What is permitted? The nature of music

1. Introduction

Our perception of musical tones depends on the intensity, frequency, and waveform of the original physical stimulus to our ears. As you might expect, our subjective sense of loudness is based mostly on the intensity of the objective source: larger variations in air pressure in the vicinity of our ears translate to more nerve impulses sent to the brain. (The familiar decibel system for measuring sound intensity levels reflects this: it expresses ratios of the quantity of energy passing through a given surface area compared with the same surface in conditions of effective silence.) The effect is not a linear one, however, since doubling the intensity of the incoming tone does not lead us to think the sound is twice as loud. That is why the decibel scale is conveniently chosen to be a logarithmic one. City traffic measuring 70 dB represents ten times more sound energy than a quiet conversation measuring 60 dB, but we only perceive this to be roughly a doubling or tripling of loudness. Or in a more musical context, ten people singing the same note will sound about twice as loud as one person singing that note at the same intensity, while a chorus of one hundred people will sound about four times as loud as the soloist. Even with this crudest aspect of musical perception —loudness— the creation of musical affect in our minds is already a complicated matter.

This is all the more true with timbre, our perception of the "texture" of a tone, i.e., the difference between a trumpet, an oboe, a violin, and the human voice when they are all producing the same note. In the first instance, the perceived differences in timbre can be attributed to the different shapes of the elaborate (but still periodic!) waveforms that make a given note in each instrument. Yet this isn't the entire story. Timbre also has to do with the ear's ability to detect tiny irregularities at the beginnings and endings of these sustained waveforms. These so-called transients range from the 20 milliseconds it takes for the blown oboe reed to settle into a steady oscillation, to the 70–90 milliseconds it takes for a flute or a violin bow attacking a string to do the same. Since the time from wave peak to wave peak for the notes above middle C ranges from about 2 to 4 milliseconds, it can thus take several dozen vibration periods for the tone to be established clearly. The ear takes considerable training to learn how to "separate" different notes of identical timbre, and it relies heavily on those little irregularities in order to do so successfully. Just for entertainment purposes, listen to how Béla Bartók uses timbre to both sustain and subtly modify a simple but intriguing melodic line from a late work, Concerto for Orchestra (1943). In the second movement ("Game of Pairs"), he links together in succession pairs of bassoons, oboes, clarinets, flutes, and muted trumpets. In each case the paired instruments move at fixed intervals with respect to each other, and because they share the same timbre, it is easy for our ears to let the combined chord fuse into a "single," slightly exotic voice.

 

Most importantly there is the problem of pitch—the location of a sound along the tonal scale, depending on the frequency of vibrations reaching the ear, fast ones producing a high pitch and slow ones a low. Since the middle of the 20th century the conventional "concert pitch" is tuned so that the A directly above middle C on the piano has 440 vibrations per second (440 Hz). The nineteenth-century standard was more often 435 Hz, and in the early modern era it ranged as low as 415 Hz. Whether the ear is presented with a smooth sinusoidal wave, or a very complex waveform, so long as both repeat themselves at the same intervals, our ears will perceive them as sharing the same pitch. Pitch can vary continuously: just squeeze or stretch the wave. So now comes the crucial musical question: how does one pitch relate to another? It turns out that in one crucial respect there is a perfect match between an objective regularity about frequency and a subjective regularity about pitch, namely, our perception of the octave. The octave is both the name we give to a frequency ratio (double the frequency and you rise one octave), and to a psychological quantity (pitches separated by octaves sound like the "same" note). When you realize that the octave frequency ratios also match up to the regularities observed in sound production by musical instruments (double the length of an organ pipe and the frequency of the note produced drops by half), the octave looks very much like an entity dictated by nature.

2. Tuning

Unfortunately, how other kinds of pitches relate to each other is not so easily anchored to natural categories, but also has to do with our acquired perceptions of consonance and dissonance. Consonance is the subjective perception that two or more notes fit together well or not, whether in sequence (melody), or simultaneously (harmony). At a more complex level this includes our ability to fuse some sets of overtones together and perceive them as "one" pitch. In the simplest instance, we often hear octave notes (say, C₅ and C₆ played together) as a single note. Whether we hear a chord as consonant or dissonant also depends a great deal on context. A nice C-major triad C-major triad with an extra C an octave below (rather than completing the octave above) sounds perfectly consonant by itself. 

 

But to give a simple example, put it in the context of a hymn in G major, where the standard "Amen" ending has that same chord in the next-to-last position. 

 

This time it sounds "dissonant" in the sense that you expect a further resolution (this can't be the final chord). Or more interestingly, consider this closing sequence of chords from Duruflé's Requiem (1947). First the final chord is spelled out, and sounds dissonant, but then the sequence is constructed in such a fashion that it feels like it can serve as the final resting place.

 

Pitches are related horizontally in the form of melody, and vertically in the form of harmony. The distances between them, however, are strictly a matter of convention (and therein lies a great deal of history which we shall pass over in silence). The human ear can just barely distinguish a difference in frequency of 1 Hz (cycle per second). Since the octave above middle C ranges from about 262 Hz to 523 Hz, we could conceivably fit more than two hundred distinct tones in that same interval if we wanted to. (We will see in a later section how Czech composer Alois Hába (1893-1972) employed quarter-tones in his work. Later 20th-century composers have toyed with other microtonal scales). But for a variety of reasons, classical music has settled pretty firmly on the twelve notes of the octave that you find on any piano today, and deviations from those individual pitches are considered "out of tune."

The fixed set of "allowed" notes is called a scale, and the step between each of the adjacent keys in the octave is called a semitone (or half-step). How you perceive this interval depends on a great number of factors. The standard intervals you find on today's pianos only came into use in the eighteenth century, and became dominant only in the second half of the nineteenth century. This particular arrangement is called equal temperament, because it insists that each semitone should be 1/12 of an octave in perceived pitch difference. But keep in mind that the octave effect is achieved by multiplying frequencies, not by adding them, so you cannot just take A₄=440 Hz and A₅=880 Hz and divide the difference by 12 (about 37 Hz) to tune the remaining intervals. The relationship is actually a logarithmic one, and this implies that you can generate each successful interval from the previous one by multiplying it by 1.05946. So the first semitone above A₄ goes up in frequency by about 26 Hz, while the last semitone taking you to A₅ and completing the octave spans about 49 Hz.

Why haven't musicians and composers been using equal temperament all along? Because it turns out upon closer examination that it involves some rather significant compromises in our choice of harmonic possibilities. 

"12" is not a magic number. Listen to these equal-tempered scales equal-tempered scales, first in six whole tones, then twelve semitones, and then a division into nineteen that has been toyed with by a few composers. 

 

From Pythagoras to Kepler, people have been fascinated by harmonic ratios which would appear to be every bit as natural as the 2:1 octave frequency. Why not 3:2 or 4:3 or 5:4 or 6:5 as well? For centuries the 3:2 ratio was what defined the perfect fifth, for example, the interval C₄-G₄ (nominally corresponding to 7 semitones). Likewise the major third was identified with the 5:4 ratio (say, C₄-E₄). Similarly for the fourth (4:3), the minor sixth (8:5), the minor third (6:5), and the major sixth (5:3). The fundamental problem is that not all the various harmonic combinations of these ratios can be reconciled with each other in a single harmonic series such that, for instance, perfect fifths can be obtained for every chosen diatonic scale diatonic scale, whether starting at C or at A or at F^#, etc.

The use of all twelve semitones in succession to form a set scale is a chromatic scale:

 

while the diatonic scale is made up of the combination of whole tones and semitones we have chosen to encompass the major and minor scales. 

 

Thus in the C major scale the sequence TTSTTTS uses only the white keys.

What composers and performers had favored earlier were tuning systems that permitted the exploration of key color. This involved concentrating tuning discrepancies in intervals that were seldom used, but it let musicians make use of true perfect thirds for many of the white-key scales, with only slight compromises in most of the fourths and fifths. (The one exception was the famous "wolf," usually G^#-E^b). Each key would have its own color, and composers could experiment with modulation between keys to obtain contrasting effects. To get a sense of the limitations this involved at the same time, listen to a later tune, "My Country 'Tis of Thee" played on a harpsichord tuned to just intervals (and not equal temperament). Listen carefully for the awkward tuning relationships in the second portion, after the key change.

 

To identify the tuning discrepancy more clearly, let's adopt a finer scale for our semitone intervals. We want to be able to talk about some semitones being larger than others, so we define the cent as a unit for which the 2:1 octave ratio is always 1200 cents, and for which multiplication of frequency ratios must always translate into addition of the corresponding number of cents. In other words, no matter where you are on the scale, each semitone is still going to be roughly 100 cents.

You may define sequences of whole tones and semitones in nearly arbitrary fashion to form your mode. Since the sixteenth century European musical convention has settled on the major (Ionian) and minor (Aeolian) modes. It doesn't matter which octave you have chosen—mode is only the sequence of tones. But how would you derive a suitable diatonic scale from the various harmonic ratios you happen to favor? It turns out to be an impossible task, for the following reasons.

Choose a convenient octave, say, between C₄ (i.e., middle C) and C₅. With your perfectly tuned octave in place, how would you specify the remaining notes experientially? You might insist on a pure third from C₄to E₄, which would be precisely 386 cents above, i.e., (E₄=329.6 Hz)/(C₄=261.6 Hz)=1.25=5:4. You know the ratio is pure because you tune it until you hear no "beats," that is, until there are five E₄ waves in perfect lockstep with every four C₄ waves. Then to get a nice E major triad, you tune up from E₄ to G₄^# in a perfect third as well. And then through various combinations of thirds and fifths up and down, you could finish the sequence (I'll spare you the details). But what if you wanted to play a piece in A^b major? You would need a nice clean third from A₃^b to the already-established baseline of C₄, which is easily accomplished. Remember that on the standard modern keyboard, A^b and G^# are equivalent. But the 1200 cents from A₃^b to A₄^b that would complete an octave are not the same as adding those three successive thirds you tuned: 386+386+386 leaves you about 42 cents short of an octave. Even without musical training, you can easily hear the discrepancy.

 

There have been numerous schemes proposed to deal with these problems (and you encounter similar discrepancies if you make fifths, rather than thirds, your priority in tuning). The usual compromise is to push the 386 cents of the pure third closer to the 400 cents that would help you keep to your 1200 cent octaves. The eventual dominance of equal temperament represents a long-term compromise that gave free rein to musical modulation in the Romantic era. The brief explanation of these dilemmas here is mainly meant to suggest how tightly structured musical practices have been throughout the modern era, so that we as non-musicologist historians are not overly tempted to bracket distinctions among modernist composers as mere matters of "individual style," but as highly systematic responses to a long series of historically-posed problems. Tonality as commonly understood in the eighteenth and nineteenth centuries involved much more than constructing pitches around a dominant "center." It also meant the whole apparatus of rhythms and formal structures that had co-developed alongside it. Drawing on the class readings, we can discuss the relationship between the musical and extra-musical sources for stylistic innovation in more detailed cultural context.

3. Music as psychology and physiology

During the nineteenth century the sciences were both a resource for those who wanted to make the study of musical systems more rigorous, and a foil for the Romantic cult of solitary genius. Fourier showed in 1822 that any arbitrarily-shaped wave form can be represented as the superposition of a number of simple harmonic (sinusoidal) curves. Two decades later the German physicist Ohm demonstrated that this applied to sound waves as well: complex sounds composed of many frequencies could be understood as an appropriate set of simple frequencies, amplitudes, and phases added together. Various machines for generating sounds with definite combinations of frequencies were also invented in the early decades of the century. Scientists established that humans can hear tones ranging from roughly 20 Hz at the low end to around 20,000 Hz, and also showed that the threshold of audibility varies with frequency.

Late in the nineteenth century music theorists aspired to decipher the code of music in their theories of harmony and meter. (The first chair of musicology was established at Vienna in 1870, but imitators followed only gradually.) This vision of scientific study of music encompassed more than music aesthetics and practical music theory; it also extended to acoustics, and to the audiophysiology and psychology of perception. Not least, it would benefit from the study of music history. One of the aims of this science of music, wrote Hugo Riemann in 1913, would be to move interest away from "the life stories of the great masters towards the development of tonal forms and stylistic features." Appeals to musical "genius" would be rendered unnecessary as scholars approached ever closer to the underlying principles of music through scientific means. As you might expect, this nonetheless led to tension between what harmony as natural phenomenon might permit in theory, and what composition had permitted as historical practice.

Music theory may have aspired to scientific status, but this was not simply evidence of a generalized positivistic spirit. Physicists in particular made contributions to music theory from 1850 onward that directly shaped the terms of debate. The most influential of these studies was made by the physiologist and physicist Hermann von Helmholtz in his On the Sensations of Tone of 1863. Helmholtz sought to ground "the entire theory of musical harmony on natural scientific principles." He was anything but hostile to aesthetics per se—he simply found it more plausible than we perhaps do today that science could serve as the foundation for aesthetic culture. The assumption he shared with music theorists—an assumption widely held since Descartes’ day—was that the laws of nature should also be the laws of music.

In classical compositional practice the major triad: 

 

and the minor triad: 

 

are essentially equivalent, occupying symmetrical aesthetic positions. Helmholtz declared somewhat regretfully that the upper harmonics (or overtones: the multiples of a fundamental frequency which are also produced in most musical instruments; the lowest-frequency whole wave that fits a given violin string when plucked will be accompanied by fainter waves whose lengths are 1/2, 1/4, 1/8, 1/16 the length of the original, and whose frequencies are thus 2, 4, 8, 16 times higher) did not match up mathematically and empirically quite as neatly for minor harmonies as for major harmonies, and thus they had to be regarded as "less consistent." This pronouncement had greater significance than you might think, once you remember that an empirical psychology of music began to develop at around the same time as the discipline of psychology itself, spurred on by Helmholtz's work. The hope was that it would bridge the gap between physical and physiological acoustics on the one hand, and musicology and aesthetics on the other. So when the authority of science suggested that the acoustical inferiority of the minor harmonies did not justify their aesthetic symmetry with the major harmonies, several late-nineteenth century musicologists took the challenge very seriously, and they struggled mightily to come up with "empirical" counterdemonstrations. There are earlier examples of attempts to construct musical theories justifying this duality, but it is only in the wake of Helmholtz’s reluctant challenge that scholars try to identify its basis in Nature. (Helmholtz himself found ample room for "artistic invention," provided one did not insist that the natural function of the ear directly dictated the construction of harmonies.)

4. Limitations

We mustn't be too quick to simply affirm or deny direct mappings between compositional forms, the production of sound, and the brain's "subjective" processing of musical pleasure or displeasure. Many aspects of subjective musical perception at the level of the brain remain opaque to this day, but for Helmholtz and his contemporaries, the point of entry for scientific study, so to speak, was the ear. The tiny and delicate mechanisms of the inner ear were not readily susceptible to direct investigation, but Helmholtz offered one of the first physically plausible models for how the ear might process sound. In the simplest instance, you might think of the ear as merely a microphone that turns sound waves directly into electric signals for further processing by the brain. It is more rewarding, however, to ask if the ear itself might play a more complex role in filtering and analyzing the sound as it passes from outer ear through the eardrum and middle ear, and on into the intricate coils of the inner ear. Here the so-called basilar membrane detects incoming vibrations. The Organ of Corti rides loosely on the inner part of this membrane and contains more than 20,00 hair cells. Different hair cells are disturbed when different parts of the basilar membrane move, and they initiate signals on the individual fibers of the auditory nerve.

Helmholtz could not replicate this series of mechanisms, and it is extremely difficult to determine the dynamic relations between the parts of the ear through direct examination of human corpses. But he understood the physiology well enough to suggest that different sensations of pitch might be associated with different places on the basilar membrane: one frequency stimulates one place on the membrane and thus a particular nerve ending, another frequency stimulates another place and nerve. Direct experimental confirmation of Helmholtz's theory came only in the late 1920s with the work of the Hungarian-born Georg/György von Bekesy. While working for the Hungarian Telephone and Post Office Laboratory in Budapest, he figured out how to map out basilar membrane response using fresh cadavers, and successfully designed mechanical models to mimic the physiological mechanism. For these investigations he won the Nobel Prize for physiology and medicine in 1961.

It turns out that the ear and brain can play further tricks, like "hearing" missing fundamental tones suggested by a series of overtones but not actually present in a given pitch. In other words, we can also "hear" a frequency that never actually stimulated the appropriate location on the basilar membrane. More complicated theories of pattern recognition relying on the inner workings of the brain have only been available since the 1970s, though Helmholtz's original model still retains some utility.