The direct examination

of consonance

The nature of tone


Scales versus scales
Data tables

A few more considerations

This is not a translation of the corresponding French page. It is a new version. Divergences are, I dare to hope, in proportion with corrections and improvements.

This article begins with fairly standard introductory material concerning single tones before moving on to tone combinations. Before stumbling across the drawing featured in the previous page where pitch is handled as a continuum, I first studied several familiar scales using Fourier synthesis as the basic tool. This page focuses on my initial purpose which was to find out just what kind of physical consonant properties each different scale might exhibit. The goal of course, is to use physical data to support opinions received as ancient wisdom, that is, classical theory. Nobody will be too surprised to hear that most, it not all, classical ideas are borne out here. On the other hand, classical tonal theory deals mostly with "what?" and "how?". My subject here is "why?".

The nature of tone

The tone of a string (or a tube) is characterized by a discontinuous spectrum. The individual lines in the spectrum are called harmonics. The first harmonic, the line representing the lowest perceived pitch, is called the fundamental. Following lines have frequencies which are all integer multiples of the fundamental. This closely shadows Fourier's theorem whereby any periodic function can be put together in this way. A periodic phenomenon, such as a vibrating string, is a different species altogether but it is nevertheless subject to Fourier's theorem (or should I say "law" in this case?) because it is periodic. Periodicity isn't some obscure numerical property: it is the very essence of pitch or tone.

A few experiments

This isn't as mysterious as it sounds. Periodicity is a natural property shared by many objects of regular shape which are longer than they are thick. Take a rope for instance. Children enjoy facing each other, each one holding the end of a 10 foot rope almost touching the ground in the middle and wiggling it in the air. A slow wiggle causes the rope to turn slowly, tracing out a simple stable three-dimensional bean shape.
When the rope is wiggled much faster, the rope changes into two smaller shapes similar to the first. Wiggle it faster and you again get three even smaller versions of the initial shape. There are no in-between stable motions. The periodic motion is somehow constant and this requires stability which is only acheived through division of rope-length by whole numbers. This is one simple aspect of Fourier's theorem. This observation may be a bit surprising.

Replace the rope with a string fastened to a guitar and tightened and even more astonishing things begin to happen!

Pluck the string and it makes a constant tone. The tone's "height" or pitch is always the same regardless of where you pluck the string. The tone is somehow "richer" when you pluck it in the middle near the sound-hole and "tinnier" when you pluck it near the bridge. But the pitch never changes. The constant pitch is a property of the string. [A column of air vibrating in a longish tube such as a flute displays the same property.] The pitch is sponaneously offered by the string. Keeping tension and thickness the same, if you shorten the string by fingering it, the pitch increases, it gets higher.

This peculiar property isn't shared by all objects. Let us consider a wooden table to which a tuning-fork is applied. The table suddenly starts vibrating at the same frequency as the tuning-fork. It is possible to force any object to vibrate at any frequency imposed from without. As soon as you lift the tuning-fork, the table vibration immediately ceases. It is not spontaneous. It is not a characteristic of the wooden table.

In order to have a spontaneous and lasting vibration of constant pitch, an object must move according to a constant trajectory. This is called a "standing wave". The taught string has two fixed points and the slowest standing wave displays maximum displacement in the middle of the string. Another standing wave shows an additional fixed point in the middle of the string and two "bellies" instead of one. Yet another has two additional fixed points dividing the length into three equal sections etc.

These modes of vibration can be crudely "revealed" by sounding the "harmonics" of open strings. If you touch the string very lightly in the middle (at the 12th fret), forcing that to be a fixed point, while plucking the string near the bridge, you hear a fundamental frequency which is twice that of the open string. Prevent motion with a touch at the third of the length (near the 7th fret) and you hear a fundamental frequency which is three times as high. And so on. I hope you can sense intuitively that the only stable frequencies which the string is capable of producing spontaneously are of this nature. [The word "harmonic" for this sound is something of a misnomer: one actually hears a tone, not a harmonic.]

points of light touch

It isn't much more difficult to demonstrate that the vibration we actually hear is the result of all the different vibrational modes happening all at once. Pluck the open string and then gently press a fingertip against the string above the 12th fret in the middle. The sound suddenly changes to the octave, with a fundamental at twice the original perceived frequency. No new impulse is given to the string. When you press it, you are merely filtering out vibrations with bellies in the middle of the string. One can only conclude that the vibration we actually hear is the result of the combination of all possible vibrational modes all happening at once. In other words, a tone we hear as "A" contains a fundamental with a frequency of 440Hz but it also contains resonances at 880Hz, 1320Hz, 1760Hz etc. This is amazing but it seems to be true.

Fourier's formula

Here is a brief outline of Fourier's method.

The general form of the sine wave is:

y(t) = a0 + a*sin(v*t + p)

where "t" is time and the result "y(t)" is the wave's amplitude at that particular time. The constant "a" is the factor which scales the amplitude and "v" is the angular velocity. If time is expressed in seconds, the angular velocity is given in radians per second. The initial value of the angle or phase is given in radians. The initial amplitude is "a0". The units of amplitude depend on the phenomenon under study: millimeters would be a good choice if one is examining a speaker coil and volts would be good if one is looking at the output from an amplifier.

It is more intuitive in tonal studies to use the frequency "f" rather than the angular velocity "v". A trivial substitution:

v = 2*pi*f
y(t) = a0 + a*sin(2*pi*f*t + p)

According to Fourier's theorem, any periodic wave, that is, a wave that is indefinitely sustained and always the same, is an algebraic sum of such terms:

y(t) = a0 + a1*sin(2*pi*1*f1*t + p1)
+ a2*sin(2*pi*2*f1*t + p2)
+ a3*sin(2*pi*3*f1*t + p3)
+ ...

The frequency of each term is always an integer multiple of a constant, "f1", the fundamental, also the lowest in the series. The fundamental frequency is roughly equivalent to what musicians call the "note" or the "pitch" of the tonal sound. The series of integer multipliers of the fundamental could be called "harmonic degrees". The series can be infinite but this is not a requirement.

This formula is most useful if you want to synthesize actual sounds. Our present purpose, to find out more about how tones work, is better served by the "Fourier transform", a.k.a. "the spectrum".

In the spectrum, the amplitude scale factor "a" of each term is plotted versus its frequency "f".

I have chosen the sawtooth wave as a model for all illustrations. It is known for its musical tone-like qualities and its terms are very easy to calculate:

for each degree n = 1, 2, 3...


Of course, relationships may be much more complex for other waveforms.

Frequency is the subject

It is convenient to restrict a general formula for building a computer model. Indeed it is perhaps the very essence of model-building to watch the behaviour of a particular version of a formula. The sawtooth seems to be a good choice. One can further restrict the model to follow only a single parameter: frequency. Amplitude will not be listed in the data on this page. Some reasons for so many restrictions should be given. I will try to be brief.

Assumptions, allowances, and guesses

Amplitude tapers off

The relationships among the constant factors "a1", "a2", "a3", etc., which determine the weight of each term in the Fourier sum, are characteristic of the timbre of any tone. The amplitude's tendancy in the sawtooth to decrease with higher harmonics could be shared "real" tonal instruments. How representative is the sawtooth of musical tones, generally? Returning to our experiments with grazing touches at specific points of the plucked string, we can note that the volume of the sound decreases dramatically as we move to higher harmonics. The sound of the 11th harmonic, for example, is so faint that I can barely hear it. This suggests that the weights do indeed decrease along the series, at least on the guitar. More importantly, they are perceived as decreasing. There is an even simpler observation which also suggests the same thing: the fact that the isolated fundamental harmonic (a sine wave made with an audio synthesizer) is perceived as having the same pitch as the corresponding musical tone. This might not be the case if the fundamental weren't also the loudest harmonic. I haven't investigated the "musicality" of series with random weights but it might be interesting to do so. In the meantime, I take the sawtooth's (decreasing) inverse law as being a reasonably acceptable model of tonal sound as I observe it with the guitar.

The (however fragile) assumption that harmonic weights decrease with degree is significant in several ways. Our subject here is the consonance of 2 tones. Consonance is the result of different harmonics of different tones having the same frequency. If the tones are of very different pitch, consonant effects will be weakened because the meeting of harmonics of very different degree and (hence) weight will be dominated by the stronger of the two and the weaker sound will merely be drowned out if the tones are of equal strength. Data of this sort plays a role in the picture featured on the previous page. Here, I ignore it for the most part, unless it be to relegate it to the end of each data list. This study focuses mainly on the "nearness" of frequencies in harmonic encounters.

Pitch and timbre

We don't hear an infinite series. Our hearing vanishes at high frequencies and we are completely oblivious to harmonic degrees higher than, let us say, 100, for a guitar string. We can ignore these in any model. Degrees higher than, let us say, 30, are beyond the realm of "pitch" and are perceived subliminally as contributions to the timbre, that is, to the tone "quality". As we will see, interactions among harmonics are more dramatic, that is, more perceptible, for lower degrees.

Harmonic degrees higher than 12 are separated by less than a semi-tone. As degree increases the harmonic frequecies appear to move closer together in terms of musical pitch. This doesn't appear in the spectrum. It is the result of the well-documented ability we have to compare frequencies as their ratio. Thus harmonics 1 and 2 are visually separated by the same distance in the spectrum as 7 and 8. On hearing frequencies, we compare them as 2/1 and 8/7, hardly the same thing! Harmonics greater than, let us say, 30, merge into a wash. Indeed, according to this model, harmonics 30 and 31 of a single tone would produce an "internal consonance" and we hear no such thing.

The first few harmonics are enough to get a general idea of what is going on

These are a few good reasons for limiting studies of this kind to the first 5 (the basis of conventional tonal theory) or 12 (just to be on the safe side) harmonics without fearing artificial over-simplification of the model.

sawtooth waveform sawtooth waveform sawtooth waveform
Sawtooth: first 6 harmonics first 12 harmonics first 100 harmonics

The staircase effect

The most important thing to remember here may be that the "staircase" of integer harmonic degrees is due to nature, not to the model. The following tables examine consonance exactly as outlined in the previous page by generating a spectrum for each tone involved in the chord and looking for harmonics of identical or near-identical frequencies.


A musical scale is a specific set of fundamental frequencies one uses in melody and harmony. A scale can be anything you like.

Several common musical scales behave in different ways. We will study three kinds of scale which differ first in their construction method. We will determine the advantages and drawbacks of each one.

Here is a summary of well-known attributes of each.

Construction Avantages Drawbacks
equal temper
  • is transposable without restriction
  • consonances are unstable but peaceful
  • has no perfect consonances
  • has a few perfect consonances
  • is transposable under certain conditions
  • has a few very unstable consonances
ideal* or diatonic or natural
  • has as many perfect consonances as possible
  • has a few very unstable consonances
  • is not transposable
*The term is Jean-Jacques Rousseau's. He deplored the absence of an "ideal" scale in the musical usage of his time. Today, we tend to consider that the many variations on "wrong" harmony offered by various scales too numerous to mention are an attractive playground. I prefer the neutral (but perhaps improper) term, "diatonic".

One should perhaps be reminded that scale as such isn't applicable to singing or to any instrument such as a string bass where it is not built in. The singer places his tones where he chooses. Refering to the picture featured on the previous page, one can imagine that a singer can move as close as he wants to the "hot spots". The hot spots defined by a "fifths scale" will be at slightly different frequencies from those defined by an "equal-tempered" scale. The singer makes his choices with respect to whatever scale is being used by the accompanying instruments, not to some yet-to-be-found "ideal". In other words, a singer can form a perfect consonance with an equal-tempered instrument. The "has no perfect consonances" disadvantage of the equal-tempered scale is self-referential: an instrument of equal temper can have perfect consonances in combination with other voices external to itself.

The rational construction of scales

I use three very different methods here. All are "rational" since they use ratios to define progressions from one note to the next. Each method produces the same series of 12 notes but not in the same order. The guitar method is the only one in the familiar order of the ascending chromatic scale. This is why I use it for labeling purposes.

The guitar uses an equal-tempered chromatic scale. The fundamentals are proportional to:
2^(0/12) 2^(1/12) 2^(2/12) 2^(3/12) etc.
The reference scale for the piano is a fifths-based chromatic scale. Its fundamental scale factors are:
(3/2)^0 (3/2)^1 (3/2)^2 (3/2)^3 etc.
A version of the diatonic scale (a.k.a. ideal or natural scale) can be devised where notes are derived using 2 distinct numerical progressions for scale factors. I will select a scale from the many possibilities. Here are the scale factors:
(3/2)^0 (3/2)^1 (3/2)^2 (3/2)^3 etc.
(5/4)^0 (5/4)^1 (5/4)^2 (5/4)^3 etc.

Scale families

All of these scales can claim to be rational inasmuch as they use mutiplication and division as the only operations in all calculations. Here is a table of the frequencies for the different scales used here, just to give an idea of what is sharp and what is flat in what respect:

name     degree   equal temper      fourths           natural
c        0        130.812783        130.812783        130.812783
c#       1        138.591315        137.810997        139.533635
d        2        146.832384        145.183602        147.164380
eb       3        155.563492        155.037372        156.975339
e        4        164.813778        163.331552        163.515978
f        5        174.614116        174.417044        174.417044
f#       6        184.997211        183.747996        183.955476
g        7        195.997718        193.578136        196.219174
ab       8        207.652349        206.716496        209.300452
a        9        220.000000        217.775403        218.021304
bb      10        233.081881        232.556058        232.556058
b       11        246.941651        244.997329        245.273967

One shouldn't read too much practical sense into such lists of "absolute" frequencies. The fourths scale, constructed starting with c and ending with g is identical with a fifths scale starting with g and ending with c. The natural scale is based on perfect major triads G, C and F. There are many other possibilities. My values are adequate for theoretical uses but chances are they won't match actual implementations of fifths and natural scales you might run across in real life.

The reference frequency A 440 is used throughout this data. Again, the values here may be adequate for synthesis but they probably won't match some values produced by singers or players who are sensitive to absolute pitch. In the strictest sense, I would understand absolute pitch as the ability to produce equal-tempered intervals at will or, indeed, whatever intervals seem to be most desirable in the context where they occur. For singing, especially, absolute pitch is very important because the trick is to tighten your muscles so as to produce a certain tone reliably before you hear it. This becomes increasingly easier to do and more accurate with practice if you always use the same reference tone to begin with. In other words, absolute pitch has to do with day-to-day sameness. The sameness can be of any temper you like. It should be safe to assume that the high degree of sameness brought about by the constant reference A 440 makes it easier to control tone with enough accuracy to also control temper.

In any case, musical comparisons of exact note frequencies aren't always simple. The notion of "inversion" is familiar in actual music: a fourth is an inverted fifth and vice versa. Moving up a fourth or down a fifth from C gives us an F. Up a fifth from C and down a fourth, we get G. When calculating fifths scales (as opposed to using them), there is also the question of "nearness" in the sequence. The cycle of fifths can't be used as a reference because it isn't true. Aside from that, inverting the direction of movement is still a matter of indifference. The fourths and fifths directions produce the same note values exactly. The same values obtain with

fourths    : C  F  Bb Eb Ab C# F# B  E  A  D  G
and fifths : G  D  A  E  B  F# C# Ab Eb Bb F  C

As long as the final G in the fourths scale is the same as the initial G in the fifths scale, the two series will be identical. An entirely different set of notes obtains however with

fifths     : C  G  D  A  E  B  F# C# Ab Eb Bb F

Taking C to be the same as in the first two scales, the new G will be different because C and G are now close neighbors instead of being at opposite ends of the scale.

This is the reason for my caveat : "the scales in the data are OK for theory but probably won't match real scales". This is not to say that there are inexplicable approximations or inaccuracies. It is first of all a matter of picking a starting point. The question of notes being higher or lower compared to the reference is brushed aside by saying that, once we have the next note in the fifths scale, we can move it around freely from octave to octave. The cycle of fifths isn't really an exact cycle and the final note in the series clashes with the first when brought back (accurately) to the proper octave. The ear can be very accurate, more so, I believe, an electronic tuner, which is why people who tune pianos use their ears. As far as I know, "real" piano tuners often use fifths to start with and then lie, cheat and steal their way to "equalized" temper. The diatonic scale bathes in even greater uncertainty. As defined here, it is a piece of a fifths scale: "F C G" (or fourths: "G C F"). In other words, you shouldn't think of C in the middle with a "higher" F and a "lower" G. The initial definition of the diatonic scale is one where G, C, and F form a compact grouplet in the series. None of the series I just outlined above will suit our purpose. Here is one that would:

fifths     : F# C# Ab Eb Bb F  C  G  D  A  E  B

Now, major thirds are considered. They could be seen as a new thirds scale draw vertically, rooted in the fifths definition.

Bb-F -C -G -D -A -E -B -F#-C#-Ab-Eb
|  |  |  |  |  |  |  |  |  |  |  |
F#-C#-Ab-Eb-Bb-F -C -G -D -A -E -B
|  |  |  |  |  |  |  |  |  |  |  |
D -A -E -B -F#-C#-Ab-Eb-Bb-F -C -G
The alterations used in this particular data set are highlighted. It should now be clear why there can be many ways to define an altered diatonic scale. Had I drawn this diagram first, I might have chosen a different Bb (also highlighted) but I didn't. The obvious pictorial reasons for choosing an alteration may or may not make musical sense.

Terms and definitions of the tables in the following section

Numbering systems

The conventional note names have been replaced here for my own convenience by numbers which I arbitrarily call "scale degrees" as shown in the table of scale frequancies (see above). The numbers are incremented following the chromatic tempered scale starting with c=0.

Harmonic degrees are numbered as usual. Lowest harmonics and their multiples are closely linked to familiar "perfect" intervals:


Chord construction method

All chords reported are constructed by selecting notes of different names from a single octave. Thus the chord C is "0 4 7" starting with the tonic, whereas A is "1 4 9", an inversion beginning with the major third. [This leads to subtle anomalies in the data but doesn't seem to disrupt most patterns too much. For example, the chord "0 2 4 7 10" contains all the notes of C9. It won't sound exactly like C9 but nevertheless "C9-like". This should be corrected but I decided to live with it for now.]

Estimating the beat rate

I use a rough formula which is reasonably accurate when the harmonic frequencies (Hz) are "very close". Let us call the higher frequency "hi" and the other "lo".

beat frequency (Hz) = (hi^2/lo) - hi

Derived from first principles, the formula is obviously inexact since the actual frequency of the combination of harmonics of equal amplitude is the mean (hi+lo)/2 and this value is not used in the formula. My excuse is that the beat rate also becomes inaudible as harmonics drift further apart. The error is tolerable, usually on the order of 1%.

Colomn headers

chord the chord name in standard notation
scale temp=equal temper, quar=fourths (i.e. fifths), diat=diatonic
T1 equal-tempered degree of the first tone in the consonance, ex. 5=fa, 0=do
H(T1) consonant harmonic in the first tone, ex. 1=fundamental, 2=octave
T2 equal-tempered degree of the second tone in the consonance
H(T2) consonant harmonic in the second tone
freq frequency factor: the frequency of the tone is freq*110*2^(3/12) Hz
ratio The ratio of the 2 frequencies, ex. 1=stable consonance, 1.030= quarter-tone separation
beat freq estimation of the beat frequency in Hertz (Hz)in the lower guitar register, ex. 0=stable consonance

Scales versus scales

Links to data tables:
(program output)
Common chords with harmonics 1-10
Common chords with harmonics 1-6
Consonant intervals with harmonics 1-6
Consonant intervals with harmonics 7-10

The major triad and the seventh harmonic

from chord table (h<=10) 
chord         scale   T1  H(T1) T2  H(T2)  freq    ratio    beat freq
Fa             temp    5    3    0    4   4.0000   1.0011   0.5919
Fa             temp    9    3    0    5   5.0000   1.0091   5.9900
Fa             temp    9    4    5    5   6.6742   1.0079   6.9844
Fa             temp    5    6    0    8   8.0000   1.0011   1.1838
Fa             temp    9    6    0   10  10.0000   1.0091  11.9799
Fa             temp    9    7    5    9  11.7725   1.0205  32.1725
Fa             temp    9    8    5   10  13.3484   1.0079  13.9688
Fa             diat    5    3    0    4   4.0000   1.0000   0.0000
Fa             diat    9    3    0    5   5.0000   1.0000   0.0000
Fa             diat    9    4    5    5   6.6667   1.0000   0.0000
Fa             diat    5    6    0    8   8.0000   1.0000   0.0000
Fa             diat    9    6    0   10  10.0000   1.0000   0.0000
Fa             diat    9    7    5    9  11.6667   1.0286  44.8501
Fa             diat    9    8    5   10  13.3333   1.0000   0.0000
Fa             quar    5    3    0    4   4.0000   1.0000   0.0000
Fa             quar    9    3    0    5   4.9944   1.0011   0.7385
Fa             quar    9    4    5    5   6.6591   1.0011   0.9847
Fa             quar    5    6    0    8   8.0000   1.0000   0.0000
Fa             quar    9    6    0   10   9.9887   1.0011   1.4771
Fa             quar    9    7    5    9  11.6535   1.0297  46.6732
Fa             quar    9    8    5   10  13.3183   1.0011   1.9694

The seventh harmonic beats wildly in all cases

The frequency discrepancy underlined in this manner is roughly a quarter of a tone. The sound might be intolerable if it weren't for the presence of lower-harmonic responses which are much louder.

This demonstrates that the expectations of zero-beat consonances for just intonation and perfection are only met by placing limits on the classical model. The expectation is the trivial product of a tautology. Reality is otherwise. The definition of these scales involves limited harmonic series:
scale defining harmonics lowest harmonic mismatch
equal temper 1 2 3
fourths (or fifths) 1 2 3 5
diatonic (or natural or ideal) 1 2 3 5 7

Prime harmonics are "wild"

It is tempting to guess that "near-miss" harmonic encounters (non zero-beat) will occur for the lowest prime number harmonic not used in the scale definition. Harmonics 11, 13, 17, 19 etc. aren't included in this data but they introduce very nervous instabilities just like we see here for harmonic 7.

Divisible harmonics 6, 8, 9 and 10 show the "perfectness" expected of 2*3, 2*2*2, 3*3, and 2*5. In other words, any harmonic degree which can be factored into prime numbers no higher than the largest one used in the definition of scale may display zero-beat consonance with another having the same property. This is true if interesting. I digress.

Harmonics >6 will usually be left out from now on, since their general characteristics already seem to be adequately described. Harmonics >6 are crucial for the timbre and it wouldn't be advisable to leave them out of a synthesis intended to be heard. I keep them out of most tables to improve readability for the loudest consonances.

"Perfect" isn't always "best"

from chord table (h<=6) 
chord         scale   T1  H(T1) T2  H(T2)  freq    ratio    beat freq
Fa             temp    5    3    0    4   4.0000   1.0011   0.5919
Fa             temp    9    3    0    5   5.0000   1.0091   5.9900
Fa             temp    9    4    5    5   6.6742   1.0079   6.9844
Fa             diat    5    3    0    4   4.0000   1.0000   0.0000
Fa             diat    9    3    0    5   5.0000   1.0000   0.0000
Fa             diat    9    4    5    5   6.6667   1.0000   0.0000
Fa             quar    5    3    0    4   4.0000   1.0000   0.0000
Fa             quar    9    3    0    5   4.9944   1.0011   0.7385
Fa             quar    9    4    5    5   6.6591   1.0011   0.9847
Do             temp    7    2    0    3   2.9966   1.0011   0.4434
Do             temp    4    4    0    5   5.0000   1.0079   5.2324
Do             temp    7    4    0    6   5.9932   1.0011   0.8868
Do             temp    7    5    4    6   7.4915   1.0091   8.9748
Do             diat    7    2    0    3   3.0000   1.0000   0.0000
Do             diat    4    4    0    5   5.0000   1.0000   0.0000
Do             diat    7    4    0    6   6.0000   1.0000   0.0000
Do             diat    7    5    4    6   7.5000   1.0000   0.0000
Do             quar    7    2    0    3   2.9596   1.0136   5.3541
Do             quar    4    4    0    5   4.9944   1.0011   0.7385
Do             quar    7    4    0    6   5.9192   1.0136  10.7083
Do             quar    7    5    4    6   7.3991   1.0125  12.2499

This comparison reveals a serious defect in the fourths scale. In the C chord, the third harmonic consonance of "c" is noticeably mis-matched with an octave of "g" with a beat frequency of 5.35 cps, unusual (and hence disturbing) for a perfect fifth. This is beacuse the fifth, "g", is the last interval of 11 to be calculated, starting with "c". There is an accumulation of discrepancies leading to this unusually high number.

This subject is usually handled by focusing on how miraculous it is that the final 12th step in the calculation "almost" produces the note "c", the starting point. You can't beat a miracle. So I won't knock it. The cycle of fiths is one of the great wonders of the world. OK. But let's get back to mere practicality. The usual way to use fifths-tuning as a device for tuning the piano is to tune the twelfth (in the octave of the third harmonic) and then tune the fifth an octave below. This way the reference doesn't go rushing off the edge of the keyboard and the information is distrubuted around the entire piano using octaves. The general form of the procedure for finding a frequency f based on some reference is:

f_fifth = f0 * 3^h/2^i

and the frequency of any octave of the reference fundamental is:

f_octave = f0 * 2^j/2^k

where h, i, j and k are integers. Leaving miracles aside for the time being, and leaving aside the trivial case where h=0, f_fifth cannot equal f_octave. It won't happen despite there being quite a few apparent degrees of freedom in the formula. Stepping through fifths really means incrementing h by 1 until the final revelation: for h=12, (3/2)^12=129.7 is close but does not equal 2^7=128. This means that increasing the stepwise distance separating notes in the cycle of fifths progressively increases the expression of the expected flaw.

There are of course advantages. Take for instance the remarkably good match in the F chord for the major third "a" beating against the fifth "c" at a rate of 0.7385 cps. Since such intervals aren't considered in either the fifths or the equal-tempered scales, one would expect a value closer to the tempered 5.9900 cps.

F chord = f   a   c (notes)
          5   9   0 (scale degrees)

The order of the fourths scale used here
c  f  bb  eb  ab  c#  f#  b   e   a   d   g    (notes)
0  5  10  3   8   1   6   11  4   9   2   7    (scale degrees)

The "a" is 9 steps away from the "c" in the fourths scale, enough to pick up the slack and bring their 3rd and 5th harmonics close to perfect consonance. One cannot point the finger at the fifths scale and say it is increasingly flawed with distance. More to the point, it is increasingly hard to predict!

The diatonic scale suffers similar defects for the same reasons. This will be shown later: Pico's chords provide a good example. This particular data set was designed using the diatonic scale as a point of reference. The chords are chosen for the most part so that perfect intervals do indeed appear to be perfect for the diatonic scale.

The question this data set was intended to answer was: How imperfect are other scales under the same conditions? The tempered scale is known to be always wrong with respect to consonance (excepting octaves) and I specifically want to know: By how much?

Speaking very broadly, the data on this page show that the most striking mis-matches are found for the diatonic and the fourths scale. The tempered scale is always a bit off-center but never the complete nuisance the others are with their shift accumulation effects. The tempered scale is more predictable. Like the fourths scale, it always has a better match for the perfect fifth than for the major third. The particular quality of equal temper is that the same interval always sounds the same, no matter where it is transposed.

The unusual case of Pico's fourths

Fioriano Pico's book of guitar chord charts is one of the oldest of its kind (1608). The novelty of such books at the time was that you could strum along with any song once you became acquainted with just a few standard fingerings. This remains one of the attractions of the guitar: you don't have to read music. I digress.

Pico's complete chord list is made up with 12 major chords, 12 minor, and a few extended chords which we would usually write as the name of the major chord followed by the symbols "4/5" to show that they contain a tonic, a perfect fourth, and a perfect fifth. This kind of chord was more frequently used then than now.

Example: G4/5 is made with g, d, and c.
There could be three distinct uses for such a chord:
root related modern usage
D D7/4sus, D11
G G4sus
C C9

from chord table (h<=6) 

chord scale T1 H(T1) T2 H(T2) freq ratio beat freq Sol4/5 temp 7 2 0 3 2.9966 1.0011 0.4434 Sol4/5 temp 7 3 2 4 4.4898 1.0011 0.6644 Sol4/5 temp 7 4 0 6 5.9932 1.0011 0.8868 Sol4/5 diat 7 2 0 3 3.0000 1.0000 0.0000 Sol4/5 diat 7 3 2 4 4.5000 1.0000 0.0000 Sol4/5 diat 7 4 0 6 6.0000 1.0000 0.0000 Sol4/5 quar 7 2 0 3 2.9596 1.0136 5.3541 Sol4/5 quar 7 3 2 4 4.4394 1.0000 0.0000 Sol4/5 quar 7 4 0 6 5.9192 1.0136 10.7083 Ré4/5 temp 9 2 2 3 3.3636 1.0011 0.4977 Ré4/5 temp 7 3 2 4 4.4898 1.0011 0.6644 Ré4/5 temp 9 4 2 6 6.7272 1.0011 0.9954 Ré4/5 diat 9 2 2 3 3.3333 1.0125 5.5187 Ré4/5 diat 7 3 2 4 4.5000 1.0000 0.0000 Ré4/5 diat 9 4 2 6 6.6667 1.0125 11.0373 Ré4/5 quar 9 2 2 3 3.3296 1.0000 0.0000 Ré4/5 quar 7 3 2 4 4.4394 1.0000 0.0000 Ré4/5 quar 9 4 2 6 6.6591 1.0000 0.0000

This chord is the most consonant of all tempered chords in these data. It is more consonant even than the major triad. One can witness fourths-scale and diatonic-scale harmonies breaking down at the same time, another instance of "perfect" isn't always "best".

This kind of chord is reputed to be "unstable" in classical harmony. Descartes, speaking of the harmony of the perfect fourth interval, called it "the most unhappy of all". The present data would appear not to support this opinion, unless it be to say that you may be taking a big risk if you play this chord on anything but a guitar. Young René played the lute, a fretted instrument tuned very nearly like the guitar, and he may have merely stated a commonplace. "Unhappy" despite "perfection"? This is a valid metaphysical idea. The 4 chord has some kind of built-in tension: it is almost never used as a final chord but always as a build-up to a final chord. Its role in music is linked with the dominant [like the dominant 7 chord which is ubiquitous today but was not included in Pico's 1608 list!]. I will return to this question later on.

A survey of chromatic intervals

from interval table (h<=6) 

chord scale T1 H(T1) T2 H(T2) freq ratio beat freq fa-sol# temp 8 5 5 6 7.9370 1.0091 9.5085 fa-sol# diat 8 5 5 6 8.0000 1.0000 0.0000 fa-sol# quar 8 5 5 6 7.9012 1.0125 13.0813 fa-la temp 9 4 5 5 6.6742 1.0079 6.9844 fa-la diat 9 4 5 5 6.6667 1.0000 0.0000 fa-la quar 9 4 5 5 6.6591 1.0011 0.9847 fa-sib temp 10 3 5 4 5.3394 1.0011 0.7901 fa-sib diat 10 3 5 4 5.3333 1.0000 0.0000 fa-sib quar 10 3 5 4 5.3333 1.0000 0.0000 fa-do temp 5 3 0 4 4.0000 1.0011 0.5919 fa-do diat 5 3 0 4 4.0000 1.0000 0.0000 fa-do quar 5 3 0 4 4.0000 1.0000 0.0000 fa-do# temp 5 4 1 5 5.2973 1.0079 5.5435 fa-do# diat 5 4 1 5 5.3333 1.0000 0.0000 fa-do# quar 5 4 1 5 5.2675 1.0125 8.7209 fa-ré temp 5 5 2 6 6.6742 1.0091 7.9956 fa-ré diat 5 5 2 6 6.6667 1.0125 11.0373 fa-ré quar 5 5 2 6 6.6591 1.0011 0.9847

This shows that intervals smaller than a minor third cause no consonance (a quarter of a tone or better) on the first 6 harmonics. One may similarly note the absence of consonance on the tritone or flatted fifth.

In other words, f displays the following consonances:
g#, a, bb, c, c#,and d, that is, minor third, major third, perfect fourth, perfect fifth, augmented fifth, and sixth.

Using higher harmonics, one can "find" more consonances.

from interval table (6<h<=10) 

chord scale T1 H(T1) T2 H(T2) freq ratio beat freq fa-sol temp 7 7 5 8 10.4881 1.0182 25.3819 fa-sol temp 7 8 5 9 11.9865 1.0023 3.5533 fa-sol temp 7 9 5 10 13.3484 1.0102 18.0205 fa-sol diat 7 7 5 8 10.5000 1.0159 22.1482 fa-sol diat 7 8 5 9 12.0000 1.0000 0.0000 fa-sol diat 7 9 5 10 13.3333 1.0125 22.0747 fa-sol quar 7 7 5 8 10.3587 1.0297 41.4873 fa-sol quar 7 8 5 9 11.8385 1.0136 21.4166 fa-sol quar 7 9 5 10 13.3183 1.0011 1.9694 fa-si temp 11 5 5 7 9.3439 1.0102 12.5354 fa-si temp 11 7 5 10 13.2142 1.0102 17.7278 fa-si diat 11 5 5 7 9.3333 1.0045 5.4749 fa-si diat 11 7 5 10 13.1250 1.0159 27.6852 fa-si quar 11 5 5 7 9.3333 1.0033 4.0809 fa-si quar 11 7 5 10 13.1102 1.0170 29.6859 fa-mib temp 5 7 3 8 9.3439 1.0182 22.6127 fa-mib temp 5 8 3 9 10.6787 1.0023 3.1656 fa-mib temp 5 9 3 10 11.8921 1.0102 16.0545 fa-mib diat 5 7 3 8 9.3333 1.0286 35.8801 fa-mib diat 5 8 3 9 10.6667 1.0125 17.6597 fa-mib diat 5 9 3 10 12.0000 1.0000 0.0000 fa-mib quar 5 7 3 8 9.3333 1.0159 19.6873 fa-mib quar 5 8 3 9 10.6667 1.0000 0.0000 fa-mib quar 5 9 3 10 11.8519 1.0125 19.6219

The "new" consonances are all a bit raunchy. They are, for f:

g, b, and eb, that is, ninth, flatted fifth (tritone), and dominant seventh.

This leaves 2 tones which appear to produce no consonances of any kind with f:

f# and e, that is, flatted ninth and major seven.

This observation eventually led to the development of the picture featured on the preceeding page with its "wells of no consonance" surrounding prominent "hot spots".

A few more considerations

Does the guitar really have equal temper?

The entire point of this study is that it is relevant to real music. The guitar, for instance, is known to have equal temper along the neck because that is really how is is constructed.

The actual guitar scale may differ from equal temper because string-to-string tuning may be acheived by comparing the 4th harmonic of the lower string with the 5th of the next one. This separates them by a perfect fourth. There is one major third to break the pattern. There are 2 common tunings which differ by where they place the break:

The modern Spanish guitar
e    a    d    g        b    e 
  4/3  4/3  4/3           4/3
The Renaissance lute
e    a    d       f#    b    e 
  4/3  4/3           4/3  4/3

With modern Spanish tuning, the consonances on the third string (g) relate to the "bass" side of the neck. With Renaissance lute tuning (f#), they relate to the "treble" side. This is one reason why Renaissance music doesn't sound right when performed in modern Spanish tuning.

This mixed tuning of the guitar isn't describable with the present simple model. There are several different frequencies for each note, depending on the string where they are played. Some string-to-string harmonies on the guitar can be perfectly consonant. The differences between equal and mixed temper should be slight. The slow accumulation of slight discrepancies on the perfect fifth should be small with mixed tuning: 2 (Renaissance) or 3 (Spanish) steps. The noticeable cumulative offset highlighted earlier for the pure fifths scale required 11 steps!

One cannot say which method of tuning the guitar is "superior". A matter of taste, no doubt. One cannot fail to notice that the instrument has a completely different sound character with these two methods. Character is timbre. It may be helpful to remember that the art of tuning the guitar generally isn't a matter of matching strings one-to-one. It is a matter of obtaining a happy timbre for the instrument.

A simpler model

Let us take one assumption a step further : that lower pitches and harmonics provide the better points of reference in study of consonance. Let us look at only the lowest possible harmonics. The particular combination of harmonics which produces consonance may be thought of as a property of the interval. The combination can be converted to a number by multiplying harmonic degrees. As we will see, the numbers range from 1 to 72. Quite a spread!

This approach ignores quality and considers only the strength of the consonance. It is reasonable when dealing with tonal reality to assume that one will first notice strong features. One may then proceed to consider quality when deciding what to do musically with each feature. An interval which is both strong and of poor quality will end up being understood in purely metaphysical terms. Here is the list of chromatic intervals in order of noticeability.

guitar symbol interval harmonic 1 harmonic 2 product = h1*h2
unison 1 1 1
octave 2 1 2
perfect fifth 3 2 6
major tenth
i.e. octave + major third
5 2 10
perfect fourth
i.e. fifth - octave (inversion)
3 4 12
minor third - octave (inversion)
3 5 15
major third
a.k.a. ditone
5 4 20
octave + perfect fourth
8 3 24
dominant seventh 7 4 28
minor third 6 5 30
flatted fifth
a.k.a. tritone
7 5 35
ninth 9 4 36
minor sixth
major third - octave (inversion)
5 8 40
minor tenth
octave + minor third
12 5 60
a.k.a. tone
9 8 72

The consonance "hit-parade" highlights several ideas.

A closer look at beat frequencies

We haven't tried very hard to compare the different kinds of consonance. To see this, we will consider only equal temper and look at the way each kind of consonance varies with transposition. Each prime number harmonic represents one "kind". We will further limit ourselves to the consonance at the pitch of each harmonic. Here is a map of the specific locations for each consonance in the picture which follows:

harmonic of reference tone type of interval for second tone octave where consonance occurs
3 perfect fifth 2
5 major third 3
7 dominant seventh 3
11 perfect fourth or flatted fifth 4

Example: the 5th harmonic of the reference tone happens to find a match with a multiple of the fundamental of the major third as the second tone. The harmonic for the second tone is transposed by octaves to meet the reference harmonic in the 3rd octave. This doesn't mean that the second tone is supposed to be played in the 3rd octave.

The x-axis in the picture shows the reference tone's fundamental frequency. The y axis shows beat rate for each prime harmonic. The tempered interval with a fundamental closest to the prime harmonic of the reference tone is used to determine a beat rate. The beat rate increases steadily with the pitch of the reference tone and passes through some "interesting territory". The 1st (unison) and 2nd (octave) harmonics are exact, show no such dependance, and are not included here.

Harmonics 7 and 11 are disturbing. The harmonic frequencies go rocketing off far above the tonal range of the guitar and into the area of timbre. The beat rates go wandering off above the range of tonal instability and into frequencies normally characteristic of tone itself. Effects which might be felt should be faint and perhaps even subliminal. Qualitative borders are being passed here. This is a possible (if unlikely) explanation for the time-tested "reputation" of the dominant seventh, the perfect fourth and the flatted fifth, all somehow linked to the notion of "tension".


On odd-numbered calendar days, I believe this is all a lot of hogwash and a waste of time. On the other hand, I feel the need for some kind of data to refer to when trying to understand some of the finer points of tonal theory. Theory is often seen as somthing to be taught. Teaching works best by making ideas seem engaging. Data is always dull and is always ignored in teaching material. This needs correcting: data can't be made exciting but a presentation must nevertheless be attempted at some point.

The data set in this presentation is but a tiny speck in the grand scheme of everything tonal.

It may be that zero-beat consonance is thought to be desirable because it is possible to set up a rational tonal system to make it happen. I believe to have demonstrated that this doesn't account for all the sounds we hear in the "natural" scale. We can acheive zero beats for the prime harmonics 1, 2, 3, and 5, but harmonic 7 is so uncooperative that we merely declare it to be irrelevant. In fact, there is no essential physical difference between harmonics 5 and 7 and the ear should handle both in a similar manner. This is not to say that you can't play music based on the natural scale. In fact it may be the sharp contrast between the intended and the inevitable that make the natural scale usable in the first place. A flaw in the theory means an endearing flaw in the scale.

My favorite conjecture: the ubiquity of the sus4 chord in Renaissance lute music and the over-use of the dominant 7 in XIXth century guitar music, both in connection with underlining tension, can be "explained" by the dismal behaviour of harmonics 11 and 7, respectively. This, only for want of another plausible explanation.

It is possible to set up an equal-tempered tonal system which just happens to offer low beat rates very much in the same range as what obtains with the natural tone system. The tempered scale is linear (in the appropriate representation) and this is a considerable advantage for the musician. The fact that it comes miraculously close to zero-beat positions for harmonics 1, 2, and 3 makes it immediately most endearing. The fact that harmonic 5 is not very close to a point on the scale really puts us back in the same position we were in with harmonic 7 in the diatonic world. The equal-tempered scale has a different character but it is similar enough that we can play the same as with other scales and at least hope to recognize the music.

My final conclusion will have to lean towards an alternative perspective. Harmonics 5 (in higher registers), 7, and 11 can deliver very fast beat rates. These higher harmonics are also softer. If we are aware of them at all, they are possibly on the fringe of perception. Lower harmonics, 1, 2, 3, and 5 (in lower registers) result in beat rates somewhere within the range of ordinary tone duration at the slow end and vibrato and trill-like gestures at the fast end. The actual separations are very small, almost insignificant, for instruments such as woodwinds and strings which can bend tones. We would not be aware of this but it is at least plausible: we can acheive any beat rate we wish by the very slightest bending of tones under unconscious control. Many years ago I heard an excellent blues musician play with just intonation on an out-of-tune guitar. The guitar was my own and I knew it had serious intonation problems. When he gave it back to me (with many compliments), it was immediately obvious that the tuning was much much worse than usual. I need no further proof of the notion that unconscious adjustment of intonation on the fly is very real. While I am on the subject of anecdotes, I once tried a newfangled guitar synthesizer. It was the first (and last) time I ever actually played in tune. The device converted the analog signal from each string to a midi code which was in turn converted to an equal-tempered synthetic tone. I acheived perfect intonation in a busy and distracting environment with the sloppiest of fingerings. I acheived something which I can only strive to acheive with an ordinary guitar with a great deal of concentration and effort. The striving means more to me than the acheiving so I went straight back to the ordinary guitar and never looked back. The experience remains a remarkable one.

It must be an odd-numbered day. One very good reason to take all of this with a grain of salt is that "music is not logical". The mere mention of the word "number" sets up the expectation that something "logical" will happen. I have been fighting this impression from the start, with mitigated success, I confess.

Randy Ayling
guitar-wielding fantasist
July 2002 - revised March 2003