I. "JUST" TUNING - THE PERFECT SCALE :

Let A = 440 Hz (or cycles per second--to use the older term). From that, how do we construct a "perfect" C-major scale, starting with middle C? Recall that the most harmonious intervals have low-order partials, or harmonics, that coincide--or match. This means that the intervals on our scale, which are frequency ratios, should cancel down to ratios of small, whole numbers. Why is this so, you ask? Because any pitch sounded on any instrument consists of an infinite series of partial pitches--all exact multiples of the fundamental pitch or first harmonic or first partial. These pitches are called a "harmonic series," with the relative strengths of the harmonics above the first one determining the "sound character" of the instrument.  They decrease in intensity and get closer together as you go up.  Such as: 2/1 (octave, e.g. 440hz/220hz), 3/2 (fifth), 4/3 (fourth), 5/4 (major 3rd), 5/3 (major 6th), 6/5 (minor 3rd), 8/5 (minor 6th), 9/8 (major tone), 10/9 (minor tone), and 16/15 (semi-tone).  The pitch intervals--fifth, minor 3rd, etc.--are named from the scales natually evolving from this process.

If you check the frequency of middle C on a modern, equal-tempered keyboard (based on A - 440), you get about 261.626…Hz. In fact, the latter frequency is 440.00 over 2 to the 9/12 power (A is the 9th tone up from C in our 12-tone chromatic scale--use your calculators).The interval is a major 6th. Can you see why this is so? (Recall that an equal-tempered half tone is exactly 2 to the 1/12 power times the next lower key.)

So, to get a "perfect" middle C, we happen to discover that 440 / (5/3) = 264--very close to our 261.626. This means that the 5th harmonic of 264 equals the 3rd harmonic of 440--or 1320 Hz. So 264 Hz will be our middle C. And the frequency ratio 5/3 is a perfect major 6th, as given above. A perfect 5th happens to have a ratio of 3/2, so that 264 x 3/2 gives us 396 Hz--a perfect G above middle C. We get our D a whole tone above C by dropping G by 4/3, the ratio of a perfect 4th.  396 x 3/4 = 297 Hz, a perfect D. We happen to notice that the ratio of D to C is 297/264 = 9/8--yet another small, whole number ratio. This is called a "major" tone.

A perfect major 3rd has the ratio 5/4, so that we generate our E above middle C by multiplying 264 x 5/4 = 330. In like manner we get our F by dividing A - 440 by 5/4: 440 x 4/5 = 352. But we quickly see that the ratio F to C is 352/264 = 4/3, another "automatic" perfect 4th! So what do we have left? The B above A - 440. If we take the G = 396 x 5/4, we get 495, which is our perfect B. We also notice that the ratio of B to E is 495/352 = 3/2, which, eureka!--is another perfect 5th. And also the ratio of B to D is 495/297 = 5/3, which, voilà!--is another perfect major 6th. Doubling our C = 264 gives us 528, which we'll call C'--an octave up.

So what about the remaining interval ratios? Well, G to E is a minor 3rd: 396/330 = 1.2 = 6/5. C' to A is another minor 3rd: 528/440 = 6/5 (boy, are we living right!). C' to E is our first and only minor 6th (within one octave, anyway). 528/330 = 1.6 = 8/5--a perfect minor 6th--an inversion of a major 3rd = 5/4 (makes sense). What about the half-tones: F to E is 352/330 = 16/15 (the matching partials are getting pretty high, as you would expect). By the same token, C' to B is 528/495 = voilà! 16/15 again. Amazing!

Let's look at our whole scale now: C - 264, D - 297, E - 330, F - 352, G - 396, A - 440, B - 495, C' - 528. This is as perfect and harmonious a major scale as we can get. And amazingly--as well as coincidentally--the frequencies themselves all turn out to be whole numbers of Hertz--with no fractions. Seems like "divine" intervention! If we had picked any A other than 440 in that region, we'd be having decimals in our scale--which is only an "on paper" phenomenon, of course (the intervals would sound just as pure).

But wait! Before we bask too long in this Nirvana, let's look a couple of intervals I've deliberately not discussed yet: What about the fifth A to D? The ratio is 440/297 = 1.481481481…. Whoops! This one is unarguably flat from our expected 3/2 or 1.5. How flat is it?  Well, the 3rd partial (or harmonic) of 297 is 297 x 3 = 891 and the 2nd partial of 440 is 880. So that interval is flat by 891 - 880 or 11 very audible beats per second! Not so perfect after all.

What about the minor 3rd F to D? Do the calculation: 352/297 = 1.185185185…. not the expected 6/5 or 1.2. But the beating harmonics here are so much higher and softer that the interval isn't nearly as distressing as D to A--which essentially "ruins" the scale for music, unless you can somehow avoid that interval ever being played. Very restrictive! This is where the first "practical" compromise had to be made.
 

II. PYTHAGOREAN TUNING - THE MEDIEVAL COMPROMISE :

With "Just" tuning of the C-major scale we had perfection--till we discovered the "wolf" interval A to D--right there on the main scale where it can't be avoided. What would we have if we deliberately made all the fifths (and thus the corresponding 4ths) perfect? If we started again at A - 440, then our D below it would be 440 x 2/3 = 293.3333. The G above D would then be 293.3333 x 4/3 = 391.11111, and our middle C would then be 391.1111 x 2/3 = 260.740740 (it was nice when we didn't have decimals!). The E below A would be the same as in "Just" tuning: 440 x 3/4 = 330. The F above middle C would be 260.74074 x 4/3 = 347.654. And finally the B would be E times 3/2: 330 x 3/2 = 495, as in the "Just" scale above. But then all the fifths and 4ths would be perfect--on the white keys. Our Pythagorean scale would then be: C - 260.740740, D - 293.3333, E - 330, F - 347.654, G - 391.1111, A - 440, B - 495, C' - 521.481481--a bit of a change from our so-called perfect scale.

But what about the other intervals? The major 3rd, for instance, now has a constant ratio of 330/260.740470 = 1.2656, clearly wider than the "just" interval 5/4 = 1.25. Let's check the beats between C and E: the 5th harmonic of C is 260.74074 x 5 = 1303.7 whereas the 4th harmonic of E is 330 x 4 = 1320. This major 3rd is wide by 1320 - 1303.7 = 16.3 beats per second, and it's a definite buzz. But. . . medieval music didn't like 3rds very much and didn't use 'em. Recall the Gregorian "organum" was a chant with parallel 4ths--all in perfect tune. So this tuning served that music quite well.

But let's see what happens to the fifths as we "circle" them up through the black keys: Starting at C, we progress as follows: C - G - D - A - E - B - F# - C# - G# - D# - A# - E# - B# - F##. . .let's stop there, and progress down through the circle: C -F - Bb - Eb - Ab - Db - Gb - Cb - Fb - Bbb. . .equally a good place to stop. If we assume perfect fifths going in either direction, it's safe to assume that E# and B# aren't the same pitches as the F and C in the same octave--above where we started. To get from C to B# through the circle, we had to traverse 7 octaves, and 12 different notes--as you would on a complete chromatic scale (that's why it's convenient to speak of a "circle of fifths"--or "fourths"). That means we have to raise the perfect fifth ratio 3/2 to the 12th power to get to the high B#, and the octave ratio 2/1 to the 7th power to get to the high C. Well, 2 to the 7th is 128, whereas 1.5 to the 12th is 129.746. It works the same, of course, going toward the flats. B# is above C in freq. ratio by 129.746/128 = 1.01364, and the latter is called the "Pythagorean or ditonic comma." (Notice that if we were to flatten [i.e. temper] each fifth by the 12th root of that comma, i.e. 1.01364, we'd get 1.00113. Dividing that into our perfect 5th: 1.5/1.00113 = 1.4983, we'd have a circle of fifths exactly equal to seven octaves, and equal to each other--in other words, modern-day 12-tone equal temperament!)

Therefore, on a fixed-key instrument--keyboard or fretted--where the sharps and flats are covered on five (black) keys per octave, A# must equal Bb, B# must equal C, and so forth. Ergo, for Pythagorean tuning the Pythagorean comma must be inserted across one 5th in the circle, creating the narrow wolf interval, somewhere where it's least likely to be "crossed" in a piece of music. It’s often placed between Ab (or more correctly G#) and Eb in Pythagorean tuning. This interval is even narrower (by a small amount) than the D to A is in "just" tuning--and is an enharmonic fifth, a definite "wolf." It can also be shown that in this tuning that any sharp is above its enharmonic flat (e.g. G#, Ab; F#, Gb) by a Pythagorean comma. On a stringed or other instrument (incl. voice) with infinite pitches, you can play and sound those different intervals--if you play "pure," with no vibrato.
 

III. MEANTONE TEMPERAMENT - A FIX FOR THE RENAISSANCE AND EARLY BAROQUE:

As music evolved into the complex polyphony (using lots of counterpoint) of the Renaissance, it became apparent that the need for using 3rds and 6ths in parallel harmony forced a reconsideration of six centuries of Pythagorean tuning of fixed-key instruments (like the organ). Many theorists noticed that if they "temper" (i.e. deliberately mis-tune) each of the fifths in the circle to narrow them (all) equally, such that the major 3rds get squeezed from being too wide to where they are again "just"--i.e. having a perfect 5/4 ratio, the result remains rather harmonious over all the keys except for the "wolf" interval--now way too wide--still kept in the black keys. Each fifth, moreover, is only flattened by a small amount since the whole circle is sacrificed to get each major 3rd to be pure. And the beauty of those perfect 3rds and near perfect 6ths are heightened in all that Renaissance choral polyphony, and in the subsequent rise in the use of keyboard instruments in the early Baroque. All one had to avoid was shifting to certain "dark key" intervals.

How much should the fifths be tempered, in the "circle," to get perfect major 3rds? Starting at middle C, we go up by fifths till we get to a high E:   C - G - D' - A' - E''.  The latter E'' should be exactly the 5th partial of our middle C--but we need to space the fifths in between equally: let X equal the unknown fifth spacing & we notice there are 4 of them: then X to the 4th = 5N, where N is our starting frequency; solve for X in terms of N: X = 4th root of 5xN = 1.49535 x N. Since the fifths are all equally spaced no matter where we start, let N = A - 440. Then D below A is 440/1.49535 = 294.2457. The G above D is (294.2457/1.49535) x 2 = 393.548. Hence our middle C is 393.548/1.49535 = 263.18. In like manner our F is: (263.18/1.49535) x 2 = 351.999, or 352. But hell!!!  We knew that since we have perfect 3rds, and 352 is 4/5 of 440--from our "Just" scale above. Consider this a check on our calculations. In like manner E = middle C x 5/4 = 263.18 x 5/4 = 328.975. B is G times 5/4 = 393.548 x 5/4 = 491.935.  As a check, multiply E by our fifth interval: 328.975 x 1.49535 = 491.933--close enough (for government work)!!

So our Meantone tempered scale is: C - 263.18, D - 294.2457, E - 328.975, F - 352, G - 393.548, A - 440, B - 491.935, C' - 526.36 Hz. Now look at this: In the whole tone succession between C and E, F and A, or G and B; the respective middle tones: D, G, and A represent "mean" values between the 3rds: As a test, let's multiply C times E: 263.18 x 328.975 = 86,579.6405. Taking the square root we get: 294.244--our D. You guessed it; that's where the term "Meantone" comes from. And when you think about it, you knew it had to be so--given the equally spaced fifths.

Let's look at the beats we'll encounter in the usable intervals: We know the major 3rds (and hence the minor 6ths) are perfect. What about those narrowed fifths? Using our test above for C to G: The 3rd partial of C is 263.18 x 3 = 789.54, while the 2nd partial of G is 393.548 x 2 = 787.096. The fifth is narrow by 789.5 minus 787.1 = 2.4 beats per second. (NOTE: The number of beats are exactly proportional to the two frequencies involved, with a fixed interval--as should be obvious.) Well, this is a slow, unobtrusive beat that still sounds quite euphonious. What about the major 6ths, which are pure at a 5/3 ratio? Here we go again, with C to A: The 5th partial of C is 263.18 x 5 = 1315.9, while the 3rd partial of A is 440 x 3 = 1320--about 4 beats per second, a slow vibrato rate and still pretty good. We care less about the minor 3rds (6/5), but they're pretty good too. In fact, everything is quite harmonious a ways into the sharps and flats--till we encounter the "big wolf."

Let's see how bad the "big bad wolf" is: Taking our Meantone fifths around the "circle": 1.49535 to the 12th power = 125.001; let's round it off to 125. Recall that 2 to the 7th is 128. Our Meantone comma is 128/125 = 1.024. Ergo our wolf fifth is 1.49535 x 1.024 = 1.53124, placed again between Eb and G# (that's a fourth, which you recall is just an inversion of the fifth--so we can stay in our Middle C octave). To avoid calculating those black-key frequencies, let's put this wolf between D and A, just to test the beats: again A = 440, D = 440/1.53124 = 287.35--quite a bit lower than our former 294.2457. So, as measured against our perfect 3/2 ratio for a fifth: The 3rd partial of D is 287.35 x 3 = 862 (rounded off), whereas the 2nd partial of A is again 880. This wolf fifth is wide by 18 very obvious beats per second!

But that's not all. Any major 3rd that "crosses" that wolf will have a wolf of its own: For example, keeping our wolf fifth between D and A, the F to A major 3rd has the wolf: A - D - G - C - F in descending fifths (or ascending fourths) clearly shows F to A crossing the D to A wolf. So what do we have? A is still 440, F has now been dropped by our Meantone comma - 1.024. Ergo the old F - 352 becomes 352/1.024 = 343.75. So, as measured against our perfect 5/4 ratio for a perfect major 3rd: The 5th partial of F is 343.75 x 5 = 1718.75, whereas the 4th partial of A is 1760: about 42 beats per second plus sounding very sharp to our ears makes it grating to even hear. That's why Renaissance and early Baroque music avoided modulating into the black keys, and when they did, they were ve-e-e-ry careful. To make one more salient point: The sharps in Meantone are now lower in pitch than their enharmonic flats--by the 1.024 comma. E.g. G# is below Ab--the opposite of what you found in Pythagorean tuning.  To play any piece requiring Ab but not requiring any G#, you have to tune all the G#s up, by the meantone comma, to Abs. Retuning to taylor what you're playing to euphonious intevals therefore becomes crucial in Meantone Temperament.
 

IV.WELL TEMPERAMENT - HAVE YOUR CAKE AND EAT IT TOO:

In the late Baroque, i.e. Bach's time, the desire to change keys became more intense--not so much to "modulate" within a piece as to write and play sets of pieces covering a wide variety of keys--without the above referred inconvenience of retuning. Yet the composers still liked the very euphonious intervals they were getting from the Meantone tempered instruments--in those "preferred" keys. Recall that in a circle of perfect fifths the "end" goes past 7 octaves by a Pythagorean comma (1.01364, from above). Various theorists, like Thomas Young in England, proposed sharing the comma equally among half the circle, where we want our smoothest 3rds and 6ths, and keep the remaining half pure. Each fifth in the white-key intervals would then be narrowed by 1/6th of a comma, and all the associated 3rds would be squeezed closer to their perfect 5/4 (1.25) state. The black-key fifths would then be pure (3/2) and those 3rds that started to cross them would get wider as more are crossed--till they reached the maximum of a Pythagorean 3rd--i.e. 1.2656--when all are crossed.

This created an aesthetic of different key colors when playing 3rds and 6ths in different keys. But they were all viewed as usable--no wolf intervals; hence a well-tempered keyboard is an unrestricted one. Bach wrote his two famous sets of 24 to show off the aesthetic effects of this temperament after he was introduced to it. He didn't invent it!!!  The tradition stayed through the end of the Baroque (1750), throughout the Classical (1750 - 1828), and the early Romantic (1814 - 1850s). Let's look at some of the properties of Thomas Young's then popular well-temperament.

Young narrowed the following fifths, each by 1/6th of a P. comma: C - G - D - A - E - B - F#. Correspondingly the remaining fifths are pure: Gb(F#) - Db - Ab - Eb - Bb - F - C--back to where we started. (Recall that it takes 7 notes to create 6 intervals.) To get our 1/6th comma, we take the 6th root of the P. comma 1.01364 = 1.00226. Our first set of fifths will then be narrowed to: 1.5/1.00226 = 1.496617. By now you know the drill: A = 440. D = 440/1.496617 = 293.996, let's say 294.  G = (294/1.496617) x 2 = 392.88. Middle C = 392.88/1.496617 = 262.513. E = (440x1.496617)/2 = 329.256. B = 329.256 x 1.4966117 = 492.77. Note above that the interval F - C is pure, so that F = 262.513 x 4/3 = 350.017. So let's list them again: C - 262.513, D - 294, E - 329.256, F - 350.017, G - 392.88, A - 440, B - 492.77, C' - 525.026.

Let's check the purity of these fifths with D to A again: 294 x 3 = 882 and 440 x 2 = 880. 882 minus 880 is 2 beats per second--not bad, actually a little better than the Meantone interval. How about the major 3rds? Checking C to E: 262.513 x 5 = 1312.6, 329.256 x 4 = 1317.02. The difference is about 4 1/2 beats per second--still very good for a 3rd. But in our complete circle of fifths, let's look at how many of these 3rds are that close to pure: C - G - D - A - E - B - F# are the flattened intervals, as you recall. The 3rds contained within this half circle are C - E, G - B, and D - F#, ergo they are equally spaced and the most harmonious.

Now let's extend the circle all the way: C - G - D - A - E - B - F# - C# - G#(Ab) - Eb - Bb - F - C. The pure fifths are the last 6 intervals in the circle, as we said above. Looking at the 3rds beyond D - F#, we can see that they must gradually widen as we progress: A - C#, E - G#(Ab), B - D#(Eb). Now B - D# has only one flattened fifth in that part of the circle: B - F#. All the rest leading to Eb are pure. Obviously this 3rd (B - D#) is nearly a Pythagorean 3rd wide. The next 3rd in the series: F# - Bb is a full Pythagorean 3rd since it traverses all pure fifths. And, for symmetry's sake, there must be three of them: F# - Bb, C# - F, and Ab - C--all Pythagorean 3rds that have "buzzy" beats in the middle C octave. As we continue with Eb - G, Bb - D, and F - A, those 3rds get progressively narrower toward our most harmonious three given above.

The value of a composition's key signature is clearly quite high in this temperament as it affects the "color" of all the intervals appearing therein. From Bach through Chopin, the choice of keys was made with a well-temperament in mind. But by the mid -19th century, harmonies were getting progressively thicker, plus string and vocal vibrati were getting wider, which reduced or eliminated the audibility of the beats in the varying intervals. Since these nuances had disappeared in currently composed music, the pressure to adopt a tuning standard increased. The easiest standard was to equally temper all fifths, so that the Pythagorean comma was equally shared around the entire circle.
 

V. EQUAL TEMPERAMENT - A COMPROMISE WE'VE GOTTEN ATTUNED TO:

By now the calculations should be old hat. From above, the 12th root of the Pythagorean comma, 1.01364, is 1.00113. Each fifth will be narrowed to: 1.5/1.00113 = 1.4983, the closest we've gotten yet in any tempered tuning to a pure fifth. Let's see how close it really is: D = 440/1.4983 = 293.66. The 3rd partial of D is 293.66 x 3 = 880.99. The 2nd partial of A - 440 remains 880. So between D and A we have about one beat a second. But let's look at the price we've paid in loss of purity for the 3rds and 6ths: First, G = (293.66/1.4983) x 2 = 391.99. C = 391.99/1.4983 = 261.626, as given way above. E = (440x1.4983)/2 = 329.626. So, testing the 3rd interval C - E: The 5th partial of C is 261.626 x 5 = 1308.13, and the 4th partial of E is 329.626 x 4 = 1318.5. Thus we have nearly 10 beats per second in the middle C to E major-third (compared with 16.3 beats, given above, for a full Pythagorean 3rd). This interval has that width in all the major 3rds in the chromatic scale, of course (with the number of beats going up with increasing frequency). Let's check the major 6th C - A: the 5th partial of C is 1308.13, as above, and the 3rd partial of A is 440 x 3 = 1320. Hence the 6th is too wide by 1320 minus 1308.3 = about 12 beats per second.

We have gotten used to these non-pure intervals because that's all we've heard since we were old enough to take any note of keys and pitches. In any music where intervals are sounded separately, it makes no difference. Sounding them together to create a "harmony" has been where the issues have historically arisen. Equal temperament has given us practically pure fifths and fourths, wide major 3rds and 6ths, and narrow minor 3rds and 6ths. A 12-tone scale evolved naturally because it provides the opportunity to get closest to the lowest ideal harmonic ratios: 2/1, 3/2, 4/3, 5/4, 5/3, 6/5, and 8/5. Notice that the 7th harmonic is missing in these intervals. It's one that's missing in our use of harmonic intervals, and in our creation of scales. Anything higher than the 8th partial is usually too high and too weak to matter.

Another point is that all tempered tunings have equally spaced whole tones: i.e. the interval C - D is the same as D - E, is the same as F - G, G - A, and A -B. So that D, G, and A are all "mean" tones, as in the Meantone temperament. In fact the only tuning where this isn't so is the "Just" intonation scale we discussed first: D to C had the ratio 9/8, whereas E to D had the ratio 10/9. A perfect major 3rd is generated from a "major" tone (9/8) and a "minor" tone (10/9)--where the product is obviously 10/8 or 5/4. A perfect minor 3rd (e.g. E - G) is generated from a "semitone" (16/15) and a major tone (9/8), giving us our 18/15 or 6/5 interval.
 

VI. LET'S MAKE "CENTS" OF ALL THIS:

We've done a lot of math in the above analysis involving powers and roots, multiplication and division. Wouldn't it have been simpler to have dealt with logarithms instead of ratios, where we'd be multiplying and dividing--and adding and subtracting?  Answer: Yes of course it would. And it's already been done and is based on equal temperament. Let's define an octave as containing 1200 "cents." Each half tone in our equal-tempered chromatic scale would then be 100 cents wide. We could then have discussed all the above tunings and temperaments on the basis of so many cents wide and narrow. While the handling of the calculations using cents is easier (in that it avoids manipulating powers and roots), the mathematical reasoning or purpose behind the necessity of these procedures is, I think, a little more obscured in transferring frequency ratios to logarithms. It introduces an extra step, which may draw many readers away from this reasoning.  If you understand multiplication and division--as well as taking powers and roots, and have almost any of today's available pocket calculators--you can readily follow the procedures I've presented above.  And perhaps better understand why you had to.