Non-Twelve on the Cello

The bowed instruments of the violin family have no frets. Physically, that is. Psychologically, the frets have been installed in the players' minds by indoctrination and what now would be called Programming by the computer-people.

These psychological frets are: in the 12-tone equal temperament and are part of the Pianolatry Religion which came to full flower in the 19th century and is now slowly ebbing away as it spends its force and new scales and new instruments gradually make the scene. In. most cases, the psychological frets are about as hard to pry out as brass frets would be on a real guitar fingerboard.

We take up the cello first because in the lower positions there is more room on the fingerboard than there is on the violin and viola -- but most of our remarks will apply anyhow to the treble and alto instruments, because that more compressed fingering scheme is used in the higher register of the cello. Contrabass players will be able to use these ideas very well.

Our first task is to clear the air, so to speak -- there is so much misinformation, and so much suppression or at least ignoring of important facts, that we cannot just barge into our subject without some preliminaries.

The general run of l)ooks on music treat the 12-tone equal temperament as though it had always been there since there was a Universe, created along with atoms and molecules and light and sound waves. Of course this is not true, but we have to make it clear that keyboards came after other kinds of instruments, not before them; and frets weren't always placed 12 to the octave, and permanently fastened in and set to the logarithmic scale known as twelve-tone-equal. Viol frets are adjustable!

The cello is tuned in fifths, like the violin and viola, and this presupposes a theoretical tuning system. This happens to be what some authorities call Pythagorean Intonation, based on an infinite string of perfect fifths, thus:

That is, there is, in correct Pythagorean intonation, no end to the number of fifths before C and no end to those after it; they come from infinity and go to infinity, forever. They never recur, never get back to the same note because no power of 3 can ever equal a power of 2. Of course, points are reached again and again where the failure to coincide is small, and later on where it is inaudible. Best-known such near coincidence is at the twelfth fifth Dbb vis-a-vis C and then B#. This is called a Pythagorean comma, nothing to do with the 81:80 comma of Just Intonation; the Pythagorean comma has the ratio 53144l:524288 -- there you can see a power of 3 not equalling a power of 2. It is 23.5 cents or about one-eighth of a whole-step. This near-coincidence suggests distorting all the fifths by flattening them about 1/600 octave adn then thecircle closes and C = C# in the 12-tone temperament.

So, far a considerable amount of purely melodic music, which does not involve remote modulations nor fuss over subtle harmonic relations, and this will be typical of cello or violin solo literature, the discrepancy of the Pythagorean B# being sharper than the starting piont C by an eighth of a tone will never come up. With the piano, the tones die away and mellowed down by heavily-felted hammers so that 12-tone equal-temperament can be used on pianos an pianos lock it in place by sounding the way they and pianos lock it in place by sounding the way they do. Textbook writers focus on this almost perfect fifth and carefully adn studiously ignore the damage done to the real major third of ratio 5:4.

That is, they ignore Just Itonation and its imitations. If the series of perfect fifths is carried to 41 or 53 or certain larger numbers, better coincidences occur which reduce the error of the distorted fifths to close the line to a circle, and as an extra bonus, do not distort the true major third of 5:4 so much either.

Our ordinary naming-system, used above for the chain of fifths, conceals important facts, indeed, it is why the late Harry Partch would not use it at all, substituting a formidable array of numbers and fractions. In this article I am addressing cellists and people writing for or dealing with the cello, so if I went all-numbers like the Phone Company, I would only confuse you.

What our naming system (or the Latin countries' use of do re mi for our CDE) does, is to mix up the just and Pythagorean major thirds, who ratios are 5:4 and 81:64 repsectively, separated by a comma 81:80 which is slightly smaller than the Pythagorean comma discussed already. This causes endless confusion among musicians about 12-tone equal temperament, meantone temperament, Pythagorean, and Just Intonation. We will use ALexander J. Ellis's scheme here, and call the true major third above C, E1 (which can be read E-sub-one or E-one-comma-down) the 5:4 to C, and the Pythagorean major third which is 81:64 to C, just plain E without a number.

Now, some things can be set out to try on the cello. The series of fifths carried upward would look like this in standard staff-notation:

A five-string cello would be so tuned. Now the same notation woul dbe used for E1 as for E, since standard naming and standard staff notation ignores the small interval 81:80. But in truth and in fact, the two pitches are different and the difference is big enough to hear, and in harmony a mistake would be clearly audible. On the cello, the E1 which is the fifth harmonic of the open C string is a comma flatter than the plain unmarked E which is the fifth above the open A string or the E an octave higher which is the third harmonic of the open A string.

If your memory for pitch is excellent you may be able to hear this difference; if not, there are easier things to hear which we will give now:

Make sure the fifths C G and G D and D A are accurate and beatless.

Then violate the Establishment Rules and put your bow upside-down UNDER the C and A strings between them and the body of the instrumnet, and you will be able to bow the C and A strings together without the G and D sounding since they will now be above your bow-hair level. This will be a sharp-sounding octave-plus-major-sixth:

Then, to remove the octave, touch the C string as the mid-point to get the Tenor C an octave higher, and keep on bowing from underneath:

This Pythagorean major sixth of ratio 27:16 is even harsher than the 12-tone tempered major sixth on the organ and piano. Now if you are good at double-stops, play the smae tenor C and a comma-flatter A (flatter than the open A string by one-ninth of a whole-tone) as a double-stop on the G and D strings: (returning to normal orthodox above-board bowing of course for this)

I put a little arrow there to show the flatter A, which is a true 5:3 just major sixth to C.

Now, something else which does not require bowing between strings and body, but just normal technique. Check the fifths again to be sure they are accurate and beatless. Then play this with the open G and E on the D-string, and probably you will have to move your finger a bit to flatten the E until it is a smooth beatless 5:3 just major sixth:

If you can sto the beats ( it is difficult at first because of Pythagorean and 12-tone-equal-temperament indoctrination) and enjoy the just major sixth, the Real Thing, then check that E1 which you will have them attained, against the open A:

and that fourth which is a just "wolf" will be out-of-tune enough to make your Aesthetic Sense itch. (Ratio 27:20)

Then, move your finger to sharpen the E until it is a perfect 1:3 fourth to the open A string and then sustaining it move the bow over to play a Pythagorean sharp icky stretched major sixth 27:16:

So there are two distinct pitches in just intonation both commonly subsumed under the name E, or some other name such as A in one example above.

Of course I realize that nobody can do accurate just intonation with commas and play Allegro Vivace or Presto or the Flight of the Bumblebee or somebody's Tarantella. But at Andante or Adagio, the comma is audible. It matters in a cello-ensemble or when overdubbing to 4 parts.

There are other tiny intervals we could discuss, but they are releveant to the solo cellist, so let's bypass them for other matters which are useful and interesting. (No point, for example, my discussing the Pythagorean comma because it would be impossible for even expert cellists to take a series of fifths all the way from C to B#, so we can leave this to electronic instruments etc. No point either fussing over the skhisma which is 1/614 of an octave and requires special techniques to hear it.)

If you want to pursue accurate just intonation beyond what the cello can realistically do, special electronic organs etc. will be available later on for this purpose.

Since Partch, the well-known non-twelver, has made many people aware of the harmonic series, let's take that up next. The G string is the best one on the cello to begin with, then you can try it on the others. The modern metal strings are more unifrom than old-fashioned gut strings, and this matters very much for a long series of harmonics. Here is the harmonic series up to the 18th harmonic on the G string:

I use various special marks to show deviation from customary pitches. Note particularly that the 7th harmonic is about 1/6 tone flat of what we generally are accustomed to: while the eleventh harmonic lies almost on the quartertone between two orthdox semitones. C-semisharp in the case of the G string.

It is easy to elicit high harmonics on the A string but they get so shrill and lie beyond our accustomed melodic range. So in demonstrating the harmonic series for others, better stick to G and D strings.

The 7th and 11th harmonics involve intervals not in the 12-tone equal temperaments, so in this way you can become familiar with some things the piano and organ cannot do but the cello can do and become aware of how shamelessly we have been wasting our musical resources by neglecting these exotic tervals. Then you can learn to play as double stops the subminor or harmonic sevenths, ratio 7:1. Start with:

Approximately one-sixth-tone flatter than orthodox minor sevenths. Contrast them by alternating them and comparing them; they have different flavors. Why are you never allowed to hear them when studying the cello? Your cello was not made in a piano factory nor tuned by a piano tuner! This is only the beginnning of a long infatuation with overtones. You can go further if you can meet other non-twelve musicians.

Now let's expore Quartertones and similar intervals.

The quartertone system, taken strictly as being the 24-tone equal temperament, seems to double the harmonic resources of 12-tone and to contain it, so many people have experimented by tuning two pianos a quartertone apart. This has been pretty much a a dead end -- the melodic resources of 24-tone are what really matter and thus are more the province of solo melodic instruments such as the violin family. Besides that, the cello has neither keyboard nor frets, and more fruitful melodic uses can be made of quartertones if you are free to bend them at will. No more point being tied down to two pianos than to one!

Furthermore, there is another temperament of 22 tones per octave which can be approached by dealing with regular quartertones first.

There are dozens of ways to notate and name quartertones. The one I use, the Couper system, is as follows:

double-flat sesquiflat flat semiflat natural semisharp

sharp sesquisharp double-sharp.

Investigate this and other systems and choose whatever you like best. Julian Carrillo used numbers from 0 through 23, to get rid of names.

The eleventh, twenty-second, and thirty-third harmonics generate intervals very close to quartertones, so that is sufficient theoretical acoustic justification for them. But there is something else: the Enharmonic Genus of the Ancient Greeks, embodied in a tetrachord pattern as follows: (they reckoned in descending fashion)

Given those two motivators, plus the continued use of similiar intervals in the Near East, you should now have enough ammunition to confront all your opponents with and be able to sail on confidently to new unehard-of tonal worlds to conquer.

There is a vast territory to explore which would take a whole Cello Method book in itself. One help is to play quartertones ont he A string using the D string open as a drone. Then play quartertones on the D string, using A and G strings in alternation as drones. That will help you become accustomed to the new intervals.

Now let's get on to other possibilities, which we have to skim through due to lack of space.

Seventy-five years ago, Ferruccio Busoni proposed what he called "the tripartite tone" or in the usual vernacular, the third-tone system. This means 18 equally-spaced pitches in each octave. Unfortunately, this 18-tone scale has no real fifths nor fourths. Thus it is not suited to the bowed instruments because the fifths we tune on them would bet between its standard pitches. Haba and Carrillo wrote in third-tones. Busoni's thought was to embed his 18 pitches in 36 sixth-tones, then to accompany 18-tone melodies with 12-tone harmonies -- you can see he was a little timid about his discovery.

Since a fifthless scale runs counter to the genius of the cello, I wouldn't even mention it if that was all one could do. There are two excellent escapes from that dilemma now available: the 17 and 19 tone equal temperaments. Both of these do have fifths. In 17-tone the fifths are tuned slightly sharp so that the circle closes at Ax and E# is the same note as G-flat. It is even more brilliant than Pythagorean intonation and so ideally suited tothe solo cellist. Many brilliant melodies exist in the literature which would be enhanced by doing them in 17.

In 19-tone, the fifths are tuned noticeably flat so that th3e circle closes at Bx rather than B-with-a-single-sharp. Then E-sharp is the same note as F-flat and hte contrast between consonances and dissonances is much greater than it is in 12-tone temperament, giving new harmonic as well samelodic resources.

Now it is rather difficult for the unaided cellist to learn and use these new scales, so the best course is to team up with a guitarist who has a guitar refretted to 17 or 19. Then you can follow along easily enough and soon will be expert in either or both scales.

The advantage of 17 and 19 is that both scales make efficient use of conventional notation and can be used, with vastly different mood effect, to play existing sheet music without requiring any new signs. Such pairs as G-sharp and A-flat become different pitches and thus the written distinction gets usefully HEARD.

Beyond 17, 19 and 22 already mentioned, lies 31 notes per octave -- again, better to learn it from another instrument built for that pitch -- a 31-tone guitar is now possible, later on 31-tone organs will be more available than the very few now in existence.

This by no means exhausts the unheard potentials of teh cello. But without producing a set of tapes or disk records, we cannot profitably discuss nor present other scales. To mention someo f them: 5-tone and 7-tone, found in many exotic cultures; 10 and 14, well worth hearing; 9, 11, and 13, the really rebellious non-twelves; 15 which is 3 x 5; 8 which is a subset of 24; and numerous unequal scales slowly emerging from the theorists' ivory towers. Most of these require some kind of auxiliary instruments to learn, but that problem will be solved soon enough. The point we wish to leave you with here, is that the surface has barely been scratched, and the cello must not remain a chained slave of the piano keyboard any longer.