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There is no harmonic or melodic combination
that is not implied in the very structure
of a single sound.
In order to accurately represent the differences in pitch between sounds, a graphic system must be used that takes into account the way the human ear perceives these differences.
The image below is a musical number line. It is an accurate graphical representation of the harmonic series, both ascending and descending. Every mark on the line represents a sound; the distance between sounds reflects the way we hear them.
The number line used to represent sounds has no direct connection with the divisions on the fretboard of stringed instruments.
This linear graph is just a simplified chart, a mathematical representation of the way we perceive differences in pitch, and does not represent the lengths of strings or pipes used to generate those pitches.
The fundamental or first harmonic and its oscillating frequency is the "1" in the middle. All other harmonics are multiples or submultiples of it.
For example, harmonic 2 of the ascending series has a multiplication sign before it "×2", because it oscillates two times faster than the fundamental. Undertone 2, being a harmonic of the descending series, has a division sign in front of it "÷2", because it oscillates two times slower than the fundamental.
Under the number line, the same values are gived using the mathematical concept of raising a number to a power, called exponentiation.1
The undertone series features negative exponents; the overtone positive. Normally, positive exponents do not have a "+" sign before them, but I'm writing it anyway, just to defy "normality".
Notice how the distance between 1 and 2 (in both directions) is the same as the distance between 2 and 4, the same as between 4 and 8, and so on.
The distance between 1 and 3 is the same as the distance between 3 and 9, the same as between 9 and 27. This is also true for multiples of 5: the distances between 1 and 5, 5 and 125 and so on, and also for all other numbers - both under and overtones.
The pitches were represented on three different levels to avoid confusion. They are part of the same group, although harmonics of such big numbers are unlikely to be observed naturally.
No matter how far they progress, harmonics of 2 will never add up to other harmonics - in fact, no series of a single pure harmonic interval will precisely match up at any point with another series of a single pure harmonic interval. 15
The numbers 1, 2, 4, 8, 16, 32, 64, ... form a series of doubles/halves of the same tone, while 3, 6, 9, 27, 81, 243, ... form successive triples/trisections.
The figures grow at an increasing rate, [soon becoming enormous,] yet musically we feel each interval to be the same. The rate of increase however is the same. 13
With other words, successive increments of pitch by the same interval result in an exponential increase of frequency, even though the human ear perceives this as a linear increase in pitch. 16
The exponential increase in frequency is evident when writing the numbers using successively higher powers. The linear increase in pitch is shown by the equal spaced marks on the number line.
This musical number line is called "logarithmic". The difference between a standard number line and a logarithmic one is this:
On a standard number line, the distance between each succesive value is the same.
On a logarithmic number line, the distance between each succesive exponentially increased value is the same.
The logarithm, invented by John Napier in 1614, is a mathematical operation that turns a multiplication into an addition, and by the same definition, raising to a higher power into multiplication. 17
A logarithm is simply an exponent that is written in a special way. We'll take a look at the formula in a moment.
According to Dr. Albert A. Bartlett, The Greatest Shortcoming of the Human Race is Our Inability to Understand The Exponential Function.
The Exponential Function is used to describe the size of anything that is growing steadily. The natural logarithm of 2 is used to calculate the doubling time of any growth. If you wanted to calculate tripling, you would use the natural logarithm of 3. 18
The natural process of doubling describes the genesis of life itself. Think of it. At the very beginning of our lives, we were only one cell. The first thing that cell did was to doubled itself. One became two, then two became four, then four became eight, and the doubling continued untill the numbers became enormous. We came into this world by doubling.
Notice that even if the logarithmic number line has as base the number 2 (meaning it progresses in succesive doublings and every interval based on the powers of 2 is equal), every interval based on the powers of 3 is again equal. Same is true for all numbers. The only difference between different bases is when we compare them with a standard number line.
A tone of any frequency will have all intervals of 2 equal, just as it will have all intervals of 3 equal, no matter in which base the number line is drawn.
Watch the progression of exponents from the picture above. The numbers 1, 2, 3, 4 and 5 are constant: the base 2 and 3 logarithmic number lines have these numbers in the same place as the standard number line, but they are up as exponents (or powers) of their respective base.
Thus the following formulas are the same thing, written in two different ways.
The formula containing the term "log" returns the value for the standard number line. It basically takes the exponent down, to the other side of the equation.
This is useful when you want to draw the musical number line yourself, using standard units of measurement. The doubling distances are all the same, but to figure out where overtone 3 fits in for example, you'll have to calculate log23.
The formulas and their meaning remain the same when numbers are written as fractions.
The same is true for negative exponents. These mathematic concepts applied to music are all you need in order to understand what music really is.
Logarithmic measurements reflect the way we hear pitches relative to one another. This is also true for the way we hear volume changes in sound.
I realized this while fading out songs. When using a linear fade, the sound seemd like suddenly coming to a halt. With a logarithmic fade, the song felt like smoothly going away into silence...
The three pictures and sounds below are a little song played on a piano tuned to harmonic overtones, from 1 to 16. The first is the initial, unmodified sound. The second has a linear fade out applied, while the third a logarithmic fade. There's no other processing so you may want to turn up the volume.
I'm really glad for arriving at this section. I never liked math and it took me a while to figure out all that stuff about logarithms and the exponential function. So let's cut right to the chase.
The Harmonic Series Is Nature's Tuning System.
The songs of birds and whales and the lullaby a mother sings to her baby are made of intervals derived from the harmonic series. The great living orchestra of Mother Nature is tuned according to the harmonic series and that's why the harmonic series is (or should be) the basis for every single note, on any musical instrument.
But is it really impossible to have an instrument tuned precisely according to the harmonic series? Not for me.
This family of musical notes is being used by every sound making life form on earth. The fact that such a system can be considered impracticable has nothing to do with its nature.
This is how it looks and sounds like:
The Undertone Series: starting from 1/1, the fundamental, and going down to the left.
The Overtone Series: starting from 1/1, the fundamental, and going up to the right.
The first note, the one, among the ancients is not yet even considered a number but rather a principle. The two, also not yet a number, is another mystery – one of the mysteries which make music possible. In one sense, it only duplicates itself and brings nothing new. Thereby however it can always provide another double to bring more tones to birth.13
I first heard about tuning to the overtone series from Sergio Aschero, doctor in musicology. Dr. Aschero calls this "Afinación Armónica" - spanish for Harmonic Tuning and also . His theory and the harmonic (overtone) model are presented in detail in his work "Opus ad Libitum".
The process of expressing these ratios in number of oscillations per second called Hertz is a very simple math multiplication.
Choose your fundamental frequency (in my audio examples 1/1 is 256 Hz) and multiply that with all the ratios to get the complete system.
One, the fundamental tone, contains within it the possibility for every other tone.
The process through which one becomes many starts by doubling. Every further successive double creates more tones - actually, two times more every time. This can be seen on the number line, by counting every colored number between the green ones - including only the first green. (Between 1 and 2 there is 1 - total 1; between 2 and 4 there is 2 and 3 - total 2; between 4 and 8 there is 4, 5, 6, and 7 - total 4; and so on.)
While every harmonic from the harmonic series is related to the Fundamental, they are also related to eachother. Each of them can be, at the same time, a harmonic of the ascending series, a harmonic of the descending series (having their own fundamental), and a fundamental having its own harmonic series.
Still, inside the first double/half - which is a rather large interval - there are no in-between tones.
The main question being answered here is how are created the tones inside the interval between the fundamental and the second harmonic. Together, they will form different "tuning systems" - different ways of arranging intervals in a kind of abstract ladder of sounds called a scale.19
There are many ways of creating these systems; parallel paths of arriving at the same result. I'm going to present all of them - basically, saying the same thing with different words, or looking at the same process from different perspectives.
No matter how you might interpret a fractio used in music to describe, among others, the pitch of a tone, its implied harmonic genesis will always be a ratio between an Overtone - above - and an Undertone - below the fraction bar.
The interaction between the harmonic series can also be described linearly, although this process is a bit more complicated.
The idea is to brake again the line at its midpoint, and slide it sideways in both directions for a distance equal to the interval between the first and second harmonics.
Consider the undertone series depicted on the number line above sliding to the right. The Fundamental is now the second overtone.
Because the whole undertone series relates to the Fundamental, and the Fundamental is now the second overtone, every element of the descending series will now relate to the second overtone.
Same is true for the overtone series that slides to the left, until the Fundamental becomes the second undertone. Every element of the ascending series will now relate to the second undertone.
Let's see how the tuning derived so far would look on the strings of an incomplete imaginary piano. It is imaginary because it has never been built, although the possibility of building it is absolutely real, and incomplete because only some of the keys have tuned strings.
Standard pianos are different, in that they are tuned in a highly practical but slightly out-of-tune system. 20
All Tones Are Really Undertones and Overtones and all ratios express that.
Yet another way of interpreting ratios becomes evident from the above image.
A fractio used in music to describe among others the pitch of a tone, also describes, in its inverted form (or if read from the bottom up), the length of string or pipe used to produce that tone. This length has meaning only when related to the length of the fundamental.
For example, the fractio 3/2 read from the bottom up is 2/3, meaning that the string used to generate this tone will be 2/3 meters long (=0,666...) - if the fundamental string has 1 meter.
String lengths / intervals and oscillating frequencies / harmonic genesis exhibit a reciprocal relationship, that is, the top and bottom numbers involved in the fractions switch places.
This can be confusing. In order to keep things clear, I leave the fractions unchanged and interpret them like this: whenever I think of harmonic genesis and oscillating frequency I read the fractions just as they are. When I think of string or pipe lengths and intervals, I read them from the bottom up.
For example, 2/3 - two thirds - read the other way around, from the bottom up (or right to left), is three halves (using the customary english fraction pronounciation). If the fundamental string has 1 meter, then the string used to generate the 2/3 tone will be three halves, or one and a half bigger: 1,5 meters.
Another way of looking at the tuning process is by interpreting every fraction as an interval. A musical interval is the sonic distance between any two harmonics - either of the ascending or the descending harmonic series.
If starting from the fundamental, harmonics were added one at a time, once the third harmonic is introduced, the system expands with one extra interval: the one between the second and the third harmonics.
The distances between sounds are now as follows:
The fractio 3/2 means "the interval between the second and third overtones".
The fractio 2/3 means "the interval between the third and second undertones".
Intervals from both series are expressed as a distance from left to right. The inverted form of the fractions is used again.
This is due to the fact that all sounds on the right are harmonics of 1, while 1 is, by turn, harmonic of all sounds on the left.
The intervals thus derived are really the already existent intervals between the second and third harmonics, having as starting point the fundamental.
Different ways of representing the intervals between harmonics 2 and 3:
Historically, the interaction between the ascending and descending harmonic series has been explained through the process of "justification" - a process through which a tone is divided or multiplied by 2, in order to be brought inside the limits. The logic behind this is that any doubling or halving of a tone's frequency represents maximum consonance and the resulting new tone is considered to be the same with the fundamental.
This of course limits the interaction to multiples of the second harmonic. That is, the ascending harmonic series will interact just with subharmonics 2, 4, 8, 16 and so on, while the descending harmonic series will interact just with overtones 2, 4, 8, 16 and so on.
More than that, because for a large period in history only the existence harmonic 3 was recognized, the sonic interactions became even more depleted. All these aspects will be covered as this study continues.
The problem I have when presenting such theories, is that I cannot accept something to be different, and at the same time the same. Still, for "backwards compatibility", that is, for the sake of understanding all previous studies of music, which use the same obsolete concepts, here it is:
For filling the gap between 1 and 2, there will be no interval larger than 2 taken into consideration. Intervals in the image above are possible candidates. They are by no means the only available options.
If we consider the fundamental sound - 1/1, and then look for further intervals, the first we come across will be that of the second harmonic. As strange as might sound, we'll have to advance in order to find more intervals, and the next step takes us to the third harmonic.
Although the limit imposed by the second harmonic is exceeded, there is a way to bring that third harmonic inside the main interval.
The same primary cause that made possible the existence of the second harmonic can be used to halv/double the third, making it fit in the interval between first and second harmonics .
With other words, if we look at 1/1 and consider it the result of 2/1 divided by 2 (or 1/2 multiplied by 2), while considering it the same note as 2/1 (and 1/2), then 3/1 also could be divided by 2, the result 3/2 being the same note (1/3 also could be multiplied by 2, the result 2/3 being the same note).
This of course is based on the acoustic sensation of maximal similarity between any two tones having a doubled/halved frequency. Even though they are different, they sound like the same tone, and thus are considered being the same note.
Dividing and multiplying by 2 - justification - is actually the process of taking as fundamental the second harmonic from above/below.
These new notes represent the interval between the second and third harmonics, having as starting point the fundamental.
The process described so far applies to other intervals as well. In order to create a scale - any scale, even the 12-note standard scale (or at least a harmonic, untempered version of it) - different intervals are brought together inside the interval created by the first and second harmonics.
Depending on the sonic flavour desired, the intervals can come either from the ascending or descending harmonic series, or from both.
TO SEE AND HEAR MORE EXAMPLES, PRESS HERE.
THIS SECTION HAS BEEN ENTIRELY RE-WRITTEN. ACESS IT HERE:
A New Methodfor Visually Describing Music
By tracing all the lines, the complete system unfolds. I call this THE LOGARITHMIC PYTHAGOREAN LAMBDOMA.
The difference between the Classic Pythagorean Lambdoma and my Logarithmic version, is that the classic provides no visual representation of the way we hear differences in pitch, and no way to instantly determine how any ratio or interval relates to all others.
THE LOGARITHMIC PYTHAGOREAN LAMBDOMA MAKES OBSOLETE ANY OTHER Logarithmic Interval Measures, LIKE CENTS.
Right through later Greek times into the Middle Ages, there were students who claimed to follow Pythagoras, and who spoke variously of his “canon”, “abacus”, “Lambdoma” or Pythagorean table, but no one else knew exactly what it was.
Then in 1868-76, Baron Albert von Thimus published a monumental two-volume work: Die Harmonikale Symbolik des Altertums. His starting point was an overlooked passage in Iamblichus’ 4th century Commentary on the Lesser Arithmetic of Nicomachus of Gerasa, a 1st century Pythagorean. The passage runs as follows: “...”
Von Thimus found that if we make such a diagram and fill in the “interplay”, then add to the numbers musical values, according to the series described above, we get a scheme as follows (in one of its variants): [...]
It can be rather startling then to realize it is a field entirely filled with interlocking over- and undertone series. All the diagonals in one direction give overtone series, all the diagonals in the other, undertones. The central axis maintains always the same note.
The diagram can be continued theoretically ad infinitum, and demonstrates the network of connected harmonies we mentioned above–even though triads as such were not used in music until the Middle Ages.
This discovery of von Thimus was elaborated and developed further to a remarkable extent by chiefly one man, Hans Kayser (1891-1964) who devoted his life to it in Switzerland, and produced 13 books, only one of which is translated as yet from the German.
He demonstrated this diagram to be an almost inexhaustible treasure-trove of relationships applicable to mathematics, music, geometry, architecture, acoustics, biology, the forms of plant, animal and man, crystallography, theology, metaphysics, philosophy, art, etc.13
Any vertical axis on the Logarithmic Pythagorean Lambdoma has the same note.
This is because complex-number ratios get simplified like this: for example, the ratios 4/6, 6/9, 8/12 and 10/15 all get reduced to 2/3. This is a simple math operation any kid knows about.
Tunings appear on every logarithmic lambdoma diamond (quadrant or rhomb).
The one above shows a simple tuning derived from one of the second diamonds. Second diamonds are all those between the second and fourth harmonics, both ascending and descending.
The image below depicts the intervals between the harmonics derived from the tuning.
The previous tuning can be extended in the next diamond above. This can also be called a family of tunings.
Different tunings appear by going around the diamond edges....
But the best and most interesting part of the logarithmic lambdoma must be the instant recognition of pitch height, related to other pitches, provided one has enough imagination or visual creativity as to see the projections traced from every note on the number line.
It's really that simple. Can you see the projections?
If you are concerned that the number line under the diamond has dimensions that, even if proportional, do not corespond with the intial logarithmic number line from which the lambdoma was constructed, just trace the projections on the actual diamond diagonal.
Of course, it can be ANY diagonal, and projecting works both vertical and horizontal.
In my example I used the bottom number lines. Now, all you need to do is change the angle between the lines so they are one straight line again.
With the tuning developed so far we covered the first 8 harmonics and all the consecutive intervals between them, plus three more which are not between consecutive harmonics.
But there is one more interval inside the limits of a double (the limits imposed by the second harmonic), and a few more which are larger than the 2/1, respectively 1/2 intervals.
These intervals also come from what I call the third "Logarithmic Lambdoma Child".
The green diamond should be the first logarithmic lambdoma child, but its shape is just a diamond.
The area in blue represents the second child, and includes 3 logarithmic diamonds.
The third child, in pink, has 5 diamonds and the fourth, orange child, has 7.
Tuning an instrument symmetrically according to both descending and ascending harmonic series works best for keyboard-type instruments (like the piano or reed organ) which have a distinct sound producing body (string or reed pipe) for every tone.
Instruments such as the flute or monochord use only one sound producing body (flue pipe or string), whose length can be modified with the fingers in order to create different tones.
Because of their construction, these instruments cannot play more than one note at a time, as opposed to keyboards.
The guitar, together with many similar stringed instruments, integrate both concepts: it can play many tones simultaneously, each changing pitch according with the fingerings on the fretboard.
The image above is an idealized concept, that has no practical application.
Its purpose is to show that when tuning an instrument to the descendig harmonic series, holes and frets are equally spaced, and when tuning to ascending harmonics holes and frets get closer and closer.
Regular pianos encompass a wider sonic range, and their tuning is but an approximation of the harmonic series. Regular guitars have all strings tuned to an approximation of a mixed tonality of the combined descending and ascending harmonic series. Contemporary flutes follow the same abstractions.
Next chapter, Current Human Civilization and the Practice of Music, deals with these concepts in detail.
1 Shmoop Editorial Team, "Exponents and Powers - Whole Numbers," Shmoop University, Inc.,11 November 2008,
http://www.shmoop.com/number-types/exponents-powers-whole-numbers.html (accessed August 27, 2013).
02. Dale Pond - "Atlin - Knowing I AM"
03. Alain Daniélou - "Music and the Power of Sound ..."
04. Dale Pond & others - "Universal Laws Never Before Revealed - Keelys Secrets to Understanding the Science of Sympathetic Vibration"
05. John Stuart Reid - "Sound Gives Birth to Light"
06. John Stuart Reid - cymascope.com
07. Thomas Frederick Harris - Handbook of Acoustics for the Use of Musical Students (8th ed) 
08. Dale Pond or others - wave's 4 phases
09. John Stuart Reid - Harmonic Voice Mandala
10. Hermann L. F. Helmholtz - "On the Sensations of Tone as a Physiological Basis for the Theory of Music"
11. Leonard Bernstein - What is harmony YouTube
12. Skye Løfvander - overtones.cc
13. The Spiritual Basis of Musical Harmony - Graham H. Jackson 
14. Harry Partch - Genesis of a Music
15. Rex Weyler and Bill Gannon - The Story of Harmony
16. Interval (music) on Wikipedia, the free encyclopedia
17. Manuel Op de Coul - Logarithmic Interval Measures
18. Dr. Albert A. Bartlett - "Arithmetic, Population and Energy"
(lecture - http://www.youtube.com/view_play_list?p=6A1FD147A45EF50D);
"The Essential Exponential! For the Future of Our Planet"
19. Dane Rudhyar - Magic of Tone and the Art of Music
20. W. A. Mathieu - The Musical Life