The Harmonic Series and Equal Tempered Scale

[mark38655]

I'm working on a tuba instructional method book with warm-ups, tuning exercises and scale exercises. I have written this draft today of info pertaining to what happens in nature vs. what compromises we have made to make things more practical for the instruments we use in performances. I would love to get feedback from people on TubeNet about this subject. Thanks in advance.

Mark Howle

(Here is an edited version of the original post:)

The Harmonic Series and Equal Tempered Scale


Sound is caused by waves moving through a medium from a vibrating source. If the frequency of a tone is between 20 and 20,000 vibrations per second, or hertz, and if it is loud enough, it will be audible to most humans. Normally we hear sounds through the medium of air, however, because of its increased density, water is an even better medium for sound propagation. In nature, when things vibrate at a steady rate, they tend to do so in a combination of frequencies simultaneously. The combination of all the natural vibrating frequencies of an object is known as the harmonic series, sometimes called the overtone series. Each separate vibrating frequency is known as a partial and the lowest of these is the fundamental. These harmonics occur at whole number ratios to the fundamental. For example, a piano string tuned to A-110 hertz will vibrate at the following frequencies, all at the same time, when struck with a hammer: 110, 220, 330, 440, 550, 660, 770, 880, 990, … ∞. There are many examples found in nature where multiple frequencies are known to vibrate simultaneously. Whether by plucking a string, beating a drum, blowing through a hollow bone or flute, or buzzing into the small end of an animal horn or brass instrument, we hear a combination of frequencies all at the same time which gives each instrument a distinct tone quality or timbre.

The reason some notes sound more harmonious together than others is because of the mathematical relationship of their individual frequencies. The octave has a relationship of 1:2, the perfect fifth is 2:3, the perfect fourth is 3:4, the major third is 4:5, and the minor third is 5:6. A major chord, or triad has a mathematical relationship of 4:5:6. A similar way of expressing the major triad is to say that it is a combination of harmonics 4, 5, and 6 of a given overtone series. These note combinations are perfectly harmonious with each other because they are closely related in nature and therefore have a simple mathematical relationship.

By using the interval of a perfect fifth from the harmonic series with its relationship of 2:3, one can progress from A through the circle of fifths back to A in order to determine the frequencies of all the twelve notes used in the Western musical scale: A-(110), E-(165), B-(247.5), F#/Gb-(371.25), Db-(556.88), Ab-(835.31), Eb-(1252.97), Bb-(1879.45), F-(2819.18), C-(4228.77), G-(6343.15), D-(9514.73), and A-(14,272.1). However the A up eight octaves is A-14,080 hertz, compared to the A produced by the circle of fifths which is A-14,272.1 hertz. Shifting down five octaves we have A-440 vs. A-446, a difference of six hertz. This is an interesting phenomenon of nature known as the Pythagorean comma.

The simplest harmonic relationship is 1:2, but that produces an octave, which actually functions as the same note. Since the perfect fifth with a relationship of 2:3 is actually the simplest ratio between two separate notes, it is easy to understand why the musical scale is made up of the 12 separate notes produced by the circle of fifths.

We can also determine the 12 notes of the scale by taking the next simplest relationship of musical notes in nature, the perfect fourth with its ratio of 3:4. Starting again with A-110, then progressing up a perfect fourth 12 times, we arrive back at A: A-(110), D-(146.67), G-195.56), C-(260.74), F-(347.65), Bb-(463.54), Eb-(618.05), Ab-(824.07), Db-(1,098.76), Gb-(1,465.01), B-(1953.35), E-(2604.47) and A-(3472.62). Shifting down 3 octaves we have A-440 vs. A-434.08, again a difference of six hertz. We see almost the same degree of pitch shift except this time it is in a different direction.
Doubling the frequency of a note, as in a relationship of 1:2, produces a tone so very similar to the original, that it makes possible the interchanging of notes in any of its octaves. This in turn enables the use of all twelve notes within the same octave thereby creating the chromatic scale. However, because of the Pythagorean comma, the specific frequencies of some of these 12 notes must to be tempered so that they are better in tune with each other when playing the harmonies that are used in Western music.

Several hundred years ago, it was common practice to tune keyboard instruments so that they would play really well in-tune within two or three closely related keys. This worked really well with music that did not change keys nor made use of notes from non related keys. However, it seriously limited what the performer could program on a single concert, because keys that were unrelated to those the instrument was tuned to were so badly out-of-tune that they were not suitable for public performance. J. S. Bach advocated the use of a temperament where all keys were satisfactorily in tune when he wrote his famous Well-Tempered Clavier, which consisted of preludes and fugues in all 24 major and minor keys. It was a compromise that significantly contributed to the use of the equal tempered scale today as the octave is now divided into 12 equal half-step intervals for most keyboard instruments.

Starting again with A-110, the approximate frequencies of the equal tempered ascending chromatic scale can be determined by multiplying each preceding frequency by 1.059463094359295264561825294946 and then rounding up to the second decimal point: A-(110.00), Bb-(116.54), B-(123.47), C-(130.81), Db-(138.59, D-(146.83), Eb-(155.56), E-(164.83), F-(174.61), Gb-(185.00), G-(196.00), Ab-(207.65), and A-(220.00). The frequencies of notes in different octaves can be determined by doubling for each additional octave higher, and then by dividing by 2 in order to lower each note by an octave, etc.

The equal tempered scale is also utilized in the design of electronic tuners and therefore is used as a reference when tuning brass and woodwind instruments. This scale is normally utilized by all wind and string instrumentalists when playing both diatonic and chromatic melodic passages. Most large ensembles use the equal tempered scale at least as a starting point for tuning while performing most music.

It is interesting to note, however, that when playing chords, professional wind and string players alike tend to tune to pure or natural intervals with the simplest mathematical relationship, such as is found in the harmonic series. In some cases, if a composition progresses through harmonies utilizing the circle of fifths, or modulates progressively through various keys, the performers will reach a point where pitch problems can become severe, if they do not use some type of temperament. Bad pitch can actually be a result of continuing to use pure intervals in the type composition. This is because the tuning of notes progress so far from where they started that it becomes impractical, if not impossible,w to continue to match pitch on the resultant notes with the acoustic instruments as they are currently designed.

[Paul Tkachenko]

Hi,

'Well-tempered' is not the same as 'equally tempered' as I understand it. Without checking it, I think equal temperament rounds it up using 3 major thirds in an octave.

I didn't read everything in detail, but I'm not sure how useful this info is for your average player. If you're going to go into commas, you should perhaps also mention that a comma is the different between an apotome and a limma, but really, is this that useful?

Even in microtonal music where theory is taught using commas, it is rarely so precise in performance.

There are many sources for finding these things out - I'm not sure a tuba instructional method is the best home for this info.

Interesting as it is ...

[mark38655]

Paul,

Thanks for reading over my post and taking the time to offer suggestions. You are probably right about Well-tempered being different from Equal tempered. I didn't intend for my wording to equate the two, so I may want to edit that section. However, my understanding of the equal tempered scale is of an octave divided into 12 equal parts. Several sources I checked confirm this as well. The New Harvard Dictionary of Music states: “ In the 20th century, a temperament with 12-part equal distribution has predominated.” I used the Bach example because he was advocating a type of tuning that worked equally well for all 24 keys and equal tempered tuning is the most prevalent present solution to the problem caused by the Pythagorean comma. Also, the reference to Bach was intended to put into perspective that this problem has been a concern for at least several hundred years.

I agree that the average tuba player will probably not find this info useful, but I'm writing this book for students who aspire to become outstanding players---those who may one day want to play professionally. Also, as a young tuba student myself, I was curious about why some combinations of notes sounded good together and others did not. At the time, (1970s') I found it very difficult to find answers to my questions. I am also aware that this info is available in other places, but there may be a tuba player who will have a deeper understanding of the natural underlying causes to tuning tendencies because he was introduced to this information in my book.

Finally the terms “apotome” and“limma” are not essential to the point I was trying to communicate. My purpose was to point out how tuning is a natural phenomenon and also what practices are common for solving some of the problems presented when trying to play in tune. I wasn't trying to impress with technical jargon, but was trying to explain a technical problem.

[PMeuph]

I'm not sure I get what your going after in terms of feedback, and in a larger sense, how this information is really pertinent and new to the tuba? (i.e. why you are writing this)

This following booklet has a concise yet very helpful and pragmatic text to accompany the intonation drone....
http://www.dwerden.com/intonation-helper.cfm" target="_blank

There are tons of resources on acoustics out there. It is presumable that if a tubists wants to learn acoustics I'm sir he can find the acoustic section in a library.

[aqualung]

This is well written, it starts from Day One and quickly gets to the meat. I haven't found anything else that is this unified and concise.

We tend to play more in ET when we are with frets, keys, and electronics. A good windband will instinctively strive to play pure intervals.

Don't get scared by a little techtalk and simple math. This stuff is really important to the bottom part, because you are setting the tonality for the entire ensemble.

[DonShirer]

One important point. You wrote:

The combination of all the natural vibrating frequencies of an object is known as the harmonic series, sometimes called the overtone series.

The harmonic series is a set of pure tones with frequency ratios 1,2,3,....
The set of natural vibrating frequencies of a tuba (or any instrument) may be approximately harmonic but usually deviates from that integer relationship. This means that the waveform emitted is not truly periodic. Only a periodic waveform may be described in terms of components which are harmonically related.

[mark38655]

One important point. You wrote:

The combination of all the natural vibrating frequencies of an object is known as the harmonic series, sometimes called the overtone series.

The harmonic series is a set of pure tones with frequency ratios 1,2,3,....
The set of natural vibrating frequencies of a tuba (or any instrument) may be approximately harmonic but usually deviates from that integer relationship. This means that the waveform emitted is not truly periodic. Only a periodic waveform may be described in terms of components which are harmonically related.

It is important to also know that instrument manufacturers have devised methods for manipulating brass instruments in a way that alters the tuning of the notes within the harmonic series to better match the equal tempered scale currently in use.

[sloan]

One important point. You wrote:

The combination of all the natural vibrating frequencies of an object is known as the harmonic series, sometimes called the overtone series.

The harmonic series is a set of pure tones with frequency ratios 1,2,3,....
The set of natural vibrating frequencies of a tuba (or any instrument) may be approximately harmonic but usually deviates from that integer relationship. This means that the waveform emitted is not truly periodic. Only a periodic waveform may be described in terms of components which are harmonically related.

I completely agree with this comment.

This is really basic - calling the "natural vibrating frequences of an object" the "Harmonic series" is simply wrong. Equating the harmonic series with an overtone series is similarly simply wrong. If you get this wrong, nothing that follows will make sense.

I recommend a thorough reading of Benade. Pay particular attention to the chapters on wood blocks, bells, and the like. Master the vocabulary by discussing instruments where the overtones clearly do NOT follow a harmonic series. Only then can you write clearly about instruments where the overtones are "close enough to lip" to a harmonic series.

Then - go and study instrument design. When you are finally completely astounded that anyone could possibly build a tuba that plays anywhere near "in tune" (in any tuning system) - THEN you are ready to try to explain it to other people.

There are already too many simplified glosses on overtones, harmonics, and tuning systems. Most of them are simply wrong (but...on the other side of the coin...often even an INCORRECT model can be useful).
I doubt that the world needs another.

[iiipopes]

Go read Hemholz, "On the Sensation of Tone." The definitive volume on the subject matter. Then extract out what you need for tuba.

[mark38655]

I'm not sure I get what your going after in terms of feedback, and in a larger sense, how this information is really pertinent and new to the tuba? (i.e. why you are writing this)

This following booklet has a concise yet very helpful and pragmatic text to accompany the intonation drone....
http://www.dwerden.com/intonation-helper.cfm" target="_blank" target="_blank

There are tons of resources on acoustics out there. It is presumable that if a tubists wants to learn acoustics I'm sir he can find the acoustic section in a library.

Thanks for responding.

I really do appreciate all sincere feedback and hope to use it to improve the quality of the instructional material. There is a section of text that I've not yet posted that is a how-to for learning to play in tune-- specifically how to match pitch with other instruments. After reviewing the info in that section, I decided that some background regarding basic physics was needed. So hopefully it will make more sense as to why I broached the subject once all the material is compiled and released.

I opened the link you posted and read the on-line booklet on the tuning CD and found it very helpful and informative. Very good material and I recommend it highly. Although it is similar to what I wrote, I don't think it presents exactly the same information, so hopefully a person could benefit from reading both texts. He does make a statement that the 7th partial is very flat and is unusable in western music which have to comment on. Blues musicians actually do use the 7th partial in their blues seventh chord. This is because it is very harmonious and less dissonant than is the equal tempered version of the chord. He is correct that from an equal tempered perspective, that the 7th partial is very flat though.

I believe this material is just as pertinent to the tuba as understanding how to play in tune is, and may be helpful to some students who want to understand the physics of the nature of sound. I'm not sure anything I wrote is actually new, but maybe my compilation of info will be unique in some ways compared to other tuba methods. It is certainly not all original material, nor are all the actual etudes. Yes, there are tons of books on the physics of music and others on complicated tuning methods, but my objective was to describe as simply and as concisely as possible some basic concepts of the science behind the tuning.

[Paul Tkachenko]

You are not correct to assume this about Bach's work. It was written to show the different 'colours' of that particular temperament. If you listen to a recording of it, some keys some keys sound pretty 'off'.

When I mentioned dividing the octave into 3rd, I was referring to people such as Sabbatini.

The idea of some temperaments having 'colours' is in common with makam and other modal musics.

No, you don't need to mention limmas and apotomes, but I guess I'$ suggesting that your current level of detail is unnecessary.

I would have thought a tuba player striving for the level you mention may benefit from a selected bibliography ...

[Paul Tkachenko]

When I say 'listen to a recording of it', I mean one in well temperament obviously ...

[mark38655]

You are not correct to assume this about Bach's work. It was written to show the different 'colours' of that particular temperament. If you listen to a recording of it, some keys some keys sound pretty 'off'.

I didn't make any assumption about Bach's work. Again, I quote from the New Harvard Dictionary of Music by Willi Apel, (ninth printing, 1999) p, 932: Well-Tempered Clavier, The “...The title refers to the use of a temperament in which all keys are satisfactorily in tune, but not necessarily an absolutely equal temperament. These collections are the first to exploit such a possibility fully.” Apel made no mention that Bach's intensions were to show the different colors of a particular temperament. He said the title refers to a temperament in which all keys are satisfactorily in tune. For the record, I don't doubt that Bach's work shows off different colors of a temperament. However, I did reword my statement to more accurately reflect the historical facts:

“J. S. Bach advocated the use of a temperament where all keys were satisfactorily in tune when he wrote his famous Well-Tempered Clavier, which consisted of preludes and fugues in all 24 major and minor keys. It was a compromise that significantly contributed to the use of the equal tempered scale today as the octave is now divided into 12 equal half-step intervals for most keyboard instruments.”

When I mentioned dividing the octave into 3rd, I was referring to people such as Sabbatini. The idea of some temperaments having 'colours' is in common with makam and other modal musics. No, you don't need to mention limmas and apotomes, but I guess I'$ suggesting that your current level of detail is unnecessary. I would have thought a tuba player striving for the level you mention may benefit from a selected bibliography …

You seem to have an impressive working knowledge of some of the technical aspects of various temperaments. I have no doubt that if a person were to be interested in pursuing this aspect of tuning that he could benefit from it. However, it goes beyond my personal interest at this time, and as an instructor beyond the scope of what I'd consider practical knowledge of a student performing in a modern orchestra or solo recital. A selected bibliography is something that I already intend to include in my book as you are correct that students might very well benefit from going to the various sources from which I derived my material.

[Paul Tkachenko]

Hi,

In a nutshell:

J. S. Bach advocated the use of a temperament where all keys were satisfactorily in tune when he wrote his famous Well-Tempered Clavier, which consisted of preludes and fugues in all 24 major and minor keys."

Yes.

"It was a compromise that significantly contributed to the use of the equal tempered scale today as the octave is now divided into 12 equal half-step intervals for most keyboard instruments.”

Not really, equal temperament was around at the same time and before. Galileo's father advocated equal temperament (for instance) and this pre dates Bach by at least 100 years.

You see my point? Bach knew about equal temperament (and mean-tone temperament) but did not advocate the use of it. Whilst your sentence above isn't strictly untrue, it is perhaps misleading to use Bach and this famous work with regard to (the development of) equal temperament, however tempting it might be.

You wouldn't be the first ... Hence why the urban legend developed.

I'm trying not to be pedantic. If you are aiming for a higher level of academic rigour then it may be worth having a think about whether this is the best work to mention in this context.

[mark38655]

"It was a compromise that significantly contributed to the use of the equal tempered scale today as the octave is now divided into 12 equal half-step intervals for most keyboard instruments.”

Not really, equal temperament was around at the same time and before. Galileo's father advocated equal temperament (for instance) and this pre dates Bach by at least 100 years.

You see my point? Bach knew about equal temperament (and mean-tone temperament) but did not advocate the use of it. Whilst your sentence above isn't strictly untrue, it is perhaps misleading to use Bach and this famous work with regard to (the development of) equal temperament, however tempting it might be.

You wouldn't be the first ... Hence why the urban legend developed.

I'm trying not to be pedantic. If you are aiming for a higher level of academic rigour then it may be worth having a think about whether this is the best work to mention in this context.

The fact that Bach's Well Tempered Clavier was the first of many compositions to require a keyboard to use a tempered scale makes it significant and on point regarding equal temperament. The fact that Bach advocated a different temperament than what eventually became the norm is insignificant. Were it not for the compositions themselves that require the performer to use chromaticism or modulate into unrelated keys within the piece of music, there would be no significant need for a tempered scale. Also, the extant knowledge of equal temperament prior to Bach's writing of his Well-Tempered Clavier does not preclude nor does it nullify the influence the composition had on the eventual evolution of the use of equal temperament in music.

[Paul Tkachenko]

Yes, Bach's work was, of course, very important in the development of the structure of keys that we know today.

It was not the first keyboard work to require a tempered scale. Look at the following book:

http://books.google.co.uk/books?id=vA6v ... ys&f=false" target="_blank

on page vii.

The Well Tempered Clavier is undeniably a landmark work in the development you mention. I guess the only point I am making is that care might need to be taken with the wording, making it clear that Bach did not use equal temperament and that equal temperament was used before his time. Although not the first to use a tempered scale on the keyboard, as the book puts it 'Bach broght this type of presentation to its Zenith a generation later'.

Even if you disagree with using this info in your book I hope this is of use and, at least, of interest ...

[mark38655]

Paul, I really do appreciate the information and your willingness to discuss it with me. I will revisit the wording and consider your comments. The text is already better because of your suggestions. And I was in error when I said that The Well-Tempered Clavier was the first such composition. I meant to say the first significant composition, but that may not be true either. We do agree that it was a very significant composition but I will look at the info at the link you provided as a reference.

Thanks again.

[Paul Tkachenko]

I was just thinking that if even one person decided to play around with different temperaments on the tuba, that would be great. So, keep up the good work!

[mark38655]

https://rapidshare.com/#!download|776tl ... 3E4F12|0|0

The above link is to a PDF of a Finale notation of the harmonic series for a BBb tuba. (Click the link, then choose "download" and then "free download".) It has additional notes at the bottom of the page that will appear in the book just prior to the text that I posted here earlier for discussion. I appreciate the suggestions offered by others and have edited my original post above. Hopefully it is in a form that is closer to what it needs to be when I release the book.

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.

Immediately following this text is a section entitled Introduction to Playing in Tune, which goes into the practical application of matching pitch, etc.

Joe (bloke) mentioned the possibility of wind players stretching the octave like piano tuners do. I don't know about the need for stretching octaves with wind and string players, but I've read about it regarding piano tuning and saw an actual diagram of where and how much the average stretching is done on pianos. I was surprised to find that most of it is done in the last two octaves of both extreme ranges on the piano (and especially the last octave), with very little stretching in the other ranges. (I used to tune pianos to help pay the bills.)

I've read that the reason this is needed on the piano is the "Stiffness of the strings". After scratching my head on that one for several years, I realized that (it might mean) that the position of the piano wire where the node that divides it into the 2nd partial (and others as well) actually utilizes a certain (although very small) portion of the total length of the wire. This tiny amount of wire that makes up the node is not vibrating on the 2nd partial and therefore is subtracted from the total length that makes up the two parts of the 2nd partial. So the 2nd partial is actually slightly sharp to the fundamental. It is more pronounced in the upper registers because the length of the node is dependent on the thickness of the piano wire and not the total length. Since the total length of wire used to produce the extreme high notes on the piano is just a few inches, the effect would be much more pronounced than on mid range and lower notes. However, using this logic, I don't have a clue why the extreme lower range has a need to be stretched on a piano, so my theory is possibly just BS.

[sloan]

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.
b) the "harmonic series" is more of a mathematical construct than something that "occurs in nature"
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.
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.
e) the "equal tempered scale" has almost nothing to do with a particular "harmonic series". In particular, the
various notes produced by the harmonic series are "out of tune" with the equal tempered scale
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?
g) as for the footnote...I hardly know where to begin...I surrender!