Do key signatures actually matter?

[fidelio67]

Pitches, and of course by extension keys, do vibrate differently.

While transposition in 12-tone equal temperament seems to give equal distances, this is only *absolutely* true if you are dealing with pure fundamentals, or say tuning fork to tuning fork. IE: the harmonics (overtones) are going to be a different set when you transpose any pitch. This is to say that 'key of E' is not going to have the same overall 'color' as 'key of C'. The harmonic series is a pyramidic structure, and a different fundamental gives a narrower (or wider) set accordingly to moving it, up or down.

Does that make sense? When you transpose the fundamental, all the overtones get transposed as well, by the same amount, so the relationship between fundamental and overtones stays the same. So that doesn't explain why C sounds different from E at all. Roughly anyway, unless you're talking about the difference in timbre between the high and low registers of an instrument.

Anyway. The idea that keys have colours is not so important as it once was. There used to be very strict rules about what range each voice in a harmony was allowed. So you couldn't just transpose from C to D by shoving everything up a tone as this might push one of the voices out of it's range. The whole harmony might have to be rewritten. This meant that keys had very different feels even though they were being used in ET.

It still makes a difference though, especially with things like guitars. If you have a piece for bass and guitar and it's in F, then it might sound a whole lot different in F#. But if you have something in Bb it will sound very different in E.

[jopy]

The overtones are different for different keys on a piano, because the resonance of the sympathetic vibrations inside the body will be quite different, and that difference is part of the structure of the instrument and the upper/lower limits of the instrument's range. The resonant frequencies (and thus the volume of overtones) of a piece that's transposed upwards on a piano will also be different because the strings get thinner as you go up and there's less doubling on the lower notes (i.e. the top notes of a piano have three strings per note, the lowest notes only have one). It's not just a difference in pitch, it's a different in timbre, i.e. volume of different overtones.

The resonance of most acoustic instruments is also often "tuned" to certain pitches. You may have noticed that some acoustic guitars ring out, are louder, or have more pure tones in certain keys than others, and it's because the overtones are interacting with other parts of the instrument's body. The overtones and their relative volumes will also be different on an electric or acoustic guitar based on where you are on the neck of the guitar or which strings you're using to sound a chord. The closer to the nut you are on an acoustic guitar, the more overtones you get, the closer you get to the bridge, the more "pure" or sine-like the tone. Thicker strings at the bottom of a guitar have a thicker sound. On a trumpet if you transpose up the timbre of the notes is quite different because of the strain put on the players lips and, once again, the physical resonance/timbre of the instrument.

I'm less sure about electronically generated sine waves though.

[fidelio67]

Yeah, but you're talking about the difference in overtones between the low and high registers of an instrument. A middle c on a piano has pretty much the same overtones as the a 3 semitones below it, relative to the fundamental. A low C and a top C are very different, but that's not the issue.

[jopy]

I should admit that I started this based on my intuitive sense that there has to be some difference in timbre based on my own hearing and I constructed a plausible theory for why that is, so I don't have good data on the claims I'm making. I can hear a small noticeable difference in the sound of different keys on the piano, if someone else is transposing (obviously unfair if I'm doing it because my hands are moving totally different so it's obvious), but you may have a point, it could be purely psychological. I still think the sympathetic vibrations of the other strings (assuming you have the damper pedal down or are playing a whole chord) after changing key by a fifth, for example, would be subtle I'll admit, but I don't think it would be nothing. I think the key is "pretty much the same" for small transpositions on a piano. I'd agree with that. Not identical, but pretty much the same.

On any other physical instrument that I can think of the differences would be a lot more substantial though. A C vs. a G (assuming a barre chord) on the guitar will never sound quite the same as one another because you either have to move up a string or around the fretboard.

This really is an empirical question though, isn't it? One could hook up a piano to an spectrum analyzer like Voxengo's Span and solve this mystery straightaway, if only you could be sure both notes on the piano were struck at exactly the same velocity.

[fidelio67]

I think with guitars or any fretted instrument you're right. Playing in E on a guitar sounds hugely different from playing in Eb! But with any instrument once you get up to transposing by more than a few semitones it's a very real difference. I'm not sure this is to do with the overtones though. You don't need perfect pitch to tell the difference between a tune in A and the same tune E, even if it's played on a synth with identical overtones for each note.

[jancivil]

Pitches, and of course by extension keys, do vibrate differently.

While transposition in 12-tone equal temperament seems to give equal distances, this is only *absolutely* true if you are dealing with pure fundamentals, or say tuning fork to tuning fork. IE: the harmonics (overtones) are going to be a different set when you transpose any pitch. This is to say that 'key of E' is not going to have the same overall 'color' as 'key of C'. The harmonic series is a pyramidic structure, and a different fundamental gives a narrower (or wider) set accordingly to moving it, up or down.

Does that make sense? When you transpose the fundamental, all the overtones get transposed as well, by the same amount, so the relationship between fundamental and overtones stays the same.

Sure, the relationship is parallel - but bigger is bigger and smaller is smaller. A higher fundamental gives a tighter, or narrower set; THEY ARE CLOSER TOGETHER. Physically, this IS different. A closer SET of vibrations sounds, IE: has a different timbral 'color' accordingly.

[nuffink]

Hmm. I'm not sure about that. The higher frequencies only appear closer together because we arbitrarily impose a linear scale (hertz) upon what is intrinsically a logarithmic series.

[jancivil]

What would it look like if you looked at it any other way???

I really, really need to see a good argument for 'smaller is not smaller' [Arithmetically or geometrically].

I'm doubtful.

[nuffink]

Well, there's definitely more "power" in the lower octaves (as is evident in the size of speaker needed to cover the octave from 20hz to 40hz as opposed to that needed to cover 10k to 20k) so you may have a point.

However, it's very late and I'm not entirely sure what either of us is talking about. I'll have a think about it tomorrow.

[jancivil]

I think with guitars or any fretted instrument you're right. Playing in E on a guitar sounds hugely different from playing in Eb!

Well, playing a single low E STRING on a guitar thru a marshall sounds substantially different than playing any other, fundamental with this much amplitude. (especially a lower one)

So, once you start amplifying the fundamental, much less adding 'chord tones', you have a different sound.

Detecting anything by ear is down to the acuteness of that ear*. And the amplitude of that pitch or pitches. And - Note well: A lower fundamental produces more audible overtones. Why is this???

Are all of these phenomena down strictly to the structure of the human ear? I doubt it.

*: But has anyone here measured this?

[someone called simon]

Hmm. I'm not sure about that. The higher frequencies only appear closer together because we arbitrarily impose a linear scale (hertz) upon what is intrinsically a logarithmic series.

I can think of a practical example that shows that while what you said may be true, it makes a big difference to our ear, in some circumstances at least.

Play a C triad in the middle of the keyboard. its... pleasant. But play it down low, and its jut a mess... Play it really high, its still ok, but to me the notes sound too close together, moreso than in the middle of the keyboard.

[jopy]

Correct me if I'm wrong here, but I think a guitar's overtones are different depending on where you play on the fretboard because of the ability of the string to vibrate freely, which modifies the waveform being generated--differences in waveform are differences in harmonics, and harmonics are so closely related to overtones that they're often used as if they were the same thing. I think the same thing would hold for a trumpet--the player's lips vibrate differently based on the muscle control being exercised pitches so a middle C is not exactly E down a major third, but it also has a looser embouchure so it will have more complex harmonic content, whereas when you get high up and the trumpet player is really straining, you get more of a pure sine type of tone. I'd guess the same thing would hold for reed instruments.

Okay, I just cheated and used google--this came up:
http://www.whc.net/rjones/jlynch/natres.html

[fidelio67]

What would it look like if you looked at it any other way???

I really, really need to see a good argument for 'smaller is not smaller' [Arithmetically or geometrically].

I'm doubtful.

It's not about size it's about shape! fnarr fnarr.

Say you've got an A at 220hz. The first few overtones will be:
440, 660, 880, 1100

And now say you have a C at whatever C is. 260 or something. You get:

520, 720, 1040, 1300,

Now it seems to me that you're saying that because the A's harmonics, (880 and 1100) are not the same distance apart as the C's (1024 abd 1300) the note will have a different 'colour'
But the waveform will be exactly the same shape. What your describing is not a difference in colour, but a difference in pitch, which is where we came in in the first place!

[fidelio67]

Hmm. I'm not sure about that. The higher frequencies only appear closer together because we arbitrarily impose a linear scale (hertz) upon what is intrinsically a logarithmic series.

I can think of a practical example that shows that while what you said may be true, it makes a big difference to our ear, in some circumstances at least.

Play a C triad in the middle of the keyboard. its... pleasant. But play it down low, and its jut a mess... Play it really high, its still ok, but to me the notes sound too close together, moreso than in the middle of the keyboard.

But that's exactly the point! A triad is not the same as a series of overtones. The relaionship bewteen the notes in a triad is geometric. The harmonic series is logarithmic (I think I have the terminology right) Anyway. The fact is that as you move a triad up and down it's colour does change, but with a single note it doesn't (ignoring timbre differences)

To preserve the real logarithmic relationship between the notes in the triad you need to space them out further as you go down. When you go higher up you can get away with 9ths and things that would sound muddy in the middle and a complete mess farther down.

In other words, it's the fact that the overtones are closer together as you rise in pitch that is responsible for preserving the colour, not for changing it!

[BertKoor]

... whatever sounds good ...