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acoustic8719
07-04-2003, 08:13 PM
I hope I dont get in any trouble for this but I really need help understanding this paragraph from a book I recently purchased. So here it goes...


From: Melody in Songwriting
by: Jack Perricone

"Harmonic Series: The harmonic series is our guide to what is natural, and therefore, is the best place to begin to study tonal music.

Every tone, with the exception of a pure sine wave, is made up of a composite of tones. These tones are called overtones, partials, or harmonics . The strength or amplitude of a partial is usally determined by its placement within the series; the closer to the fundamental, the stronger the partial."


My thoughts:


So basically, from what I understand is this... A note is made up of several different factors, which are called overtones, partials, or harmonics.

Is this right, wrong? also, "The harmonic series is our guide to what is natural": can someone reword this so maybe I could understand?

Michel
07-04-2003, 09:52 PM
A note is made up of several different factors, which are called overtones, partials, or harmonics.

YES...Hit any note call it the root ok?...you always get the same overtones pattern it goes
Root always up to
Root (octave up) Up to
The 5
Root
3
5
b7
root
9
3
#11
5
13
b7
root
So we get 1 3 5 b7 9 #11 13
in C it would go like
C E G Bb D F# A .... When you hear a C you hear all those notes a C is made of all these notes

..to make the scale C D E F# G A Bb C so you get the overtone dominant scale (Mixo #11)
If you are still whit me and not sleeping
:D
If we start the scale the 5 ...G
we get G A Bb C D E F# g and i'ts the G melodic minor
Fascinating
I dont no a lot more about this subject i just take it as it is
a C is a C and it's ok

Bye...And welcome

Michel
07-04-2003, 09:59 PM
The harmonic series is our guide to what is natural": can someone reword this so maybe I could understand?

i forgot to say the melodic minor scale is the "natural "minor scale

Bye

Bongo Boy
07-05-2003, 06:17 AM
Originally posted by acoustic8719 "Harmonic Series: The harmonic series is our guide to what is natural, and therefore, is the best place to begin to study tonal music.

Every tone, with the exception of a pure sine wave, is made up of a composite of tones. These tones are called overtones, partials, or harmonics . The strength or amplitude of a partial is usally determined by its placement within the series; the closer to the fundamental, the stronger the partial."
Let me take a little different approach to explaining. Most all objects that vibrate and create a tone do so at a frequency that is the objects 'natural' frequency. Simply because of the object's physical shape and because of how it was caused to vibrate, it vibrates best at some particular frequency, The easiest examples are strings (such as instrument strings) and tubes or bars--such as wind chimes, bells, pipe organ pipes, and tuning forks.

But, these physical objects also vibrate at other frequencies too--AT THE SAME TIME that they vibrate at their natural frequency. These other frequencies are all of less amplitude than the natural frequency, and there may be hundreds of these different frequencies, or tones. But, the strongest of them will be very close to small rational multiples of the strongest, natural frequency. A 'rational multiple' is a multiple that can be expressed as the ratio of two whole numbers, such as 2/1 (or 2), 3/2 (or 1.5) and 5/4, as examples.

What the author is saying, basically, is that when any physical thing vibrates, the vibration is actually the main, natural vibration, with many higher frequency vibrations also taking place. The higher these other vibrations are, the weaker they are relative to the 'fundamental' or natural frequency.

Are you familiar with setting up a harmonic in a guitar string by gently touching the string at say the 12th fret and plucking it to get a one-ocative higher tone? This is in effect changing the string to behave as though it were really two strings, each half the length of the original string. But, even when you pick the open string normally, without touching it at the 12th fret, it still also vibrates at that higher octave frequency. You may not hear it, but it's there.

What makes one 'E' string sound different than another 'E' string on the same guitar is all those other frequencies that one string vibrates at that the other does not--or with a different mix of amplitudes. It's why two opera singers singing exactly the same 'A' note sound like different voices--it's the huge mix of all the other stuff that is not 'A' that gives each voice its own character.

Maybe a picture will help. Below are three lines. You can think of the lines as the magnified or exagerated shape of a guitar string, or of a sound wave itself, it doesn't matter.

The first line represents the shape of a 'perfect' string at an instant while it vibrates at its first, fundamental, natural frequecy. The entire string is in motion, with the greatest motion in the center of the string.

The second line represents the shape of that same string, but with a 'node' at the center of the string--the node you create with you finger when you touch the string at its midpoint as you pluck it. Perfect, theoretical nodes don't move at all--real nodes on real strings are simply close to motionless. Each half of the string now vibrates--moves up and down--as though it were a separate string, separate from the other half. It's natural for each of these halves to vibrate twice as fast as the whole string would as in the first line.

The third line represents more of what a real string would look like if you greatly exaggerated the displacement of each location on the string. Notice much shorter, tiny 'ripples' in the string. They represent even higher frequency vibrations--with many nodes on the string.

The first two strings represent 'simple' shapes, or simple waveforms if you were to imagine they represented the sound wave produced by the string and not the string itself. Such simple shapes don't actually occur in physical objects--although they are the predominant shapes, and therefore represent the predominant sound you hear.

But it's the real shape represented by the third drawing that gives the string its unique sound--and that shape is determined by stuff like: is the string nylon, bronze or steel, was it plucked by a pick or a finger, or was it being damped by the player's hand, etc. Lots of things.

BUT...that complex wave shape in the third drawing can be approximated by ADDING a number of simple wave shapes, such as those in lines 1 and 2. And that's what your author means by a tone being a "composite of tones". Wave shapes--their frequencies and amplitudes--are in effect added together or superimposed on one another to produce a single complex tone.

Hope this helps a bit--I wish I could think of a better example, but think of a huge suspension bridge that gets hit by a ship. You might imagine the bridge would vibrate at its natural frequency that's so low it can't be heard--maybe one cycle every 4 seconds. But, you might also imagine the bridge ringing like a bell or a gong. BOTH the extremely low vibration (the one that will break the bridge) and the ringing tone are occuring at the same time, and add toghether to form one single 'sound' wave.

Doug McMullen
07-05-2003, 04:33 PM
also, "The harmonic series is our guide to what is natural": can someone reword this so maybe I could understand?

Ok, you understand that overtones are created when a string, or column of air, or what-have-you oscillates. The specific overtones created are quite predictable. There is a fundamental tone and then a series of additional tones that come along with it -- these additional tones are proportional to the original tone... if we know the note we are playing, we can calculate the overtones created.

The formulae for calculating the overtones are very simple --

They look like this (these are NOT the actual formulae):

X times 5/4
X times 4/3
X times 3/2

etc.

So in my example if you play an A 440 then an A note would have overtones of 440 x 5/4 and 400 time 4/3 etc. (These numbers are made up by me for ease of comprehension... but the real overtone reseries is similar ... I also don't have the real series handy or remember what it is... I do think X times 3/2 is one of the real partials)

The bottom line is whenever you play a single note you are playing within that single note several overtones, which are notes ... so you are playing a submerged chord. If you could balance the sound to hear the fundamental note and the first few partials at equal volume, you would hear a chord.

This chord generated by the overtone series is sometimes called the "chord of nature." Much has been made of this "natural" chord (and how all harmony must be based on it etc.) and in turn the theory that all harnmony is based on the chord of nature has been debunked about a zillion times.

In western music theory, traditionally, notes are added to chords in a series of stacked thirds that, at first, seems in keeping with the way notes occur in the harmonic series.

This is how traditional western harmony is based on sonic principles upheld in nature itself... except of course this is just cultural bias and complete horsegack and what about the X chord and Y chord and Z chord which sound great when they shouldn't or sound lousy when the shouldn't etc etc.

It is a very old fashioned debate... Is the harmonic series the foundation of harmony is as current a question as: "Must the well-dressed man wear sock-garters?"

At any rate, the overtone series is very pertinent to acoustic engineering. IMO the overtone series has nothing to do with practical harmony for a student musician.

Starting a book on harmony for songwriting with a discussion of the overtone series is probably just an example of pedagogical puffery. It's a way for the author to make himself look smart and in command of his subject. In other words... it's pretty much a load of bullsh#t. Which doesn't mean the book is bad or full of incorrect stuff... good authors as well as bad sometimes engage in this stuff.

There's certainly no harm in knowing about the physics of sound and music... but in terms of practical consequences for a musician, the harmonic series is a complete non-factor. If you were going to re-tune the musical alphabet or design a musical instrument the harmonic series might well figure into your thoughts.

Doug.

szulc
07-05-2003, 05:00 PM
Gee Doug,
I hand my Curmudgeon Crown over to you and bow!


Much has been made of this "natural" chord (and how all harmony must be based on it etc.) and in turn the theory that all harnmony is based on the chord of nature has been debunked about a zillion times.

Can you elaborate on this topic?
Do you mean because we use Mean or equal temperment systems?
What chord is supposed to sound good that doesn't or sound bad that sounds good?
Isn't that pretty subjective anyway?

Zatz
07-05-2003, 07:59 PM
Doug, Bongo Boy,

great insight into overtones!

To me the "natural" chord has always been a simple major triad in which I saw a philosophical aspect being that nature is joyful, full of harmony and consonance. Then goes dominant 7 chord. But the m7 is slightly out of equal temperment frequency but still fits OK on top of major chord.

Overtones theory gave me mostly the understanding of P5 interval importance, clue to the reason why power chords sound so raw and natural. As to dominant 7 - don't we know that we can use it on about any degree in progression instead of the chord of different quality but with the same bass without much harm to overall harmony? There might be different applications of overtone theory - but they will surely rather quench one's musical curiosity that push one forward in his playing.

But who knows... Maybe this topic is worth thinking over once again...

Zatz.

Doug McMullen
07-05-2003, 08:43 PM
Curmudgeon Crown? I thought it was more of a torch (of the eternal flame variety) -- it just gets passed from hand to hand...



Can you elaborate on this topic?
Do you mean because we use Mean or equal temperment systems? What chord is supposed to sound good that doesn't or sound bad that sounds good?
Isn't that pretty subjective anyway?


Tempered tunings cause problems for the chord of nature, so do Debussy/Evans style chords built in fourths.... but I'm no expert on any of this stuff and shouldn't talk about what I don't know about. IMO the "subjective" comment hits the nail directly on the head.

Harmony is a subjective field. Anytime the terms good and bad are involved we are in a subjective place... Physics can explain a vibrating string... but the goodness or badness of the sound is determined culturally.

European culture at one time liked to assert itself as the 'highest, most advanced culture.' The Arts of Europe were the Crowning Glory of Human Endeavor ... . And musical art was the greatest of Europe's arts its most "universal" triumph.

Given the then (mid 19ty century let's say) prevailing notions of European culture superiority -- it's easy to understand why European thinkers were interested in theories that validated the European approach to music objectively. The harmonic series gave empirical evidence of the rightness -- the mathematical universality -- of European harmony. As a corollary, anyone +not+ following european harmonic schemes was making a substandard music.

Now, in the era of world music, and nearly a hundred years or so past the acceptance of jazz and blues harmony and the atonal experiments of Schoenberg, the scientific basis of harmony looks sort of silly.

There's no arguing that there is a "chord of nature" ... but how much involvement that chord has in determining what chords "work" and "don't work" for contemporary tastes-- is a question that just doesn't need answering, or at least, one that makes no difference to a musician. What works and what doesn't work for the western harmonic sensibility has been in constant evolution since the medieval period.

Doug

Schooligo
07-05-2003, 10:16 PM
Wow,

Doug, I respect the way you replied to Szulc's question, IMHO your response was well thought out, intellectual, & Professional!

"Anytime the terms good and bad are involved we are in a subjective place... Physics can explain a vibrating string... but the goodness or badness of the sound is determined culturally."

I will agree that Culture is at least one significant Factor to consider, and a very important Factor at that!

"In truth what works and doesn't work for the western harmonic sensibility has been in constant evolution since the medieval period."

Some extreme Music Theory historians may debate that this is true, for the intent of this topic once again I agree!

Excellent response, Well Said!!!

szulc
07-05-2003, 10:53 PM
I especially like this line!

In truth what works and doesn't work for the western harmonic sensibility has been in constant evolution since the medieval period.
I guess this is what I was gettting at by subjective.

In general I do think it is a good Idea to learn about the harmonic series, so you have some historical information about where western concepts of harmony are from. I do think you were a little harsh on the author (but who am I to talk about that!)
I also am partial to things in fourths and fifths!

I like your answer and it does bring up the whole idea of Eurocentricism, which has dominated so called intellectual and artistic pursuits for centuries.

How civil we are all being here!

Cool!

Bongo Boy
07-06-2003, 04:10 PM
I think you guys have added an entirely 'new' topic to the thread--at least as I see it. The original question had more to do with what it means to say a real tone is composed of a fundamental and overtones. At least I interpreted the problem that way--IOW, what does it mean to have two tones 'superimposed' over one another?

The topic I think you added that may really complicate things for the original poster is the difference between natural harmonics and those of tempered tuning, PLUS the implications for 'psychoacoustics' or the human response.

What I wonder is, did we provide an understandable and correct description of how a set of frequencies, such as a fundamental and its overtones, occur simultaneously in a real tone?

Also,


The specific overtones created are quite predictableIf the definition of 'overtones' is that set of overtones that are rational multiples of the fundamental, then okay. But isn't it also true that for any real vibrating object, such overtones will not be the only additional vibrations that accompany the fundamental? You would also expect quite a bit of 'noise' as well--although I guess what makes a great instrument is that these non-harmonic components are acceptably inaudible relative to harmonics. Am I right?

szulc
07-06-2003, 05:27 PM
If you are involved in systhesis you will be used to the concept of adding components together to create the various sounds.

Your fundamental tone building block is a sine wave, you add them together to create various wave forms to emulate different instruments. For instance a flute or piccalo has a mostly sinusoidal wave form with litte harmonic content (Except at the attack and release). But a cello or saxaphone have many harmonics.

White noise is equal amounts of every frequency component.
Pink noise is White noise that has been filtered to sound to your human ears as thouh all the frequency componants are equal.

If you add all of the harmonics together along with the fundamantal you get a square wave. (As you keep adding harmonics the corners full in) Distorted guitar at its best has mostly even harmonics. When using a solid state stompbox of the 60's or 70's variety your wave becomes almost square. That is why a chord sounds like pink noise through one of these.

Schooligo
07-06-2003, 09:46 PM
Has anyone read this book?

If so does the Author sufficiently explain the quote above & answer Acoustic8719's question, so he/she can understand the importance the Author is placing on this quote and topic?

My experience is that Author's who write like this, and will not answer this question sufficiently throughout the rest of their book (after stating the above quote in question), use this quote as a way to gain credibility with the reader so that he/she respects the Author's knowledge of the subject.

Unfortunately there is a "downside" to using a quote in this manner and that is, that it confounds the reader so that they don't understand the material &/or the importance of Priority the Author places on this topic.

Thereby hindering the teaching process, since understandable communication to the reader is so important when teaching a concept.

Scenario:
A musician who may have Musical theory knowledge(maybe by studying Guni's Musical Theory articles, or studying in College Music Theory classes, etc.) or may have the "bare minimum" amount of knowledge to create music,

decides he/she would like to learn more about writing songs and creating melodies.

To gain knowledge in order to meet this goal he/she buys a book about the topic of writing melody's in songwriting.

He/she decides to purchase
"Melody in Songwriting
by: Jack Perricone"

During the 1st chapter the Author quotes this statement

"Harmonic Series: The harmonic series is our guide to what is natural, and therefore, is the best place to begin to study tonal music.

Every tone, with the exception of a pure sine wave, is made up of a composite of tones. These tones are called overtones, partials, or harmonics . The strength or amplitude of a partial is usally determined by its placement within the series; the closer to the fundamental, the stronger the partial."



the reader after reading this quote doesn't understand this quote, but considers it significant enough(because the Author considers it significant) to warrant research on this topic. Even at the expense of frustration & may choose not to continue to read the book until he/she get's enough of a satisfying answer to importance of this quote.

So he/she creates a Thread in IBreathe to get an answer &/or at least to help explain the quote so he/she can understand it enough to attach an importance for Priority and if this knowledge is useful at present.

Harmonic Analysis with Harmonic Series & the Overtone series is one of many HUGE topics(that includes Mathmatical Analysis & Application, Physics, Physical Science, Music Theory, etc.)

based on the INTENT of this Thread, AND that the reader is trying to learn about Melody in Songwriting,

and that many of the IBreathe Community contributed important insights,

IMHO I believe (for the majority of Musicians as well as Acoustic8719) the question has been answered sufficiently.

P.S. Bongo,

I don't think this thread was off topic for any significant amount of time since the importance of the Priority of much of this topic can be SUBJECTIVE, yet I think it's cool how your concerned about the integrity of this thread & that acoustic 8719's question is satisfactorily answered.

Doug McMullen
07-07-2003, 03:26 AM
hey Bongo --

You are IMO absolutely right about there being different things being talked about in this thread concerning sound and overtones. And yes it has confused the issue a bit. Respectfully though, I submit that you are the guilty party in going "off-topic" and confusing the issue.

The acoustic property your post addressed is the one that is involved in the production of "timbre" ... what makes a trumpet playing C sound distinctly different from a flute from a guitar from a clarinet? ... it is the distinctive timbre of the sound... which is what most of those waves within waves you described (and pictured) produce.


You would also expect quite a bit of 'noise' as well--although I guess what makes a great instrument is that these non-harmonic components are acceptably inaudible relative to harmonics. Am I right?

No not right. Synthesizers produce 'pure' waves. They sound interestingly unnatural, but also rather generic. What makes a great instrument great are the unique impurities.

None of which has anything to do with harmony. Timbre has _very_ little to do with harmony. I can think of no likely reason why the author of a songwriting manual would begin a discussion of harmony with a discussion this kind of acoustic phenomenon.

The overtone series is a different thing and _does_ have to do with harmony... although as I mentioned in my earlier posts I believe much too much has been made of this possible relationship.

By the way, Szulc... you mentioned you thought I was too harsh on the author and I completely agree. Yes, that was curmudgeonly of me. I mean _maybe_ the author was being a poseur, but maybe not... maybe it is a brilliant bit of exposition that I should read and learn from .... since I haven't read his book how on earth would I know what he's really about?

At any rate, I really should learn to stay away from offering these arm-chair psychology based ad hominem attacks (I've done it before, and have always gotten in trouble for it). I've got nothing against Jack Perricone, whoever he is -- All I really wanted to suggest was that IMO the overtone series is not necessarily the fundamental fact of harmony and one needn't begin the study of harmony from this "natural" chord.


Doug.

metaljustice83
07-07-2003, 03:31 AM
My head hurts :( j.k j.k cool stuff....wait my head does hurt ;) cool stuff

Bongo Boy
07-07-2003, 04:06 AM
You are IMO absolutely right about there being different things being talked about in this thread concerning sound and overtones. And yes it has confused the issue a bit. Respectfully though, I submit that you are the guilty party in going "off-topic" and confusing the issue.

The acoustic property your post addressed is the one that is involved in the production of "timbre" ... what makes a trumpet playing C sound distinctly different from a flute from a guitar from a clarinet? ... it is the distinctive timbre of the sound... which is what most of those waves within waves you described (and pictured) produce.Exactly. I didn't think the original question was about harmony so much as it was about how components can combine to form a tone. I see your point though--by trying to clarify something I did indeed confuse the issue. Probably nothing new there :)

Bongo Boy
07-07-2003, 05:14 AM
Going back through this thread I still feel the number of topics is rather large, and the relationship between them isn't clear to me. Here's what it seems we're talking about--all at the same time.

First, there's the idea of natural harmonics. Overtones that are produced when we play a single fundamental note on an instrument. Nature, if you will, decides what these overtones are, and they have little to do with tunings or musical harmonization.

Second, we're talking about the ideas that musical harmony is, in the view of some, based the natural harmonic series. I interpret this to mean that, at least at some point in history, harmonization was felt to be 'correct' only when done using tones within the harmonic series of some fundamental, or root, tone.

Third, the idea of tempered tuning was brought up. The comments seemed to suggest that tempered tunings led to harmonization NOT based on the harmonic series. I interpreted the comments that way because they were made in the context of historical preferences for some harmonizations over others. The idea of acceptable harmonization over the course of history came up as well, and this was also in the context of the need (or lack of need for) a scientific (or mathematical) basis for harmony.

I thought one purpose of well-tempered tuning was to provide notes that form a close approximation to the natural harmonic series, but which have the advantage that all notes are higher in pitch than their next lower neighbors by exactly the same factor. I'd say this tuning has as its basis the harmonic series--no question--but, as mentioned earlier, that 'natural chord' won't be generally possible in a tempered tuning, I guess.

But whether harmonization uses notes from a tempered scheme or from a natural harmonic series--either one--it will still be subject to contemporary 'quality' evaluation from a completely unrelated perspective (such as consonance/dissonance). And both acceptable and unacceptable harmonies are still possible with either scheme. Is that a reasonable interpretation?

Because we use a system of pitches that are NOT exactly equal to the natural harmonics, when we harmonize on an instrument, the chord tones we play will seldom exactly 'match' any of the overtones of the instrument, correct? But, are they not just as close as we would ever hope to get them even using 'perfect' pitches (pure harmonics)?

Doug McMullen
07-07-2003, 03:28 PM
Because we use a system of pitches that are NOT exactly equal to the natural harmonics, when we harmonize on an instrument, the chord tones we play will seldom exactly 'match' any of the overtones of the instrument, correct? But, are they not just as close as we would ever hope to get them even using 'perfect' pitches (pure harmonics)?

This is a complex subject... A recently published book, and rather a good read, is "Temperament" by Stuart Isacoff (hope I got the name right .... I'm looking for my copy of the book and can't seem to find it) which gives a history of tuning and the struggle to adopt tempered tuning in the west.

The best minds, artistic and scientific, of Europe struggled with the question of tempered tuning (is it musical!?) for several hundred years... if folks like Isaac Newton and Leodardo Da Vinci had their moments of confusion regarding tuning I suppose we can be forgiven for making a stew of the conversation, here.

But... we are making a real hash of it!

When you say: "when we harmonize on an instrument, the chord tones we play will seldom exactly 'match' any of the overtones of the instrument"

Well, I'm not sure we are in real agreement on some of these terms... and we are also getting into acoustics. Acoustics is an _extremely_ complex field.

At any rate, one of things that makes an instrument, an instrument, is that the fundamental tone and meaningful partials are amplified. If the tone were too pure you would have a synth tone... too impure and you have noise. But somewhere inbetween pure and noise is the tone of a Guarneri cello.

Tempered and untempered tunings are a related but different issue from the harmonic series. A fretless instrument like a cello can play any tuning, tempered or untempered. Whatever scale the fretless instrument plays each individual note will have, zinging around inside it, the upper partials (and any other stray tones reflected from the wood and it's peculiar resonances).

I don't know what else to say or how to bring this conversation to a close.

Doug.

Bongo Boy
07-08-2003, 12:44 AM
I agree with all you've said. I think any further attempt to discuss this without a whiteboard may lead toward violent argument! :D

szulc
07-08-2003, 01:13 AM
But somewhere inbetween pure and noise is the tone of a Guarneri cello. I believe that this (a Guarneri cello, in the hands of a master like Yoyo Ma) is the eptiome of tone! It is the perfect balance of sine waves and noise!
I would never have picked up a guitar had I started on Cello instead of violin! (and If my bowing tone was sound!)

Koala
07-08-2003, 04:12 AM
Id have to say its either that, or a distant oboe, it just sounds so tight...

But if any of you dont agree we can just settle it right here right now

jk.-

Koala
07-08-2003, 04:14 AM
Oh yeah i forgot to ask acoustic8719 if he finally got a grasp of what he was reading.

Bongo Boy
07-10-2003, 01:32 AM
I didn't have the courage to ask...;)

WaterGuy
07-10-2003, 05:19 PM
As a latecomer to this thread, I would just like to tip my hat to all of you. Bongo, Doug, szulc – all of you have done an outstanding job of describing the overtone series and how overtones effect (or don’t effect depending on your view) everything from tambre to theory. All of the tangents were appropriate in my opinion. This has been one of the best threads to read in a long time. And Doug gets the award for using “pedagogical puffery” effectively outside of a university.

To perhaps tie all this back in for acoustic8719… I agree that what sounds “good” or “bad” is largely a matter of culture influence and personal taste, and as such, the Perricone’s use of the harmonic series as the “guide to what is natural” can be misleading. However, a study of the harmonic series, and in particular frequency ratios, can be useful to explain in mathematic terms why any interval sounds the way it does by relating the sound to the complexity of the ratio and/or how the notes relate to each other through the harmonic series. For example, the octave and the perfect fifth are the first intervals encountered in the series, and the frequency ratios are fairly simple numbers. This helps explain why those intervals are so stable but uninteresting. The major third is further down the series and the frequency ratio is slightly more complex. This helps explain why, although stable, the third injects more flavor. This sort of rationale can continue through the series all the way to the tritone, which is quite complex all the way around. Ascribing judgments like “good” or “bad” to any of these intervals (in either just-intonation or equal-temperament) is again more a matter of what you are used to, so this sort of analysis doesn’t help much with music theory. However, as an abstract exercise, it can be very helpful in ear training, which has great benefits for all around musicianship.

Hopefully that helps… or maybe I just opened a whole other can of worms…
:p

Shaun
07-11-2003, 01:51 AM
Wow~! I absolutely got no idea where you guys learn all this from. But I may have understood at least a bit. Throughout this entire discussion (which has been a very informing one) do we actually get down to the fact that all intervals sound the way they are because of these overtones that are produced when playing simple note?

Bongo Boy
07-11-2003, 11:14 PM
Originally posted by Shaun
Throughout this entire discussion (which has been a very informing one) do we actually get down to the fact that all intervals sound the way they are because of these overtones that are produced when playing simple note?What's the connection between intervals and overtones? I don't understand your comment.

Shaun
07-14-2003, 12:37 AM
Earlier in this discussion, i think Doug or Bongo Boy said something like chords sound or maybe are constructed the way they are because of the overtones that are produced when playing a simple note. Either one of guys said that the overtones cant be heard with the naked ear but there *ARE* overtones. So, from what i understood from this discussion is : intervals in chords or scales, are they the reflection of the overtones? Or, do the overtones determine the intervals? And also, something came to me recently, do overtones have both major and minor qualities?

Bongo Boy
07-14-2003, 04:38 AM
Okay, I understand.

The first 15 harmonic overtones of a fundamental tone, along with that fundamental, form the partial tone series, or simply the partial series. These 16 tones form the basis for theory of harmony--subject to all the comments others have made above. Each of the 15 overtones in this series has a frequency that is an integer multiple of the fundamental--the nth overtone is n times the fundamental's frequency.

So...if you were to choose as your fundamental tone C = 64Hz, then the first partial would be the C appearing two lines below the bass clef at 64Hz--the partials would be:

1. C = 64
2. C = 128
3. E = 192
4. C = 256
5. E = 320
6. G = 384
7. Bb = 448
8. C = 512
9. D = 576
10. E = 640
11. Gb = 704
12. G = 768
13. Ab = 832
14. Bb = 896
15. B = 960
16. C = 1024

This particular list illustrates that, while the overtones for the chosen fundamental may form a common triad (for example, C256, E320 and G384), those overtones are not overtones of the ROOT of that triad--that is, E320 and G384 are not members of the partial series for C256, as I understand these definitions.

All I can say is that, within a scale or a particular chord, the overtones of the chord's root MIGHT be exactly the same tone as other voices in the chord--they might not be. Conversely, some overtones of a note that is in a particular scale will definitely be members of that scale (any overtone that's an even multiple 2n of the fundamental will be n octaves above the fundamental), and some will not be members of that scale.

As far as major and minor intervals--you can see from the table above that the fundamental and a given overtone may form any one of a number of interval types.

Someone else may be able to draw a conclusion or some significance between these related topics (overtones and intervals), but I can't--I just don't have that depth of understanding. I would expect very large books have been published on this 'simple' topic alone.

Bongo Boy
07-14-2003, 05:39 AM
Originally posted by Bongo Boy Because we use a system of pitches that are NOT exactly equal to the natural harmonics, when we harmonize on an instrument, the chord tones we play will seldom exactly 'match' any of the overtones of the instrument, correct? But, are they not just as close as we would ever hope to get them even using 'perfect' pitches (pure harmonics)?What I was getting at in this obtuse comment may be illustrated in the table below, where I'm depicting a triad CEG. Across the top row of the table are the partial numbers 1 thru 10 for each of the three tones. In each row to the right of the note names are the frequencies of those partials. I've put all overtone frequencies in bold that appear more than once. So...these are overtones that exactly the same as overtones for other notes in the chord.

Now, these numbers are all based on the frequencies of the natural partial series originating from C = 32, which puts middle C at 256. What would the situation be if you played this same chord on a real instrument tuned 'normally'? That is: with C = 261.626, E4 = 329.628 and G = 391.995? In that case, can you expect ANY overtones of one note to also be overtones of any other in the chord?

I have no point here guys--I just thought it was interesting.

Bongo Boy
07-14-2003, 05:49 AM
While the above table might be called 'Natural Chord Partials', one might refer to this one as 'Tempered Chord Partials'. In this concept the same triad is played on an instrument with the same 10 perfect overtones, but the instrument is tuned to a tempered scheme. This might be one (rather goofy) way of showing just one reason why you might expect tempered tuning to change how performed music would sound--matching overtones are mismatched by a few beats per second.


Doug had saidTempered and untempered tunings are a related but different issue from the harmonic series....and this post and my previous one form an attempt to understand one relationship between the two (even if my attempt fails or is outrageously naive). As always, comments are strongly solicited.

magus1234
07-15-2003, 05:51 AM
man this is really cool stuff,im really learning alot!Thx bongoboy for you large contribution!

Bongo Boy
07-18-2003, 08:24 PM
Thanks...if you see any relevance to anything important, be sure to let me know! :)

jazz_cat
07-24-2003, 03:30 PM
Hi everybody,

I'm very interested in music as well as in mathematics and physics so I hope to bring
some light into this discussion by pointing out the mathematical
nature of this whole tempered/natural tuning thing...
I don´t want to confuse anybody, but this is not an easy topic
and simplifying things too much might not help either.


Intervals
=========

Definition
----------

What´s an Interval? In music we say that it´s the "distance"(see note*) between two notes.
In physics it would be more accurate to say that it´s the ratio between the two frequencies:

Interval = frequency of higher note / frequency of lower note.

Example: f1 = 440Hz, f2 = 880Hz, Interval = f2/f1 = 2.000
That´s the Interval of an Octave, as you might know.
The ratio may also be expressed as a fraction: 1:2 or 2:1, depending on which note you put first.
But keep in mind that the interval is 2.000 (2:1) and not 0.500 (1:2), so when calculating the actual number
always think "bigger divided by smaller".

2:1 is the simplest way of writing the interval as a fraction, but
4:2, 8:4, ... would also be possible. Sometimes it´s possible to "chain" these fractions to express the intervals
between more than 2 notes. The intervals between a note and all it´s octaves can thus be written as:

1 : 2 : 4 : 8 : 16 : 32 ....


(note* in fact it´s correct to call it a "distance" because musical notation is sort of a
logarithmic scale with the base of 2, where ratios appear as distances)


"Stacking" Intervals
--------------------
Now suppose we have 3 notes at the frequencies f1 = 220Hz, f2 = 330Hz and f3 = 440Hz
The interval between the first two is f2 / f1 = 1.500
The interval between the second and the third is f3 / f2 = 1.333
Two calculate the interval between the first and the third note we have 2 possibilities:

a) as usual, calculating the ratio f3 / f1 = 2.000
b) multiplying the 2 intervals calculated before: 1.500 * 1.333 = 2.000
If an interval is composed by several smaller ones, it can be calculated by multiplying these.



The harmonic series
===================

The theory behind the harmonic series is quite complicated(see note**), but the result is easy to remember:
In our fractional notation the frequencies of a note and it´s harmonics relate like

1 : 2 : 3 : 4 : 5 : 6 : 7 : 8 : ...

So a note of 100Hz has harmonics at 200Hz, 300Hz, 400Hz, 500Hz, 600Hz, 700Hz, 800Hz, ...
Note that all the octaves of the note occur in the series:


1 : 2 : 3 : 4 : 5 : 6 : 7 : 8 : ...
^ ^ ^

Let´s calculate the intervals between adjacent harmonics (think: "bigger divided by smaller")!

2 : 1 = 2.000
3 : 2 = 1.500
4 : 3 = 1.333
5 : 4 = 1.250
6 : 5 = 1.200
7 : 6 = 1.167
8 : 7 = 1.143
.
.
.


What´s the names of these intervals? How do they relate to the interval names we know (second, third, fourth,...)?
Up to now we only know that the interval 2 : 1 is an octave.
To answer this question we have to have a look at the system we are using all the time, namely the

(note** wave equation and fourier series)


Well tempered tuning
====================

I don´t want to (nor am I able to) talk about the historical development and motivation
of this tuning but will give only the "hard facts".


What is "Well tempered tuning"?
-------------------------------

The interval of an octave (2 : 1 = 2.000) is "divided" into 12 equal parts, i.e. 13 notes are placed
within an octave, the intervals between any 2 adjacent notes being the same:

13 notes with frequencies f1, f2, f3, ....., f12, f13

f13/f1 = 2.000

f2/f1 = f3/f2 = f4/f3 = ... = f13/f12

Don´t be confused by the presence of 13 (instead of 12 notes). The last note is an octave away from
the first and thus has the same name, so we only have 12 different notes, the 1st occuring also an octave higher.

The interval between 2 adjacent notes is called a "half step" (HS):
HS = f2/f1 = f3/f2 = f4/f3 = ... = f13/f12
If we stack (multiply!) a half step 12 times, the resulting interval must be an octave:

HS * HS * HS * HS * HS * HS * HS * HS * HS * HS * HS * HS = 2.000

A friend of mine who is good in mathematics calculated

HS = 1.059

We are now able to calculate all the ratios of all intervals we know:


# of half steps interval ratio

0 1 1
1 b2 1.059
2 2 1.122 (= 1.059 * 1.059)
3 b3 1.189 (= 1.059 * 1.059 * 1.059)
4 3 1.260 .
5 4 1.335 .
6 #4/b5 1.414 .
7 5 1.498 .
8 b6 1.587 .
9 6 1.682 .
10 b7 1.782 .
11 maj7 1.888 .
12 8 2.000 (= 1.059 * 1.059 * 1.059 *
1.059 * 1.059 * 1.059 *
1.059 * 1.059 * 1.059 *
1.059 * 1.059 * 1.059)



But how do these ratios relate to the harmonic series?



Well tempered tuning and the harmonic series
--------------------------------------------

We already calculated some of the intervals in the harmonic series:

2 : 1 = 2.000
3 : 2 = 1.500
4 : 3 = 1.333
5 : 4 = 1.250
6 : 5 = 1.200
7 : 6 = 1.167
8 : 7 = 1.143

Do you see any connection to the well tempered system?
Well, there is hardly any!!!
The ONLY thing they have IN COMMON is the interval of the OCTAVE, which has a ratio of 2.000.

The other intervals in the harmonic series can only be APPROXIMATED by the well tempered system:

3 : 2 = 1.500 approx.= 1.498 (7 half steps, perfect fifth)
4 : 3 = 1.333 approx.= 1.335 (5 half steps, perfect fourth)
5 : 4 = 1.250 approx.= 1.260 (4 half steps, major third)
6 : 5 = 1.200 approx.= 1.189 (3 half steps, minor third)
7 : 6 = 1.167 approx.= 1.189 (3 half steps, minor third)
8 : 7 = 1.143 approx.= 1.122 (2 half steps, major second)


Note that the approximation becomes less and less accurate, so I wouldn´t dare
approximating more intervals than shown (starting from C):

C C G C E G bB C

Of course you could continue with the famous "beginning of lydian scale":

C C G C E G bB C D E F#

but then you have to keep in mind that the intervals between C D E and F#
are

8 : 9 : 10 : 11.

So you state

9 : 8 = 1.123 approx.= 1.121 (2 half steps, major second)
10 : 9 = 1.111 approx.= 1.121 (2 half steps, major second)
11 : 10 = 1.100 approx.= 1.121 (2 half steps, major second)

When it comes to half steps it gets even worse. The well tempered system is just not the proper tool
to describe the harmonic series over more than 3 octaves.


Still there? Hey, come on, WAKE UP!!!
Hope you enjoyed my contribution!

cu jazz_cat

Doug McMullen
07-25-2003, 01:36 AM
Hey Jazz Cat.

Great post. The clearest description of much of this stuff I've ever encountered. I'm embarassed to say I never knew or understood that the harmonic series broke down this way:


"In our fractional notation the frequencies of a note and it´s harmonics relate like

1 : 2 : 3 : 4 : 5 : 6 : 7 : 8 : ...

So a note of 100Hz has harmonics at 200Hz, 300Hz, 400Hz, 500Hz, 600Hz, 700Hz, 800Hz,"

Great post. The only thing that had me scratching my head a little was not mentioning that 1.059 is an approximation of the 12th root of 2.

Doug

Bongo Boy
07-25-2003, 02:00 AM
Very nice. Interestingly, the error between the integer ratios of the harmonic intervals and the non-integer ratios of the tempered approximation to that series does not simply increase with higher-order harmonics.

The chart below plots the ratio of the harmonic interval ratio to the tempered approximation of that ratio. So for example, the interval between 3rd and 2nd octaves is 3/2 (harmonic series), and 1.498307077 (tempered 'series'). So, at x = 3 on this chart, I'm plotting 1.500000/1.498307077; and so on for each of the other octaves indicated. It should probably be plotted as a histogram, not a continuous function, but it just looks so much cooler that way, how could I resist? How much fun is that?

Fortunately for all of us, you can see that the maxima/minima up to 4 or 5 octaves is less than about 2%--close enough for music and government work. Now, if I could play the freakin' guitar. :)

WaterGuy
07-25-2003, 02:16 PM
Bongo - That is awesome, and I'm glad to see that I'm not the only Excel geek around here. jazz-cat, nice job on breaking down the numbers.

We're not the only ones thinking about this stuff. The new issue of Guitar World has an interview with Jeff Beck who mentions how he's doing microtonal and Indian-influenced stuff with the wammy bar, and thinks that Western music is moving in that direction. Who knows, maybe we'll all be savoring the sweet harmonies found outside equal temperment in the 21st century...

Bongo Boy
07-25-2003, 02:52 PM
Yeah...just got the mag out of my box this am. So I guess we're in good company, then! I'd think he could afford a little better hair colorist, though. :)

Jago
08-01-2003, 07:32 PM
Hi, Just Joined here, so far ive found this thread, and indeed this site highly informative!, i'd like to thank all of the above guys for their contribution to my small knowlage of music theory, really very interesting stuff.thanks:D