WHAT IS A MEAN TONE?

by

Ivor Darreg

My Webster's Collegiate Dictionary (older edition) has at least 18 meanings for the word tone. It also has about 18 meanings for the word mean! Under these circumstances, it is no wonder that everybody gets confused. Now couple the two words together and you have a real mess. Then use the compound term, with or without hyphen, or "solid" as the printers say, and nobody agrees on that either: mean tone mean-tone meantone all are found in books.

Much the same situation about these words exists in foreign languages. It all started by the Ancient Greeks having too many meanings for their word from which our tone was derived. It's had 2000 years to get worse!

However, once we decide to explore beyond 12 equally-spaced pitches per octave, we have to deal with this confusing term. During the 17th and 18th centuries, many keyboard instruments, especially pipe organs, were tuned in some kind of meantone temperament, and with the current revival of the harpsichord, tuners are just beginning to be asked for it. This is one of the few places where the interests of music history buffs and contemporary composers coincide, and if we play our cards right, we could effect an agreement.

The phrase meantone temperament can mean a range of systems, or a family of tunings; but ordinarily it is understood to mean the 1/4-comma temperament, so let us describe that first. Unlike other writers on this subject, my interest is the future rather than the past, so what it will mean is more important here than what it may have meant in 1700 or so. In particular, I strongly disapprove attempts to use meantone temperament with only 12 pitches per octave. The antiquarians and traditionalists will hate me to pieces for this, but let 'em scream their heads off! Most of the prejudice against meantone temperament stems from that very restriction to 12 tones/octave, which has crippled musical progress. If all you want is nostalgia, seek it elsewhere.

Our ordinary nomenclature for musical pitches is based on an infinite series of fifths, usually reckoned up and down, i.e. sharp- and flat-wards, from C as zero or starting point.

If the fifths are just, i.e. exactly 3:2 ratio, then this series extends forever in both directions. The customary distortion of this state of affairs is the cheapest one that is musically of any value or use: viz. to remove a six-hundredth of an octave from each fifth, so that now the twelfth fifth up, B#, or the twelfth fifth down, Dbb, will exactly coincide with the C we started with. This is the 12-tone equal temperament in common use today. This creates a circle of twelve fifths. It also means that whereas an infinite series of fifths fits our conventional notation and nomenclature perfectly, now with only 12 pitches we have lots of redundant names for the same pitch:

These redundant names, such as A### B# C Dbb Ebbbb and so on forever, are usually called enharmonics, which is a very serious deadly-confusing misnomer. We need "enharmonic" to describe the small intervals used by the Ancient Greeks. Why not call them equivalents or synonyms in systems like 12, 17, 19, 31 etc.?

Back to that infinite line of fifths which does not close a circle at all. Don't criticize me for infinity I realize just as you do, that only the fifths from about Abb to Gx will appear in any ordinary music -- indeed some people start "respelling" with those equivalents-in-2-tone before they get to double sharps or flats. So let's focus down now on a few fifths: say Ab Eb Bb F C GD A E G F# C# G# D#.

If they were just, G# would be about 1/8 tone D1 up there. The interval between C and D is sharper than Ab. That's fine for melody. Violinists use this kind of a scale very often. Now let's sound C and E of this series together. It will not be a just major third with 5:4 ratio, but a Pythagorean or sharp major third of 81:64 rati&and quite harsh in normal qualities of tone. Otherwise stated, Pythagorean intonation arid its close imitation, 12-equal, favor melody and spoil harmony, sacrificing major and minor thirds for the sake of good fifths. The worshipper of 12-equal says: "Look at our excellent fourths and fifths! Only 1/600 of an octave out of tune and nobody can hear that in any real music." You are never allowed to learn that there are such things as smooth restful just major thirds, with ratio 5:4.

Suppose now that we take that line of fifths, or a small portion of it, and set the new notes that do not belong to it, major thirds of true 5:4 ratio away from it, forming other lines of fifths:

The sub- and superscript numerals there are approximately ninths of a tone, and usually called commas above and below the pitches with the same letter-name in the next line of fifths, and this comma is that of Didymus, not of Pythagoras, so don't get them mixed up.

Ouch! A double infinity! A plane surface of tones instead of just a thin line. That's too many. Such is the problem raised by just intonation. If we want pure thirds and pure fifths we demand an impossible number of tones per octave for a keyboard instrument, beyond a narrow range of modulations. But it sounds so much more smooth and soothing and restful than 12-tone-equal. (Music therapy, since it is never done in just intonation, has never been tried. The poor patients get only restless 12-tone-equal, which disturbs them all the more.)

12-tone sacrificed thirds for the sake of fifths. What if we take the converse position? What if we sacrifice fifths for the sake of thirds? How about that?

Meantone temperament to the rescue. Look at Fig. 3 again: look at the pair of tones called D and D1 up there. The interval between C and D is usually called a whole-tone. Or a major second. In just intonation, there are at least two kinds of whole-tones. C:D1 has the ratio 10:9, while C:D has the ratio 9:8. If we now look at the other part of a major third, in this case D1 :E1 and D:E1, we get the same two kinds of whole-tones; i.e. it takes, in a justly-intoned scale, one of the narrow and one of the wide whole-tones to make up the major third. To play, even in the scale of C major, we need two kinds of D, a comma 81:80 apart. Otherwise the chord D F A will be horribly out of tune.

Only recently has it been feasible and affordable to build electronic organs or prograrn computers to give us enough just pitches to produce worthwhile music, free to modulate over a range of keys. It is impossible and too expensive if it were possible, to build pianos with that capability, and nobody would have the patience to tune them. This is why meantone temperament as well as just intonation have been streng verboten for some two centuries. Musical instrument manufacturers know only two clefs, which look like this: $ and

The question of how closely "free "instruments, such as voice, violin, cello, etc. approach just intonation is too big a subject to digress here. It can be taken up in other articles, or better by recordings.

Other than resigning ourselves to the ordinary 12 equally spaced pitches, how can we reduce the number of notes that just intonation requires. Many alternatives exist, but here we are concerned with the Meantone Family of Temperaments.

The interval 81:80, the comma of about 1/9 tone, which appears in even a small just-intonation array, is rather small for a melodic interval. If we choose the perfect major thirds of 5:4 ratio, and elect to sacrifice the fifths for their sake, then we take the line of fifths C G D A E (open strings of violin and viola, for example, give those fifths) and rob each fifth of 1/4 comma, say 1/37 ordinary whole-tone. 1/222 octave is down to the microscopic, surely. So two of these shrunk fifths will give a new kind of whole-tone, C to D, and since it is halfway between the two just whole-tones in size, it is called a MEAN tone. This is the so-called Geometric mean, not the average of arithmetic. But as far as musicians need be concerned, it is half of a just major third of 5:4 ratio. So a meantone-tempered instrument will have perfect octaves and just major thirds, and some other intervals will be distorted to make this come out right. Now our Web of Fifths and Thirds in Fig. 3 shrinks back to a single line of fifths in Fig. 1, although now these are meantone-tempered fifths, and will beat if sustained on an organ in a normal tone-quality. Their error is roughly 2 1/2 times that of the 12-tone fifths, in the same direction, i.e. too flat by a tiny amount.

We get rid of those commas completely by distributing their error among the notes being tuned. In so doing, we gain restfulness and serenity, but lose the punch and zing in melody. (Well, that's not entirely so.) Instead of a doubly-infinite plane of pitches, we have a singly-infinite line of them. Our name-system and staff accommodate them perfectly. No problem. The Pythagorean difference between C-- and Db is much smaller than the mean-tone difference between those pitches. So now, in meantone temperament the difference is big enough to be a usable melodic interval. About a fifth-tone.

It cannot be ignored. The circle of fifths does not exist in meantone. It fails to close at the twelfth fifth by this much larger amount. C-sharp is not the same as D-flat; they are now independent and the term Augmented Seventh becomes a new interval instead of 12-tone utter nonsense.

Before we go on: in the line of just or pure fifths, the Pythagorean system, B# is sharper than C by 1/8 tone; but in meantone temperament, B# is FLATTER than C by 1/5 tone.

I know it's confusing, but that's the truth. Both systems fit present notation perfectly, and that makes the confusion worse. Don't let this purely visual problem of names for notes keep you from enjoying the non-12 systems. I never composed for eyes. Maybe other people do.

Now let's explore other meantone systems. It is possible to use other fractions of a comma as the amount by which the fifth is flattened. For practical playing of worthwhile music, perhaps anything from 1/3 comma to a small fraction (theoretically, flattening by about one-tenth comma will get us close to the ordinary 12-tone system). "Meantone" used without any qualifications is generally understood to he the 1/4-comma system just described. The 1/3-comma system has the minor third just instead of the major third, while the 1/5-comma meantone has the major seventh just instead of the major third. The 1/6-comma system would have its augmented fourth just, and so on to less-important intervals that would be just, so we don't have to bother for the moment. <