"Tuning is the secret lens through which the history of music falls into focus, " says Kyle Gann. Yet in Western circles, no other musical issue is so ignored, so taken for granted, so shoved into the corners of musical discourse.
A classroom essential and an invaluable reference, The Arithmetic of Listening offers beginners the grounding in music theory necessary to find their own way into microtonality and the places it may take them. Moving from ancient Greece to the present, Kyle Gann delves into the infinite tunings available to any musician who feels straitjacketed by obedience to standardized Western European tuning. He introduces the concept of the harmonic series and demonstrates its relationship to equal-tempered and well-tempered tuning. He also explores recent experimental tuning models that exploit smaller intervals between pitches to create new sounds and harmonies.
Systematic and accessible, The Arithmetic of Listening provides a much-needed primer for the wide range of tuning systems that have informed Western music.
Audio examples demonstrating the musical ideas in The Arithmetic of Listening can be found at: https://www.kylegann.com/Arithmetic.html
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Sound is the ear's perception of a vibration. Generally speaking, the human ear perceives a vibration as sound when it vibrates between about 15 times per second and 23,000 times per second. (The upper theshhold decreases as people age, often descending as low as 8,000 vibrations per second.)
A regularly repeating vibration is perceived as a pitch. For example, the A above middle C is often defined as vibrating 440 times per second, or at 440 cycles per second (cps, or HertzâHz). That pitch is referred to as A 440. It is a common tuning standard for instruments in the Western world. In Western twentieth-century tuning, if that A is 440 cps, then our lodestar middle C vibrates at 261.626âŚcps.
We perceive two pitches as being strongly congruent, almost as identical in a certain way, when one vibrates twice as fast as the other. For example, the pitch at 220 cps is strongly congruent to, and identified with, the one at 440 cps. We call both of those pitches âA.â We also call the pitches at 110 cps and 880 cps âA.â In a tuning sense,
110 cps = 220 cps = 440 cps = 880 cps = 1760 cps
because we call them all âA.â We find it strange to think that 110 = 220 = 440 and so on, and yet musicians are quite used to looking at eight different, evenly spaced notes on the piano keyboard and calling them all âA.â It's the same phenomenon.
Because it is so easily intelligible, the octave is the interval that most offends the ear when it is out of tune. There is also the interesting fact that it is much easier for most people to sing an octave accurately than to sing slightly smaller intervals such as minor sevenths, major sixths, and so on. Our very larynxes can calculate the ratio 2:1âand unconsciously (as they do when you sing the first two notes of âSomewhere Over the Rainbowâ or the sixth and seventh notes of âWay Down Upon the Swanee Riverâ).
The perceived pitch distance between any two pitches is called an interval. And for somewhat accidental historical reasons, we call the interval between a pitch and the twice-as-fast pitch above it an octave. Since one of these vibrates twice in the time the other beats once, we could just as easily call it (as Harry Partch insists on) a 2/1. âOctaveâ is a convenient term largely because it is the term in current use among musicians. (The term evolved from a Latin root connoting the number eight, because in our common major scale the octave, or 2/1, is the tone we reach after ascending through eight steps.)
Every octave, or 2/1, is perceived as the same size interval. That is, the octave from 110 to 220 cps is perceived as being the same size as the octave from 440 to 880. It can be seen from this that the perceived size of an interval depends on the ratio between the two vibrations, not the difference between the numbers in cycles per second: 220 minus 110 leaves only 110, while 880 minus 440 leaves 440. But they are still the same interval.
Because of this strong identity of two notes an octave or 2/1 apart, the octave, or 2/1, has been the most basic interval among most musical systems of the world. We say that 2/1 is the most consonant interval possible, except for the unison, or 1/1, which refers to two identical pitches.
One definition of consonanceâthe one used in this bookâis that two pitches are consonant when their vibrations can be related by small numbers. The smaller the numbers, the more consonant the interval, and there are no natural number ratios smaller than 1/1 and 2/1. Consonance is commonly thought of as a sweet-sounding quality, but it is more accurate to think of it as intelligibility to the ear. The more consonant an interval is, the easier it is to tune by ear.
We will find, however, that the human ear can accept a certain fuzziness of approximation, recognizing the meaning of the interval even if the ratio is not exact. Thank goodnessâotherwise musical performance would be extremely straitjacketed.
CONSONANCES OTHER THAN 2/1
From 2/1 we can proceed to other intervals. Adding 3 to our repertoire of numbers, we now have the intervals 3/1 and 3/2.
If an A vibrates at 220, then the pitch needed to form a 3/1 interval will vibrate at 660 cps. The pitch needed to form a 3/2 interval will vibrate at 330 cps. Both of those pitches happen to be ones that we call âE.â We now have the following group of pitches:
Pitch
Frequency
Ratio
A
880 cps
4/1 (4/1 Ă 220 = 880)
E
660 cps
3/1 (3/1 Ă 220 = 660)
A
440 cps
2/1 (2/1 Ă 220 = 440)
E
330 cps
3/2 (3/2 Ă 220 = 330)
A
220 cps
1/1
Notice that a second octave exists between E 330 and E 660, because 330 Ă 2 = 660.
The ratio 3/2 defines an interval that musicians commonly (again through accidental historical reasons, because it is the distance from the first to fifth notes of the scale) call a perfect fifth. (If you are unfamiliar or rusty with the names of musical intervals, a chart in chapter 2 will refresh your memory.) The ratio 3/1 defines an octave plus a perfect fifth, called a perfect twelfth.
Notice that a third interval exists between E 330 and A 440, which can be defined as the ratio 4/3. Musicians call this ratio a perfect fourth. The same interval can also be found between E 660 and A 880. We can also construct a 4/3 interval above A 220â220 Ă 4/3 = 293.333âŚâwhich happens to be the pitch D.
We've now used all permutations of the numbers 1, 2, 3, and 4. It is not necessary to include the ratio 4/2, because, just as in common fractions, 4/2 reduces to 2/1.
Let's go up to the number 5. This adds the ratios 5/1, 5/2, 5/3, and 5/4.
If A vibrates at 220, then those â5â ratios give us the following notes:
Integrating these notes into our collection, we get:
C#
1100
5/1
A
880
4/1
E
660
3/1
C#
550
5/2
A
440
2/1 = 4/2
F#
366.666âŚ
5/3
E
330
3/2
D
293.333âŚ
4/3
C#
275
5/4
A
220
1/1
This encompasses all possible ratios among the numbers 1, 2, 3, 4, and 5 with respect to the fundamental A 220. The same ratios, of course, could be rebuilt on any other pitch.
The ratio 5/4 corresponds to the interval that musicians call the major third; 5/3 corresponds to the major sixth; 5/2 is an octave plus a major third (major tenth); and 5/1 is two octaves plus a major third.
Note that in tuning theory we use a ratio to refer to both an interval and a pitch within a specified tonality. The interval 5/4 refers to two pitches whose frequencies are related by a ratio of 5 to 4. The pitch 5/4 is the one whose frequency is 5/4 times that of whatever pitch has been designated, for this context, as 1/1. Until some pitch (and it can be any pitch: E
, C#, F
) has been defined as 1/1 for contextual purposes, no other pitch can be named by an interval. We have also surreptitiously introduced the minor third, whose ratio is 6/5: C# to E (275 Hz to 330 Hz), for instance, and also F# to A (366.666âŚto 440).
For now, these are all the intervals we need in order to plunge into our discussion of tuning. In fact, over the course of the sixteenth and seventeenth centuries, the music theorists of Europe decided these were all the consonant intervals we would ever need, and in general Europe has spent four hundred years working with them and not caring much about all the others. (I like to tell students that once they know what sounds the intervals of 1, 2, 3, 4, and 5 make, they can figure out the rest of the history of music by following the implications. I am perhaps given to hyperbole.)
CENTS: A UNIT OF PERCEIVED PITCH SPACE
Since we aren't yet used to talking about intervals as ratios, we need a unit of measurement that will allow us to specify how large each interval we discuss is.
In the late nineteenth century, Alexander Ellis (1814â1890)âa Cambridge-educated mathematician and philologist who, late in life, made the measurement of absolute pitch across Europe and throughout documentable history his particular studyâdevised a unit of measurement for perceived pitch space. He set this unit at 1/1200 of an octave, or 2/1. He called that 1/1200 of an octave a cent.1 There are, by definition, 1200 cents in an octave or 2/1.
There's an obvious reason to call 1/1200 of an octave a cent. Musicians of the Western world in the last few centuries have divided the scale into twelve pitches per octave. The distance between any consecutive two pitches in that scale is called a half step. Therefore, if there are 1200 cents in an octave, there are (on the average, at least) 100 cents in each of the twelve half steps in that octave. A cent in music is 1/100 of an average half step, the way a cent in American money is 1/100 of a...
