The Harmonic Series and Equal Tempered Scale

[mark38655]

Professor Sloan:

At any given point in history, there is usually an accepted set of beliefs on any subject, whether scientific or not. There are always people in position of authority because they are, in fact, “authorities” on the prevalent belief system, but not necessarily authorities on the facts of the subject.

I believe I have something to contribute to this topic as a tuba performer and educator. As such, I have no problem being challenged, or myself challenging accepted authority. As a general rule, when asserting something that is not yet accepted as true by the majority of students of a particular subject, I've tried to back it up with a logic that is both clear and concise. I had expected disagreement as I was fully aware that my thoughts don't match popular beliefs. Otherwise, I would have no need to write about the subject in the first place.

I do appreciate the challenge of defending my position. I have chosen to respond to each of your specific comments separately. Regarding your final comment, “as for the footnote...I hardly know where to begin...I surrender!”, it appears obvious that you found a need to publicly make fun of my work but lack the courage to justify your snide response to it. I find it both inappropriate and distasteful on your part.

If you would be willing to state facts and then back up your position with logic or specific sources then your adverse position to my thoughts would actually be an asset to the people reading Tubenet who are interested in the subject.

A few notes:
a) the notation gives a harmonic series starting at Bb (well...a crude approximatin, anyway).
It is not specific to "BBb tuba". It's the same for any instrument. The notation is only correct for
a listener with a very high tolerance for deviations.

As I stated: “Each of these notes can only be approximately notated using the musical scale which in modern times is tempered away from nature.” It seems obvious that the harmonic series is not specific to BBb tuba, but the one that I notated is. Since this is a method book for the BBb tuba, it seems a good choice to me. (I think we agree that there are problems with notating the harmonic series using the Western musical scale. If there is a better way to notate it on the bass clef, then your suggestions would be appreciated. I want to notate it the best/most practical way for the BBb tuba student.)

b) the "harmonic series" is more of a mathematical construct than something that "occurs in nature"

But, the harmonic series most assuredly does occur in nature—often and in various ways. Otherwise the mathematical construct used to express it would be useless in the practical application of the construction of the musical scale. However, I have shown how the chromatic scale is produced using the harmonic series, (first with the relationship between harmonics 2 & 3 and again with harmonics 3 & 4). Do you want to argue that when a string is plucked that it does NOT vibrate simultaneously at several of its multiple harmonics? Or do you want to say that the multiple frequencies and resulting timbre of that string are are made up of frequencies that are NOT related harmonically to the fundamental? If so, I'd love to read the details of your reasoning on the subject. (If it is your understanding that brass instruments only approximate the harmonic series, then it would be helpful to to say why you think so.)

c) a "partial" is a concept best applied to particular instruments, and NOT a concept that is useful to relate to a harmonic series. As a general rule, "partials" are *not* "perfectly in tune with any other partial". That's a general engineering goal when designing instruments - but it is seldom (if ever) achieved. I'm not aware of ANY instrument where the "partials" precisely match "the harmonic series" without manual adjustment.

I have a working knowledge of your use of the word “partial”. I was taught all about the concept when I was a student. However, when studying the physics of music, I was taught that a partial was a specific harmonic from a series of partials that occur in nature. If a partial is “NOT a concept that is useful to relate to a harmonic series”, then why not?? And why do you say "a "partial" is a concept best applied to particular instruments, and NOT a concept that is useful to relate to a harmonic series"?

I understand that most people believe that partials are not in tune perfectly with each other. I know we are taught the 5th partial is flat, the 6th partial is sharp, the 7th partial is very flat, etc. My point is that this is only ACTUALLY true as compared to the equal tempered scale. For example, partials 4, 5, 6, and 7 make a very good and perfectly harmonious dominant 7th chord. This version of the chord is used in blues music and is known as a “blues 7th”. If you can give useful information to show that I am wrong on this point, then I will first, thank you, and then change what I wrote.

d) BBb tubas can usually play a note or two NOT shown in your notation. In particular, most BBb tubas are capable of playing a low Eb (in fact...some make it EASIER to play a low Eb "without valves" than a "fundamental" Bb.

Yes, you are correct, and I realized that I left out the false tone Eb right after I posted. I will make note of it. However, I will not include it with the harmonic series since it does not belong there. Instead I will remove the statement that these are the first 16 open notes on BBb tuba.

e) the "equal tempered scale" has almost nothing to do with a particular "harmonic series".

Well, equal temperament is a form of the 12 note-per-octave chromatic scale, and the chromatic scale is as extension of the circle of 5ths (harmonics 2:3) and the circle of 4ths (harmonics 3:4). So, saying it has "almost nothing to do with a particular harmonic series" is a bit of a stretch in my opinion. (Unless you think the decision to divide the octave into 12 parts was arbitrary or for cultural reasons ---12 tribes of Judah, or 12 disciples or Jesus, or 12 signs of the zodiac?)

In particular, the various notes produced by the harmonic series are "out of tune" with the equal tempered scale

Yes, and it is also true that notes of the equal tempered scale are “out of tune” with the harmonic series.

f) I'm curious about these methods that instrument manufacturere have devised to manipulate brass instruments to alter the instrument away from the natural tuning...can you elaborate on that, please?

I spoke with Mr. Walter Lawson, the famous maker of (French) horn lead pipes a few years ago. He told me that he (and other instrument manufacturers) knew how to alter the shape of the tube of a brass instrument to either raise or lower the pitch of specific notes. In particular, the shape of the lead pipe is altered at specific node points. This can be with a different diameter at the point of alteration, or a different shape which is not always perfectly round.

g) as for the footnote...I hardly know where to begin...I surrender!

Again, what I thought was understood when I made my post was that I wanted constructive criticism. The impression you left was that my work was so beneath your level of understanding that you wouldn't even discuss this point.

[Amilcare]

Equal temperament is a yardstick. Most ensemble players drift toward Just Temperament. Pianos are still using a form of Well Temperament that deals with the peculiar acoustics of the instrument. THE ART OF MUSICAL TEMPERAMENT from Cherry Music is the best and most technical explanation. Better is for a student to have singing and solfege training. Also, most pre-adolescent students cannot hear low pitches, even trombone range.

[mark38655]

Better is for a student to have singing and solfege training. Also, most pre-adolescent students cannot hear low pitches, even trombone range.

Arnold Jacobs once told me that although there were no shortcuts in music, that if there were one, it would be solfege.

Do you mean that most pre-adolescent students cannot hear pitch problems on low pitches?

[Amilcare]

Nope, can't match pitches when singing. Male elementary music teachers must sing falsetto for students to match pitches. Psycho-acoustics here.

[mark38655]

Interesting.. What do you think causes them to not hear pitch problems in that range?

[iiipopes]

iiipopes (Mike?) suggested reading Hemholz's, "On the Sensation of Tone" and then extracting out what you need for tuba. Although I've not read that work, I could surely agree that reading a whole book on the subject should give more detailed info than what i offered on the subject. I should probably refer the readers of my work to more detailed sources. However, my purpose for discussing the subject is not to give exhaustive details for the tuba student, rather to provide an introduction in order to better understand what is required to play in tune

Um, it's Scott, not Mike. Anyway, the reason I suggested reading it is because if a person completely understands the depth (pun intended in this context) of the subject matter, then it is easier to summarize or extract and summarize what is necessary for a given audience.

[sloan]

iiipopes (Mike?) suggested reading Hemholz's, "On the Sensation of Tone" and then extracting out what you need for tuba. Although I've not read that work, I could surely agree that reading a whole book on the subject should give more detailed info than what i offered on the subject. I should probably refer the readers of my work to more detailed sources. However, my purpose for discussing the subject is not to give exhaustive details for the tuba student, rather to provide an introduction in order to better understand what is required to play in tune

Um, it's Scott, not Mike. Anyway, the reason I suggested reading it is because if a person completely understands the depth (pun intended in this context) of the subject matter, then it is easier to summarize or extract and summarize what is necessary for a given audience.

Besides - that Helmholtz was one smart dude.

[iiipopes]

And yes, I've read it. Unfortunately, my college library went to digital scanning checkout, so I can't show you the card in the pocket in the front of the book where I signed it out, um, er, more than once.

Here's the bottom line: for horns, strings and voices, the harmonic series has value to get chords "dead" in tune for maximum chordal and melodic solidarity and clarity. Equal temperament was developed so a keyboard can play in any key with no more "out-of-tuneness" than any other key.

I've played both. When playing primarily with horns or singing, I have to readjust my ears, especially to the major thirds, so I tune them properly. Otherwise, a "pure" major third, after playing a lot with keys and/or fretted instruments will sound flat to me, and tempered thirds, after singing or playing a lot otherwise, will sound bright or harsh to me.

The debate over which tuning is proper for keyboard instruments, especially organs, became especially pointed in England. As late as 1857, S.S. Wesley and the builder "Father" Willis on the meantone side, and the builder William Hill and W.T. Best on the other, debated the issue hotly. The Making of the Victorian Organ, Nicholas Thistlethwaite, Cambridge University Press (paperback edition, 1999, at page 143, of which I have a copy).

[iiipopes]

Better is for a student to have singing and solfege training. Also, most pre-adolescent students cannot hear low pitches, even trombone range.

Arnold Jacobs once told me that although there were no shortcuts in music, that if there were one, it would be solfege.

Do you mean that most pre-adolescent students cannot hear pitch problems on low pitches?

I completely disagree. One reason I play bass was when I was a boy, my father showed me the bass lines in hymns on our piano. He sang in the church choir, and before he died when I was nine years old, instead of the melody, I would sing along with him an octave higher on the bass lines of hymns, looking forward to when my voice would change and I'd get to sing them with him in the proper octave. Of course, that didn't happen, and I'm looking forward to when my son's voice changes, but I digress.

Moreover, even as a boy, I was enthralled with such music as the pedal parts of pipe organs, and Paul McCartney's bass lines on the Beatles tunes when I was three and four years old watching the Beatle's cartoons in 1965-1966.

I could definitely hear the lines, and when they were in tune and not.

[Amilcare]

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

[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??