Thanks, Ridseard.
What about this comma issue with more than one octave in just intonation?
My understanding has been that you can maintain just intonation across as many octaves as you like and it will always work, but Sturob has a different point of view.
Can you shed some light?
Best wishes,
Jerry
Whistles Made With "Just Intonation"?
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Jerry Freeman
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StevieJ
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There will be always be a comma if you follow the cycle of fifths. This is completely irrespective of key. There is, in fact, is a divergence between the perfect mathematical relationship of the octave (2:1) and the fifth (3:2). What I call God's mistake!Jerry Freeman wrote:Without looking into it further, it's been my impression that the comma pertains to what happens when you try to apply a non-tempered or other than equal tempered system to more than one key at a time, following the cycle of fifths. I was not aware that there would be any error, or comma, if you stayed in a single key using just intonation.
It only becomes noticeable over a span of several octaves, so on a whistle it wouldn't matter. But if you formed a whistle consort (heaven forbid) with say contrabasses and sopraninos, and you tuned them to each other from bottom to top in just intonation, you'd surely run into a problem.
Perfect fifths and octaves don't get on in the long run, or over the big span if you like. And since we can't stand octaves being out of tune, the fifths have to be fudged... An example - the strings of the violin family are tuned in perfect fifths. In a quartet, everyone tunes to the leader's (first violin's) A. But, I am informed by a skilled quartet cellist (my sister) the cello's bottom string has to be fudged - tuned up by almost a quarter tone - to ensure correct intonation with the violins.
It's all very weird I know but remember that clever people have been looking for a solution since Pythagoras...
I think the question of temperament for whistlers is like hyphens for editors. If you take it seriously, you will quickly go insane.
Have a look at this page:
http://www.rdrop.com/~tblackb/music/temperament/ for a balanced view of the subject.
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Jerry Freeman
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StevieJ,
Please read carefully. The cycle of fifths is irrelevant to what I'm talking about because I'm talking about an instrument that's only played in one key.
This whole idea of a cycle of fifths where you can use the same instrument and the same sharps and flats across various keys is a fairly modern scheme involving compromises that don't come into play at all with an instrument that's only played in one key.
It's my understanding that there's never a comma with a just intonation instrument that's only going to be played in one key, regardless of how many octaves the instrument spans.
Best wishes,
Jerry
[edit] I've gone to the link you've suggested, and I believe I see the problem. First, there are two ways the cycle of fifths is used. One, to describe the relationship between various keys as you add sharps or flats; and two, to attempt to produce a single scale based on notes related by intervals of pure fifths.
The assumption inherent in the comma debate as it pertains to a single scale, is that the only way to produce a scale that's based on harmonic relationships is to use intervals of pure fifths.
Just intonation doesn't try to do that at all, so there's no comma. As far as I'm concerned, there's no problem to be solved and no danger of driving myself crazy trying to solve anything.
Just intonation has both a third and a fifth that are in perfect harmony with the tonic, and all the other notes are in exact mathematical relationship with the tonic as well.
To my ear, that seems a much more logical way to tune an instrument that's only going to be played in one key. That, incidentally, is how I was taught to sing, by a choir master who was also an accomplished violinist and understood the difference between intervals and chords with true harmonies, and the approximations and resultant lack of true harmonies created by equal temperament.
Please read carefully. The cycle of fifths is irrelevant to what I'm talking about because I'm talking about an instrument that's only played in one key.
This whole idea of a cycle of fifths where you can use the same instrument and the same sharps and flats across various keys is a fairly modern scheme involving compromises that don't come into play at all with an instrument that's only played in one key.
It's my understanding that there's never a comma with a just intonation instrument that's only going to be played in one key, regardless of how many octaves the instrument spans.
Best wishes,
Jerry
[edit] I've gone to the link you've suggested, and I believe I see the problem. First, there are two ways the cycle of fifths is used. One, to describe the relationship between various keys as you add sharps or flats; and two, to attempt to produce a single scale based on notes related by intervals of pure fifths.
The assumption inherent in the comma debate as it pertains to a single scale, is that the only way to produce a scale that's based on harmonic relationships is to use intervals of pure fifths.
Just intonation doesn't try to do that at all, so there's no comma. As far as I'm concerned, there's no problem to be solved and no danger of driving myself crazy trying to solve anything.
Just intonation has both a third and a fifth that are in perfect harmony with the tonic, and all the other notes are in exact mathematical relationship with the tonic as well.
To my ear, that seems a much more logical way to tune an instrument that's only going to be played in one key. That, incidentally, is how I was taught to sing, by a choir master who was also an accomplished violinist and understood the difference between intervals and chords with true harmonies, and the approximations and resultant lack of true harmonies created by equal temperament.
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Ridseard
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Here's the problem. Let's say you start with D and construct the circle of fifths until you get to D again. The problem is that with just intonation the frequency ratio between fifths is 3/2. If you go around the circle multiplying by 3/2 for each fifth, when you have gone full circle and reach D again, it will be some power of 3/2 times the frequency of the original D, but it wil be out of tune. Mathematically this is because the ratio between different octaves of the same note is some integer power of 2, but the integer powers of 3/2 can never be equal to an integer power of 2 (basically because 3 and 2 are prime numbers).
As long as you stick to octaves and don't worry about the circle of fifths, there's no theoretical problem, just an esthetic problem.
I can't help but draw an interesting mathematical oddity from all this. If the circle of fifths worked the way it "should" work, the prime factorization theorem would be violated, and practically all of number theory would be destroyed. This in turn would invade and invalidate almost all of mathematics.
As long as you stick to octaves and don't worry about the circle of fifths, there's no theoretical problem, just an esthetic problem.
I can't help but draw an interesting mathematical oddity from all this. If the circle of fifths worked the way it "should" work, the prime factorization theorem would be violated, and practically all of number theory would be destroyed. This in turn would invade and invalidate almost all of mathematics.
Last edited by Ridseard on Sat Mar 22, 2003 11:46 am, edited 1 time in total.
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sturob
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Jerry,
This is a great discussion! Wow!
The cycle of fifths doesn't work for the octave; that's the comma/'God's Mistake' to which we've alluded. Even an instrument which can "only" play in one key (discounting all harmonic minors, blah blah blah) would need tempering if you got to a second octave, because the pure intervals are out of tune with themselves. I'm trying to find a good reference for this.
You mention the fact that interval tuning seems, "to your ear," to be a much more logical way to tune an instrument that's only going to play in one key . . . and you mentioned choir. With voice, you bring up an interesting phenomenon. Again, I'll find you a good reference if you're interested. Voice is unlike violin, even, in that an a capella choir is truly infintely tempered, and will sing intervals which don't fit into ANY described tuning system. String players, like let's say a group of violins, violas, and cellos, will play in tune with themselves and play at pitch (until their instruments start to detune, or something weird). Voices a capella tend to go flat because of the way the human ear adjusts the voice to be in tune. Choirs really only go sharp when they're very excited AND some lead-type person goes sharp and everyone else adjusts.
Also, the point (to my mind) of the idea of comma is that the purely mathematical intervals are not in tune. Sure, you can tune an instrument to a purely mathematical interval system, but it won't be in tune with itself. At the octaves, in particular. You can't even go a to A to A' with the "perfect intervals" and have them all be consonant.
Stuart
This is a great discussion! Wow!
The cycle of fifths doesn't work for the octave; that's the comma/'God's Mistake' to which we've alluded. Even an instrument which can "only" play in one key (discounting all harmonic minors, blah blah blah) would need tempering if you got to a second octave, because the pure intervals are out of tune with themselves. I'm trying to find a good reference for this.
You mention the fact that interval tuning seems, "to your ear," to be a much more logical way to tune an instrument that's only going to play in one key . . . and you mentioned choir. With voice, you bring up an interesting phenomenon. Again, I'll find you a good reference if you're interested. Voice is unlike violin, even, in that an a capella choir is truly infintely tempered, and will sing intervals which don't fit into ANY described tuning system. String players, like let's say a group of violins, violas, and cellos, will play in tune with themselves and play at pitch (until their instruments start to detune, or something weird). Voices a capella tend to go flat because of the way the human ear adjusts the voice to be in tune. Choirs really only go sharp when they're very excited AND some lead-type person goes sharp and everyone else adjusts.
Also, the point (to my mind) of the idea of comma is that the purely mathematical intervals are not in tune. Sure, you can tune an instrument to a purely mathematical interval system, but it won't be in tune with itself. At the octaves, in particular. You can't even go a to A to A' with the "perfect intervals" and have them all be consonant.
Stuart
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Ridseard
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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 wrote:Also, the point (to my mind) of the idea of comma is that the purely mathematical intervals are not in tune.
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Jerry Freeman
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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
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
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sturob
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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
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
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Jerry Freeman
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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
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
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sturob
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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
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
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boyd
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- Tell us something.: Sets in D and B by Rogge and flute by Olwell, whistles by Burke and Goldie. I have been a member for a very long time here. Thanks for reading.
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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
www.strathspeyinmay.com
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
www.strathspeyinmay.com
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Jerry Freeman
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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
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
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dabooks
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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
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
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Jerry Freeman
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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 ... 7&start=15
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 ... 7&start=15