Just intonation intervals really sweeter than 12et?

[Lorenzo]

Talking of which, Lorenzo I don't understand your apparent problem with moving your finger to obtain a slightly different tuning on a note on the fiddle deliberately.

Oh, is that what you meant! I can do that, I just can't place my finger in the correct second postion on the A string and reach over with the other hand and tune the note to the B on the piano!

But, you probably know that if your bridge is set too high...the note will go sharp, even with the finger in the correct position, when you stretch the string down to the fingerboard.

[Lorenzo]

I've never read anything indicating that equal temperament was developed so the music would sound better. Only that it would work at all to be able to modulate across various chords and keys on a single, fixed tuning instrument.

It's probably just an accident that ET gives the mid-range major 3rd the same kind of tremolo aeolian effect you hear in some voices...which some people, such as my uncle, also think is beautiful.

Isn't the whole idea of equal temperament that performers and audiences were willing to accept some compromise in the quality of the tonality for the benefit of greater flexibility?

Partly, but of course no one could accept pure 3rds throughout the temperament octve, because of the last interval. Even choirs have to adjust away from Just at this point because a ¼ of a whole tone is too far off for any human ear to accept.

[Jerry Freeman]

Isn't the whole idea of equal temperament that performers and audiences were willing to accept some compromise in the quality of the tonality for the benefit of greater flexibility?

Partly, but of course no one could accept pure 3rds throughout the temperament octve, because of the last interval. Even choirs have to adjust away from Just at this point because a ¼ of a whole tone is too far off for any human ear to accept.

This makes my brain hurt.

I'm pretty sure you've got something seriously screwed up here, but I don't have the patience or energy to figure out where you've gotten a principle from piano tuning confused with how an ensemble with infinitely adjustable pitches at its disposal functions. Can anyone help sort this out?

Best wishes,
Jerry

[StevieJ]

It sounds as though Lorenzo is conceiving of an octave constructed by adding intervals together. Whereas I've always assumed that, in both theory and in the human brain, the octave comes first. It is _divided_ into intervals. The whole question is exactly how you divvy it up. No?

[Jerry Freeman]

Here's a just intonation scale:

Tonic = 1/1
2 = 9/8
3 = 5/4
4 = 4/3
5 = 3/2
6 = 5/3
7 =15/8
Octave = 2/1

Here's how this just intonation scale
compares to an equally tempered scale:

tonic 0 cents
2 +4 cents
3 -14 cents
4 -2 cents
5 +2 cents
6 -16 cents
7 -12 cents
octave 0 cents

There's no 1/4 of a whole tone comma, nothing to compensate for, that I can see here. This scale is not built by stacking perfect thirds and then adjusting for the fact that they don't add up.

Best wishes,
Jerry

[Lorenzo]

I'm pretty sure you've got something seriously screwed up here, but I don't have the patience or energy to figure out where you've gotten a principle from piano tuning confused with how an ensemble with infinitely adjustable pitches at its disposal functions. Can anyone help sort this out?

You've already nailed it. You say the ensemble adjusts pitches...and that's exatly what they'd have to do to keep the upper octave in tune with the lower, otherwise they go off the map. By singing in more than one octave, they've adjusted to ET. (if you don't temper the first octave the second octve will go way sharp)

Have your ensemble try this experiment (this will take six people): starting in the 4th octave up from the bottom of a piano, have each person hum one of these notes. F4, G#4, B4, D4, F4 / G#5. Well, if Just Intonation were maintained throughout this series of minor 3rds, this 6-note chord would be a dischord. The upper G# would end up about ¼ step sharper than the lower G#. These notes all played (or hummed) together at the same time, in ET, make a wonderful chord, a diminished chord.

To get an idea of what this chord would sound like without tempering each inerval -- on the piano play these six notes at the same time: F4, G#4, B4, D4, F4 / A5. Sounds horrible right? (only slightly exagerated) That's because the upper A5 is a ½ step higher than it should be. It should be a G#. Well, the right note in this series of minor 3rds should G# but in Just intonation it ends up ¼ step high (rather than the ½ step we tried -- but still just as sour). So, we know the ensmble has tempered their intervals somewhere between the first and second octave in order to keep the wolf from howling.

[Jerry Freeman]

Lorenzo,

Is this a chord that someone would actually use in a choral piece? It looks like an intellectual trick, rather than something one would expect to encounter in real life.

I think you're fixated on the idea that you build scales by starting somewhere and then stacking perfect intervals until they approximate notes you want to use in various keys, then making adjustments away from the perfect, stacked intervals so the notes are exactly an octave apart (plus the octave stretch where needed, which is a separate issue).

That's how you tune a piano, but it's not how you construct a scale when you're not constrained to a fixed tuning instrument.

You can keep your wolf notes. They're an artifact of the interval stacking to construct scales approach, and they're not a feature of just intonation. Seriously, you're not understanding how this works.

Best wishes,
Jerry

[Lorenzo]

It's not that difficult. As I've said before, you can tune a piano (or sing in a choir) in Just Intonation if you stay within the confines of the first 11 semitones and limit yourself to certain keys and certain chords. But if you want to go beyond that, into the next octave, or other chords, something has to give somewhere. The experiment I gave is only done to demonstrate the problem of 3rds, Just 3rds vs. tempered 3rds, because 3rds are the main component of an ensemble.

[Jerry Freeman]

It's not that difficult. As I've said before, you can tune a piano (or sing in a choir) in Just Intonation is you stay within the confines of the first 11 semitones and limit yourself to certain keys and certain chords. But if you want to go beyond that, into the next octave, or other chords, something has to give somewhere. The experiment I gave is only done to demonstrate the problem of 3rds, Just 3rds vs. tempered 3rds.

Lorenzo, I think the problem may be that you're assuming one constructs chords by building on the previous note in the chord. So you would have us start with the base note of the chord, then go up a third, then go up third from that third, etc.

However, this isn't how chords are built.

In reality, every note of the chord will be derived from the base note. If you want to build a major triad, you'll start with the base note, then add the perfect third from the base note, than add the perfect fifth from the base note, which will be different from the perfect third from the perfect third from the base note. When you construct your chords this way, deriving every note in the chord from the base note, you can go from octave to octave with no dissonance, no wolf notes, no adjustments.

Using the just, whole number intervals, a major third is 5/4 the tonic frequency. A major fifth is 3/2 the tonic frequency.

If you try to arrive at a major fifth by taking a major third from the tonic and then taking a major third from the major third, you'll end up with 25/16 of the tonic frequency. That will still be a just intonation interval, but it isn't a major fifth. It would be applicable to some kind of microtonal scale, but not anything that would sound familiar. The correctly derived, just intoned note will be 1.5 of the tonic frequency, but the note derived by your method will be 1.5625 of the tonic frequency.

Best wishes,
Jerry

[BoneQuint]

It's not that difficult. As I've said before, you can tune a piano (or sing in a choir) in Just Intonation if you stay within the confines of the first 11 semitones and limit yourself to certain keys and certain chords. But if you want to go beyond that, into the next octave, or other chords, something has to give somewhere.

But of course that doesn't mean you have to go to equal tempering, or any other fixed tempering system. It means you're choosing which notes to "lock into." You're singing in a just scale in the context of what's going on around you, with different singers locking into different things. I don't sing in choir, but I imagine it's a pretty intuitive, subconscious process, depending on what's "musically important" at that time in relation to your note. It would be very difficult to make a set of rules to define what "sounds good" (although it might be very interesting to try). Which is why I wrote earlier in this thread:

I think good musicians get to be good at figuring out by instinct which notes most need to be "tweaked" to a just interval, and in relationship to what other notes. Computers are trying to adjust ALL the intervals, and you can't do that in most music. Maybe programmers haven't yet hit on an algorithm that prioritizes as well as a well-trained human.

Edit: Jerry replied above while I was typing, we're saying pretty much the same thing, although at times there may not be a single root to a chord to lock into, in which case that's where the "musical intuition" comes in.

[Lorenzo]

As long as an ensemble stays with the four regular notes in a chord, not six etc., in different octaves, it can move together in JI from one octave to another, ever adjusting (a piano can't do this), but if it is hitting these four notes in two or three separate octaves, and is still Just, it would be out of tune with itself somewhere. It would have to temper some notes to sound good, which it probably does. The fact that any of these four notes in different octaves remain in tune with a piano, more or less (rather than ¼ or ½ step out of tune) shows the ensemble has adjusted to the piano--ET.

Try it. Have four different people hit the four separate notes of a chord in two or three different octaves (like F3, A4, C5, F5) and check it with your ear, then with a piano that's been tuned by a good technician. It's hard to demonstrate without an instrument or a good tuning machine that flashes the beats.

[Jerry Freeman]

As long as an ensemble stays with the four regular notes in a chord, not six etc., in different octaves, it can move together in JI from one octave to another, ever adjusting (a piano can't do this), but if it is hitting these four notes in two or three separate octaves, and is still Just, it would be out of tune with itself somewhere. It would have to temper some notes to sound good, which it probably does. The fact that any of these four notes in different octaves remain in tune with a piano, more or less (rather than ¼ or ½ step out of tune) shows the ensemble has adjusted to the piano--ET.

Try it. Have four different people hit the four separate notes of a chord in two or three different octaves (like F3, A4, C5, F5) and check it with your ear, then with a piano that's been tuned by a good technician. It's hard to demonstrate without an instrument or a good tuning machine that flashes the beats.

I think you're still not getting it.

Please provide an example of a chord with four notes, all derived from the base note of the chord. From there, it should be simple to show that any note in that chord will be 1/2 the frequency when struck one octave lower and twice the frequency when struck one octave higher. There will be no need to temper the pitch. (There may be some octave stretch, but this is a different issue and not relevant to this discussion.)

However, as long as you continue to use the piano tuning approach to approximating pitches by stacking intervals on top of intervals, you'll run into trouble when you cross over from octave to octave.

Best wishes,
Jerry

[Lorenzo]

It's easy for a trio to keep their triads pure as long as they all stay within an octave of each other, up or down the scale. A choir would find that easy too. The bass note can always adjust no matter what octave it needs to be in.

However, as long as you continue to use the piano tuning approach to approximating pitches by stacking intervals on top of intervals, you'll run into trouble when you cross over from octave to octave.

It's been tried. I was recently reading about an orchestra where the bass instruments tuned to JI, and the highest instruments tuned to JI also. When the high instruments played alone with the bass instruments, they were way off from each other, like a whole step. It doesn't really matter if the instrument is a piano, human voices, or a variety of instruments. It's important that 3rd intervals are kept pure all the way up and down through four or five octaves, using the 12 semitones chromatic scale, if one claims to be singing or playing in JI. Of course not all the 4ths and 5ths will be pure then.

[Jerry Freeman]

OK, here we go, leaving out octave stretching (see quote below).

F3, A4, C5, F5.

F3 = 174.614

A perfect major third above F3 = 218.2675 times two = 436.535 (not 440) which is the just pitch based on F3 that's closest to A4

A perfect fifth above F3 = 261.921 times two = 523.842 (not 523.251) which is the just pitch based on F3 that's closest to C5

F5 = 698.456 (this is exactly four times F3)

All the relationships are intact.

The just pitch corresponding to your major third plus an octave is exactly twice the major third in the same octave as the base note, or 5/4 the base frequency times two.

The just pitch corresponding to your fifth plus an octave is exactly twice the fifth in the same octave as the base note, or 3/2 the base frequency times two.

The note two octaves above the base note is four times the base frequency.

Again, the problem only arises if you try to derive pitches from other pitches than the base frequency. I realize that's how you tune a piano, but it's not how chords are performed in real life. There's no problem when the bass section sings a pitch in a different octave from the sopranos to create a chord. It's easy, and there's no adjustment to be made.

The reason it's easy is that there's a PHYSICAL relationship between the pitches. You can literally feel it in your throat when you're spot on pitch and the sound waves of your own voice are in phase with those of the others singing the complementary notes of the chord. And this is true all the way across the range from bass to soprano, across octaves.

http://en.wikipedia.org/wiki/Piano_key_frequencies

This is a virtual piano with 88 keys tuned to A440, showing the frequencies, in cycles per second (Hz), of each note (i.e. Note frequencies of each note found on a standard piano). This distribution of frequencies is known as equal temperament, i.e. each successive pitch is derived by multiplying the previous by the twelfth root of two. For other tuning schemes refer to Musical tuning.

This list of frequencies is for a theoretical ideal piano. On an actual piano the ratio between semitones becomes slightly larger due to string thickness which causes inharmonicity due to the nonzero force required to bend steel piano wire even in the absence of tension. This effect is sometimes known as stretched octaves, and the pattern of deviation is called the Railsback curve.

Best wishes,
Jerry

[Jerry Freeman]

I was recently reading about an orchestra where the bass instruments tuned to JI, and the highest instruments tuned to JI also. When the high instruments played alone with the bass instruments, they were way off from each other, like a whole step.

This doesn't make sense.

It would happen with equal temperament too, then. An equal octave is the same as a just octave. If the distance between octaves is the issue, then the problem is octave stretch. If you're trying to argue that an adjustment needs to be made to accomodate octave stretch, well and good, but it has nothing to do with the relationship between just and equal tuning.

Best wishes,
Jerry