The Many Meanings of Every Ratio:

How to İnterpret the 9 Different and Non-exclusive Meanings
of Every Musical Fraction

part of an ongoing research by

Bo Constantinsen

Section Contents

— content is being edited and added —

This entire chapter is still a draft.


The 9 Meanings

Every musical ratio, written as a fraction of two integer (rational, whole) numbers, has at least nine different and non-exclusive meanings. With other words, every ratio can be interpreted in nine different ways, and these interpretations do not cancel each other. Every meaning remains true, no matter how you look at it.

The number nine represents the understanding I have so far; there might be more interpretations of which I’m not aware at the moment.

1. Harmonic Genesis

How the tone was generated: Dual Harmonic Genesis (double non-exclusive meaning).


All Tones Are Really Undertones:
undertones of overtones (of the fundamental).

All Tones Are Really Overtones:
overtones of undertones (of the fundamental).

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These examples serve as model for the genesys of any ratio, including those having 1 above or/and below the fraction bar. When the number 1 is present in a ratio, it never represents the absolute fundamental 1, but the undertone 1 or overtone 1 of that series.
TO DO: reformulate.

The ratio 12 is undertone 1 of overtone 2 (of fundamental 1) or overtone 2 of undertone 1 (of fundamental 1), because 1 ×2 ÷1 = 1 ÷1 ×2 = 12 . Can you figure out the harmonic genesys of ratio 21 ?

The ratio 11 is undertone 1 of overtone 1 (of fundamental 1) or overtone 1 of undertone 1 (of fundamental 1), because 1 ×1 ÷1 = 1 ÷1 ×1 = 11 .

No matter how we might look at it, every musical fraction represents the harmonic genesis of a tone and is written as a ratio between an overtone, over the fraction bar, and an undertone, under the fraction bar, implied (but not depicted) as related to the absolute fundamental 1.

2. Sonic Distance

Harmonic interval: the sonic (aural, acoustic) distance (space, remoteness) between any two harmonics from the same series (whether descending or ascending).

a) Where (in the harmonic series) is the source of the ratio (to the left or right of / lower or higher than the fulcrum) and
b) what the size of it is (larger or smaller than 1).

Before trying to interpret ratios as sonic distances, we first must be able to tell which harmonic series is the source of the sonic distance. The two options are the ascending and descending series.

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If the number above the fraction bar
is smaller than
the one below it (denominator),
the sonic distance belongs
to the descending series.
If the number above the fraction bar
is bigger than
the one below it (denominator),
the sonic distance belongs
to the ascending series.

Every musical fraction represents the harmonic interval,
or sonic distance, between any two harmonics −
either of the ascending or the descending series,
and is written as a ratio between them.

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When looking at a vertical number line and think about ascending harmonics, the natural progression of these starts from the bottom up. So for example, when talking about supra-harmonics 2 and 3 in this context, we read first 2, and then 3. This is the exact same order in which the numbers of the vertical fraction 23 should be read, when interpreted as sonic distance: from the bottom up. 23 is the sonic distance between harmonics 2 and 3 of the ascending series.

When considering the descending series, the natural progression on the vertical line is from the top down. So for example, when talking about sub-harmonics 2 and 3 in this context, we read first 2, and then 3. This is the exact same order in which the numbers of the vertical fraction 32 should be read, when interpreted as sonic distance: from top to bottom. 32 is the sonic distance between harmonics 2 and 3 of the descending series.

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where the visual element is discarded and only numbers are kept together with a small line signifying the ticks on the number line, squeezed together.

We know that 23 is implied as related to the absolute fundamental 1 (because this ratio is in fact 1 ×3 ÷2 or 1 ÷2 ×3) with other words, there can be no 23 without a 1. So in interpreting 23 as a sonic distance (or interval) means that the sonic distance between 11 and 23 is the same as the sonic distance between harmonics 2 and 3.

The same sentence makes more sense when written with horizontal ratios: the sonic distance between 1|1 and 2|3 is the same as the sonic distance between harmonics 2 and 3.

Yet another way of saying the same thing with different words is: the new ratio(s) represent the sonic distance between the second and third harmonics, having as starting point the fundamental.

doesn't necessarily have to be consecutive harmonics

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3. Physical Length

How to generate it physically (on musical instruments)

in inverted form, the length of string or pipe and formula for dividing it: #/№ = №/#
? formula is based on #1 above
- divide the string and bow the other end.

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.

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TO DO: ALL images alt text

Every musical fraction represents,
in its inverted form
(or read from the bottom up),
the length of physical oscillator
(like string or air inside pipe)
used to produce that tone.

bla bla

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TO DO: change colors?

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.

4. Oscillating Frequency

What does it sound like

About frequency as an attribute of anything cyclical
ANY RATIO IS A FREQUENCY - it measures HOW MANY TIMES a certain event takes place in the same UNIT (=1) of time. Above fraction bar: cycles (or repetitions); below fraction bar: time units.
3/2 = 3 cycles in 2 time units, which is 1,5 cycles in 1 time unit.
2/3 = 2 cycles in 3 time units, which is 0,666 cycles in 1 time unit.

The oscillating frequency of that tone: f=x/x.
the ratio between the oscillating frequency of the fundamental to the frequency of an overtone to the frequency of an undertone: 1 ×3 ÷2 is one cycle per unit of time, spun 3 times faster then 2 times slower - both in the same amount of time.

the ratio between: the number of oscillations (or vibrations, or cycles or repetitions) performed by the musical entity, and the amount of time it takes to perform them.
or how many times it vibrates faster/slower than the (implied) fundamental

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draw 0,66 and 1,33 on the blue helix?
add time axis label (what about other axis?)
description: 3/2

The new simple tone performs 1,5 (one and a half) oscillations for every 1 of the fundamental. Another way to write 1,5 is 1 21 , or simply 23 , meaning that the new simple tone performs 3 oscillations in the same amount of time the fundamental needs to reach 2, developing a periodicity recognized instantly by the acoustic processing centers inside the brain.

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draw 1,33 on the red helix
description: 2/3
add time axis label (what about other axis?)

The new simple tone performs 0,6 (repeating pattern of decimals) oscillations for every 1 of the fundamental. Another way to write 0,6 is 32 , meaning that the new simple tone performs 2 oscillations in the same amount of time the fundamental needs to reach 3, developing a periodicity recognized instantly by the acoustic processing centers inside the brain.

what does a negative exponent mean? Negative seconds means going back in time! If going forward grows by a scaling factor, going backwards should shrink by it.
2^(-1) = 1/(2^1)
The sentence means “1 second ago, we were at half our current amount (1/2^1)”.

That's because 2^1 means “after 1 second, we will be at double our current amount”


Yet another one:


5. Harmonic Alignment

The fundamental alignment of resonant harmonics (generating two reciprocal child series)

The alignment between the (ascending & descending) harmonics of that tone, and those of the implied fundamental

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to optimize


1. musical fractions written as
ascending progression or archaic ratios
(describing also accurately
sonic distances between ascending harmonics)
describe sub-harmonic alignment
in this exact form.

2. musical fractions written as
descending progression
(describing also accurately
sonic distances between descending harmonics)
describe supra-harmonic alignment
in this exact form.

3. The place where harmonics coincide
is the begining of a new series in the same direction,
whose complete harmonic series will coincide
with harmonics of all initial sonic entities.

6. Scaling

every number can be considered a scaled version of 1

scalar waves refer to harmonic cycles or the periodically occuring patterns of movement which define resonance: vibration in usion, or at a scaled/harmonic frequency. both terms, “scaling” and “harmony” imply natural laws of relationships mathematically based on whole/rational/integer numbers.

learning to count is learning to add. it is physical.
learning to sing is learning to multiply. it is metaphysical. becoming
learning to listen is learning to exponentiate. it is. being

7. Rhythm

Rhythm is the pulse of music.

Somewhat similar to the wave-particle duality theory, so is the tone-rhythm duality a mere perceptual one. This doesn't change the fact that rhythm comes from the descending harmonic series (sub-harmonic unde-rtones), while pitch from the ascending harmonic series (supra-harmonic over-tones). More details about this soon.

A ratio is a beat, when played at low frequencies. A beat turns into a tone when played at high frequencies.

The beat is the precursor of tone. Play a constant beat at a higher frequency (scale it) and it will become a tone. If the beat is composed of at least 2 simultaneous patterns, then raising the entire system in frequency, thus keeping the ratio between the components constant (scaling it), will sound a harmonic sonic distance (or interval) in the hearing range.

A clear demonstration of this is under preparation. Until it is ready, check out this video:

This spectrogram shows a three over two polyrhythm being sped up into the audible range
where the polyrhythm becomes a 2|3 Harmonic Sonic Distance.

Sorry for the western music theory nomenclature; it seems the title of the clip cannot be removed from the player. "Fifth" is a very confusing term which shouldn't be used at all in music. Unfortunately, it became entrenched in musician's minds and there's no way of freeing them from it. Rest assured, I will keep my promise and COMPLETELY AVOID such confusing and irrelevant terminology. The What Music Really İs Manual shall remain free from such historical accidents and culturally defined misaprehensions, in order to trully have an universal book suited for the needs of the free-thinking 3rd Millennium Musician.

8. Tension

Frequency is proportional to the square root of the tension.

The speed of propagation of sonic energy in a string is proportional to the square root of the tension of the string and inversely proportional to the square root of the linear density of the string. (source: wiki)

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9. Thickness

Frequency is inversely proportional to the square root of the string thickness (diameter or 2 x radius).

Other Ways of Representing Musical Entities

tables here

  undertones / sub-harmonics absolute fundamental overtones / supra-harmonics
Harmonic 27 9 3 1 3 9 27
Division/Multiplication ÷27 ÷9 ÷3 1 ×3 ×9 ×27
Fractions 1/27 1/9 1/3 1/1 3/1 9/1 27/1
Interpretation 1÷27            
Exponents 3⁻³ 3⁻² 3⁻¹ 3⁰ 3⁺¹ 3⁺² 3⁺³
Monzos (5-limit)              
Fractions & Exponents 1/3³ 1/3² 1/3 1/1 3/1 3²/1 3³/1
Decimals 0,037 0,1 0,3 1 3,0 9,0 27,0
Percent 3,703% 11,1% 33,3% 100% 300% 900% 2700%


TO DO: Ancient Harmonics (archaic ratio math)

Besides the archaic forms of music having an upper fundamental and a descending scale, a peculiar way of employing numbers has been preserved from the beginning of written history, which has been used to convey an ancient form of harmonic knowledge. The secret language of ratios encoded in this kind of mathematics should not be limited to the interpretation of physical properties of sound and the laws of acoustics alone.

It seems like our ancestors made exclusive use of integer numbers to express any tuning (sequence of ratios), and – an important fact overlooked by many scholars – these numbers are all harmonics from the harmonic series.

Let’s take half of our previous arbitrary-chosen tuning as a first example. The ratios 11 34 35 12 can also be written, using unsimplified fractions, as 33 34 35 36 which then become 3 4 5 6 by overlooking the common number below the fraction bar (denominator), which is the fundamental tone of the set. In the above example, this fundamental is the third undertone. Mathematicians skip the second step and reach directly the result by calculating the least common denominator of the series and discarding it.

TO DO: (logarithmic) number-lines graph

If however we are faced with a sequence of rational numbers from which we want to deduce the ratios, the series 3 4 5 6 can be written as fractions by dividing every member by the first, resulting in the set 33 34 35 36 whose simplified version is 11 34 35 12 .

The resulting series of numbers 3 to 6 is not, and should not be interpreted as 13 14 15 16 because in this way, even though the acoustical and mathematical relation between tones remains the same, the starting point or fundamental of the series is different. For the above set the fundamental is the absolute fundamental of the master set, 11 , while in the proper interpretation giving the ratios 33 34 35 36 (no matter if simplified or not) the fundamental is the third undertone 13 .

The Tilted HarmoniComb shows us there are many tunings that have the same structure, where the ratio between members is constant from one level to the other, but whose starting point differ according to the undertone they are tuned to.

HarmoniComb (logarithmic number-lines graph) version.

It follows that the ancients understood each figure from different sets of rational numbers as having an implied relation to the first figure in the set – the fundamental of that set. Our example shows how the sonic intervals between any two members of the set are the exact same intervals as between the members on the outer edges (pure harmonics).

So in answering the question “How acoustically large is the sonic distance between 35 and 12 ?” first we notice that 12 is the simplified fraction of 36 . Then we realize it is (the same as) the sonic distance between harmonics 5 and 6.

But this is just half of the picture. Thorough studies of ancient writings conclude that our predecessors had a mature philosophy which they encoded in numbers using a simple and rather primitive science of music and number. The most important idea we need to understand in order to grasp the ancient way of thinking is the notion of reciprocity. In ancient times integers functioned both as multiples and submultiples, and every tuning set of such numbers had two implied relations to the first degree: one through multiplication, the other through division.[TO DO: add endnote and reference this to McClain] As we’re about to see, these two reciprocal meanings describe in fact one and the same tuning system.

In the example above harmonics 3 to 6 ascend, giving pure harmonic but unsymmetrical intervals. The numbers function as multiples only; the set has no arithmetical reciprocity. In order to obtain tonal symmetry, the tuning must be extended in both directions. This we achieve by reintroducing harmonics 3 to 6 descending into the set.

Once this is done, the 5+1 tone (pentatonic) scale 11 56 34 23 35 21 which is the same with its reciprocal 21 53 32 43 65 11 because all sonic distances between musical entities are identical, can be written as 30 36 40 45 50 60 and also as its reciprocally symmetric 60 50 45 40 36 30 using only rational, integer numbers. These simple representations imply, and are to be interpreted as related to the smallest number in the set: 3030 3036 3040 3045 3050 3060 respectively 6030 5030 4530 4030 3630 3030 .

TO DO: HarmoniComb (logarithmic number-lines graph) version.

TO DO: optimize, fix (PA)HW

In this way all integers function both as multiples and submultiples. These are harmonics from the harmonic series; every tuning system can be written this way – except temperaments.
whole number multiples

Given this reciprocal set of seemingly irregular integers, we can find out the ratios or how the numbers relate to one another by:

Simplifying the fractions is optional. The reverse process is simple, and turns any tuning system of ratios into a seemingly irregular sequence of integers by finding the least common denominator.

TO DO: (logarithmic) number-lines graph

TO DO: update picture to complement text below like in

Since the numbers below the fraction bar (denominators) are overlooked when considering the set of large integers as ascending, while those above the fraction bar (numerators) are overlooked when the set is considered descending, the end result from both operations is the same – at least so it seems. The difference between ascending and descending is given by the direction in which the numbers are written.

In order to fully understand that, we must give up all preconceived notions according to which writing has to be in a certain direction like for example left to right, from the top down. Imagine writing each letter of each word from right to left, beginning from the bottom of the page and continuing with each new row upwards. The fact that we use one certain convention, is just… a convention. It has no relevance.

That is why, no matter the language in which this work is translated and no matter the direction in which the words proceed on paper, the ascending harmonic series will always be written small-to-large numbers from left to right (as in 1 → ∞), while the descending the opposite way: small-to-large numbers from right to left (as in ∞ ← 1).

Last updated: 16 january 2017 

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