Just intonation intervals really sweeter than 12et?

[dwhite]

Thanks for all for your replies! I would still be interested to hear from any JI die-hards if you prefer anything whatsoever about the ET versions. Try the ones on this page to make sure:
http://www.skytopia.com/project/scale/third

I actually heard equal temperament as hopelessly out of tune. It's built in.

I would be grateful if you could listen to the tests shown in the link above to explore different aspects of why you found/find equal temperament to be out of tune.

If you find that the Saw wave tests sound out of tune in ET, then do you find the tests with octave partials also sound out of tune?

I must say, your "The 12 Golden Notes" is one of the best analysis I've seen on the subject. At the bottom of the page you have a survey with examples of Just vs. Equal Temperament. I love that "blind" test re the Triad Options!!!!!!! That's exactly what I've been thinking--that the difference is so small, unless you know which you're hearing in advance you may not be able to tell which is which.

Glad to hear! Yes, you're right - with vibrato, and/or with simpler timbres such as sine waves and octave partials, and at least for me, it's easy to confuse them.

I also like your samples of the fugue played on the harpsichord in JI and ET. You say you "knocked up this tune in about an hour." So, that's that you playing!?

Haha, no I composed the music, but even though I play the piano, it's the computer that's playing the music.

Great web site you've got there Dan! Tell us a little about yourself! Are you student or faculty? If student, is your major music, math, or other?

Yep I'm a student currently in my last at university, and just about to study tuning and temperament even further for my dissertation. Although I took a keen interest in tuning and temperament years before I went.

It's easy to hear when the strings come into tune with each other - the "beats" disappear and there is no clash. This is a perfect fifth, a natural and very sweet interval. Any minute movement of either string away from this interval will produce a "clash" - beats. In equal temperament, perfect fifths have to be sacrificed in order to preserve perfect octaves.

Again, I count the perception of beats to be a different type of consonance to the 12 basic interval types. The first is relating to how harmonic two pitches will sound to form a single timbre. The second type I call 'chromatic consonance', and is related to how 'sweet' intervals will sound. How "chromatically consonant" an interval will sound varies from person to person, from what I've gathered.

I'd be grateful if you could also try the Sample A/Sample B test that Lorenzon gave above and then maybe visit my third comparison page. If you find that the Saw wave tests sound out of tune in ET, then do you find the tests with octave partials also sound out of tune?

My subjective impression is that it's mostly keyboard players who fret (!) about academic questions of intonation.

Being a pianist myself, you could well be right! However, the subject is interesting anyway, because it forms an important root of music, at least its harmony (no pun intended!).

But that "plain" sound is what is nice about just intonation, at least with real instruments, as opposed to synthesized ones used in your test.

I agree, and I also love the droning sound that just intonation provides. However, I am not looking for that kind of consonance (what I call 'harmonic consonance'), but instead for what I call 'chromatic consonance' which is related to how sweet an interval sounds (and this type varies from person to person). I had to make the JI interval sound timbrally/'harmonically' the same as the equal tempered version, so as to make it fair comparison. It's just unlucky that we have separate the phenomena like this.

The synthesized ones are simple and "artificial" enough to sound plain when the harmonies are too pure.

With real instruments, intonational errors and other complicated phenomena start to creep in, so 'synthesized' sounds are necessary to keep the tests as objective as possible. However, I do agree that the just intonational, droning, timbral effects can be even nicer with real instruments, but I'm pretty sure this is mostly thanks to their almost-but-not-quite-spot-on JI intervals, creating phase-like effects. See my new M.third comparison page, and listen to the "Almost Just Intonation" examples.

The other important thing to say is that even with vibrato, the JI sound is still very much in the JI pitch interval range, and far away from ET.

That's interesting.
I listened to those and thought, "Neither sample sounds like just intonation." I could hear the beats, and I didn't hear the characteristic richness.

That could well be due to the tiny vibrato. Listen to the appropriate tests on this page, and you'll hear the characteristic droning sound of JI:
http://www.skytopia.com/project/scale/third/

I don't know how to explain why, but I think maybe Dan can if we can get him away from his computer games!

Lol! Funny you should say that as I have been playing a few more as of late Anyway, yeah it's down to mathematics in the end. Obviously, JI intervals can be represented by close, but different ratios. At a basic level, with JI, you need two slightly different whole tones (10:9 and 9:8 ) so that you can obtain the different intervals of the octave properly.

I think one of the simplest examples is this: Take the JI major third which is 1.25. Now modulate to E major, and take the major third again. In equal temperament, this is G# or 800 cents. In JI, it falls significantly short of this at 772 cents, and 42 less than the usual JI minor sixth. Then if we ever want to play a minor sixth with C as the bass note, we have a problem houston, as we need two different notes. See this pic, and you can see some of the differently sized gaps between each interval:
http://www.skytopia.com/project/scale/eq-ji.png

There are now electronic keyboard instruments that can provide "automatic just intonation," which adjusts pitches in real time to create accoustically perfect chords according to whatever base note is used. That's rather similar to what a highly trained chorus does.

Even then, you'll find interval contradictions popping up, as the music will often find it itself slightly transposed up or down a little compared to when it started (at least when changing keys). Also, according to this site, mixing harmony with melody, you can't make all intervals Just with dynamic JI:
http://www.music-cog.ohio-state.edu/Mus ... uning.html

[BoneQuint]

With real instruments, intonational errors and other complicated phenomena start to creep in, so 'synthesized' sounds are necessary to keep the tests as objective as possible. However, I do agree that the just intonational, droning, timbral effects can be even nicer with real instruments, but I'm pretty sure this is mostly thanks to their almost-but-not-quite-spot-on JI intervals, creating phase-like effects.

When you're singing in a choir, you try to "lock in" with the other singers as closely as possible, which provides that resonance or "richness" that Jerry describes (and which you seem to call a "droning" quality). True, some of the harmonics won't be as precisely matched up, and that may add to a pleasing sound or timbre, but the chord itself I think is mainly defined by the fundamentals (again, especially in the real world with acoustic instruments). I've never heard of singers trying to "almost match" to creating beating or phase effects.

It appears you like phase effects, which is a matter of taste. That's why accordions are available with differing amounts of "tremolo" for different sounds. And it's not only personal taste, different genres of music stongly favor distinctly different amounts of tremolo in accordions as part of their "signature sound."

Using the word "sweet" as you do is misleading to me -- I think of "sweet" as something that's as pure and smooth as possible, even if that might not be my favorite sound. I think a word like "pleasing" would better match what you're talking about and avoid confusion.

As far as melodies go, I'd say I have very little preference whether it's equal tempered or just or whatever -- and I could rarely even tell. It's only when chords come in that it's much of an issue.

[Jerry Freeman]

There are now electronic keyboard instruments that can provide "automatic just intonation," which adjusts pitches in real time to create accoustically perfect chords according to whatever base note is used. That's rather similar to what a highly trained chorus does.

Even then, you'll find interval contradictions popping up, as the music will often find it itself slightly transposed up or down a little compared to when it started (at least when changing keys). Also, according to this site, mixing harmony with melody, you can't make all intervals Just with dynamic JI:
http://www.music-cog.ohio-state.edu/Mus ... uning.html

The article you reference says:

A third approach is to continuously adapt the tuning as the music unfolds. This can be done with a computer, where the tuning of each successive note in a sequence is adjusted so that just intervals are always used. Unfortunately, this adaptive approach causes the "tonic" pitch to vary over the course of a melody: the "doh" you end with will not necessarily match the "doh" you begin with. For many melodies, most listeners find such adaptive tuning to be unpleasant. Another problem with this approach is that it only works for single-note melodies. Once you add harmony, it is impossible to ensure that both the harmonic and melodic intervals are just. A final disadvantage is that adaptive tuning is really only practical using a computer. It would be a significant challenge for human performers to adopt adaptive tuning.

To begin with, I don't believe it's a significant challenge for human performers to adopt adaptive tuning. As we've discussed elsewhere, it's what highly trained musicians do all the time.

From that thread:

Orchestra players do not use equal temperment. I have actually been in situations where I was required to change the pitch of a sustained note midstream, becuase the surrounding harmony changed, and my note changed its function (e.g., from the 3rd of the chord to the 5th of the chord) in the harmony.

Of course, if you're playing with an instrument that can't be tuned (or can't be readily tuned on the fly), like a piano, harp, or percussion instrument, you use whatever tuning that instrument is using, which is probably tempered.

From Wikipedia:

In practice it is very difficult to produce true equal temperament. There are instruments such as the piano where tuning is not dependent on the performer, but these instruments are a minority. The main problem with equal temperament is that its intervals must sound somewhat unstable, and thus the performer has to learn to suppress the more stable just intervals in favour of equal tempered ones. This is counterintuitive, and in small groups, notably string quartets, just intonation is often approached either by accident or design because it is much easier to find (and hear) a point of stability than a point of arbitrary instability.

Based on what little I've read about automatic just intonation, I'm not convinced that the statement you've referenced about its drawbacks (aside from requiring a computer) is correct.

Best wishes,
Jerry

[BoneQuint]

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.

Jerry, I think the "adaptive tuning" mentioned in the article is a very specific method, not the sort of tweaking a musician does. It seems to be to try to adjust as many intervals as possible, even at the expense of changing the pitch of the whole scale -- it appears you could start out at A440, for example, and wander around until you reached A460 or something. That would be tough for humans. That the article doesn't even mention what people actually do when adjusting is quite an oversight (common to the academic, theoretical point of view).

The article seems to miss the point in a few other ways. It mentions even professional musicians aren't good at hitting exact pitches, which is true -- in isolation. But the issue isn't about singing a melody in exact just intonation or whatever. It's mostly important when singing against another note, especially in a simple relationship such as an octave or a fifth, and then their accuracy will be MUCH higher than mentioned in the article, because they can hear the resonance, the blending of the pitches. That's when temperament is a real issue anyway, so arguing about pitch inaccuracy in a vacuum is almost meaningless.

[Yuri]

Well, it is a can of worms, isn't it?
First, a quick reply to steel v nylon: yes, it is the very different overtone content. Believe me, it really requires a library's worth of stuff to really understand the whole process. What complicates the problem even more with strings that the overtone that in theory should be produced are not actually sounding at their theoretical value, because we live in a very untheoretical world, and such things as material resistance, material impurities, and other factors all knock the theoretical values off to a larger or smaller degree.
But in a nutshell. If two waves are an exact fraction of each other, they merge into one very clearly defined wave. That's when we hear them as pure. When there is distuning, they set up complex patterns, that are heard increasingly out of tune, the larger the mistuning is. In all 12-note systems there is plenty of distuning. Just temperament, by the way is not used for keyboards, as it is a diatonic system, as soon as you add semitones, you leave it. Equal temp. is the only system where all intervals are mistuned, by the way, except for the octave.
But I seriously suggest that anyone taking it really seriously should go to the local university library and start from there. It's a huge subject, unfortunately a hell of a lot of musicians are not even aware of it's existance.

[Lorenzo]

-- it appears you could start out at A440, for example, and wander around until you reached A460 or something. That would be tough for humans.

It's not that difficult for some choirs and some barbershop quartets to wander off that far from where they started!!!

[Yuri]

Second installment, steering clear of physics and maths.
Different instruments have a very different overtone content. It follows that some are very much more tolerable than others in certain temperaments. Hence the piano was habitually tuned in equal temp as early as the beginning of the 19th cent, while at the same time organs (pipe) have not. Equal temp sounds quite painful on them. In some places, mainly Scandinavian, they never really switched to equal temp for them. Since the 60-es organs are once again increasingly tuned in non-equal temperaments.
Another aspect is this. Temperaments do matter a lot in polyphonic/harmonic instruments, that's keyboards and fretted strings, mainly. With monodic instruments the whole tuning issue is much less crucial, as there is no immediate clash of dissonant overtones, which is what mainly causes the whole problem. And, yeah, good musicians on woodwind can compensate to some extent, even it is not necessarily a straightforward thing, either.

[Yuri]

Third installment.
Good singers, when there is no fixed pitch to rely on, tend to sing naturally in just temp. Hence you sing a (pure) third up, (pure) fifth down, third up, fifth down, you'll be shifting your pitch very very quickly. When you think you should arrive at the starting note, a couple of octaves higher or lower, if you measure it, it will be probably more than a whole tone off.

[Jerry Freeman]

Scroll down to "Listen to Samples" and you can hear some clips.

http://www.amazon.com/Perotin-Hilliard- ... F8&s=music

Best wishes,
Jerry

[Jerry Freeman]

Third installment.
Good singers, when there is no fixed pitch to rely on, tend to sing naturally in just temp. Hence you sing a (pure) third up, (pure) fifth down, third up, fifth down, you'll be shifting your pitch very very quickly. When you think you should arrive at the starting note, a couple of octaves higher or lower, if you measure it, it will be probably more than a whole tone off.

I sang for a year in a sacred music choir that performed almost entirely acapella. I am not an experienced musician, by any means. The director allowed me in because there was only one other singer for the bass section. That was my good fortune, and I was able to learn to sing reasonably accurately on pitch and did make a useful contribution.

This director was extremely fussy about pitch accuracy (or to say it more precisely, interval accuracy). He discussed and demonstrated the difference between just and equal tuning numerous times.

At the end of a practice run through a piece, he would take his pitch pipe and say, "Let's see where we ended up." I think he would not be upset if we were within a semitone or so of the original tonic, but we were often much closer than that.

The idea that a choir might end up somewhat off the original tonic note, to my thinking, isn't an especially meaningful piece of information.

As we've discussed, the pitch relationships that matter most are the simultaneous ones, within chords. Arpeggios might be second in importance to those, and the precise accuracy of intervals between successive pitches in melodies, not even very important at all in this discussion.

So the fact that the ending tonic pitch, performed several minutes after the first tonic note, might have drifted somewhat, would not spoil the effect at all.

Best wishes,
Jerry

[BoneQuint]

-- it appears you could start out at A440, for example, and wander around until you reached A460 or something. That would be tough for humans.

It's not that difficult for some choirs and some barbershop quartets to wander off that far from where they started!!!

Just to be clear (I think you already know what I mean): it'd be difficult for humans to adjust the tuning of the scale at certain times to help the tuning match just intonation better, all at once and precisely. Slowly wandering a bit off pitch together is a different issue, and I agree with Jerry, it's not a big deal.

[Mitch]

What a fantastic thread!

This has answered many of my musings on temperament. I'd like to add the following:

Pianos are not strictly even tempered. Most of the notes are articulated using 3 strings - these 3 are not all tuned to the same note, but spread to introduce a phase-beat. This assists somewhat to resolve the dissonances when changing key.

Wind and voice instruments have a great advantage in that you can feel the harmony in your throat - in fact, when playing with others, I find the ambient vibrations cause my whistle to harmonise without a lot of conscious effort - it tends to "slot-in" to whatever is going on. I know my whistle is out of tune when this does not happen.

Modern music is in love with key changes - there are a great number of nuances that can be achieved doing this, but singularities lurk in this practice. Just and Even temperament - even Dynamic Just Temperament are creatures of this complexity. When framed in the proviso that a tune does not change key, Just temper is the better choice due to the natural at-rest harmony.

Lastly I'd like to point out that music is less a math question and more a function of the psycho-emotional state of the performer and listener. This continuum has more to do with communication and survival of a, so far, outlandishly successful species of homonid. It is probably not cogent to be laying the flyscreen of numbers accross this continuum or the singularities will give rise to endless eddies of noise producing reason. The falacy in the numbers is that the key changes are natural - one could say they are transitional - meaning that the at-rest state is migrated. Once the reference point is violated a new vista is mapped - in other words, it dosn't really matter if the choire finished the song a whole semitone sharp - what was important was the journey experienced by the audience.

[Yuri]

Choirs and pitch.
When the last pope had his funeral service, I by chance happened to watch it. (stranded in a motel between performances, with nothing else to do). Well, the boys' choir alternated with the organ. Every time the choir stopped and the organ fired up, there was a very audible difference in pitch.

[Jerry Freeman]

Choirs and pitch.
When the last pope had his funeral service, I by chance happened to watch it. (stranded in a motel between performances, with nothing else to do). Well, the boys' choir alternated with the organ. Every time the choir stopped and the organ fired up, there was a very audible difference in pitch.

They need to do something about that organ.

Best wishes,
Jerry

[Lorenzo]

Just a little thread house cleaning...

I read earlier on this thread that with a violin, try tuning the B. I'm wondering how that could be done, or how one could place a finger back in the exact same spot (I've played fiddle for years). Normal stretching of the string, by placing a finger down on it, means when the tension is taken off, the open string would be at a different tension (not differnt sounding length). This would seem to be impractical.

Pipe organs. Someone (Yuri) said pipe organs were and may still be tuned Just. 12 note chromatic reed instruments have no different needs than 12 note chromatic stringed instruments when it comes to pitch, intervals, chords, etc. It could be those pipe organs were tuned to be played in certain keys? Keyboard accordions and pump organs are tuned to ET.

Regarding pianos having 3 strings per note (on most of the central and upper keys). These are always tuned in unison, as perfectly as can be gotten. The old square grands had 2 strings per note, but lacked the punch and power felt and heard in modern pianos.