Examples: How All This Fits into Practice

The Accidents of Human History and Culture
which We Call “Music”

part of an ongoing research by

Bo Constantinsen


Section Contents

— content is being edited and added —

This entire chapter is still a draft.


TO DO: reformulate musical terms and western nomenclature

The intonational pluralism of ancient Greek culture can be divided into three competing philosophies:1

  1. the conservatives: 3-limit (and 5-limit comma-shifted) rational intonation, commonly attributed to Pythagoras without the evidence to back up this limitation imposed on "the father of Western culture";
  2. the highly progressive: no restriction on higher prime numbers (n-limit just intonation), started by Architas and the Harmonists with the acceptance of 7-limit harmony and taken further to n-limit by Ptolemy, and
  3. the irrational camp: geometric divisions of the monochord introduced by Aristoxenus, the star pupil of Aristotle.

The fight between these 3 camps started as early as the 4th century B.C., had been going ever since, and it's still not settled today.

Although it might seem the What Music Really İs philosophy is framed around the second camp, this is not so. I salute and respect the Harmonists, but I am not one of them. What Music Really İs ackowledges the deviation from perfect harmonic numbers (a.k.a. "inharmonicity") due to physical properties (like stiffness) of acoustic oscillators and resonators, and uses pure harmonics as a generalized approach to mapping musical entities and their interactions by taking these deviatons into account.

This approach accounts for perfectly consonant musical instruments which are in tune with themselves, a concept not unique to this author but very new in the tormented history of music.

The Scholars’ Approach to Tuning

Although historically extremely few have been aware of the “tone-number” matrix I call HarmoniComb (and as far as I know only one in the logarithmic form presented here), for most cultures — and this is especially true for West Europeans — established musical theory was concerned with abstract principles that for the most part had no connection to nature, the natural harmonic principles of sound and the perfect numerical relationships of these.

In order to understand mainstream music and the scholars' approach to tuning throughout history, we'll have to reduce the two-dimensional matrix needed for the proper functioning of musical entities to a one-dimensional number line.

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Next, we'll have to understand how the tuning theorists of the past approached numbers, ratios and musical entities. This is by no means a universal approach, as for most cultures music was transmitted orally. It is mainly the West European... from Greeks...
Today we understand that sonic distances comes from the harmonic series, that is, from the audible distances between harmonics. As we're about to see, not all musical practices endorse the perfect and the natural, but even the music built on artificial ....

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Music is as old as any human art or craft. Knowledge of pure harmonics is at least 4,500 years old, perhaps much older. Most early human literature is song, the history of tribes passed from generation to generation. [...] Early musicians discovered that certain tones, when sounded together, created pleasing sounds, and these relationships became known as harmony. Therefore, in a very real way, melody evolved from harmony, although the process was more likely auditory and instinctual, rather than theoretical. Sound's inherent harmonies were simply strung out into scales and melodies. [...]
It is certain, however, that the earliest recorded knowledge of human sound indicates an understanding of a fundamental tone to which other tones were related. We tend to think of harmony evolving after melodic forms, but the historical record indicates that the earliest melodic scales were derived from a recognition of the harmonic qualities of sounds. [...]
[p. 19 (2 The first musical scale)]
[All text from "The Story of Harmony" by Rex Weyler and Bill Gannon]

Early musicians used numbers to denote string or pipe lengths. As soon as the technology permitted us to measure frequency in cycles per second (which year?), musical fractions gained another meaning: that of frequency ratios. But behind these two meanings, a third was implied since the beginning of written history: that every ratio is a sonic distance from the harmonic series.

Although not described as such, and at times even against number and quantization, the truth is that the pitch processing centers in the human brain have always been capable of distinguishing harmonic relations, and use them to make music.

Radial Tuning

Radial tunings are those who start from a center (the fundamental) and "radiate" in one direction. All musical tunings of the world for the past ~5000 years, except artificial temperaments, are radial. As we will see, a special case of radial tuning is "cyclic" tuning, presented below.

A radial tuning is nothing more than a slice of the harmonic series: ascending or descending, but never both at the same time. In its simplest form, this slice contains a complete series of numbers, which makes all sonic distances of the scale related by consecutive harmonics.

A good example of such scale is harmonics 7 to 14 of the descending series

TO DO: ...

Add fixed arrow pointing the radial direction.

The first row of ratios under the sonic number line displays the natural progression of the scale from right (1|1, highest tone) to left (2|1, lowest tone): 14|7 13|7 12|7 11|7 10|7 9|7 8|7 7|7. The second row is an alternate way of interpreting the scale, when it is played the other way around: from left to right, that is, from low (this time taken as fundamental 1|1) to high (1|2), giving 14|14 13|14 12|14 11|14 10|14 9|14 8|14 7|14.

But all that is just a study of the scale, its properties and mathematical permutations. What interests us is what this scale really is: where it comes, how and why is formed, and so on.

1. (consecutive) harmonics, one after another: relative to one another.

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2. 7 and its relatives: a true uninterrupted slice: relative to the fundamental.

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Add all other non-consecutive ratios.

In order to use this slice in practice, it will have to be scaled to fit inside the limits defined by the first and second harmonics. In the West they call this transposing, but the fact is, this is juast a harmonic combination from the HarmoniComb. There are two options: the limits of the descending, or the ascending series.

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TO DO: fix arrows and optimize

Or, if scaling doesn't cut it for you, consider the absolute fundamental of the series somewhere high, and the tuning slice in the desired frequency range.

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TO DO: fix this mess,
add frequencies in cycles per time unit

We could proceed to transposing this number-set into the opposite part of the spectrum, and rewrite the ratios as sonic distances of the ascending series. Unfortunately for this example, that would generate extremely large and impractical numbers between harmonics 51.480 and 102.960. The next example will illustrate this better.

TO DO: show the HarmoniComb: "This last fidus actually means the harmonic combination of the series with the 7th ascending harmonic."

Permutations of One Radial Tuning

A permutation consists of several steps (corresponding to the number lines below):

  1. take a sonic space from one end and move it to the other
  2. slide the modified tuning (permutation) so it starts on the origin
  3. the colors are wrong because they reflect the positions of the initial tuning
  4. re-paint colors (representing frequencies) so they correspond with the new positions
TO DO: ...

TO DO: merge arrow head into arrow;
TRANSFORM TEXT (tick and x) INTO PATHS and optimize

The math by which the permutation ratios are derived is boring when presented, but interesting (and sometimes frustrating) when calculated. All music scholars and theorists give the numbers and sometimes a poor and abstract representation that cannot convey meaning because it's linear instead of logarithmic (sonic, or what-you-see-is-how-you-hear). You now have an excellent starting point so have fun with the numbers!

Continuing to take the left sonic space from the last permutation and repeating the process results in 7 total permutations.

TO DO: ...

TO DO: TRANSFORM TEXT (tick and x) INTO PATHS and optimize

There are only 7 because the 8th is the same as the 1st, and so we're back where we started. This is a direct consequence of having 7 tones in a scale. If we had only 5 tones, there would have been only 5 possible permutations. The colors in the above image are not correct, but the visual effect describes beautifully the process of permutation.

Below are the re-painted frequency colors. The permutation process is not so obvious anymore, but a correct representation of the difference in ethos between permutations is gained.

TO DO: ...

TO DO: TRANSFORM TEXT (ticks) INTO PATHS and optimize

Ultimately, every such permutation is in itself a tuning which abides to the single acoustic law of music: that of being part of the harmonic series. Aside from the intial segment 7 to 14, all other permutations come from a slice of the harmonic series, where not all the harmonics are taken to be part of the scale.

Each of these being a TRUE MODE / RĀGA / ECHOS / MAQĀM, show them as SUB SPECIES of one (TRUE) MODE.

Combined Radial Tunings

A distant slice may be used where not all harmonics are taken to be part of the tuning.

Let's take as example the previously studied scale 11 56 34 23 35 12 from the What “Tuning” Really İs: Subsets of the HarmoniComb section of The Fine Art of Combining Harmonics chapter. When written like this, every simplified fraction points to a specific sonic distance, implied as relative to the 11 of the scale. This becomes obvious when looking at the horizontal version of the same ratios: 1|1 5|6 3|4 2|3 3|5 2|1.

However, this interpretation is not evident when taking into account the genesys of the scale: a symmetrical structure generated by the interaction of all harmonics inside the sonic distance created between the 3rd and 6th harmonics of both series.

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The image above shows the sonic spaces that make up the scale. These are all consecutive, and not all relate to the fundamental (exception to this is the small number line at the bottom, where all sonic spaces habe been calculated to relate to the fundamental). The image below displays one way of interpreting the sonic spaces thus created, this time related to the fundamental of the scale.

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Replace horizontal ratios below with vertical ones to make obvious the difference?

When looked at from this perspective, it will be noticed that the first three sonic spaces span between consecutive harmonics, while the fourth (5|3 respectively 3|5) doesn't. In re-wrtining the scale using archaic ratio math as 30 36 40 45 50 60, a rather excessive example of non-consecutive harmonics in sonic distances can be observed.

TO DO: (logarithmic) number-lines graph

TO DO: fix thin line oddity
give up the subharmonics (if properly presented in archaic r.m.)?

It's rather obvious why simplification of fractions works: because no matter whether reduced or not, the sonic distances are the same! For example, it can be seen — and this is visually demonstrated — that 3|4 (the sonic distance between harmonics 3 and 4) has the same size as 30|40 (the sonic distance between harmonics 30 and 40).

The genesys of this tuning scale can be explained through scaling like this:

TO DO: (logarithmic) number-lines graph

TO DO: fix arrows and optimize

There are two variants because historically, it seems that before the modern era, scales were treated like in the first example: the fundamental was the highest tone of the series and the scale was descending. The second example is how people started thinking about scales after giving up the previous way: the fundamental is the lowest tone of the series and the scale is ascending.

Having (the absolute fundamental of the series somewhere high or low, and) the tuning slice in the desired frequency range doesn't work as an alternate genesys viewpoint when considering this scale as a result of the interaction between harmonics 3 to 6 of both series.

However, when looking at it as a remote slice of non-consecutive harmonics from one series or the other, or even both when symmetrically reciprocal like in this case, this process works as described above for the undertone slice example.

Let's look at the ascending series this time, knowing that the same applies, in inverted form, to the descending.

TO DO: (logarithmic) number-lines graph

TO DO: fix funny looking thin lines

Permutations of One Combined Radial Tuning

!!! TO DO: MODES !!!

modes and mode species

Cyclic Tuning: Another Type of Radial



now there's no space to see how far the 27th harmonic stretches. making the number line smaller...

giving up the rainbow colors to make a better point. this is just temporarely.

shrinking even more the number line.

representing only harmonics 2 & 3
using 2 separate number lines
using simple (harmonic) numbers instead of ratios.
left and right are opposite, even if numbers don't reflect it.

the result: "cyclic" tuning system (needs to be colored)

the right part of the above set up, the overtones of 12 "cyclic" 3/2's are the starting point for messing around with the 12 notes we all know from the piano, guitar, and basically all modern instruments.

Permutations of One Cyclic Tuning


The European (Western) System

Western music has been under the dictatorship of the zebra-colored 7-white/5-black piano keyboard for hundreds of years. This single musical instrument dictates everything in the music of many nations across continents: the single tuning system used and the notation.

TO DO: WMRI-Making of a Keyboard-01
Halberstadt classic piano keyboard
— all .svg

TO DO: (logarithmic) number-lines graph

TO DO: optimize

If the sonic (logarithmic) number line is bent into a circle, this is how it would look like (magnified 5 times):

TO DO: (logarithmic) number-lines graph

TO DO: remove extra data and optimize

After 12 steps in one direction,
3¹² SHOULD be equal to 2¹⁹.
Because it obviously IS NOT,
another number MUST be,
one that REPLACES 3 and makes
the following equation true:
x¹² = 2¹⁹.

The solution is:
or x=2^(19/12)

Now all there's left to do is disfigure every sonic space
by replacing 3 with 2^(19/12).

TO DO: (logarithmic) number-lines graph

Voilá! Equal Temperament! Can you notice the ticks on the first sonic line are not equally spaced, as opposed to the ones on the second? If your eyes can see it, your ears can hear it.

TO DO: introduce symmetry - progressing in both directions

TO DO: (logarithmic) number-lines graph

TO DO: optimize

Perhaps the main characteristic of all the musical practices in the world is finding a set of tones, a tuning scale, compact enough to serve the different purposes music was given throughout time. It had to incorporate as many tones as possible, in order to support artistic expression, but as few as possible in order for the scale to be used on physical musical instruments.

A common theme encountered in many works is the alleged universality of the hable, or sonic distance between the first and second harmonics, misleadingly called “octave”. Supporters of this theory seem to have an arguably convincing body of evidence according to which the hable is common to all cultures.

Not only common, but the claim is made that all the world's cultures consider a tone whose frequency is halved or doubled to be the same tone. Mathematically, this would mean that 1÷2=1 and 1×2=1, and even after many repetitions 1×2×2×2=1. True or false as it may be, the very fact of repeating this theory for long enough led to the assumption that it is true: an undeniable fact based on the reality of human perception and musical evolution.

As such, the particular set of tones Western musicians were after needed to sit inside the hable: the sonic space defined by the frequency of the oscillator (string) in its entire length, and the doubled value of this frequency, arrived at by reducing the length of the oscillator exactly in 2. Halving the frequency would mean extending the oscillator by doubling its length.

TO DO: put a simple ruler and measurements
put one on top of others, with fundamental in middle, aligned right.
flip to vertical?

Western "intervals" are not sonic distances, not harmonic (quote rudhyar); only string lengths and intervals, because they could not measure frequency although they could isolate harmonics on a string/monochTO DO: ...ord

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TO DO: make it 5 tones, write on sides "ancient" and "modern"

The following is not how theoriests of the past approached the subject, but a translation of what they really meant to convey interspersed with the way it was usually presented. The presentation is not cronological. Their exact methods and thinking, ambiguous as they might be, can be found in historical works.

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TO DO: image has extra drawing (vertical ratios)!

But instead of recognizing this harmonic sonic distance for what it is and naming it accordingly, Western scholars refer to it as “fifth”: a tone derived from the division of one string in three equal parts, and use a peculiar concept called “justification” to “bring the tone inside the octave”, that is, inside the sonic distance between the first and second harmonics.

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TO DO: break into 2 drawings; put ascending guitars on top of each other;
put a simple ruler and ratio measurements

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TO DO: fix arrows and optimize

This would account to using the other side of the string to produce sound. TO DO: DRAW IMAGE WITH STRINGS ONLY, NO GUITARS


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TO DO: image

But the process of “justification” cannot explain this new tone, this frequency ratio which is a sensory consonance: it sounds pleasant and is of musical value — even if subjectively. Perhaps at one point far back in history the true origin of this sonic distance was known, but the knowledge was lost or perverted in time. As a result, the explanations given by theorists to justify its presence range from the strange to the absurd and ridiculous, though all have one thing in common: they make no sense and are at odds with the natural principles and physics of sound. Here are a couple of them:

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TO DO: fix arrows and optimize

The Byzantine (Eastern) System


Last updated: 28 october 2016 

The What Music Really İs Square Spiral Logo

References (TO DO):

1 I am indebted to Siemen Terpstra for making available this knowledge in condensed form through his writings: Siemen Terpstra .com