Equal Temperament

[michael sam wild]

Hi

I've just been reading a novel, The Piano Tuner by Daniel Mason, pub. Picador 2003

 

It's a bit pretentious and far fetched but he uses some interesting material

 

He describes Bach's pieces .The Well Tempered Klavier'. I was totally lost but felt I intuitively knew what he meant but couldn't quite get it. The tuner divided the octave into equal intervals so he could play in all keys.

.

As kids we used to twang bits of wire between two pegs and run a stick along them, a bit like a dulcimer and there was always a bit of a variation each side when we went into other keys to do the Doh Re Mi, . I asked at Secondary school but my Physics teacher just told me to shut up and listen to his boring formulae so I switched off. 'You are merely a receiver' -- so it was ' OK - Over and Out!' and that was at Manchester Grammar School in the 50s !, supposedly a top school. The music master just whacked you with a board ruler! Some joined the school skiffle and jazz and rock groups with the musical drop outs who lurked in the remedial room with a grand piano where we could pick the lock and have lunch hour sessions.

 

Is that string thing what Pythgoras was playing with. Can anyone give me an easy explanation, without pointing me to Wikipedia or some other geeky site? Why do people want to make it appear complicated, is it because Maths and Music are linked and that limits easy communication or is it elitism? Birds seem to do it intuitively and so did Dinosaurs apparently.!

 

I will recommend the best answer for a Clarity Award

Thanks

Mike

Edited February 12, 2009 by michael sam wild

[fiddlerjoebob]

It's moonlight refracting off pockets of swamp gas. Its nothing to worry about. Return to the safety of your homes, there's nothing more to be seen here.

 

 

..............

[Greg Jowaisas]

Fools rush in....

 

So here is one fool's take on what is happening:

 

Your do, re, mi scale might start on your instrument of fixed pitch (concertina, fretted instrument) with C.

 

You work out a major scale which goes:

C, D, E, F, G, A, B, c,... Sounds good in C as you subtly adjust by ear for good nice third, fifth. seventh intervals.

 

Now the rub. If you play in a different key, say G, you start in a different place.

G, A, B, c, d, e, f, g'....

 

Problems occur because the relationship of a do to mi in the C scale is an interval of a third. But starting on G (using the adjusted C scale) the relationship of do to re (G to is really the relationship of so to ti (4th to 6th).

 

The relationships are subtly different relative to the starting point.

 

Equal temperment is a compromise to insure none of the interval relationships are too jarring to our ears in any key. The compromise also leaves not too many sweet, perfect intervals. For whatever reason 3rds seem to get the worst of it; 5ths seem to fair among the best.

 

In laying out the notes on a fretted guitar or dulcimer (which on a linear instrument is based on a logarithm; the Pythagorus connection) a compromise is necessary to play in different keys. Take a little from the third so the fifth over here is acceptable; compromise the 7th to octave in C because that will be the 2nd to 3rd in G.

 

One folk musician's take.

 

Greg

[david fabre]

As a physics teacher I can't resist the challenge...

 

Point 1 : for some reason related to the way the ear functionates, two notes are heard to

form a well-sounding interval if the ratio of their frequencies is a simple fraction. Most important

ones are the octave (ratio 2/1), fifth (ratio 3/2), major third (ratio 5/4).

 

Point 2 : when tuning an instrument you can select seven notes to form a "just" scale whose notes

form such ratios together. This is probably what you did intuitively in your youth. If considering a C

scale, the natural choice is to tune the notes such that C-E-G, F-A-C and G-B-D form perfect major

chords, with just thirds and fifths.

 

At this point you can already notice that some notes of your scale will not sound right together.

In particular, the D and A you have will not form a just fifth but a so-called "Woolf". In effect, a little

arithmetics will show you that the frequency of the D is 9/8 times the frequency of the C, and the

frequency of the A is 5/3 times the frequency of the C. The ratio of these two frequencies is 40/27 = 1.4815,

instead of 3/2 = 1.5 for a just fifth. This problem may give you a first idea of the problems you

will face when you will want to play in a different key.

 

 

Point 2bis :

Then, you can complete this scale with five addidional notes which fill the largest spaces in the scale.

You will most likely choose these notes as forming just intervals with the notes you already have.

For instance you will tune F# a just third above D. For the other notes there are several possibilities,

but this point is not very important here. You end up with a 12-note scale.

You will then notice that the 12 notes are almost equally spaced but not exactly,

some of the intervals (called semitones) will be slightly larger or smaller.

 

Point 3 : if you have an instrument tuned this way and try to play a scale starting from any other note than

the one you chosed as the root of the primary scale, the sequence of inequal semitones will be modified,

and some of the intervals between the main notes of the scale will be alterated and will now sound wrong.

You can convince yourself of that by experiment, but if you really want to understand the reason I'm afraid

you have to do a little mathematics.

 

Point 4 : musicians have seached various ways to overcome this problem. The most largely used today is

to make all semitones equal. The result is that all intervals are more or less false (the worst being the thirds)

but all scales sounding the same (main advantage).

 

I add two "historical" notes :

1/ about Pythagoras and the ancient greeks : they actually did not use quite the same scale as

the one I explained because at this time the major third was not considered as a consonant interval at all.

So in the greek scale the notes A, E and B were tuned by superposition of just fifth above D. Contrary to what

I pointed out above in this case all fifth sound rigt (by construction) but the thirds such as C-E sound bad

(but anyway the greeks never sounded these notes together)

 

2/ about Bach : contrary to what is often thought, he apparently did not used equal temperament

but a so-called "well temperament" (whose precise nature is still the object of debate). This kind of temperament

corresponds to another kind of compromise in which some of the intervals are retained just, the others

are altered in various maneers in a way that they are inequally good but none of them is as bad as a "woolf"

as I described above. With such temperaments each scale was playable but each of them had its own feeling.

 

 

Hope this is clear enough for you...

 

[edited to clarify a few points]

Edited February 13, 2009 by david fabre

[Hereward]

As a physics teacher I can't resist the challenge...

 

Point 1 : for some reason related to the way the ear functionates, two notes are heard to

form a well-sounding interval if the ratio of their frequencies is a simple fraction. Most important

ones are the octave (ratio 2/1), fifth (ratio 3/2), major third (ratio 5/4).

 

Point 2 : when tuning an instrument you can select seven notes to form a "just" scale whose notes

form such ratios together. This is probably what you did intuitively in your youth. You must have

proceeded this way : start from C, tune G up one fifh, E up one third, F down one fift,

and then B, D forming just third and fifth with G and finally A forming a third with F.

 

At this point you can already notice that some notes of your scale will not sound right together.

In particular, the D and A you have will not form a just fifth but a so-called "Woolf". The reason

of this can be easily explained using basic arithmetics.

 

 

Point 2bis :

Then, you can complete this scale with five addidional notes which fill the largest spaces in the scale

(but the way to do this is not unique) to form a 12-note scale.

You will then notice that the 12 notes are almost equally spaced but not exactly,

some of the intervals (called semitones) will be slightly larger or smaller.

 

Point 3 : if you have an instrument tuned this way and try to play a scale starting from any other note than

the one you chosed as the root of the primary scale, the sequence of inequal semitones will be modified,

and some of the intervals between the main notes of the scale will be alterated and will now sound wrong.

You can convince yourself of that by experiment, but if you really want to understand the reason I'm afraid

you have to do a little mathematics.

 

Point 4 : musicians have seached various ways to overcome this problem. The most largely used today is

to make all semitones equal. The result is that all intervals are more or less false (the worst being the thirds)

but all scales sounding the same (main advantage).

 

I add two "historical" notes :

1/ about Pythagoras and the ancient greeks : they actually did not use quite the same scale as

the one I explained because at this time the major third was not considered as a consonant interval at all.

So in the greek scale the notes A, E and B were tuned by superposition of just fifth above D. Contrary to what

I pointed out above in this case all fifth sound rigt (by construction) but the thirds such as C-E sound bad

(but anyway the greeks never sounded these notes together)

 

2/ about Bach : contrary to what is often thought, he apparently did not used equal temperament

but a so-called "well temperament" (whose precise nature is still the object of debate). This kind of temperament

corresponds to another kind of compromise in which some of the intervals are retained just, the others

are altered in various maneers in a way that they are inequally good but none of them is as bad as a "wholf"

as I described above. With such temperaments each scale was playable but each of them had its own feeling.

 

 

Hope this is clear enough for you...

 

I would dearly like to understand this but right now I feel more baffled than trying to fathom quantum mechanics makes me.

 

Ian

[Bill N]

I would dearly like to understand this but right now I feel more baffled than trying to fathom quantum mechanics makes me.

 

Ian

 

 

Never mind quantum mechanics and equal temperament. This is what truly baffles me:

 

Wheatstone Memorial 2009 Mornington Crescent Game

[michael sam wild]

I would dearly like to understand this but right now I feel more baffled than trying to fathom quantum mechanics makes me.

 

Ian

 

 

Never mind quantum mechanics and equal temperament. This is what truly baffles me:

 

Wheatstone Memorial 2009 Mornington Crescent Game

 

 

Thanks for thr time and trouble. Actually so far it's beginning to make sense to me ( Not Mornington Cresecent tat never will)! Apart from the man from the swamp! What do you grow over there Bob, or is it cabin fever?

 

Mike

[Steve Mansfield]

Thanks for thr time and trouble. Actually so far it's beginning to make sense to me ( Not Mornington Cresecent tat never will)! Apart from the man from the swamp! What do you grow over there Bob, or is it cabin fever?

 

Mike

 

If you're interested in finding out more, have a read of Ross Duffin's book on the subject with the splendid title

 

'How equal temperament ruined harmony: and why you should care'.

 

It's part musical history, part social history, and part scientific/musicological treatise: but even if, like me, you've not got the academic musical knowledge to follow all of the more musicologically technical parts of the book it's still a fascinating read and entertainingly written.

 

Mr Duffin also restricts his examples and discussion to Western classical music (personally I'd love to know more about the temperaments used by the Breton bagad bands or indeed by Southern English fiddlers like Jinky Wells), but he is the professor of music at some-prestigious-American-university-or-other so the classical leaning is inevitable.

 

You can get the paperback off of Amazon for under a fiver, and it will answer all your questions (including many you didn't even realise needed asking!). Recommended.

[david fabre]

I edited my first post to clarify a few points (not sure it is really simpler though)

[michael sam wild]

Just ordered Duffin book from Guardian Books 9.95 GBP incs p&p

 

I googled him and got a nice review too

[Hereward]

Just ordered Duffin book from Guardian Books 9.95 GBP incs p&p

 

I googled him and got a nice review too

 

I ordered a copy too. A little light reading never did anyone any harm. This however...

 

Ian

[Theo]

(personally I'd love to know more about the temperaments used by ................. by Southern English fiddlers like Jinky Wells),

 

John Dipper is your man for that topic

[michael sam wild]

Hi Theo

Any furtehr sources from him?

Mike

[Theo]

A few seconds with Google will find links to his website.

[michael sam wild]

Already done that but nowt on the temperament thing sadly

[Chris Drinkwater]

(personally I'd love to know more about the temperaments used by ................. by Southern English fiddlers like Jinky Wells),

 

John Dipper is your man for that topic

 

 

John Boden, fiddler wiith Spiers and Boden and Bellowhead, tells a lovely tale about Jinky Wells. Some posh bloke came to hear Jinky play. After hearing Jinky play a tune, he asked Jinky if he could borrow his violin it for a moment. Jinky obliged, whereupon the chap promptly checked the tuning on it and discovering it was 'out', retuned it and gave it back to Jinky. Jinky placed the violin under his chin, played a few bars of some classical piece of music, then retuned the violin himself to its original tuning and began to play another of his favourite folk tunes. It was obviously tuned to his preferred temperament, what ever that was, specially suited to his folk tunes!

 

Chris

[michael sam wild]

Just got the Duffin book from Guardian Books, , my head's hurting but I will persevere and survive and report back!

 

Mike

[Hereward]

Just got the Duffin book from Guardian Books, , my head's hurting but I will persevere and survive and report back!

 

Mike

 

I have it too. Hard going for someone like me who didn't know a concertina from an onion a few weeks ago and not much about music in general either. Blame it on the UK school system. Terrible now and limited in my days.

 

Ian