What is permitted? The nature of music
The materials in the music modules of the course are intended to supplement the texts we will be discussing in class. The selection and presentation are neither systematic, nor are they intended to satisfy the standards of musicology. I have made every effort not to assume any previous musical training, and chosen the musical excerpts with the thought of creating a series of linked impressions, rather than providing any systematic account of compositional developments. The selection also presupposes that you do not want to spend more than an hour with each section, which is why I have been so philistine about taking pieces of music outside the context of the total work.
If you do not know how to read musical notes but are willing to devote a half hour or so to understanding music as a semiotic system, a very efficient way to get started would be at musictheory.net. See if you can get as far as generic and specific intervals under "Lessons." Those with some musical knowledge might spend a few minutes toying with the interval ear trainer under "Trainers."
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#-Eb). 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 Ab major? You would need a nice clean third from A3b to the already-established baseline of C4, which is easily accomplished. Remember that on the standard modern keyboard, Ab and G# are equivalent. But the 1200 cents from A3b to A4b 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.