I see your point, but there’s almost a paradox here, since the just intonation ratios are derived from the series of harmonics. My gut feeling is that using these ratios for only a couple of octaves would not be far enough “out of tune” to be a problem.
Sturob, et. al,
Again, looking at the intervals of just intonation and forgetting the cycle of fifths entirely, no matter how many octaves you span, in just intonation every D will be consonant with every other D, every E will be consonant with every other E, etc.
The math indicates that the octave relationships will be preserved no matter how high or low you go. If there’s some other dynamic that throws that off, I’m not yet sophisticated enough to understand it (though perhaps you’ll be able to make it clearer to me).
The cycle of fifths logic doesn’t apply to just intonation because it makes no attempt to create perfect fifth relationships between notes other than the tonic and the fifth above the tonic.
For example, the fifth is 3/2 of the tonic. A fifth above the second will be 27/16 of the tonic. However, the sixth above the tonic in just intonation is 5/3 of the tonic. So a fifth above the second isn’t the sixth. They’re close but not the same, and the comma doesn’t come into play because nothing’s pushing the octaves apart.
What comes to mind is that, because it’s an instrument with more or less fixed tones, there are compromises that will show up when one attempts to play chords (as in our whistle consort) or intervals built from other notes than the tonic.
However, I can’t imagine that the compromises would be greater than those already made in equal tempered music, and at least there would be some perfect harmonies instead of none at all (except octaves) as in equal tempered music.
It does sound as though the choir or string scenario, where every chord or interval can be built from scratch, allowing every tone to shift subtly to allow for perfect harmonies (ideally) would be the best. But we’re only working with a bit of pipe with six holes, after all.
Best wishes,
Jerry
Just call me Belabor-the-Point-Boy.
The only thing that’s confusing the issue now, as I see it, is that the “scientifically” perfect intervals are not in tune.
There’s actually a ton written about this. You CAN do the math and even do the intervals for octaves up and down. Yes, you just do something like A(n)=440*2^(n), and you get intervals that OUGHT to be consonant but aren’t. For whatever reason (that’s not been explained), other than the fact that we’re dealing with human hearing and not with math, the mathematically perfect intervals, even for the octaves, very quickly do not SOUND right. That’s where the whole “comma” phenomenon comes from.
I can’t explain it. Maybe for the first octave it makes no difference. But, depending on how good your hearing is, it will make a difference quickly.
So maybe in a whistle it doesn’t matter. Renaissance instruments, such as consort instruments (recorder, crumhorn, etc.) tend to be tuned to meantone temperament . . . which gives you the wolf tones (sic.) mentioned in earlier messages. There are so many different tuning systems it’s not even funny . . .
For solo play, it doesn’t matter at all. Solo or unison. It’s when you start trying to do harmony that things don’t work. Or, when you play with an instrument that’s tempered differently. Fact of the matter is, regardless of what the fiddler thinks, s/he should be able to adjust to whistle temperament, or flute temperament without difficulty. It may not in a pure sense sound right, but s/he could then get his/her instrument in tune with the other person. It’s when you put several temperaments (pipes + flute + fiddle, for example) that things can get nasty.
Stuart
Let’s see if I understand you.
If I get what you’re saying, if you tune a series of A’s:
55, 110, 220, 440, 880, 1760
The adjacent octaves will sound OK, but the further apart they get, they’ll sound out of tune?
How would they be tuned to sound OK?
If you tune them to sound OK, would there be no “beats” when you play the two notes together? And how can there be no beats if the sound waves aren’t exactly in phase? Etc.
Interesting discussion.
Best wishes,
Jerry
Those are the right questions, of course!
By sounding “right” I mean minimizing (or obliterating) the beats, you are quite correct. And it seems strange to me that the waves would have to be a little out-of-phase to “sound” right.
I think, if memory serves, that you have to get relatively sharper as you go up and flatter as you go down. Things shift up slightly. By what degree, though, meaning how many cycles, I can’t tell you off the top of my head. I’ll have to look it up and get back to you. It may be fractions of a cycle, but I think as you get further and further up (or down) the correction becomes even greater.
The other extremely important information I can’t recall is whether or not these distinctions are purely human-hearing-related or if they’re easily analyzed. Stupid, I know, but there are a lot of things I can’t remember.
Now I have homework!
Stuart
Uilleann pipe drones give you an insight into all of this.
Some of the chanter notes do not harmonise with the drones if the chanter is in tune with an electronic tuner. [or vice versa…or if drones are all tuned into an elec. tuner, the chanter may seem to be out of tune].
The tuner is a handy gadget, but will not take you to musical paradise if it’s Irish Trad that you play!! :roll: 
Boyd,
For clarity, do I assume correctly that you mean the chanter would have been tuned to an electric tuner’s settings for just intonation (not equal temperament), but would still sound out of tune to the drone? In what way would it be out of tune and what adjustments would make it sound correct?
Best wishes,
Jerry
Dear Folks,
What needs to be added here, of course, is that dissonance (a note relationship which isn’t quite right, or is even jarringly not right) has a powerful role in all music. If the “wrong” notes are played judiciously, they highlight how “right” the next note is.
When I fit the above discussions into my basic piano tuning training, it’s interesting how much fudging is involved in ear tuning. I learned a disdain for any tuning machine, and to set a temperence octave using A440 as its standard. When we followed the “circle of fifths” to set that all-important first octave, we fudged the fifths royally in order to equalize the intonation. It was explained to me that, if you tuned perfect, beatless, fifths, one of the keys used in composition would be rendered so out-of-tune as to be nearly obscene (seems to me I was told it used to be called the “devil’s key”).
However, when tuning octaves above and below the temperence octave on an instrument as widely ranging as a piano, one thing is easily noticed: the more extreme the note, the more difficult the dissonance is to hear. As much as two octaves higher or lower than A440–and definitely in the greater extremities than those–the beats become extremely difficult for even a professional tuner to distinguish.
With this in mind, I wonder if worrying about the “commas” building up between octaves to the point of unplayability is just an exercise in academics?
Additionally, in ITM, are not harmonizations kept to a minimum? With the exception of a guitar accompaniment (which can be tuned to JI) and the occasional drone, the point may become a bit moot.
Awaiting constructive destruction,
David
Whether or not we come to any consensus or well-defined conclusions, this discussion has been a wonderful exercise for me. I very much appreciate everyone’s willingness to participate.
Best wishes,
Jerry
P.S. If you’re interested in scientific advancements, you might want to visit this thread where, for the first time, I have published my spore-germination theory of cat origination:
http://chiffboard.mati.ca/viewtopic.php?t=10737&start=15