The Harmonic Series and Equal Tempered Scale

[PMeuph]

Also, most pre-adolescent students cannot hear low pitches, even trombone range.

This statement is quite misleading an erroneous on many levels. If you change the word "hear" to recognize, then it makes more sense.

It's related to their development. As for the guy who could sing the bass lines up TWO octaves; that's quite exceptional.

Development, training, ability, environment... I find your claim to be blanket statement. Non-musical children surely do not have the ability to match pitches from octaves lower. The same surely goes for non-musical adults. My mother could not carry a tune with a bucket before a year ago. She started clarinet lessons about three years ago and since then, partially because of theory, solfege and rhythm exercises are hearing and singing has greatly improved.

Back to the kids, I have taught many children and have been in a conservatory. Most of the kids who started lessons as 5 year olds could hear the grass grow. They could sing back anything... I don't believe there is anything that proves that kids can't recognize low pitches, other than lack of experience which such exercises. Given proper training and time, kids and adults can learn to hear and recognize higher or lower pitches. I just have the strong suspicion that most people only concentrate on the one that pertains the most to them, usually their voice.

[Amilcare]

Sorry, I am the spouse of a great Kodaly- trained music teacher. She has also taught many student teachers as well. The info on children's pitch perception is the result of the doctoral work of one of the leading Kodaly educators.

Why do you folks get so upset when you know NOTHING about a subject except your narrow little world?

[Amilcare]

OK, try this. Play or sing a pitch below middle C on the piano. Ask the child to sing the pitch back to you in their own voice.

This is why MALE elementary music teachers must sing in falsetto.

[imperialbari]

Thank you for putting a declaration of contents on your postings.

Also:

Thank you for only showing the first half of the declaration!

[hup_d_dup]

Why do you folks get so upset when you know NOTHING about a subject except your narrow little world?

Why not confine yourself to challenging the argument instead of introducing a nasty personal comment?

Hup.

[Amilcare]

I ran out of patience. People who pretend that their experience trumps all are trying.

[PMeuph]

Why do you folks get so upset when you know NOTHING about a subject except your narrow little world?

Why not confine yourself to challenging the argument instead of introducing a nasty personal comment?

Hup.

I ran out of patience. People who pretend that their experience trumps all are trying.

If you quote the passage that upsets you, and challenge it directly, you will not end up looking like "the pot calling the kettle black."

________

If you are going to make vague brash statements, like:

Also, most pre-adolescent students cannot hear low pitches, even trombone range.

Make sure that the one meaning you want to convey is absolutely clear. Your one-line snippets are surely backed up by some evidence, but since we don't have it they come off as irresponsible and sloppy.

The ambiguous nature of the word hear is a pet peeve of mine. Musicians use it as a synonym for "recognize," "comprehend," "perceive" and "discern." Heck, the current wording of your statement *almost* implies that "most pre-adolescent students cannot hear their own fathers speak." (I am being obtuse, purposefully, but my intent is to clarify what is a sloppy statement that seems to come out of no-where...)

[sloan]

OK...Enough NIGYSOB...

' time for practical application:

the final A♮ in the second movement of the Vaughan Williams...

When playing that pitch with a piano accompaniment, should you play it

- in tune (ref: equal temperament, A-440)...??
...or...
- flat (ref: equal temperament, A-440)...??
...or...
- sharp (ref: equal temperament, A-440)...??

What is/are the reason(s) for your choice?

I don't know - because you haven't told me how the piano is tuned (and...I don't know offhand what notes the piano is playing
before, during, and after that note).

The answer is: I will try to play it in tune with the piano. I don't think temperment or A-440 enters into it.

I'd like, of course, for the piano to play in tune with me - but even Carol can't do that.

[PMeuph]

the final A♮ in the second movement of the Vaughan Williams...

When playing that pitch with a piano accompaniment, should you play it

- in tune (ref: equal temperament, A-440)...??
...or...
- flat (ref: equal temperament, A-440)...??
...or...
- sharp (ref: equal temperament, A-440)...??

Since the piano is playing that same pitch two octaves higher on the second beat I would play it in tune (with the piano that is)....

_____
I know that usually fifths of chords are usually a tad sharp. I also know that pianos usually go flat after the chord is struck. I still think that I would in tune on the initial solo attack.

[mark38655]

OK...Enough NIGYSOB...

' time for practical application:

the final A♮ in the second movement of the Vaughan Williams...

When playing that pitch with a piano accompaniment, should you play it

- in tune (ref: equal temperament, A-440)...??
...or...
- flat (ref: equal temperament, A-440)...??
...or...
- sharp (ref: equal temperament, A-440)...??

What is/are the reason(s) for your choice?

I don't know - because you haven't told me how the piano is tuned (and...I don't know offhand what notes the piano is playing
before, during, and after that note).

The answer is: I will try to play it in tune with the piano. I don't think temperment or A-440 enters into it.

I'd like, of course, for the piano to play in tune with me - but even Carol can't do that.

I'm with the professor 100% on this one. .... well then there is the fact that the tuba has to play the A one beat before the piano plays the A major chord. But still, I'd have to play an A, then adjust to the piano if I were out-of-tune with it when it comes in on the final chord, and it wouldn't matter one bit if it were tuned to A-440 or some other pitch, except that I'd have had plenty of time to try to match whatever it was the piano was tuned to earlier in the piece. If I was still hunting for the pitch when I got to the last note, it really wouldn't matter would it??

[PMeuph]

.... well then there is the fact that the tuba has to play the A one beat before the piano plays the A major chord. But still, I'd have to play an A, then adjust to the piano if I were out-of-tune with it when it comes in on the final chord, and it wouldn't matter one bit if it were tuned to A-440 or some other pitch, except that I'd have had plenty of time to try to match whatever it was the piano was tuned to earlier in the piece. If I was still hunting for the pitch when I got to the last note, it really wouldn't matter would it??

....you mean D major chord, don't you?

-I Assumed that bloke was referring to a piano that was tuned to A=440Hz

To me, Bloke's question raises the following issues:

-How do you line up this pitch vertically? (It's the fifth of a chord but its also the lower pitch and puts the chord in 2nd inversion)
-Fifths are usually tuned sharp, but when it are the bass pitch (i.e in a 6/4 chord) isn't easier to play it at it's position on the piano?
-The pitch of the piano will inevitably drop, so are you best to drop a bit with the piano?
-In the context of the preceding two measures, it seams like the tuba outlines a half-diminished seventh-chord. I would have reflex to play the 7th of such a chord, flat.
-

[mark38655]

.... well then there is the fact that the tuba has to play the A one beat before the piano plays the A major chord. But still, I'd have to play an A, then adjust to the piano if I were out-of-tune with it when it comes in on the final chord, and it wouldn't matter one bit if it were tuned to A-440 or some other pitch, except that I'd have had plenty of time to try to match whatever it was the piano was tuned to earlier in the piece. If I was still hunting for the pitch when I got to the last note, it really wouldn't matter would it??

....you mean D major chord, don't you? -

oops-- I thought it was A major but you are right. I think I would tune the A to the A in the chord down how ever many octaves it is. (I don't remember and am too lazy to go get the piano acc.) It is easier to hear pitch discrepancies on octaves than on 5ths or major 3rds.

Putting all of those factors in the mix (and assuming a well-tuned piano), I would tend to try to play the final A of that movement right at a 220 bps frequency.

Since an equal tempered piano tuned to A-440 would likely have the A actually tuned to 440 vps, then I believe my A matching the one in the chord shifted down to the correct octave would also be same pitch as your A-220.

[PMeuph]

OK... At least people aren't butting heads anymore.

I think the OP, would be all the wiser the use the approach you did (ie. case studies of repertoire) as examples in his book...

[Amilcare]

Quoting on an iPhone is hard.

I find a lot of this concern misleading. Solfege is the real skill we must all acquire. Concerns about tuning systems is not germane to our skill.

There is a new system for tuning out that is quite limited in its information. Jay Friedman talks about it at his website.

Chris Leuna's THE ART OF MUSICAL INTONATION is available at Gordon Cherry's CHERRY MUSIC as a PDF. It is the most complete and reasonable approach there is. Understanding tuning is important. Doing it is moreso. Arnold Jacobs' six years of Solfege at Curtis were central to his craft.

This can be complicated. The problem is that we must simplify everything at the end. Jacobs always put SONG first. Who are we to doubt him?

Our technology is actually limiting us. The only way to excellence is to exceed its limitations by our musicianship. Anything else is a distraction.

[mark38655]

Quoting on an iPhone is hard.

I find a lot of this concern misleading. Solfege is the real skill we must all acquire. Concerns about tuning systems is not germane to our skill.

There is a new system for tuning out that is quite limited in its information. Jay Friedman talks about it at his website.

Chris Leuna's THE ART OF MUSICAL INTONATION is available at Gordon Cherry's CHERRY MUSIC as a PDF. It is the most complete and reasonable approach there is. Understanding tuning is important. Doing it is moreso. Arnold Jacobs' six years of Solfege at Curtis were central to his craft.

This can be complicated. The problem is that we must simplify everything at the end. Jacobs always put SONG first. Who are we to doubt him?

Our technology is actually limiting us. The only way to excellence is to exceed its limitations by our musicianship. Anything else is a distraction.

Yes, but solfege is useful mainly in melodic intonation (notes in succession), which is certainly important. But my focus is mainly on harmonic intonation (multiple notes or instruments playing at the same time). Learning to match pitch is simple in that there aren't a whole lot of different skills required to do it, so I agree that getting too much technical info to the student can be more of a hindrance that a help.

There are ways our technology is limiting us--any time we become dependent on it, we are setting ourselves up for future problems. However, my use of SmartMusic for rehearsals actually helps me to develop the art of matching pitch. This is a skill that, once learned, helps me be LESS dependent on technology than perhaps using a visual tuner would in the long run.

I'm with you that I wouldn't want to disagree with Arnold Jacobs. He changed my whole concept of playing and teaching and I consider my four lessons with him among the most important and positive events in my career.

[mark38655]

The reason I wrote the post on the Harmonic series vs. equal temperament is because you can't always achieve the desired goal of playing really well in tune without understanding the basis for both of them.

Playing in equal temperament seems to work well for tutti chords, but not so much for a small group of say three or four players on a major chord.

Matching pitch using pure intervals is a tendency of the very best players and singers. However, with some music it is impossible to do this without having serious problems as a piece progresses through several keys. An example would be from the CSO recording with Solti of Wagner Overtures (from the 1980's I think). I believe it was the Overture to Tannhäuser that the musicians were matching pitch so well that when the chords progressed through about the 4th different (key signature), the pitch had moved too far away from where it started that it became too difficult to continue matching pitch. So there was a momentary lapse in pitch matching for a chord or two.

Technically it is possible to perfectly match pitch on any piece no matter the complexity of the harmony progressions. However, our acoustic wind instruments would have to be altered to achieve this on a piece like a typical Wagner overture.

[mark38655]

First and second trombone players deal with this stuff much more than to tuba players, but usually don't play in pure temperament, as the "consensus" (as opposed to "science") of modern ears now prefers that interval far closer to the "equal temperament" version.

It is interesting to listen to a pure tone major scale and realize that it just sounds "wrong" because we have been conditioned to hear that the equal tempered scale is "right" with its imperfect intervals.

I think however that in a major chord the trombone player will tend to match closer to a pure interval than an equal tempered one if given the chance. (if he plays an exposed long note) If its a tutti major chord then the equal tempered version may very well be more desirable.

I keep having to edit my poor writing. I think Joe is correct in everything he says though. In just about every live performance I've heard of a major symphony I noticed at least one note that was further out-of-tune than could be attributed to the imperfections of equal temperament. But when I heard notes that were not perfect but seemed to be within the tolerance of equal temperament, I didn't see musicians reacting like something needed adjusting on their instrument.

[mark38655]

real question: Why is the third in a major chord played flat in pure temperament?

In addition to what Bloke said--

A major third is played lower in pure "temperament" than equal temperament because it is actually sharp when played in equal temperament. The pure temperament one sounds the best because (as Bloke said) there are no interference beats that make it sound out-of-tune.

The first time the major third occurs in nature is with the relationship between harmonics 4 and 5. Based on A-110 that would be (4th harmonic) A-440 and (5th harmonic C#-550). This interval will sound perfectly in-tune. However when you tune the C# to equal temperament, you get C#-554.36. In that octave you would be hearing interference beats at the same rate you tongue the low brass eighth notes in the break-strain of the Stars and Stripes Forever, (about 4 per second). That is pretty far out-of-tune in my book. And when I play that same major third on my exactly equal tempered electronic piano keyboard it sounds both bad and familiarly acceptable at the same time.

This is proof to me that humans can be conditioned to accept all sorts of undesirable things.

[EstroArmonico]

Just have to chime in with one more correction here:

"Pure temperament" is a contradiction in terms. If intervals are pure they are, by definition, not tempered. Temperament means a system of compromising pure intervals to solve various comma issues in various ways. Meantone temperament, well temperament, and so on. There is no such thing as pure temperament.

~Luke

P.S. In the original post it would be clearer to describe an equal-tempered half step as 1 plus the 12th root of 2, rather than 1.059... That way it will make sense that the half steps come from 12 equal divisions of the octave (2:1).

[mark38655]

P.S. In the original post it would be clearer to describe an equal-tempered half step as 1 plus the 12th root of 2, rather than 1.059... That way it will make sense that the half steps come from 12 equal divisions of the octave (2:1).

I described it that way so that the average person could understand it. I think the 12th root of 2 is probably more accurate than even the long decimal number I used, but most people who play tuba (or other musical instruments) simply don't understand the terminology. (But yes, if I didn't say in the original post that the octave was divided into 12 equal half-steps, then I should have.)