All About Musical Scales (and How to Tune Them)

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Salmoneus
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All About Musical Scales (and How to Tune Them)

Post by Salmoneus »

So, as some of you may remember, some time ago I said I would post a series of posts on How Music Works, which I thought may be of general interest, and could be specifically of interest to conworlders thinking about how music works in their own societies.

That, as you may have noticed, hasn’t happened – I’ve started writing the series up several times, and do still hope theoretically to finish it, but... not yet.

In the meantime, however, Solarius wrote a little on his own conmusic, and that made me want to explain a few things about one specific (though very important) area of music. So I thought I’d write up a quick post on the topic, in lieu of the whole giant series I’m theoretically planning to write one day.
...and the sheer ridiculously humoungus project that this has turned into kind of illustrates why I’ve never yet written up the full series (that would be probably at least twice this size again).

So, I appreciate that this may be TLDR, and I’m splitting this up into multiple posts, spread across multiple days to allow chance for feedback (also because I haven’t actually finished it yet – but don’t worry, I’m close enough to the end that I’m confident I’ll get there soon).

Do feel free to ask tangential questions, and please do query anything that’s unclear. It’s difficult to find a balance between “racing through so quickly that nobody understands” and “taking an eternity and making everyone feel patronised”, particularly because, as someone who knows a certain amount, it’s hard to know which bits other people do and don’t know. [particularly because I also know that there’s a lot I don’t know myself, some of which I probably don’t realise I don’t know]. I'm going to give a proposed table of contents, so if you're thinking of asking a question that might help you to decide when the question should best be asked.

I’d also like to be clear from the start about a massive caveat: I’m not a musician, a musicologist, an ethnologist, a historian, a physicist, a mathematician, an organologist or craftsman, a psychologist or neurobiologist or any of the other things I ought to be to in order to explain this topic with any authority. I’m just a layman with an interest in music and an access to the internet. But I think I’ve found out more than many other layman I’ve encountered have found out yet, and I hope that putting some of it on paper in one place may save other people some time and provide some starting points for their own reading.

All that being said, let’s get started! Or, at least, let's move on to the next stage of preamble!
Last edited by Salmoneus on 18 May 2023 16:40, edited 1 time in total.
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Re: All About Musical Scales (and How to Tune Them)

Post by Salmoneus »

Why are there twelve notes in an octave? Why does a scale only include seven of them? Why do many cultures only use five notes in their music? Why those five? What on earth is ‘temperament’? A lot of things in music appear arbitrary and confusing. Partly because some of them are. But a lot of things DO actually make a lot of sense when you understand their basis. So today we’re going to talk about scales!

[I’m mostly going to try to avoid talking about what people DO with their scales – modes, tonality, anything to do with melody and harmony – although occasionally I will have to briefly refer to these things. Do stop me if you get lost]

I’m currently planning to discuss these topics, in probably this order (some topics may end up split up or joined together):

1: Introduction (this bit!)
2: Octaves, Tuning in Fourths, Ditonic Scales
3: From Tritonic to Pentatonic
4: Tangent: Transposition, Permutation, Tuning in Fifths, and Tuning in Different Directions
5: Heptatonic Scales and Gamuts
6: Dodecatonic Scales
7: Just Intonation
8: Intentionally Dissonant Scales
9: Heptadecatonic Scales
10: Scales With More Than 17 Degrees
11: Extended and Adaptive Tuning
12: Meantone and Other Tempered Scales
13: Equitonal Scales
14: Imperfect Octaves and Non-Octave Scales
15: Harmonic Scales

Do ask if there's some specific additional topic you want me to touch on.

[One caveat/warning/admission: I'm only going to be looking at things at a fairly basic, reality-centred level. I'm not going to go any distance into the mind-flaying quicksand of modern online microtonal/xenotonal/etc fandom; hopefully this thread will give you some hopping-off points if that's a direction you want to go in yourself, but it's too big a topic to meaningfully talk about here and not one that I understand very well]

But, for now, let's actually get into it!
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Re: All About Musical Scales (and How to Tune Them)

Post by Salmoneus »

Octaves, Tuning in Fourths, and Ditonic Scales.

Let’s assume that we’re trying to play different notes on a string instrument. The easiest (at least, conceptually simplest) way to do this is to add an additional string. So let’s use this as our thought experiment.

[the same logic also applies to finding new notes on an existing string (eg by adding frets), or to instruments based on different lengths of pipe, but we’ll stick with this thought experiment, which directly describes a harp or lyre]

Without precise tools, we can’t guarantee perfect replication of pitches or intervals. However, we’re probably going to want to have some way of keeping the intervals between string pitches fairly consistent, both between instruments and every time we have to restring our own instrument, because otherwise popular tunes will sound very different. [two instruments only need to have the same specific pitches if they’re going to play together, but they need roughly the same intervals between pitches in order to recognisably play the same tunes]

We could make use of a measuring stick, to measure string lengths (all else (i.e. composition and tension) being equal, string length determines pitch). This is a little awkward, particularly if we want to make our next instrument slightly bigger or smaller – since what we really care about is the ratios between string lengths, not their absolute lengths.

So the easiest way? Just fold a string. That’s easily replicable, doesn’t require specialised tools, and works with any length of string. It also has the benefit of inherently creating consonant intervals [for reasons that are too fascinating to go into here!], which is advantageous for three reasons: first, because they inherently sound nice [for reasons that are too fascinating to go into here!]; second, because they are immediately recognisable by ear [for reasons that are too fascinating to go into here!]; and thirdly [for reasons that are too fascinating to go into here, but relating to ‘sympathetic resonance’] they make the instrument itself sound more ‘alive’, and sometimes louder. So folding a string will immediately give us more or less a replicable interval that sounds pleasant in tunes or chords, and it will be close (if not exact) to a highly recognisable interval that we can easily fine-tune by ear (and teach our students about without much confusion, because it’s so recognisable). That sounds like a win!

So, we start out with one string, with length X. First plan, fold that string in half as a guide and cut a second string to that length (X/2). Or, equivalently, start with two strings the same length, but fold one in half and cut it (throwing away one half). What interval does that give us between the two strings? We call this ‘an octave’ (for historical reasons that will eventually become apparent).

Octaves are great. They’re almost perfect: play an interval of an octave as a chord and [for reasons far too fascinating to go into here] it barely sounds different from playing one note (it sounds richer, rather than sounding like a different note; an octave on one instrument might not immediately be distinguishable from a single note on a richer-sounding instrument). This effect is so strong that we commonly say that notes separated by an octave interval are ‘equivalent’. If we call our first note A, we’ll call our second note... also A. Or A’, to indicate that it’s a different instance of A, but it’s still in some important way ‘the same’.

This isn’t just a human thing, by the way. Studies show that even rats recognise octave equivalency!

But music consisting only of A (and A’, and A’’, etc) is kind of dull. We’re going to want more notes that that.

So, let’s fold our next string in half TWICE. The new string length gives us... A’’, two octaves up. That’s no better. But what if, when we cut our second string, rather than keeping the short length (1/4 of X) we instead keep the long length (3/4 of X)? Now we’re getting somewhere!
The note this string gives us we conventionally call (for reasons that will eventually become apparent, honestly) D. And again, by halving that we could easily create D’, and D’’, and so on.

In fact, once we ‘create’ a note, we can easily ‘have’ the same note at octave intervals, into any octave. And given how recognisable an octave is it’s barely a chore to find a longer string to have the same note an octave (or more) below as well. Octave equivalence is so basic that we generally ignore it altogether, and assume that when we say we ‘have’ D, we also mean we have, or could easily get, all the other Ds in every octave as well [there are some rare exceptions to this that we won’t worry about yet, and maybe never]. So we don’t have to worry about listing every string, we’ll just talk about the strings within a single octave span as representative of all possible octaves (though of course in practice no one instrument will actually have all octaves). We’ll call this representative list of available notes within the span of an octave a ‘scale’. So far, our scale is A-D (and A’ and D’, etc etc).

[talking about scales thus requires a sort of clock logic, as they go around in a circle]
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Re: All About Musical Scales (and How to Tune Them)

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From Tritonic to Pentatonic

So, A-D. Well, it’s more interesting than just A. Entire cultures survive on this, and even in many cultures with more complicated music some instruments may have only these notes. But... we can do better. We can tune up by a fourth again, the same way as before! This will give us a new note, which we will (bafflingly, for the moment) call G, and a new, “tritonic” scale: A-D-G.

A three-note scale gives us more possibilities than a two-note scale, particularly because the interval between G and A is very different from that between A and D – it’s not really all that consonant at all, which gives our music more variety (whether we use this interval in sequence or simultaneously). But here’s an interesting thing: we have more notes now, and hence more intervals, but how many TYPES of intervals do we have? Specifically, how many intervals between adjacent notes?

Let’s call intervals between adjacent notes “steps” for simplicity! We have three steps in our scale: A-D, D-G, G-A’ (when I said before that we’d only talk about the notes in one octave span, I was counting inclusively; we don’t have to list A’ when we spell out the scale, because it’s implied, but it it is important for intervals, because the scale goes round like a clock; it doesn’t just stop at G). But we still only have two TYPES of step: our original fourth (A-D, D-G) and a new smaller, less consonant interval we’ll call a “tone” (G-A’). This isn’t really important right now, but it will become significant in a moment.

But let’s assume for the sake of argument that we’re not happy with this “tritonic” scale (which many cultures are!). We want more variety, more compositional possibilities. So let’s add a new note, in the same way. Going up a fourth again from G, we get our new note, C. Actually, in a physical sense it’s C’, because we’ve now gone up more than an octave from where we started, but remember, we’re using octave substitution (i.e. in practice ‘finding a string half or twice the length’) to just talk about a single octave span, so if we have C’ then we also have C, and we’ll just talk about C for simplicity. So now we have a tetratonic scale, A-C-D-G.

[real-world housekeeping: annoyingly, in European tradition, note names clearly start from A, but octave labelling bizarrely starts from C. So if we’re writing European music we would actually call the strings we have so far A-C’-D’-G’-A’-C’’, etc. Which is worth knowing, but needlessly confusing for our purposes, so forget about that for now].

But wait! Something’s weird here! In our ditonic and tritonic scales, there were only two step types, a big step and a little step. But in our tetratonic scale, we have three types: a fourth (D-G), two tones (C-D, G-A) and also something we’ll call a “minor third” (A-C).

Is that a problem? Not necessarily. Some cultures do use tetratonic scales. But not very many. And that’s because we’re about to see what looks at first like a coincidence... which is that one of our step types (the fourth) happens to be exactly equal to the sum of the OTHER two step types (the tone and the minor third). And if we think about it that’s not a coincidence at all. When we had three notes, we had an interval of a fourth between A and D; adding a new note has, as it were, ‘split’ that interval into two parts, a minor third and a tone. And that kind of puts a thought in the back of our minds: why can’t we do that with the other fourth as well? If we play around with a tritonic scale, and then with a tetratonic, it becomes very obvious to us that the remaining fourth is ALSO only the sum of two smaller intervals, just like the other one was... we just don't have the string that would let us play the note ‘in the middle’. Once we feel that we can just split fourths up and add a note in the middle, we’re going to feel limited that there’s this big interval in our scale, the size of any two other intervals combined, and we just have to leap over it. It feels like there’s a note missing on our instrument!

And, lo and behold, that note is right within our grasp. Because if we just tune up ANOTHER fourth, we get a fifth note, which we’ll call F. And this note, by marvellous coincidence, exactly splits the D-G step (our remaining fourth-step) into two parts that are exactly the same as the two parts we split our first fourth-step into: a tone and a minor third. Well that simplifies things! Now our scale, our pentatonic, A-C-D-F-G, has only two types of step again: the tone (C-D, F-G and G-A) and the minor third (A-C and D-F). It doesn’t feel as though there’s a missing note anymore. And once we’ve tried tuning up to get this fifth note, and realised that it perfectly fills the D-G gap in a way that makes the scale sound like it doesn’t have any missing notes anymore, it seems unnatural to us NOT to do that.

This thus gives us the most common type of scale in world music: the pentatonic. And a specific, almost universal type of pentatonic, one made up of this specific sequence of steps: minor third, tone, minor third, tone, tone, (minor third, tone, minor third, etc). For simplicity, we’ll call this pattern 3-2-3-2-2.
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Re: All About Musical Scales (and How to Tune Them)

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And I think that's enough for today! Back with more tomorrow!
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Re: All About Musical Scales (and How to Tune Them)

Post by Creyeditor »

Nice, nice, nice! In general, I will read through the whole thread and I like your style and structure so far. More specifically, I read through this part and I already knew the individual facts but I wasn't aware of the connections between them. So good job [:)]
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Re: All About Musical Scales (and How to Tune Them)

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Creyeditor wrote: 18 May 2023 19:37 Nice, nice, nice! In general, I will read through the whole thread and I like your style and structure so far. More specifically, I read through this part and I already knew the individual facts but I wasn't aware of the connections between them. So good job [:)]
Thank you.
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Re: All About Musical Scales (and How to Tune Them)

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Tangent: Transposition, Modes, Tuning in Fifths, and Tuning in Different Directions

Let’s have a bit of housekeeping at this point. To define a scale in real terms, we actually need two things: we need to have a pattern of relative pitches (our 3-2-3-2-2), but we also need a specific pitch to ‘anchor’ those relative pitchs to. For instance, we might declare that our A has a pitch of (approximately) 440 Hz. From this, plus our pattern, we can produce the rest of our scale (A-C-D-F-G) and find the pitches of each. [Of course, in real life nobody in a pre-modern society can know the exact pitch of any note, so this is always going to be approximate; the key is to decide a pitch for one string on one instrument, then tune the rest of the instrument, and any instruments it will be playing with, to fit with that first string.]

We can move our scale up or down in pitch without altering its basic relational properties – we call this ‘transposing’ it. If we want, we can keep using our old note names for the notes of this new scale – this is how the old ‘do, re, mi’ (etc) note names work – and just specify the new pitches of our new ‘do’. Alternatively, we can reserve the names we originally used for specific pitches (ie. A is still 440 Hz) and rename all the notes in our transposed scale – this is how the modern A-B-C European note names work (it’s more convenient when you’re dealing with lots of instruments playing in multiple scales, from written notation). So, for instance, if we transpose our original pentatonic up a minor third, we get a scale we can call C-Eb-F-A-B. This scale has different notes from our original pentatonic on A, but it has the same relationships between its notes – it’s the same TYPE of scale. But it’s a different scale because it has different notes (at some points here I’m probably going to forget this pedantry and refer to two instances of the same type of scale as being ‘the same scale’, but bear in mind that when Ido that I’m speaking generically, using ‘scale’ to stand for all actual instances of a certain type of scale).

Remember, however, that our scale pattern – our 3-2-3-2-2 – is repeating, and hence conceptually infinite, and thus the choice of where to ‘start’ that pattern is arbitrary. We could start at a different point in the cycle instead. In this sense, the notes D-F-G-A-C and A-C-D-F-G are the same scale: they have exactly the same notes (though one has the pattern 3-2-2-3-2 and one has 3-2-3-2-2). We say that these are different ‘modes’ of the scale. If we arbitrarily define one mode as the ‘first’ mode (say, 3-2-3-2-2), we can number the other modes in order (so, the D-mode is the third mode in this case). Another way to describe this is to say that the first mode is built on the ‘first degree’ of the scale (A) while the third mode is built on the ‘third degree’ of the scale (D).

For a real-world example: in Western music, what we now call a ‘C major scale’ is physically the same set of notes as the ‘A natural minor scale’ – they are simply different modes of one another.

[What does “starting on” a certain degree of the scale actually mean in real music? Well, that’s a much broader question, with answers varying from “very little” to “absolutely everything” depending on musical culture. For now, just bear in mind that how you ‘use’ a scale, in terms of which notes you treat as more fundamental, does change how it sounds (A minor and C major give quite different impressions) but doesn’t change the actual mechanics of which notes are present]

In a pentatonic scale, there are only five modes – one for each degree (note) of the scale. There are in theory an infinite number of transpositions of that scale, however – up or down by any pitch ratio you choose. [In practice, of course, once our instrument is tuned to one scale, it will be more convenient and easier to understand and talk about if we transpose our scale to start on another degree of that original scale... but we don’t have to!]
Now, if you’re still with me, let’s take a step back and answer a lingering question: what if we don’t tune up in fourths in the first place? Well, that’s an open-ended question. But let’s look at three of the most obvious alternatives.

First, what if we tune up in fifths instead? To make a fifth, we have to fold our first string into three, and cut a second string that equals two thirds of the first string. That’s kind of a bit more complicated, but the upside is that fifths sound even more recognisable than fourths, so are a bit easier to tune by ear. Starting with A again, we get: A-E-B-F#-C#, which we can rearrange as an A-B-C#-E-F# scale. That’s a tone, a tone, a minor third, a tone, and another minor third – or 2-2-3-2-3 for short. But wait! That pattern’s just the fourth mode of our original pentatonic! If we modulate it to start on C# (C#-E-F#-A-B) then it’s just our original “natural pentatonic on A” but transposed up a bit! So tuning up in fifths gives specific different notes, but the same fundamental type of scale (but in transposition).

Now, what if we try tuning DOWN in fifths? We do this by folding our string in half, and then finding a new string exactly three times that folded length, and so on, so that our new strings get lower in pitch rather than higher. Starting on A, this gives us A-D-G-C-F. Which we can rearrange into a scale, A-C-D-F-G. Hang on, that’s the scale we started with! It turns out, tuning down in fifths gives the same notes as tuning up in fourths. And similarly it’s easy to demonstrate that tuning down in fourths gives the same notes as tuning up in fifths. So whether we go up or down, in fourths or fifths, we get the same natural pentatonic scale type.

And indeed we could go up AND down in either fourths or fifths, or both. Babylonian (and likely other Mesopotamian) instruments were tuned in this way, alternating downward fourths and upward fifths. Starting on F, this gives F-C-G-D-A, which unscrambles to A-C-D-F-G (again). [this method also removes the need to worry about octave substitution at all]

The ‘natural’ pentatonic, in its various modes and transpositions, is therefore very easy to discover – indeed, it’s hard (albeit not impossible) to escape, if you only tune in fourths and fifths, which are the easiest (both mechanically and aurally) to tune in.
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Re: All About Musical Scales (and How to Tune Them)

Post by sangi39 »

I've been looking at scale-building on and off for ages now, and I've only just worked out the whole "build on relative pitches" step you mentioned (3-2-3-2-2 as your example) and man that's helped understand, I think, a little bit better what makes a scale "good" (in terms of, like, someone sitting down, messing about with strings or weights or whatever, and noticing changes in pitches)

So like, you've got the 2/3 and 3/4 ratio, and they both give you these nice, incremental, well-spread series of notes from one note to the next octave, in regular steps (2-2-3-2-3 and 3-2-3-2-2 respectively, where those gaps don't feel too big, even between "big" and "small" intervals), but then when you start messing round with other ratios (I've been working along the "fold length of string in half, then again, then again, and so on, and keep various sections of those strings back, e.g. 13/16ths, 7/8ths, etc.), and I've found the only two ratios after 2/3 and 3/4 that are built on the "halving and keeping" method that really give you any immediate results for little effort are 7/8ths and 13/16ths

(assuming A=440 and working from there)

7/8ths gives you whole steps right up the octave of ~231 cents (a little over a whole tone), but then between what's sort of a sharp F# (19 cents higher than the F# you built using perfect fifths) and the next A, you jump up by ~275 cents instead which feels a little unbalanced

13/16ths gets you back onto regular relative pitches 122 cents. vs. 238 cents arranged 2-1-2-1-2-1-1, which feels a little more balanced, but you need to do a little more work with the "halving and keeping" method to get there (going up into heptatonic scales), and I imagine without notes breaking up those 238 intervals, those gaps might feel quite large? Like that's going from an A, to something almost halfway between a B and a B#, then to something halfway between a C and a C#, then right up to a D#, quickly followed by a slightly sharp E)

5/8ths, 15/16ths and 11/16ths* bunch up too much, so you have to work on them even more to get regular intervals. 5/8ths, for example, has intervals of 41 cents (less than half a semitone) and 346 cents (almost one and three-quarter whole tones) (~17:2 respectively), so you need to repeat the process more to get those relatively intervals nice and spread out. 15/16ths has the same problem that 7/8ths has (I assume for the same reason, e.g. (n-1)/n being the ratio), and you get four intervals of 112 cents followed by a massive jump of 753 cents to the next octave (almost 7:1)

(EDIT: I did just have a run through 15/16ths and, yeah, it basically seems to divide that last interval up into 112 cents plus whatever remainder, so you only get a "well-spread" interval with 10 or 11 notes to the octave, but as with 7/8, the final interval is irregular, being either too small or too large, as compared to the regular intervals up until that point)


So, yeah, 3/4 and 2/3 ratios are like "hey, here, have these two really nice pentatonic scales almost for nothing" which I imagine, as you point out, Sal, is why they keep popping up everywhere (easy to make, nice and spread out, and balanced, and you can do a lot with them)



*I had a quick run through with 11/16ths, and this one seems to balance out by basically dividing the octave into "short" intervals of 97 cents and "long" intervals of 162 cents (2-2-2-2-2-3-2-2-2-2-3), with 11 tones to the octave, which I can't imagine a culture just stumbles its way into in the same way you might see with 3/4 and 2/3



EDIT2: Sorry for stepping on your toes at all there, Sal. Your posts just massively helped me make a big leap in understanding tones, and I started rambling, haha
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Re: All About Musical Scales (and How to Tune Them)

Post by Salmoneus »

sangi39 wrote: 20 May 2023 22:09 I've been looking at scale-building on and off for ages now, and I've only just worked out the whole "build on relative pitches" step you mentioned (3-2-3-2-2 as your example) and man that's helped understand, I think, a little bit better what makes a scale "good" (in terms of, like, someone sitting down, messing about with strings or weights or whatever, and noticing changes in pitches)
It's weird how it seems to work - music seems to have a lot of "ah-HA! now I understand that!" moments in my experience. A lot of this post series is because a while back I found a website using scales as an example of mathematics, that showed how tuning in fifths, with clock mathematics, would lead to these scales with only two interval types (in specific sequences) - with helpful diagrams. Unfortunately, I can't find it again. But reading it the first time I felt I suddenly understood things that had been a mystery to me before!
So like, you've got the 2/3 and 3/4 ratio, and they both give you these nice, incremental, well-spread series of notes from one note to the next octave, in regular steps (2-2-3-2-3 and 3-2-3-2-2 respectively, where those gaps don't feel too big, even between "big" and "small" intervals), but then when you start messing round with other ratios (I've been working along the "fold length of string in half, then again, then again, and so on, and keep various sections of those strings back, e.g. 13/16ths, 7/8ths, etc.), and I've found the only two ratios after 2/3 and 3/4 that are built on the "halving and keeping" method that really give you any immediate results for little effort are 7/8ths and 13/16ths
Interesting!

FWIW, the scales you're building here (and that I was building with fifths/fourths) are known as linear tunings, and they're a subset of regular tunings. Regular tunings, in this sense, can be 'generated' through the repetitive application of a finite number of set intervals, the 'generators'. Linear tunings are specifically those in which only two generators are required - in my example, the fourth and the octave (or the fifth and the octave, or the fourth and the fifth, they're all equivalent descriptions). Linear tunings are the musical expression of, apparently, the mathematics of free abelian groups of rank 2. I don't know what that means, the wiki entries for those are frightening.

Anyway, unless the non-octave generator is a rational factor of the octave itself, no number of applications of the generator will ever equal a finite number of octaves; and no consonant interval can be a rational factor of the octave. EITHER your generator can be a rational fraction of the starting frequency, OR it can be a rational fraction of the octave, but not both. However, whatever your non-octave generator you'll always be able to produce scales with only two interval step types - though they may be so lopsided in sizes that they aren't great for music!

[incidentally, I don't think there's an obvious word for "scale with only two step types". I've seen both "ditonic" and "diatonic" used in this sense, but neither is great, as they both primarily mean something unrelated!]
(assuming A=440 and working from there)

7/8ths gives you whole steps right up the octave of ~231 cents (a little over a whole tone), but then between what's sort of a sharp F# (19 cents higher than the F# you built using perfect fifths) and the next A, you jump up by ~275 cents instead which feels a little unbalanced

13/16ths gets you back onto regular relative pitches 122 cents. vs. 238 cents arranged 2-1-2-1-2-1-1, which feels a little more balanced, but you need to do a little more work with the "halving and keeping" method to get there (going up into heptatonic scales), and I imagine without notes breaking up those 238 intervals, those gaps might feel quite large? Like that's going from an A, to something almost halfway between a B and a B#, then to something halfway between a C and a C#, then right up to a D#, quickly followed by a slightly sharp E)

5/8ths, 15/16ths and 11/16ths* bunch up too much, so you have to work on them even more to get regular intervals. 5/8ths, for example, has intervals of 41 cents (less than half a semitone) and 346 cents (almost one and three-quarter whole tones) (~17:2 respectively), so you need to repeat the process more to get those relatively intervals nice and spread out. 15/16ths has the same problem that 7/8ths has (I assume for the same reason, e.g. (n-1)/n being the ratio), and you get four intervals of 112 cents followed by a massive jump of 753 cents to the next octave (almost 7:1)

(EDIT: I did just have a run through 15/16ths and, yeah, it basically seems to divide that last interval up into 112 cents plus whatever remainder, so you only get a "well-spread" interval with 10 or 11 notes to the octave, but as with 7/8, the final interval is irregular, being either too small or too large, as compared to the regular intervals up until that point)
As I say, you'll always have an irregular interval, because your generator will never divide equally into the octave. However, rather than having all big intervals and one small remainder, you can keep on adding your generator to break up the big intervals.

In this case, if you have a 112 cent generator, your first 'ditonic' (in this sense) scale will have 10 big intervals (111.73) and 1 small one (82.7). But if you keep going 'around the clock', as it were, those big intervals all break up and you're left with 11 intervals of 82.7, and 10 intervals of 29.03.

This 21-note scale would actually have one big thing going for it: the 14th note of the scale would be almost exactly a perfect fifth! However, the thirds would be really bad, and the 'perfect' fourth would be awful.

FWIW, ratios of the form (n-1):n are known as "superparticular" ratios, and they're incredibly important in music, for reasons too fascinating to go into here. The 16:15 ratio crops up a lot in music as it's usually one of the semitones in a just intonation scale.
EDIT2: Sorry for stepping on your toes at all there, Sal. Your posts just massively helped me make a big leap in understanding tones, and I started rambling, haha
Not at all! I'd much rather have some discussion than just me rambling into the void by myself!
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Re: All About Musical Scales (and How to Tune Them)

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@Sal: Could you fix the quotes in the second part of the post? I can't see them and I would like to follow this intriguing discussion.
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Re: All About Musical Scales (and How to Tune Them)

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Heptatonic Scales and Gamuts

Right! Now that all that’s out of the way, let’s get back to the fun bit: adding notes!

Pentatonic scales are great. But let’s say that we’re not satisfied with them. We want more notes. So, we keep on adding strings. For ease of notation, I’ll actually tune the next note DOWN a fourth from A rather than up – but, as just discussed, this doesn’t affect the type of scale we generate. So, down a fourth from A is a note we call E. And we have a hexatonic scale. But wait!

This new note, E, slots into a gap in our pentatonic scale between D and F – in effect, it has split one of our big steps (a minor third) into two smaller steps. One of these steps is a tone, like most of the other steps in our scale; but, annoyingly, the remainder is not a tone but something a bit smaller than half a tone, which we’ll call a ‘limma’. We could encode our new scale as 3-2-2-1-2-2 So our hexatonic scale is a lot like our old tetratonic was: one the one hand, we now have three step sizes rather than two; and, on the other, we have a single big step (the minor third between A and C) that’s obviously conceptually divisible into the two smaller steps (the tone and the limma), but that our scale pretends is indivisible: we have a missing note again. So the hexatonic scale feels inherently imperfect: there’s a missing note, and there’s one step bigger than all the others (and one step smaller than all the others as well). Hexatonic scales are, accordingly, relatively rare, and most of them are generated in an entirely different and less common way (which I’ll briefly describe in an appendix here, but for now it’s beyond our topic).

Fortunately, you may not be surprised to realise that if we tune down another fourth, to B, we place a new note right in the middle of that weird big step. Now we have again created a scale with only two step types, with no big gaps that stick out. Our new scale is heptatonic, with the step pattern 2-1-2-2-1-2-2. The notes in this scale are now A-B-C-D-E-F-G. This should look familiar to you!

Yes, the heptatonic is incredibly common; most cultures that don’t use pentatonic scales use heptatonic ones, including (sort-of! back to that in a moment!) European classical music. It should also be pointed out that heptatonic and pentatonic music aren’t incompatible: a lot of Chinese and Japanese music, apparently, is essentially pentatonic, but also uses the ‘extra’ two notes from time to time, in auxiliary functions. [specifically, the Chinese heptatonic scale is derived by tuning up in fifths, giving what for ease of notation we could call F-C-G-D-A-E-B, but with the final two notes, E and B, only being used in auxiliary functions – that is, the scale F-G-A-(B)-C-D-(E)].

[NB. as a result the often-cited distinction between heptatonic European music and pentatonic Chinese music is a little over-stated: Chinese pentatonic music does sometimes use two additional notes, while many tunes in ‘heptatonic’ European music, particularly those deriving from folk music, are actually pentatonic at their core. The distinction is therefore to a large extent that of positions on a continuum rather than the true binary dichotomy that it is often presented as]

Indeed, even if a person wishes only to play purely pentatonic music, a heptatonic instrument may be useful, as it allows limited transposition of music (for instance, to more easily fit the comfortable singing range of a second singer, or to bring two instruments into tuning alignment with less effort). Using the pentatonic scale A-C-D-F-G, for instance, a melody starting on A cannot simply be transposed up to start on any other note, without fundamentally altering its character; but on an instrument strung to the heptatonic scale, A-B-C-D-E-F-G, more or less the same melody can be played beginning on A (using an ACDFG scale), on B (using a BDEGA scale) or on E (using an EGACD scale). There is thus a practical advantage to having more strings on your instrument than you actually need for your tunes.

At this point, relatedly, we can define a new concept: the gamut. A gamut (and I’m not 100% sure I’m using this term in its most common contemporary technical meaning, although it’s a useful meaning that fits with historical usage) is a complete set of notes that, for the purposes of a musical culture, exist. A gamut contains all the notes of all the scales used by that culture, but most scales will not contain all the notes of the gamut; in turn, some melodies may not make use of all the notes available to them in the scale the melody is conventionally said to belong to. Once a gamut is defined, a scale can be defined as a subset of that gamut. Having a gamut makes it a lot easier to talk about different scales! [however, the concept of a gamut may not be useful for all musical cultures].

A further possibility opened up by gamuts is the possibility of scale-changes: a single piece of music can now contain sections in different scales, something impossible on an instrument with no more notes than the primary scale it is intended for. By starting to use certain chromatic notes intensively, and certain diatonic notes little or not at all, a piece can shift its scale midway, adding considerable musical interest. So, if the first part of your song heavily uses ACDFG, with occasional use of B and E, but the second part of our song heavily uses ABDEG with occasional use of C and F, your song has effectively changed scale halfway through. This is an effect impossible to achieve in ‘pure’ pentatonic music, where only a single scale is available. Scale-changing of this sort is known as ‘modulation’, although this term can be a little confusing (as it also refers to using different modes (key notes and the resulting orders) of a single scale).
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Re: All About Musical Scales (and How to Tune Them)

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Gamut! That's new to me and definitely a useful concept.
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Re: All About Musical Scales (and How to Tune Them)

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Creyeditor wrote: 27 May 2023 14:37 Gamut! That's new to me and definitely a useful concept.
Grove defines it as the hexachord system (which I'll briefly touch on in a bit) or by extension 'any system'. Given that the hexachord system included (when multiple hexachords were taken into account) more than 7 notes, and that non-hexachord notes were considered "fictional", I think it fits my meaning.


It's an English-only word, apparently, from "gamma ut", i.e. "the G that's our first/lowest note" (which was the G a tone below the A a tenth below middle C). From there it came to mean all the notes from gamma ut upward. In other languages apparently they just used versions of "gamma", without the "ut".

It's also, if you're not aware, an ordinary English word now too. Meaning "all possible settings for this thing", sort of. Often found with, for instance, emotions: the full gamut of emotions is all the emotions you can have. Wiktionary also tells me it has a more specialised meaning of all the colours a computer monitor or printer is able to produce, which again does fit with my meaning (but in a visual rather than auditory form).
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Re: All About Musical Scales (and How to Tune Them)

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Both the general and the specific version(s) seem very useful. There is a German version <Gamut> but it's mostly used in the specific color/computer sense, less commonly in the musical sense (for medieval music) and not at all in the general sense. At least that's what the German Wikipedia tells me. (It also links to an article about the Guidonian hand, which I don't really fully understand from reading it.)
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Re: All About Musical Scales (and How to Tune Them)

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Don't worry, nobody understands the Guidonian hand!

There's two confusing things about it.

Firstly, it's about hexachords. The thing to bear in mind with hexachords is that music is complicated, scales are complicated, and people had no easy way to talk about it all. Even modern musical notation didn't exist until Guido, and the Guidonian hand (in some versions, if not its most famous form) actually predates him. It was hard to explain to singers in practical terms - most of whom couldn't even read, of course - about different notes and intervals and scale patterns and different versions of scales and notes.

So Guido invented the 'hexachord': a set of six notes, with the memorable interval pattern 2-2-1-2-2 - that is, all the intervals were the same, except the small interval in the middle. He then assigned each note a syllable: ut, re, mi, fa, sol, la. Singers could learn melodies by singing these nonsense syllables, and they could remember that there was always a smaller interval between 'mi' and 'fa'. To make the full heptatonic scale, you had to overlap hexachords. But depending on where you overlapped them, you changed the scale and got new notes. Specifically, hexachords started on C, F and G; the choice between the F hexachord and the G hexachord gave you a choice between what we now call B natural and what we call B flat (the sharp, natural and flat symbols today are all just ways of writing 'b'), which lets you, in modern terms, play in either C major or F major. [so they are first saw the two notes, B natural and B flat, as versions of B occuring in different hexachords]

The second confusing thing is what the point of the hand itself is, since it doesn't seem to actually do anything. It doesn't, for instance, give you an intuitive reminder of where the mi-fa intervals are, which it feels like it should. But we have to remember that they didn't have any much notation of any kind, so the hand didn't have to do much to be valuable by comparison. As I understand it, it was just an attempt to provide a physically-anchored mnemonic for the hexachords: if you remember the route, and where each hexachord starts, you can remember where the semitones are. [remember that, for example, there's no actual marking on the stave to show that the E-F interval is any different from the F-G one.]
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Re: All About Musical Scales (and How to Tune Them)

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Intriguing [:)] Looking forward to the next post [:)]
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Re: All About Musical Scales (and How to Tune Them)

Post by sangi39 »

I'm very much enjoying this look at scales, that's for sure. It's definitely nice seeing them built up rather than presented as something that just is. A lot of videos I've seen talking about scales and tuning treat scales as basically "here are the notes*, and here are the notes in this scale" (*I guess the gamut), treating them as basic right off the bat, as opposed to something that's constructed. Ssimilarly, for example, the circle of fifths is often just presented as "a thing that exists" with little to know explanation of where it comes from, so seeing how this sort of thing arises in musical traditions, and how things can be done differently

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Re: All About Musical Scales (and How to Tune Them)

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Dodecatonic Scales

But back to adding notes! You’re probably thinking: can’t we just do the same thing again to produce a 9-tone scale? Well, not so fast. Yes, we can keep on tuning up or down in fourths, and doing so will again create an ‘imperfect’ scale, with three step sizes and some seemingly missing notes. If we go back to tuning up from F, for example, (because I find it a bit more intuitive than tuning down), we’ll get Bb. This involves creating a new step type: the gap between A and B gets split into a limma and another step type a little bigger than our ‘limma’, which we’ll call an ‘apotome’ and notate as ‘a’: so our octatonic pattern (starting from A) goes a-1-1-2-2-1-2-2. But this time it’ll take more than one extra step to return us to a ‘perfect’, two-step-size scale. That’s because, in our heptatonic scale pattern (2-1-2-2-1-2-2), it’s now the bigger step that’s more common, and once we start splitting up those bigger steps (the tones) we’ll have to split all of them before we’re back to a perfect scale. The heptatonic scale has five tone-steps in it, and therefore five more strings have to be added to it to return to a ‘perfect’ scale: that is, after the 5-note and 7-note perfect scales, the next perfect scale is the 12-note (dodecatonic) scale. This scale contains seven limmas and five apotomes.

The idea of a 12-note scale should ring a bell for you! European music is traditionally dodecatonic (although modern European music does not use the ‘natural’ dodecatonic described here, for reasons we can talk about in a bit).

Let’s remember, for a moment, how some East Asian music is primarily pentatonic, but can still make use of two additional notes. Well, most European music is primarily heptatonic, but can still at times make auxilliary use of five additional notes. In the European context, this is defined as a contrast between consistently-used “diatonic” notes (notes within a given heptatonic scale) and sporadically-used “chromatic” notes (notes outside that scale, but within the gamut). Another way to say this is that the primarily heptatonic music ‘borrows’ some notes from a dodecatonic “chromatic” scale. However, the scope of “chromatic” notes need not be as wide as the gamut. Chinese music sometimes has this contrast of diatonic (in this case pentatonic) and chromatic (in this case heptatonic), but Chinese music theory is still based on exactly the same dodecatonic gamut that European music uses (this becomes relevant when comparing different songs, or different instruments).

[it’s worth pointing out here that, despite this big thought experiment, the dodecatonic in European music was probably not actually created simply by people randomly adding strings, and then discovering that this enabled them to change scales. Instead, it was discovered by trying to change scales, and in the process inventing new notes. A particular culprit was the technique of having two singers singing the same melody in parallel, one higher than the other by a fixed interval (this is a gross simplification!). This effectively means transposing the melody, which requires more notes. To give a simple example: if your song goes “A-B-C”, and you want your second singer to sing the same melody transposed up a fifth, what notes do they sing? They start on E, as it’s a fifth above A, but then what? If their second note is F, that’s no longer a (perfect) fifth above the B in the lower melody, because A-B is a bigger interval than E-F. Instead, they need to sing ‘F#’, a note that doesn’t exist in our heptatonic scale. For hundreds of years, European musicians were culturally forbidden from explicitly mentioning this and other improper notes outside the heptatonic scale, even though they composed music containing them: they were known as ‘ficta’, the ‘imaginary’ notes. In effect, however, these composers and performers were using a dodecatonic gamut.

In any case, let’s remember that when we talk about “adding a string” a fifth higher (etc) this goes for any occasion when a performer or composer needs a note a fifth higher than a certain note they already have. The practice of trying to sing in parallel fifths is effectively adding an extra note to your scale a fifth up; and then when you become sufficiently used to THAT note to use it in your melody, singing a parallel fifth above it is, again, adding another note, and so on. Practical and theoretical note requirements tend to develop in parallel, but not necessarily simultaneously; theory is more likely to catch up quickly when fixed-pitch instruments are used, as here a non-intuitive question (how long do I make this string?) is posed; whereas variable-pitch instruments and in paticular singers may be able to just wing it on instinct for longer before theoreticians manage to catch up and describe what they’re doing.]

There are many advantages to 12-note music. An instrument that can play all 12 notes can also play 5- or 7-note music, and a pentatonic or heptatonic melody on A can be transposed, more or less (more on the ‘less’ later!), to any other note in the gamut without hugely distorting its shape. It should be no surprise, therefore, that both European and Sinitic musical traditions have historically found it excellent. In Europe, incidentally, the theory of this music is traditionally ascribed to Pythagoras and his followers, while Chinese theorists discussed the same principles a few centuries later (having probably developed them independently). Other Greeks ascibed the Pythagorean tuning itself to Eratosthenes (while granting that Pythagoras ‘s mathematical and physical work was the basis for all tuning theory); however, the practice of Pythagorean tuning of the pentatonic and heptatonic scales is now believed to have originated in Sumeria, at the dawn of civilization, and certainly by the time of Babylonia.
The ‘natural’, ‘Pythagorean’ 5-, 7- and 12-note scales we have been discussing (i.e. built from stacked fourths or fifths and containing exactly two step types), however, are not universal in music, even in theory. So now we’re going to talk about six (related) ways to extend or distort these scales: through just intonation; through what we may call ‘unjust’ intonation; through scales with more than twelve notes; through temperament; through equitonalism; and through octave stretching and asymmetry. Together, these approaches describe most human music! I’ll then in an appendix describe an (almost) entirely different way to build a scale.

First up: justice!
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Re: All About Musical Scales (and How to Tune Them)

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Just Intonation

Let’s recap. By stacking fifths or fourths, we have produced 2-tone, 3-tone, 5-tone, 7-tone and 12-tone scales that are very easy to tune (both mechanically and by ear) and that deliver nice-sounding fourths and fifths (the most important intervals). The scales with more degrees, in this system, contain within them the scales with fewer degrees, so that tuning an instrument to one of the scales with more degrees but playing music primarily written for a scale with fewer degrees both permits occasional ‘chromatic’ notes for spicy flavour and facilitates easy transposition both of pieces as whole and within a piece (either to create contrasting sections, or to allow strict parallelism at a non-octave interval). That’s a great basis for interesting music.

But there are some problems. To illustrate, let’s assume, for sake of argument, a heptatonic scale built upward in fifths from F (or, equivalently, built upward in fourths from B). Then let’s say that we’ll treat C as the starting note (that is, we’re in the fifth mode of the scale if we built it from F, or the second if we built it from B; to avoid the ambiguities of numbering, we can use the traditional European name for this, “the Ionian Mode” – or, as we now call it, “C Major”). By multiplying together the frequency ratio of each fifth (i.e. 3:2) and dividing by the appropriate number of octave intervals (i.e. 2:1), we can calculate the ratio of each note’s frequency relative to that of C:

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

Unfortunately [for reasons far too fascinating to get into here!], the human ear is designed to both discern and enjoy simple frequency ratios. A ratio (i.e. interval) of 3:2 (AKA “a fifth”) is consonant: it’s both pleasant to hear and easy to recognise. A ratio of, say, 5:4 is also quite consonant. But a ratio of 81:64... is not. And that’s not just an aesthetic problem. Your stringed instrument, strung in advance, may be able to produce that note, but good luck getting a singer to imitate it precisely without a lot of work! And if your instrument slides out of tune, good luck correcting it again without having to restring all the strings on your instrument one by one!

[an additional reason to prefer consonant intervals is mechanical: on many instruments, not only the intervals themselves but the timbre of the instrument as a whole just ‘sounds better’ (or ‘sweeter’), or indeed just louder, when the notes are in consonant ratios. This is because, for example, strings in consonant ratios with one another are more likely to be excited to resonate sympathetically when one string is plucked; strings in dissonant ratios, by contrast, will tend to remain ‘dead’, and to deaden the sound of the instrument as a result]

The easiest way to avoid this problem is simply not to ever deal with the C:E interval in your music (particularly when heard simultaneously, as a chord). Or F:A, or G:B, which have the same problem (all three of these are “major thirds”). Similarly, the D:F, E:G, A:C and B:D intervals (“minor thirds”) have the only slightly less difficult ratio of 32:27. and since a third is just a sixth ‘upside down’ (that is, C:E, a third, is in some ways the same, via octave equivalence, as E:C, a sixth), you can’t really use sixths either. And you’re DEFINITELY going to want to avoid using the major seventh (243:129) and minor second (256:243), while you’re not even really going to want to spend too long dwelling on the minor seventh (16:9) or major second (9:8). As for the “tritone” between B and F, with a ratio of 1024:729, that’s just RIGHT out!

None of this is a fatal problem, per se: European music throughout the Middle Ages mostly used this tuning, and as a result simply avoided simultaneous seconds, thirds, sixths and sevenths, and reviled the hideous tritone as “Diabolus in Musica”, Satan’s own personal representative in music. It’s also possible, by tuning strings in different orders or using different modes (i.e. starting notes) of the scale, to move the ‘bad’ intervals elsewhere in the scale, and avoid those notes instead (indeed, since the biggest culprits here are the sixth and seventh notes added to our scale, avoiding these nasty intervals is a good reason for preferring to stick to pentatonic music!).

Nonetheless, a musically adventurous society will eventually perhaps become frustrated with limiting their harmony to fourth ands fifths, and gingerly tiptoeing over half the notes in the scale, and they’re going to want to do something about this problem. Indeed, if music is performed on instruments with flexible pitch – including human voices! – then performers are likely to have taken matters into their own hands and surreptitiously ‘improved’ at least some of the notes themselves, well before theorists get to work on the issue, not only because a performer who ‘improves’ the intervals will be more popular, but also because accurately playing the ‘correct’ (ugly) intervals can be genuinely difficult! So, eventually, some compromise between performers and theorists (and between players of flexible-pitch and fixed-pitch instruments) is going to have to be made.

There’s no single ‘right’ way to do this. But an extremely common way to do it is the way described by Claudius Ptolemy in the 2nd century CE: supplement the old, Pythagorean way of tuning (in fourths and fifths from existing notes) with tuning by thirds directly. Specifically, a new interval of exactly a 5:4 frequency (or string length) ratio, a “major third”, can be built above the existing C, F and G, to give new, nicer-sounding and more easily replicated values for E, A and B (this is equivalent to defining E as a major third above C, and then defining A and B as a fourth and fifth respectively above E). Doing this replaces the old ratios for these notes – 81:64, 27:16 and 243:128 – with the much neater 5:4, 5:3, and 15:8. All the intervals to the root note of C are therefore relatively pleasant (I mean, C:D and C:B still aren’t beautiful, but there’s nothing you can really do about that) and also relatively easy to tune (tuning a third by ear, or physically (shortening a string by 1/5th of its length), is a little trickier than tuning a fourth or a fifth, but still pretty straightforward with practice). And most of the other intervals of thirds and sixths between notes in your scale are now improved as well.

[sidenote: another way to describe the shift from a Pythagorean to a Ptolemaic tuning is to say that you have lowered the third, sixth and seventh degrees of the scale by a ratio of exactly 81:80. This isn’t important in a practical sense, but I mention it just to introduce the recurrent ratio of 81:80, which will come to haunt your nightmares if you look into tuning in too much depth!]

Ptolemy’s scale – the “Intense Diatonic” – was immediately recognised as a great idea. Indeed, it was probably already unofficially being used at least some of the time before Ptolemy. In the following centuries it came to be widely used (as at least one common option) not only in the later Roman Empire and Byzantium, but in Europe, the Middle East and Persia (where theorists had access to Ptolemy’s work) as well as in India (where they may not have done). In Europe, the Ptolemaic scale is the scale most commonly known as “Just Intonation” – ‘just’ here meaning simply right, proper, correct. Because that’s how it sounds: moving to the Ptolemaic scale makes the notes just sound right, damnit!

[tuning up in a true major third will also come naturally to people playing, or playing alongside, instruments that generate their notes through harmonics, such as ‘brass’ instruments, for reasons we won’t get into]

[here it should be said that Ptolemy’s scale is not the ONLY ‘just’ scale. A scale has ‘just intonation’ whenever a large number of its intervals have simple integer ratios with one another; Ptolemy’s is just one way of accomplishing this, though it may be the most ‘natural’ way. But, for example, as Ptolemy’s scale has two sizes of tone (10:9 and 9:8), other scales may allocate these two tone sizes to different degrees of the scale]

At this point, though, you may be wondering, though: if Ptolemy has solved the problems of tuning (or at least been the first to write the solution down on paper), why isn’t this the end of the story? Why doesn’t everyone just use Ptolemy’s scale?

Well, there’s kind of a big problem here. In fact, there’s a couple.

First, let’s point out something odd: this is no longer a scale with only two types of step between notes. Instead, there are now three: adjacent notes can differ by ratios of 16:15, 10:9, OR 9:8. That’s not inherently a fatal problem, but it does sound weird: in this scale, the interval between F and G is noticeably larger than that between G and A. And unlike our earlier scales with three step types (like the Pythagorean tetratonic), here the largest step is NOT just the sum of the other two types.

Second, this has a knock-on effect for the concept of the gamut. Now remember, if we define a scale with more notes as our gamut, we can select scales with fewer notes from that gamut, allowing multiple scales to be played on the same instrument. So, with an instrument tuned to a dodecatonic scale, we can play heptatonic scales built on either C or, say, G (with one specific but enormous caveat that we’ll discuss later). But if we’re using Ptolemy’s heptatonic scale, we cannot define a single dodecatonic scale that allows us to do that. A single 12-note instrument cannot play a Ptolemaic heptatonic scale on both C and on G, because in the Ptolemaic scale the interval between C and D isn’t the same as that between G and A – in other words, the A demanded by a Ptolemaic scale on C isn’t the same as the A demanded by a Ptolemaic scale on G. So the instrument would have to be able to play two different versions of A, depending on the scale. We can of course define a scale on G that uses the A from the scale of C, but it won’t sound the same as the scale on C, which is a problem if we’re trying to simply play the same tune transposed.

[reminder: we need to distinguish a SCALE, a set of notes with a precise relationship between adjacent notes, from a MODE, in which the same set of notes are played ‘starting on’ a different note. In Pythagorean tuning, different modes sound different – the intervals are in a different order – but different scales (in a given mode) sound (with one caveat to be discussed later!) the same, just moved up or down, because the intervals are in the same order. With a 12-note Pythagorean gamut, you can play the same music in different scales; but with Ptolemaic tuning, and only 12 notes, you cannot simply play the same heptatonic scale but transposed up or down, because the order of intervals will be different, effectively creating a new mode and making the tune sound different]

And finally, the big one: although the Ptolemaic scale improves all the intervals in relation to C, and most of the others throughout the heptatonic scale, it does not improve ALL the intervals. In fact, it makes some worse. A LOT worse. The minor third between D and F (and the sixth from F to D) is still bad. But more importantly the fifth between D and A (and the fourth between A and D) is now relatively horrible: 40:27 (and 27:20). Given that the fifth and fourth are “meant” to be the most consonant of all intervals (other than unison and octave), this is a glaring and unbearable problem, which effectively makes these intervals unusable in harmony, and makes it hard to use the modes built on D and A at all – or to use parallel motion in fifths or fourths between two voices if the melody will pass though D or A. Horrible fifths or fourths are traditionally known as “wolves”: the interval lurks in the scale, unseen, until the music happens upon it, and the wolf devours the tune, abruptly biting the listener’s ear.

Now, these problems aren’t fatal. You can simply avoid the wolves (and ideally the dud third and sixth) – although it’s worth noting that you can’t do that if, like mediaeval Europeans, you have a musical culture in which parallel motion in fourths or fifths is an important feature. But taken together, these two problems – the impossibility of transposing without changing mode (within a 12-note gamut), and the impossibility of using certain modes, or even certain harmonies within a single mode, without being eaten by a wolf – pose serious difficulties, not only for composers looking to exploit the full possibilities of their scales, but also in a practical sense whenever musicians are trying to come together with multiple instuments. With Pythagorean tuning, one instrument tuned in a 12-note scale on C can play easily with a second instrument tuned in a 12-note scale on, say, G: the same heptatonic scales on C and G can be played on both; with Ptolemaic scales, however, this is no longer possible.

The Ptolemaic scale, therefore, has its own serious problems, and we’ll look at one way to try to solve them in a moment. But first: what if we’ve been going about this in entirely the wrong way altogether?
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