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There is no connection between music and math. There is, and can never be made no connection between sound and numbers. To connect means to join, link, fasten, unite or bind together; to associate or consider as related, to establish a rapport or relationship, and this is exactly the problem: music and math are not two separate things, having perhaps some elements in common, allowing therefore yet another external force like the human intellect to draw connecting lines between them.
«We will do well to remember that the theorists [...] only calculated and confirmed the musical ear of the singers. They did not invent the pure intervals out of numbers. The numbers were revealed to them when they calculated precise methods of tuning the harmonies that singers discovered naturally.»1 «Harmony was not a rational calculation, but was rather a natural observation discovered by the acute ears of musicians. Yet these perfect harmonies were shown to have exact numerical relationships. [...] Music and numbers are therefore eternally linked, not due to convention or theory, but because of the very nature of sound.»2
Thus there can be made no connection between music and math, since the two are one and the same. Mathematics is a branch of Harmonic Theory, all math is in fact music. All numbers and number systems, all equations are sounds. Contemporary mathematics describes sound in terms of numbers and equations, forgetting that these come from the properties of sound in the first place. This is (in part) what the Advanced chapter is all about.
The perfect math of true harmonic music is a natural property of sound. It is the perfect tool to actually understand the physical relations between tones, as opposed to learning by rote conceptual abstractions which fail to give insight into the wonderful acoustic world we are immersed in.
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 sub-multiples 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 given 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 successive value is the same.
On a logarithmic number line, the distance between each successive 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 until 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 successive 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 seemed 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.
In proceeding with this subchapter, it is assumed the reader has knowledge about The Physics of Sound, with emphasis on The Physics Behind Music — The Physics of Harmonics and Natural Vibration Modes of Perfect Numbers.
Any two sonic entities, consecutive or not, of any of the two series, will have reciprocals in the other series. The numbers need not be converted, as the sonic distance between them is always the same.
To make that clear, let's say we want to find where supra-harmonics 2-3 (the sonic distance between them, to be more precise,) is located in the sub-harmonic series. That's a no-brainer, the answer is sub-harmonics 3-2.
TO DO: fix arrow, set alt & optimize + PAHW
This also works for non-consecutive values, like 3-5, and is valid both ways for transposing not only overtones to undertones but also undertones to overtones.
Any series of more than two sonic entities, consecutive or not, will have correspondent sonic disances, not consecutive, in the reciprocal series. In simple terms: any number of harmonics from the Descending Series can be written as harmonics of the Ascending Series and the other way around. Though these numbers need not follow one after another (like 4, 3, 2), if they do, the converted values won't (as in 3, 4, 6).
Visually, the procedure is straight forward: we slide the sonic entities from one side of the what-you-see-is-how-you-hear number line to the other, until we find a match.
In order to calculate that, we have to:
The resulting numbers are the respective harmonics from the other series.
When calculating, the fundamental (and its implied division or multiplication) must not be taken into account. Use only integers (like 4 , 3 , 2 or 3 , 4 , 6) and not operations (like 1÷4 , 1÷3 , 1÷2 or 1×3 , 1×4 , 1×6) nor ratios (like , , or , , ).
In the example from the image above, for the consecutive undertones 4 3 2÷:
So the corresponding overtones are ×3 4 6.
This works the other way around in the same way, and the beauty of it is the math is the same — as long as the fundamental is not introduced in the calculations. So if for example we were to transpose the consecutive overtones ×2 3 4, we'd get the same results (LCM is 12, and 12÷2=6; 12÷3=4; 12÷4=3) and the corresponding undertones would be ÷6 4 3. But if we already did the math for one side, calculating again for the other is superfluous. It doesn't really matter to which Series the Harmonics belong to, the transposition algorithm is one.
Trick: to transpose fast undertones to overtones, find the Smallest Common Undertone (= Least Common Denominator or LCD) of the series by using an automated tool like http://www.calculatorsoup.com/calculators/math/lcd.php and discard it (get rid of the numbers below the fraction bar). The resulting set is the respective overtones. [And to do the reverse, find the Smallest Common Overtone (= Least Common Numerator) and discard it]
To do the same with musical ratios (like 1/1, 4/3, 3/2), first convert them to integers and then apply procedure. In order to convert any set of ratios to Harmonics of the Ascending Series, find the LCM of undertones (denominators) and discard it. To converti it to Harmonics of the Descending Series, find the LCM of overtones (numerators) and discard it / replace by 1.
Any number of ascending harmonics will cycle, returning to the same common (in-phase) starting point, after a number of oscillations that will always fit the cycle length of the fundamental. While the fundamental performs one complete oscillation, each ascening harmonic will perform a number of oscillations equal to its harmonic number.
For the descending series, one full in-phase cycle will take place after a number of cycles of the fundamental, equal to the Least Common Multiple of all subharmonic values in integer form, and not 1/n reciprocal. (When ratios are used, then the Least Common Denominator of the fractions will return the period.)
From the pictures, it becomes clear how the subharmonics translate into supraharmonics. Mathematically, the operation needed is finding the Smallest Common Undertone (Least Common Denominator) of the series. For example, the SCU (LCD) of 1/1, 1/2, 1/3 is 6, and the equivalent fractions are 6/6, 3/6, 2/6. The numbers over the fraction bar are the overtones.
Harmonics of the Descending Series as Under-tempos,
depicted using series of ticks as cycle measurement (as opposed to Under-tones, depicted as string lengths).
Important! The grey number line and its ticks are depicted the way we measure (NOT the way we hear).
This representation is different, and IT IS NOT a sonic number line where what-you-see-is-how-you-hear.
Last updated: 15 december 2015
References (TO DO):
1 “The Story of Harmony” by Rex Weyler and Bill Gannon (Justonic Tuning Inc., 1995); Chapter Three: Pythagoras: music & numbers; pages 27 to 28.
2 Same as Note 1 above; page 25.