Consonance of simple ratios: physical or metaphysical?

The page From monochords to musical scales briefly reviews the origin of various musical scales. The determination of the structure of these scales in various eras and civilisations has been based more on "aesthetic" considerations, established by practice and judgement of the consonance of intervals, rather than considerations taken from a real understanding of the sound phenomenon. Even Pythagoras's research, which some consider to be an early version of the experimental method, was not such, since the experimental recognition of "simple ratios" of consonant sounds was used to develop a type of metaphysical (almost numerological) picture. For the modern scientist, for whom Galileo is the paradigm, the discovery of a link between "simple ratios" of the lengths of the parts of a monochord and the consonance of the two sounds produced, once we have established that sound is a physical entity, must have an origin connected to the modality of production and perception of sounds. Let us see what Galileo himself said^[1]

Galileo Galilei, from Discourses Relating to Two New Sciences, First Day (page 100 in the original Italian text)

...since the explanations hitherto given by those learned in music impress me as not sufficiently conclusive. They tell us that the diapason, i. e. the octave, involves the ratio of two, that the diapente which we call the fifth involves a ratio of 3:2, etc.; because if the open string of a monochord be sounded and afterwards a bridge be placed in the middle and the half length be sounded one hears the octave; and if the bridge be placed at 1/3 the length of the string, then on plucking first the open string and afterwards 2/3 of its length the fifth is given; for this reason they say that the octave depends upon the ratio of two to one and the fifth upon the ratio of three to two. This explanation does not impress me as sufficient to establish 2 and 3/2 as the natural ratios of the octave and the fifth... [Ed. Note: bold text is ours]

What determines the pitch of a note?

Galileo's explanation

Continuing in the dialogue, Galileo clearly showed that the length of a string cannot be the only aspect for "sufficiently establishing" the pitch of a note. With a series of experiences, he showed that "...There are three different ways in which the tone of a string may be sharpened, namely, by shortening it, by stretching it and by making it thinner...".

At the same time, Galileo reported that, when sounding a glass immersed in water, the ripples formed on the surface of the water become denser, each one splitting in two, when the sound jumps to the upper octave (the "diapason", in the musical language of the time). This observation caused Galileo to think that the fundamental aspect for determining the pitch of a sound is the frequency of its oscillations. He was tempted, albeit aware of the difficulty of counting the vibrations of an oscillating string (note that this number is of the order of hundreds of vibrations per second!), to jump to conclusions and say that the sound in an octave is due to double the number of vibrations. However, being faithful to his experimental method, he could not accept claims that were not confirmed by experience. He devised a method to accurately "measure" the ratio of the frequencies of two sounds that still amazes us with its ingenuity to this day. Once again, we quote Galileo directly:

Galileo Galilei, from Discourses Relating to Two New Sciences, First Day, (page 101 and 102 in the original Italian text)

... The device is one which I hit upon by accident; my part consists merely in the observation of it and in the appreciation of its value as a confirmation of something to which I had given profound consideration; and yet the device is, in itself, rather common. As I was scraping a brass plate with a sharp iron chisel in order to remove some spots from it and was running the chisel rather rapidly over it, I once or twice, during many strokes, heard the plate emit a rather strong and clear whistling sound; on looking at the plate more carefully, I noticed a long row of fine streaks parallel and equidistant from one another. Scraping with the chisel over and over again, I noticed that it was only when the plate emitted this hissing noise that any marks were left upon it; when the scraping was not accompanied by this sibilant note there was not the least trace of such marks. Repeating the trick several times and making the stroke, now with greater now with less speed, the whistling followed with a pitch which was correspondingly higher and lower. I noted also that the marks made when the tones were higher were closer together; but when the tones were deeper, they were farther apart. I also observed that when, during a single stroke, the speed increased toward the end the sound became sharper and the streaks grew closer together, but always in such a way as to remain sharply defined and equidistant. Besides whenever the stroke was accompanied by hissing I felt the chisel tremble in my grasp and a sort of shiver run through my hand. In short we see and hear in the case of the chisel precisely that which is seen and heard in the case of a whisper followed by a loud voice; for, when the breath is emitted without the production of a tone, one does not feel either in the throat or mouth any motion to speak of in comparison with that which is felt in the larynx and upper part of the throat when the voice is used, especially when the tones employed are low and strong. At times I have also observed among the strings of the spinet two which were in unison with two of the tones produced by the aforesaid scraping; and among those which differed most in pitch I found two which were separated by an interval of a perfect fifth. Upon measuring the distance between the markings produced by the two scrapings it was found that the space which contained 45 of one contained 30 of the other, which is precisely the ratio assigned to the fifth... [Ed. Note: bold text is ours]

The modern explanation

From a physicist's point of view, a musical instrument is a system that generates sound waves and radiates them into the environment. In particular, it is composed of (at least):

1. a vibrating element (string, membrane, plate, air, etc.) tuneable to various frequencies;
2. a resonator that amplifies the vibrations and gives a shape to the sound waves;
3. an impedance adapter between the vibrating system and the surrounding air.

• The formation of stationary waves in the vibrating parts of musical instruments is explained in detail on the pages Interference between reflected waves and Sound reflection . The geometric characteristics of these parts (string length, sound tube length, etc.) determine the selection of specific wavelengths.

• However, knowledge of selected wavelengths is not enough to explain the pitch of emitted sounds. To do this, as Galileo intuited (from the preceding text emerges Galileo's extraordinary ability to investigate nature and to grasp its unifying aspects), it is necessary to know the oscillation frequency of the vibrating parts.

As explained on the page How is a wave described?, the calculation of the frequency of an emitted sound also requires knowing the speed with which the wave is propagating along the vibrating element based on the formula:

${\displaystyle f=\frac{v}{\lambda }}$

This speed depends on the physical characteristics of the vibrating element; in particular:

• in the case of strings, both the string tension (which is what we vary when we "tune" a guitar) and "thickness" (the thinner the string, the faster the wave travels). The guitar's first E string generates the highest sound among the open strings (about 330 Hz, two octaves above the lowest string) because the generating string is the thinnest and is very taut. When Galileo spoke of ways to "sharpen the tone", today we could say that he actually meant to change the propagation speed of the mechanical waves of the vibrating element.

• in the case of sound tubes (in which the air particles are made to vibrate by the musician), the pressure and density of the fluid contained in the sound tube.

When a musician plays an instrument, they generate desired sounds by modifying the geometric characteristics of the vibrating element (except in wind instruments, where the musician can select more notes by modifying the pressure of the air flow without modifying the sound tube length):

• a violinist shortens and lengthens a part of the vibrating string creating a fixed end with their finger;
• a flautist modifies the "actual length" of the sound tube by opening and closing specific holes.

It is preferable that the physical characteristics remain unaltered; however, we often see violinists retuning their instruments "on the spot" by finely adjusting string tension (the quality of an instrument is also measured by its ability to stay in tune). In the case of monochords, tuning is obviously stable (i.e. the speed does not change for at least the time of the execution of the two sounds) and therefore

The ratio between the sound frequencies is completely determined by the ratio of the lengths of the vibrating parts that generate it or as Galileo said "...I assert that the ratio of a musical interval is not immediately determined either by the length, size or tension of the strings but rather by the ratio of their frequencies, that is, by the number of pulses of air waves which strike the tympanum of the ear.

[Ed. Note: bold text is ours] In particular, as there exists a direct proportionality between the wavelength and length of the vibrating part (i.e. λ=k l ), we can write:

${\displaystyle \frac{f_1}{f_2}=\frac{\frac{v}{{\lambda }_1}}{\frac{v}{{\lambda }_2}}=\frac{\frac{v}{k\cdot l_1}}{\frac{v}{k\cdot l_2}}=\frac{l_2}{l_1}}$

Theory of consonance

Similitude of Galileo's pendula

Having established that the ratio between sound frequencies is completely determined by the ratio of the lengths of the vibrating parts that generate it, Galileo tried to justify the reasons for consonance in the simplicity of ratios between frequencies of emitted sounds. In a very famous passage from Discourses Relating to Two New Sciences" (1638), Galileo attributed the origin of consonance to the fact that when two sounds of commensurable frequencies (according to small ratios) reach the ear, the latter perceives a regularity of the phenomenon that repeats in a period of brief duration. In this passage, Galileo strengthened his argument with a mechanical similitude: two pendula that were made to oscillate in synchrony and completed an oscillation in 2 and 3 seconds respectively would quickly recover their synchrony (after six seconds, the minimum common multiple between two and three). If the ratio of the frequencies is not commensurable (inconceivable for the first Pythagoreans but not for the last!) or if it is expressed by fractions containing large numbers (e.g. 9/8), recovery of synchrony will happen only after nine pulsations of the more frequent note rendering the ear unable to recognise the regularity (and, therefore, the consonance) of the phenomenon. Let's see what Galileo said about this:

Galileo Galilei, from Discourses Relating to Two New Sciences, First Day, (page 103 and 104 in the original Italian text)

...This fact established, we may possibly explain why certain pairs of notes, differing in pitch produce a pleasing sensation, others a less pleasant effect, and still others a disagreeable sensation. Such an explanation would be tantamount to an explanation of the more or less perfect consonances and of dissonances. The unpleasant sensation produced by the latter arises, I think, from the discordant vibrations of two different tones which strike the ear out of time. Especially harsh is the dissonance between notes whose frequencies are incommensurable; such a case occurs when one has two strings in unison and sounds one of them open, together with a part of the other which bears the same ratio to its whole length as the side of a square bears to the diagonal; this yields a dissonance similar to the augmented fourth or diminished fifth. Agreeable consonances are pairs of tones which strike the ear with a certain regularity; this regularity consists in the fact that the pulses delivered by the two tones, in the same interval of time, shall be commensurable in number, so as not to keep the ear drum in perpetual torment, bending in two different directions in order to yield to the ever-discordant impulses. The first and most pleasing consonance is, therefore, the octave since, for every pulse given to the tympanum by the lower string, the sharp string delivers two; accordingly at every other vibration of the upper string both pulses are delivered simultaneously so that one-half the entire number of pulses are delivered in unison. But when two strings are in unison their vibrations always coincide and the effect is that of a single string; hence we do not refer to it as consonance. The fifth is also a pleasing interval since for every two vibrations of the lower string the upper one gives three, so that considering the entire number of pulses from the upper string one-third of them will strike in unison, i. e., between each pair of concordant vibrations there intervene two single vibrations; and when the interval is a fourth, three single vibrations intervene. In case the interval is a second where the ratio is 9/8 it is only every ninth vibration of the upper string which reaches the ear simultaneously with one of the lower; all the others are discordant and produce a harsh effect upon the recipient ear which interprets them as dissonances. [Ed. Note: bold text is ours]

To give strength to his argumentations, Galileo introduced his famous similitude of pendula to explain how even our vision can be confused by the observation of the oscillation of two pendula that do not have commensurable frequencies according to simple ratios.

Von Helmholtz's consonance theory

However inspiring and wisely argued Galileo's theory may be, it is incomplete for three reasons:

• the first is that the perception of consonance between two sounds does not only depend on the ratio of their frequencies but also on the absolute value of the frequencies themselves. For example, in instruments with a wide range, an interval of a minor third (ratio 6:5) appears to be less consonant in the lower range of the keyboard (i.e. at lower frequencies);
• the second is that, in Galileo's reasoning, sound is characterised by a single frequency (to put that in modern terms, a pure sound). As discovered after Galileo (visit the page Fourier's Theorem), when a string vibrates (if you want to see how, visit the page on wave synthesis), it emits a composite sound composed of a principal wave (called the fundamental, which determines the perceived frequency) and waves (called partials) having frequencies that are all multiples of the frequency of the fundamental. This is a consequence of the selection mechanism of wavelengths;
• the third is physiological: attributing the sensation of consonance to the regularity of the vibrations of the tympanum is evidently an extrapolation, if we do not know how this mechanical vibration is processed by the brain.

Hermann von Helmholtz created a theory of the perception of sounds that allows us to partially overcome the three abovementioned difficulties.^[2]

• As explained on the page How is a musical scale constructed?, at low frequencies, sounds with the same frequency ratio are very close in frequency. Von Helmholtz claimed that sounds with very close frequencies induce dissonance because they cause very close nerve fibres to generate confusion in brain processing. This is the phenomenon of beats described from a physiological point of view on the page regarding critical bands.
• Von Helmholtz attributes the emergence of dissonance to the beats between high order partials if they fall within a critical band. The harmonics of a note are always the same distance apart, equal to the frequency of the fundamental, whereas the width of the critical band increases with frequency: beyond roughly the sixth or seventh harmonic, two consecutive harmonics fall within the same critical band and produce beats. This is why sounds with a large content of high order harmonics can be self-dissonant. You can experiment with this fact directly by synthesising sounds of this type in our virtual laboratory.
• Galileo's intuition to attribute consonance to the regularity of the vibrations of the tympanum was actually not too far from the truth, even in more recent explanations. Sounds with frequency ratios expressed by simple numbers, have frequent coincidences in their high order partials (e.g. the third harmonic of C is identical to the second harmonic of G, which is one fifth higher) and activate the same receptors. It is as if the nerve fibre "map" activated by C were also partially contained in the nerve fibre "map" activated by G. This fact is confirmed by comparing pure tones and sounds rich in harmonics^[3]. With two pure tones, the interval sounds rough only if the two frequencies fall within the same critical band; for wider intervals the frequency ratio matters little, and a perfect fifth does not sound very different from a nearby interval with a complicated ratio. With sounds rich in overtones (e.g. the piano) the difference becomes clear: in the perfect fifth many harmonics coincide and do not beat, whereas a slight mistuning of the interval is enough for the nearby harmonics to produce beats. It is as if the Galilean "synchrony" were strengthened by the coincidence of the higher harmonics.

A possible conclusion

However, we must not believe that von Helmholtz's theory offers an objective criterion for establishing a hierarchy of consonant sounds. Today, physicists, musicians and acoustic engineers are trying to find the laws that unequivocally establish this hierarchy. However, the underlying problem that renders much of this research vain or questionable is that the amount of major or minor consonance depends on so many parameters that they cannot be kept under scientific control (the musical context, the register in which the musical instruments are playing, the relationship between instruments with different timbres, etc.) making any type of univocal determination almost impossible (perhaps luckily!).

Check your understanding

In-depth study and links

• The admirable method with which Galileo experimentally controlled his intuition to link the pitch of a sound to the frequency of the sound itself, shows us that one of an experimental physicist's concerns is this: how to measure "something" when the technology for that "something" does not yet exist? If you want to learn about the ingenious method that Newton used to measure the speed of sound, visit this page.

• If you want to read a musicians point of view on the issue of consonance-dissonance, visit this page. On the page about Psychoacoustic and musical aspects, you will discover that the issue of consonance-dissonance is part of the greater issue of the creation of a harmonic syntax. In this wider context, the issue cannot be addressed without referring to the "functional" aspects that the various groups of notes (chords) play within the modality chosen for their concatenation.