The Fine Art of Combining Harmonics

The HarmoniComb Matrix: A New Method
for Visually Describing Music

part of an ongoing research by

Bo Constantinsen

whatmusicreallyis.com

Section Contents

Introduction

The visual representation described in this paper is the result of my own research and work, but does not constitute a novelty in its entirety. The basis, or underlying principle of this mathematical graph is a matrix of numbers with ancient history1 called “Lambdoma”, also known as the Lambdoid by Nicomachus, the Pythagorean ChiX, the Tetraktys by Plato, the Lambdoma Table by von Thimus, and the Cantor array.2 Other names include Mensa Pythagorica3 and combinations of the above terms, like Pythagorean Table. Whole-number systems that bear a one-to-one relationship with the Lambdoma matrix are the Farey series of gear ratios generated by a Diophantine equation4;5 and the Stern–Brocot tree. Related concepts include the Archetypal Splay–Tree or Co–Prime Pattern6 mapped by Ervin M. Wilson inside this grid of harmonic ratios, and the Partch–Meyer–Novaro tonality diamond structures which constitute a confined development of the original concept.

The novelty of my version consists in the accurate representation of musical entities, by using a graphical system mathematicians call “logarithmic”, which I call “sonic”, where the eye sees the distribution of tone-spaces in the same way as the ear hears them.

After finishing research, while working on the presentation of the construct I came to call “HarmoniComb” (short for “Matrix of Harmonic Combinations”), which could also be called “Logarithmic Pythagorean Lambdoma”, I found in a book called “Textbook of Harmonics” a presentation of the same concept, studied as a peculiar application of logarithmic measurement to the linear Lambdoma matrix of musical ratios. Its author, Hans Kayser, writes:

«In contrast with the equidistant coordinate system used previously, we use a logarithmic system here, i.e. one that diminishes perspectively in its ratios from the middle outwards. We use a diagrammatic inventory of the tone-values, meaning that the tones, in a certain sense, look the way they sound (not the way we measure or count them).»7

Although Kayser’s work contains a two-dimensional logarithmic construct and a three-dimensional linear one, he never joined them into a three-dimensional logarithmic matrix – at least as far as I know. He also never looked at the ascending—descending reciprocals from the temporal perspective of tempo or rhythm, his work remaining confined to the tonal realm of pitch. Nevertheless, he took the study of harmonics to extreme, and wrote a lifetime’s worth of books on the subject.8

As it turned out, at least one person before me has independently discovered a part of the same (harmonics–related) basic concepts, and this confirmed and reinforced my own work. A work based on the fact that visual perception is the most absorbed method of understanding, and maybe the only proper way to learn about the unseen, formless world of sound–relations.

This paper is part of the “What Music Really İs: The Manual for The 3rd Millennium Musician, Spiritual Seeker and Free Energy Discoverer”. The information presented here, although in a very simple and intuitive manner, might be of little value to the innocent music lover when taken out of its original context. That’s why in continuing, it is assumed that the reader has knowledge about the true harmonic series in its completeness: what it really is, where it can be found in nature and how different harmonics can be isolated and emphasized on instruments that allow it; also that the reader is well acquainted with “The Math Behind Music”.

Whenever the term “harmonic series” will be used herein, it implies both the ascending and descending series, unless otherwise stated. Whenever the term “harmonics” is used, it implies both supra-harmonics (overtones or harmonics from the ascending series) and sub-harmonics (undertones or harmonics from the descending series), unless otherwise stated. These are simple tones: manifestations of sound in their purest form, exactly as they appear in nature inside complex tones, and their use throughout this work is implied in the precise numerical form given. The use of temperament or other artificial contrivances and approximations is totally excluded.

All the ratios presented here refer to simple sound tones. It will be shown that complex tones are generated from these simple tones, as depicted in the conceptual matrix presented further down. By extension, these harmonics can be used as template for practical tuning systems applied to compound tones. In this case, it is only their fundamental which is taken into account in computation, while their respective harmonic series is implied as being related to the fundamental in whole (rational, integer) number relations.

Main Tonal Components
and The HarmoniComb Tonal Matrix

Musically, the harmonic series is nature’s single perfect tuning system and rhythmic pattern. Most of the cultural tunings and rhythms currently employed over the world were developed according to arbitrary social constructs and are part of the subjective experience of musico-social conditioning. Many are based on the true harmonic series and represent a distant slice of it with irregularly missing members. Others, like temperaments, are totally artificial despite any attempt to make them seem as extensions of natural harmonics. Excluding equal and unequal temperaments, any other conceivable tunings based on rational numbers (and their combinations in form of fractions) can be traced back to a pattern inside the matrix presented here.

Mathematically, the harmonic series is the formula for nature’s laws of relationships. These laws are not reflected only in music, as harmonic genesis of timbre, rhythms and pitches, as sonic distances between harmonics, frequency relations between tones, and size proportions of bodies generating sound, but also in all numbers and number systems in use, as the pattern of true prime numbers, inherent in the geometry of architecture and visual art, at the core of trigonometry, in the colors of the rainbow, and generally in all arts and sciences.

The Ascending Harmonic Series: Vertical

The ascending harmonic series is graphed vertically on the “what-you-see-is-how-you-hear” sonic number line of frequency perception, to convey the resolution of this series unto itself. This simply means that all overtone components fit perfectly within the fundamental, and also with each other – as long as the same fundamental remains their common reference.

Vertical “what-you-see-is-how-you-hear” sonic number line of frequency perception (red) depicting the ascending harmonic series through numbers (frequency multipliers) and outlines of oscillating strings (actual physical dimensions).

The sonic number line depicts differences in pitch exactly as they are perceived and interpreted by the human brain. The numbers represent frequency multipliers, as related to the absolute fundamental (the second supra-harmonic ×2 has a frequency 2 times higher than the fundamental 1; supra-harmonic ×3 sounds 3 times higher than 1, and so on). The outlines of oscillating strings represent actual physical dimensions of each harmonic’s sound producing source.

The vertical arrangement also means that any or all supra-harmonics can be played simultaneously (at the same time), the resulting sound being perceived by the ear as harmonically consonant. This is in fact the definition of consonant harmony: simple tones whose frequencies are integer multiples of the fundamental, playing simultaneously with the fundamental.

This definition obeys one rule: any number of overtones are harmoniously consonant when played together, as long as the fundamental is also sounded with them. When the fundamental is not present, the further we ascend, the lower the perceived consonance. For example, sounding together supra-harmonics 2 and 3 will appear more pleasing to the sense of hearing than sounding together supra-harmonics 8 and 9. (But if we introduce the fundamental 1 into the picture, then 8 and 9 will resolve perfectly into it giving the sensation of consonant harmony.)

These supra-harmonics are simple tones. Playing them together creates a compound (or complex) tone, and the specific volume (intensity or power) of each overtone in the mix accounts for timbre: that unique characteristic which makes a piano sound different from a sitar, when both instruments play a (compound) tone having the fundamental vibrating at the same frequency. This also accounts for the difference between two human voices pronouncing the same vowel, and the difference between different vowels chanted by the same person – all under the same condition: that their fundamentals have the same frequency.

Overtones could also be played consecutively, but let’s leave that aside for later and stick with the basics for now.

The Descending Harmonic Series: Horizontal

The descending harmonic series is graphed horizontally on the “what-you-see-is-how-you-hear” sonic number line of frequency perception, to convey the fact that the fundamental fits perfectly within all undertone components (which in turn don’t always fit with each other).

Horizontal “what-you-see-is-how-you-hear” sonic number line of frequency perception (blue) depicting the descending harmonic series through numbers (frequency divisors) and outlines of oscillating strings (actual physical dimensions).

The sonic number line depicts differences in pitch exactly as they are perceived and interpreted by the human brain. The numbers represent frequency divisors, as related to the absolute fundamental (the second sub-harmonic ÷2 has a frequency 2 times lower than the fundamental 1; sub-harmonic ÷3 sounds 3 times deeper than 1, and so on). The outlines of oscillating strings represent actual physical dimensions of each harmonic’s sound producing source. The amplitude (power, volume) of these is not properly depicted.

The horizontal arrangement also means that any or all sub-harmonics can be played consecutively (one after another), the resulting melody being perceived by the sense of hearing as musically valid. If however more than two are played simultaneously, the resulting compound sound will be perceived as harmonically dissonant. This is in fact the definition of dissonant harmony: simple tones whose frequencies are integer sub-multiples of the fundamental, playing simultaneously with the fundamental.

As a side note, non-harmonic dissonance or out-of-tune-ness is created by the simultaneous sounding of complex tones that are not in tune with themselves. This simply means that any number of complex tones whose fundamental frequencies do not exhibit relationships taken directly from their own harmonic series will sound out-of-tune when played together. That’s because their fundamentals and complete harmonic series will combine into non-repeating patterns: non-harmonic movement that does not repeat the same cycle in exactly the same time. Physically, this translates into outlines of oscillating strings that do not fit perfectly into each other.

The simplest frequency-relationships (ratios) account for purest consonances. With other words, harmonic relations of very large numbers tend towards higher states of harmonic dissonance, while unmusical, seemingly unresolvable and out-of-tune dissonances are generated by unmusical numbers having non-recurring and never-ending decimals (irrational numbers). There is of course a questionable degree of dissonance that can be tolerated, but the most obvious fact is that the brain can well be conditioned to accept and even enjoy these dissonances, as long as they are part of the only music it has ever experienced.

The use of undertones could be extended to simultaneous playing, but this is part of an ampler talk about expanded musical possibilities which currently doesn’t serves us in laying out the foundations.

The Same Fundamental

The vertical and horizontal tonal aspects of music being defined (through simultaneous and consecutive sounding of tones), we can proceed by extending the two reciprocal harmonic series from the same fundamental.

HarmoniComb (logarithmic number-lines graph) depicting the harmonic series extended from the same fundamental: the ascending series vertically from the bottom up; the descending horizontally from right to left.

Although extending much higher in reality, and into infinity conceptually, only the first 5 harmonics have been depicted here, for the purpose of clarity. The associated colors are arbitrary.

It can be seen from the picture that the fundamental, marked 1, has a series of harmonic overtones rising higher and higher in pitch, progressing vertically from the bottom up, and the intervals between these get smaller and smaller as pitch increases. This is in fact a conceptual representation of a compound tone, which sounds the fundamental and all vertical harmonics simultaneously (in a chord, if you will). Volume or intensity is not depicted on this graph.

The undertone series of harmonics displays an exact opposite and symmetrical structure. As the series decreases in pitch, progressing horizontally from right to left, the intervals get smaller and smaller. The difference is, this series does not represent the structure of a compound tone, but rather the steps to be melodically taken by such tones. This literally means that every undertone will have its own series of overtones.

So what does this actually mean? It means that frequency grows or diminishes (exponentially), multiplying or dividing its value by the number after the operator (× or ÷ signs), and is always related to the fundamental of the series. Say this fundamental 1 gets assigned to a frequency of 300 events per unit of time; that would make the ×2 overtone vibrate 600 times, the ×3 overtone 900 times and so on, while the frequency of the ÷2 undertone will be 150 events per unit of time, that of the ÷3 undertone 100, and so on.

That’s why, for the values on the outer edges, even though we write “×5”, “÷3”, etc., the implied interpretation is “1×5”, “1÷3”, etc.

One Way of Looking at İt

Every undertone will have its own series of overtones, identical in structure with that of the fundamental, but shifted altogether in frequency. Taking as example the second undertone, a compound tone tuned to this frequency will have its own shifted series of overtones as depicted below.

HarmoniComb (logarithmic number-lines graph) depicting a compound tone tuned to the second undertone, with no values associated to the ticks.

The numerical values of every new member of this child-series are given by the interaction between the fundamental of the series, which in this case is the second undertone, and the original overtone series. Every new tick on the graph is in fact an intersection point resulted from a parallel extension and the tracing of projections. Every new value will contain the numbers that intersected to create it, together with their respective operators.

For example, the second overtone ×2 of the second undertone ÷2, at the exact point where they intersect, is the result of the interaction between ÷2 and ×2, which is ÷2 ×2 (and whose implied interpretation is 1 ÷2 ×2). This simple math continues in the same manner for the entire child-series of ÷2, multiplying its value by those of the ascending series of harmonics, one by one.

HarmoniComb (logarithmic number-lines graph) depicting a compound tone tuned to the second undertone, having just the first value associated to its tick. HarmoniComb (logarithmic number-lines graph) depicting depicting a compound tone tuned to the second undertone, having the first five values associated to their ticks.

Thus a complex tone having as fundamental (or which is tuned to) the second undertone will exhibit its own complete structure of overtones, of which just the first five are depicted here.

We can continue like this and represent the natural harmonic overtones of every other compound tone tuned according to the sub-harmonic model.

HarmoniComb (logarithmic number-lines graph) depicting compound tones tuned to the undertone series, having the first five values associated to their ticks.

Thus we have depicted five different compound tones, all related on the basis of the natural descending harmonic series, each having its own series of ascending harmonics.

The fact that they are all shown on the same graphic at the same time doesn’t necessarily mean they can be played together (simultaneously). On the contrary, by the universal laws of acoustics, these tones should be played one after another (consecutively). The order in which they are to be played, the length of each tone through time, its intensity, and most important, the amount of overtones each compound tone will sound and the intensities of these, rests in the hands of the 3rd Millennium Musician.

The Other Way of Looking at İt

Every overtone will display the possibility of having its own series of undertones, identical in structure with that of the fundamental, but shifted altogether in frequency. As a first example, the second overtone of a compound tone (having the fundamental 1), together with its own series of shifted undertones, is depicted below.

HarmoniComb (logarithmic number-lines graph) depicting an undertone series tuned to the second overtone, with no values associated to the ticks.

The numerical values of every new member of this child-series are given by the interaction between the fundamental of the series, which in this case is the second overtone, and the original undertone series. Every new tick on the graph is in fact an intersection point resulted from a parallel extension and the tracing of projections. Every new value will contain the numbers that intersected to create it, together with their respective operators.

For example, the second undertone ÷2 of the second overtone ×2, at the exact point where they intersect, is the result of the interaction between ×2 and ÷2, which is ×2 ÷2 (and whose implied interpretation is 1 ×2 ÷2). This simple math continues in the same manner for the entire child-series of ×2, dividing its value by those of the descending series of harmonics, one by one.

HarmoniComb (logarithmic number-lines graph) depicting an undertone series tuned to the second overtone, having just the first value associated to its tick. HarmoniComb (logarithmic number-lines graph) depicting an undertone series tuned to the second overtone, having the first five values associated to their ticks.

Thus the second overtone, which is part of another complex tone, will exhibit its own complete structure of undertones, of which just the first five are depicted here.

Continuing in this manner we can represent the natural harmonic undertone series of every other supra-harmonic of the main compound tone.

HarmoniComb (logarithmic number-lines graph) depicting undertone series tuned to the overtone series, having the first five values associated to their ticks.

We have thus depicted five different harmonic overtones of a single compound tone (fundamental included), each having its own series of descending harmonics.

The One and The Other Are The Same

If the two constructs get overlapped, all intersection points overlap and their values correspond. They are no different from each other; their values coincide because – for example: (1) ÷2 ×3 = (1) ×3 ÷2. The two operations are the exact same thing, giving the same result, arrived at in two similar ways. This dual genesis of every musical entity will be further discussed later on.

Two HarmoniCombs (logarithmic number-lines graphs); one depicting five compound tones tuned to the undertone series, the other five undertone series tuned to the overtone series, both having only one inner value associated to one tick as an example.
HarmoniComb (logarithmic number-lines graph) depicting the overlapping points between the overtone series tuned to undertones and vice-versa: undertone series tuned to overtones.

For the values inside the HarmoniComb matrix (generated by the interaction between those on the outer edges), even though we write “÷2 ×3”, “×3 ÷2”, etc., the implied interpretation is “1 ÷2 ×3”, “1 ×3 ÷2”, etc. In this example, the two expressions are equal and can be written in simple form as one fraction:

1 ÷ 2 × 3 = 1 × 3 ÷ 2 = 23

Generally, for any harmonics b and c having 1 as absolute fundamental:

1 ÷ b × c = 1 × c ÷ b = bc

Ratios are the best way of representing musical entities (sonic distances) because every musical ratio, written as a fraction of two integer (rational) numbers, has at least five different and non-exclusive meanings. With other words, every fraction can be interpreted in five different ways, and these interpretations do not cancel each other. So far we covered one of them, and we’re not quite finished.

HarmoniComb (logarithmic number-lines graph) depicting

The special case of the original harmonic series (on the right and bottom outer edges) which have the number “1” above and below the fraction bar will be discussed in the next chapter.

Notice how horizontally, all numbers above the fraction bars (numerators) are the same – these are all ascending harmonics previously marked with “×”; vertically, all numbers below the fraction bars (denominators) have the same value – these are all descending harmonics previously marked with “÷”. Therefore, if taken out of this context, just by looking at a ratio (or set of ratios) we can instantly tell how they were generated and where they belong on the graphic.

Depicted below is a HarmoniComb Matrix of 1024 harmonics extending in both directions and interacting with each other. In its completeness, it is of little use musically – especially since many of its pitches fall outside the human hearing range. But any part of it can accurately represent different tuning systems, as we shall see in the following chapters.

HarmoniComb (logarithmic number-lines graph) depicting

Every intersection point on this diagram is a ratio that represents the frequency of a simple tone, and music can be made out of it by choosing a tuning (abstract scale of sounds) from the descending series and by constructing the compound tones of this tuning from the ascending series. Sounding a compound tone is the definition of a chord.

If however every ratio is taken to represent the fundamental of a compound tone, the principles described hitherto still apply and a third dimension would have to be added to the HarmoniComb in order to properly depict that. Any music using this system will be harmonically consonant, as long as consecutive tones are to be played horizontally on the descending series and simultaneous ones vertically on the ascending series. Sounding harmonic chords of compound tones means emphasizing overtones by amplifying their volume. This process gives birth to other sounds: some real, and some products of human perception, but this falls outside our current concerns right now.

There is of course no reason to stop at a specific number of harmonics. The HarmoniComb matrix can be extended up to any figure, and doesn’t have to necessarily include the beginning. Any portion of it will do just fine, as long as it serves the divine inspiration of the 3rd Millennium Musician.

The Tonal Concept, in İts Simplest Version

By simplifying (reducing) the fractions it can be observed that all ratios meeting the same diagonal (at half of a right angle) have the same value. The mathematical operation of reducing a fraction to its lowest terms consists of dividing both the numerator (above fraction bar) and denominator (below fraction bar) by the highest number that can divide into both numbers exactly, called greatest common divisor or factor.

TO DO: HarmoniComb (logarithmic number-lines graph) depicting

img to do: bold script 1:1's; tilt ratios?; PAHW

All ratios intersecting a (diagonal) dotted line have the same value; those who seem to be standing by themselves are also included, and in order to find matching values for them, the current HarmoniComb of (arbitrarily chosen) 9 harmonics would have to be extended. To make this alignment obvious, a rainbow-type spectrum of colors has been placed in the background. Any continuous color line represents one pitch only.

One particular application of this observation is that the complete HarmoniComb projects down to a single sonic number line. This simplification from bi- to uni-dimensionality has as drawback the tightening of sonic distances: on the projected number line, all distances between harmonics get shortened by a ratio of √2:1. This shows that the square root of 2 is a physical distance of diagonal projection. It is not a musical ratio because it can’t be found as such in the harmonic series, no matter how much the HarmoniComb gets extended.

The alignment of same-value ratios may not be so obvious from this perspective, because although we perceive straight angles innately, in most humans orientation is given by horizontal versus vertical placements – pretty much like holding our balance and keeping a straight, vertical posture at a right angle with the seemingly flat, horizontal earth. The best reference we intuitively seem to have about right angles and projections is vertical; this sense of orientation is not by chance connected to the mechanics of the inner ear.

By tilting the HarmoniComb Matrix half of a right angle clockwise, the sense of under- and overtones as related to horizontal and vertical space is slightly compromised. However, this elegant tilted perspective places the combined progressions of pitch horizontally, low to high from left to right, while emphasizing constant pitch vertically.

In its Tilted version, the HarmoniComb retains its most important feature: the accurate representation of musical entities, where the eye sees the distribution of tone-spaces in the same way as the ear hears them.

TO DO: HarmoniComb (logarithmic number-lines graph) depicting

TO DO: PAHW

The lower, outer edges of the Tilted HarmoniComb represent the harmonic series progressing from the absolute fundamental 1. By tracing projections from any intersection point down on these edges, all derived ratios are visually integrated in the acoustic context defined by the original harmonic series. All fractions that simplify to the same value intersect the same projection.

There is only one vertical dotted line and only one color for the fundamental, representing the pitch space of 11 situated at the center. There is only one dotted line and color for 32 , only one for 23 and only one for every other value, regardless of the dimensions of the matrix. Wherever one might wonder on the Tilted HarmoniComb, when coming across a vertical projection, there’s only one pitch (color, nuance, tone) there.

Example: we want to find out whether 34 sounds higher or lower than 23 , whether one or both fit inside the sonic distance between the first and second harmonics, and how far apart are they from the fundamental? All we have to do is trace projections from these points all the way down on the original edges of the graph and, alternatively, on a number line placed under it. We then notice the following: 34 is lower in pitch than 23 ; they both fit inside the first double and are symmetrically spaced around a theoretical midpoint between 11 and 12 .

Issues like “Which is highest: 79 or 57 , 53 or 85 ?” or “How does my set of ratios compare to a certain aboriginal tuning?” are easily and elegantly addressed. There is no confusion. This is also true when considering ratios to be a universal musical language and the most accurate way of representing pitches – a subject that will be expounded in the next chapter.

Real-Life Applications

The change in perspective by tilting is extremely useful if the graphic becomes the playing surface of an actual musical instrument. On every “\” diagonal there is a harmonic series of descending undertones well adapted to the natural proportions and tilting range of the human left hand. On every “/” diagonal there is an ascending harmonic series of overtones well adapted to the natural proportions and tilting range of the right hand − provided the instrument is built using the right proportions.

HarmoniComb (logarithmic number-lines graph) depicting

TO DO: start from 1/1 with both hands?; PAHW

Raise your hands in front of your eyes with the back of your palms facing you, and look at them. The distance from thumb to index is the same for both hands, but the direction is opposite and the angle mirrored. The idea is to have the left-hand tones mirror-imaged to those on the right and vice versa, to match the natural design of the human hands.

The implication is not that the Tilted HarmoniComb somehow fits the geometry of the human hand, only that it’s mirrored symmetry fits perfectly the mirrored symmetry of human hands. The left hand should not play a pattern devised for the right hand, and neither the other way around.

The central line of fundamental pitch divides the matrix area into two mirrored sections of reciprocal frequency ratios. Every ratio on the left can be found in inverted form on the right, mirrored around the same spatial coordinates relative to the fundamental line, and vice versa. All tone-values on the left side sound lower than the absolute fundamental; those on the right side sound higher. That's because all ratios on the left represent sonic distances (intervals) from the descending series while those on the right, from the ascending. This will become clear in the next chapter, “The Many Meanings of Every Ratio”.

Think of the Tilted HarmoniComb as a two dimensional surface projected as a hologram on top of a polyphonic Theremin pitch field, which is constant vertically but increases exponentially in frequency from left to right. (If you fail to see that, go with a large touchscreen with the picture below displayed on it, which generates sound when any area is touched.) Put your finger somewhere on the left and move it slow to the right. The pitch increases steadily. Stop and move it up and down. The pitch is constant.

HarmoniComb (logarithmic number-lines graph) depicting

TO DO: PAHW

Switching between tuning systems as-you-play (also known as “modulation” and “changing key”) is a simple process of vertical finger-sliding − which keeps all sounding pitches constant − and finding a new tuning. This is extremely easy especially in the upper regions of an extended matrix, and with practice the visual cues needed to identify musical relationships could be rapidly absorbed. Conversely, the Tilted HarmoniComb could be engineered to slide up-and-down under one’s fingers, at the action of pre-defined tuning pedals.

This is what makes the Tilted HarmoniComb different from all other known versions of the Lambdoma. It is a logarithmic construct, a perceptual representation that graphically maps a human sensation (perception of pitch) onto a diagram by the use of symbols. These symbols are the points of intersection formed at right angles between two sonic number lines of frequency perception. Musical entities are visually distributed along these lines and marked by the intersecting points exactly as they are heard by the ear. What You See İs How You Hear.

What Tuning Really İs:
Subsets of the HarmoniComb

A tuning, also known as musical scale, is an abstract ladder of tones, a set of numbers with musical significance and precise mathematic definition. Although harmonic combinations and patterns could be found in one form or another in nature, most tunings employed in music by different cultures all over the world and from all time-periods are patterns created by the human intellect: more or less arbitrary, and conditioned by social and cultural factors. These are often inspired, or meant to imitate, natural-occurring phenomena or theoretical postulates thereof.

Any scale based on integer numbers and fractions of these can be traced back as originating form the HarmoniComb, or, depending on the perspective of approach, any rational scale can be mapped inside the HarmoniComb.

The use of the HarmoniComb is not limited by the master edges. Subsets can be found in any area of the extended diagram. Let’s take as an example the symmetrical structure generated by the interaction of all harmonics inside the sonic distance created between the 3rd and 6th harmonics of both series.

HarmoniComb (logarithmic number-lines graph) depicting

Just by looking at each ratio’s vertical projections we instantly notice there are no duplicate tones, except for the absolute fundamental present four times. This can be verified by simplifying the fractions. The outer edges of this (and any other) subset act as generators for the inner structure, just like the pure harmonics on the outer edges of the master set act as generators for the entire HarmoniComb. All subsets follow the same rules and patterns of the master set.

If the inner ratios are discarded, we end up with two relatively and reciprocally identical 5-tone (pentatonic) tuning sets, each inside its own hable sonic distance. Hable is a word made up from “half” and “double” and refers to the sonic distance between the first and second harmonics of both or either series. (That’s because “half” limits the meaning to sonic distances from the descending series, while “double” to those from the ascending.)

HarmoniComb (logarithmic number-lines graph) depicting

Left- and right-side ratios are reciprocal (inverted form) reiterations of each other, symmetrically mirrored around the fundamental. If the graph is folded like a book, the hinge being the vertical line of the fundamental, then the left and right sides overlap perfectly. If working just with the projections on the sonic number line below the subset, we can fold once more every side at the theoretical midpoint and still get overlapping symmetry. And if we place these tonal number lines on top of each other, the direct correspondence becomes obvious. Because of this, the mirrored symmetry of any subset relative to others or to the master set is fractal in nature.

HarmoniComb (logarithmic number-lines graph) depicting

The only difference between the values of left- and right-side ratios is multiplication/division by a factor of 2, reflecting the arbitrarily chosen sonic distance limits of the hable: 3 is the half of 6, and 6 is the double of 3. This means that if we multiply the sequence 21 53 32 43 65 11 by 2, we get 11 56 34 23 35 12 and of course the reverse is true: divide the last set by 2 to obtain the values of the first.

It seems like in ancient times, musical scales were defined by the left side. The fundamental was the highest tone in the set and all other descended according to their harmonic position in the scale. Modern practice reverses the direction, so now the fundamental is the lowest tone from the set and all others ascend (or should ascend) accordingly to their predefined harmonic structure.

HarmoniComb (logarithmic number-lines graph) version.

Besides tightening of sonic distances, another drawback of simplifying a two dimensional matrix to a single dimensional number line is losing the perspective over the harmonic genesis of the tones, with consequences for the creative process. Let’s suppose we’re composing a musical passage using the previously (arbitrary chosen) 5-tone scale, where we start from the absolute fundamental and want to reach its double in three steps, then descend back using the same number of steps.

On the Tilted HarmoniComb the obvious choice for ascending is on the ascending harmonic series, while for descending on the descending series. The letters from A to G are meant to describe each step of the progression and have nothing to do with the names given to “intervals” (sonic distances) in certain musical systems.

On the one-dimensional number line both series get mixed up and the sense of natural harmonic progression is lost. On the Tilted HarmoniComb the ascending and descending series have their own diagonal slope and direction, and their combined pitches are integrated through vertical projections.

One notation system that can visually represent the above passage is depicted below. All tones have the same length and are played one after another; the last tone is silent. An extensive explanation of this system follows in the chapter about notation.

Numerofonía Geometrica de Sergio Aschero - simple notation system.

More about alternate views over scale genesis, and how sets of numbers like these were arrived at and interpreted by theorists and mystics across history, as well as what they really represent, in the section The Scholars' Approach to Tuning of the chapter Examples: How All This Fits into Practice, which will be understood only by going first through the next chapter, The Many Meanings of Every Ratio. Especially relevant for this tuning scale example is the Ancient Harmonics: Archaic Ratio Math section from the last mentioned chapter. This particular (and arbitrary) set of numbers is retaken as example throughout this work.

The Complete 2D Model

If we would give up the concepts of horizontal and vertical, and represent both series on one number line (just like the one projected down from the Tilted HarmoniComb), then 2 complete harmonic series of both under- and overtones could be mapped horizontally, and also vertically, each on its own line.

TO DO: HarmoniComb (logarithmic number-lines graph) 2D version.

All there's left to do is finding all intersection points generated by the interaction of the represented harmonic series with one another.

TO DO: HarmoniComb (logarithmic number-lines graph) 2D version.
TO DO: HarmoniComb (logarithmic number-lines graph) 2D version.

So far, we have been describing only a quarter of this entire construct, namely the upper-left “quadrant”.

The complete two dimensional HarmoniComb projects down to a single logarithmic number line. Only some of the main harmonics of a (randomly chosen) 6×6 matrix have been depicted below.

TO DO: HarmoniComb (logarithmic number-lines graph) 2D version.

description

Reversing the HarmoniComb

Given a complete 2-Dimensional Matrix of Harmonic Combinations limited in this example to Harmonics 1 to 5,

TO DO: HarmoniComb (logarithmic number-lines graph) 2D version.

Regular HarmoniComb 5 ÷ 1 × 5
Size of fractions determined by placement, unrelated to their respective values.

here's how its reverse would sound like:

TO DO: HarmoniComb (logarithmic number-lines graph) 2D version.

Reversed HarmoniComb 1 ÷ 5 × 1
Size of fractions determined by placement, unrelated to their respective values.

Steps needed to reverse a HarmoniComb (or any other type of bi-axial sonic matrix):





The Complete 3D Model

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

TO DO: enlarge text

The intersections of the 3 dual harmonic number lines with each other have been mapped, resulting in 3 complete 2D identical matrices entangled at the origin of the absolute fundamental. Their intersection results in 12 smaller planes of which 3, 3 and 6 are respectively identical.

Three dimensional moving image

HarmoniComb 3D - 16 harmonics
(three dimensional logarithmic pythagorean lambdoma)

Every harmonic and harmonic combinations with a unique pitch has its own two dimensional sound plane. The three dimensional HarmoniComb is the totality of these planes stacked on top of one another. Below, the plane of the absolute fundamental 1 of a 16-harmonic matrix is represented (very large image, it make take some time to load).

Three dimensional moving image

HarmoniComb 3D - 16 harmonics and the hexagonal plane of the absolute fundamental 1
(three dimensional logarithmic pythagorean lambdoma)

The complete three dimensional HarmoniComb projects down to a single logarithmic number line. Only some of the main harmonics of a (randomly chosen) 16×16x16 matrix have been depicted here.

TO DO: HarmoniComb (logarithmic number-lines graph) 2D version.

Below, the sound planes of a few random pitches inside a 6-harmonic matrix are represented in 3D: 361 201 121 81 51 31 21 11 12 13 15 18 112 120 136

Three dimensional moving image

In 2D (or 3D viewed from one side) these pitches are:

TO DO: HarmoniComb (logarithmic number-lines graph) 2D version.



Last updated: 12 january 2016 

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References (TO DO):

1 Barbara Ferrell Hero and Robert Miller Foulkrod - The Lambdoma matrix and harmonic intervals [1999]; http://www.org/

2 The historical context of the lambdoma matrix and its applications - Hero, B.F.; Int. Lambdoma Res. Inst., Wells, ME, USA; Martinez, E.; http://www.org/

3 MUSICA UNIVERSALIS: from the Lambdoma of Pythagoras to the Tonality Diamond of Harry Partch – K. Scott Eggert, Independent Study, Theory/Composition; http://www.org/

4 http://www.anaphoria.com/lamb.PDF; http://www.anaphoria.com/lamb.PDF

5 Barbara Ferrell Hero and Robert Miller Foulkrod - The Lambdoma matrix and harmonic intervals [1999]; page 4, low left; http://www.org/

6 http://www.anaphoria.com/splay.PDF; http://www.anaphoria.com/splay.PDF

7 “TEXTBOOK OF HARMONICS” by Hans Kayser (1891 – 1964); §36. Logarithmic Arrangements, pages 326 to 338; Translated by Ariel Godwin. Edited by Joscelyn Godwin. German Edition 1950. English Translation 2006. 630 pages, 485 illustrations. Translation Society Edition. CAT#385. Available through the Sacred Science Institute online book store. I am grateful to W. Bradstreet Stewart for making the section available for personal study.

8 Hans Kayser produced thirteen books on the Science of Harmonics during his lifetime, the fourteenth being published after his death. Bibliography of Hans Kayser’s Publications, as published online on the official Hans Kayser (1891 – 1964) website administered by the Sacred Science Institute, under the “Bibliography” section.