All About Musical Scales (and How to Tune Them)

[Creyeditor]

Thank you for this post. Yesterday, I was trying to re-enact the Pythagorean tuning as you described it and became really frustrated because of the ratios 64:81 (or 81:64) and thought I was doing something wrong. And here your new post comes along and just takes it up from there. One minor question: Is it possible to construct a Ptolemaic gamut and if yes, how many tones would there be in an octave? (I hope this question makes sense.)

[Salmoneus]

A good question! One that I should have answered in a later post... but won't have done. So thanks for asking it!

19.

That is, if you want to play a ptolemaic heptatonic scale on your instrument, AND you want to play six more ptolemaic heptatonic scales, one starting on each degree of the first scale, then your instrument needs to be able to play 19 notes to each octave.

Interestingly, I'm not aware of this being super-widespread in practice, although iirc it may have been used at times in the Islamic world (they mainly historically - at least in theory - used 17-note gamuts, for a reason I'll get to in a bit). Three different 19-note tunings were historically used in Europe, for reasons I'll get to in a bit, but none of them were, to my knowlege, this 19-note ptolemaic tuning!

I think that one reason for this is that the Ptolemaic gamut isn't as great an idea as it might at first appear, for reasons I'll mention when I get to adaptive tuning (but basically: such a scale always provides a suitable note value to yield a consonant interval, but it can't make all the intervals consonant at the same time, so you have to pick which ones to prioritise when choosing which note to use [eg do you want the A that's a nice fifth above D, or the A that's a nice fourth above E? Because they're not the same - unless the E isn't a nice third above C... unless you move the C, in which case... etc], and if you're not careful you can end up down a rabbit hole where your short-term decisions are incompatible with the long-term structure of a piece. Most importantly, this can lead you into comma pumps (which I'll explain later). So someone using this system needs to be really musically aware, and must either be a mathematician or else must rehearse their pieces a lot to make sure they actually don't go disasterously wrong. There's also a problem with multiple performers (you all have to make the same choices), basically requiring complex notation for all music.

[Creyeditor]

Thank you, that sounds interesting. Reminds me a bit of what I read on enharmonic keyboards, except that the Pyt. gamut is about 'proper' notes. I wonder of there is a quick formula that connects number of scales or number of notes in a scale and the way the scale comes about with the size of the gamut. But maybe this is not actually interesting for other kinds of intonation.

[sangi39]

C – 1:1
D – 9:8
E – 81:64
F – 4:3
G – 3:2
A – 27:16
B – 243:128

Oh my actual god, I feel so dumb Like, I still don't understand the difference between major, minor, perfect, augmented and diminished, but this is the first time I've looked at intervals and gone "oh that's why a "third" is called a third!" (and then same for seconds, fourths, etc.). I've only just literally now understood that it means "up to the third note in the heptatonic scale" (for the third), or "the fourth note" for the fourth, and so on. Not going to lie, feeling pretty dumb about that

[Salmoneus]

C – 1:1
D – 9:8
E – 81:64
F – 4:3
G – 3:2
A – 27:16
B – 243:128

Oh my actual god, I feel so dumb Like, I still don't understand the difference between major, minor, perfect, augmented and diminished, but this is the first time I've looked at intervals and gone "oh that's why a "third" is called a third!" (and then same for seconds, fourths, etc.). I've only just literally now understood that it means "up to the third note in the heptatonic scale" (for the third), or "the fourth note" for the fourth, and so on. Not going to lie, feeling pretty dumb about that

Well then I'll complete the revelation for you!

"Major" just means "big"; "minor" means "small". So a major third is a big third and a minor third is a little third. Why is there a difference? Because if you look at the heptatonic scale and pick pairs of notes a third apart (that is, two notes with one note between them, because intervals are double-inclusive - eg C-E, D-F, etc) you'll see that there's two different types. The C-E interval is bigger than the D-F interval. The 'correct-sounding' or 'just' values for these are respectively 5:4 and 6:5. The former is bigger than the latter, so we call it a 'major' third, and the latter is smaller than the former so we call it the 'minor' third.

Similarly, the gap between E and F is maller than the gap between F and G, so the former is a 'minor second' while the latter is a 'major second'. In a Pythagorean heptatonic scale, these are respectively the limma and the tone. In Ptolemaic tuning there are TWO types of tone (9:8 and 10:9), but they don't get 'major' and 'minor' names as seconds because the difference doesn't exist in the Pythagorean scale; however, we do call them the major and minor tone.



"Augmented" means bigger than major. "Diminished" means smaller than minor. In the Pythagorean heptatonic scale, these only applied to fourths and fifths that were 'wrong'. Whereas thirds can be 'major' or 'minor' because they both sound equally OK - like two alternative versions of one another - fifths and fourths have a 'perfect' version and 'augmented' or 'diminished' versions because the former sound great and the latter sound appalling.

When you then start having more than seven notes, it becomes possible to create intervals that can be described as augmented or diminished versions of other intervals.

It's worth remembering - although I'll admit it doesn't come naturally to me - that European terminology is all based on this seven-note scale. This is how, for instance, you can tell whether something is a flat or a sharp - because every heptatonic scale 'should' have each member of the 'real' heptatonic scale, an if it doesn't then the 'wrong' notes must just be warped (flattened or sharpened) versions of them.

So, for instance, a fifth above B is F#, not Gb. That's because if you go up from B you need a version of C (C#), a version of D (D#, assuming we're in a major key), a version of E (E) and then a version of F (F#). If we called the note Gb, we'd be missing an F. Conversely, a fourth above F is Bb, not A#: F, G, A, Bb.

Finally, why do we now talk about 'major' and 'minor' keys at all?

Originally, we had the heptatonic scale: ABCDEFG. In the middle ages, you could compose music in any permutation of this scale, called a 'mode'. Over time, two modes became more fashionable: the one 'starting on' A and the one 'starting on' C. These are traditionally known as the Aeolian and Ionian modes respectively, which is a little confusing as these 19th century names don't actually match the use of the names in the middle ages, which in turn have no relation to any ancient greek scales or places in Greece, despite having the same names. The Aeolian mode, however, happens to have a minor third between its first and third notes, so is called the 'minor' key, and the Ionian has a major third there instead, so is called the 'major' key!

[Creyeditor]

I actually head a similar revelation. I was thinking that in English terminology, a third should be an interval based on literal 'third parts' e.g. 3:2. That's maybe also something peculiar about English but it feels like an octave shpuld be called 'a half'. German uses Latin-derived terms all the way, e.g. Terz, so this was even more opaque to me in school.

[Salmoneus]

I'd actually briefly considered avoiding using these terms, just because they can be confused with regular english (i.e. the general adjective rather than the jargon noun), but the alternatives (eg calling a fourth a 'diatesseron') just seemed unwieldy and unnecessarily jargony.

Perhaps I should have defined my terms better when I started, though!

[Creyeditor]

For me at least, you are going at exactly the pace I need and introducing everything in the order I need it to be introduced.

[Salmoneus]

Thank you, that's great to hear!

[Salmoneus]

“Unjustly Intoned” Scales

(yes, that’s just my own term, I’ve not seen an official term with the meaning I’m aiming at here)

We’ve been talking so far as though every interval should sound nice. That, after all, seem intuitively a good thing in music. But in fact, most music relies not on permanent euphony, but on contrasts between more and less ‘pleasant’ or ‘calming’ sounds and other, more ‘stressful’ or ‘tense’ sounds. And indeed, that’s a big part of why the heptatonic is a fruitful basis for music, because it contains both highly consonant intervals (like the fourth and fifth) and relatively dissonant intervals (like the tone). Even pentatonic scales allow contrast between consonant fourths and fifths and a less consonant minor third. So why is it a problem that the Pythagorean scales have dissonant thirds and sixths as well as dissonant seconds and sevenths? Why do we have to ‘improve’ them? Why can’t we, perhaps, even intentionally make them sound worse?

Well, perhaps people don’t usually expressly ask that question out loud itself – but sometimes the musical scales they use do seem to follow this logic. That is, having given themselves some ‘fixed points’, in the form of consonant intervals to anchor their scale, they fill out their scale with intentionally dissonant additional intervals.

We can imagine this by looking at a possible pentatonic scale. First, we might give ourselves our root note (and the octave above) and the fourth and fifth above it, collecting all the really consonant intervals. Let’s call them C, F, G (and C’). As we discussed (far) above, this tritonic scale has two large steps (fourths: C-F, G-C) and one much smaller step (a tone: F-G). If we want to add two new notes, the logical thing is to stick one somewhere in the middle of each of those two big fourths. And that’s what Pythagorean tuning does: if we tune up in fifths we add D and A, or if we tune down in fifths we add Bb and Eb (both are examples of Pythagorean pentatonic scales, but effectively in different modes and transpositions of one another).
But what if we put those notes somewhere else? We could put them right in the middle of the big interval. Or we could put them right at the top or bottom of that interval – exaggerating the asymmetry in the Pythagorean scale even further. In particular, there’s a cross-culturally-common interest in what we might call ‘sharp leading tones’ – that is, in having a note below a ‘stable note’ be particularly close to it (and hence in a rather dissonant relationship to it), as this produces a strong listener expectation for the note to ‘resolve’ up to the stable note. This can encourage players of an instrument with Pythagorean tuning to ‘sharpen’ their ‘dissonant’ notes – for instance to raise a Bb to something in the vicinity of a B, or even higher, to emphasise the discordant relationship with C.

One convenient way of thinking about a scale like this is as two three-note scales put together (with a tone between them (or at the top or bottom of them, depending on where you start counting)). Each ‘trichord’ consists of a stable interval of a fourth, which can easily be tuned to be consonant, and an additional unstable note somewhere in the middle, which the peformer can tune to their own personal preferances. Changing the trichords involved (i.e. lowering or raising that unstable middle note) can significantly change the feel of the scale. Using the same upper and lower trichord automatically puts an interval of a fifth between the two unstable notes, which as well as creating a musical possibility (an additional consonant interval) also assists in the tuning: if upper and lower trichords match, a range of wildly different pentatonic flavours can be created by changing only a single ‘parameter’ (the tuning of the lower unstable note, with the upper unstable note automatically a fifth higher). Having said this, those seeking even more possibilities can use non-matching upper and lower trichords.

Similarly, a Pythagorean heptatonic scale can likewise be interpreted as two ‘tetrachords’ and one tone. Each tetrachord is, again, the stable interval of one fourth, with two further, unstable notes that can be tuned to preference in between them to create heptatonic scales with distinctly different flavours.

And this is exactly how the music of Ancient Greece in fact worked. Tetrachords were seen as two ‘fixed’ notes a perfect fourth apart, and two ‘moveable’ notes – the lower ‘parhypate’ and higher ‘lichanos’ – between them. Tetrachords were divided conceptually into three ‘genera’, by the position of the lichanos relative to the upper note of the tetrachord (the mese): ‘diatonic’ tetrachords where the upper interval is approximately a tone; ‘chromatic’, where the upper interval is approximately a minor third; and ‘enharmonic’, where the upper interval is approximately a major third. [why yes, the fact that modern music theory since the Renaissance has just randomly stolen words from classical music theory while randomly assigning them completely unrelated random meanings IS potentially confusing!] In effect, the chromatic and enharmonic genera (contrary to my earlier suggestions of sharpening the leading tone) flatten the note below the stable note, and in the process compress the other two intervals in the tetrachord (i.e. the lowest two), exaggerating the difference between the larger upper interval and the smaller lower intervals to create more characterful music.

The Greeks believed that the diatonic species was the oldest, with the chromatic and enharmonic being later variations on it. Within each genus, there were recognised distinct ‘species’ or ‘shades’, with different regional or emotional connotations (although all are united by the rule that the lowest step cannot be bigger than the middle step of the tetrachord). In the earliest accounts, these shades are loosely defined in a continuum; in later accounts, they are more theoretically defined, as discrete alternatives with specific, mathematically-defined intervalic values, and also become more numerous. The Pythagorean scale (which, at least in one mode, breaks up the tetrachord into intervals of 256:243, 9:8 and 9:8) was recognised as a shade of the diatonic, although not the most commonly used in practice – that would be Archytas’ diatonic, which instead used intervals of 28:27, 8:7 and 9:8 (i.e. narrowing the smaller interval, the limma). This, in fact, is where Ptolemy’s scale that we have been discussing comes from: as one of several different possible shades of the diatonic (the “intense diatonic” he called it) described by Ptolemy (it is likely that Ptolemy was creating specific numbers to try to match the actual tuning practice of people he was listening to; because his numbers sounded good and were easy to tune, later musicians intentionally copied him).

Different tunings/scales could be used in different regions (Ptolemy describes one scale as having a distintly ‘rustic’ feel), by different musicians, or by the same musician in different songs, to give a different feel to the music. So long as the key harmonic relationships of the octave, fourth and fifth were stable and perfect, the notes in between could be modified freely to taste. Indeed, although I don’t know whether the Greeks themselves did this, a musician with more strings than notes (or using a flexible-pitch instrument) could tune multiple ‘movable’ notes within one tetrachord – not to use the intervals between two variants of the same note, but to form a larger gamut, from which different scales could be selected even within a single song.

A simple version of this practice is still used today! The so-called “melodic minor” of modern European music is really a convention of playing two different scales in different situations. When melodies rise up through the top of the scale (i.e. in A minor when they approach A’ from below), the leading note below the root note is sharpened (in a way that has been incorporated into the dodecatonic gamut - i.e. sharpened enough to match the next note up in the gamut) to increase melodic interest, and the note below that has likewise been sharpened to avoid the step of a minor third being generated (as this is larger than the steps conventional in modern European music); but when the melody is descending, the sharpened leading note is not required, and therefore nor is the sharpened sixth note, so the plain ‘natural’ minor scale is used. [A third form of minor, with a sharpened seventh but not a sharpened sixth, is often used for harmonic purposes, because the sharpened seventh allows a particularly nice chord (the “V7”)]
Using intentionally ‘unjust’ scales, therefore – scales intentionally distorted away from the ‘natural’ Pythagorean note values – can enable performers to inject more personality into their music, and to exaggerate particular relations that they or their listeners enjoy. Indeed, SOME amount of semi-intentional deviation from theoretical note values is probably inevitable for any variable-pitch instrument in practice (including the human voice) no matter what theoretical values are employed.

[although it’s worth pointing out again that the scales closest to the Pythagorean values are believed to have been the oldest, and are also the only ones to have survived into the middle ages in Europe, suggesting some degree of universal popularity with listeners and/or performers]

We should also say at this point that trichords and tetrachords are just a very intuitive way for people to think about scales: the tetrachord actually precedes the theoretical concept of the octave scale (though not its practice). Many musical cultures see their scales as composed of sets of known sequences of notes, such as tetrachords: remembering the distinctive relationships within a set of four or five notes seems to have been highly intuitive for people, much more so than remembering the relationships among seven or twelve notes. Middle-Eastern music today, as I understand it, is still largely composed of a Greek-derived system of tetrachords (supplemented by trichords). Even in Europe, throughout the later Middle Ages and the Renaissance music was conceived of not in terms of octave scales but in terms of “hexachords” – each hexachord being a distinctive tetrachord pattern plus the interval of a tone (9:8) on either side. The octave was therefore thought of as a pattern of ‘mutation’ from one hexachord into another.

[specifically: the hexachord was a pattern of notes named ut-re-mi-fa-sol-la, where the mi-fa interval was a limma (or ‘semitone’) and the other intervals were all tones. A heptatonic scale, A-B-C-D-E-F-G, therefore, could be seen as two overlapping hexachords – one on D and one on G. The entire dodecatonic gamut could be seen as seven overlapping hexachords.]

So this approach to music is intuitive, flexible and powerful, and as a result it is widespread and enduring. And yet... it’s not perfect. In fact, by some lights, it’s fatally flawed. Unjust scales tend to yield distinctive melodies – but perhaps TOO distinctive, too determined by the asymmetries of the underlying scale, making originality difficult. As a practical point, an instrument strung to such a scale makes it difficult or impossible to simply transpose notes up or down, as the scale interval patterns do not repeat – even stringing to a broader gamut is limited in its effectiveness, because it is hard to generate multiple versions of the same highly-asymmetrical scale simultaneously. This difficulty with transposition also applies to parallel voices at non-octave intervals, and to switches to a new scale within the same piece. Indeed, any sort of extensive harmony can be challenging, because the ‘moveable’ notes are intentionally dissonant with the fixed notes, and may (if different upper and lower tetrachords are chosen) be dissonant even with one another. And above all, the reliance on ad hoc tuning by ear to the taste of the performer becomes exponentially more problematic as more performers are added – if each and every note on each instrument has to be independently tuned by ear to match every other instrument, this is time-consuming and frustrating. And on flexible-pitch instruments each performer must constantly check themselves, to ensure their own vaguely-defined choices for each moveable note are remaining identical to those of every other performer in their group. Such tunings, therefore, are much more appealing for music with one or two performers – otherwise people will eventually say “why don’t we just use some sort of fixed and easily-recognisable ratios to tune all our strings? It would be so much easier to avoid being out of tune that way!”
So this approach isn’t ideal either.

Now, let’s recap. We built Pythagorean heptatonic scales, and a Pythagorean dodecatonic scale that let us play multiple heptatonic scales on the same instrument. This gave us a lot of flexibility, and pleasing intervals of a fifth and a fourth, but it left thirds and sixths sounding rather unpleasant. We tried just accepting this. We also tried fixing it with the ‘just’ Ptolemaic scale, which gave us a single heptatonic scale that sounded great, including great thirds, but it took away our ability to transpose music easily and left us with some fourths and fifths that were no longer usable. And we tried going the other way, not only accepting the limitations of the dodgy thirds but amplifying them to buy more character in an ‘unjust’ scale – but this left us with the problems both of the Pythagorean system (lack of good thirds and sixths) AND of the Ptolemaic system (lack of flexibility in transposition).
But what if there were a solution that solved all these problems at once? What if we... just added more notes!?

[sangi39]

For me at least, you are going at exactly the pace I need and introducing everything in the order I need it to be introduced.

Honestly, same. I think it's been presented in just the right way that even when something hasn't been explicitly defined, or a label not been explicitly explained, it's allowed for a fair few "realisation" moments. Like in the post about Ptolemaic tuning, it was easy to start going "okay, but if you change 81:64 to 5:4, that would change this ratio to that, right?" then a couple of paragraphs down, it was brought up that, exactly, yes, that's what happens! So it feels just well paced enough that even if you start thinking ahead, or falling behind, eventually you're brought back on track. And so easy to follow too!

[Salmoneus]

Adding More Notes (Again)

Let’s start again. We built up our scales originally by going up (or down) in fourths (or fifths). If you have a keyboard to hand you can retrace our progress – for convenience, this time we’ll start on F and go up in fifths:

F – C – G – D – A (in order, ACDFG, the conventional pentatonic)
F – C – G – D – A – E – B (ABCDEFG, the conventional heptatonic)
F – C – G –D –A – E– B – F# – C# – G# – D# – A# (the conventional dodecatonic)

As you’ll remember, we tried stopping on five, seven and twelve notes to the octave because these feel like natural stopping points, where our scale has only two types of steps between notes, with no ‘missing’ notes or uniquely small or large steps. But what happens if we continue this process? Well, your keyboard gives you an answer: go up a fifth from A# and you’re just back to F. It’s a circle, the “circle of fifths”. We can’t go any further; we have all the notes. Right?

Well, no. Because your keyboard is lying to you. And we’ll talk more about the consequences of that later. But fundamentally, the problem here is that your keyboard is made by people used to European music, which never consistently (until the 20th century, at least) advanced beyond the dodecatonic scale. [Europe did not as a whole follow the ‘solution’ we’re about to discuss, although Europeans did experiment with it]. In reality, there is no circle. Having gone up twelve fifths, you can just go up another fifth and get a new note.

What note? Well, following our naming logic above, it’s “E#”... but in reality, it’s actually slightly higher in pitch than F!

And what has the addition of “E#” done to the step pattern within our scale?

Let’s recap (again, sorry). Our pentatonic had a pattern we’ve called 3-2-3-2-2 (or 2-3-2-2-3, etc – it doesn’t matter where you start), with a small ‘tone’ interval and a larger ‘minor third’ interval. Our heptatonic ‘split’ the two large intervals into tones and a new, smaller interval, which we called a ‘limma’: 2-1-2-2-1-2-2. And our dodecatonic split the larger tones into limmas and “apotomes”: a-1-1-a-1-a-1-1-a-1-a-1 [this is why I’ve shied away from saying ‘semitone’ as much as possible, since both the limma and the apotome could be called semitones]. Adding a thirteenth note, accordingly, splits the first apotome into a limma and an even smaller interval we can call a “comma”: c-1-1-1-a-1-a-1-1-a-1-a-1

As you have probably long since worked out, we can go through the whole of the dodecatonic scale splitting up these ‘big’ apotomes into limmas and commas, simply by adding perfect fifths, until we have split up all five apotomes, and we are left with a scale with twelve limmas and five commas: c-1-1-1-c-1-1-c-1-1-1-c-1-1-c-1-1. This is a heptadecatonic or 17-note scale.

This 17-note scale has some interesting properties. First, let’s name the notes. Our new notes are only very slightly above our old notes, so we’ll use the old notes with a ‘+’ sign (goodbye, E#, hello F+!): A-A+-A#-B-C-C+-C#-D-D+-D#-E-F-F+-F#-G-G+-G#-A. And our old Pythagorean heptatonic scale is, of course, still included in that: A-B-C-D-E-F-G. As is our old dodecatonic if we want it. But what if, for instance, we take the old heptatonic but replace some notes with their ‘+’ neighbours? What if we start on C+, while keeping as close as possible to the old heptatonic on C that we used to have?

That gives us: C+-D+-E-F+-G+-A+-B. Now let’s have a fit of curiosity and lower the penultimate note as well: C+-D+-E-F+-G+-A-B. How does this differ from the usual heptatonic? Well, the first, second, fourth and fifth degrees are raised a little bit. Or, equivalently, the third, sixth and seventh degrees are lowered slightly. Hang on, haven’t we said that about something before?

Oh yeah! That’s the Ptolemaic scale, isn’t it?

No, no it’s not. Of course it’s not. This is tuning, where NOTHING EVER WORKS PROPERLY.

But how much is it not working by? Well, in our 17-note gamut, these degrees of the heptatonic are lowered, in effect, by the comma – more specifically, by one ‘Pythagorean comma’. And in the Ptolemaic scale they are lowered by what is known (among other names) as one ‘Ptolemaic comma’. The Ptolemaic comma, as discussed, is a ratio of 81:80. The Pythagorean comma is a ratio of 531441:524288. Different! And yet... when you do the maths, and the the biology, the difference between these two commas (about 2% of the difference between E and F on your keyboard) is actually far smaller than the smallest difference that can reliably be heard by human ears and brains. [This difference is called a ‘schisma’, fwiw]

This isn’t the Ptolemaic scale... but it’s indistinguishable from it in practice. And unlike the Ptolemaic scale standing by itself, this 17-tone gamut allows this pseudo-Ptolemaic scale on C+ to be transposed up a fifth to G+; in other words, not only can you now play your Ptolemaic tune a fifth higher (useful if you want to accompany a woman or a young boy, for instance), but now every note in your Ptolemaic tune can if necessary be given a consonant fifth chord, making the scale much more harmonically useful! [remember how we said you couldn’t play a C tune on G in a Ptolemaic heptatonic scale (or a dodecatonic one) because, amongst other things, the A you’d need in scale wouldn’t be the same as the A you’d need for the other? Well, with a 17-tone gamut, a scale on C+ can use A while the scale on G+ uses A+ – problem solved!]

The 17-note gamut also solves some problems for people trying to play Pythagorean tunes. Remember how I pointed out that the circle of fifths didn’t really work in dodecatonic music? How the final fifth didn’t actually quite land back where you started? And do you remember how I earlier almost glossed over the fact that the Pythagorean scale doesn’t QUITE enable limitless transposition? Remember how I kept mentioning a big ‘caveat’? Well, that’s why: because although twelve of the intervals of a fifth in the Pythagorean 12-tone scale are identical to one another, the thirteenth fifth is not. It’s not the same, it’s not consonant, it’s hideous. It’s a wolf. Historically, in fact, it’s THE wolf, the original musical wolf after which other lesser nasty wolf intervals are named. The monster lurking in the woods to leap upon unsuspecting musicians over-casually transposing. [it’s actually called that because this chord supposedly sounds so awful that it “howls like a wolf”]. The wolf isn’t a massive problem in conservative Pythagorean music, because it can be safely hidden on the ‘far side’ of the circle of fifths, far away from the most common heptatonic scales (it’s often put between C# and Ab), but it’s always lurking. As a result, a 12-tone gamut cannot allow each of the 12 natural 7-tone scales to be identical, so cannot really allow unlimited transposition. In fact, only 5 of the 12 possible heptatonic scales are actually entirely free from the wolf (though several more have the wolf present but in a position where in European music it doesn’t get played very often).

One way to look at this is to say that, in conventional European notation, as we go further around the circle of fifths our scales get more and more sharps, and as we go the other way around we get more and more flats, and in our 12-tone gamut we assume that the scale with six flats (the scale built on Gb) is the same as the scale with six sharps (the scale built on F#). In reality, however, Gb (the note six fifths below C) isn’t the same as F# (the note six fifths above C) – they differ by a Pythagorean comma. As a result, all those sharp notes in the F# scale aren’t really the same as all the flat notes in the Gb scale. But on a dodecatonic keyboard, they look the same – because there’s only one black key between each white key. And if you forget this, the wolf will eat you.

...but wait a moment! In our 17-tone scale, suddenly there’s TWO black keys between each white key! We can now ditch our ‘+’ notation, and talk about flats and sharps. Our gamut now goes:

A-Bb-A#-B-C-Db-C#-D-Eb-D#-E-F-Gb-F#-G-Ab-G#-A

...and this lets us play all 12 of the heptatonic scales we wanted to play, all perfect transpositions of one another. Hooray!
[yes, flats of the note above are below sharps of the note below; this is NOT the case in some other tunings!]

Finally, what if we want to play intentionally unjust scales? Well, this system doesn’t have any explicitly ‘moveable’ notes, or any notes selected by ear for their unique melodic flavour. But what it DOES do is expand our choices. Now when we try to select two notes to fill the perfect fourth – two moveable notes to plunk in the middle of our tetrachord – we have not two notes to choose from (as in a purely heptatonic gamut) nor four (as in a purely dodecatonic gamut) but SIX. Put another way: a heptatonic gamut permits only one tetrachord; a dodecatonic gamut permits six different tetrachords; but a heptadecatonic gamut gives you a choice of FIFTEEN different tetrachords. Does anybody really need to be able to play fifteen different scales on one instrument? If you allow your upper tetrachord to differ from your lower tetrachord, that means your instrument can now play TWO HUNDRED AND TWENTY-FIVE scales! And whichever 7-note scale you choose, you’re still able to play another TEN extra chromatic notes if you feel you need more flavour! If you can’t find a scale to your liking in among all that lot, what more DO you need!? And because some of these notes are so close together that they sound more like versions of a note than independent notes in their own right, a performer can even use different versions of the note in different contexts – they can, for instance, use F# when the melody is leading up to G, but Gb when the melody descends, or to exploit a certain harmony.

So, a 17-note gamut has a great deal of potential. And it should therefore be no surprise that it’s had a fair amount of use: from the Middle Ages through to the modern era, the 17-note gamut was the theoretical foundation of Middle-Eastern and North African music (notes only a comma apart were never melodically contrasted; the 17-tone scale was used as a gamut from which other scales could be chosen for melodic purposes). In practice it’s likely that many performers deviated from this theory in their practice, altering some notes and ignoring others altogether; but nonetheless the Pythagorean 17-note gamut provided a conceptual basis and a practical starting point for tuning. 17-note gamuts even received a decent amount of use in Rennaissance and Early Modern Europe – there were even keyboards built to accomodate the extra notes, though they look rather odd to modern eyes.

But wait, you’re saying. We’ve gone from a 5-note scale to a 7-note scale to a 12-note scale to a 17-note scale, and they just keep getting better and better! Why stop when we’re on a roll!?

[sangi39]

I was just wondering, is it possible to say that "adding more notes" (using various systematic approaches) is a response to 1) wanting to allow more more instruments/voices to come together in a single performance, and/or 2) wanting the ability to transfer more smoothly from one key(?) to another more smoothly (while still being able to play/sing well alongside other instruments/voices) without having to either retune or pull out a pre-tuned instrument? Or would it be safer to say that those two things come from experimentation of adding more notes, and those new scales/systems being picked up and becoming more common?

(or is it a bit of both? Or neither?)

[Salmoneus]

I was just wondering, is it possible to say that "adding more notes" (using various systematic approaches) is a response to 1) wanting to allow more more instruments/voices to come together in a single performance, and/or 2) wanting the ability to transfer more smoothly from one key(?) to another more smoothly (while still being able to play/sing well alongside other instruments/voices) without having to either retune or pull out a pre-tuned instrument? Or would it be safer to say that those two things come from experimentation of adding more notes, and those new scales/systems being picked up and becoming more common?

(or is it a bit of both? Or neither?)

Well, when mediaeval and renaissance europeans began to talk about gamuts of more than seven notes (mostly twelve), it was (AIUI) primarily in response to harmonic practices - primarily, trying to accompany one singer a fifth higher or lower than them meant using notes that shouldn't have existed, which required them to conceptualise these new notes. Conceptualising them helped to standardise them, whether by getting different instruments in tune or by getting more tuned notes onto individual instruments.

AIUI, however, when Arabic theoreticians began to talk about seventeen-note gamuts, it was primarily in response to melodic practices - performers were using notes that didn't exist in their melodies (that is, they were tuning a lot of their notes 'wrongly' by pythagorean standards). Rather than just say "all you musicians are tuning all your notes totally wrongly!", theoreticians conceptualised a new framework that allowed them to explain (at least something closer to) actual practice, and this conceptualisation, again, eventually helped standardisation, to some extent (although as harmony and large ensembles are less of a thing in the arabic music tradition, less standardisation was wanted).

So in both these cases - again, in my admittedly limited understanding - the driver was musical experimentation, leading to theoretical explanation, leading to standardisation and more extensive use.

What, though, lead Sumerians to originally have seven note scales, rather than (as is more common) five note scales? As what point were those additional notes added, and why, and how? We just don't know. At least, I don't, and I don't think anyone else does either!

It is worth mentioning, though, that when musicians are exposed to instruments and traditions with more notes than they're used to, they seem happy to explore them, even if it doesn't fundamentally change their musical culture. And at least two major tuning reforms - Sumerian pythagorean tuning of heptatonic scales, and modern European 12-tone equal temperament - seem to have spread rapidly and extensively.

It's also worth saying that while I'm not aware of people consciously saying "it would be really convenient if we had more notes in our gamut so that we could play a wider range of music together more easily, let's invent another note!", I do think that once those notes were 'invented' (experimented with and then theoretically conceptualised) musicians did recognise the practical advantages of incorporating more note possibilities into their instruments, particularly with fixed-pitch instruments.

The organ, for instance, for hundreds of years (if not a thousand or more) had only seven notes to the octave. At some point an eighth was added, largely to enable the organ to play music in transposition (specifically, music written to be played on a mode starting on G could then instead be played up a fifth on the note that we would call C*, which saved them from needing to build big pipes all the way down to G). But in the early 13th century organ keyboards across Europe were rapidly rebuilt to the modern twelve-tone scale (by simply adding the additional notes in between and above the white keys they already had), since this was cheaper and less confusing than moving all the existing keys to new places. This shift was apparently a response to the development of polyphony, since having more voices with more intervals between them greatly increased the 'fictional' notes being used.


*on early organs, however, this note was instead called A. Fixed pitch notation took a while to develop! It's worth remembering that the pattern of the intervals between note in a mode/scale was for a long time more important than what those notes were called - whether you called the note C, G or A didn't actually matter. The mode on G had the penultimate note a full tone below G, but a keyboard designed to play a mode on C has the penultimate note of the octave only a semitone below C; but if you add an additional key to your keyboard a semitone lower than that (our 'Bb'), you can play your G-mode music just transposed up a fifth, and probably think of yourself as now playing on G. Or B, as I think an organist would have called the note.

[Salmoneus]

Adding EVEN MORE NOTES!!!

So, you know where we’re going with this. What happens if we continue around the circle of fifths another time? Well, if we keep tuning up in fifths (/fourths), we’re going to, as you can probably guess, split up the ‘big’ limmas into commas and a new interval that... I don’t actually have a name for (weirdly, given how many specific word there are for small musical intervals). Semi-limma? Let’s just call it that. There are 12 limmas in the 17-tone scale, so the next ‘round’ of cutting gives us a 29-tone scale, which... I’m not aware of anybody ever using? Although for what it’s worth, the semi-limma is almost exactly one third of a tone.

Maybe people don’t use this scale just because... well, having gotten this far, why stop there!? The semi-limma is still bigger than the comma, so we can just cut the semi-limmas up into commas and... demi-semi-limmas? [apparently some modern tuning enthusiasts call this the “mystery comma”] This gives us a 41-tone scale, which has had occasional use – 41-tone keyboards and guitars have been built, though not actually with Pythagorean tuning. At this point, though, the intrepid interval-slicer has almost reached their great prize: one more division (replacing our 12 demi-semi-limmas with hemi-demi-semi-limmas) and we get... 53-tone scales!

Why is that exciting? Because, if you’ve not followed along with the arithmetic, the hemi-demi-semi-limma is smaller than the comma... but by a very, very small amount. The difference between the two is not reliably perceivable to the human ear and brain (although some studies apparently show that it is discernable on some occasions to some listeners). The next round of tuning, therefore, which would divide the five commas into near-commas and this tiny, tiny interval (which some modern theorists call ‘Mercator’s comma’, although there’s another interval also known by this name) is probably impossible for humans to reliably accomplish. Following Pythagorean principles, therefore, 53-tone scales are probably the end of the line; anything beyond this is likely to be purely theoretical. [Having said this, other, non-Pythagorean scales with more degrees, without the tiny Mercator’s comma as a step, have been used; 72-tone scales have had some use in the Middle East]

The 53-tone scale itself, however, has a long history. Because its two steps (41 Pythagorean commas and 12 slightly smaller ‘hemi-demi-semi-limmas’) differ by an interval that’s not reliably distinguishable even in isolation (and would certainly be indistinguishable in the midst of actual musical complexity), it is effectively the same as equally dividing the octave into 53 notes (“53-tone equal temperament” or “53tet”). It is believed that the Greeks believed the octave did indeed consist of 53 commas; if they believed that these commas were all equal in size, these would be what are today called “Holdrian commas”. Certainly, the 53-tone scale (whether equal or not is unclear) was known at least by the time of Boethius in the 6th century. In China, the difference between the two steps of this scale (or, equivalently, between 53 perfect fifths and 31 octaves) was calculated to a six-digit fraction in the 1st century BC. In Europe, the scale was rediscovered in the 17th century (and the size of the discreprancy calculated precisely), and gradually popularised. In the 19th century, it became the standard tuning system in Ottoman music, and also found use in Arabic and Persian music; it never became standardised in Europe, although in the 19th century some instruments were built to accomodate it.

The 53-tone gamut, by having (almost) equal steps, is (almost) perfectly symmetrical, allowing (almost) perfect transposition between any contained scales on any notes. This includes allowing Ptolemaic scales on every note, and other scales with just intervals, and all the smaller Pythagorean scales, and a wide range of alternative scales with more characterful intervals. And the Pythagorean version (which is virtually identical to the perfectly symmetrical equitonal version) can still be tuned straightforwardly with fourths and fifths. Basically: it’s amazing!

So what’s the catch?

Well... who has time to tune 53 strings just to end up with an octave span? Worse, who wants to put 53 frets on a guitar, or 53 finger-holes on a pipe? Sure, the Pythagorean method makes each new tuning physically and aurally easy, but... 53 times? For each octave? And remember, instruments don’t just have to be tuned once and forgotten – string instruments in particular need to be retuned continually (either tension is released for storage, or tension is not released and gradually warps the instrument and its strings – not to mention the effects of heat and humidity). That’s just a lot of work. I’m not saying it’s not feasible, for the particularly dedicated with plenty of time on their hands. But for the ordinary musician down the pub (or equivalent), they’re surely going to think about taking shortcuts. The 53-tone scale may be useful for thinking about music, and talking about music – and not only for academic theoreticians but even for ordinary practitioners – but I suspect that in actual practice there will often be a lot of deviation from this model, particularly for vernacular instruments that need frequent retuning.

[And then again, there’s an even bigger problem for singers, who don’t have the benefit of physical indicators of pitch – asking a singer to sing a pitch cold, when they have 53 different pitches to choose from in every octave... that’s a lot of notes to learn, many of which may not frequently be used. Again, it’s not impossible, for expert professionals... but it’s a major challenge, I think.]

Given the practical problems with these ‘microtuned’ scales, it’s perhaps no surprise that they have been used in practice less than their repeated and cross-cultural theoretical discovery might suggest. It’s just much easier to stick with, say, a 17-tone scale... or, simpler yet, 12-tone!
First, though, let’s give this “add more notes” idea one last twist...

[Creyeditor]

I know this is probably a dumb question, but why do people distinguish between the 53tet and the 'Pythagorean'? Is that a purely theoretical difference thst cannot be perceived or does it indirectly influence other stuff that we can actually hear? Similarly, wouldn't such a system be perfect for an instrument with lots of buttons/keys and no strings, e.g. Accordions or Organs?

[WeepingElf]

I am chiming in late, but I have just read through the entire thread, which explains these things in an understandable way. Great work, Salmoneus! In fact, I have now decided to toss the bizarre harmonic-scale-based tuning that I imagined for my Elves on this page because it apparently just doesn't work the way human music works (and those "Elves" are actually humans, so they have no business doing things humans don't do!).

[Salmoneus]

I know this is probably a dumb question, but why do people distinguish between the 53tet and the 'Pythagorean'?

Well, to be fair, it's not always clear whether people have. As I mentioned, Classical civilization theorised the 53-tone octave, but we don't actually know if they only meant the Pythagorean scale (which presumably they did originally) or whether they actually thought that that was equivalent to the 53tet scale, or whether they realised they were theoretically different but didn't think it was worth making the distinction. And iirc the earliest explicit surviving early modern discussion of these scales was talking about how they were indistinguishable in practice.

That said, they are different scales. Just (probably) not audibly different!

Is that a purely theoretical difference that cannot be perceived or does it indirectly influence other stuff that we can actually hear?

I suppose that theoretically the differences would add up. For instance, if you worked out the size of one division of the 53-tone pythagorean scale, assumed that all the divisions were that size, and made an octave of 53 of them, your octave would be audibly slightly dissonant.

Historically it would probably have been easiest to use the pythagorean scale and just pretend it was equitonal. In the modern age, on the other hand, where we have better understanding of logarithms, it's probably easier to just use the equitonal scale and pretend it's pythagorean if that's what we wanted.

Similarly, wouldn't such a system be perfect for an instrument with lots of buttons/keys and no strings, e.g. Accordions or Organs?

The problems again are with the practicalities. Firstly, the tuning - that's a lot of pipes and/or reeds to hand-tune to very precise ratios. And while, yes, pipes don't wear as much as as strings, they do still have to be periodically retuned (with organs, this was traditionally done by hitting them with a hammer, gradually shortening the pipes and leading organs (and church music in general) to gradually increase in pitch over the centuries!). I'd think that metal reeds would be even more robust... but accordions are still sold advertising the ease of access to the reeds for retuning purposes, apparently, so it's still an issue.

Second, the size. Imagine an accordion - the standard piano accordion, apparently, covers three and a bit octaves on the treble keyboard. That's 41 keys. Now let's imagine three octaves in a 53-tone scale: that's 159 keys. That's nearly twice as many keys as on a standard piano! Maybe you can cheat - perhaps have 12 keys per octave plus additional buttons to raise or lower pitches. But even that adds complexity (and hence cost!) and difficulty of playing. And you then have to find somewhere to put all the reeds. It's actually easier for the accordion, because the reeds are quite small; for organs, that's a LOT of new gigantic pipes to fit somewhere. With all these considerations, remember the cost. For most of history, even a two-string lute instrument has been expensive. Accordions, with their complex internal workings and countless moving parts, didn't become a thing until after the industrial revolution!

And third, the player. That's a LOT of different keys to have to learn to play!

So... there are strong impulses to having fewer notes per octave, from the practical side, despite the theoretical advantages. And much of the theoretical advantage is... well, theoretical. Most players of most music aren't going to ever use most of those potential notes!

So the biggest use of the 53-tone scale is conceptual: it's a way to describe the tuning of instruments tuned to other, actual, more manageable scales. Having said that, 53-tone instruments are possible. They're just really inconvenient. If a culture has them, it's probably because it has a lot of a) mathematicians, b) wealth (to pay someone to make them and someone else to play them), and c) a sophisticated culture of 'art music'. It's less likely that they'll be used by people playing in a band down the pub. That said, iirc there have been baglamas (simple lutes, basically) with 34 frets to the octave, so it's maybe not impossible!

[Salmoneus]

I am chiming in late, but I have just read through the entire thread, which explains these things in an understandable way. Great work, Salmoneus! In fact, I have now decided to toss the bizarre harmonic-scale-based tuning that I imagined for my Elves on this page because it apparently just doesn't work the way human music works (and those "Elves" are actually humans, so they have no business doing things humans don't do!).

I don't think it necessarily is unrealistic. It's an understandable basis for a scale, and weird music does exist on earth (hello, Indonesia!). It's worth pointing out that five of your notes are actually the same as the in the Ptolemaic scale, and even the septimal minor seventh shows up in various places - its just your 4th and 6th degrees that are weird! In real life, plain horns have been used quite a bit, and will play your scale - apparently there are compositions for alphorn that uses these notes. It's just that generally people find a way to make a scale that's less weird (having keys, valves, hand-stopping, using pitch-bending, or just having multiple pipes).


Harmonic scales do actually exist on earth! I'll be mentioning them briefly at the end. However, the only ones I'm aware of (as the main musical system of a culture, rather than just something that shows up on a weird instrument like an alphorn or military bugle) have fewer notes - 4 or 6, iirc. However, that's probably because these scales (which occur in Africa) are based on the musical bow, rather than the alphorn, so it's harder to select higher harmonics. But if your music culture is based on, say, the drum and the carnyx, it might make sense to use your scale.

It would be regarded as an oddity, I think, but it wouldn't be the only odd scale in the world, by far.

[Creyeditor]

@Sal: Thank you for your detailed answer, again. Of course, space and expertise are right. I really value your more conlanger-y perspective. Now I have a rough idea of where to place ridiculously complex pipe instruments. (Right now I'm thinking maybe a massive derivative of Sheng that uses keys/buttons and some additional mechanism for air pressure.)

Edit: Maybe similar to a Melodica. Also Wikipedia has a (small) list of fictional musical instruments.