Treatise on Intervals
With a Review of Fractions and Decimals and
Recipes for Some Keyboard Temperaments
(c) 2004 David Schulenberg
Pitches are commonly expressed in terms of either (a) the frequency or (b) the length of the sounding object.
Common units for (a) frequency are Herz (vibrations per second); for (b) length are feet.
An interval is a relationship between two pitches.
Intervals can be expressed as ratios, and ratios can be expressed either as fractions or as decimal numbers. [See below on fractions and decimals.]
Just intonation is any system in which the intervals can be expressed as simple whole-number ratios such as 1/2, 3/4, and so on.
Pythagorean intonation is a system of just intonation traditionally ascribed to the mythical ancient Greek mathemetician Pythagoras in which the most common intervals are defined as follows:
Unison: 1/1 Octave: 1/2 Fifth: 2/3 Fourth: 3/4 Major third: 4/5
Minor third: 5/6 Major whole step: 8/9 Minor whole step: 9/10
We can express each of these ratios in decimal form [see below on converting fractions to decimal numbers]. This produces the following results. We shall find that decimal numbers are more easily compared than fractions, and they are useful for certain calculations as well.
Unison: 1.000 Octave: .500 Fifth: .667 Fourth: .750 Major third: .800
Minor third: .833 Major whole step: .888 Minor whole step: .900
These ratios correspond with those theoretically produced by notes in the harmonic series of an acoustically perfect string or wind instrument as follows:
Pitch: C c g c‘ e‘ g‘ bb-‘ c” d” e” f ”+
Number in series:
1 2 3 4 5 6 7 8 9 10 11
Ratio to previous pitch
– 1/2 2/3 3/4 4/5 5/6 6/7 7/8 8/9 9/10 10/11
Ratio expressed as a decimal:
– .5 .667 .75 .8 .833 .857 .875 .888 .9 .909
[Note: numbers 7 and 11 in the series are not used in Western music, even though they form just intervals. The notes produced by these intervals, shown above in brackets, are considered to be out of tune; for example, the seventh note in the series is too low to form a pure minor third to g‘ or a pure major third to d”.]
The ratios in the above chart can be thought of as applying to the sounding lengths of pipes or strings that produce the successive notes. For example, an organ pipe sounding e‘ will, in Pythagorean intonation, be 4/5 as long as a pipe sounding c‘, all other things being equal.
If, then, the pipe sounding c‘ is two feet long, the length of the pipe for e‘ will be 4/5 of that. (“4/5 of ” means 4/5 times the given length of two feet. To calculate this, you can convert 4/5 to a decimal number and then multiply that result by 2.) The result is 1.6 feet.[See below on converting this to inches.]
What if we want to talk about pitch as a function of frequency instead of sounding length?
Frequency and length are in inverse proportion; that is, as pitch rises, the sounding length decreases but the frequency increases (and vice versa). Applied to the above chart, this means that for each fraction we must substitute its reciprocal:
Pitch: C c g c‘ e‘ g‘ bb-‘ c” d” e” f ”+
Number in series:
1 2 3 4 5 6 7 8 9 10 11
Ratio to previous pitch:
– 2/1 3/2 4/3 5/4 6/5 7/6 8/7 9/8 10/9 11/10
Ratio expressed as decimal:
– 2 1.5 1.33 1.25 1.2 1.167 1.143 1.125 1.111 1.1
These relationships hold for any notes, not just those shown in the above table. Thus, in any octave, the frequency of the higher note is twice that of the lower one: if the first a‘ on the flute has a frequency of 415, the note an octave below it has a frequency half that, or 415/2 = 207.5. If the lowest pipe on a particular organ stop is 8 feet long, then the pipe an octave above that is 4 feet long, and the pipe sounding a fifth above that is 2/3 as long, or 2/3 x 4 = 8/3 or about 1.667 feet long. [To convert this to inches, see below.
Suppose now that we want to know the relationship between g‘ and the note one major whole step above it, that is, a‘. The frequency of the latter will be 9/8 that of g‘.
Of course, a‘ is today normally defined as the note whose frequency is 440. So what would be the frequency of g‘ in Pythagorean intonation? We know the ratio between the two frequencies; it is 8 to 9. The latter number represents the higher frequency. So the frequency of g‘ must be 8/9 that of a‘, or 8/9 x 440 = 391.1111…. (391-1/9).
What would be the frequency of b‘? Since we consider the latter to be one whole step higher than a‘, then its frequency is 9/8 that of a‘, or 495. (This result is obtained by multiplying 440 by 9/8.)
We now have ratios and frequencies for the two whole steps above g‘. What, then, is the relationship between g‘ and b‘? To find this we must add the two whole steps. We can think of this as adding one ratio, 9/8, to another, also 9/8. To add ratios, we multiply the fractions that represent them. Thus the interval from g‘ to b‘ is 9/8 x 9/8, or 81/64. [See below on multiplying and dividing fractions.] The frequency of this b‘is therefore 81/64 that of g‘, or 495.
The ratio 81/64 is often referred to as that of the Pythagorean third. Technically, this is incorrect; the interval g‘ to b‘ as described above is actually a ditone, the sum of two whole tones, or more properly, two major whole tones, each formed by frequencies in the ratio 9/8. In just intonation, a major third is actually formed from two different-sized whole tones, as in the interval c” to e” in the tables shown above. The smaller whole tone, in the ratio 10/9, is a minor whole tone.
Suppose that we decided to define b‘ not as the sum of two whole steps above g‘, but rather as a pure or just major third above g‘. In that case, the frequency of b‘ would be 5/4 that of g‘, or 5/4 x 391-1/9 = 488-8/9. This number is significantly lower than the one we obtained above (495).
These considerations are important when we proceed to our next problem, the definition of the half step. Suppose we wish to express the interval from b‘ to c” as a ratio. We could think of this interval as the difference between the just fourth g‘/c” and the just third g‘/b‘. That is, we could take the difference between the ratios of the perfect fourth (4/3) and the major third (5/4). To find the difference between two ratios, we divide the larger by the smaller. Division by a fraction is the same as multiplying by its reciprocal. Thus, in this case the ratio representing the half step would be 4/3 ÷ 5/4 = 4/3 x 4/5 = 16/15. This is the value usually given for the diatonic or major half-step in just intonation. Expressed as a decimal, it is equivalent to about 1.067.
We could find other half steps as well. Suppose, for example, we define bb’ as the note that is a minor third above g‘. In that case, the frequency of bb’ would be 6/5 that of g‘. What is the interval between this bb’ and the b‘ that forms a pure third to g‘? We must subtract a just major third (6/5) from a just minor third (5/4): 5/4 ÷ 6/5 = 4/5 x 5/6 = 25/24. This is the chromatic or minor half-step. Expressed in decimal terms, it is about 1.042, which is significantly smaller than the diatonic one.
What if, in order to calculate our diatonic half step from b‘ to c”, we defined b‘ not as a pure third above g‘ but as a so-called Pythagorean third above that note? In that case we would be subtracting a ditone (81/64) from a perfect fourth (4/3). The result is 4/3 ÷ 81/64 = 4/3 x 64/81 = 256/243 or about 1.053. This is a significantly smaller number than 16/15, although larger than 25/24. In other words, the “Pythagorean third” results in a significantly higher b‘ than does a just third.
The difference between these two intervals—the ditone and the just major third—is known as the syntonic comma. It is equal to 16/15 ÷ 256/243 = 16/15 x 243/256 = 3888/3840 = 81/80 = 1.0125. [See below on Quantz and the comma.]
Now for some practical problems.
Suppose that we have two flutes of slightly different lengths. We suspect that the difference in length represents a difference in the pitch standards at which they were built. How can we determine the interval between those two pitch standards?
Let us imagine that the sounding length of one flute is exactly two feet; that of the other is an inch longer. To compare them, we might first convert both lengths into inches. In that case the length of the longer flute, in inches, 2 x 12 = 24; the other is 25. The sounding lengths of the two flutes are thus in the ratio 25/24, or a chromatic semitone. If the frequency of a‘ on the shorter flute is 440, then that on the longer one is 24/25 x 440 = 422.4 (remember that length and frequency are inversely proportional, so if the length of one flute is 25/24 that of the other, its frequency will be 24/25 that of the longer one).
Now suppose instead that we know two frequencies and wish to determine the interval between them. If one flute sounds a‘ at 405 and another at 430, then the interval between them is 430/405. We can reduce that fraction to 86/81, but clearly this is not a small whole-number ratio, so it does not correspond exactly to any interval in Pythagorean intonation. Nevertheless, as a decimal, this fraction is equivalent to about 1.062. Comparing it to the decimal values for the common intervals, we see that it compares fairly closely to the diatonic half-step (1.067).
One final problem: how can one divide an interval in half? For example, what note is at the exact midpoint of the octave a/a‘? We would probably call it eb’, but what is its frequency?
If the frequency of a‘ is 440, that of a will be half that of 220. We might imagine then that the note halfway between them will have a frequency that is midway between 440 and 220. The difference between the two is also 220, and if we split this difference evenly might imagine the note halfway between them to have a frequency of 220 + 110 = 330.
But 330 is equal to 2/3 of 220, and 2/3 is the ratio of the perfect fifth. The note a perfect fifth above a is e‘, which is obviously not the same as eb’. So one cannot simply split the difference of the frequencies between a and a‘ in order to divide the interval between them in half.
In order to divide an interval in half, it is necessary to remember that intervals are expressed as ratios, and that to add ratios you must multiply the fractions that express them. To subtract ratios, you divide the fractions. Thus, when we speak of dividing an octave in half, we are imagining it to be the sum of two equal intervals. But if the ratio that represents the octave is the sum of two equal ratios, then to find their value we must find the square root of the fraction that represents the octave.
Since the octave can be represented as the number two, half of the octave must be represented as the square root of 2. This is a so-called irrational number roughly equal to 1.414; looking at our second chart, we see that this corresponds to an interval somewhere between a perfect fifth and fourth. In fact, the interval in question is an equal-tempered tritone. But no interval represented by an irrational number can be expressed as a fraction, and therefore such intervals have no place in just intonation. Tritones in just intonation are always just a little different from half an octave; their ratios can be determined through the same sort of process we used to measure the halfstep. (Thus the diminished fifth is a fifth (3/2) minus a diatonic half-step (16/15), or 45/32 = 1.40625; the augmented fourth is a fourth (4/3) plus a chromatic half-step (25/24), or 100/72 = 25/18 ≈ 1.389.)
The so-called Pythagorean third can be easily divided in two, since it is defined as the sum of two major whole steps. But what if we try to divide the pure major third? Earlier we found that if g‘ had a frequency of 391-1/9, then b‘ would have a frequency of 5/4 that, or 488-8/9. If we split the difference, we find that a‘ = 440, as we would expect. But, again, this frequency corresponds not to the midpoint between g‘ and b‘ but to a note slightly higher. This is because a‘ forms a major whole step to g‘ but only a minor whole step to b‘.
Like the octave, the major third can be divided equally only when it is defined as the sum of two equal intervals, as in the case of the Pythagorean ditone. To divide a pure major third (5/4) in half, one would need to find the square root of 5, which is another irrational number. An equal division of the pure major third g‘/b‘ as defined above would yield an a‘ of frequency about 437, which is audibly lower than a‘ = 440.
To put it another way, when adding or subtracting intervals, it is important to multiply or divide ratios properly. If modern pitch is defined as a‘ = 440, then a major whole step beneath that is 8/9 x 440 = 3520/9 or about 391.111. The halfstep above that must be defined as either g#’ or ab’, the former forming a diatonic halfstep to a‘ (frequency 15/16 x 440 = 412.5), the latter a chromatic one (24/25 x 440 = 422.4). Equal-tempered g#/ab’ lies between them at about 415. The latter, morever, is about 25 Hz below 440 and about 23 above 392—that is, not equidistant in terms of Hz.
Some of the distinctions noted above are audible; others involve immeasurably tiny differences that are meaningful only on paper. But it is necessary to do the arithmetic in order to determine which of these distinctions might involve differences of practical significance.
To convert a fraction to a decimal number, divide the top number (numerator) by the bottom number (demonimator). For example, to convert the fraction 9/8 to a decimal number, divide 9 by 8; the result is 1.125.
To convert a decimal number to a fraction, understand the decimal portion of the number as a fraction over 10, 100, 1000, etc. Thus .65 = 65/100; .375 = 375/1000. If the top and bottom numbers have a common denominator, then you can “reduce” the fraction. For example, in 65/100, both 65 and 100 are divisible by 5 (that is, 5 “goes into” both 65 and 100). 65 divided by 5 is 13, and 100 divided by 5 is 20. Thus 65/100 = 13/20. In 375/1000, both numbers are divisible by 125, and the fraction is equal to 3/8.
Some fractions cannot be expressed as exact decimal numbers. For example, if you divide 2 by 3 you get a never-ending series of digits: 2/3 = .66666666666666666666666666666666666666… Numbers of this type are called repeating decimals. Calculators and normal human beings usually round them to a simpler non-repeating decimal which is not exactly equal to the fraction but close enough for practical purposes. Thus 2/3 is approximately equal to .67, or to .667, or to .66666667. The degree of precision is arbitrary: you round the number to as many or as few decimal places as you need to, depending on the accuracy of your measuring device. Rarely, however, will it be necessary or desirable to express decimals more accurately than to the nearest thousandth (three decimal places).
It is not easy to convert repeating decimals to fractions. The simplest method is simply to memorize the most common ones, such as: 1/3 = .333… 2/3 = .666… 1/6 = .166… 5/6 = .833… 1/9 = .111… 2/9 = .222…
Some decimal numbers cannot be expressed as fractions. For example, pi, the ratio between the circumferance of a circle and its diameter, is a non-repeating decimal number that begins 3.1416… and continues infinitely without forming any repeating patterns. Such a number is calledirrational. The square roots of prime numbers such as 2 and 5 are also irrational numbers. Most of the intervals of equal temperament can be expressed only as irrational numbers; many calculations with such numbers require the use of logarithms, which were not invented until around 1600 and not applied to music until around a century later.
An English inch is 1/12 of a foot. In calculations involving both inches and feet, it is necessary to express all measurements in one or the other unit, choosing whichever unit yields the most useful results. It may be necessary to experiment. For example, if the pipe for c” is one foot long and we want to know the length of the pipe for d”, it doesn‘t help to know that the latter should be 8/9 of a foot in length; we want to know how many inches. Therefore start by converting one foot to 12 inches, then multiply the latter by 8/9 to get 10.666… or ten and 2/3 inches.
To multiply two fractions, multiply the numerators and then the demoniminators separately. Thus, to multiply 3/4 by 5/6, first multiply 3 x 5 = 15, then 4 x 6 = 24. The result is 15/24. Because 15 and 24 are both divisible by 3, 15/24 can be reduced to 5/8.
According to Quantz and others, there are nine “commas” in a whole tone. Is he talking about the syntonic comma? We would have to add nine of these intervals, each 81/80, and compare the result with the value for a major whole tone, or 9/8.
The following are simple pragmatic schemes for tuning harpsichords and other keyboard instruments. Although they correspond more or less to historically documented methods, I cannot cite specific sources for any of them.
5. Test the fifth a/e‘; it should beat at about the same rate as the fifth g/d‘. If it beats too fast, lower the a slightly and adjust the other intervals accordingly. If it beats too slowly, raise the a.
8. Tune these perfect fifths and fourths pure: b–f#’, f#’–c#’, c#’–g#’. Also tune the octaves f#’–f# and g#’–g#. Test each of the triads that incorporate these notes. The triads should sound increasingly harsh as you move toward “sharp” keys, but all should be usable. If not, go back to step 7 and raise the note b slightly in relation to the g.
9. Tune these perfect fourths pure: f–bb, bb–eb’. Test the fifth ab/eb’. It should sound pretty bad. Lower the eb’ until the fifth becomes tolerable; also test the fourth bb/eb’, which should also be tolerable. If not, adjust the eb’ further. It may be necessary to lower the bb as well. When done, test the major triads on f#, g#, bb, and c#’. All should be bearable, although only the one on bb will be close to being pure.
4. Tune the perfect fourth c‘–g and the perfect fifth g–d‘ as in 1/4-comma meantone (temperament no. 1, steps 3–4), but with both intervals beating slightly less quickly. Then check the fourth a/d‘ and the fifth a/e‘; these should beat like the fourth g/c‘ and the fifth g/d‘, respectively. If all is well, also tune the octaves f–f‘, g–g‘, and a–a‘, and test all the resultant triads, which should sound close to pure.
5. Tune pure these fourths and fifths: e‘–b, b–f#’, f#’–c#’; and f–bb, bb–eb’. Also tune the octave f#’–f#. Test the resultant triads. Some will be fairly strident, but because you will have tuned are no more than three perfects in a row, none of the thirds will be as wide (impure) as a so-called Pythagorean third (which is the product of four perfect fifths).
6. Next tune the fifth eb’–ab so that the ab is high and the interval beats at roughly the same rate as the fifth a/e‘. Then test the fourth ab(g#)–c#’, which should be wide and beating at about the same rate as the fourth a/d‘. Also tune the octave g#–g#’ and test the fifth c#’/g#’, the fourth eb’–ab’, and the thirds ab–c‘ and e‘–g#’. You may need to adjust the notes ab (g#) and ab’ (g#’) up or down a bit in order to get all of the intervals tolerably in tune.
4. Tune the perfect fourth d‘–a and the perfect fifth a–e‘ as in 1/4-comma meantone (temperament no. 1, steps 3–4), but with both intervals beating slightly less quickly. Then check the fourth b/e‘ and the fifth b/f#’; these should beat like the fourth a/d‘ and the fifth a/e‘, respectively. If all is well, also tune the octaves g–g‘, a–a‘, and b–b‘, and test all the resultant triads, which should sound close to pure.
6. Next tune the fifth f‘–bb so that the bb is high and the interval beats at roughly the same rate as the fifth b–f#’. Test the fourth bb–eb’. You may need to adjust the bb a little upward to get the third bb/d‘ more in tune (especially if you will be playing in “flat” keys), downward to tune the third bb(a#)/f#. But any large adjustments force you also to adjustments of f/f‘ and/or eb’ (d#’), so avoid this if possible.
3. Optionally—especially if playing pieces in “flat” keys—check the major thirds d‘/bb and g‘/eb’. If these are too wide, raise the note g so that it forms a pure major third b–g. Then raise the notes c‘, f‘, bb, and eb’ by the same amount. Also retune the octaves g–g‘ and f‘–f. This has the effect of transforming the temperament into something close to tempérament ordinaire, but it should work. Check the resultant triads, including the ones on ab and b.