The equally tempered scale is dirty

Chords, scales, harmony, melody, etc.
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Sendy wrote: There is a solution - dynamic tuning.
Yes, the Waldorf Q synth, for example, attempts to do this if you use setting HMT tonal:

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I never bothered using it because I'm not that interested in tuning and I don't trust it to make correct tuning decisions on the fly.

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jancivil wrote:I am failing to feature situationally what you mean, really, with 'every time there is a new chord'.
You'll notice I was talking about Western instrumental music - perhaps I could have been a bit more specific, but Just Intonation requires a certain pitch flexibility that can only generally be achieved with voices, or certain string instruments. In other words, if a fixed-pitch instrument is used, it will require constant re-tuning (which is extremely impractical).

As an example, let's assume for the sake of argument that we set A=440Hz and we tune in pure fifths (3:2) above that:
A-E, E=660Hz
E-B, B= 990Hz
B-F#, F# = 1485Hz
F#-C#, C# = 2227.5Hz

So my final C# is 2227.5Hz.
And that's in the third octave after the original A=440Hz.

But hang on a second... We know that an octave is 2:1, so that would mean the second octave starts with A=880Hz, the third octave starts at A=1760Hz, and the C# in question would be a pure major third (5:4) above that, which would work out as 2200Hz.

So there is a discrepancy. C#=2200Hz or C#=2227.5Hz. In one chord you might want the pure (compound) fifth, in another you might want the pure third, in which case you'd have to re-tune during the piece.
jancivil wrote:And then you do this //It doesn't make music sound any "better"//
I should have said "it doesn't necessarily make it sound any better".
And indeed several blind tests have been done where people actually prefer the sound of ET to Just Intonation (largely because they're so used to it). You might not (because of your wider experience), and that's fine - as I said, it's subjective. I'm not for one minute saying ET is perfect or necessarily better than anything else, but nor is it necessarily inferior (overall). As with a lot of things context will have to be the definitive guide. Either way, there is no one system that is perfect in all situations.
Unfamiliar words can be looked up in my Glossary of musical terms.
Also check out my Introduction to Music Theory.

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JumpingJackFlash wrote:
jancivil wrote:I am failing to feature situationally what you mean, really, with 'every time there is a new chord'.
You'll notice I was talking about Western instrumental music - perhaps I could have been a bit more specific, but Just Intonation requires a certain pitch flexibility that can only generally be achieved with voices, or certain string instruments. In other words, if a fixed-pitch instrument is used, it will require constant re-tuning (which is extremely impractical).

As an example, let's assume for the sake of argument that we set A=440Hz and we tune in pure fifths (3:2) above that:
A-E, E=660Hz
E-B, B= 990Hz
B-F#, F# = 1485Hz
F#-C#, C# = 2227.5Hz

So my final C# is 2227.5Hz.
And that's in the third octave after the original A=440Hz.

But hang on a second... We know that an octave is 2:1, so that would mean the second octave starts with A=880Hz, the third octave starts at A=1760Hz, and the C# in question would be a pure major third (5:4) above that, which would work out as 2200Hz.

So there is a discrepancy. C#=2200Hz or C#=2227.5Hz. So if you want both, you'd have to re-tune during the piece.
No. Your first premise is simply faulty. Your subsequent work relies on confusing the two systems. I stated something which happens to be true, about a just intonation. You've moved the goalposts. All I see is "Here is a discrepancy between your ratios and 12tET".

The ratios are true per se. You can perfectly well fix an instrument to any of them. In fact, the oud is all about that. My friend Hansford Rowe has electric bass guitars built for JI.
Here's how it works:

1:1 = C.
'Db': 16:15 x 5:4 = 4:3. 4:3 x 6:5 = 8:5.
D: 9:8 x 5:4 = 45:32. 45:32 x 6:5 = 27:16 [I gave 8:5 for that so we do have this problem of a difference from the completely pure chord @ 81:80, ~21.5¢.].
'Eb': 6:5 x 5:4 = 3:2. 3:2 x 6:5 = 9:8.
E: 5:4 x 5:4 = 25:16 [I gave 8:5, so again a compromise of nearly exactly the above.]. 25:16 x 6:5, though, gives 15:8.
and so forth.

So, I have to repudiate one claim of mine above, by particulars there are results here that are more variant from the simplest intervals by the syntonic comma (which is half again as 'bad' as a 12tET M3rd).
There is no retuning, we make a compromise per the limitation of it {by applying the syntonic comma to the basic JI 12, as I'm showing (giving a 24 tone 'scale'), we can obtain the perfect concord at each of the 12.}. Do the multiplication of the ratios. Don't be confusing the issue, that's not even coherent.

There is no reason you can't fix this system to that or to a keyboard instrument. I believe there are keyboard instruments in history and probably in use for historical purists today that go back a bit further than 12tET. The reasons for more and more tempered equality belong strictly to movement in terms of a new tonic. The ultimate outcome of this equality is dodecaphony.
Last edited by jancivil on Wed Apr 09, 2014 11:53 pm, edited 2 times in total.

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jancivil wrote:You've moved the goalpost and are mixing in ratios with ET.
No, none of that was related to ET. It was all pure ratios. The point, in a nutshell, is that you can't have all pure fifths and all pure thirds, they simply don't work.

This example might be better:

Let's say I want to start with a second inversion D minor chord. So A is in the bass and I want the D a perfect fourth above it. Let's assume that my bass A=110Hz. That means my pure fourth (4:3) above that gives D = 146.66667Hz. Now let's assume that I want this to be an octave higher (so it can be sung comfortably by a tenor). That means I need to double the frequency to get D = 293.33333Hz.

My next chord is E major in root position. I set the bass to a pure fifth (3:2) above the A I already tuned to, meaning E = 165Hz. And I want a B a perfect fifth above that, so I use the pure 3:2 again to give me B = 247.5Hz.

No problems so far, but now suppose my third chord is a diminished triad on B. I'll start with the B as I just tuned it (247.5Hz). Now I want the D a minor third (6:5) above that. That would give me a D of 297Hz.

But hang on a minute. I already tuned that D to 293.33333Hz in the first step, so now I've got to re-tune it!
Unfamiliar words can be looked up in my Glossary of musical terms.
Also check out my Introduction to Music Theory.

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JumpingJackFlash wrote:
jancivil wrote:You've moved the goalpost and are mixing in ratios with ET.
No, none of that was related to ET. It was all pure ratios. The point, in a nutshell, is that you can't have all pure fifths and all pure thirds, they simply don't work.

This example might be better:

Let's say I want to start with a second inversion D minor chord. So A is in the bass and I want the D a perfect fourth above it. Let's assume that my bass A=110Hz. That means my pure fourth (4:3) above that gives D = 146.66667Hz. Now let's assume that I want this to be an octave higher (so it can be sung comfortably by a tenor). That means I need to double the frequency to get D = 293.33333Hz.

My next chord is E major in root position. I set the bass to a pure fifth (3:2) above the A I already tuned to, meaning E = 165Hz. And I want a B a perfect fifth above that, so I use the pure 3:2 again to give me B = 247.5Hz.

No problems so far, but now suppose my third chord is a diminished triad on B. I'll start with the B as I just tuned it (247.5Hz). Now I want the D a minor third (6:5) above that. That would give me a D of 297Hz.

But hang on a minute. I already tuned that D to 293.33333Hz in the first step, so now I've got to re-tune it!
I did not say that through that JI, or any other 12 note JI that every triad is going to be the same quality and I showed my work. Why would you think I need all that??? You're just skimming, aren't you.
It's clear from the very beginning that 3:2 doesn't agree with 2:1, FFS. I've done extensive work towards designing a fretted instrument, 22 to the octave following Danielou et al.
What IS true is that sitars have movable frets per a given raga. I was interested in seeing if something that obviated that was workable. I am sure it is.

There are, in fact, fretted instruments built to use JI. There is no retuning, 'retuning' is not good. It's going to lead people astray, it's not a factual thing.
JumpingJackFlash wrote:Just Intonation requires a certain pitch flexibility that can only generally be achieved with voices, or certain string instruments.
(so now I've got to re-tune it!)
No, it really does not. You simply take the problem under consideration and temper the thing, just as w. all tempered instruments.
Last edited by jancivil on Wed Apr 09, 2014 9:28 pm, edited 1 time in total.

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jancivil wrote:There are, in fact, fretted instruments built to use JI. There is no retuning, there are just these choices made.
Which is what I said; JI only really works with voices, certain stringed instruments or any other instrument that can make real-time adjustments during the performance of music. On fixed pitch instruments (such as the piano), it doesn't work.
jancivil wrote:Your word 'retuning' is not good. It's going to lead people astray, it's not a factual thing.
Well that's just semantics.
Whatever you call it, if I set D=297Hz for one chord (for example), then I'll need to change that to D=293.33333Hz for another chord.
jancivil wrote:You simply take the problem under consideration and temper the thing, just as w. all tempered instruments.
But then you're not using pure ratios anymore which I thought was the whole point!
Last edited by JumpingJackFlash on Wed Apr 09, 2014 9:31 pm, edited 1 time in total.
Unfamiliar words can be looked up in my Glossary of musical terms.
Also check out my Introduction to Music Theory.

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JumpingJackFlash wrote:
jancivil wrote:There are, in fact, fretted instruments built to use JI. There is no retuning, there are just these choices made.
Which is what I said;
what you said and what I said are two different things.
JumpingJackFlash wrote:JI only really works with voices, certain stringed instruments or any other instrument that can make real-time adjustments during the performance of music. On fixed pitch instruments (such as the piano), it doesn't work.
jancivil wrote:Your word 'retuning' is not good. It's going to lead people astray, it's not a factual thing.
Well that's just semantics.
Semantics is looking at meaning. You are quite simply mistaken. You have a bad premise that's not going to give you good results. You are saying 'retuning' as something adjusted 'in real time', I am saying 'No, you just temper it, arrive at a system, and go with it'. Through any kind of instrument, not excluding fixed/fretted, keyboard/hammers, it's a matter of defining the thing.

There is nothing about JI that *requires* making adjustments. What would it have to adjust to? 12tET?? I have, in fact, in reality, used a rational intonation that was good throughout the composition. Just like you did with a piano. THINK!!! You're just replying immediately and I don't think you've read what I have for shit.

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jancivil wrote:There is nothing about JI that requires making adjustments,
Yes there is. I thought I had shown that with the previous examples?! - Otherwise please explain how the above example can work using nothing but pure ratios and without ever making any adjustments.

Ultimately, in a 12-note octave, pure ratios don't work. 12 pure 3:2 fifths does not give you the same as one pure 2:1 octave, neither do three pure 5:4 major thirds.

It might not be an issue in other cultures who divide the octave differently, but in Western music it just doesn't work. - If it did, why would we ever need any kind of temperament in the first place?!
jancivil wrote:I am saying 'No, you just temper it, arrive at a system, and go with it'.
If you temper it, it ceases to be pure ratios!
Unfamiliar words can be looked up in my Glossary of musical terms.
Also check out my Introduction to Music Theory.

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JumpingJackFlash wrote:But then you're not using pure ratios anymore which I thought was the whole point!
Not at all the point. The point is that we can obtain a 5:4 M3 and this is a good thing. We could adjust to everything and obtain that kind of purity, which the 22-tone system in my wiki charts comes close to! People with 12tET instrument adjust all the time to the better concords.

Last night I fixed a thing where an Ab (per C) was quite closer to G than ET, in itself not at issue but there was a bend up to Bb, and the Bb was too flat for me. I didn't mind it that much even as off as it was, it kind of lent a live aspect to it but for me, I wanted a copy of it that didn't do that, so I made a matrix with a differently intoned Ab and switched to it. THAT is 'retuning'; however I could have made an intonation that was a compromise. THAT is 'temperament'.

In the tutorials for VI Pro, they show how if you want this more concordant result but you want to modulate, set up matrices of your articulation dimension that give you the best of both worlds. You seem to think it's exotic, but it really isn't.

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JumpingJackFlash wrote:If you temper it, it ceases to be pure ratios!
I have shown in detail exactly what I mean. If we find that 5:4 x 5:4 gives a result, 25:16, that isn't in the original set, we say 'oh well, here's 8:5, close enough for rock n roll'. Maybe this happens a lot, you define that as 128:81 instead. If it's really prevalent... you make a choice, maybe 25:16 is the least compromised. I changed that to 8:5 last night for the one passage.

You are arguing with yourself through a misconstruction of what I have said. How many times do I have to tell you, I did not say 'the reason for using a JI is to make a statement about pure ratios'. I would NEVER proceed by "first principles rule ok". I would have whopping errors, not very unlike your idea that JI requires fluidity of pitch production as you go.

I showed my work, it's perfectly clear with that JI that there will be compromises. I'm growing weary of you trying to hammer me with something really basic I've known for decades.
However a rational intonation that provides a syntonic comma as a fix does obtain perfectly pure triads from your basic 12 tones in the 2:1. You have to provide an extra 10, or 11, or 12 choices. I looked at what had to be done to obtain 10 more, cf. Danielou and Hindustani theory, by spacing of frets.
Hansford has a bass that guarantees much of these 'purer' results through a different theory. Both of these are complex things to play, but look at the demonstration in the vid. It's a fretted instrument.
And note well, even as I said that a sitar has movable frets, no one moves them during the performance. You set an intonation for the composition and there you are.
Last edited by jancivil on Wed Apr 09, 2014 10:17 pm, edited 1 time in total.

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FYI

http://en.wikipedia.org/wiki/Just_intonation
For many instruments tuned in just intonation, one can not change keys without retuning the instrument. For instance, a piano tuned in just intonation intervals and a minimum of wolf intervals for the key of G, then only one other key (typically E-flat) can have the same intervals, and many of the keys have a very dissonant and unpleasant sound. This makes modulation within a piece, or playing a repertoire of pieces in different keys, impractical to impossible.

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No shit Sherlock. My first reply to JJF said that bit. He said that you have to retune it for every new chord.
Last edited by jancivil on Wed Apr 09, 2014 10:18 pm, edited 1 time in total.

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jancivil wrote:You are arguing with yourself through a misconstruction of what I have said... I'm growing weary of you trying to hammer me with something really basic I've known for decades.
I think you'll find that you started it.
I was talking about a system using nothing but pure ratios - that is, where every harmonic interval throughout the music is pure. You then jumped in wanting to talk about the experiment you did last night... But I'm not the least bit interested in that, it wasn't (and isn't) relevant to what I was (and am) talking about.
jancivil wrote:You seem to want to get rid of your mistake, you believe that JI requires instruments that are not fixed. You have no experience musically with the thing, I do.
Wow, I didn't realise that you were psychic!

Just Intonation, that is a system that calls for harmonically pure intervals at all times, requires that notes have the flexibility to vary in pitch according to the needs of the harmony at any given moment, and thus is only possible with voices or on instruments with the capacity to make real-time adjustments as the music is being performed.

Fact.
Unfamiliar words can be looked up in my Glossary of musical terms.
Also check out my Introduction to Music Theory.

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jancivil wrote:No shit Sherlock. My first reply to JJF said that bit. He said that you have to retune it for every new chord.
Sorry, I'm at work and don't have time to follow the discussion in detail. The limited reading I have done mostly agreed with what you are saying. On the other hand, I had a book by Yehudi Menuhin that spoke about a Russian choir which sang in JI and how they could make adjustments as needed. I don't know whether they were supposed to be adjusting per chord or per key change.

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