djm, in your calculation, you asked if going the other direction would work. Yes, but
you meant G#, not G.
440
divided by 1.0594631 = 415.305 c.p.s. for G#.
Joseph, that's why I consider all this fuss about Just vs. Equal to be a little silly, at least for the UP. Other variances have more of an affect than the difference in temperaments...like weather, reed scrape, staples, cane sources, bag preassure, bore design, fingering alternative (closed/open holes), etc., etc. Imagine counting 1 beat in 5 seconds vs. O beats in 5 seconds. Eventually there's going to be some beat...even where you think its been tuned Just. Pressure will do it. That what you look for in Equal 5ths vs. Just 5ths. 4ths are a little more, not much.
Just to eleaberate a little, there's several ways to tune an octave (the 12 semitones) on any fixed chromatic instrument. On stringed instruments you can usually hit 2 or 3 notes at the same time (more with piano). On flute type instruments, where you can only hit one note at a time, you'd space and size the holes to a tuner, or to another fixed instrument like a piano. You can use a machine that has already been calibrated, or tune by ear, listening to the natural intonation and the beats. When setting the octave scale on a piano, I use the F to F method around middle C. Some use A to A.
Whatever, there are a series of 3rd, 4ths, and 5ths in this octave. This is called the Temperament Octave. Starting with middle C, I could tune it to a fork or a machine. After that, I use a series of 5ths and 4ths, back and forth, all the way up the octave. So the lower F tunes to middle C to, then G to C, then D to G, then A to D, then E to A, then the B to E, then the low F to the high F. At this point I cheat and tune the Bb to the high F since it's an easy 5th. This completes all the white keys, and one black. On to the rest of the black keys. The white B has already been tuned, so the F# is tuned to the B, then the C# to the F#. The G# is tuned to the C# and the Eb tuned to the G#.
Now the big momet!!!! The Bb has already been tuned to the high F...it can't be changed. The moment of truth has arrived. I hit the Eb to the Bb. It had better be right using Equal. If it isn't right (sounds sour or painful), the tuner must start all over. The most common problem why it doesn't come out right is because the tuner tuned the 4ths and 5ths too pure.
The system of tuning 12 semitones jumps back and forth between 4ths and 5ths. They are used because they are the purist and are the notes with the widest spread in the octave...where the ear hears tuning problems easiest. The 6ths are related to the 3rds. A 3rd above is a 6th below...same note only an octove lower/higher. Like the 3rds, 6ths are the most impure in the Equal Temperament Octave Scale. The 7ths just aren't used in tuning, nor are the 2nds.
Guess what? If I had tuned every 4th and 5th pure, with no beats nor compomise, the last interval of Bb to Eb would have been off by 24 cents. That ΒΌ of a semitone!!! Best to have every 4th and 5th off a little than to have this one off a huge amount. It's this way with any chromatic fixed instrument. And...this 24 cents...that's why the 12 semitones are off by about 2 cents. 2x12=24.
But, in another system of tuning, I could have started out with the low F and tuned the octave in pure 3rds (major and minor 3rds). After completing all these intervals up through the octave, the last one would come out 42 cents off. There is no way to tune a fixed chromatic instrument, with 12 semitones, strickly to Just.
It can't be done! Even with a flute, oboe, sax,
or even a chromatic uilleann pipe chanter -- that would be a lot of key levers!! Tuning the last interval on any of these woodwinds would present the same problem. With chromatic, every tonehole has to be adjusted a little to sound better against a growing number of notes. But a whistle, with only six toneholes can be pure like the diatonic accordian. The other notes are corrected by alternate fingerings. I has only a few fixed notes that have to sound good against each other. The F# is the biggest problem. It's a 3rd up from D. It's off the most. It also is subject to greater pressure differences. Either the upper or lower F# sounds best, usually not both. Any 3rd is off the most, even on a Equal tuned piano or guitar. The less strings, like a violin, the less you have to worry. The other notes get corrected by a fretless board. Not so with guitar (six strings and fixed frets).