres of the instruments that are playing. This can create a special kind of coherence in the music. This once again is in keeping with Bill Sethares’ investigations of the relationship between tuning and timbre as footnoted above. Admitedly, for harmonic series scales (as well as some other types of scales), the relationship between tuning and timbre turns out to be more complicated in practice than it might appear at first blush: while the overtones of an instrument playing the tonic note of the scale will indeed align with the scale tones, the overtone content for other scale notes will introduce many other tones only much more distantly related. Nonetheless, the sense of coherence is real in music in which tuning and timbre are integral to one another.
And then there’s the inversion: just as you can create scales based on the harmonic overtone series, you can also base scales on its inversion, the subharmonic series. One could argue whether the subharmonic series as a theoretical construct has any real meaning in the physical world, but the scales you get this way do, subjectively speaking, feel musically meaningful to the ear.
Scales Based on Other Mathematical Series
Beyond the harmonic series, there are other sorts of mathematical series that can plausibly form a the basis for scale intervals. Some might be purely mathematical abstracts; others are known to arise here and there in nature. A couple of examples that musicians have explored are Fibonacci sequences and Aliquot sequences — look them up if you’re curious and not familiar with them. For that matter, nothing is stopping you from devising some other mathematical logic of your own, using it to generate a numeric sequence, and finding a way to map it into the frequency relationships of a scale.
One can raise a question and a follow-up question here: do scales built this way have intuitive or felt musical meaning to the ear, or do they remain as Platonic abstractions? And if the latter, does that matter? It may be seen as inherently valuable that the music reflects a coherent underlying rationale whether or not anyone can intuitively hear it. One may even have faith that the underlying logic gives the music a quality of coherence that can be heard or felt even when the underlying rationale is not recognized or understood. The contrary perspective would be: That’s silly. If people can’t hear or intuit the relationships you’re building your scale on, what’s the point?
Scales Based on the Overtone Content of Inharmonic Instruments
A moment ago I noted that scales based on the harmonic series have a kind of built-in integrity when played by instruments like strings and winds whose overtones are harmonic, because the overtone recipes we hear in the timbres of those instruments are in accordance with the frequencies of the scale. Many other instruments, on the other hand, produce inharmonic overtones. These include bar instruments (marimbas, vibes, xylophones) lamellaphones (kalmibas, mbiras), membranophones of various sorts, and many more.* Would it be possible, in an analogous way, to create scales based on the overtone intervals of these instruments?
[*For some of these inharmonic instruments the maker may in some cases carefully reshape the sounding elements to bring the most prominent overtones in line with tones of the harmonic series, but even then plenty of inharmonics remain.]
For instance: in bar instruments such as marimbas with simple rectangular bars, the first overtone above the fundamental appears (ideally) at an octave plus 555¢, which is an octave and a very sharp fourth. The next overtone is at 2-8ves + 521¢; then 3-8ves + 191¢, 3-8ves + 8853-8ves + 191¢ … and so forth through as many more overtones as you wish to observe. You could make a scale, albeit a peculiar one, by taking these frequencies, transposing them all down into the same octave, and arranging them in a scalewise sequence. We can speculate that this scale would have a kind of musical coherence when played with rectangular bar instruments, and indeed there seems to be some experiential evidence that something like this happens. (But keep in mind the caveat mentioned earlier near the end of the section on harmonics scales having to do with the additional frequencies introduced in tones other than the tonic.) The work of Bill Sethares is yet again foundational here.
You could do this sort of scale-building based on any number of instruments of inharmonic timbre, or for that matter, with any sound you may encounter with an intriguing overtone blend — sample the sound, analyze it, build a scale, and then compose and perform music in the scale using the samples for your instrument sounds. Offhand I can’t think of anyone actually doing this, but I bet that someone somewhere has pursued this idea.
To take the idea one step further: without any reference to the natural world, you could play around in digital synthesis in search of find some unique timbre that like — some unique overtone recipe that appeals to your ear — and then build the scale to match, built on the the overtones that make up your invented timbre, octave-displaced as needed to make the scale. This would be an intriguing sort of musical world-building in the digital realm.
Scales Based on Mappings from Light Spectra and Other Natural Sources
Several composers have found their scale material in areas outside of the audio realm. The number one candidate for this sort of treatment is light. Like sound, light has frequency. Where in the audio realm frequency corresponds to pitch, in light it corresponds to color. The frequencies for electromagnetic radiation, both within the range of humanly visible light and beyond, are generally much higher than the frequencies of humanly audible sound, so to map light frequencies to the range of audible frequencies typically calls for downward transpositions of many octaves. For instance, Google’s search engine AI tells me that a nice violet would have a frequency somewhere around 440 terahertz (440 billion hertz), while 660 terahertz would be a color the AI describes as “red-flourescent.” In musical terms, we could say that these two colors are a fifth apart, since the musical fifth is defined as a frequency ratio of 3:2. It’s easy to imagine how you could have a great old time with these ideas when it comes to scale-making based upon your preferred color palette. We can ask: Is there any real meaning in these correspondences between light colors and musical tones? E.g., is there really some musically meaningful sense in which we can say that violet and red-fluorescent are a musical fifth apart? The evidence for this is shaky, as evidenced by the fact that historically people have been be inconsistent in how they make and interpret mappings of color to pitch. (If their really was some universal intuitive correspondence between color and pitch, you’d expect that people would be more consistent in how they make the mappings.) Still, over the years people have found such synaesthetic explorations to be irresistible; some fine music has been generated this way, and who can argue with that?
Another way to approach mappings from light to tone is to look at the light spectra for specific molecules or atoms. Each type of molecule and atom emits or reflects light at specific characteristic frequencies, and these frequencies can be transposed to the hearing range in hopes that they’ll present interesting scale relationships. At least a couple of composers have fruitfully explored this idea.
Beyond light, you could look for promising number sequences in any number of other places in nature, even including things that aren’t rooted in frequency. You could, for instance, look at a photograph of a city’s skyline, and assign frequencies based on the heights of the buildings … or do the equivalent with the form of a shapely tree, or a mountain range. Once again we come up against the question, can we actually expect that harmonious visual relationships will somehow translate into meaningful musical relationships? Then again, perhaps we don’t really need to make such a claim. We need only let the composer take the material where she will, and let the listener find in it whatever musical value may be found.
Found Scales and Random Scales
At the beach, I like to make and play driftwood marimbas. In doing so I’m at the mercy of whatever driftwood pieces happen to be lying around, with whatever tones they happen to make when struck. Yet most of the time I find I’m able to come up with appealing scales for seaside improvisations.They ay bde semi-random, but somehow my ear finds ways to make sense of them.
If you make a simple kalimba, you might choose to first put all the tines in place untuned, expecting to go back later to and properly tune them as a final step in the building process. But of course that initial random untuned tuning is in itself a tuning. It might, by chance, be a very nice one. Maybe you don’t need to go back and retune.
The collection of unused clay flower pots in the yard … you may have noticed that they work nicely as bell forms, either hung like bells or just standing upright and side-struck. If you line them up from largest to smallest, their pitch relationships will be an unplanned case of take-what-you-get, but I bet you can make some nice music with them.
It is surprising how often some scale that you’ve randomly stumbled-upon turns out to be very nice.
Sliding Pitch
In the physical world we hear sounds with steady pitch and we also hear sounds whose pitch changes through time. Humans have a long history of seizing on the steady-pitch sounds for music making, as reflected in the opening lines of this article where I talked about selecting specific pitches from the continuum. But can you make music without fixed pitches, using only glissando? Well of course — in music you can do anything you want! Some composers have explored this idea, Henry Cowell being perhaps best known. But I was surprised just now, in doing a quick search for sliding pitch music, to find less discussion of this idea than I had expected. (Maybe I just need to do a more diligent search.)
My Own Take
In this article I’ve described (very superficially) several different approaches to scale-making. Certainly many more are possible than those touched on here. I’ll close with my own thoughts on scale-logic, focusing on the question of what I think makes scales meaningful to the ear. These thoughts are subjective, not rigorous; you may find they ring true with your own perceptions and experience or you may not.
We know that people are able to make very fine distinctions of pitch, especially with training. Without some faith in our ability to recognize and respond to microtonal distinctions there would be no reason to care about the differences between closely related just scales, or between just scales and their tempered counterparts, or about any of the other things that make intonation theory worth studying. And you can learn from listening experience that there’s a special place in heaven, or at least in our ears, for exquisitely tuned just or microtonal music. But in what follows I’m going to go in the opposite direction. I’m going to talk about hearing pitch and scale relationships in ways that are not so finely calibrated, but rather are freer, more flexible, less exacting.
When we speak of a particular musical interval we can speak precisely by specifying a frequency ratio or multiplier, but we can also speak more loosely. For example, we can speak of a minor third without necessarily specifying whether we’re talking about a just minor third or a tempered minor third. Both of those, it seems, fall into a region of the scale in which some sense of minor-thirdness pertains. The same can be said for other intervals: there is a region over which we hear a given interval type, even though subtle pitch inflections within the region may imbue the interval with different qualities. Whether this is inborn or learned I cannot say, but it does seem that, within a given musical context, the ear readily hears and classifies some number of musically meaningful pitch regions within the octave. In some musical contexts there may seem to be five such regions, as in pentatonic tonalities. In other contexts, for people who’ve grown up with western chromaticism (which is almost everyone now), there may be closer to twelve. Within those regions there is room for subtle gradations and expressive inflection, while still retaining the broader musical sense and meaning of the interval. This is one of the important lessons, for instance, of blues: there is a range of expression residing in nuanced pitch inflection within the region of the third or fifth or seventh, yet these intervals retain their identities as thirds and fifths and sevenths. This is also true, or at least has the potential to be true, for other styles as well.
The idea here is that meaningful intervals are recognized over their respective regions, while within those regions there is room for inflection and modulation. I suspect that this kind of thinking is what typically allows people to effortlessly make at least superficial sense on first hearing of most of the scales, that one might either encounter or dream up, be they familiar or exotic. It’s part of what makes it possible to find musical meaning, for instance, in the random scale of a driftwood marimba. This kind of musical response can be very much in-the-moment and unstudied, and it can be part of a very open-minded sort of listening. At the same time, it doesn’t preclude further study and refinement if one cares to explore within the regions. For myself, the main reason I like thinking this way is, it allows for a kind of creative sloppiness that I much enjoy and am at home with. The most interesting sounds are often in the interstices and anomalies, and I’m happy not to try to legislate them away in the scale-making process.
With that I close, wishing smooth scaling for all on musical seas exotic or familiar.