All About Musical Scales (and How to Tune Them)

[Salmoneus]

@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.

A massive derivative of the sheng that uses keys and some additional mechanim for air pressure is called a harmonium, or reed organ!

[a melodica is just a harmonica (mouth organ) with keys. In the harmonica, each reed is blown on individually; in the melodica, the player blows into an air box, while keys open up access from that common air box into a specific reed. Accordions/concertinas and harmoniums (/melodeons, etc) add a belows system so that the air is provided from a more powerful source than the player's breath. The sheng is actually a little different in mechanism: each reed is attached to a resonating pipe. The pipes are open, but this leads to a clash between the frequency of the reed and that of the pipe, resulting in no sound playing - so all the reeds are blown on simultaneously, and you CLOSE a pipe to activate that reed, which is sort of cool, since it's the opposite of how pipes usually work!]

[the weird thing about free reed instruments, incidentally, is that they've been massively popular on the pacific coast for three thousand years, but otherwise basically didn't exist until about 1800, when suddenly they exploded in popularity and diversity in the West, and from there were introduced into India, where they made their own innovations. The explosion in the West is generally ascribed to contact with China... but the weirder thing is that they'd actualy already existed in Europe for at least a hundred years by that point, and appear to have been independently developed out of organs, via the regals, an instrument that's basically a kind of miniature harmonium but with beating reeds instead of free ones, and was itself derived from the reed stop pipes on organs. So why, after hundreds of years, did Westerners suddenly realise these things they had lying about were actually amazing? And why, for that matter, did all these instruments suddenly go from respected instruments - harmoniums had a huge repertoire and university degrees, they were hugely important - to being a laughing stock nobody takes seriously, or at best an instrument repurposed for playing 'folk' music? It's all a bit weird!]

Anyway, you'll be pleased to know that the 53-tone harmonium was built at least twice (in England and independently in Germany) in the 19th century! If you look up 'orthotonophonium', wikipedia has a photo of a 72-tone variant, if you want an impression of how one might look...

[WeepingElf]

Anyway, you'll be pleased to know that the 53-tone harmonium was built at least twice (in England and independently in Germany) in the 19th century! If you look up 'orthotonophonium', wikipedia has a photo of a 72-tone variant, if you want an impression of how one might look...

Ah, the humungous orthotonophonium!

[Creyeditor]

Such a beauty

[Torco]

boooiiiii am i gonna have a fun afternoon reading this.

edit: indeed I did. just one addendum, from the very subjective and vibes-based perspective of someone who, besides being interested in this from a conworldological perspective, also enjoys playing music.

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?...

one way to understand it is that it means the note which you treat as the tonal center: the note around which you're playing, so to speak. whether you're playing a sequence of single notes, i.e. a melody, or something more complicated, when you noodle around on an instrument you often feel like you're, well, starting a melody, then doing the melody, then finishing the playing of the melody, and closing it with a thump. people speak of this special note as a note (or notes, sometimes) that feels "at rest", or "final" or "stable" or is "home". this is one of those things that lend themselves more to experiential learning than to theory, I think: just sing whatever with your human mouth. notice how some notes, as you sing them, feel like you could stop there and others not so much? that's the kernel of the concept of a tonal center.

for example try singing "rule britannia, britannia rules the waves. britons never never never shall be slaves", when you're singingin the first "ruuuule" you're starting the thing, thing? it'd be weird to sing starting from the first "tannia", and not just because we know the song and that's not how it goes: musicians often say things they're "departing" in this beggining part. but then, at the end, at "will be slaves", it feels very different, like you're closing a sentence or something: you're finishing the melody and, even though you could repeat it, or make up some variation, or go on to sing something else you probably feel, too, as if if you were to stop playing you'd stop in the "slaves" part. you *could* stop at "waves", or at "tannia", but it wouldn't feel as conclusive, it'd feel somehow less sayisfying, no? (*) but in this sense we say that the note you play when you say "slaves" is your tonic, or has tonic function, or is your melodic center, or your tonal center, or the "home" of the scale. I myself don't immensely enjoy this metaphor of "home", I like to think of it more as a pivot, a hinge, the thing you play around, but hey, whatever metaphor makes you grok it is good.

In principle, you can take any set of any number of notes and decide that any one of them is the tonic: in practice, this is quite difficult to do, and for any set of notes some of them will be easier to tonicize than others: just try to, on a piano, noodle on the white keys but treating B as the center: part of this is, of course, habit, but this happens, as well, with weird scales.

* part of the "finality" of the "waves" part is not melodic, but rather rhythmic: having to do with the repetition of beat patterns and the "question and answer" structure of melody and oh so many other things, but ryhthm is its own dealio and I don't want to the flanges of this train to hit the dirt too hard.

[Salmoneus]

boooiiiii am i gonna have a fun afternoon reading this.

edit: indeed I did. just one addendum,

I'm glad you enjoyed it.

Yes, sorry, I wasn't setting out to explain the whole of music - just scales. Explaining the difference between two modes in this sense - permutations of a scale - is a massive whole thing in its own right, because it requires a discussion of modes in the broader sense, as melody types. And if we'e talking about tonal centres, that means tonality, so this discussion requires explaining both melody and harmony, probably via discussions of texture, polyphony, and the history of ostinato. That just seems like much too much to get into in what's already a grossly inflated brief discussion of what notes to tune your strings to!

one way to understand it is that it means the note which you treat as the tonal center

I really wouldn't want to tie it to tonal centres, because that commits us to talking about Common Practice music and its derivatives. Tonal centres didn't exist prior to around 1600 and basically don't exist outside the European tradition.
[Melodic centres, of course, absolutely do! But melody types/modes don't always feature clear melodic centres that you're 'playing around'. In a church mode, for instance, where is the centre? Is it the final, where the tune ends? Is it the lowest note of the mode, which usually isn't the same as the final (even three of the four authentic modes have subfinals, and of course in plagal modes the final is in the middle of the ambitus, not near the bottom)? Is it the tenor, which is the note most of the tune is actually on or around? Or is it even the medial, which, being between the tenor and the final, is objectively the note the tune is played "around"?]

for example try singing "rule britannia, britannia rules the waves. britons never never never shall be slaves", when you're singingin the first "ruuuule" you're starting the thing, thing? it'd be weird to sing starting from the first "tannia", and not just because we know the song and that's not how it goes: musicians often say things they're "departing" in this beggining part. but then, at the end, at "will be slaves", it feels very different, like you're closing a sentence or something: you're finishing the melody and, even though you could repeat it, or make up some variation, or go on to sing something else you probably feel, too, as if if you were to stop playing you'd stop in the "slaves" part. you *could* stop at "waves", or at "tannia", but it wouldn't feel as conclusive, it'd feel somehow less sayisfying, no? (*) but in this sense we say that the note you play when you say "slaves" is your tonic, or has tonic function, or is your melodic center, or your tonal center, or the "home" of the scale. I myself don't immensely enjoy this metaphor of "home", I like to think of it more as a pivot, a hinge, the thing you play around, but hey, whatever metaphor makes you grok it is good.

To be clear, you're talking here about modern European music. Cross-culturally, it is not the case that the final is necessarily the same as the melodic centre, and neither are necessarily the lowest note of the mode. And, also to be clear, there is no tonal centre without tonality.

[what's more, WITH tonality the tonic isn't always the final; because tonality is harmonic, and so long as the harmony ends on the tonic (or the parallel major, in baroque music in particular) it doesn't matter where the melody ends. Probably the majority of melodies do end on the tonic - particularly with older melodies that predate tonality - but a huge number don't - many classical melodies will end on the third or fifth degree, and once you get into the 20th century you'll even have them ending on the seventh (or ninth, or even sixth, etc), provided the harmony ends on the tonic. And of course in pop music a lot of harmony doesn't even end on the tonic (a lot of pop songs end on the dominant).]

In principle, you can take any set of any number of notes and decide that any one of them is the tonic: in practice, this is quite difficult to do, and for any set of notes some of them will be easier to tonicize than others: just try to, on a piano, noodle on the white keys but treating B as the center: part of this is, of course, habit, but this happens, as well, with weird scales.

FWIW, the Gregorian Hypophrygian does indeed 'start on' B, in the sense of B being the lowest note and the scale running up from there (though it's plagal, so the final is actually E).
By contrast, throughout the whole of the middle ages nobody had the idea of basing a tune on C: neither the mode with C at the bottom nor the mode with C as the final existed. [though the gregorian Lydian does have C as the tenor]. And yet today that's the mode that feels most 'natural' to us!

Judgments of naturalness in music are heavily dependent on which musical language you've learned...

[Fluffy8x]

Nice work so far; I’m looking forward to the rest as someone in the xenharmonic fandom, even if it’s all old news to me.

[Creyeditor]

Just wanted to link to an older thread, because I understand the discussion much better now that I have some background: viewtopic.php?t=7702. Also, I am patiently waiting for the next post, provided there will be one.

[Salmoneus]

Extended and Adaptive Tunings

OK, so, here’s an idea: if we want the flexibility provided by having lots of possible note pitches, but we don’t want the hassle of keeping track of 17 or 29 or 53 notes for each and every octave... can’t we just use the notes we need as we need them? Why don’t we start out with an ordinary heptatonic scale (or indeed pentatonic) and then just fix the problems as we come to them?

If that seems a little naïve... well, I guess it is. But it can also work surprisingly well. The human brain isn’t (for most people) very good at recognising absolute pitches, only relative intervals between adjacent or simultaneous notes. That means that, for example, an A at one point in your song doesn’t actually have to be identical to the the A at a different point in your song – the same general area, sure, but not necessarily identical. So let’s look again at some problems in the Ptolemaic scale: for instance, the A you’d need for a scale on G isn’t the same as the A you’d need for the scale on C – and the latter C clashes when you try to play it alongside a D (it should be a consonant fifth, but it very much isn’t). But... is that REALLY a problem? Play one value of A when it’s in some way relative to C, and a different value when it’s relative to G or D. Problem solved! And so on: where one value of a note is better for one purpose but worse for another, just play (or sing) the value that works best in the context you use it in.

In fact, in some ways, that’s what people do naturally anyway. Certainly singers do, and to some extent other players of flexible-pitch instruments (a lot of reed and brass instruments, for instance, are ‘out of tune’ much of the time anyway, and require the performer to “bend” their notes to the right pitch... and their instinctive sense of the ‘right’ pitch varies with the musical context). Sure, this strategy is really inconveniencing players of fixed-pitch instruments... but even then, maybe not so much? Most instruments have some at least potential way of bending pitches (eg a harpist can use one finger to still the string a fraction below the end of the string, or just apply pressure to it to change its tension). It’s really only some of the keyboard instruments that are completely bound to their fixed pitches.

And yet... there are serious problems with this ‘adaptive’ tuning. For a start, it requires performers to have an excellent grasp of musical theory and tradition – they can’t just read a score, they have to understand what each note is doing to judge what note value is best. For another, it only really works when music is structurally simple, so that it’s clear what each note IS doing. And sometimes that’s just not possible even in the simplest music. Taking our previous example: what value do you give the A in a sequence G A C D? Should it be justly intoned relative to the C... or to the D? It can’t be both! And then again, while we could say “well, let the musician decide, it can just be part of their interpretation”... that works fine with one musician, but gets tricky if you have, say, ten musicians playing a piece and suddenly five of them assume A should have one value and five assume that it should have another!

Yet there’s also a more fundamental, mathematical problem with adaptive tuning: the “comma pump”. Let’s take this simple sequence: C G D A E C. We start on C, and go up to G: a perfect fifth, easy. Down a fourth to D, easy. Up a perfect fifth to A, no problem. Down a fourth to E, no problem. And then down from E to C, a major third, easy enough. All very consonant intervals. Except... check the maths. Multiplying two fifths (3:2) and dividing by two fourths and a third (4:3 and 5:4) does not give 1:1. It gives 81:80. We’ve somehow ended up from where we started by one comma. And that’s not even the problem. It’s not just that the second C is dissonant to the first. It’s that we can keep doing this sequence and we’ll keep going up one comma at a time until we fall off the top of our instrument. And if the pitch of the entire song is gradually creeping upward, good luck having any sense of long-term pitch awareness! And pity the poor musician who enters halfway through the song with their own C without realising that the song is now half an octave higher in pitch than written! And it’s not just this sequence – it’s easy to construct comma pumps both long and short, with longer pumps in particular creeping up on musicians more insidiously.
In theory this can be solved by being careful. In this example, the E-C third just has to be replaced with the Pythagorean third instead of the just/Ptolemaic third. But, as I say, sometimes it’s not so obvious. The musician performing adaptive tuning has to be constantly alert, constantly aware of where the mathematics of their tune is headed – and that includes long-term and surreptitious tonal paradoxes that may not immediately be obvious. And just as importantly, if adaptive tuning is ALL we have, then we don’t even have any language – mental or verbal – to codify and track the concepts of absolute pitch that we need in order to prevent things going very wrong.

Plus, you know, if we’re going to just throw in an ugly Pythagorean third whenever we run into trouble, why are we bothering to call this adaptive tuning instead of, just, you know, Pythagorean tuning? Maybe we’re using Ptolemaic thirds sometimes and Pythagorean ones at other times... but the Pythagorean ones still sound bad, and frankly probably just sound even worse if they’re sticking out like sore thumbs from a bunch of nice Ptolemaic ones.

So adaptive tuning is a useful – perhaps inevitable – way for performers to make specific short, simple melodies sound a bit nicer. But it becomes dangerous to rely on it when music gets structurally or melodically complicated, or when multiple musicians are involved – and the more dangerous it gets, the more desperate we get for some robust conceptual framework to anchor it to. And the less nice it actually sounds, as we make more and more compromises to avoid comma pumps.

In which case, could we just take the general idea of adaptive tuning – only worry about what’s in front of you, don’t get bogged down in unnecessary theoreticals – and just apply it a bit more systematically? In the case of Pythagorean tuning, how about we just don’t say that how many notes are “in” the octave at all, but just tune as many as our actual piece of music requires?

We can call this an extended tuning. It’s intuitive, and in a way it’s actually more objectively TRUE than limited-cycle Pythagorean tunings like 5-, 7-, 12-, 17- or 53-tone scales. In reality, Pythagorean tuning doesn’t give us a circle, ever – it gives us a spiral (albeit one that becomes indistinguishable from a circle to human ears eventually). So why not just use the bit of the spiral that we need?

This, in fact, is at the core of European musical notation. Remember when we said that having distinct sharps and flats made sense in 17-note scales? It’s not because Europeans actually “used” specifically 17-note scales. It’s because they just didn’t specify what they were using. When they were writing music that needed notes not found in their 7-tone scale, they created them anyway, by extended tuning (creating what we call the sharp and flat notes). But for them (at least at one point) a Db wasn’t the same as a C# – not because these had different specific values in a 17-note scale, but because, conceptually, one only showed up in certain 7-note scales, and the other only showed up in other 7-note scales, and each scale was created as needed. Likewise, if they had bothered to go exploring the far reaches of their scales, they’d have come across (occasionally DID come across) new notes, and described them as “double sharp” or “double flat” (or indeed as ‘E-sharp’ or ‘C-flat’) – notes that don’t appear as distinct values even in a 17-tone scale. European musical notation, and to some extent Renaissance musical practice, was conceptually based on an octave containing an INFINITE number of notes, all generated as and when required. They just, for practical reasons, didn’t bother using very many of them! But if music called for the key of Gb, they could work out the values for all the notes in that scale. If it had called for the key of Gbb... well, it would probably have confused most of them, because they’d never have seen it, but they could have worked it out logically. Not by reference to a 53-tone scale, but by starting with C and building out in fifths until eventually they got to where they needed to be. Or just, you know, pretending to have done that and actually played in F because it’s not like the audience could tell the difference. But that’s not the point: the point is that they only used what they needed, so they didn’t need to get bogged down in theory.

But, as you’ve probably guessed: there are problems with this approach too. Again, it works fine for flexible pitches, within reason. A singer, for instance, given a set note, could immediately sing you a heptatonic scale based on that pitch, no matter what name you gave it. But the poor fools with fixed-pitch instruments... what were they to do? If your harp had only 12 strings, you didn’t want to have to physically restring half of them because some arsehole composer had decided to write for scales built on C# instead of on C!

Having said that: in the Renaissance European instrument makers, particularly makers of keyboards, actually DID add extra strings to each octave to accomodate annoying scales. These were occasionally true heptadecatonic instruments (i.e. with a 17-tone gamut), but more often had either 13 (with separate D# and Eb strings) or 14 (with additional separation of G# and and Ab strings) notes to the octave. These gamuts were therefore in a sense ‘asymmetrical’, because they were ‘missing notes’ – but that’s because they only included notes they actually needed, and ignored the others that in some conceptual sense existed but weren’t needed. Having different strings for D# and Eb lets you play E major and C minor properly, without the wolf eating you – and since these seemed like useful scales to have, the strings were worth including.

But still: it’s kind of a lot of faff. It meant including extra strings and reshaping the keyboard interface – wouldn’t it just be easier to just avoid those scales entirely? It’s not as though it even solves the problems entirely anyway – it adds a couple more usable scales, but not all of them. Even if you had a true 17-note keyboard, you’d still run into the wolf eventually. It’s all very well talking about the theory of an infinite scale (Isaac Newton, for instance, was a big fan of extended tuning), but you can’t have an infinite keyboard! In any case, while we’ve been talking about extending Pythagorean tuning, that still leaves us with all those ugly Pythagorean thirds (and sixths). If we switch to Ptolemaic tuning, it’s not really clear how to ‘extend’ the tuning infinitely at all, since the tuning is essentially ad hoc and not automatically ‘generated’, at least in theory, by a single (infinitely repeatable) mathematical principle the way the Pythagorean tuning is – the less automatic the process, and the more active decisions the musician must make, the less feasible it is to leave the details of the scale unspecified, and the more likely it is that careless extension of the scale will lead to unexpected clashes.

In short: none of these solutions are really satisfactory. We need to take more drastic action: so let’s try taming us some wolves!

[Creyeditor]

Thank you again for the post
A note/question/idea: A lot of the tunings you mentioned seem to work well for solo performances of flexible pitch instruments (maybe with some percussion). Incidentally, I have come across several descriptions of flexible pitch instruments that are said to be played solo or acconpanied by drums, etc. Could one say that there are two different contexts with different requirements for tunings? And maybe flexible pitch solo instruments fit with e.g. adapative tuning, but group performances or fixed pitch instruments need a different tuning?

[Salmoneus]

Yes, absolutely. As I think I mentioned, it was common in Europe at some point to use entirely different tunings for flexible and fixed-pitch instruments: flexible instruments being closer to Ptolemaic, fixed being closer to Pythagorean, and the conflict between these being a major reason why experiments in tuning were attempted in the first place.

It should probably be said that the dichotomy isn't that clear-cut really, because a lot of instrument are what we might call semi-fixed to various degrees: pipe instruments, for instance, have fixed pitches, but the performer can 'bend' them either to remediate an error in the tuning or to better harmonise with other instruments and their tuning.

An extreme example of this I haven't mentioned yet: many European lutes for a long time used a different tuning system from all other instruments, as they were often fretted in something approximating equal temperament. However, the thin, light strings of the lute and the nature of the frets (soft string wrapped around the neck) were particularly conducive to pitch-bending by the performer, so it's thought that in practice the instruments were played in a more mainstream tuning. [it may be that part of the reason for the equitonal tuning was to make it 'wrong' wrt both ptolemaic singers and other, pythagorean instruments, so that the performer could adjust their tuning depending on what/who they were playing with].

There also should be a distinction between instruments where the built-in tuning can actually be adjusted for each performance - like most string instruments - and those where the tuning is more or less fixed, like harpsichords and organs (which can be retuned, but it's way too much of a hassle to want to have to do it before every performance). If your music is based on, say, the lute, which enables both easy retuning (there's not many strings and they have easily-accessed tuning devices) and easy pitch-bending, you're going to have less problem with nuanced and variable tunings than if you music is based on the organ, where both retuning and pitch-bending are extremely difficult so you're kind of stuck with what you're given.

[WeepingElf]

In short: none of these solutions are really satisfactory. We need to take more drastic action: so let’s try taming us some wolves!

So, one could call a tempered interval a 'dog'? ;)

[Salmoneus]

Tempered Tunings

Let’s step back for a second and look at the situation Western Europe found herself in in the 16th century.

Thanks to an ancient chain of inheritence dating all the way back to ancient Sumer, European music was conceptually founded on Pythagorean tuning. Melody and harmony primarily relied upon the “diatonic” natural Pythagorean heptatonic scale (some folk music was even principally pentatonic), but adventurous musicians were also making use of a “chromatic” dodecatonic gamut, also Pythagorean in concept. This had traditionally been considered a closed, 12-note circle of fifths, but prominent musicians, mathematicians and physicists (three groups that overlapped considerably in this time period) were aware that this was not the case (the 13th note differed from the first by a Pythagorean comma), and so cutting-edge music theory relied on the notion of a theoretically infinite spiral – from where we get our modern ‘sharp’ and ‘flat’ terminologies, including ‘double sharps’, ‘double flats’ and so on.

However, practical limitations did not allow an infinite scale, and so the 12-note gamut was used as an approximation, sometimes with ad hoc extensions (including some built into the instrument) of a note or two as required, sometimes up to a full 17-note gamut. These extensions, however, were seen as rather inconvenient for both the instrument-builder and the performer. As a result, most musicians playing fixed-pitch instruments stuck, at least in theory, to the 12-note gamut, which resulted one terrible ‘wolf fifth’ where the comma difference betwee the 13th note and the 1st was simply ignored. This led musicians simply to avoid certain musical areas.

Meanwhile, a much more pressing problem was the ugliness of thirds and sixths in the Pythagorean system, which greatly limited musical possibilities. More generally, many note values even in the 7-note Pythagorean scale (even leaving aside the 12-note and extended scales) had counterintuitive pitches that were difficult for singers and players of variable-pitch instruments to reliably select. These performers therefore preferred scales with just intonation, and specifically the Ptolemaic heptatonic scale, which delivered mostly pleasant thirds. This, however, inherently created some wolf fifths and fourths elsewhere, which again limited musical possibilities. These could be tiptoed around by adaptive tuning by the performers (altering tuning on the fly to avoid clashes) but this required great caution, particularly in complex or ensemble music.

Most pressingly, there was inherently a clash between variable-pitch performers (primarily singers and to some extent players of wind instruments (which had built-in fixed pitches, but considerable freedom for players to bend these pitches in performance)), who performed with Ptolemaic scales, and fixed-pitch performers (primarily players of harps, organs and early string keyboards, and to some extent players of other fretted or many-stringed instruments like lutes, guitars, citterns, psalteries, dulcimers and so forth), who performed principally with Pythagorean scales. Small clashes could be smoothed over by the singers (etc) using adaptive tuning but, again, this took a lot of effort and discouraged ambitious music.

Enter: tempering!

The general principle behind tempering is to identify potential unpleasant intervals in the scale and make them less unpleasant by ‘tempering out’ the dissonance by spreading it somewhat to other intervals instead, so that several intervals are ‘wrong’ but none are horrible. This was generally done to fixed-pitch instruments, particularly keyboards, to bring them closer to the just intonation of the singers. Nobody knows when this practice started, but it was mentioned in writing in the early 16th century; however, the first extended and detailed discussion of the practice is by Venetian musician Zarlino in the middle of that century.

Specifically, Zarlino discussed forms of “meantone” temperament. There is not one single meantone (Zarlino described three as being usable), but one of Zarlino’s meantones, “quarter-comma meantone”, would go on to be the European standard tuning for around two hundred years (although I have also read that 1/6-meantone eventually became more popular).

To explain this (don’t worry, it’s simple!), remember how Pythagorean tuning works. We tune up (or down) in fifths (or fourths) and gradually fill out an entire scale in this way. C G D A E B, etc. An interval like C-E, for instance, is actually generated by stacking four fifths and substracting two octaves (or, equivalently, stacking two upward fifths and two downward fourths, etc). And this gives a third that doesn’t sound right.

But if the third is generated from the fifths, we can change the third by changing the fifths.

Specifically, the Pythagorean third is “wrong” by 81:80, the Ptolemaic (or in this context more often called the “syntonic”) comma. This third is generated by a stack of four fifths, so if we simply reduce each fifth by ONE QUARTER of a syntonic comma, we end up with a justly-tuned third.

This, unfortunately, means that ALL our fifths and fourths are now theoretically bad. However, each fifth is now bad by only a quarter of a comma, which is a difference that is JUST ABOUT imperceptible to the human ear.

MISSION ACCOMPLISHED!

We now have a single scale that makes all thirds just AND keeps all fifths and fourths perfect, or at least as perfect as we can hear. It’s a bit annoying for instrument tuners to have to shrink each fifth by one quarter of 81:80, but not impossible mathematically. And the singers are going to be annoyed because their instinctive fifths will be slightly out... but imperceptibly out, so so long as they keep their other notes in tune with the keyboard (etc) nobody’s going to really care if they actually just sing a perfect fifth here and there. Problems all solved!

...except...

...when I said that it makes all thirds just? It doesn’t. It makes MAJOR thirds just (the interval between C and E). MINOR thirds (the interval between A and C) still sound noticeably wrong. They are, it must be said, much LESS wrong than in pure Pythagorean tuning, but they’re still not right.

Couldn’t the same method be used to make minor thirds right? Yes, yes it can. A minor third can be built from three stacked fifths (granted octave substitution), and the Pythagorean minor third differs from the just third (which has a 6:5 frequency ratio) by, you’ve guessed it, a syntonic comma (81:80), so shrinking each fifth by one third of a comma gives you just minor thirds...

...but then you no longer have just MAJOR thirds. The major thirds aren’t terrible, but they’re not quite right. You therefore have a choice as to which interval you want to favour. Zarlino proposed both meantones as acceptable; historically quarter-comma meantone was preferred, but there’s no major (no pun intended) reason why 1/3-comma meantone couldn’t have been chosen instead.

[other meantones are also possible, shrinking the fifth by different amounts to make different compromises. They all, like Pythagorean tuning, have only one value of the whole tone, whereas just intonation has two, a bigger and a smaller; quarter-comma meantone (QCM) in particular has a tone that is exactly equal to the mean of the two Ptolemaic tones, hence the name ‘meantone’. Meantones also have no more than two values of ‘semitone’, unlike just intonations, which can have many.]

Now, to reiterate: meantone has a big advantage over just intonation (eg Ptolemy’s scale) because the 12-tone gamut, tuned in meantone, gives a lot more scope for transposition and modulation, and throws up fewer wolves to catch you out. It also has a big advantage over Pythagorean tuning, not only because, in the abstract, the thirds are better, but also because in practice they’re a closer approximation to the just intonation instincts of singers (and other variable-pitch performers). Singers may still have to bend some of their notes to avoid clashes, but the bending can be much smaller – this is easier to accomplish on some instruments, conceptually less stressful (it feels more like changing intonation, rather than outright changing the note), and leads to less incongruity if some non-clashing intervals AREN’T bent (just and meantone thirds occuring in the same piece will be accepted as the same, unless actually heard side-by-side, whereas just and Pythagorean thirds may not be, so you sometimes have to pick one or the other).

It should be no surprise, then, that meantone was widely (if not universally) accepted across Europe for hundreds of years. True, it brought a conceptual confusion with it: meantone did not offer a single correct and obvious tuning, but rather a collection of related tunings with advantages and disadvantages – alongside QCM there was also some use of 1/3-meantone, 1/6-meantone, and 2/7-meantone (which establishes a single, just value for the tone itself, and in the process splits the difference somewhat between major-favouring 1/4 and minor-favouring 1/3 tunings). But this was a minor inconvenience, something for theoreticians and bandmasters to debate over.
Unfortunately, however, there’s still a great big elephant in the room. Or, more accurately, a Big Bad Wolf.

If you attempt to portray a 12-tone meantone scale as being the entire gamut of available notes, you necessarily conflate the 13th and 1st notes in the circle of fifths – just as in Pythagorean tuning. The final fifth in your circle, therefore, will still be horribly wrong. In Pythagorean tuning, the circle of fifths slightly ‘overshoots’ (13 fifths is slightly bigger than 9 octaves), but meantone shrinks the fifths so much that the circle now ‘undershoots’ – 13 fifths is now smaller than 9 octaves. A lot smaller. As a result, the wolf fifth that fudges this gap is even futher from being consonant than the one in Pythagorean tuning. We’ve just made the primordial wolf even bigger and badder!

Now, there’s an obvious solution to this: one that mathematicians like Newton, Huygens and Mersenne all suggested: ADD MORE NOTES!

Specifically, they advocated for combining extended tuning with meantone – extended tuning (that is, acknowledging the infinite spiral of fifths rather than an artificial 12-note circular approximation to it) would solve the problem of the wolf fifth, while meantone would solve the problem of the wolf thirds.

Unfortunately, as discussed before, extended tuning is a faff. And, in a sense, impossible, at least on fixed-pitch instruments: you can’t have an infinite number of strings, or of frets, or even of tone holes. Huygens and Mersenne both designed practical harpsichord variants with more than 12 notes to the octave (31 and 19 respectively, for reasons we’ll discuss in a bit), but even these are substantially more awkward than a simple dodecatonic instrument, particularly when you consider that these ‘additional’ notes would have been used only rarely, as the 12-tone wolf was already positioned as far as possible from the most-used patterns of notes – do you want to deal with a whole 7 or 19 extra notes on your keyboard just so you can deal with a theoretical problem that popular music of the day already largely deals with simply by avoidance? And what’s more, after centuries of European music written and performed with the assumption of a 12-note octave, such extensions simply ran headlong into a conceptual barrier. Everyone knows there’s only 12 notes, why are you making me play an instrument with 31 notes!? Most of these notes are clearly just the real notes but slightly out of tune! How do I even know which version of a note to play – my stave only has five lines, how the hell am I meant to depict 31 different note values on it!?

In any case, theoreticians and instrument builders realised: none of this is really needed! Becausehere’s a far simpler solution that neatly solves all of our problems and makes our music immeasurably richer. And it’s called... well-temperament.

[Creyeditor]

Thank you for the new post. Maybe this is the right time to link to an old post: viewtopic.php?p=316240&.

[sangi39]

So just to go back to the Slendro and the Pelog, I can't seem to find any information about how these are actually tuned beyond "they vary from region to region" or something to that effect (which... fair, I wasn't expecting some sort of standardisation), but I was just wondering if any of the tuning traditions are built up in a similar way to Pythagorean tuning, but using a different ratio, or if the tunings are based on something else

[Khemehekis]

So just to go back to the Slendro and the Pelog, I can't seem to find any information about how these are actually tuned beyond "they vary from region to region" or something to that effect (which... fair, I wasn't expecting some sort of standardisation), but I was just wondering if any of the tuning traditions are built up in a similar way to Pythagorean tuning, but using a different ratio, or if the tunings are based on something else

If you think that's hard to find information on, try tuning or playing a crwth! No one alive today is old enough to remember how the crwth is played.

[sangi39]

So just to go back to the Slendro and the Pelog, I can't seem to find any information about how these are actually tuned beyond "they vary from region to region" or something to that effect (which... fair, I wasn't expecting some sort of standardisation), but I was just wondering if any of the tuning traditions are built up in a similar way to Pythagorean tuning, but using a different ratio, or if the tunings are based on something else

If you think that's hard to find information on, try tuning or playing a crwth! No one alive today is old enough to remember how the crwth is played.

Makes sense, I guess. If no-one's taken the time to record the information, or it's been recorded in a form that's not readily available to a wider audience (say, for example, orally, or written down in a language that might otherwise not be the subject of translation attempts all to often), then finding anything might be tricky

I did find one PDF while I was quickly looking around online at work a couple months ago about Slendro and Pelog, but it was honestly beyond me, and from what I can remember it mostly described the ratios between each note in various scales, but not how those ratios were derived

[Salmoneus]

Huh, this kind of... just stopped, didn't it? Sorry about that! Don't worry, I'll get back to it...


So, Indonesia. South East Asia more generally, really, but Indonesia is the most extreme part. Soo.... yeah.

To an outside observer, the key thing in Indonesian music is to sound as unpleasant as possible at all times.

More specifically/accurately, music in this region has come to be dominated by gongs. Gongs are very strange instruments, because they are neither harmonic instruments (like flutes, violins, etc) nor inharmonic (like drums). Instead, they're semi-harmonic, which essentially means that they always sound slightly out of tune, even with themselves. Although, fortunately, to a less exteme extent than conventional bells. [the weird sound of conventional European church bell is because its harmonic include simultaneous major and minor triads on the same tonic but an octave apart; but we're not getting into harmonics in this series!]

Now, in most of the world, the problem of semiharmonicity is mostly solved by eliminating gong-like and bell-like instruments, or at least relegating them to minor genres (like church bell-ringing) or to minor roles in major genres (like the use of a gong or bell as a special effect, or to signal important moments in a piece, rather than as full melodic instruments).

In SEA, however, and particularly in Indonesia, gongs have been accepted, which has required coming to terms with and even embracing their "lack" of true full harmonicity. Because gongs never sound really in tune (and as a result are also quite hard to tune to a specific pitch!), there's no point trying to maximise consonant intervals in your scale, because they'll never be purely consonant anyway. I imagine the practicalities of gong tuning are also an issue here - acoustics aside, they're physically hard to fine-tune (compared to a string or pipe that just has to be lengthened or shortened to change the pitch predictably), can't really be fine-tuned before each performance, and I imagine probably (like organ pipes) change pitch over time due to stress on and distortion of the metal.

Tuning difficulties are probably a big part of the variety of tunings: each village reportedly has its own ideas on tuning, with every single gamelan ensemble being tuned slightly differently.

Within a gamelan, attention is paid to exact and specific tuning. However, this is not an attempt to maximise consonant intervals, which are unachievable with gongs. As I say, instead different types of dissonance and near-consonance are embraced. Famously in Bali even unisons and octaves are intentionally tuned out of tune - even with non-gong instruments.

------------

All that said, however, there's reason to think the origins of the systems are still similar to those elsewhere. Slendro is a pentatonic scale with no small intervals, just like a pentatonic scale elsewhere in the world. Pelog is a heptatonic scale that does incorporate small intervals, just like most heptatonic scales elsewhere in the world.
Moreover, pelog is actually much like Chinese pentatonic music - in that it's primarily five-note music, but with two additional notes that can be used in certain ornamental ways (or sometimes not at all, on some instruments). And which notes are only ornamental (or missing entirely) is telling.

Remember constructing a scale by going up in fifths, and where necessary down in octaves. If we start on C, we get C G D A E... and at this point we have a nice consonant pentatonic scale of the sort we find all around the world. If we continue doing this, however, we get B and then F.

B and F have two key characteristics in this system. Firstly, it's the addition of B and F that introduces the first semitones into the system. And secondly, because the pythagorean method of stacking fifths gradually moves us further from small number intervals, it's B and above all F that sound most dissonant.

In most pelog modes, it's B and F that are missing, suggesting to me that pelog originated in a system derived from stacking fifths (or fourths) and that was trying to avoid dissonance. When it's not B and F that are missing, it's C and F, essentially the same system but using B instead of C and preserving similar overall consonance (the C-G fifth is replaced by the E-B fifth, and you've essentially just cycled through the scale and transposed). In some places in Indonesia B and F are also given distinct names ('deviating tone' and 'wailing tone' respectively) and recognised as additions to break up the largest intervals in the five-tone scale; the five-tone slendro is traditionally considered older than seven-tone pelog.

So everything's OK, right?

Well, except that no.

Going back to the slendro scale, it seems as though, in the absence of a strong pressure to maintain harmonic consonance (which is unachievable with gongs), melodic symmetry (or perhaps transpositional convenience) has been prioritised. Remember, the pythagorean pentatonic scale has five notes, but two types of interval, a big and a small one (the whole tone and the minor third). This makes melodies 'uneven' because some steps are bigger than others. It also means that direct transpositions are extremely noticeable. Slendro appears to have dealt with these problems by allowing the intervals to drift over time until the five notes are approximately equally apart. [5-tone equal temperament!]

In pelog, however, the difference between big and small intervals has been maintained and exaggerated, maximising asymmetry. It's said that the pelog scale tends to approximate 9-tone equal temperament (with two notes never played at all, and two more only being ornamental).

It's not clear why that is. One explanation is that, in the absence of the ability to clearly and precisely tune consonant intervals, Indonesian tuners have resorted to equal division as the next-best most psychoacoustically possible option. Another explanation is the the 9tet model is a conscious emulation of the 5tet model; certainly this seems to be true of influential 20th century theorists. Another explanation is that there was once, historically, an actual 9-tone scale in use that became equally tempered just like slendro, but, just as pelog's two ornamental tones are absent from many instruments entirely, the final two tones of the 9-tone system were simply absent so often that they've been lost entirely. And another explanation is that the 5-from-7-from-9tet system is an illusion in the minds of theorists attempting to find reason in the insanity of pelog, and that in reality the tones have simply drifted apart in a way that maintains the idea "have different interval sizes but not too different", and that have simply coincidentally strayed toward 9tet as a result.

It's important to reiterate however that all these tuning ideas are only approximations; actual slendro and pelog scales differ between every gamelan and every village and every region of Indonesia, and probably none of their intervals are ever precisely equally tempered. This is partly because of the difficulty of tuning gongs, partly because of a low priority placed on precise tuning... but also I suspect because of an active effort to avoid regular tuning of any sort.

I found one paper giving specific measurements of one particular gamelan ensemble.

This gave the slendro notes at 0, 237, 489, 712 and 962 cents - that is, with intervals of 237, 252, 223 and 250 (and then 238). The 'correct' 5tet interval is 240 cents, giving notes at 0, 240, 480, 720 and 960. So that's pretty close - some of the notes would be noticeably 'wrong', but only just. I do wonder though whether the slight alternation of large and small intervals is intentional - or, indeed, just historical, since it puts the large and small intervals in the right places to match the presumed 'original' minor third and whole tone intervals respectively of a primordial pythagorean scale. But you'd need to have measurements of a lot more gamelans to know whether that was meaningful or just a coincidence!

For the pelog, it gave notes at 0, 116, 281, 550, 669, 769 and 961. That is, intervals of 116, 165, 269, 119, 100 and 192 (and then 239). The 'correct' 9tet interval is 133.3, which would give 0, 133, 267, [400], 533, 667, 800, and 933 [and missing 1067]. This is... less good a fit. It's worth reiterating that while slendro is immediately recognisable as an attempt at 5tet, the average '9tet' nature of pelog is something that is allegedly statistically significant following testing of many ensembles. [relevant: that paper was actually about tone recognition testing of western and indonesian musicians, and found that several of the indonesian musicians drew reference lines on an octave graph equally dividing the octave into five parts and saying that these were the slendro notes; so even if slendro isn't actually pure 5tet, it does seem that musicians broadly think of it that way. It's less clear whether they think about pelog as 9tet].

Also worth pointing out that of the six pelog intervals there, only two intervals would sound similar to one another (116 and 119) - the others would all be recognisably distinct.


-------

finally, two other points of reference:

- nearby Vietnam shares the indonesian convention of having two entirely independent scale systems that never mix

- for some random reason, Uganda is also a hotbed of 5-tone equal temperament. I've never heard of it occuring anywhere else in the world.

[Creyeditor]

To an outside observer, the key thing in Indonesian music is to sound as unpleasant as possible at all times.

Can confirm. For me personally, it feels similar to what certain kinds of jazz music to with rhythm.
I was also wondering if the strong stratification in these cultures (espicially Javanese culture) has some influence, since gamelan music is (historically) strongly associated with nobility, religious ceremonies, and royality, IINM. 'Popular' music seems to be different and use either imported scales or at least different ones. I was wondering if the stratification maybe allowed the drifting to sound more 'artsy' and less 'common sense'.

- for some random reason, Uganda is also a hotbed of 5-tone equal temperament. I've never heard of it occuring anywhere else in the world.

Do you have some further reading on this? It sounds very intriguing and I couldn't find anything with my modest google skills.

[Salmoneus]

To an outside observer, the key thing in Indonesian music is to sound as unpleasant as possible at all times.

Can confirm. For me personally, it feels similar to what certain kinds of jazz music to with rhythm.
I was also wondering if the strong stratification in these cultures (espicially Javanese culture) has some influence, since gamelan music is (historically) strongly associated with nobility, religious ceremonies, and royality, IINM. 'Popular' music seems to be different and use either imported scales or at least different ones. I was wondering if the stratification maybe allowed the drifting to sound more 'artsy' and less 'common sense'.

In terms of tuning specifically, I can't find any infomation on this, unfortunately. Except that I don't think old Indonesian popular music would have been much more approachable for western ears. I see mention of heavy use of tritones in some areas, and of "harmony" in seconds in others, both of which would sound highly discordant to westerners. Heterophony seems the norm (as to be fair it is in most of the world), and this tends to imply a huge amount of dissonance. Additive rhythms were also present in some places. And gong ensembles existed independently in many areas independently of the large-scale gong ensembles of the Javanese court. Which, incidentally, seem to have developed rapidly just as Europeans were arriving - some use of chimes is known from the 14th century, then by 1600 the court was using a dozen or so gongs, while half a century later had developd into full ensembles of up to 50 gongs.

However, it's maybe worth saying that the gongs aren't the original courtly music and originally had a popular character: court music itself used softer, quieter instruments (played indoors), while gong music was used in outdoor settings, including courtly events like processions but not limited to those. [a parallel can pehaps be drawn to the mediaeval and renaissance outdoor ensembles in Europe, which are almost as aggressive to modern European ears (bagpipes, shawms, etc).]

Javanese popular music is apparently entirely imported, and mostly quite recent.

Among the rich prior to 1900, European music was found in the form of christian music and of military and dance music, which entered mainly through the aristocracy, and in a form (kroncong) deriving from Portuguese music, which was popularised via slaves and became somewhat indonesianised.

Shortly after 1900, hwoever, European music exploded in popularity with the development of theatre, which used music derived from or borrowed from opera and operetta, but in local languages, and intermixed with kroncong. Crucially, this music was popular for not being Indonesian, and Indonesian elements - anything to do with gamelan - was intentionally shunned and kept separate from it, leading to the gradual elimination of Indonesian elements in popular music. Since then, American music has continually flooded in, replacing almost all other popular music, other than, apparently, a genre of popular gong music among the sundanese. Kroncong has also been revived and developed and mixed in with more recent styles. And among the urban poor Indian and Malaysian film music has developed into its own genre, later modified with the addition of Arabic religious music and American rock and country music.

So although all this music has been to varying extents 'indonesianised', actual indonesian popular music prior to 1900 is almost unrecoverable, it seems. Even relatively 'remote' and 'unsophisticated' groups have apparently imported forms of Western music. And even a lot of the explicitly 'native' music was invented during WWII, when the Japanese outlawed all Western-influence music and pressured local intellectuals to rediscover their own national heritage.

Such, at least, is my understanding, mostly via Grove.

- for some random reason, Uganda is also a hotbed of 5-tone equal temperament. I've never heard of it occuring anywhere else in the world.

Do you have some further reading on this? It sounds very intriguing and I couldn't find anything with my modest google skills.

I coincidentally came across this a while back, reading something about african harps, and made a mental note to mention it in this series. And then I had it coincidentally confirmed while reading that paper with the pelog note values, Perlman & Krumhansl, which in passing mentioned "quasi-equitonic pentatonic scales, such as those found in Uganda and Java". Their citation is to Cooke (1992), "Report on pitch perception experiments carried out in Buganda and Busoga (Uganda)". They also compare Indonesian tuning variability - where different instruments just have totally different tuning, rather than different performers bending tuning due to melodic or harmonic context as found in Europe and northern India - to practices in Uganda, saying "these practices [of contextual pitch variability] seem qualitatively different from the context-independent variability of fixed-pitch instrumental tunings in Java and certain other cultures (eg. the harp and amadinda xylophone music of Uganda; cf Wachsmann, 1950, 1967)". Those being to "An equal-stepped tuning in a Ganda harp" and "Pen-equidistance and accurate pitch : A problem from the source of the Nile".

Anyway, Cooke's proposal is that these two Ugando-Javanese traits (5tet and wide pitch variability) may be linked psychologically - that having only five pitches and not having them linked by specific and immediately audible harmonic relationships leads people to see pitch in fairly broad terms and 'hear' quite divergent pitches as being the same note. This makes sense and Perlman and Krumhansl find some support for this, because Indonesian and Western musicians both tended to have their pitch perception distorted by the preconceptions of their native tuning systems, but it doesn't seem ALL that convincing because they also found that some musicians in both cultures are actually quite accurate in their pitch perception.

Grove's discussion of the court music of Buganda says simply "the melodies are pentatonic, probably of the pen-equidistant variety, which tends to adapt to Western diatonic tuning" and cites Cooke and Wachsmann again. It also mentions that all traditional Ugandan music is pentatonic, with the exception of the music of the Konzo and Masaba (heptatonic and hexatonic respectively), but says nothing about the tuning outside of Buganda.

Wachsmann's other work may be relevant - he was curator of the Uganda Museum and wrote an important survey of all Uganda's instruments, as well as recording a lot of Ugandan music. Also of interest may be Kyagambiddwa, who published staff notation and analysis of Ganda music.

It's worth noting that, as with Javanese court music, Bugandan court music appears to have been complex ensemble music.

[Creyeditor]

Thanks again, Sal
I agree that most modern Indonesian popular music is 'Westernized', 'Indianized', and/or 'Arabiced'/'Persianized'. And probably you are right that it's hard to know what was there before. From my own personal experience, even 'ethnic' music of the Indonesian part of the island of Papua is heavily influenced by Melanesian pop music, which in turn is influenced by Western/American music.
And thanks for the info on Ugandan music, I got some basic info and more sources and keywords to continue.