230 Richards ." [J. V. 1.,

APPENDIX.

T h e Goldschmid t T h e o r y of H a r m o n y .

BY JOSEPH W. RICHARDS, PH.D.

(Read at the meeting o f the Physical Seclion o f the Institute, January v2, I9o2.)

Dr. Vic tor Goldsehmid t is professor at the Unive r s i ty of He ide lbe rg , and one of the bes t .known erystal lographers in Europe. His " Index of Crystal F o r m s " and his " Angle T a b l e s " are m o n u m e n t s of indus t ry and research ; he is a fre- q u e n t con t r ibu to r to Groth 's Zeitschrif t f i i r Krystallographie und Mineralogie. His deve lopment of the me thods of gno- monic project ion, and of the appl icat ion of the " two-circle" or " t h e o d o l i t e " g o n i o m e t e r to crystal measuremen t s , has m a d e his unpre ten t ious labora tory , of bu t three small rooms, the Mecca of numerous Amer ican s tuden t s of crystallog- raphy.

Less than a year ago, the genial professor surpr i sed his f r iends no less than the scientific publ ic b y pub l i sh ing an oc tavo book of I36 pages ~ on the sub jec t of musieal har- mony , and the appl icat ion to it of the law of inter-relatioti ~rhieh governs the d is t r ibu t ion of planes in zones on the .~urfaees of erystals. Th is law was d iscovered b y Gold- ~sehmidt in his crys ta l lographic s tudies , and is called by ,him " Das Gesetz der Complicat ion," which we may trans- l a te ra ther freely as " T h e Law of Inter-relat ion." Th is law is the most general law of c rys ta l lography ; it explains the d i s t r ibu t ion of all possible planes be tween two pr imary planes. I t is so fundamenta l tha t i t includes as corollaries the formerly recognized fundamen ta l laws of crystal logra- phy, viz., the law of the ra t ional i ty of the indices, the law of zones, and the law of cons tancy of the angles. But no t only is this law the p r imary law of c rys ta l lography: Go ldsehmid t bel ieves he has proved tha t the same funda- menta l law governs, in musical harmony, the ha rmony of no tes in a chord, the sequence of related notes, and the rela-

* " U e b e r Harmonie und Complicat ion." Springer, Berlin, I9OI.

2 3 I

f 6

tions b e t w e e n a sequence of chords. In fact, " D a s Gesetz der Complicat ion " is also the fundamenta l law of musical harmony.

Fur the r than this, a l though foreign to our present sub- ject, an appl icat ion can be made of the same law in discuss- ing the harmonies of colors, the historical deve lopment or evolution of the color sense, and also to the ~esthetic sense of proportions. Goldschmid t makes some very sugges t ive iuquiries in these directions, which will be of par t icular interest to the psychologis t . If these general izat ions are true, if it can be proved even in the one case alone, of mu- sical harmonies , tha t wha t sounds well to the ear, and there- fore commends itself to the mind as agreeable , is tha t which

Sept.. t9o3. ] Goldschmidt T/,eory of Harmony.

Fig, x.

corresponds to ha rmonious deve lopment in the objec t ive realit ies of crystal forms, then we have a pure ly objec t ive explanat ion of our sub jec t ive discr iminat ions. W h a t sounds agreeable to our minds is tha t which has its real harmonious counterpar t in ob jec t ive nature , and thus the final source of mental sa t isfact ion is tha t one finds his mental processes in accord wi th the grea t harmonies of Nature .

T H E L A W OF I N T E R - R E L A T I O N IN C R Y S T A L L O G R A P H Y .

A m o n g the planes occurr ing on one crystal, and which may be grea t in number , certain planes are par t icular ly im- portant because of f requency of occurrence and size, o thers occur seldom, while o thers are qui te rare. The rarer the plane, in general , the smaller it is, and the less certain is its identification. The feebler planes are ar ranged in series

232 Richards : [J. F. I,

be tween the principal or p r imary planes, in zones, with parallel intersect ions.

Let A and B (Fig. z) be two p r imary faces (K may be a third). Then, by different iat ion there is first formed be- tween them the plane C, t runca t ing with parallel edges the edge be tween A and B. If the .differentiat ion proceeds further , we m a y have D and E t runea t i ng the edges between A C and CB respectively. The planes D and E are sub- ordinate to C. By still fu r the r different iat ion there are formed the sti!l feebler t runca t ing planes F, G, H, L We thus have, so far, th ree stages of different iat ion or develop- men t in the h is tory of this zone.

N o = A B N I = A c B & = A o C A B N 3 = A p D a C H E I B

Usual ly the deve lopment stops at N1, often it goes to N2, seldom to N3, and very rarely to N 4 or over. The first derived plane, C, the most impor tan t next to the p r imary planes, is called the "dominan t . "

A similar deve lopment can take place On the edge be- tween A and K or B and K. We would call these the " p r i m a r y zones" A B, A . K , B K, be tween the pr imary planes or nodes A, B, K. F u r t h e r different ia t ion be tween one of the p r imary planes, as K, and the " d o m i n a n t " of the o ther two, as C be tween A and B, gives us " seconda ry zones," such as CK, each with a " s e c o n d a r y dominant ." Differen- t iat ion be tween two " p r i m e d o m i n a n t s " would give ter t iary zones, wi th " te r t ia ry dominan t s ; " while different iat ion be tween a pr ime dominan t and a secondary dominan t would yield " q u a t e r n a r y zones" wi th " q u a t e r n a r y dominants ." Zones, therefore, have thei r orders, t hough very seldom is the order above secondary. This scheme of deve lopment gives a rich mass of zones, especial ly if the p r imary faces are well developed and the different ia t ion be tween them goes up to N2 or N 3.

In every zone, on every crystal, the a r r a n g e m e n t of the planes relative to each other and to the p r imary planes of

Sept., Ic~o3.] Goldschmidt Theory of Harmony. 233

that zone follows a de te rmined law of numbers . If, by a simple geometr ica l device, the p r imary planes be represented in posit!on by o and ~ respect ively, the dominan t will be represented by i, and the subordinate planes by { and 2, ~t and 3, ~ and 3, etc., fo rming as a whole a ha rmonic pro- gression such a s o ½ ½ ~ i ~ 2 3 ~ . These we may call har- monic numbers or tile harmonic series. We will proceed to show how it applies to crystal planes in a zone.

Assuming tha t the physical part icles which build up the crystal have pr imary forces of de te rmined direct ion and intensity, and tha t these are the const ruct ive forces which at tract other part icles and build up the crystal planes, we may fur ther assume tha t the plane-bui lding forces are per- pendicular to the faces formed. The pr imary crystal-build-

~o o -; ¢ . ~ "-~-B

t I I / / / !

~tr Fig. ~.

ing forces are therefore perpendicular to the pr imary crystal planes, and the secondary or der ived forces are perpendicular to the secondary or der ived planes. Considering, therefore, each plane of a crystal to be buil t by a force at r ight-angles to tha t plane, we may replace all the planes by their normals, and discuss these normals as represen t ing the plane-build- ing forces.

Tak ing the planes a l ready i l lus t ra ted in F~,. z, we may, from any point inside the crystal, drop perpendiculars to the respective faces. Let M (Fig. 2) be the point chosen ; then the p r imary planes A and B of Fig. z are here replaced or rep- resented by their normals A and B, the in te rmedia te plane C by its normal ¢, 29 and E by thei r normals d and e, etc. The replacing of planes by thei r normals is a common device in mathemat ica l crysta l lography. The p r imary forces A and B have therefore a de te rmina te direction, the angle be tween

234 Richard$ : [J. F. I.,

them being the,~mae as the angle between the planes them- seA-a~,-and the relat ive intensi t ies of these forces are as the reciprocals of the axial ratios in the respective directions. Given, then, these pr imary forces of given direction and intensi ty, the " l a w of inter-relat ion " considers the differen- t ia t ion or development of subordinate planes to have pro- ceeded in the fol lowing m a n n e r :

The forces A and B split into two halves ; one of each of these halves, a and b, uni te to form the resul tant c. The plane Cresu l t s normal to c. The process repea t ing itself,

, ½ a uni tes wi th ½ c to a resu l t an t d, and likewise ½ b and ½ c to e. By fur ther repet i t ions of the same process still fur- ther, feebler resu l tan ts form between a d, d ¢, e b, etc. So we find the directions of the derived forces, and thence the position and order of rank of the derived planes. The planes s tand normal to the forces, the order of rank corresponds to their relat ive impor tance in the order of development .

By d rawing th rough A a parallel to B, and l eng then ing the forces M a, M d, AI c, etc., unt i l they intersect A Z, the directions of the forces are characterized by the points of intersect ion A, D, C, E, B. B li.es at infinity. Calling A -~ o, the s ta r t ing point of measu remen t on A Z, and A C--~ x, because it represents equal components of a and b; then will A D ~ ½ , A C ~ x , A E - - - - 2 a n d A B-~--oo. The intersect ions o n A Z may be called the project ion points of the different planes in the zone A t t ; the project ion points characterize the posit ion of the planes; the i r place is des ignated by the dis- tances from A, by the numbers o, ½, i, 2, oo. These num- bers are the harmonic numbers , their series the harmonic series, and, if complete, the normal series. The law of development is expressed in the normal series and the har- monic numbers as follows:

P r imary planes : A No ~ Normal series o ~ o

i. Complicat ion: A N~ -~ Normal series x ~ o

2. Complicat ion: A N, ~ Normal series 2 ~ o

C I

D ~ E ½ I 2

B

B oo

E

Sept., i933. ] Goldschmidt Theory of tIarmony. 235

3. Complicat ion: A F O G C /4 E I B Ns ~ Normal series 3 -~ o ½ ½ .~ I ~ 2 3 Cx~

It is clear how the h igher series would look, but nature very ~mlflam Ae•elops p lanes .above t h e ~e"ries Ns.

To inves t iga te any series of numbers, so as to test them as to harmonic relations, it is necessary to make a transfor- mation, to subs t i tu te for the end members 0 and o~, and for the in termedia te members their corresponding values. These intermediate values are easily found by dividing the distance of the number from one end by its dis tance from the other. Thus, given the series of members ,

if we call the extremes x and 2, 0 and oo respectively, the intermediate members become--

4 i

2 4 2 a 8 ~ _ _ ~ - - I - - I

8 2 2

tl - - I

2 - - 3 and the series becomes in harmonic n u m b e r s - -

o I 2 o =W2

Or, in general, call ing Z and Z2 the two "extremes of any series of numbers , Z any in te rmedia te number , and p the corresponding harmonic number of a harmonic series, t h e n ~

p _ Z ~ Z 1 f o r Z ~ o a n d Z ~ = o o Z~ - - Z

The s tudy of crystal forms, par t icular ly by the method of gnomonic projection, in which all zones are projected in s t ra ight lines, and the posit ion of planes in series can be directly measured or calculated, has subs tan t ia ted the " law of i n t e r r e l a t ion" as being the law governing the distr ibu- tion of planes in zones. W e will now s tudy the diatonic scale in the l ight of the same law, t r ans forming its mem- bers into harmonic series, and therein mak ing clear the fun- damenta l relat ions of the various notes to each other.

[To be concluded.]