Plato

[dincz]

Slightly off track. I've often heard of different keys expressing different "moods". With the tempered scale, surely transposing from one major (or minor) key to another major (or minor) key would not affect the "mood".

[TimR]

[quote name='leftybassman392' timestamp='1445185195' post='2889395']
Transposing in Equal Temperament is a doddle - it's to do with...

Actually, and at the risk of sounding a bit patronising, a lot of this stuff really will make more sense if people take the time to read my articles on the subject.

Essential Tension linked it a while ago but here it is again:

[url="http://basschat.co.uk/topic/59011-ancient-greek-music/"]http://basschat.co.u...nt-greek-music/[/url]

The video is very good by the way - it skirts over one or two issues a bit too easily for my liking, and the narrator's tone is a bit patronising, but apart from that it goes through it very nicely (and, annoyingly, is more interesting than reading through my articles as well... )
[/quote]

It may well be a doddle but in your article that you link to you explain that you sacrifice pure tones and all notes become a compromise that is close enough for us to live with.

What happens is that in some keys the perfect 3/2 ratio is closer than in other keys. The do sound slightly different. In a C major chord the ratio for the 5th is not as close as the ratio in an A major chord. It's subtle and we can hear it, it doesn't sound wrong, but it does sound different.

[leftybassman392]

[quote name='TimR' timestamp='1445246964' post='2889825']
It may well be a doddle but in your article that you link to you explain that you sacrifice pure tones and all notes become a compromise that is close enough for us to live with.

What happens is that in some keys the perfect 3/2 ratio is closer than in other keys. The do sound slightly different. In a C major chord the ratio for the 5th is not as close as the ratio in an A major chord. It's subtle and we can hear it, it doesn't sound wrong, but it does sound different.
[/quote]


With respect I think you may be quoting me out of context (and in the process have misinterpreted what I wrote).

Quick summary:

Pythagorean tuning gives a series of harmonically pure notes, but eventually falls foul of the Pythagorean comma. If you stick to the notes in the diatonic (major) scale the problem doesn't arise (unless you try to get too cute with the harmonies of course...). The problems start when you try to transpose or modulate to a different key (not a problem for early musicians - much less the Pythagoreans - as they didn't really do harmonies - much less modulation - in the way we think of them).

Equal Temperament tuning sets up the notes so as to be harmonically precisely equidistant from each other (you may recall the narrator in the video talking about the 12th root of 2 - personally I'd have preferred that he expand on that idea to show how the Equal Temperament scale is constructed, but he didn't so you'll have to make do with my explanation! ). I went into some detail as to how this is done in the articles so if it's all the same to you I won't regurgitate it en masse here.

However - and this is the important bit - in order to create a scale in this way one has to give up most of the harmonically pure ratios that would come from the Pythagorean tuning method. In harmonic terms it's a compromise, so that (to put the text you quote [i][b]into[/b][/i] context) the notes are a little way of their Pythagorean equivalents but are close enough so we can live with it.

Here's a quick example:

Take concert A at 440Hz (yes, I know not everybody uses 440Hz...).

In Pythagorean tuning the fifth above (E) it is at 440 x 1.5 = [u]660Hz[/u] This is the harmonically pure perfect fifth.

In Equal Temperament tuning E is the 7th note of the Chromatic scale starting at A and is therefore 2[sup](7/12)[/sup] x 440 = 1.4983 x 440 = [u]659.2551Hz[/u]

(Results correct to 4 d.p.)

As I said, slightly off the pure pitch but close enough for people to not notice.

If you pick any pair of notes from the Equal Temperament system that are a perfect fifth apart, the higher not will [i][b]always[/b][/i] be 1.4983... x the pitch of the lower. That's how the system works. (Please don't quibble this point - the Maths is correct, o.k.?). The pitch relationship, not the actual pitch values, is the determining factor.

Looked at another way, Pythagorean tuning gives you a whole load of harmonically pure but mutually incompatible scales: Equal Temperament gives you a single, homogeneous scale that goes to the limits of human hearing in both directions.

The genius is that we can move around the scale pretty much anywhere we choose and be confident that the pitch relationships are always [i][b]exactly[/b][/i] the same. As such, transposition is a simple matter of choosing your new starting point and away you go: simples!

The downside is that the notes are a tiny bit off-pitch in any given key in any given register. But we can live with it...

Edited October 19, 2015 by leftybassman392

[TrevorR]

[quote name='dincz' timestamp='1445244080' post='2889781']
Slightly off track. I've often heard of different keys expressing different "moods". With the tempered scale, surely transposing from one major (or minor) key to another major (or minor) key would not affect the "mood".
[/quote]

Unless you transpose to D minor which, as we all know, is the saddest of all possible keys.

[TimR]

[quote name='leftybassman392' timestamp='1445249919' post='2889860']



With respect I think you may be quoting me out of context (and in the process have misinterpreted what I wrote).

Quick summary:

Pythagorean tuning gives a series of harmonically pure notes, but eventually falls foul of the Pythagorean comma. If you stick to the notes in the diatonic (major) scale the problem doesn't arise (unless you try to get too cute with the harmonies of course...). The problems start when you try to transpose or modulate to a different key (not a problem for early musicians - much less the Pythagoreans - as they didn't really do harmonies - much less modulation - in the way we think of them).

Equal Temperament tuning sets up the notes so as to be harmonically precisely equidistant from each other (you may recall the narrator in the video talking about the 12th root of 2 - personally I'd have preferred that he expand on that idea to show how the Equal Temperament scale is constructed, but he didn't so you'll have to make do with my explanation! ). I went into some detail as to how this is done in the articles so if it's all the same to you I won't regurgitate it en masse here.

However - and this is the important bit - in order to create a scale in this way one has to give up most of the harmonically pure ratios that would come from the Pythagorean tuning method. In harmonic terms it's a compromise, so that (to put the text you quote [i][b]into[/b][/i] context) the notes are a little way of their Pythagorean equivalents but are close enough so we can live with it.

Here's a quick example:

Take concert A at 440Hz (yes, I know not everybody uses 440Hz...).

In Pythagorean tuning the fifth above (E) it is at 440 x 1.5 = [u]660Hz[/u] This is the harmonically pure perfect fifth.

In Equal Temperament tuning E is the 7th note of the Chromatic scale starting at A and is therefore 2[sup](7/12)[/sup] x 440 = 1.4983 x 440 = [u]659.2551Hz[/u]

(Results correct to 4 d.p.)

As I said, slightly off the pure pitch but close enough for people to not notice.

If you pick any pair of notes from the Equal Temperament system that are a perfect fifth apart, the higher not will [i][b]always[/b][/i] be 1.4983... x the pitch of the lower. That's how the system works. (Please don't quibble this point - the Maths is correct, o.k.?). The pitch relationship, not the actual pitch values, is the determining factor.

Looked at another way, Pythagorean tuning gives you a whole load of harmonically pure but mutually incompatible scales: Equal Temperament gives you a single, homogeneous scale that goes to the limits of human hearing in both directions.

The genius is that we can move around the scale pretty much anywhere we choose and be confident that the pitch relationships are always [i][b]exactly[/b][/i] the same. As such, transposition is a simple matter of choosing your new starting point and away you go: simples!

The downside is that the notes are a tiny bit off-pitch in any given key in any given register. But we can live with it...
[/quote]

That's exactly what I said.

[leftybassman392]

[quote name='TimR' timestamp='1445250472' post='2889869']
That's exactly what I said.
[/quote]

No it isn't! If you can't see why then you're not looking hard enough. As far as I'm concerned this particular conversation is at an end.

Edited October 19, 2015 by leftybassman392

[TimR]

[quote name='leftybassman392' timestamp='1445250671' post='2889873']


No it isn't! If you can't see why then you're not looking hard enough. As far as I'm concerned this particular conversation is at an end.
[/quote]

Yes. The earlier poster was right. You are condescending.

"Not always 1.4983..."

.

Edited October 19, 2015 by TimR

[Guest bassman7755]

[quote name='TimR' timestamp='1445251652' post='2889890']
Yes. The earlier poster was right. You are condescending.

"Not always 1.4983..."

.
[/quote]

Given the original post said:

[color=#282828][font=helvetica, arial, sans-serif]"If you pick any pair of notes from the Equal Temperament system that are a perfect fifth apart, the higher [s]not [/s][u]note[/u] will [/font][/color][i][b]always [/b][/i][color=#282828][font=helvetica, arial, sans-serif]be 1.4983... x the pitch of the lower."[/font][/color]

[color=#282828][font=helvetica, arial, sans-serif](I am assuming "not" should have read "note" )[/font][/color]

[font=helvetica, arial, sans-serif][color=#282828]I guess the clue is in the name - the [i][b]even[/b][/i] tempered scale, ergo its sounds the same wherever you start.[/color][/font]

Edited October 19, 2015 by bassman7755

[DavidMcKay]

Guys - let's not have a falling out over an argument that very few us of understand (making a big assumption here I know).

[TimR]

Ok. I've done the maths now and we're cleared up the typo/misread issue.

The key is understanding that the frequencies are not just evenly spaced across the 440hz to 880hz range in 12 equal gaps, the 2^(x/12) ratio is key.

[BassTractor]

[quote name='dincz' timestamp='1445244080' post='2889781']
Slightly off track. I've often heard of different keys expressing different "moods". With the tempered scale, surely transposing from one major (or minor) key to another major (or minor) key would not affect the "mood".
[/quote]

One would think it wouldn't change the mood indeed, but people with different types/grades of perfect pitch (I'm lumping all types together here to avoid another lengthy post) still report that each key has its own character or mood.

I can't for the life of me remember whether they all agree on the different moods, nor whether they need physical instruments (typically orchestra instruments) for this rather than electronic instruments.

[Happy Jack]

[quote name='DavidMcKay' timestamp='1445254815' post='2889929']
Guys - let's not have a falling out over an argument that very few us of understand (making a big assumption here I know).
[/quote]

Bloody hell! Talk about a poacher turned gamekeeper ...

[Dandelion]

I think in D minor.

[TimR]

[quote name='BassTractor' timestamp='1445256387' post='2889946']


One would think it wouldn't change the mood indeed, but people with different types/grades of perfect pitch (I'm lumping all types together here to avoid another lengthy post) still report that each key has its own character or mood.

I can't for the life of me remember whether they all agree on the different moods, nor whether they need physical instruments (typically orchestra instruments) for this rather than electronic instruments.
[/quote]

There must be something psycho-aural going on. Transposing tunes from a sharp key to a flat key definitely affects the sound. Even if its not backed up by the maths/physics.

Although I'm not sure pianos are actually tuned by computers. Doesn't a man tune them to a reference tuning fork and his ear, listening for beats? Hence there's a fair bit of 'whether it sounds good' going on.

[ras52]

[quote name='TimR' timestamp='1445269515' post='2890101']
There must be something psycho-aural going on. Transposing tunes from a sharp key to a flat key definitely affects the sound. Even if its not backed up by the maths/physics.
[/quote]

You don't even have to transpose... I for one would think of a piece in D flat minor as being somehow "softer" than the same piece in C sharp major.

[TimR]

[quote name='ras52' timestamp='1445270046' post='2890109']


You don't even have to transpose... I for one would think of a piece in D flat minor as being somehow "softer" than the same piece in C sharp major.
[/quote]

That's a completely different scale not just different key.

[ras52]

[quote name='TimR' timestamp='1445270185' post='2890114']
That's a completely different scale not just different key.
[/quote]

Ah, you spotted the deliberate mistake! I meant D flat major and C sharp major...

[BassTractor]

[quote name='TimR' timestamp='1445269515' post='2890101']
Although I'm not sure pianos are actually tuned by computers. Doesn't a man tune them to a reference tuning fork and his ear, listening for beats? Hence there's a fair bit of 'whether it sounds good' going on.
[/quote]

Funny you should write this, as I'd deleted a chunk about piano tuning before posting.
Electronic tuning machines do exist, but I've seen one in use only once.
My experience is with people who do the tuning, and contradictory to what I'd assume is the impression in the general public, the pianos aren't exactly tuned in equal temperament, but the tuner person would ask what music you tend to play most and/or like most.
Concert grands are often tuned fitting to the material of the next concert, in cooperation with the solo pianist, and I've seen one master pianist using two differently tuned pianos during the concert, one for music from the early 19th century and one for newer music.

BTW, I was in a tuning project & concert once where we had four differently tuned harpsichords on stage. Great experience to hear the same piece four times with different tunings, and the effect the tuning had on the piece/listener.

Edited October 19, 2015 by BassTractor

[DavidMcKay]

[quote name='Happy Jack' timestamp='1445260048' post='2889988']
Bloody hell! Talk about a poacher turned gamekeeper ...


[/quote]

I'm all about the love!