Let’s get to the bottom of this.
Well, it’s my education, too, please!
I think that there has been a fairly good discusion of this at the thread, 8-key vs. Boehm, at least, I learned quite a bit.
I don’t propose to get to the bottom of the topic, but I have written the following, which is on a page at my website.
A Word About Tuning: In western music (as opposed to world music) the octave is divided into twelve notes or half tones. 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, with each of the half-tone intervals being further broken down into 100 units or cents. Therefore, the octave is divided into 1200 cents, with half-tone intervals being 100 cents and whole-tone intervals being 200 cents. This system of intonation is called equal-temperment, and it is the standard for all electronic tuners, except for the more expensive tuners that allow tuning to the exact frequency.
Human beings have an innate sense of relationship, both visually and aurally. For example, without using any measuring devices we are able to divide a pie into eight equal pieces. Similarly, with regard to music we can sense certain basic musical intervals in the octave. It turns out that the basic intervals that we sense are related to rational numbers or fractions. The simplest relationship is the octave interval (twice the frequency) or 2:1 ratio. The perfect fifth interval has a ratio of 3:2, and the fourth interval has a ration of 4:3. As the integers become larger, it becomes more difficult to sense the interval. For example, the interval of a seventh is 15:8. A system of intonation for intervals arrived at in this fashion is called Just intonation. However, with Just intonation the intervals are not all the same size, and in the past this caused many technical problems in tuning, so this system was abandoned in favor of equal-temperment..
Our current equal-temperment system of intonation is obviously convenient, but this convenience has been bought at a price. The price is that the scale doesn’t sound right to a musically sensitive person. For example, it is nearly impossible for a musician to tune a piano, because the “correct” notes don’t sound exactly right. In fact, three of the eight notes in a one-octave diatonic, equal-temperment scale vary between 14 and 19 cents from the same notes in the musician’s scale of Just intonation. Also, as an orchestra tunes by ear to an A440 pitch on the oboe, the string section tunes their instruments by ear (no tuners) by listening for perfect fifths between the strings, and somehow the whole orchestra comes together and sounds in-tune to the listener. But actually there are many variations in pitch that are not perceived by the listener. Because of variations in attack strength and embouchure, the side-blown flute is especially vulnerable to frequency fluctuations, and frequency fluctuations up to 4 per cent have been observed, even among concert musicians. I can play my Irish flute in front of an electronic tuner and have the needle swing from 20 cents below to 20 cents above standard pitch just by the way I blow and the way I shape my lips (embouchure).
And lastly, quoting from the classic book “Music, Sound, and Sensation” by Fritz Winckel: “Although completely pure intonation is most difficult on the flute, it is also of least importance, for the (flute) tone is poor in overtone content, and a slight distuning can have a positive equalizing effect.” In other words it may be a good thing to be slightly off-pitch. And that is my main thesis and purpose for writing the above comments about tuning. I believe that some people have unreasonable expectations about pitch and intonation. They think that their $20 electronic tuner set to equal-temperment is the gold standard and that any variation from “dead-on” pitch accuracy is not acceptable. To the contrary, my attitude with regard to flute performance is to lay perfectionism aside and accept some variations in intonation as inescapable and perhaps even a good thing.
Whaaa…it’s easy …when you are playing with a flute or a fiddle and you match perfect 5ths, 3rds etc…you are playing just…if you are stuck w/ an instrument that is equal tempered you can’t do it…
Edit: Hey Doug…we crossed in posting…good points!
For what you have just said, I am tempted to from here on refer to you as Doc_Tipple. You said it very well!
Just intonation is what one’s ear hears as musical truth, no less than that!
Yes, we agree.
Thanks, Jack.
Doc_Tipple, Jack, and I appear to agree, yet what of the Irish flute?
The “Irish” flute is a little easier to play just, as the Boehm, w/ larger holes, can’t adjust quite as easily…
Of course, if by “Irish” playing, you mean all played monotonically (assuming that’s the right word), it makes no difference…
First of all, brilliant post, Doug. It actually made sense to me.
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.
–Sincerely, a computer working trapped forever behind a screen…
I believe you have just coined a new word, but I am convinced that you know whereof you speak.
But where could the modern Irish flute fit in?
Anybody?
Why oboe?
'cause it doesn’t tune as well as anything else…
Because, because, because, because, becaaause (to the Wizard of Oz tune).
Apparantly, oboes have a very stable pitch response..ie. they probably can’t be retuned…I remember my roommate in the early 60’s, making a xtal controlled oscillator just so he could show his critics in the orchestra that the oboe realy was a pitch standard…
Apparently there is some agreement, that just intonation and equal temperament are not the same, yet, the question remains, as just where the Irish flute could be tuned.
Just intonation, equal temperament, or both?
Well both, of course ! The “Irish” flute, like the fiddle is capable of both…it’s amazing to me that so many flute players have never, NEVER, ever tried pulling the flute into pitch ! I think it’s a result of them being taught that it’s a fixed pitch instrument..ala Boehm…
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
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.
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.
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.
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