Just intonation, Equal temperament, and Irish flutes...

[peeplj]

Ok, what's been written here is good, but incomplete.

Here's some more info:

Every note has a frequency, measured in Hertz, or vibrations per second. The note A is usually defined now as 440 Hz, and the note a an octave above it is 880 Hz.

Note: octaves have a 2:1 ratio. That means if you look at their waveform on an oscilloscope, every other wave lines up.

Octaves when in tune are an "open" invterval, which means that they are beatless. What you hear as "beats" when two notes are out of tune is actually their waveforms marching in and out of line with each other, creating interferance patterns...which are the "beats" you hear.

Perfect fifths, when perfectly in tune, have a 3:2 ratio. They are also a beatless, "open" interval.

Ok, so you start at a note, say C, just to have a point of reference. And you go to the fifth above it, which is G, and you tune them until there are no beats, and they sound absolutely open and perfect.

Then you go to the fifth above the G, which is D, and you tune them to be open and beatless. The tuning you are creating is called "just temper."

Then you tune D to A, and A to E, and so on, all around the "Circle of Fifths," until you finally find yourself having just tuned B-sharp.

B-sharp and C are "enharmonic," which means they are different names for the same note.

Or are they?

The b-sharp you just tuned will be about a fourth of a semitone sharper than the C you started out with. This is about a 24 cent difference, and this difference is called the Pythagorean Comma.

This poses a problem.

If you take the last key you tuned, and just basically pull the last fifth 24 cents flat out of tune, you've created what's called a "wolf fifth"--and the key you are in when you do that just became horrible sounding and unusable.

So now you can't play in all the keys.

Well, maybe that's a big deal to you and maybe not.

Different tuning schemes have been used through the years to arrive at different compromises to handle that pesky comma.

The one that finally got settled upon is known as equal temper...you divide the comma up equally among all the fifths, so that each one is just a little bit out of true. (Actually, not just the fifths--it gets divided out among the intervals within the scale, and it's divided out exactly the same way in every key.) This makes all keys equally usable.

It also makes all keys sound the same, and you lose the glorious interval of the open, beatless fifth, which is major sucky.

Now tuning a flute requires its own set of compromises. One key on a simple system flute will be diatonic...that is, its tone holes will form the notes of a D scale if opened one at a time. But these diatonic notes also have to work in other keys, and the top tone hole also has to be the octave vent for the D and it also tunes your cross-fingered C-natural.

Also the octaves have to be in tune. Just two octaves isn't too bad, but when you add the third, things get very dicey, and more serious compromises have to be made or the third octave notes will just be too horrible to ever use.

Finally, the tone holes need to be where the fingers can reach them, which is really not the best place for them to go, so they wind up being different sizes, which creates yet another set of problems that have to be worked around.

So the tuning of a simple-system flute isn't really any traditional tuning, per se.

Some makers tune closer to just, others closer to equal, yet others closer to some of the older, more antiquated compromise tunings like "quarter-comma meantone," which is a scheme that allows you to keep your open thirds, which is another open interval in just tuning.

Some folks are surprised by that, because the major third in equal temper (like on a piano or Boehm-system flute) is anything but beatless. In fact, in equal temper the third is very sharp. Pull it down a bit and it'll get more and more open until it's finally beatless.

Hmmm...the bell note on a flute is D and the F-sharp is one major third up from there. Maybe tuning that F-sharp "flat" compared to a Boehm-system flute or a piano isn't a mistake after all, hmmm?

These are deep waters, and I've only gone just slightly past the surface in this long post. There is a lot more to it.

But it's pretty cool stuff to know, because it explains a lot of the "whys"--why a flute is tuned like it is, why on a Baroque flute B-flat and A-sharp aren't fingered the same, why it just may be a bad thing that on the Boehm-system flute every key sounds like every other key.

Fascinating stuff. And lots of lovely math!!!

--James

[jemtheflute]

Brill post, James! I've read about this stuff before - Rockstro, Quantz et al; I know about it. Now I understand it rather better. (The maths is beyond me, though! ) Thanks.
I like your point on F#! Of course, flutes only play one note at a time, so the tuning criteria/requirements are not the same as for chord-playing instruments.

[Cork]

Ok, what's been written here is good, but incomplete.

Here's some more info:

Every note has a frequency, measured in Hertz, or vibrations per second. The note A is usually defined now as 440 Hz, and the note a an octave above it is 880 Hz.

Note: octaves have a 2:1 ratio. That means if you look at their waveform on an oscilloscope, every other wave lines up.

Octaves when in tune are an "open" invterval, which means that they are beatless. What you hear as "beats" when two notes are out of tune is actually their waveforms marching in and out of line with each other, creating interferance patterns...which are the "beats" you hear.

Perfect fifths, when perfectly in tune, have a 3:2 ratio. They are also a beatless, "open" interval.

Ok, so you start at a note, say C, just to have a point of reference. And you go to the fifth above it, which is G, and you tune them until there are no beats, and they sound absolutely open and perfect.

Then you go to the fifth above the G, which is D, and you tune them to be open and beatless. The tuning you are creating is called "just temper."

Then you tune D to A, and A to E, and so on, all around the "Circle of Fifths," until you finally find yourself having just tuned B-sharp.

B-sharp and C are "enharmonic," which means they are different names for the same note.

Or are they?

The b-sharp you just tuned will be about a fourth of a semitone sharper than the C you started out with. This is about a 24 cent difference, and this difference is called the Pythagorean Comma.

This poses a problem.

If you take the last key you tuned, and just basically pull the last fifth 24 cents flat out of tune, you've created what's called a "wolf fifth"--and the key you are in when you do that just became horrible sounding and unusable.

So now you can't play in all the keys.

Well, maybe that's a big deal to you and maybe not.

Different tuning schemes have been used through the years to arrive at different compromises to handle that pesky comma.

The one that finally got settled upon is known as equal temper...you divide the comma up equally among all the fifths, so that each one is just a little bit out of true. (Actually, not just the fifths--it gets divided out among the intervals within the scale, and it's divided out exactly the same way in every key.) This makes all keys equally usable.

It also makes all keys sound the same, and you lose the glorious interval of the open, beatless fifth, which is major sucky.

Now tuning a flute requires its own set of compromises. One key on a simple system flute will be diatonic...that is, its tone holes will form the notes of a D scale if opened one at a time. But these diatonic notes also have to work in other keys, and the top tone hole also has to be the octave vent for the D and it also tunes your cross-fingered C-natural.

Also the octaves have to be in tune. Just two octaves isn't too bad, but when you add the third, things get very dicey, and more serious compromises have to be made or the third octave notes will just be too horrible to ever use.

Finally, the tone holes need to be where the fingers can reach them, which is really not the best place for them to go, so they wind up being different sizes, which creates yet another set of problems that have to be worked around.

So the tuning of a simple-system flute isn't really any traditional tuning, per se.

Some makers tune closer to just, others closer to equal, yet others closer to some of the older, more antiquated compromise tunings like "quarter-comma meantone," which is a scheme that allows you to keep your open thirds, which is another open interval in just tuning.

Some folks are surprised by that, because the major third in equal temper (like on a piano or Boehm-system flute) is anything but beatless. In fact, in equal temper the third is very sharp. Pull it down a bit and it'll get more and more open until it's finally beatless.

Hmmm...the bell note on a flute is D and the F-sharp is one major third up from there. Maybe tuning that F-sharp "flat" compared to a Boehm-system flute or a piano isn't a mistake after all, hmmm? :wink:

These are deep waters, and I've only gone just slightly past the surface in this long post. There is a lot more to it.

But it's pretty cool stuff to know, because it explains a lot of the "whys"--why a flute is tuned like it is, why on a Baroque flute B-flat and A-sharp aren't fingered the same, why it just may be a bad thing that on the Boehm-system flute every key sounds like every other key.

Fascinating stuff. :) And lots of lovely math!!! :twisted: :party:

--James

Brilliant post, James!!! To be honest, I've been playing flute(s) since 1962, and I am absolutely sincere in saying that I have never heard and/or seen a better dissertation, absolutely brilliant.

My hat goes off, to you.

:-)

[I.D.10-t]

For a time I was taking lessons from a baroque piccolo player. He was reluctant to teach an instrument that he did not understand (the fife). Later I gave him a CD of fife music. His comment later in a lessen was "They are in tune with each other, now play like a fiffer!"

It seems to me, if you play in tune constantly with your surroundings, it will benefit you and the listener.

[peeplj]

Brill post, James! I've read about this stuff before - Rockstro, Quantz et al; I know about it. Now I understand it rather better. (The maths is beyond me, though! ) Thanks.
I like your point on F#! Of course, flutes only play one note at a time, so the tuning criteria/requirements are not the same as for chord-playing instruments.

Thanks! Although I hope you got the part that what I wrote is way oversimplified for the sake of keeping the post "short."

What you wrote about flutes playing one note at a time is also true, but flutes routinely play alongside other instruments like boxes. You don't want to hear a flute out of tune with a box...it'll make your teeth grind.

Also, in other styles of music, flutes routinely play in harmony together, so the inverval tunings still need to be reasonably close to true.

--James

[Dana]

When playing in an ensemble, any good player can and must markedly adjust each note to be "in tune". So however your instrument's scale has been tuned, much of intonation is the player's responsibility. (except for certain string instruments that aren't quickly tunable like piano).

I play in orchestras. There's a real art to getting woodwind chords tuned, and it's one of the major challenges. Each member is constantly aware of pitch, and where his/her note needs to fit in. You can't base your note placement on a tuner, because it will often sound "out of tune" with everyone else. Some of this pitch adjustment is based on knowledge of the chord, but for me, most of it is instinct - getting the sound and feel just right.

Then there are the issues of tuning to the strings...

Dana

[tin tin]

I can't get it to open at the moment...but the article "Playing In Tune on a Baroque Flute" is a very interesting read about temperament/tuning issues.
http://traverso.baroqueflute.com/

[Jack Bradshaw]

It's at:

http://www.flutehistory.com/Resources/D ... RAV101.pdf

http://www.flutehistory.com/Resources/D ... RAV101.pdf

[david_h]

I can't get it to open at the moment...but the article "Playing In Tune on a Baroque Flute" is a very interesting read about temperament/tuning issues.
http://traverso.baroqueflute.com/

Try here for (I guess) the same thing.
http://www.flutehistory.com/Resources/D ... RAV101.pdf
Fascinating.

Edit - Crossed post - should have posted before I got engrossed in the article !

[rama]

hey james nice job! you pulled together alot of important stuff, seems to explain alot that we experience in our flute journeys. i learned alot - thanks.

[Jumbuk]

Although it is contrary to what a musically sensitive person hears, to simplify the tuning of instruments (especially keyboard instruments) it was decided to make the interval between these twelve half-tone notes exactly the same size...

But I think the above quote is sad. Another example of how we bowed down to the electronic gadgetry that's taking over our lives.

I think that is being a bit harsh on Bach et al. Equal temperament scales were a prerequisite to being able to modulate from key to key, and the whole edifice of western classical music (not to mention jazz, pop etc) is built on that foundation.

An interesting point is that modern electronic keyboards have actually given us more, not less flexibility. Many keyboards allow the selection of different scale types (just, mean, pythagorean, quarter tone etc).

Even though we tend to think of celtic dance music as essentially diatonic (ie using the same pitch set throughout), in fact many tunes modulate between major keys. This is particularly the case with hornpipes, which often modulate into the dominant key.

What would also be interesting is to do some measurement of say a vocalist or a solo flute or violin playing a straight air to see whether the natural tendency is to play just or equal-tempered intervals when there is no context of a harmonic accompaniment.

[Denny]

now Jumbuk, I think you're overlooking Bach's love of his electronic gadgetry.

[Jumbuk]

now Jumbuk, I think you're overlooking Bach's love of his electronic gadgetry.

Yes, I forgot about that photo on the cover of "Switched on Bach"

[Denny]

now Jumbuk, I think you're overlooking Bach's love of his electronic gadgetry.

Yes, I forgot about that photo on the cover of "Switched on Bach"

It was a pretty big seller for him ya know.

[tin tin]

Thanks for finding the article, Jack and David.
Now, any good ideas for drone/tone generators against which to practice pure intervals?