Acoustic methods of work in relation to systematic comparative musicology
Thorvald
Kornerup
lit38535
ACOUSTIC METHODS OF WORK
IN RELATION TO
SYSTEMATIC COMPARATIVE MUSICOLOGY
INCLUDING SOME ACOUSTIC TABLES
BY
THORVALD KORNERUP
COPENHAGEN F.
COPENHAGEN MCMXXXIV
PRINTED BY J. JORGENSEN & CO.
Translation from the Danish by Marie Baruël.
Copyright 1934 by Thorvald Kornerup, Copenhagen F., Denmark. All rights of translation reserved by the author
Published by the author.
Copies are obtainable, post free, on application in writing to the author, Th. Kornerup, Holger Danskesvej 82, Copenhagen F., Denmark, either by Post Office Order for 4 Danish Kroner for each copy, or against cash on delivery.
In Memoriam
FREDERIK LASSEN LANDORPH
3rd February, I860—5th March, 1923
Chief-administrator for
The Amsterdam Deli Companie, Sumatra 1896—99
4
Th. Kornerup:
PREFACE
The aim of this treatise is:
1. To propose a common international terminology of music; thus, for the white keys of the piano the names
and for the black keys
1 c d e f g a b
1 bis fes eis ces
J cis dis fis gis ais
1 des es ges as bes J
and for double ffff and [?[? the terminations isis and eses; for syntonic »Comma-tones« the signs »+« and »—« placed after the name of the tone to indicate increase
81
or diminution of a Comma = ^., as, for instance, in Indian music: a = 884cents and
80 ’
a -f = 906 cents, or 22 cents more.
Only Ellis’s 1200 cents are used (never 1000 millioctaves, or 100 Octav-Zentimeter oz, or 600 centitones), as the number 1200 can be divided evenly by 3, 4X4 and 5X5 and their multiples (as 720 particles can be divided on a string by 3X3 and 4X4 and 5).
Below are given some further proposals concerning terminology; for instance, the expression »triple chord« is used for »triad« in opposition to the »Greek Triad« (3 neighbour-tones: des—, des and d+), and »quadruple chord« for »chord of the Seventh«; »Tonic« or starting-point for »key-note«; »Phrygian-Doric« for the Aèolian scale-type, and »Lydian-Phrygian« for the Ionic or Iastic scale-type, according to the succssion of Tetrachords, Scheme 17; in other words, a difference is made between »secondary partition in particles« in the formula X, and »tertiary division« in the formula Y; further: quartary and quintary instead quaternary, etc.
- The Greek spelling of names is also used, e. g. Arkytas, Plutarchos, Ptolemaios etc., and the spelling of the terms: Trichord, Tetrachord, Tritonos, Medium represents international spelling; and capital letters for »the tone Octave«, but small letters for »the interval octave«.
2. To draw attention to 3 principles which have lately been set forth concerning systematic comparative Musicology:
a) that in the development of tone-systems from the earliest times to the present day there has, without a doubt, been a much greater stability than has hitherto been supposed, viz. Professor v. Hornbostel’s demonstration of the stability in the length of the flute (and the gold measure) from 2000 up to 3000 B. C. till far down through the ages (Note 1). This stability we here transfer to the proportion between the pitch of the tones;
Acoustic Methods of Work.
5
b) that in mentioning the ancient tone-systems it is only possible within very narrow limits to rely upon abstract Fifth-steps, but that one can rely in the highest degree on the concrete shortening of a string (secondary »partition into particles« formula X) cf. v. Hornbostel’s remark: »One of the fundamentals of the Indian tone-system, the partition of a string, is evidently primitive« (Note 2); and v. Hornbostel’s and R. Lachmann’s observation in 1933: »Breloer sought to deduce Bharata’s (Indian) system solely from this principle (the pure Fifths); he arrived at the 22 Sruti through a chain of pure Fifths. But it is not clear why the advance of the Fifths should stop short at 22 Sruti.« (Note 3).
c) that the principle of Relativity can be recognized also in relation to the tone-intervals, steps, cf. v. Hornbostel’s declaration: »The investigations of recent years have .... led to surprising consequences. As is known, melodies can be displaced in their pitch as optical figures in space, without changing their shape; they may also be enlarged or diminished as figures in space without alteration of their shape if only the relative interval-distances are kept. In this way the structure-building intervals, as the Fifth and the Octave, also keep their function in the new formation.« (Note 4).
Indeed these 3 principles can also operatein a wider sense, as the above-mentioned authors also point out: »Thus the comparison of musical styles as well as the comparison of other musical functions — tones and instruments — might prove a valuable aid in the investigation of the history of civilization«. (Note 4).
For example: In his investigation of the length of the flute (and of the gold-measure) Professor v. Hornbostel has built up a hypothesis of an ancient culture-stream from China across the Pacific Ocean to Central America.
3. To add to these 3 principles the under-mentioned system of acoustic methods of work:
Chapter I: 5 kinds of exact interval-calculation, and:
Chapter II: 7 hypothetical principal-rules for the origin (genesis) and structure of tone-systems.
4. To make plain the advantage of having an objective means of valuation for ancient as well as modern or future tone-systems, which, mathematically speaking, can only occur by means of the authoritative golden system formed by the golden cut
(super division) of the octave according to the formula : — s® -— or
co 1 — CO
’ ^°o - = 03 (Omega),
Formula: CO2 -j- CO =1
in decimals: 0,381.966 + 0,618.034 = 1
Luca Pacioli’s »Divina proportione«, Kepler’s »sectio divina« and »gemma« i. e. precious stone (Note 13), the everlasting fundamental formula for universal power of adaptation, — here the formula for universal sense of harmony.
6
Th. Kornerup:
CHAPTER I
FIVE KINDS OF EXACT INTERVAL-CALCULATION.
Scheme 1 shows five methods of calculation of the intervals:
Group A. Quantitative string-partitions (Nos. 1-2), the musical faculty of vision (musikalisches Sehen).
1. Primary (Flageolet)-string-partition, i. e., Partial- or Flageolet-tones are created by partition of a string (or flute) into several equal parts, which all vibrate at the same time, e. g. Natural triple chord c e g c’, the Partial-tones Nos. 4, 5, 6 and 8.
The difference in the first grade between the cents of the adjacent Partial-tones approximates to the golden tones (Chapter IV).
The difference in the second grade approximates to the golden Molecules:
Partial-tone Cents 1st grade Golden tones: 2nd grade Golden molecules :
No. 3 g 701.955
498.05 503.79 f
— 4 c’ 0. 111.7 118.9 des
386.31 384.86 e
— 5 e’ 386.314 70.7 73.5 cis
315.64 311.36 es
~ 6 g’ 701.955 48.8 45.4 deses
266.87 265.93 dis
— 7 ais’ 968.825
2. Secondary (ornamental, decorative) string-partition: only part of the string vibrates; we divide the string into 720 particles = 3X3X4X4X5.
The tone-fractions are indicated in the following »stable formula X :
the whole string as the common numerator) the vibrating part as the denominator f
never inverted.
Plutarchos’ median point, the secondarily halved octave, is the Fourth, the
Mese, Indian Mahijama: = y in formula X: with the common numerator 4.
The octave secondarily 3-parted is the small Third es = ~j = -~- and the Fifth
g =. — A Plutarchos’ Paramese, Indian Pancama, (»para« i. e. »beyond« the Mese)
480 2
in formula X:
6
6. 5. 4. 3.
with the common numerator 6.
Tautological: the Fifth secondarily halved is the tone es: ^
Acoustic Methods of Work.
7
Group B. Qualitative divisions, (Nos. 3-5).
3. Tertiary (arithmetic) division of the decimals of the tone-fractions (as decimal-fractions), or division of the difference between the vibration-numbers, also indicated in the following »stable formula Y«.
3
The octave 2,0 tertiarily halved is just the Fifth — = 1,5 with the common deno-
2 3 4
minator 1, or in formula Y: -J—r:—1 with the common denominator 2, see Scheme 19.
J
4 o 6
The Fifth = 1,50 tertiarily halved is e, either 1,25 (0.50 halved) or in Y: —^—1 with the common denominator 4.
The number of vibrations (by Dr. Illo Peters called objective tone pitch) increases proportionately to the given decimal-fractions.
The difference between the decimal-fractions (or vibration-numbers) of 2 adjacent
3 5 1
Partial-tones is constant 0,25; tt minus — = —; or g’ minus e’ = C, compare Rule 5, d.
J 4 4
4. Quartary (geometrical, neutral) logarithm-division, or division (and transposing) of tone-intervals, steps; subjective tone pitch.
The product of 2 tone-fractions is consistent with the sum of their logarithms;
3 3 9
for example: 2 Fifths-steps • — = — agree with f2 X 176.0913 = 352.1826 as logarithm,
12 X 701.9550 = 1.403.9100 as cents, minus 1200 = 203.91 = d +
Professor Joseph Yasser, New York, remarks »that the New York Public Library possesses a very rare and original manuscript: The Geometrical Scale in Musick, or Gam-Ut reduced to Geometrical proportions, and according to the statement of its author (whose name Gaudy is merely guessed from indirect indication), was written some time before the year 1705« (Note 5), — he was thus the pioneer before the Swiss mathematician Leonhardt Euler (1707—83), who in the year 1729 limited the common logarithms of tone-fractions to one octave (2) by dividing them by the logarithm of 2 = 0.30103.
In order to facilitate the transition to Ellis’ cents as an international method of counting, an easy and practical method of logarithm-reckoning was introduced by me in 1929 by using the Constant K, i. e. (Scheme 22):
Log. 1,2=............... 0.079.1812
minus log. of (log. 2 = 0.30103) = minus 0.478.6098 -f 1 Difference K = 0.600.5714
Thus an addition and a subtraction are merged into one single addition = K.
5. Quintary structure is used in connection with the stretching of a string, see Scheme 1 above, (j/2)2 = 2 (or, the octave: 22 = 4).
By pressing the string down on the finger-board (e. g. on a violoncello) the string will be somewhat tightened, and, accordingly, the pitch of the tone will also be somewhat raised (quintarily).
Th. Kornerup:
Résumé: a) The methods of work set forth in Nos. in the Greek splitting of fractions, formula Z, e. g.,
4 12 12 11 10 „ , .
3 IT ~ Tl X ÏÔ X 9 of whlch:
2, 3 and 4 will be recognised the Fourth divided by three:
Form. Cents
X 1st fraction 12 — is secondary 3-partition common numerator 12. 150.6
2nd fraction 11 , . — equals in cents: 165.0
10 u
498 only little below quartary 3-division =——= 166.0
Y 3th fraction 10 — is tertiary 3-division 182.4
9 3 common denominator 9.
The quartary cents are but little below the average number of the secondary and tertiary cents, 166.5 cents.
Should the Fourth be divided by 4, the result will be:
4 16 16 15 14 13
Formula Z. —- == — XttXt-X-
3 12 15 14 13 12
resp. Formula X Y ’ of which :
Cents 111.7 124.5 ! 138.5
1st fraction — is secondary 111.7 cents, common numerator 16, X,
1 ö
13
4th — — - tertiary 138.5 — , — denominator 12, Y,
Whereas the average number between — and ^ comes close to the quartary cents 124.5.
b) Twofold symmetrical calculations. Pythagoras may have established »his small 32
Third« es — =^= = 294.1 cents by means of
secondary 8. partition 32 our X.
32 27 24
Spaces 8 = 5 + 3
tertiary 9. division 27 32 36 our Y.
27
Spaces 9 = 5 + 4
Acoustic Methods of Work.
9
Or by means of particles and decimals respectively:
secondarily Our particles c es — f 720 60772 540
Spaces 11272 + 6772 § 180
abbreviated 5 + 3 = 8
tertiarily c es — f
Decimal-fractions 1,0 1,185 1,333
Decimals 0,185 + 0,148 = 0.333
abbreviated 5 4 = 9
c) The »Natural quadruple chord« :
Partial-tones Nos. 4—8 Primarily : Fractions c e g [aisj c’ 1 + 72 74 2
Secondarily : our particles, quickly decreasing spaces 720 576 480 4117t 360 144 + 96 + 684/7 J|7 51 7t = 360
Tertiarily : numerator in Y 4 5 6 7 8 Nos. 4—8
Quartarily: our cents, slowly decreasing intervals 0 386 702 969 1200 386 + 316 + 267 + 231 = 1200
Quintarily; stretching, 2nd power of No. Increasing tension Arithmetical progression, Andr.Kornerup’s observation 16 25 36 49 64 9 11 13 15 | 2 Sgi 2 + 2 OO 1 1
10
Th. Kornerup:
CHAPTER II
SEVEN HYPOTHETICAL PRINCIPAL RULES.
The following rules are proposed in the hope that a cooperation between philologic-historic inquiry and instrument-investigation on the one hand and theoretic-acoustic calculation on the other will lead to a clearer understanding of the origin and development of the ancient tone-systems than has hitherto been the case. Naturally, these rules are partly hypothetic, and are only advanced as a basis for discussion.
Group A. Presumable Construction of Scala-types and Tonics, (Rules Nos. 1—3).
Principal Rule No. 1. The stepwise progression of a scale-type is formed by secondary partition (on the string). 5-partition of the whole string gives thus (by three successive
halvings) »the ornamental pentatonic scale«: c d = ^ e (ges—) = y a c':
c a e” 0
e e 1
d (ges — ) c’ a’ !
720 648 576 504 432 360 288 216 144 0
Secondary 20-partition of the whole string, then partly secondary 5-partition of the Fourth, gives also 5 ornamental Tetrachords in oriental tuning:
des —
Doric
es —
Primitive
Phrygian
Greek before Pythagoras ?
Lydian
Tritonos
Harmonic, oriental.
des —
for instance: the Greek »scale-types«,...... Scheme 17, Nos. 1—7
Jewish Hedjaz-Ivar, double harmonic.... — —, — 8,
— Ahavad-Rabbah, harmonic-Doric. — —, — 12,
according to some indications of A. Z. Idelsohn (Note 6).
Acoustic Methods of Work.
11
Principal Rule No. 2. The oriental sequence of Tonics (key notes) can be formed by tertiary division. Three consecutive halvings of the Octave, Fifth and Third give thus the presumed primitive Greek (before Pythagoras) as well as the Indian funda-
mental Tonic-sequence: c
e g c’, (only Partial-tones):
1 9/s 74 72 27l6 “/« 2
1st halving c g
2nd halving c e g b d-r
3th halving c <1 —e g al§£ b d +
The result, the oriental (Indian) Tonic-sequence:
Roman numerals: Oriental Tonics. I II III.IV c d -)H e. f. V VI VII. I. g a + b. c’
Indian Sruti-No. 0 4 7.9 13 17 20, 22
Scale-Structure : 4 3 2 4 4 3 2 Ma-Grama
Scheme 2, J : 9 4 9. = 22
In any case it agrees, mathematically, exactly with what Victor Mahillon (cf.
E. v. Hornbostel and R. Lachmann’s report, Note 7) has reported (without mentioning
the source) as an old Indian prescription for the partition of an F string, so that
seven tones can be made in the middle third part of the string corresponding to the
number of particles on a whole string as given below (transposed from an F string
3 \
to a C string by multiplving by— J:
F. 480 particles 4267s . . 384 .. 360 .. . . 320 . . 28479 .. 256 . . 240
C. 720 transposed 640 . . 576 .. 540 .. . . 480 . . 42673 .. 384 . . 360
C. Ma-Grama, Major: d+ e f g a +7- b c’
F . Particles 288 270 240
C. transposed 432 405 360
C. Sa-Grama, Lydian-Phrj'gian: a bes — c
Fractions: W/9 2
The difference between Ma and Sa has reference to the structure of the two Indian scale-types respectively, see Rule 4.
12
Th. Kornerup:
Principal Rule No. 3. The Pythagorean and Persian Tonic-sequence is naturally formed quartarily, 6 Fifth-steps (Q Fifth, q Fourth):
Pythagorean Lvdian scale 0 2Q 4Q c d + e - - iq f. IQ . . . £ 3Q a + 5Q b+ c’
23 Pythagorean Instrument-tones: Structure 4 4 1 5 4 4 1
19 — Song-tones 3 4 1 3 1 3 4 1
Compare: 17 Persian Sruti 3 3 1 3 3 3 1
Group B. Probable Construction of Ancient Tone-Material
(Rules Nos. 4 — 7).
Principal Rule No. 4. Permanent and variable tones.
a) The transposing of secondary (ornamental, decorative) scales on the Tonics (key notes) mentioned may give several tone-systems with not a few common permanent tones, for instance the presumed 13 oriental permanent tones, and the 17 Persian tones with the exception of d, e, a and b:
Oriental Tonic No. I II III IV ges —
Name c des — d + es — e e + f ges 2 —
Oriental Tonic No. V VI VII I
Name g as — a + bes — b b + c’ Scheme 2.
The tone »ges 2—« almost fis-)-.
The 13 Persian tones (with the exception of d, e, a and b) are presumably the common foundation for primitive Greek, Persian, Indian and the later Arabian tone-material. Continued transposition on other Tonics or with other scale-types gives new tones, varying both for the different tone-systems and within the same system, called »variable tones of the system«. The following are cited as examples:
13 + 4—17 Persian Sruti Nos., of which 13 are permanent, Scheme 2, P.
13 + 9 = 22 Indian Sruti Nos., of which 17 are permanent, Scheme 2, J.
22 + 2 = 24 medieval Arabian tones.
The 4 variable tones in Persian music are presumably the Persian Sruti Nos. 2,5, 12 and 15; the 5 variable tones in Indian music: the Indian Nos. 1, 6, 10, 14 and 19.
The structure of the Tetrachords;
C. Lydian C. Phrygian
Indian Ma: g a + b c’ Sa: g a bes— c’
Intervals .. . 204 182 112 182 112 204 = 498
Sruti 4 3 2 3 2 4 = 9
Acoustic Methods of Work.
13
b) The idea in itself of a few permanent Sruti-distances as a structure cannot be thought of »as rising through a simple advance« of a Second only, i. e.
Indian Nos Ma, Lydian. . . . Sa, Phrygian. . . c d-f- e f g 0 4 7 9 13 4 # 3 + 2 !8 3 + 2 + 4 = 9 = 9
Persian Nos. Doric 3 4 . 6 7 10 ....... 13 e+ f g a+ = 7 7
Transposition. . d-f- es— f ...... g
but tertiary 9-division of the Fourth gives also the Indian Sruti-numbers for the upper Phrygian Tetrachord in Sa, Lydian-Phrygian.
Common denominator 48: | d-f'. 7s e 74 f Vs g sh
Numerators . Distances 54 6 60 + 4 64 + 8 72 — 18
abbreviated, Indian Sa ... . 3 + 2 + 4 = 9
corresponding to the tertiary 9-division.
c) The number of touches: Scheme 2 P, in all 7X7 = 49, and Scheme 2 J, in all 5 X 7 X 2 = 70 disperse naturally very unevenly (the columns in the middle).
The number of Molecules (intervals of equal value) is, according to all Sruti:
in Scheme 2, Persian 12 X 4 Commas 5X1 — 1 I Total I 53 1 Commas | f In Scheme 2, Indian 7X4 Commas 5X3 — ( 10X1 —
Intervals 17 2 Molecules Intervals 22 3 Molecules
This condition can naturally be changed by continued transposing. The expression »Molecule« for equal intervals between tones in stepwise progression was suggested in 1923 by Professor Chr. Kroman, (1846—1925), Copenhagen.
Principal Rule No. 5. Chromatic and Enharmonic.
a) In the practice of Music even Pythagoras has not been able to carry through the »consequent« Pythagorean Fifth-system, but he has had to content himself with the 7 fundamental diatonic Tonics, and has presumably formed the intervening chromatic and enharmonic tones by secondary interpolation, for instance, the above mentioned chromatic des —, es —, etc. by secondary halving of the small Second c. .d, or d+. .e, whereby with great approximation the same interval is obtained from des— to d113.7 cents, and from e to f = 111.7 cents — inaudible difference (Formula X):
14
Th. Kornerup :
1)1 ■“ {20 11 ,8} , 20 des— = —— = 88.8 cents Inaudible difference 1.4 cents, nearly k. Scheme 14.
2) Pythagorean: 5 Fifths below , 256 = des— = = 90.2 » 243 J
3; c . . d + { 18 j ' \l8 17 16/ / , 1 \ 18 = ^des — j = = 98.9; c . . es 3-parted
4) syntonic, difference, Scheme 24, between the Partial-tones Nos. 16 and 15 = deS= 15 = 1117 Ct,| Inau i fn dible Scisma-dif- rence : 2.0 cents.
5) Pythagorean : 7 Fifths up te = cis2+ =113.7—J
6) Golden des = 5 X 503.8 cents = des =118.9 — »c . . d« super divided.
7) c .. e ( 15 ) 115 14 13 12/ = (des) = — = 119.5, c..e 3-parted.
The Greeks Eratosthenes and Plutarchos used the secondary fraction 20/19; Arky-tas, Didymos and Ptolemaios 16/15; Plutarchos also 15/14 in Kroma malakon (soft, i. e. diminished Kroma).
When construing tone-systems one can for convenience overlook the small inaudible differences and, on paper, calculate by theoretical fractions. The Greeks themselves have to a large extent made use of the splitting of fractions, Formula Z, i. e.,
, 20 20 19 f the first is secondarv halving
d= = = = X = , °f which { ~
18 19 18 { and the second a tertiary one.
By this primitive method of work, splitting of fractions, the legitimate tone systems have been far outstripped, by continual partition, i. e., secondary interpolation; thus
Eratosthenes and Plutarchos halve des — = jjj = ^ X of which the first is a
secondary enharmonic deses = = 44 cents, while Didymos halves the syntonic
16 32 32 31 32
des = Î5 ~ 3^ ~ 3Ï^30’ which = 55 cents also is the secondary enharmonic deses,
28
just 11 cents larger.
Curiously enough Arkytas makes use of the fraction ~ = 62.9 cents, very nearly
It
the Molecule in the 19-toned temperament 63.16 cents, just the Fourth tertiarily 9-divided (Formula Y):
c des — es — f
27 28 32 36
27
28 32
with 97 = des — = 62.9 cents and ^ = 294.1 cents, see Chapter I, Resume, h.
Plutarchos has a similar fraction ~ = 58.7 only a little lower, and (des) = ~ ~
~ 14 28
= 119.5 cents, syntonic d secondarily 3-parted, Formula X:
30
30 29 28 27
Ptolemaios uses the fraction ^ = 38.9 cents, very nearly the Molecule in the 31-
44
toned temperament, 38.71 cents, the Pythagorean d -f- secondarily 5-parted (Formula X):
Acoustic Methods of Work.
15
45
45 44 ........... 40
b) Interpolation within the Tetrachords is also presumably to be found to a great extent among other nations than those mentioned. Thus Helmut Ritter quotes from Rauf Bey (Note 8) for the Turkish scale-types »Suznak« and »Neu eser« one and two Tetrachords respectively of two unusual forms which we shall call
Scheme 17, No. 13.
cents
I »uneven Tritonos-Tetrachord« 204 -j- 120 -j- 267 = 591, Medium 111 I »uneven harmonic«................... 120 -)- 267 • j- . . .... Ill
Scheme 17, Nos. 10 b and 13.
total 702 total 498
which however, can easily be explained in the following way:
The interval on the string »df^fs... fis -j-« (in Tritonos) is parted secondarily into
3 parts or the Fifth »d+ ... a+« into Sparts in:
d -f- es -j- Vs Comma fis + a -f
Particles: 640 597 Vs 512 4267s
Spaces 42 7s + 85 Vs + 00 II to CO
Abbreviated 1 2 + 2 = 5
Thus the tone - ^. ■ — —— = 323.4 cents, 1/s Comma over es, which is 315.6 cents. The oy / /3 Hä
two scales can hereafter be characterised as follows:
1 c d -j- e f Suznak i 1 Lydian Major cents 204 g (as) b c’ uneven harmonic Scheme 17 No. 10 b.
Si fc d + ies) fis M Neu eser < ( uneven Minor Tritonos 111 g (as) b c’ uneven harm, repeated No. 13
c) In the same way some five- or six-toned scales, as for example some Indian scales may be explained by Trichords:
Secondary partition: Middle tone:
Trichords : Particles Abbreviated
c d f 72 -f 108 2 + 3 = = 5 10/9
c d+ j f 80 + 100 4 4* 5 = 9 9/8
c (es—) f 108 ■ 72 3 -f- 2 - = 5 20/17
c es - f 1127 * H" 6772 5 + 3 = 8 32/27 *)
c es f 120 + 60 H = 3 6/5
c e f 144 n 36 4 + 1 = = 5 5/4
*) see Chapter I, Resumé, b.
16
Th. Korneküp:
And it may be supposed that many other riddles of a similar type can be easily cleared up as being something quite natural by secondary interpolation alone — without fantastic Fifth-steps.
d) We might probably solve Professor v. Hornbostel’s theory concerning Comma-Fifth and Comma-Fourth, probably g— and f-j- respectively, by means of
1) secondary interpolations, »4-partition of 2 Fifths , d . . a and d + .. a-f respec-
tively:
Formula X:
| 36 30 27 24
g— 40 — = Comma-Fifth Particles d 648 f 540 g— 486 a 432
3 proportionally 2 1 1
abbreviated spaces 2 1 1
f+ • Particles 27 — = Comma-Fourth 20 640 d + 1 533l/s f + 480 g 4267s a ~f~
I f 27
Formula X: J-----------------------------
I 24 20 18
Then Comma-Fifth: d . . a 4-parted or: f ....a j
— -Fourth: : d+.. .. g 3 - or : d -|- . .. . a+ j
16
halved
2) Further Tertiarily following divisions, see Scheme 19:
( 4 8
3-divided < 1
10 5
Formula Y 30 32 36 40 45 48
Denom. 27 : d es — f g — a bes — b
10
8
Halved j
3-divided 18 36
5- — 18 27
Formula Y 90 96 108 120 135 _ m )
Denom. 80 : d + es f + g a + bes j
Halved.............. 12 12
Acoustic Methods of Work.
17
3) Further »bes ... d + ... f + « and »g— ... b — ... d« are Major triple-chords:
I f + is deep difference-tone between d+ and bes 1 (g— — — — d and b— f
9 72 Super-Tonic bes = — = — = 1,800 9 45 minus Third d+ = — = — = 1,125 50 resp. b— = — —- 1 85185 Third v 27 30 minus d = — = 1,11111 Sub-Fifth
27 Deep difference FEg = — = 0,675 G— = ^ = 0,74074
27 The Fifth f+ = — = 1,350 g— = — 1 1,48148 Tonic 2 7
0. 0,675 1 1 1,125 1,350 1,600 2
F + d+ f+ bes 1 ' .1 ' = =3
I 0,74074 G- J 1,11111 1,48148 1,85185 a g- b- [ I I
1 i
Finally it may be proved by means of other similar difference-tones. Prof. v. Hornbostel however, gives as a physical explanation of this problem (Note 8), (f+) = 521.5 cents, (g—) = 678.5 cents, »die Blas-quarte und Blas-quinte« in decimalfractions: 1,3515 and 1,4798 respectively.
Principal Rule No. 6. The intervals between the tones in stepwise progression can often be grouped in several »Greek Triads« (3 neighbour tones) of two dimensions in two directions:
great »Triad« [ direct 4 + 1 Comma retrograde 1+4 Commas, Pythagorean small »Triad« | » 3 1 » » 1 + 3 » Arabian?
As it happens the octave comprises 53 Commas of 22.24 cents (about midway between the syntonic and Pythagorean Comma, respectively 21.50 and 23.46 cents); from Scheme 3 can, for example, be constructed Scheme 4 as an experiment: the »direct Greek Triads« with 53-toned tone-No. (Nicholas Mercator, Dane, 1675).
Perhaps the structure of the »Greek Triads« has had some importance in Melody formation.
18
Th. Kornerup:
Principal Rule No. 7. Javanese salendro and pelog Gamelan were interpreted by F. Lassen Landorph (Note 9) in 1923 as »Two tone-systems, in which two Gamelans (adapted to two different systems) cannot be played together«; we assume that they have both been built up by quartary interval-division, but in two different ways. During his stay in Sumatra (1877—99) Lassen Landorph recorded the vibration-numbers (oscillations) for the tones, and arrived at the conclusion that
1) the older form, the Salendro-octave, has 5 tones (Pentatonic) »with a distinct tendency towards five equal (quartarv) intervals* within the octave, thus a 5-toned
temperament; whilst
2) the latter form, the Pelog-octave, is also »called 5-toned, as the whole material of 7 tones in several places is seldom or never used, but on the other hand, various combinations on 5 tones only«.
Following the vibration numbers recorded by Lassen Landorph, Scheme 5 has been constructed showing the Salendro-octave as a 5-toned temperament and the 7 tones of the Pelog as tone-material, i. e.
partly »a harmonic Tritonos«, in quartary 4-division with omission of the central tone: »c cisis ... eis fis -f«,
partly a Pythagorean harmonic Tetrachord, »g as— .... b -f- c’«, in quartary 5-division with omission of the central tone. The interval »as— ... b + « is the syntonic es == 6/5.
Other intervals:
Nos. Interval Nos. Inter. Nos. Inter. Nos. Inter, j Nos. Inter. 1
1 — 2 cisis 4—5 des 5—6 des — 6—7 es 4—6 d +
3—4 cisis 5—6 des — 6—7 es 7—8 des — 6—8 e -f
2—3 Sum es — Sum d + Sum e + Sum e H- Sum ges —
Lassen Landorph is of opinion that the Salendro is reminiscent of the very ancient Chinese Tone-system, and »in its most simple composition, with reference to the Java-tradition, it is supposed to derive from the oldest Hindu period, or, more correctly, to be traced to this period, which means, from the beginning of the Christian era«; — while Pelog must be presumed to be later, as it, with its various tones and varied musical instruments, is more fully developed«. Both are, however (partly) built up on quartary interval-division, — naturally secondary in practice.
The scale: »g as— b+ c fis-f- g« may be presumed to be pure Pythagorean (fis-f = ges 2—).
In the pelog Gamelan the tone Pelog, 512 particles = 590 cents, may however easily be found on a string (or on the flute) with great approximation by the 5-partition of the octave, and the Fifth g exactly by halving, while analogously the other tones may be found b}r secondary 10- and 5-partition respectively, as shown in Scheme 5.
Acoustic Methods of Work.
19
CHAPTER III
THE PYTHAGOREAN SYSTEM IN THEORY AND PRACTICE.
That the pure, i. e. consistent Pythagorean system can only be taken symbolically is evident partly by the fantastic tone-fractions formed in the outer circle of the svstem. Dartlv bv the Greek Schisma, for instance
Syntonic tones :
6
but es = — = 315.6 cents
5
9
» bes = — = 1017.6 »
5
27
» ^ = 519.6 »
.... natural.
Only 6 Fifth-steps give the same tone as »2 Fifth-steps and in the opposite direction 1 syntonic Third«, that is:
Pythagorean tones :
9 Fifths is »dis 3SB
10 — »ais 3 4-*
11 — »eis 3 IB
19.683
16.384
59.049
32.768
177.147
65.536
The fractions : overstated
6 Pythagorean Q minus 3 octaves == 4211.730 — 3600 = 611.730
2400 minus (2 Q -j- 1 syntonic Third) = 2400 — 1790.224 609.776
Difference, a Greek Schisma = 1.954
m fis 2 -j-= ges — cents.
After Pythagoras having altered the presumed primitive Greek
Tonic-sequence (key-notes) : c d-j- e f g a b-j- c’
to his Diatonic (Persian) : c d-f9 e-b f g a-b b-r c’
which is also the authorised Greek Tonic-sequence, he hereby reaches his limit, and beyond that he only creates syntonic tones with the exception of one Schisma, so that the Pythagorean exaggeration recoils, in that, beyond 6 Fifths, all other Fifths return like a boomerang, as syntonic interchangeable tones. It is the playful way of Nature that all exaggeration corrects itself.
The Arabian-Persian musicians probably knew, before 636 A D. that, for instance, 8 q Pythagorean = fes 2 — are almost identical with the syntonic e, (according to Helmholtz, as quoted by Jonquière, Note 10).
If to this be added that the 4 chromatic tones des—, es—, as— and bes— are so near to the primitive secondary tones that the difference of 1.4 cents is inaudible
20
Th. Kornerup:
all Pythygoras’s exaggerated tone-fractions may be changed to reasonable human fractions.
The Pythagorean system can thus (with the exception of the inaudible Schisma) be indicated as quoted in Scheme 6:
Song
Lexis
Instrument
Krusis
Scheme 14:
— 3 enharmonic interchangeable tones 1 stage No. 6
2 2 chromatic — /
5 5 — — — I y
7 7 diatonic Fifth steps 1
5 5 enharmonic interchangeable tones \ — - 8
— 1 — — — 1
19 23 Tones in all 3 stages
(c)
8 (es)
The tones, however, are not used with equal frequency; if all the Greek scale-types be played on the following 7 Tonics (key-notes): cd+e+fga+ and bes— for song, the number of touches will be 7 X 7 X 7 = 343, distributed as the numbers in Scheme 6, with the maximum 37 on g, which is the middle one of the 7 Tonics selected, arranged according to Fifths.
The number of Molecules for 19 song-tones will be 10 X 90.2 + 2 X 66.7 + 7 X 23.5 cents = 1200 cents.
The names of the song-tones are thus given in the Greek Triads from above downwards; for the instrument-tones a name-No. was originally given to the uppermost in the Triad (Note 11) following our hypothesis as indicated in Scheme 7 according to triple-chords (chords of Thirds, English triads).
Scheme 17 Nos. 1—7, shows the 7 Pythagorean scale-types in syntonic or golden
(Nos. 1 2 3
\r /
tuning, 6 paired off symmetrically
(and 7 6 and 5.
Acoustic Methods of Work.
21
CHAPTER IV
THE AUTHORITATIVE TONE-SYSTEM, THE GOLDEN SYSTEM AS AN OBJECTIVE MEANS OF VALUATION.
However, in order to pass judgement on the various historical tone-systems, the chief defect of which is the displacement of Commas between for example »d and d-j-«, »des— and des« etc., it is necessary first to construct an authoritative tone-system without these Comma-displacements, which claims that many tones with plus would be made much lower, and many tones with minus much higher.
In August 1930 I found the new Fifth in two ways which gave the same result.
Group A. Arithmetic Series I—VI, retrograde and direct,
The sum of 2 golden adjacent Nos. forms the following No., and the sum of the cents of two golden adjacent tones forms the following cents in the same golden Series.
1. The astronomer J. Kepler (1571—1630) combined tones with the sum of the numerators and the sum of the denominators respectively of 5 tone-fractions:
1 2 3 5 8 13
T T 2 t T Partial-tone 840 5 cents
c c g a as gisis
The golden gisis........... 843.2 »
In the 19-toned and the 31-toned golden section respectively we transform the
Fourth-Series I retrograde :
19 toned 31 —
50 —
Double super division
1 cis 2 des 3 d 5 es 8 f 13 as
1 deses 2 cis 3 des 5 d 8 es 13 f 21 as
1 bisis 2 deses 3 cis 5 des 8 d 13 es 21 f 34 as
m
22
Th. Kornehup:
Ludwig Sonnenberg, principal teacher in Bonn (1820-88) called the sequence 1, 2, 3, 5, 8, 13: »Kepler’s Series« (Notes 12—13), our golden Fourth-Series I retrograde, in 1844 extended up to No. 17 by Gabriel Lamé, in 1929 up to No. 40 by L. Kaiser.
2. Dr. Ludwig Kaiser, the mathematician, found by pure mathematics (Note 14) (without reference to tones) a similar sequence; we call the latter »Kaiser’s Series«, our Fifth-Series II, which we use in the following manner:
Fifth-Series II retrograde
Double super division.
19-toned 4 dis 7 fes 11 g 18 ces
31- - 7 dis 11 fes 18 g 29 ces
3. In May 1930 Andreas Kornerup, engineer, directed my attention to the fact that
the Fifth in these systems has tone-Nos. which form similar sequences of Nos. for
, Fifth 7 11 18 29 47
example 7r— = tt,’ ïïT’ fä’ etc-
Octave 12 19 31 50 81
We name the sequence 12, 19, 31, 50, 81 the Octave-Series III, Andreas Kornerup’s Series, the denominators of the fractions.
a) It occurred to me to calculate these fractions in cents, which I did, August 15th 1930. The value moved like a pendulum quickly approaching the point of balance,
where the seventh decimal would be stable at the Fifth -696.2145 cents
61oo
(Acoustic statics).
The temperaments, the octaves of which are the denominators of these fractions, 1 called the organic temperaments, and, at the same time, it occurred to me to construct an authoritative »tone-system of the Fifth with this Fifth which, however, also generally appears by means of the super-division (golden cut) of the octave, Scheme 10, Series III. I calculated the Fifth on August 17th 1930, as shown below:
b)
The fraction of super-division............... Cents
» 0,618.03398 X 1200 gives 741.64078
1 — co 0,381.96602 X 1200 ...... 458.35922
... Series III :
... direct ases retrograde eis .
Total 1,0
1200 cents
c’
Further
ases is 11 Fourths from c eis is 11 Fifths —
and 4 octaves back then f - 503.7855 — 6 — — I — g= 696.2145
Total - 1200 cents
In 19-toned and 31-toned golden sections respectively we use »Andreas Kornerup’s Series« in the following manner:
Octave- Double super division
Series III retrograde 19-toned Oc 7 eis 12 ases 19 c’
31-toned 0 c 12 eis 19 ases 31 c’
The difference between golden eis — 458.4 and syntonic — = 457.0
Schemes 14 and 15: nearly k = V15 Comma =
1.4
Acoustic Methods of Work.
23
4. Out of the calculated golden system I have formed some other Series for instance 31-toned see Scheme 20:
Double super division.
Great Third-Series IV: 4 cisis 6 eses 10 e 16 ges 26 bes
Tritonos — V : 9 feses 15 fis 24 beses
Great Sixth- — VI: 9 disis 14 geses 23 a
Group B, Geometrical Constructions of golden Tones.
The golden tones and intervals can also be constructed geometrically as shown in Scheme 8, which indicates:
1) above, part of a regular Pentagon, the angle of which =108° is divided into 3 parts by means of 2 chords; if the chord be equal to a Tetrachord— the interval C-F, (the Fourth f), this interval will be super-divided in »great double super-division«
I C------D........Es--------F = Series I 1
m { 192.43 ; 118.93 + 192.13 = 503.79 cents J
2) below: a square placed inside a semicircle, whereby the diameter is super-divided into a »small double super division«
in cents
corresponding with:
Cis .... D------E ....... F
D ....Es--------F ....... Ges
B .... C ------- D ...... Es
118.93 + 192.43 jjl 18.93 = 430.29
C .... D--------F ....... G
192.4 + 311.4 H- 192.4 =696.2
3) further: within the square we can form a »great double super division«
. j D ------ Dis .... Es----- E )
m 1 73.50 -B 45.43 + 73.50 = 192.43 cents j
Scheme 9 shows the golden system constructed by means of the Pentagram with the same chord C .. F, so that the side is = C .. Es in the Pentagon.
Fourth-Series I: C, Deses, Cis, Des, D, Es, F and further As, Des’.
C (Es) es
Further: — = Cosinus 36°, the relation between small and great Third.
• • G 0
Scheme 10 shows »double super-division« in 4 Series; the upper and lower edges of a square are divided:
1 from c upwards into ases, direct,
\ — c’ downwards into eis, retrogade.
24
Th Kornerup:
If we continue a single super-division,
1) direct, upwards, we will get decreasing intervals upwards,
2) retrograde, downwards, we will get decreasing intervals downwards, from the corner upwards on the right \
— downwards — left I
f fron
s\ -
the oblique lines divisions, which give the result :
We use forming 8 double super-
upwards, retrograde: Cents downwards, direct, Scheme 10:
^ Series I c es., f as 815 e g . . a c’
— IV c e .. ges bes 1008 cis f . . gis c’
— II c fes . . g ces 1126 d— fis . . as c‘
— Ill , y
We continue double super-division only in Series I:
I c---des .. d ----es 311
Series I: c---cis .. des---d 192
[ c---deses .. cis---des 119
a -----bes .. b------c’
bes----b .. ces---c’
b -----ces .. bis----c’
The tones are in pairs supplement-tones (making an octave together) for instance:
J retrograde deses cis des d es f as c’
( direct bis ces b bes a 0 r> e c
Scheme 20 shows golden tones in 6 Series formed geometrically by means of parallel lines in a Pentagon, for instance:
The tones : Eis F Fis Ges G As A C’ Des’
Series I-III: - IV-VI:. III. I. ... ... V. IV .. II. I. ... .. VI. . . .. III. 1.
Scheme 21 shows cents (to 4 decimals) for an organic 19-toned section of the infinite golden tone-system with two Molecules:
I t = 73.501 cis, which is the smallest interval in this section,
I v = 45.426 deses, the greatest interval in the next organic section on 31 golden tones.
v. t. cents
The Molecules in 12-toned golden section 5 X cis -)- 7 X des = 1200
19 — - 7 X deses -(- 12 X cis = 1200
31 — — 12 X bisis -|- 19 X deses = 1200
v. outside the section.
Acoustic Methods of Work.
25
Group C. All Comma-displacements disappear.
Example: d = ^ and d+ • ; 'J-y o
merges into the golden tone d = 192.43 cents, approximating to the secondary halving of the string between d and d+, which in the Formula X gives:
180
162 161 160
d.............d+
Particles 648 644.3 640
cents 182.4 193.1 203.9
180
or — - = 193.12 cents, the difference is inaudible, 161
so that the golden tone can easily be found approximately on the string. The tertiary halving gives a somewhat higher tone (Formula Y):
160 161 144
162
^ = 193.20 cents. 144
The structure of the triple chord is often the golden cut directly put to use (x = great Third, y = small Third, (2) and (3) formations):
Triple chord: Symbol Intervals Small Sixth with golden cut:
Maior 2nd form ......... xy (2) yx (3) e g c retrograde: 311 -4- 504 = ) } 815 cents direct: 504 + 311 = )
Minor 3rd — g c es
For teachers of harmony the golden system can thus also be of use through its clear and logical construction; thus xyy (1) the Dominant quadruple chord c e g bes will, on the 1st step, be resolved into xy (3) ®f the tonic triple chord on the 3rd step: (c) f a c’ (on the Tonic f) by which means the tones e, g and bes glide either 119 or 192 cents, »c des« and »c d« respectively, systematically up or down, with the intervals in cents:
From
o
c
385
e
696 1007 1200. Symbols
g bes c’ xyy (1)
Bi —»
Gliding in cents 119 i ■4r-v- -> 119
192 192 192
to
I 0
f
504
xy (3)
888
1200
26
Th. Kornerup:
Since all Comma displacements disappear in the golden system, one can therefore transpose any scale whatever on any golden tone as Tonic (key-note) with the smallest possible number of tones; the authoritative Golden tone-system represents in this field the formula for the principle of the smallest activity, Nature’s economic minimum principle. It is the system with the smallest possible number of tones (acoustic Okologv), an outcome of Nature’s wonderful power of adaptation, — the supporting principle of all life in Nature.
Andreas Kornerup has called the fraction of the golden cut 0,618.034 Omega, which is recommended as an international expression in the formula — = . 03— or: or — a> — 1
CO 1 CO
or: eis + ases = c’, Series I, or 458 + 742— 1200 cents, which we here designate, once and for all, as the authoratitive principle of relativity in the field of acoustics: The essential formula for the universal sense of harmony.
Even if the composers do not know of this formula, the Danish Physicist H. G. 0rsted (1771—1851) is indeed right in saying: »The work of the composer is based on mathematics although in a deeper measure than has ever dawned upon us.« (Note 15). It will be the task of the future student of the theory of music to get to the bottom of this deeper-lying law of Nature, the golden cut, the super division — the basis of the future renaissance of harmonics.
Acoustic Methods of Work.
27
. CHAPTER V TEMPERAMENTS.
Tone-systems with 1 Molecule only.
The Octave-Series III, Andreas Kornerup’s Series, marks the boundary of the organic temperaments, i. e. the only rational, the only temperaments fit for use, namely the sequence: 12, 19, 31, 50, 81 — with 19-toned as the practical and with 31-toned temperament as the Standard-temperament, see Schemes 12 and 18.
How many tones will be required for pianos, organs etc. is indeed a question; but, for practical measures only the organic 19 and 31 are efficient; all the others are unworkable. This is the authoritative judgement passed on the matter in question.
»Ceterum censeo: If we wish to abandon the 12-toned pianoforte we can in no circumstances choos any other than the practical 19-tonic temperament, and for finer requirements the Standard 31- (or the 50-) toned temperament.
All the inorganic temperaments ought to be excluded.
Scheme 11 shows, by way of example, how the Fourth-Series I, Kepler’s Series, is carried through logically, in 19- or 31-toned temperament, where the sum of cents for two neighbour tones gives the next tone in the Series, but is split in, for instance, the 24-tone system, which is therefore authoritatively considered to be unworkable.
Further: Dr. P. S. Wedell and N. P. J. Bertelsen, actuary, Copenhagen, in January 1915 proved, by means of »the method of the smallest squares«, that the 19-toned is better than the 12-toned temperament, and that the 31-toned, again, is better than the 19-toned. To have this judgement expressed in numbers it can easily be calculated how much the single tone in the different temperaments deviate from the golden tones, and by summing-up such deviations for 35 tones (the tones of the 7 white keys and #, % 2# and 2t?) the result is found to be as follows see Scheme 12:
Deviation
for
12-toned Temperament 1174 cents estimated at 100%
19 — 458 — thus 39
31 — 174 - — 15
50- — — 67 - - 6
In comparison it may be quoted that in the pure consistent Pythagorean system, the collective deviation for the same 35 tones is 1780 cents or 52 % greater than for the 12-toned temperament.
Among the numerous observations as to the change of the piano-temperaments we shall quote the following few selections:
28
Th. Kornebup:
Director Gotfred Skjerne says (1909): »We are indebted to the tempered tuning for
enormous musical progress ....... but the ear is, as a matter of fact, coarsened.
(Note 16).
Professor Dr. José Würschmidt showed 1920—28 »that a division of the octave into 18, 24 or 36 parts can represent no natural extension of our tone system, hut that we have before us such an extension in a division of 19 steps. (Note 17).
Professor Joseph Yasser, New York, recommends the 19-toned temperament with the purpose: »to enrich our musical language, particularly when one takes into account the growing significance of the independent twelve-tone foundation in modern music«. (Note 18).
Professor Louis Kelterborn, Neuchâtel, (Note 19) proposes, further, on practical grounds, that simultaneously with the introduction of the 19-toned temperament, the pitch of »the tone a« should be made a little lower, so that c may keep the same pitch as it has now.
Scheme 13 shows, geometrically, an equilateral hyperbole through the tones No. 5 in various organic temperaments forming the tones in the Golden Fourth-Series I, retrograde, (with approximate pitches of tone).
Scheme 18 shows lines through corresponding tones in 5 temperaments.
Acoustic Methods of Work.
29
CHAPTER VI
CHANGE OF THE SYNTONIC SYSTEM INTO GOLDEN TONES.
a) In Schemes 14 and 15 we have the syntonic system, built on Pythagorean Fifths (the Partial-tone = 3/2) in horizontal lines and the Thirds = 5/4 in diagonal lines with an angle of 60°, according to the proposal of the Japanese Shohé Tanaka in 1890.
The axis through »feses, c or gisis« in the schemes is called Zero-Axis, because the difference between the syntonic and golden feses (equalling 15 golden Fourths) is almost nil, just as is the difference between the corresponding supplement-tones, syntonic and golden gisis (equal to 15 golden Fifths), which is seen in the following table:
4 syntonic Thirds..............give 1545.255 cents
minus 1 — Fifth, the Partial-tone 3/2 = g 701.955 —
< gives syntonic gisis____ 843.300 —
15 golden Fifths minus 8 Octaves, golden gisis, is 843.217 —
inaudible difference .... 0.083 —
Further: 4 golden Fifths syntonic Third, the tone 5/4 == e minus 2 Octaves: golden e 386.314 — 384.858 -
Slight deviation 1/15 of a syntonic Comma 21.5062 cents is k = 1.456 — 1.434 —
inaudible deviation..... 0.022 —
All the tones in Scheme 14 can thus, in a practical way, be changed to golden tones by adding or subtracting a number of »k«. These numbers are regularly grouped in all the lines which can be enclosed through the tones in the Scheme, i. e. with regular rise in numbers of k, so that the Scheme will be reminiscent of mathematical number-designs, or number-figures, constructed by the Danish astronomer Thorvald Nicolai Thiele (1838—1910) in 1872 (Note 20).
Scheme 14 is thus divided into 9 symmetrical figures, each comprising 15 tones, formed on about 9 central points of, respectively -L 15 or 0 or — 15k:
30
Th. Kornerup:
+ Zero —
gisis — gisis 0 gisis +
-f- is 0 — 15
c — c c +
+ 15 0 — 15
feses — feses feses -f-
+ 15 0 — 15
resp. from 22 from -|- 7 from — 8
to +8 to — 7 to —22
Thus, outside the Zero-Axis the tones are here changed by »addition on the left side«, »subtraction on the right side« of fifteenth parts of a Comma, increasing evenly in all directions.
Axis : Direction: Result: Central-points
1) Pythagorean Fifth-Axis, horizontal . . . then one Small Third-step to the right downward o. c — 4. g — 8. d + — 12. a + - 3‘
1 — 15. c -)-
2) Pythagorean Fifth-Axis, horizontal . . . then one Great Sixth-step to the left upward 0. c — 4. f + 8.bes— + 12. es — + 3.
—f— 15. c —
3) Great Third, Axis, 60° upward then one Great. Sixth-step to the right — — left o. c — 1. e — 2. gis — 3 bis + 3
0 gisis
4) Small Sixth-Axis 60° downward then one Small-Third-step to the left — — right 0. c -f- l.as + 2. fes 1 1 -(- 3 deses — 0 feses
5) Small Third-Axis through the figure . from »-f- 15 gisis—« (through »o. e«) to »— 15 feses -j-«.
If all the Third-lines, inclining 60°, be elongated they will cut the Zero-Axis in nothing but Zero-points, — ad infinitum.
Two examples of the division of 1 Comma will finally be quoted:
d -(- sunk 8k = 11.4700 es sunk 3 k = 4.3 cents
d raised 7 k = 10.0362 es - — raised 12 k = 17.2 —
Total 15 k = 21.5062 5 X 3 k : = Total 15 k = 21.5 — =1 Comma.
The golden d = 192.43 cents is only a trifle below the secondary halving of the distance on the string between d and d +, as above mentioned (Chapter IV, Group C).
Acoustic Methods of Work.
31
32
Analogously the distance on the string between Pythagorean es — = ^ and the
syntonic es = -4- is secondarily 5-parted according to the Formula X: o
480
405 401 400
es — golden es
with the particles 607,5 601,5 600
An extreme example:
Pythagorean 6q = ges 2 — = 588.3 cents Syntonic fis -f- : - 590.2 cents
Scheme 14 : plus 24 k = + 34.4 — Minus 9 k = — 12.9 —
golden ges = 622.7 — golden fis = 577.3 —
Scheme 21 : plus Molecule v m + 45.4 —
ges = 622.7 —
b) Scheme 15 shows the exact change (to three decimals) in the middle figure (15 tones) and gisis. The number of cents diminish or increase evenly with
( 5.741 cents in all horizontal Fifth-Axis 1 11.456 — — — oblique Third — f
representing »the 2 building materials« in the syntonic mixing-system: Fifths and Thirds:
1) ^Sl.373 cents eis 2) — 2.912 gis 3) —2.912 gis 4) 5 syntonic e = 378.2 feses -f-
— 1.456 plus e + 2.829 plus # — 2.829 minus ff 5 golden e = 356.8 feses
— 0.083 gisis — 0.083 gisis — 5.741 g distance = 21.4 = 1 Comma
c) Rationa lexplanation : Super division of the octaves e ... e’ and as ... as’ (Supplement-pairs) gives »retrograde gisis« and »direct feses« respectively.
d) Similar methods (but of inferior structure) were up to 1558 used in the three famous systems of chance by: (see Scheme 16):
Zero-Axis The unit
increases decreases
(X unknown author) before 1511 a c es as — |/s e
The Bohemian, Arnold Schlick 1511 . as c e + V 4 a — ’A es
The Italian, Gioseffo Zarlino 1558 ..... ces c cis ( as Bla —-Vie — l/i es
here, Scheme 14 ... . 1930 feses c gisis as — Vi5 e
32
Th. Kornercp:
Scheme 16 shows the 3 Fifths oscillating about the golden g, »the truth :
X Zarlino Golden sj'stem Schlick
the result g: 694.8 695.8 696.2 696.6
by means of: 5/l5 V.4 Vis 4/16
in decimals: = 0,333 - 0,286 = 0,267 = 0,250
Distance from the ideal: + 0,066 -f 0,019 Ideal — 0,017
Addendum.
Scheme 17 shows diagrams of 7 Minors and 6 Majors, some of them in pairs symmetrically, see pages 10, 15 and 20.
Glareanus’s erroneous nomenclature of the medieval scale-type should once for all be obliterated from the littérature of music, and should be replaced by that recommended in Scheme 17, compare Helmholtz’s heartfelt cry: Übrigens werde ich Glareans Namen nicht brauchen .... es wäre überhaupt besser, wenn man sie vergessen möchte,« repeated by Ellis: »But I shall not use Glareanus’s names .... It would be better to forget them altogether. (Note 21).
Scheme 18 shows lines through corresponding tones in 5 temperaments.
Scheme 19: tertiary distances as arcs in a circle, see pages 7 and 16.
Scheme 20: construction of golden tones by means of parallel lines, see page 24.
Scheme 21: Golden tones in cents with 4 decimals, see page 24.
Scheme 22: examples of logarithms as training in the use of my Constant K 0.600.5714, see page 7.
In Scheme 23 is set forth a proposal for a system of notation on music-paper, staff, in all other tone-systems than the 12-toned temperament, with the suggestion, offered by K. Steensen (Note 22) for a rational arrangement of the £f and 17-
Scheme 24 shows how the cents can easily be calculated by taking the difference between the logarithms for Partial-tones, the numbers of which are respectively the numerator and the denominator in a tone-fraction, according to the table constructed by the organist Kai Kroman, Copenhagen, for example:
f Logarithm for Partial tone No. 10 = 3.986.3
oriental d = ^ - - --- 9 = 3.803.9
[ difference ... 182.4 cents
The word »Partial-tone« is preferred to »over-tone« in order to avoid the confusion, which Ellis characterises as »great confusion, Tyndall’s erroneous translation« (Note 21) between
the German ober ! and the English upper: adjective .
— über j — — over: preposition, adverb.
Acoustic Methods of Work.
33
7
10
11
15
17
19
20
22 12
Bibliography.
Professor E. v. Hornbostel: »Die Massnorm als kulturgeschichtliches Forschungsmittel«, »Festschrift P. W. Schmidt«, 1928, page 311 and »Analecta et Additamenta« : »Die Herkunft der alt peruanischen Gewichtsnorm«, pp. 255-258.
Ibid: »Musikalische Tonsysteme« in »Handb. der Physik«, Bd. VIII, Chp. 9, p. 438.
Prof. E. v. Hornbostel and Robert Fachmann: »Das indische Tonsystem bei Bharata und sein Ursprung«, in »Zeitschrift für vergleichende Musikwissenschaft«, Berlin, 1933, p. 89, Note 1.
Professor E. v. Hornbostel, Berlin, above mentioned: »Musikalische Tonsysteme«, Chp. 9, p. 429 and Prof. E. v. Hornbostel and R. Lachmann: »Asiatische Parallelen zur Berbermusik«, in »Zeitschrift f. vergl. Musikw.«, 1933,
p. 10.
Professor Joseph Yasser: »A Theory of Evolving Tonality«, New York. 1932, p. 21, Note.
Professor A. Z. Idelsohn: »The Features of the Jewish Sacred Folk-Song in Eastern Europe« in »Acta Musicologica«, Leipzig IV, 1932. pp. 22—23.
Prof. Hornbostel and Fachmann: »Zeitschr. f. vergl. Musikw.«, 1933, p. 80: Victor Mahillon: Catalogue descriptif------- du Musée instrumental du Con-
servatoire .... de Bruxelles, T. 1. 2. éd. Gand 1893, p. 93 ff.
Helmut Ritter, Istambul: »Der Reigen der Tanzenden Dervische«, in »Zeitschr. f. vergl. Musikw.«, 1933, p. 28—40 and appendices 5; and concerning Blas-quinte see: Hornbostel: Note 3 in »Handbuch der Physik«, Bd. VIII, Chp. 9, p. 431.
Administrator F. Lassen Landorph: »Javanese Gamelan« in »Annals for Music«, published by »Dansk Musikselskab«, Copenhagen 1923, pp. 7—23, confr. Prof. v. Hornbostel: »Handb. der Physik«, VIII, Chp. 9, p. 433.
Dr. phil. Alfred Jonquière: »Grundriss der musikalischen Akustik«, Leipzig 1898, p. 120, and H. v. Helmholtz: »Tonempfindungen«, 1913, p. 457.
Director Godtfred Skjerne: Danish translation of Plutarchos’s dialogue on music, with explanation, Copenhagen 1909, pp. 1—214; Prof. Joh. Wolf: »Handb. der Notationskunde«, 1913; and Professor Guido Adler: »Handb. der Musikgeschichte«, Berlin 1924.
Ludw. Sonnenberg: »Der goldene Schnitt«, Progr. des Kgl. Gymnasiums zu Bonn, 1881.
34
Th. Kornerup:
«
be
dl * z
22
13
— 14
26 15
28 16
17
18
— 19
29 20
32 21
33 22
(Pioneer) Luca Pacioli: Divina proportione , 1509, into German in «Quellenschriften für Kunstgeschichte«, Wien 1889, pg. 196; Johannes Kepler: Letter from Prague of 12th May 1608 to Professor Joachim Tank (t 1609), in »Opera omnia«, C. Frisch’s edition, Vol. I, Frankfurt 1858, pp. 140, 145 r and 375—384. Alexander Braun in »Nova acta Acad. Leop. Carol. XV 1831, pp. 195—402; L. and A. Bravais: »Lois géométriques des spirales« in »Annales des sciences naturelles«, 2. Serie, Paris 1837, Bot. Tom. 7, pp. 42—110; Gabriel Lamé: »Comptes rendus de l’academie des sciences«, 1844, vol. III, Juillet—Décembre, pg. 867—870; Prof. H. E. Timerding: »Der goldne Schnitt«, 1918; Ing. Vilh. Marstrand: »Arsenalet i Piræus og Oldtidens Byggeregler (rules of building of antiquity) Copenhagen 1922.
Ludwig Kaiser: »Über die Verhältniszahl des goldenen Schnitts«, Leipzig 1929, p. 122; Thorvald Kornerup: »Die Hochteilung der Octave«, Copenhagen, Oct. 1930; the main contents is embodied in this treatise.
Phycicist Hans Christian Ousted: »Samlede og efterladte Skrifter«, Copenhagen, B. 7. 1858, pp. 93—95.
Director Godtfred Skjerne: above mentioned »Plutarchos«, 1909. p. 77.
Professor José Würschmidt, Tucuman in Argentine: »Die rationellen Tonsysteme im Quinten-Terzen Gewebe«, in Scheel’s Zeitschrift für Physik, Berlin 1928, p. 526, (in the German text tone »b« is used for the tone l5/s); Spanish translation: »Los sistemas de sonidos racionales«, Buenos Aires, 1928, p. 3.
Professor Joseph Yasser: »A Theory of Evolving Tonality«, New York 1932, pp. 278—84.
Professor Louis Kelterborn: »Die Quinten Spirale«, Darmstadt 1929.
Professor T. N. Thiele’s »number-figures«, grafic statements by means of points of systems in »Beretning om Naturforskermodet« in Copenhagen 1872.
Hermann v. Helmholtz’s »Tonempfindungen«, 1913, p. 441 and Ellis translation, »Sensations of Tones«, 1912, p. 269, rep. p. 25, Note.
Arrangement of ff and |? according to K. Steensen: »Den musikalske Skrive-
maade«, in Skjerne’s periodical »Musik«, Copenhagen 1922, p. 46.
Structure :
35
tn
B. Qualitative divisions.
A. Quantitative partitions.
Scheme 1. Graphie sketch of 5 kinds of calculation (primary-quintary of tones.
36
17 Sruti Nos.
Z g 4 Persian
p g blank,
<zj »3 ^ variable
o Sruti-Nos.
o TH
Cents ! Cents Cents
90 17 1200
22 16 1110
92 (15)
90 14 996
22 13 906
92 (12)
90 11 792
90 10 702
24 9 612
90 8 588
90 7 498
22 6 408
92 (5)
90 4 294
22 3 204
92 (2)
90 1 90
— 0 0
13 permanent tones Persian Doric Minor on 7 Tonics (key-notes).
1088
884
386
182
bes —
bes
es — I es —
des —
d 4-
bes-
as — as —
bes— bes
4 a-
g Î
| I
fis -
f I f
e +
des-c (c)
d +
= 17
ges
f
e 4-
(c) (c) (c)
Total
1200 Total 13 + 4
Scheme 2, P. The 17-toned Persian Sruti-System; fis-)- — ges 2—.
49
37
Scheme 2, J. The 17-toned Sruti-System, extended to 22-toned Indian-system.
38
Scheme 3. Great and small retrograde Triads: Greek Triad............ 5 Commas,
and presumably Arabian Triad ... 4 —
39
Scheme 4. The 13 permanent oriental tones and 4 variable tones (d e a b). Total 17 Persian.
40
Scheme 5, see below.
41
•gopd Âjepuooas
(psuapeui) §o[9(j AaepienQ
Scheme 5, continuation. Javanese Gamelan (salendro and pelog respectively) built up upon Lassen Landorph’s vibration-numbers,
in “Annals for Music”, Copenhagen 1923, pp. 7—23.
44
Retrograde Octave = Series HI c
eis ases
c’
Fifth -Third *
Fourth =
. • /
ases
Direct Series
Scheme 10. Great double super division (golden cut) in 4 relations (Series).
A temperament I Organic | Organic Inorganic
too small temperament temperament temperament
12-tones 19-toned 31-toned 24-toned
No. 1 ( cis. 100 cis. No. 1 63.2 cis. Nr. 2 77.4 cis. 25 o o
( des. 100 | des. 2 126.3 des. 3 116.1 des. 2 ! 100
2 d. 200Ÿ d. 3 S k 189.5 d. 5 ^ r 193.5 cisis. 3 'V150 d. No. 4 200
3 es. 300 es. 5 315.8 es. 8 309.7 , es. 6 300
5 f. 500 f. 8 505.3 f. 13 503.2 f. 10 500
8 as. 800 as. 13 821.1 as. 21 812.9 as. 16 T 800
Scheme 11. Golden Series No. I used as a means of valuation for organic and inorganic temperaments. The lower c always No. 0.
43
Es
Scheme 8. Geometrical construction of great and small double super division (the golden cut), by means of a Pentagon and a square respectively.
Scheme 9. The golden system constructed by means of the Pentagram. The sides of the isosceles triangles are Molecules, for examples deses and cis in 19-toned section.
bisis and deses » 31- » »
42
Existing.
12-toned
Practical,
Standard
50-toned
6olden system, Essential
Scheme 12. Graphic description of the deviation (falsity) of 4 organic temperaments and the Pythagorean system.
s 2.
ocus.
Centre Olx
Golden Series I... des d retrograde. 1 ^
Scheme 13. An equilateral hyperbole through the tones having No. 5 in different organic temperaments: des d es f as, the golden Series I retrograde.
Stage NS
46
Tones +2 Commas
Ceniral- tanes without plus or minus
Tones
minus 1 Comma
Tones
plus 1 Comma
cisis-
CISIS+
-11
cis! gis
a ; e
as ; es
+ 2 ; ~2
deses+
-12
Scheme 14. Change of the syntonic system into golden tones by means of numbers of k(=Vi5 of a Comma), in 9 oblique figures, each containing 15 tones, formed on 9 central-points of plus 15, 0 and minus 15 k respectively.
47
Stage Zero - - Axis.
3 gisis — 0.083
4 ais -p 7.114 e + 1 is .373 b — 4 is .367 flsis -j- — 10.108
5 fis ^98.569 # = cJs + 2.829 gis — 2.912 dis 4r 1 — 8.652
6 d + 10.052 a ■f 4.285 e — 1.456 b — 7.196
7 f + 5.741 c 0 g — 5.741 Pytl îagorea n tones.
Zero — CO <5
Scheme 15. The middle figure (and gisis) in Scheme 14, showing exact change into golden tuning.
0. 0. 0. Zero—Axis.
The highest cents. ■f V 4 cis. 76.0 0 gis. 772.6
A. Schlick 1511. 0 e. 386.3
7* = 5.377 cents. 0 c. 0
Authoritative 4~ V15 cisis. 147.6 4- Vis eis. 458.4 0 gisis. 843.2
golden system, k = 2/l5 gis. 769.7
Vis = 1.434 cents. 0 c. 0 ^vT e. 384.9 —vl g- 696.2
The lower cents 0 cisis. 141.3 -2A gisis. 837.2
G. Zarlino 1558. 0 cis. 70.7 - 7* eis. 453.9 -7t gis. 766.5
Vt = 3.072 cents. 0 c. 0 - %■ e. 383.2 -7t g- 695.8
The lowest cents — Vs cis. 63.5 — 7s gis. 758.3
0 c. 0 -f Vs e. 379.1
X. before 1558. 0 es. 315.6 — Vs g- 694.8
Vs = 7.169 cents. 0 ges. 631.3
Scheme 16. Three systems oscillating around the authoritative golden system, k of a Comma.
48
1. Doric-Tritonos Minor. Tritonos — ges.
2. Doric Minor.
3. Phrygian-Doric (or Aiolian) Minor, our descending melodic Minor.
4. Phrygian Minor, double symmetric.
9. Phrygian-Lydian Minor,
our ascending melodic Minor.
11. Phrygian-harmonic Minor, Ernst F. E. Richter 1853.
13. Turkish »uneven Tritonos .. . uneven harmonic« Minor, Neu-eser (Rauf Yekta Rev).
Scheme 17.
49
7. Tritonos-Lydian Major. Tritonos = fis.
6. Lydian Major. Indian Ma.
5. Lydian-Phrygian (or Jonic, or Jastic) Major, Indian Sa.
8. Double harmonic Major, Hedjaz-Kar (A. Z. Idelsohnl
10 a. Lydian-harmonic Major, Rimsky Korssa-koff 1893.
10b. »Lydian-uneven harmonic«, Turkish Suz-nak, (Rauf Yekta Bey).
12. Harmonic-Doric Major, Ahavah Rabbah, (A. Z. Idelsohn).
Scheme 17. Names and diagrams of 13 scale-types, on the Tonic c, in syntonic or golden tuning on c; 7 Minors and 6 Majors; No. 1 — 7 Pythagorean.
50
Jib d#
685 7/
Scheme 18. Lines through corresponding tones in the Pythag. system and 5 temperaments.
Scheme 19. Two Tetrachords tertiarily graduated as a semi-circle and a sector.
51
Scheme 20. Geometrical construction of many golden tones, by means of parallel lines.
No. Cents Dec. No. Cents Dec. No. Cents Dec.
26 16 bes 1007 5711 28 17 b 1081 0724 30 18 bis 1154 5737
21 13 as 815 1421 23 14 a 888 6434 25 15 ais 962 1447
16 10 ges 622 7132 18 11 g 696 2145— 20 12 gis 769 7158
11 7 fes 430 2842 13 8 f 503 7855+ 15 9 fis 577 2868
8 5 es 311 3566 10 6 e 384 8579 12 7 eis 458 3592
3 2 des 118 9276 5 3 d 192 4289 7 4 dis 265 9303
29 18 ces 1126 4987 31 19 c 0 — 2 1 cis ■ 1 73 5013
or No. 0 Two 19-toned Molecules, j ;:! cis ) deses 45 4263
Scheme 21. The 19-toned golden section, with 19-toned and 31-toned Nos.
Logarithmic Examples:
1) The decimal fraction of the golden Fifth?
Log. 0.696.2175 (g) = ....... 0.842.7431 1
minus Constant K.. = minus 0.600.5714
Log. difference....... = 0.242.1717 1
to which corresponds c'o 0.174.6512
to which again corresponds the decimal
fraction 1,495.34
Syntonic g = 3 a = 1,5 is a little larger.
How many particles?
Log. 720 particles.............. 2.857.3323
minus the above cx>............. 0.174.6512
Difference ... 2.682.6813 to which corresponds ... 481,594
For »syntonic g« 480 particles a litlle less.
Scheme 22, compare Chapter I, B, 4.
C 5.
cA
3.
cZ.
A.
o.
t
V.
e-
I, it
o * *. x
, ï T“
#;
—©
0 o
^- cl L
-e
a ej'à»
V j
o L * V
V V h 1
!): 9 V
Scheme 23. B new 5-lined staff (1931) applicable to other temperaments than the 12-toned temperament.
53
The first octaves The 6th octave The 7th octave
Cents Cents c —g Cents g — c’ Cents
25=32 6 000 0 26= 64 c 7 200 0 96 g 7 902 0
C 1 0 0 3 053 3 5 26 8 7 19 9
c. 21 = 2 1 200 0 4 105 0 6 53 3 8 37 7
g 3 1 902 0 5 155 1 7 279 3 9 55 2
c. 22 = 4 2 400 0 d + 6 6 203 9 8 305 0 100 72 6
e. . . . . 5 2 786 3 7 251 3 9 30 2 1 989 9
g ... . . 6 3 102 0 8 297 5 70 55 1 2 8 006 9
7 3 368 8 9 342 5 1 379 7 3 23 8
c. 2s = 8 3 600 0 e. 40 6 386 3 2 7 403 9 4 40 5
d -■ 9 3 803 9 1 429 1 3 27 8 5 57 1
e. . . . 10 3 986 3 2 470 8 4 51 3 6 73 5
11 4 151 3 3 511 5 5 74 6 7 089 8
g- ■ • 12 4 302 0 4 551 3 6 497 5 8 105 9
13 4 440 5 f# + 5 6 590 2 7 520 1 9 21 8
14 4 568 8 6 628 3 8 42 5 110 37 6
b . . . 15 4 688 3 7 665 5 9 64 5 1 53 3
c. 24- - 16 4 800 0 g. 8 6 702 0 80 e 586 3 2 68 8
17 4 905 0 9 737 7 1 7 607 8 3 84 2
fU . 18 5 003 9 50 772 6 2 29 1 4 199 5
19 097 5 1 806 9 3 50 0 5 214 6
e. . . . . 20 5 186 3 2 840 5 4 70 8 6 29 6
1 270 8 3 873 5 5 691 3 7 44 4
2 351 3 4 905 9 6 711 5 8 59 2
3 428 3 5 937 6 7 31 5 9 73 8
g. . . . 4 5 502 0 6 968 8 8 51 3 120 b 8 288 3
5 572 6 7 999 5 9 70 9 1 302 6
6 640 5 8 7 029 6 90 7 790 2 2 16 9
7 705 9 9 059 2 1 809 4 3 31 0
8 768 8 b. 60 7 088 3 2 28 3 4 45 0
9 829 6 1 116 9 3 47 0 5 58 9
b . . . 30 5 888 3 2 145 0 4 65 5 6 72 7
31 945 0 3 172 7 5 883 8 7 386 4
c. 25 =32 6 000 0 26=64 7 200 0 96 g 7 902 0 128 c’ 8 400 0
Scheme 24. Special cents-table, showing 6 octaves of the Partial-tones 1 — 128, built up upon the original table by K. Kroman, Organist, Copenhagen.
Example: b, Partial-tone No............. 15 ......... 30 ......... 60 .......... 120 ..
cents .... 4.6883 - 5.888.3 = 7.088.3 = 8.288.3
54
Th. Kornerup:
Scheme-List.
<U tc No.
X CJ H Cliché Table
6 1
11 — 2
— — 2
17 — 3
4
5
20 — 6
7
23 8 —
— 9 —.
10
27 — 11
— 12 —
29 13 —
— 14 —
31 15
32 — 16
Chapter I—II.
Graphic sketch of 5 kinds of calculation (primary-quintary) of tones.
The 17toned Persian Sruti-system, 13 permanent, 4 variable tones.
— 22 — Indian — — ,17 — ,5 —
Great and small retrograde Triads«, Greek and presumably Arabian.
The 13 permanent oriental tones and 4 variable tones as »Triads«. Total 17 Persian.
Javanese Gamelan built up upon Lassen Landorph's vibration-numbers 1923.
Chapter III—IV.
The 23 Pythagorean instrument-tones, Ivrusis; 7 diatonic, 7 chromatic and 5 enharmonic — 19 song-tones, Lexis.
The Nos. for 7 instrument-tones, presumably founded on triple-chords. Geometrical construction of great and small double super division (the golden cut), by means of a Pentagon and a square respectively.
The golden system constructed by means of the Pentagram.
Great double super-division (golden cut) in 4 relations (Series).
Chapter V—VI.
Golden Series No. I used as a means of valuation for organic and inorganic temperaments.
Graphic description of the deviation (falsity) of 4 organic temperaments and the Pythagorean system.
An equilateral hyperbole through the tones having No. 5 in different organic temperaments: des, d, es, f and as, the golden Series No. I retrograde.
Change of the syntonic system into golden tones by means of xk (k — Vis of a Comma), in 9 oblique figures, each containing 15 tones, formed on 9 central-points of plus 15, 0, minus 15 k respectively.
The middle figure (and gisis) in Scheme 14, showing exact change.
Three systems formed by means of Vs, V7> ll* of 1 Comma, oscillating around the authoritative golden system, the ideal.
. Acoustic Methods of Work.
55
Text-Page No.
Cliché Table
17
— 18 —
— 19 —
— 20 —
— — 21
— — 22
— 23 —
— — 24
___ Addendum.
Names and Diagrams of 13 scale-types in syntonic or golden timing. Lines through corresponding tones in 5 temperaments.
Tetrachords tertiarily graduated as a semi-circle and a sector. Geometrical construction of many tones, by means of parallel lines. The 19-toned golden section, with 19-toned and 31-toned Nos. Logarithmic examples, compare Chap. I, B, 4.
A new 5-lined staff (1931).
K. Kroman’s special-table, 6 octaves of the Partial-tones Nos. 1—128.
a>
0) D .
fcXD 3 6 o °
C3 £ £ o cn
Contents.
Preface. The aim of this treatise:
4
17
— 1. International terminology of Music.
— 2. Three principles concerning systematic comparative Musicology
(v. Hornbostel and R. Lachmann):
1 a) Stability in the development of tone-systems.
2—3 b) Secondary partition of a string, particles (spaces).
4 c) Relativity of interval-distances.
— 3. Acoustic methods of work (Chapter I—II).
— 4. An objective means of valuation for tone-systems (Chapter IV).
6
1
7 19
9
Chapter I. Five kinds of exact interval-calculation:
Group A. Quantitative string-partitions:
1. Primary (Flageolet-) string-partition.
2. Secondary (ornamental, decorative) string-partition. (Formula X).
Group B. Qualitative divisions:
3. Tertiary (arithmetic) decimal-division (Formula Y).
4. Quartary (geometrical) logarithm-division (Gaudy, before 1705).
5. Quintary structure, stretching of a string.
1—5. Resumé: a) Greek splitting of fractions (Formula Z).
b) Pythygorean es —, twofold calculations.
c) Natural quadruple chord.
10
17
11
12
2
2
Chapter II. Seven hypothetical principal rules:
Group A. Presumable construction of Scale-types and Tonics.
6 1. The stepwise progression of the scale and Tetrachords; Jewish
scale-tones (Idelsohn).
7 2. The oriental Indian sequence of Tonics.
— 3. The Pythagorean and Persian Tonic-sequence.
56
Th. Kornerup:
V c r. Scheme No. Note No.
— 2 —
13 — —
— —
13 — —
14—16 —
1 15 — 8
15 — —
16 19 —
17 3—4
— 5 9
19 6—7 10—11
21
— 10 12—13
22 — 14
— 10
— —
— — —
23 —
/ 8-10 \
1 20-21 /
25 — 15
27 11—13 16-19
29 14 20
31 15 —
— — —
— 16 —
32 33 35 54 55 17—24 CM cja CM
Group B. Probable construction of Ancient tone-material:
4. Permanent and variable (Persian and Indian) tones:
a) Sruti-Nos. in many S37stems.
b) The idea of Sruti-distances and Indian tertiary structure.
c) The number of touches and Molecules.
5. Chromatic and Enharmonic. Secondary interpolation:
a) Greek splitting of fractions, Formula Z; Pythagoras, Arkytas, Eratosthenes, Didymos, Plutarchos and Ptolemaios.
b) Turkish scale-types (Helmut Ritter).
c) Trichord-scales, 5-toned.
d) f — 27/2o and g — — 4%7 as difference-tones.
6. Greek (and Arabian?) »Triads«: des—, des, (d?)d+.
7. Javanese salendro- and pelog-Gamelan (Lassen-Landorph).
Chapter III. The Pythagorean System in Theory and Practice.
Chapter IV. The authoritative Tone-system, the golden System, as an objective means of valuation.
Group A. Arithmetic Series I- VI, retrograde and direct:
1. Johannes Kepler’s Series I Fourth: 1, 2, 3, 5, 8.
2. Ludw. Kaiser’s Series II Fifth: 3, 4, 7, 11, 18, 29, 47. \
3. Andreas Kornerup’s Series III Octave: 12, 19, 31, 50, 81. )
a) Golden Fifth as point of balance.
b) — — by means of the golden cut, Omega = 0,618.
4. New Series: IV Third, V Tritonos, VI Sixth.
Group B. Geometrical constructions of golden tones. Group C. All Comma-displacements disappear.
Chapter V. Temperaments;
31-toned: »the standard temp.«
Chapter VI.
Change of the syntonic system into golden tones:
a) by means of k = V15 of the Comma.
b) The exact change.
c) Rational explanation.
d) Similar methods up to 1558.
Addendum.
Bibliography 24 Schemes Scheme-List.
Contents.
*1
I
;