Schoenberg's Harmonielehre

111
		F
71
		Bb
(7...n)0		E
		C	G	D

	etc.

G
^
C
v
F

Here the dependence of G on C,

may be considered like

• hanging by his hands from a beam and
• exerting his own force against the force of gravity.

He

• pulls on the beam just as
• gravity pulls him,

But the effect is that

• his force works against the force of gravity,

and so in this way

I will

• often speak of this characteristic and will
• draw a number of conclusions from it.

What is important for the moment is

• these tones are very closely related to one another, that
• they are next of kin.

G is the first overtone (after c) of C, and

c the first of F.

Such an overtone

• bears the closest similarity to the fundamental

• contributes most to
• the quality of the sound, to
• its euphony.

If

then

After all,

And thus it is explained how

These nearest relatives are

it represents the point of balance

This scale appears as

• the residue of the properties of the three factors, as
• a vertical projection, as
• addition:

(7...n)0	A	E	B
	F	C	G	D

[p 23]

Adding up the overtones (omitting repetitions) we get the seven tones of our scale. Here they are not yet arranged consecutively. But even the scalar order can be obtained if we assume that the further overtones are also in effect. And that assumption is in fact not optional; we must assume the presence of the other overtones. The ear could also have defined the relative pitch of the tones discovered by comparing them with taut strings, which of course become longer or shorter as the tone is lowered or raised. But the more distant overtones were also a dependable guide. Adding these we get the following:

Fundamental     Overtones
F            f . . . c . . f . a . c . (eb) f g a  bb* c etc. f  etc.
C                    c . . . g . . c .  e   . g . (bb) c  d e f etc.
G                            g . . . d  .   . g .  b   .  d . (f) g a b c d
----------------------------------------------------------------------------
                                        (eb)      (bb)
                                   c d  e   f g a  b   c  d e f   g a b c d										

our C-major scale.

* [note from Monzo: Carter's 1978 English translation contains a serious error at this point of this diagram: in the original 1911 German edition the 11th partial of F is shown as "b", which in German means our B-flat; Carter notates it as our "b". See the beginning of my analysis of Schoenberg's overtone-theory, Searching for Schönberg's Pantonality, for a detailed explanation. Below is an expanded version of Schönberg's diagram, giving the numerical values of the overtones which make plain the ratios involved in Schönberg's analysis.

The fundamentals are F = 4, C = 6, G = 9.
 
                                      d = 12*9 = 108
                                      c = 11*9 =  99
                                      b = 10*9 =  90
                                      a =  9*9 =  81
                                      g =  8*9 =  72
                    f = 11*6 = 66
  f = 16*4 = 64
                                     (f =  7*9 =  63)
                    e = 10*6 = 60
                    d =  9*6 = 54     d =  6*9 =  54
  c = 12*4 = 48     c =  8*6 = 48
                                      b =  5*9 =  45
  bb= 11*4 = 44
                   (bb=  7*6 = 42)
  a = 10*4 = 40
  g =  9*4 = 36     g =  6*6 = 36     g =  4*9 =  36
  f =  8*4 = 32
                    e =  5*6 = 30
 (eb=  7*4 = 28)
                                      d =  3*9 =  27
  c =  6*4 = 24     c =  4*6 = 24
  a =  5*4 = 20
                    g =  3*6 = 18     g =  2*9 =  18
  f =  4*4 = 16
  c =  3*4 = 12     c =  2*6 = 12
  f =  2*4 =  8


       (eb)            (bb)
 c   d   e   f   g   a   b   c   d   e   f   g   a   b   c   d
                       [44]            [64]
       (28)            (42)            [66]
24  27  30  32  36  40  45  48  54  60  63  72  81  90  99 108



The partial-numbers are also given for the resulting scale
at the bottom of the diagram, showing that 7th/F (= eb-28)
is weaker than 5th/C (= e-30), and 7th/C (= bb-42) is weaker
than 5th/G (= b-45).

Also note that 11th/F (= b-44), 16th/F (= f-64) and 11th/C
(= f-66) are all weaker still, thus I have included them in
square brackets.  These overtones are not even mentioned by
Schoenberg.

Below is a lattice diagram illustrating all of the pitches and relationships in Schönberg's diagram.]

An interesting by-product appears here:

The two tones e and b appear in the first octave, but e is challenged by eb, b by bb. That explains why it was once questioned whether the third was a consonance, and shows why b [bb] and h [b] appear in the German tone alphabet.

[Carter, p 25:]

[note from Monzo: Anyone familiar with historical tuning theory knows that this explanation is silly. Schoenberg is speculating wildly here, and revealing his lack of historical knowledge; but at the same time, his imaginative theorizing is impressive. It is also interesting to me that the above-mentioned error by Carter in translating the German "b" into English "b" instead of "b-flat" occured right before the point where Schoenberg's text comments on exactly this notational difference.]

Whether the pioneers of music arrived at this scale through intuition or through inference we cannot judge. It is unimportant anyway. We may nevertheless take issue with theorists who set up complicated doctrines; for, contrary to those doctrines, we have to give the discoverers credit not only for instinct, but for the ability to reason as well. It is then not at all impossigle that the right thing here was discovered by reason alone, that therefore credit is due, no to the ear alone, but in part to inference. We are not the first who can think!

[p 24]

The discovery of our scale was a stroke of luck in the development of our music, not only with regard to its success, but also in the sense that we could just as well have found a different scale, as did for example the Arabs, the Chinese and Japanese, or the gypsies. That their music has not evolved to such heights as our does not necessarily follow from their imperfect scales, but can also have to do with their imperfect instruments or with some other circumstance which cannot be investigated here. Moreover, it is not to our scale alone that we owe the evolution of our music. And above all; this scale is not the last word, the ultimate goal of music, but rather a provisional stopping place. The overtone series, which led the ear to it, still contains many problems that will have to be faced. And if for the time being we still manage to escape those problems, it is due to little else than a compromise between the natural intervals and our inability to use them -- that compromise which we call the tempered system, which amounts to an indefinitely extended truce. This reduction of the natural relations to manageable ones cannot permanently impede the evolution of music' and the ear will have to attack the problems, because it is so disposed. Then our scale will be transformed into a higher order, as the church modes were transformed into major and minor modes. Whether there will then be quarter tones, eighth, third, or (as Busoni thinks) sixth tones, or whether we will move directly to a 53-tone scale that Dr. Robert Neumann has calculated *), we cannot foretell. Perhaps this new division of the octave will even be untempered and will not have much left over in common with our scale. However that may be, attempts to compose in quarter or third tones, as are being undertaken here and there, seem senseless, as long as there are too few instruments available that can play them. Probably, whenever the ear and imagination

[Carter, p 26:]

have matured enough for such music, the scale and the instruments will all at once be available. It is certain that this movement is now afoot, certain that it will lead to something.

*) Concerning his calculations, Dr. Robert Neumann has informed me as follows (in part):

"As more and more of the possible combinations of our twelve tempered scale degrees are felt and used as harmony,

[p 25]

the stock of unused possibilities is being gradually exhausted; and the continuing need for new harmonic (and melodic) vocabulary will finally break through the boundaries of the system. Then new systems of temperament with smaller intervals might come about, later perhaps even complete independence and freedom in the use of all conceivable intervals, all conceivable frequencies and combinations. Now the division of the octave into 53 equal parts would be an example of a new temperament that could come under practical consideration: specifically, whenever music has advanced so far that there is need for a system with about four times as many tones as we now have; and when there is need (or more correctly, when there is again need) for the purest possible intonation of the basic intervals -- those determined by the first overtones -- but at the same time, no wish to do without the convenience of a temperament. As intermediate stages between the 12 and the 53-part division of the octave only multiples of 12 could be considered, thus, division into 24, 36, and 48 parts; for with any other number of parts there would be no sufficiently pure fifth.

[from Monzo: This is not correct. 29edo and 41edo, while not as good as 53edo, both offer better approximations to the 3:2 "5th" than 12edo, and 41edo also gives a better approximation to the 5:4 "major 3rd" while the 29edo "3rd" is only very slightly worse than 12edo. See my analysis of 29 and 41edo compared to 12edo.]

The first step, anyway, is to divide every semitone into two equal parts, whereby the octave would be divided into 24 parts. Dividing again by 2, one could then obtain 48 parts. ¹/₄₈th of an octave and ¹/₅₃rd are almost the same size; the 53-part temperament, however, is preferable to the 48-part because of the much greater purity of the consonances. The 48-part system has of course the same basic intervals as appear in our 12-part temperament, from which it is derived.

Now our tempered fifth is indeed already pure, but that of the 53-part system is about 28²/₃ times purer; the third of the 53-part system is about 9 times purer than ours and is even purer than our fifth, which is only 7 times purer than the third."

---

[from Monzo:] Below is an analysis of Neumann's statements:

The average musician will laugh at such speculations and will not be inclined to see their point. It is clear that, just as the overtones led to the 12-part division of the simplest consonance, the octave, so they will eventually bring about the further differentiation of this interval. To future generations music like ours will seem incomplete, since it has not yet fully exploited everything latent in sound, just as a sort of music that did not yet differentiate within the octave would seem incomplete to us. Or, to cite an analogy — which one has only to think through completely to see how very relevant it is: The sound of our music will at that time seem to have no depth, no perspective, just as Japanese painting, for example, affects us as primitive compared with our own, because without perspective it lacks depth.

[p 26]

It may be that here again many digressions and errors will have to be overcome; perhaps these, too, will lead to exaggerations or to the delusion that now the ultimate, the immutable has been found. Perhaps here, once again, laws and scales will be erected and accorded an aesthetic timelessness. To the man of vision, even that will not be the end. He recognizes that any material can be suitable for art -- if it is well enough defined that one can shape it in accordance with its supposed nature, yet no so well defined that the imagination has no unexplored territory left in which to roam, in which to establish mystical connection with th universe. And since will can still hope that the world will long continue to be a riddle to our intelligence, we can say in spite of all Beckmessers that the end of art is not yet at hand.

2002.09.14
2002.01.23
1999.06.05

For many more diagrams and explanations of historical tunings, see my book. If you don't understand my theory or the terms I've used, start here or try some definitions.