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 WEAD 
 
 Contributions to the 
 History of Musical Scales 
 
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 SMITHSONIAN INSTITUTION. 
 
 UNITED STATES NATIONAL MUSEUM. 
 
 CONTRIBUTIONS TO THE HISTORY OF 
 
 MUSICAL SCALES. 
 
 BY 
 
 CHARLES KASSON WEAD, 
 
 Examiner, U. S. Patent Office. 
 
 Prom the Report of the United States National Museum for 1900, pages 417-462 
 
 with ten plates. 
 
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 GOVERNMENT PRINTING OFFICE. 
 1902.
 
 SMITHSONIAN INSTITUTION. 
 
 UNITED STATES NATIONAL MUSEUM. 
 
 CONTRIBUTIONS TO THE HISTORY OF 
 
 MUSICAL SCALES. 
 
 BY 
 
 CHARLES KASSON WEAD, 
 
 // 
 
 Examiner, U. S. Patent Office. 
 
 Prom the Report of the United States National Miiseum for 1900, pages 417-462, 
 
 with ten plates. 
 
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 7-r/ ^- ^^"^-^^ 
 
 |per\ 
 
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 WASHINGTON: 
 
 GOVERNMENT PRINTING OFFICE. 
 1902. 
 
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 CONTRIBUTIONS TO THE HISTORY OF 
 MUSICAL SCALES. 
 
 BY 
 
 CHARLES KASS0:N^ AVEAD, 
 
 Examiner, U. S. Faterd Office. 
 
 417
 
 TABLE OF CONTENTS. 
 
 I. Introduction 421 
 
 II. Stringed instruments 424 
 
 III. Instruments of the flute type - 426 
 
 IV. Instruments of the resonator type 428 
 
 V. The influence of the hand 433 
 
 VI. Composite instruments - - - - 436 
 
 VII. Conchisions - 437 
 
 Appendix - - - 441 
 
 LIST OF ILLUSTRATIONS. 
 
 PLATES. 
 
 Facing page. 
 
 1. Stringed instruments 444 
 
 2. Flutes with equal-spaced holes 446 
 
 3. Fhites with equal-spaced holes 448 
 
 4. Flutes with equal-spaced holes - 450 
 
 5. Flutes with holes in two groups - 452 
 
 6. Flutes with holes in two groups - - 454 
 
 7. Central American resonators or whistles 456 
 
 8. Composite instruments 458 
 
 9. Pan's pipes 460 
 
 10. Scales given by resonators 462 
 
 TEXT FIGURES. 
 
 Page. 
 
 1 . European mandolin, after Viollet le Due 424 
 
 2. Greek guitar, after Drieberg 424 
 
 3. Terra cotta whistle, after INIahillon 430 
 
 4. Babylonian whistle, after Engel 431 
 
 5. Chinese resonators, after Amiot 431 
 
 6. Globular whistles, after Frobenius 432 
 
 7. Globular whistle, after Kraus - - '^^^ 
 
 8. Xylophones, after Kraus '^^^ 
 
 419
 
 CONTRIBUTIONS TO THE HISTORY OF MUSICAL SCALES. 
 
 By Charles Kasson Wead, 
 Examiner, United States Patent Office. 
 
 I. INTRODUCTION. 
 
 In the development of musical scales four stages may be recognized: 
 
 1. The stage of primitive nuisic, where there is no more indication 
 of a scale than in the sounds of birds, animals, or of nature. Students 
 of the origin of music may give free rein to their fancy in this period, 
 and the uncertain musical utterances of living primitive peoples may 
 be construed in accordance with almost any prepossession of the hearer. 
 
 2. The stage of instruments mechanically capable of furnishing a 
 scale. This stage has been almost entirely overlooked by students and 
 is the special subject of the following paper. 
 
 3. The stage of theoretical melodic scales — Greek, Arab, Chinese, 
 Hindu, Medigeval, etc. All the original treatises concerning these 
 scales imply that a stage of development has been reached far in 
 advance of the second. Thousands of pages have been written on this 
 stage, largely polemical and lacking in insight, for the subject has been 
 a dark one; but Ellis and Hipkins's work of 1885 has thrown a flood 
 of light on it. 
 
 4. The stage of the modern harmonic scale and its descendent, the 
 equally tempered scale, which are alike dependent both on a theory 
 and on the possibility of embodying it in instruments. The relation 
 of this scale to the present study will be noticed later. 
 
 These four stages, of course, overlap even in the same locality; they 
 correspond in a rough way to the recognized four culture stages, 
 namely: the savage, barbarous, civilized, and enlightened. 
 
 At the outset it should be recognized that the only working hypothe- 
 sis the physicist can use is that of the instrumental origin of scales. 
 Helmholtz's view that the harmonics in the voice and in the tones of 
 instruments were influential in settling the positions of the notes of 
 our scale is obviously consistent with this hypothesis; and his opinion 
 that this influence acted on other scales need not be wholly rejected, 
 though some of his historical authorities were untrustworthy, and 
 
 ■421
 
 422 REPORT OF NATIONAL MUSEUM, 1900. 
 
 siOiiH- of ills coincidences between other scales and the harmonic scale 
 can be explained in other and simpler ways. Writers less careful than 
 Helmholtz have made the assumption that these harmonics and the 
 constitution of the ear nuist have o'uided primitive musicians to a sub- 
 stantially harmonic scale; and one writer has even maintained that 
 instruments corrupted the taste of men. But as yet there has been 
 no such ))ody of facts collected in support of this assumption as need 
 deliiy one followinj^' out the other theory. Of course the knowledge 
 of the scales is only a step|)ino-stone to the understanding of the music 
 and something of the life of a people; so some day the materials worked 
 into shape by the physicist ma}' be built into a fairer structure by the 
 psychologist. 
 
 The ])road fact which underlies all study of scales was recognized by 
 the Greek nuisician Aristoxenus three centuries before the Christian 
 era. He pointed out that the voice, in speaking, changes its pitch by 
 insensible gradations, while in singing it moves mostly by leaps. We 
 recognize the same fact when we say that a singer follows a scale, but 
 do not say it of a speaker. The one, to use the conunon figure, ascends 
 or descends a ladder or staircase; the other follows a continuous slope, 
 and ma}^ never step twice in the same place. Now, it is quite possible 
 that in a song the voice may alwaj^s move by leaps, and in repeating 
 the song always take the same leaps as closely as can be observed, yet 
 never strike a note which it has struck before; just as one may toss 
 a stone up and doAvn on a hillside, marking each time where it lands, 
 and after a hundred tosses finds it had not landed twnce at (juite the 
 same level, or in striding up and down hill may never plant his foot 
 twice at the same level. I think this was the character of the songs 
 of the first stage and of nuich primitive song to-da}', though the 
 evidence is too scanty to be conclusive. 
 
 However this may be, it is certain that most peoples who have attained 
 any moderate degree of civilization have attempted to limit the num- 
 ber of steps to be tak<Mi by the voice in any song between the highest 
 and lowest note, and to fix these steps by rules, so that many men 
 may learn them and be in substantial agreement. Various old writers 
 give the rules in vogue among (ireek theorists; in the last century 
 Amiot descrilx'd the Chinese rides, while in th<» last two decades the 
 rules of Arab, Hindu, flapanese, and Siamese nuisicians have been 
 made accessible. The most familiar rules, as is well known, depend 
 on that law of \ ibi'ating strings which is followed by a violinist in his 
 fingering — namely, that the frecjuency of vibration of parts of any 
 stretched string is inversely as the length of the parts, provided the 
 ti'nsion does not change. Our latest rule, historically derived from 
 one of the manv (ireek and Arab rules bv subdividintr the whole tones, 
 so giving twelv(> ste])s to th(^ octave, is eml)()died on the neck of a guitar 
 or mandolin: here it is obvious that the successive stopi)ing points as
 
 HISTORY OF MUSICAL SCALES 423 
 
 marked by frets get cloiser and closer together us the pitch rises. All 
 musicians know that this number of notes, twelve, is found confus- 
 ingly great for ordinar\^ playing, and know the principles by which the 
 player selects certain notes for any tune. But this inidtiplicity of 
 notes has an important bearing on all studies on nonharmonic nuisic 
 made b}' harmonic musicians. For every sound within the compass 
 of the instrument comes very near to some one of the twelve notes and 
 may readily be represented thereby, owing to the difficulty the hearer 
 has in estimating deviations from the familiar series and in noting them 
 down. The results of this approximation are to mask all deviations 
 from the twelve-tone piano scale, whether intentionally or accidentally 
 made, and to make it appear to musicians, first, that nearly all the music 
 of the world is performed substantially in our scale; and second, that 
 any other theoretical scales, such as those found among Orientals, or 
 described b}^ our European ancestors, are merely niathematical jug- 
 glery and of as little significance as proposals for a change that occa- 
 sionallv appear in modern musical or scientific journals. 
 
 It is the purpose of this paper first to describe several types and 
 forms of instruments widely used, each emljodying a i)rinciple of scale 
 building distinctly unlike ours, though sometimes giving a result that 
 seems surprising!}- familiar. Nearly all these instruments, it will be 
 noted, belong to what was called above the second or barbarous stage, 
 though a few of them come from countries where musicians have 
 reached the third and fourth stages. A second purpose is to present 
 a new and generic principle of primitive scale-building applicable to 
 the various types of instruments discussed. 
 
 But before going further it must be recognized that the word 
 "'scale'' has many meanings. Perhaps the lowest and loosest is — the 
 series of sounds used in any musical performance, arranged in order of 
 pitch. The one that will most closely fit the present needs is — the series 
 of sounds produced upon a particular instrument; while the most 
 exact definition, but one applicable only where musical principles are 
 well developed is this: 
 
 A scale is an indepeiidently reproducible series of sounds arranged 
 in order of j^iicK recognized as a standard and iitted for musical 
 purposes. 
 
 While the last two definitions implv an instrument in which tne 
 scales are embodied, the limitation is in appearance only, for there is 
 no evidence that any musicians do have a standard series of tones, 
 unless they have one or more instruments embodying it, and have 
 learned the series directly or indirectly from such an instrument.
 
 424 
 
 REPORT OF NATIONAL MUSEUM, 1900. 
 
 II. STRINGED INSTRUMENTS. 
 
 Ill shiirp coiiti-u.st to that widrly used division of a string which we 
 know on the guitar, showing decreasing distances between the frets as 
 the pitch rises, we find many instiinces of a uniform spacing of the 
 frets through a considerable distance. Instances from four countries 
 may here be cited: 
 
 1. The well-known arcliitect, YioUet-le-Duc,^ gives a figure (tig. 1) 
 of a mandolin from the end of the sixteenth century which shows frets 
 
 for the first seven semitones pretty uniformly 
 spaced; the frets for the next five to complete the 
 octave are again imiform, though closer than be- 
 fore, and the following five are also uniformly 
 spaced and still closer. Figures in other books ^ 
 of European lutes, viols, etc., very often show a 
 similar equal spacing. These are too numerous 
 to be lighth' treated as artists' blunders. Two 
 instruments in the United States National Museum 
 are illustrated in Plate 1. 
 
 ^. Among the Greek rules given 
 by Ptolemy is one for the division 
 called Diatoncm, homalon^ in which 
 the whole string being twelve units 
 long the points for stopping would 
 be at 11, 10, y, and S, giving C, a 
 note between Dj^ and D, Ej?, F, and 
 G. Here it will be noticed the inter- 
 vals get larger and larger as the pitch 
 rises. Again, Carl Engel ^ refers to 
 Drieberg's drawing of the ancient 
 Greek guitar in the Berlin Museum, 
 which has "seven frets at equal dis- 
 tances," but objects to it as it does 
 not give a diatonic scale. The tracing of this drawing 
 furnished by Professor Howard, of Harvard, adds to 
 Eiigel's data the fact that the whole compass of the six 
 intervals is slightly more than an octave (fig. 2). 
 
 3. Among the instruments described in the Arabic treatise of the 
 
 \ 
 
 Fig. 1. 
 
 EUROPEAN MANDOLIN. 
 Alter Vio!Iet-le-Duc. 
 
 Fig. 2. 
 GREEK GI'ITAR. 
 After Drieberg. 
 
 Plates V, fig. 8; vi, 
 Rome, 177(>. Plates 
 
 ' Dictionnaire raisonn^ du niobilier franc^aip, II, 1871, ])I. m. 
 ■''M. Priotoriup, Syntajrina Mnsicuui, II, ItilS. Reprint, 1894 
 fig. 1; XVI, fig. 1; xvii, fig. 4; xx, figs. 1, 3. 
 
 Bonanni, DcscTiptinu de.s instruments harinoniques. LM. ^'^\. 
 
 LII, l.VII, I,X, LXXI. 
 
 J. Ruliliiiann, (iesehichte diT Bogeninstrumente, 1882. I'lati-s ix, figs. 2, 5, 0, 13; 
 X, fig. 1*>; XIII, figs. 3, 8. 
 'Music of tin- M(^st Ancient Nations, 1864, p. 205,
 
 HISTORY OF MUSICAL SCALES. 425 
 
 famous Al Fanibi,^ who died 950 A. D., is the short-necked tanhmr 
 of Bao-dad, usually having- two strings: on this a fret was first placed 
 at one-eighth the length of the string from the upper end, and this 
 space then divided into five equal parts. As the compass on each string 
 was but little over a whole tone, each step was about a quarter-tone. 
 These ligatures or frets are called "heathen '' or "pagan," and the tunes 
 played on them "heathen airs," clearl}" indicating that there was a 
 scale native to the people whom the Mohammedan armies had con- 
 quered, a scale utterly different from either that of the lute or the 
 tmiboiir of Khorassan, with their resemblances to Greek scales. Three 
 hundred years later, or about 1250 A. D., Safi-ed-din,^ a famous musi- 
 cian of Bagdad, wrote for his pupil, the son of the Vizier, a Treatise 
 on Musical Ratios. He based them on string lengths, and in discussing 
 instruments gives a figure of the frets on the neck of the lute, and it is 
 noteworthy that these are equally spaced over a distance of a quarter 
 length of the string. Further, he explains how of the ten frets in this 
 short distance, located by various rules, five were fixed by arithmetical 
 bisection or halving of the space between two frets already fixed; one 
 of these, midway between what we should call D and E, if the open 
 string gives C, was called the "Persian middle," and was very much 
 in use in his time. Safi-ed-din'^ further describes, in two connections, 
 a division of the Fourth, like the Greek one already quoted, where the 
 string lengths are 12, 11, 10, 9, saying it is consonant and much used; 
 in fact it is preferred to one that is substantially like the theoretical 
 diatonic scale; still it should be added that when he comes to arrange 
 intervals to make up two octaves he puts our arrangement along with 
 the most agreeable half dozen genera. 
 
 4. In India there has been in modern times a curious reversion from 
 an elaborate historical scale of twent^'^-two steps to the octave, of which 
 no modern Hindu or European knows the theory, to an equal linear 
 division;* one-half of the string on the sitar is bisected; the first or 
 end quarter-length is then divided into nine parts, each marked by a 
 fret, and the second quarter-length into thirteen parts similarly 
 marked. Out of the twenty-three tones within the octave the player 
 selects a limited number, five, six, or seven, rarelv eight, for any par- 
 ticular tune. Most of the notes used are found on calculation to be 
 deceptively close to the notes of our chromatic scale, and so may be 
 easily confounded with them by European hearers. 
 
 5. This arithmetical division has been advocated by European 
 
 ' Land's translation in Travaux de la 6*^ session du Congres internationale des Orien- 
 talistes a Leide, 1883, pp. 107-114. 
 
 ^Carra de Vaux's translation in Journal Asiatique, XVIII, 1891, p. 330. 
 
 ^Idem, pp. 308-317. 
 
 * Tagore, Musical Scales of the Hindus, Calcutta, 1884, supplement. Partly quoted 
 by C. R. Day, Music ... of Southern India, 1891, and Ellis, Journal Society of Arts, 
 XXXIII, 1885, p. 502. 
 
 NAT MUS 1900 30
 
 426 REPORT OF NATIONAL MUSEUM, li»00. 
 
 tliforists, iis l»_v Jiiiinu'd' in ti treatise of ITo!). a copv of which is in the 
 Lrnox Library. New York: and Fetis" in his brief account of this 
 author refers to others who maintained siniihir views. 
 
 III. INSTRUMENTS OF THE FLUTE TYPE. 
 
 The sinipU' Hutes arc instruint'nts of a type more primitive and more 
 widely distributed than fretted strinoed instruments. These instru- 
 Ujcnts arc sometimes si(U'-bU)wn, as is the case with the modern tiute; 
 or ^'u^\ bh)\vn. as one ]>iows into a key or pan's pipe; or bk)wn with a 
 whisth" mouthpiece, like the Hatj;e<)let; or l)lown with a weak reed, as 
 tlic oboe. For the purposes of this discussion the mode of excitintr 
 the vibration is inunaterial. All of them (Mubody the law that the 
 fre(|ueiiev of vibration of a column of air in a tube depends mainh'on its 
 IcnLTth. and the variation in length of the air column so as to produce 
 several sounds from one tuV)e is produced by opening holes in the 
 sides of the tube. In practice these hoh\s never can open so freely to 
 the outside^ air that the portion of the tube beyond them may be con- 
 sidered as icmoved (the possi))ility or necessity of cross-tingering 
 proves this to th(> })layer). so the proper location and diameter of the 
 holes to produce the notes of our scale of even quality are fixed, not 
 t)y a simple law as the frets on the guitar are located, but by laborious 
 exi)erimenting to get a standard instrument which is then reproduced 
 with Chinese hdelity. 
 
 Now. as one looks over a collection of wind instruments, like the 
 splendid one in the U. S. National Museum, or examines flutes figured 
 in books, it will l)e easy to recognize that there are two principal 
 types — (A) those having the holes spaced at sensibly equal distances, 
 and {l^) those having- two groups each of three equally spaced holes, 
 the interval l)etween the nearest holes of the two groups being" obvi- 
 ou>ly greater than that between the holes of each group. As the 
 eonunoii primitive method of making the holes is by burning, the 
 holes are generally more uniform in diameter than those on European 
 f1ut<'s of a centuiT ago. 
 
 Illustrations of flutes of type A are found in Fngers Musical 
 Instruments, some of which are copied on Plate '2. Dr. Wilson's 
 paper on Prehistoric Art'' has many more illustrations, as the figures of 
 bone flutes from Costa Rica and British (niiana, of pottery flutes from 
 Mexico and the Zuni Indians, of tubes with a simple reed from Egypt 
 and Palestin*'. of wooden llutes brought from Thibet by Mr. Rockhill, 
 and a woodi-n (lute from the Kiowa Indians. Fetis* has a cut of the 
 -t iL'-horn flute from the .stone age with three e(|uidistant holes, referred 
 
 '.Taiimnl, Kccherchcs siir la Theorie <le la Musique. 
 •Fi'tis, Hionraphit' uiiiver>»elli' <U'8 Musiciens. 
 ' Report of the V. S. National Musenni for 1896, pp. ;52.5-664. 
 * lli.>^toirt' gciu'iak' di- la iiiUi^iipK-, I, p. l'G. 
 
 I
 
 HISTORY OF MUSICAL SCALES. 427 
 
 to ))Y Wilson (p. 526). So far as is known not one of the peoples 
 from whom these instruments have come has any musical theory, but 
 some of them do have a principle of instrument construction; for a 
 parth' educated young- Kiowa Indian, in Washington a few years ago, 
 in a party under charge of Mr. James Moone}", showed the writer how 
 the holes on a flute on which he played were located by measuring 
 three finger-breadths from the lower end to the lower hole, and then 
 taking shorter ])ut equal spaces for the succeeding holes. The inter- 
 preter added that he had seen the holes spaced by cutting a short stick 
 as a measure. The late Mr. F. H. Cushing has furnished the addi- 
 tional fact that measurement b}^ finger-breadths is very common among 
 Indians; and Dr. Fewkes^ gives a figure to show how the prayer 
 sticks, used by the Hopi Indians in the Snake ceremonials at Walpi, 
 are measured off into seven parts by the distances from creases on the 
 hand to the tip of the finger. On the Kiowa flute (Plate 4, No. 2) the 
 distance between the centers of the holes is 32 mm., which is two 
 medium finger-breadths. Some instruments of this type belonging to 
 the U. S. National Museum are show^n in Plates 3 and 4. 
 
 But it is not only among primitive and prehistoric peoples that 
 such a succession of holes is found. The common military fife has it. 
 The bagpiper recently seen on the streets of Washington used a 
 chaunter (oboe), the holes of which were at sensibly equal distances, 
 so conforming to the well-known fact that the bagpipe scale is inten- 
 tionally unlike the harp scale. A Japanese Fouye with 7 holes figured 
 in the catalogue of the Kraus collection at Florence shows to the 
 eye holes at nearly equal spaces, and has, as reported, the steps of 
 the scale increasing in length as the pitch rises. From Egypt' there 
 have come twenty-five 3- and 4-hole ancient flutes, or more exactly, 
 oboes, and a few of 5, 6, and more holes. One of the 4-holed instru- 
 ments from a tomb of about 1100 B. C. shows the holes 35 mm. apart 
 and the lowest hole twice this distance from the bottom. Villoteau's^ 
 plates of modern Egyptian instruments show various types of tubes 
 with equally spaced holes. 
 
 Flutes of the second or B type with two groups of equal-spaced 
 holes were sold in quantities at the Java village at the World's Fair 
 held in Chicago in 1893 (Plate 0, No. 1). No two of the instruments 
 seemed to have the same length or location of holes, but this group- 
 ing was unmistakable. Of this type is also a curious ancient Chinese 
 instrument, the Tche, described by Amiot,* closed at both ends with 
 
 ^Journal of American Ethnology and Archaeology, IV, 1894, p, 25-26. 
 
 ^Loret, Journal Asiatique, 8th ser., XIV, 1889, pp. Ill, 197. Musical Times, Lon- 
 don, XXXI, 1890, pp. 585, 713. 
 
 •^ Description de 1' Egypt, Etat moderne, II, 1809, plate cc. 
 
 ^Memoires concernant I'histoire .... des Chinois, VI, 1780, p. 76, pi. vi, tig. 42. 
 Mahillon, Brussels Conservatory Catalogue I, No. 865.
 
 428 KEPoKr OF NATIONAL MUSEUM, liX)0. 
 
 an rmhouclicn' at the iiiiddlf and holes .syminetrically placed on each 
 side dividing'- the whoh' UMi«:th into thirds. (juartei>, and sixths; so. if 
 th(>\\h()h' length is called 12, the mouth hole is at 6 and the tinger 
 holes at 2, 8, 4, 8, '.♦. and 1(». Mahillon copied the instrument, but did 
 not close th«> ends, and reports the scale as a chromatic one from E to A #. 
 Most of the old European wood wind instruments fig-ured by Pneto- 
 rius' (Itils) are conspicuously of this type, as the appended Plate 5 
 shows without necessity of description, and v'arious similar instru- 
 ments of the Museum collections are ligured in Plate <>. 
 
 IV. INSTRUMENTS OF THE RESONATOR TYPE. 
 
 I . The next group includes a variety of instruments of the resonator 
 tvpc. a type that is widely distributed and conforms to a law hitherto 
 unrecognized as capable of furnishing a scale; though Sondhaus in 
 lsr>() stated the law and tried a few rough experiments. The mathema- 
 ticians" have provi'd that a mass of air in a contined space with a ver}^ 
 small nearly circular opening, as a short-necked bottle or a whistle, 
 has a frequency of vibration proportional to the square root of the 
 fraction which expresses the diameter of the hole divided b}' the volume 
 of the cavity; and if there are two such openings so placed that the 
 flow of air through one does not interfere with that through the other, 
 the numerator of the fraction will be the sum of the two diameters. Now 
 extend the same principle, and one may have a series of sounds rising 
 in pitch as on(» after another of several holes in the wall is opened; 
 and ])rovided the character of the vibration is not essentially changed, 
 the freijueiRy of vibration of these notes W'ill increase as the scpiare 
 root of the sum of the diameters of the holes opened. Suppose, for 
 examph'. that a vessel has one mouth-hole of diameter 2 and several 
 properly placed finger-holes of diameter 1; then on successively open- 
 ing these a scale may b(» jiroduced having vibration frequencies in the 
 ratio of the s(|uare roots of 2, 3, 4, 5, etc. A moment's consideration will 
 show that in such a scale the intervals between successive sounds 
 become less and less as the pitch rises, instead of l)ecoming greater as 
 is the ca.-e with strings or flutes Avhere the spacing of frets or holes is 
 unifoini. 
 
 I 111' iiio-t elaborate and beautiful illustrations of instruments of this 
 type a IV tioiii graves in Central and South America. (See Plate 7.) 
 1 li«' I iiitrd States National Museum lias many whistles f rom Chiriqui 
 ill Colombia, most of tliciu giving but a single high note; these differ 
 substantially, it will be noticed, from stopped organ pipes, since in the 
 latter the month extends the full width of the tube. Whistles with 
 one or two tinger-holes have come from ^lexico and San Salvador, but 
 the most coni])!.!.- and perfect are from Costa Rica. Of these the one 
 
 1 
 
 S\ iiiauiiiji Musicuni, yils. ix and x. 
 
 Kayli-ijrh, Tlii'ory <-f Sdund, II, ]S78, Chai.. ■'^"^J-
 
 HISTOEY OF MUSICAL SCALES. 429 
 
 bearing- tho catalogue number 50970 (Plate T, tig. 1) has served as the 
 type specimen, and is the instrument which led to this investigation. 
 It has a globular body with bird's head, a mouthpiece about in the 
 position of a bird's tail, and four finger-holes on the back symmetri- 
 cally placed; these holes seem to be precisely equal in diameter, and 
 equivalent in musical effect, so the order of lingering is a matter of 
 indifference, and all the tones are clear and distinct; in Dr. Wilson's 
 paper,^ Mr. Upham, who is a violinist, notes them as F, A, C, D, E. 
 On measurement the volume was found to be 36.0 cc, the equivalent 
 diameter of the trapezoidal mouth hole 1 cm., and the diameter of the 
 linger holes .65 cm.; these diameters, however, need a correction on 
 account of the thickness of the walls, since the air can not pass freely 
 through the rather thick wall. The tinal result of the calculation is 
 to give, with all iinger-holes closed, the note F on the highest line of 
 the treble staff", to within half a semitone, and on opening the finger- 
 holes in iinj order to give the succession of intervals 4, 3, 2, and 2 
 equal semitones, with a mean error of only one-eighth E. S. Accord- 
 ing to the theory the series of intervals depends only on the ratio 
 between the diameters of the holes and the mouth hole, in this case 1 
 to 1.62; so the series of tones has vibration frequencies approximately 
 as the square roots of 1.6, 2.6, 3.6, 4.6, 5.6, or of 1, 1.62, 2.24, 2.86, 
 3.48; but the pitch of all depends on the quotient of the radius of the 
 mouth-hole by the volume. Although the theoretical correction for 
 thickness of wall can not be quite precise, it affects all the holes to 
 nearly the same extent, and the greatest probable error that can be 
 assumed will not change the whole compass more than half a semitone; 
 so the calculated scale would still be substantially what the ear con- 
 firms — F, A, C, D, E, or in syllables do, mi, sol, la, si. 
 
 The Museum has several other Costa Rican instruments also of 
 pottery quite similar in appearance to this, but not capable of giv- 
 ing such clear tones, or quite so perfect in the equality of the holes. 
 If the holes are unequal in diameter, in thickness of wall, or in loca- 
 tion with reference to the vibrating mass of air, the order of pitch will 
 depend on which holes are opened instead of merel}^ on how many; 
 with five holes sixteen combinations are possible; but of the eleven 
 instruments in the Museum eight give only five notes each, two give 
 seven notes, and one gives nine notes. If the finger-holes are small 
 relatively to the mouth hole, the compass is small, so one high-pitched 
 whistle has a compass of only six semitones — G to C# — and another runs 
 from B to E; three have a compass of seven E. S., that is, a musical 
 fifth, and two each have, respectively, eight, nine, and eleven semitones. 
 
 Still other National Museum instruments, similar in principle, but 
 ruder in workmanship and more grotesque in form, have come from 
 Chiriqui, Columbia, and are figured in Dr. Wilson's report, pages 628 
 
 1 Report of United States National Museum for 1896, p. 617.
 
 4:m) 
 
 REPORT OF NATIONAL MUSEUM, ]i«)0. 
 
 to (>4<>. In othoi- iim^cuuis similar iiistniinciits are to be found. A 
 h'W from C'liiri(iui were biicHy desoi'il)ed forty yoai's ag-o as belonging 
 to tlie American Ethnological Society/ 
 
 III tlic Amci'ican Museum of Natural History in New York, as 
 icpoitcd l»v Prof. F. W. Putnam, half a dozen .such three- and four- 
 hole whistles from the i-egion of Santa Marta, Colombia, are to be 
 .seen: while under his charge at Caml»ridg-e, Ma.s.s., there are a number 
 from the I'loa ^'alley. Central America;"^ of those tigured, three have 
 three tinger-holes and are .said to give five notes each. 
 
 In the Brussels Con.servatory Collection'' there are twenty-five terra 
 cotta insti-uments fi-om Mexico: two of them are clearly of this 
 resonator tvi)e. giving five notes and having a compass, respectively, 
 
 of eight and eleven E, 8. (tig. 3). Lastly, a similar 
 instrument described and tigured by Dr. Walter 
 Hough, in the Report on the Columbian Historical 
 Exposition at Madrid. 1892-1893, has the small com- 
 pass of six E. S. The point should again be empha- 
 siztxl that with these instruments the notes get closer 
 and closer together as the pitch rises; for instance, on 
 the type instrument the successive intervals are in 
 whole numbers4, 3, 2. 2. E. S. ; on the Brussels instru- 
 ments, 3, 2, 2, 1, and 4, 3, 2, 2; on the Madrid speci- 
 men, 2, 2, 1, 1. A chart (Plate 10) will show more, 
 accurately what the four intervals are with any speci- 
 fied ratio of holes, and whether there is appre<!iable 
 error in expressing the interval in whole numbers. 
 Of course the calculations assume uniformity in the 
 l)l<»wiiig, for it is easy for the performer to vary the notes bj' a con- 
 siderable amount. Still, it is a surprise to find how well these simple 
 scales .satisfy the ear. 
 
 A sort of stone Hageoh^t from Costa Rica appears to be connected 
 with these instruments in principle (Plate T, tig. 8). This is closed 
 iJ one end and has a small mouth opening and four tinger-holes 
 arranged in pairs; its scale of .seven notes from live holes proves that 
 the holes are not acoustically e(|uivalenl. l»ul the two of each pair are 
 found to lie nearly e({ui\alent: so on trial it appeal's that the square 
 root formula may t)e ap|)lied. by giving to the mouth-hole the value 5, 
 to each of the nearer holes the \alue 1. and to the other holes the value 
 2: then the vibration fre(|uencies will be as the stpiare roots of the 
 iMunbers 5 to 11. The calculated intervals from the lowest note are 
 l.»», 2.9, 4.1. 5.1. t;.o. c.s E. S.: the ob.served intervals are 2, 3, 4, 5, 
 6, and 7 E. S. 
 
 Fig. 3. 
 
 TERRA COTTA WHISTLE 
 
 After Miiliillcm. 
 
 ' Miiua/ine of AinericaM History, IV, 1860, pp. 144, 177, 240, 274. 
 ■'Mt'inoirs of tlu' IVuImmIv Museum, I, No. 4, pi. ix. 
 '.Miihilk.irs Cutalogiir, II, N oh. 8.52, 853.
 
 HISTORY OF MUSICAL SCALES. 
 
 431 
 
 Fig. 4. 
 
 BABYLONIAN WHISTLE 
 After Engel. 
 
 2. A pottoiv whistle found in tlic ruins of Babylon, datino- probably 
 from about 5<»0 B. C, is in the Museum of the Royal Asiatic Society, 
 London^ (tig. 4). Rowbotham' says this is similar to the reindeer 
 joint used by the cave men. Its extreme length is 3 inches and it 
 has two finger' holes. The three notes are stated to be C (of 525 d. v.), 
 E, and G; but the holes not being quite 
 equal, the E from one of them is a quarter of 
 a tone flat. By blowing hard the G can be 
 carried up to A. The chart (Plate 10) shows 
 that if the interval C-G is exact, with equal 
 holes the intermediate note E will be a very 
 little sharp of the piano note, but the difier- 
 ence is only about 1 per cent, one-fifth of 
 a semitone, and so is utterly negligible in 
 notes of such uncertain intonation. 
 
 3. Striking comparisons have sometimes 
 been made, and especially by the late Prof. 
 Terrien de la Couperie, between the Assyrian 
 and early Chinese civilizations. Whatever 
 their relations may have been, it is curious 
 that the only instrument of the resonator 
 type, having several finger holes and coming 
 from a people who had a musical theory, is ' 
 the Hsuan (Van Aalst)'' or H'men (Amiot)* of the Chinese, said to have 
 been invented some 2,700 years before our era, and still used in the 
 Confucian ceremonies, though very rarely seen. It is described as a 
 hollow cone of baked clay about 3i inches high, having a mouth-hole 
 at the top, three equal finger-holes on one side, and two equal holes on 
 
 the other. The descriptions available 
 are inconsistent and incomplete,, but 
 that given by Amiot a century ago is 
 the fullest. He reports the scale as re^ 
 fa, sol, la, do^ re, and as he gives a cut 
 (fig. 5), also the diameters of holes and 
 the external measures, an approximate 
 calculation can be made of the scale b}^ 
 the laws of resonators. The pitch of 
 the fundamental comes out D above 
 middle C, and the other notes, F, G, 
 and A, for one side; then starting anew for the other side we get C and 
 D, all within a quarter of a semitone. This, it will be noticed, is a five- 
 
 ' Engel, Music of the Most Ancient Nations, p. 75. 
 
 '' History of Music, II, p. 628. 
 
 =* Chinese Music, Shanghai, 1884, p. 82. 
 
 *Memoires concernant rhistoire . . . des Chinois, p. 225. 
 
 .•^i^. 
 
 Fig. 5. 
 
 CHINESE RESONATORS. 
 After Aniiot.
 
 4:vi 
 
 KEl'oKT OF NATIONAL MUSiEUM, 
 
 1900. 
 
 Fig. G. 
 
 <;i.<(KVI.AR WUrsTI.ES. 
 After ^Yol)eIllU8. 
 
 stoj) snilc. liko most of tho thoorctical Chinese, scales. The uoTeemcnt 
 between th.' niiithcinatieul theory and observation is strikinoly close. 
 4. All the cases thus far referred to have been of prehistoric or very 
 ancient instriinients. But some curious little instruments of this type 
 aiv li^aired l.y Fiobenius' (tjo-. tl) as ^^a splendid parallel between the 
 cultures" of some West African tribes and the natives of New Pom - 
 
 crania. These are little whistles made out of 
 gourds (a, 1), d) or pottery (c). They have the 
 mouth-hole and two, three, or four Unger- 
 Ja^ \ 9 W'^^ iioles. No dimensions are given. Kraus, of 
 ..__^^ ^-^.^^J^ Florence, figures and describes" a similar 
 
 instrument from Melanesia made of a gourd 
 ♦ 1 cm. in diameter, having three finger-holes 
 close to the mouth-hole (fig. 7). The scale is 
 stated to be A, B, C#, E, F, but no further 
 measures are given. However, this series is 
 easily ol)taincd by assuming the diameter of 
 the mouth-hole to be 1.0, of one hole 0.3, and 
 of the others 0.6; apparently D# is omitted. 
 In tlM' Finsch Collection in the American Museum of Natural History, 
 ill N«'w York City, there are several similar gourds of difterent sizes 
 having three finger-holes. They are labeled "Blasekugeln," "used 
 by women." ' 
 
 r». In Europe there have ])een many instruments depending on the 
 same general principle of resonance in a nearlv closed cavity (in dis- 
 tinction from the open or closed organ-pipe principle), 
 but not conforming to the simple law already set 
 forth. Pnvtorius'' in his famous book of 1618 gives 
 figures and descriptions of several such instruments, 
 along with the recorders, fiutes, violins, etc., that one 
 reads of more frequently; for instance, he says the 
 fagotti are sometimes closed at the extreme end, but 
 have a side hole; the Cornamuse has the end closed and 
 holes in the side. Besides these he describes various in- 
 strunnMits having stopped bodies on reeds — the rankett, 
 l>ear ])ipes, «'tc., and similar forms on the organ. These things liave all 
 gone out of use along with the other delicate and weak-toned instru- 
 ments of theii" times. 'I'o-day nmsicians demaiid tones more powerful 
 and richer in harmonics than instruments of this t3'pe can give. But 
 a curious sni-\i\iilor re\i\alof this earlier type occurred in the middle 
 of thi> century, which is told of in Groves's Dictionary of Musi(\ A 
 blind peasant. nam«'d Bicco, gave pul)lic perforiuances in London on a 
 
 i »' t I i-i>ruii>j (Ut .\frikaiiischen Kulturen, 1898, p. 150. 
 •' Anhivii) ]kt L' Antropold^'ia e la Etnoloiria, XVII, 1887, pp. 35-41, fig. 5. 
 Syutaf^ina Miisicuin II, pp. 44, 4H, 85. 
 
 Fig. 7. 
 
 GLOBULAR WHISTLE. 
 
 At'tor Kiiiiis.
 
 HISTOKY OF MUSICAL SCALES. 433 
 
 flageolet 2 inches long and having only three holes. U.v partially or 
 wholly closing the end of the tube with his hand he made use of the 
 resonator principle to lower the pitch of his notes; so he obtained a 
 compass of more than two octaves. The instrument is similar to 
 Pr»torius's sclnoeigeV except that it is shorter, and the accuracy of 
 the notes performed would depend almost wholly on the performer. 
 Later a traveling troupe appeared in European cities with seven 
 instruments called ocarinas. These are familiar to us, being on sale 
 everywhere. They are properly resonators, but the holes are more 
 numerous than in the instruments already considered and vary widely 
 in size. The scale, Avhich the instruments furnish with more or less 
 precision, is not dependent on any simple principle, but is adjusted by 
 the maker by varying the sizes of the holes so as to conform to a scale 
 fixed on other instruments. 
 
 V. THE INFLUENCE OF THE HAND. 
 
 All the instruments of the three groups now discussed are ' ' fingered;" 
 that is, the acoustical dimensions of the vibrating string or mass of 
 air are varied as the player manipulates the fingers of one or both 
 hands. These instruments therefore involve a feature not associated 
 with drums and other*instruments of percussion, or with primitive 
 harps. Instead of using the hand as a whole, the more delicate 
 fingers are utilized separately; so the simple instrument becomes in a 
 peculiar sense a part of the plaj^er's means of self-expression and is 
 specially responsive to his own moods, as many legends of the power 
 of music testify. But leaving to the musical writers such compari- 
 sons between instruments, it is important to the phj^sicist to recognize 
 that the dimensions of the human hand have fixed absolutely some 
 dimensions of these instruments. 
 
 The first thing to strike one, considering the hand from this point of 
 view, IS the fact that onh' with difficulty can the five digits be brought 
 into line, so the thumb is not used on primitive instruments for finger- 
 ing, so far as observed. In the more highly developed flutes there 
 may be a hole for it on the back side, while on our own flutes, clari- 
 nets, etc., it governs one or more keys. Similarly, the little finger 
 does not readily fall in line with the three longer ones, and, besides, is 
 much weaker. The remaining three fingers on a hand of medium size 
 can be brought into a space of about 1 cm., or spread to span perhaps 
 12 cm. (5 inches). To fix one's ideas before comparing these limits with 
 measures on some actual instruments, it will be convenient to recall 
 that on piano keyboards the distance between key-centers an octave 
 apart is 165 nmi. (6^ inches), the same as on a spinet of 1602; but on 
 the physiologically designed Janko ke^^board, with the octave distance 
 
 * Syntagma Musicum, p. 39, pi. ix.
 
 484 REPOKT OF NATIONAL MUSEUM, 1900. 
 
 140 mm. (r).V inclu's). :iii ordinarv Imndciin readily span an octave and a 
 Fiftli, because the lingers are not forced into line. 
 
 Exatninin*,'' lirst some string- instruments, it is found that on a guitar 
 of New York make (No. 55<)9U, IT.S.N.M.) the distance between frets 
 ranges from '.V.\ to 14 mm. The greatest distance noticed between 
 frets is on the large Siamese Km Chapj^ce (No. 27810, U.S.N.M.), 
 where there are three spaces, respectively, of 71, 73, and 77 mm. A 
 similar instrument examined at the World's Fair held in ( 'hicago had 
 the corresponding spaces 60, 60, and 67 nnn. The string lengths to 
 the lii-st frets were, respectively, 878 and 740 mm. The smallest dis- 
 tance observed between frets is the above-cited 14 mm., except that 
 the Syrian lute, Blzug (No. 95144, ILS.N.M.), has two spaces of 
 li' and 18 mm. On most instruments the frets cease w'hen the limit 
 of 20 to 2;") nun. is reached. It is obvious that these and similar data 
 for fi'etted instruments are not of much importance unless one can 
 know that the hand was not shifted from one fret to another. 
 
 With our i ustruments shifting is notorious!}' common, but the histories 
 of the \ ioliii report that two or three centuries ago it was a notable thing 
 f ( ir a })layer to shift. The usual theory of the old manj^-stringed instru- 
 ments, of which the Arab lute is a particularly good example, required 
 the strings to be tuned in Fourths, and the stririg lengths Avere not too 
 gn^at for the four lingers to govern all the frets within this range — that 
 is. in a <|uarter-length of the string — so a shift would be unnecessary. 
 On the Aral) lute ' there were sometimes ten very unequally spaced frets 
 in this space, but for any one tune onl}^ a few of them were used, and 
 in the principal modes, '' OcJukj and Rast^ one fret each for the index 
 and ring lingers sufficed to give substantially our diatonic scale. 
 
 With sinq)le wind instruments the ease is quite different, for sev- 
 eral lingers nuist ])e used sinndtaneously to cover holes, so the hand 
 can not be shifted. In the Kiowa flute referred to above the uniform 
 distanc*^ ])etween holes is 82 mm.; in the stone whistle from Mex- 
 ico. 20 n)m.: in the four Egyptian flageolets and oboes flgured by 
 Villoteau (his Plate v c) the intervals are, respectively, 12, 15, 15, and 
 36 mm. These distances re(iuire only a convenient spread of the 
 Angers, Many otiui- measures can readily be obtained from the 
 accompanying flgures with their appended scales. 
 
 If the nuisician has a theory demanding that the holes ])e so near 
 together or so tar apart as to make direct Angering inconvenient or 
 impossible, keys with longer short levers are added, as on modern 
 flutes and clarinets, while among the Romans extra holes Avere bored 
 to provide for sev<'ral genera, the holes not needed for any tune lieing 
 closed by i)lngs oi- rotating i-ings. 
 
 In a Ifw <;is.'s wind insti-unieiits are found so long that the player's 
 
 M-m.l. Ti.i\;mx .If la ti' Cniiu'ivH .lis Oriciitalistc's, 1883, i)p. 107-114, or Ellis, 
 .Itiurrial ..i tin- Si.cifty of Arts, XXXIII. lss.->, j.. o02.
 
 HISTORY OF MUSICAL SCALES. 435 
 
 arm is too short to reach the lower end. Then, necessarily, the holes 
 to be fingered are located at the middle or upper end of the tube, but 
 the holes are so small that the pitch of the resulting notes is much 
 lower than the position of the holes would suggest, so the discrepancy 
 to the ear is not as great as to the e^^e. In other cases the leugth is 
 misleading, for the holes are bored obliqueh" or holes are bored in the 
 tube below the holes to be fingered, thereby raising and adjusting the 
 pitch of the lowest note, as Mahillon shoAvs in the Brussels catalogue 
 (Nos. 830, 1039, 1117, 1119, and 1123) and Villoteau shows on his Plate 
 c c, No. 1. This is a possible explanation of the superfluous holes in 
 the flute on the statue from the ruins of Susa (Plate 2, fig. 1), if the 
 figure be accepted as archa?ologically correct. In modern instru- 
 ments, as is Avell known, the distant holes are controlled by covers at 
 the ends of long levers. 
 
 The relation of the instruments of the resonator type to the hand is 
 too obvious to need discussion; the objects must be of such size and 
 shape as to be held by the hand or by two hands while the fingers are 
 manipulated, and the holes must be conveniently located and small 
 enough to be closed by the tips of the fingers, or in the Chinese hiuen 
 also by the thumbs. 
 
 It is rather surprising to see how little the thumb is used in plaj^ing 
 upon the instruments under consideration. Although from its anatom- 
 ical structure the thumb has a peculiar independence in its movements, 
 yet most of its services are rendered by cooperation with the other fin- 
 gers; and the natural training of these, as in grasping, sewing, weav- 
 ing, or the most delicate savage industries, appears likewise to call for 
 their cooperation, not for independent action. It is only in playing 
 instruments like the lyre and harp (whose tuning depends on princi- 
 ples outside the instrument, and so they do not belong to the present 
 discussion) that one sees a grasping action requiring two or more fin- 
 gers at once. But in the guitars, flutes, etc., under consideration, the 
 thumb is constantly occupied in merely supporting the instrument, so 
 any variation in the pitch of the sound can come only as the other fin- 
 gers become independent in action. When we remember how" diflicult 
 it is for a civilized piano-player or typewriter to-day to acquire a sat- 
 isfactory independence in movement of all the fingers, especially of 
 the third and fourth, and recall that the early instruction-books for 
 the harpsichord required the use of but tw o fingers on each hand, we 
 shall have a higher respect for the technique of primitive musicians, 
 and shall not wonder that primitive wind instruments have so few 
 holes. Presumabh' the index finger first gained independence, and then 
 it marked a long advance when two fingers could act independently of 
 one another. So the four-hole flute or resonator, requiring the action 
 of two fingers from each hand, and giving a scale of five tones, is a 
 monument commemorating an important stage both in the development 
 of the hand and in the extension of musical resources.
 
 43«> 
 
 REPORT OF NATIONAL MUSEUM, U>(H>. 
 
 VI. COMPOSITE INSTRUMENTS. 
 
 Eiii-h of the instruments thus fur exuniiued is capable of furnishing 
 several notes of approximately eonstant pitch. ))ut the general princi- 
 ])!(• Ix'fore us may 1h> eml)o(lied in composite in.struments, where each 
 note has its own vil)rating Ixxly; thus 
 
 I. Various forms of harps and dulcimers show strings of regularly 
 decreasing h'ngth; here, of course, difference of tension may nullify 
 the scale due to the lengths. One form is shown on Plate 8. 
 
 ■2. Pan's pipes arc sometimes seen with regularly decreasing lengths; 
 it is true that this regularity is not very common, but it is the only 
 
 principle of scale building (except 
 the Chinese cycle of fifths) yet I'ecog- 
 nizable in these primitive instru- 
 ments. (Phite 0.) 
 
 8. Instruments of the bar type are 
 found frequently in our orchestras 
 and l>ands imder various names, as 
 wyJaplume; they are familiar in 
 children's toys and are widely dis- 
 tri])uted in savage and half -civilized 
 lands under the names of marlmha^ 
 h(th(fong^ harmontcon, etc. (Plate 
 8 and fig. 8.) The law of the uni- 
 form bar is that the frequencies of 
 vibration of a series of bars of the 
 same material are proportional to 
 the quotients of the thickness divid- 
 ed by the square of the length; the 
 breadth is immaterial if it is uni- 
 form. So if one takes a series of 
 vniform bars of tiie sauKi thickness and i-egularly decreasing length he 
 may obtain a scries of ascending notes. Thus, let the first bar be 24 
 units long (for exam[)le 24 cm,), the successive bars decreasing by one 
 unit: the eighth bar will ])c 17 units long, and tlu; fifteenth bar 10 
 units; the scries of t'i(M|iicncies would then ])e as the reciprocals of the 
 .s(juares (.f 24, 28, etc., so giving to the ear a series of increasing inter- 
 \als; with these proportions bar No. 8 would give the Octave of the 
 lirst. luit l)ai' No. !"> would give the Twelfth of bar No. 8. The sim- 
 jdicily of tlic rule, howi'ver, fre(iuently disappears, either because of 
 variations in the thickness, as when a savage splits a bamboo stem and 
 then cuts his bars so that the shorter ones are also thinner, or because 
 of the attachment of lumps of wax or day to the bars to tune them to 
 some other insti-unient; or 1)ecause of the hollowing of the center, as 
 is done by modern Japanese; so :il pi'csent one can not affirm that this 
 
 Fig. s. 
 
 XYI.()l'II()NE.<i. 
 AftiT Kraus.
 
 HISTORY OF MUSICAL SCALES. 437 
 
 theoretical principle of scale determination is certainty and consciously 
 embodied in any instrument anywhere; but some instruments in the 
 National Museum and some drawings in books make the assumption 
 seem plausible, that the primitive type of tliis instrument is a series of 
 bars, supported at points about one-tifth or one-fourth of their length 
 from their ends, and decreasing in length by equal linear amounts. 
 
 It is evident that these composite instruments are of minor impor- 
 tance in this study; but in the light of the theoretical laws here 
 suggested perhaps travelers may learn something of the intention of 
 a savage who cuts his Pandean pipes or bars to form a musical 
 instrument, 
 
 VII. CONCLUSIONS. 
 
 There have now been considered all the types of instruments in 
 which several notes of different pitch are produced from the same 
 vibrating body — whether string, colunni of air, or mass of air. 
 
 (la.) There have been found examples from various parts of the world 
 of the intentional location of the stopping points of a vibrating string at 
 equal linear distances; since with all stringed instruments the linger- 
 ing will cause a slight increase of tension, the equivalent length of the 
 string is less than the actual length. 
 
 (1*^.) There have been found numerous examples of wind instru- 
 ments pierced with holes in one or two groups spaced at equal linear 
 distances; since these holes are never sufficiently large to allow the air 
 to flow through them with perfect freedom (unless in some Chinese 
 flutes) the equivalent length of the vibrating column of air is greater 
 than the actual distance from the mouthpiece to the hole. 
 
 (2.) There have been found instruments of the Marimba type with 
 bars of regularly decreasing lengths. 
 
 (3.) There have been found many forms of instruments of the resona- 
 tor type embodjdng a series of equal and similarly-located holes; in 
 these, thickening the wall is equivalent acoustically to making the holes 
 smaller; while locating the hole nearer the point where the vibrating 
 air has its maximum change of density is equivalent to enlarging the 
 hole. 
 
 Three simple laws give to the first approximation the scales of these 
 several instruments, namely: 
 
 (1) The law of inverse lengths. 
 
 (2) The law of inverse squares of lengths. 
 
 (3) The law of the square roots of a series of numbers propor- 
 
 tional to sums of diameters. 
 The lirst and second laws give scales whose intervals increase as 
 the pitch rises; scales based on the third law have decreasing inter- 
 vals. Some results are shown in a table in the appendix and graphi- 
 callv in Plate 10.
 
 438 REl'OUT OF NATIONAL MUSEUM, 1900. 
 
 From these it is ovidoiit that whichever type of instrument one ma}^ 
 take, there will he some intervals that very closely agree with inter- 
 vals of our familiar scale. In a few cases this comes about because 
 our scale is pi-incipaily derived from tlie Greek theorists, who based 
 their scales on proportional string-lengtlis; so, if the unit of equal 
 distance on a simple iruitar chanctvs to be an aliquot part of the length 
 of the string from the bridge to the nut, some of the resulting notes 
 will belong to our scale. (However, the divisor nuist not have a prime 
 factor greater than tive.) But whatever the instrument, on any doc- 
 trine of chaiiccs. tluM-e will l)e some approximate coincidences; and 
 these coincidences, as judged by the ear. will be found much closer and 
 more numerous than when judged mathematically oi' gra])hically; for 
 the training of modern musicians, as has often been recognized, not 
 only aUows but compels them to ignore deviations from their standard 
 scale — deviations amounting sometimes to more than half a semitone. 
 So one is forced to conclude that the recognition, even by a musically 
 educated ear. of a series of notes as agreeing substantially with our 
 diatonic scale or with any other known scale, does not aft'ord any ade- 
 quate ground for judging of the principles underlying the series: in 
 fact, the failure to note the deviation msLj prevent the recognition of 
 the underlying principle. 
 
 The t^'pc Costa Rican four-hole whistle is the most striking example 
 of a series agreeing closely with notes of our scale, yet based on an 
 al)solutely different principle: for the mean computed deviation from 
 the piano intervals is only one-eighth of a semitone. 
 
 Further, the whole discussion makes it evident that the people who 
 made and used these instruments, or any single type of them, had not 
 that idea of a scale w'hich underlies all our thinking on the subject, 
 namely: A series either of tones or of intervals recognized as a stand- 
 ard, inch'pendent of any particular instrument, but to which every 
 instrument must conform. Modern P^uropeans for the sake of har- 
 mony ha\'e nearly banished all scales ])uto!ie, and seldom know^ by what 
 rules the instruments are tuned to furnish this. Hut for these people 
 the instrument is the primary thing, and to it the rule is a^jplied, while 
 the scale is a result, or a secondary thing; and the same rule applied a 
 hiuidn-d times may possibly give a hundred ditferent scales. Natur- 
 ally one does not expect to tind nuich concerted music among people 
 in this stag«» of development. 
 
 I III- \ aiioiis rules discussed !il)o\c may !»(' united in a generic one, 
 iianifly : 
 
 77" jtrn/it/ri/ jPi'iiu'ipJt in the nuiklng of inusical instruinenU t7int 
 If II hi (I Kcalc Ix fill' rij>,'tltlon of elonenU similar- to the eye; the size, 
 nirmher^ and location of these elements heing depeiidevt m, the size of 
 111, Jill nil II ml til, illgltal expertness of the prrformer. 
 
 This principle shows itself in the occasional equal spaces on the neck
 
 HISTORY OF MUSICAL SCALES. 439 
 
 or table of a stringed instrument, and conspicuously in the series of 
 holes on flutes and primitive oboes, while a sense of balance and s} m- 
 metr}' added to the repetition appears in the two groups of holes on the 
 flutes, etc., and especially in the resonators, and appears in a dift'erent 
 way in the trapezoidal forms of dulcimers. Pan's pipes, and marimbas. 
 The pitch-determining- elements are therefore primarily decorative. 
 In fact no one can examine any collection of primitive wind instru- 
 ments, or drawings of them, without being struck b}' the way in which 
 the holes often cooperate in the decoration; while they are not found 
 interfering wnth the artistic design (see fig. 8, page -iSO; Plate 2, figs. 1 
 andi>; Plate 3, fig. .2). 
 
 Simple decoration involving only repetition and symmetrical placing 
 or grouping of similar parts is not only found among living primitiAC 
 peoples everywhere that musical instruments emliodying a scale can 
 be found, but is prehistoric. The prehistoric flutes are believed to 
 come from the neolithic age, and the potter}- from this age shows a 
 nuiltitude of geometrical designs, some of which are collected in Wil- 
 son's Plates 19 and 20. The paleolithic age has furnished few geomet- 
 rical designs and no flutes or many-holed resonators. In applying such 
 decoration to the hollow^ bones of animals or human enemies, to the hol- 
 low reeds that Lucretius says whistle in the wind, or to gourds and sim- 
 ple pottery, nothing can be more natural than sometimes to perforate 
 the walls and to get a several-toned musical instrument as the result. So 
 although no conclusions regarding the mental operations of prehistoric 
 man can be absolutely certain, one feels a strong conviction that, as 
 with immature minds among us, art appealed first to the eye and later 
 to the ear; that beauty of material form incidentalh" furnished series 
 of sounds that could be repeated, and could give to the ear and the 
 mind the idea of the definite leaps or steps that Aristoxenus, countless 
 ages afterward, called the characteristic of music. (Of course rhythm 
 in movement and in sound are independent of the structure of an 
 instrument.) An}^ influence that ma}" have been exerted on the estali- 
 lishment of scales by the songs of birds, by the recognition of over- 
 tones in the sounds of the human voice, or by the production of har- 
 monics on the horn must have been limited and trivial. The principle 
 here presented is at any rate a vera ccmsa, and explains facts hitherto 
 unexplained; further, (1) it is extremely simple both in theory and 
 practice; (2) it is flexible, allowing of multifarious results in prac- 
 tice; (3) it is suggested by prehistoric instruments, supported l)y the 
 instruments of many living primitive peoples and repeatedly con- 
 firmed by its survival in several instruments of peoples in an advanced 
 stage of musical culture. 
 
 It only remains to add, in order to prevent misunderstanding, that 
 the principle here set forth never appears as the dominating one among 
 peoples who are known to have had a theory of the scale. The Greek
 
 440 REPORT OF NATIONAL MUSEUM, IJKK). 
 
 theoretical scales, diatonic and iioiidiatonic, arc doubtless its direct 
 descendants, though at present it is not known what the influence was 
 that so. transformed them and made them depend on ratios, not on dif- 
 ference of lenoths. Possibly the theory' of luimbers bewitched musi- 
 cians then as it has sometimes since, thougfh the converse speculation 
 is a plausible one — that the recognized musical ratios gave a mj^stical 
 meaning- to numbers. It is curious to note that Aristoxenus had some- 
 how got far enough to complain that flutes distort most of the inter- 
 vals (p. 42, Mb.), and if his lost treatise on boring flutes should, be 
 found it might throw light on this history. The Arab *" step by step " 
 method is apparently a late descendant of the equal linear divisions, 
 appearing after men had lejirned to recognize the equality of intervals 
 as well as of spaces. But the Chinese cycle of lif ths must be explained 
 and determined on entirely difl:erent physical principles, and the vari- 
 ous European scales as detined by theorists or rendered by the best 
 violinists or fixed by good tuners, when properl}^ examined, reveal 
 elements as diverse as the elements of our language or our population. 
 The principle in question is therefore presented only as the simplest, 
 earliest, and most primitive principle of scale-building.
 
 HISTORY OF MUSICAL SCALES. 441 
 
 APPENDIX. 
 
 The laws briefly stated on page 437 for the several kinds of instruments discussed 
 in the paper may be expressed more accurately by the following formula;: 
 
 Let N = number of complete vibrations per second. 
 I = length of string or column of air or bar. 
 
 a = diameter of mouth-hole of resonator, corrected for thickness of wall. 
 h = diameter of finger-lioles of resonator, corrected for thickness of wall. 
 ft = number of finger-holes opened on resonator. 
 / = thickness of bar. 
 K = constant, depending on material and units of measurement. 
 Assuming centimeter-gram-second units and ordinary temperatures, 
 
 •K^' =v Tension in dynes -^ mass in grams per cm. = velocity ; e. g., in piano 
 strings 17,000 to 40,000 em. -sec; in violin strings from 13,000 for the 
 covered string to 43,000 for the gut E-string; in weak primitive instru- 
 ments probably much less. 
 K" = 34,000 cm. -sec, the velocity of sound in air. 
 K"' = 520,000 cm. -sec. for iron bars; 340,000 to 520,000 for wood bars supported 
 
 as usual in a xylophone. 
 K'^- = 5,500. 
 
 Then, corresponding with the brief laws, 
 
 (la) For strings: N = -— = - — — - v/tension -^ linear density- 
 
 ni\ T? 1 f • AT ^" 17,000 
 
 (lo) liorcolumnsof air: ^ = -^j—= — fr^- 
 
 (2) For bars: N = K"' L= 340,000 to 520,000 A. 
 
 (3) For resonators: N = K'y ^' ^^^ of b ^ 5,500^ L/l + nl^. 
 
 V volume v volume^ V a J 
 
 These constants are sufticiently accurate for the general purposes of the anthro- 
 pologist and musician. But the results should be expressed in musical terms. The 
 French standard pitch, now adopted by the Piano Makers' Association, gives 
 A = 435 d. v., or C = 258.7 d. v., and the ratio for'any interval of p piano semitones 
 is 2t5. In most cases it is much more convenient to have intervals than ratios; and 
 incomparably the most convenient unit of intervals is the piano semitone, of which 
 12 by definition make an octave; these can readily be grouped by anyone with slight 
 musical knowledge into larger intervals, Thirds, etc., and the musical value of any 
 whole numl)er of them can instantly be found on a well-tuned piano. 
 
 Since the reduction of ratios to intervals can not ordinarily be done without 
 logarithms, a short table has been calculated and is appended by the use of which 
 the reduction may be done by inspection in most practical cases. This table gives 
 the logarithm of every whole numlier from 1 to 40, and the product of these by 40, 
 less one three-hundredth, together with the successive differences; these are in 
 semitones; for the factor is so chosen that when the logarithm of the ratio 2:1 is 
 multiplied by it the product will be 12, which is the number of semitones corre- 
 sponding to the ratio of the octave. Much more elaborate tables, but without the 
 column of differences, have been published by Prony and by Ellis. In using the 
 table it is well to remember that the average uncertainty in pitch of public per- 
 formers in Berlin was found to be about one-tenth of a semitone. 
 
 NAT MUS 190U 31
 
 440 REPORT OF NATIONAL MUSEUM, 1000. 
 
 ,he Hi,„U. .*,.,, tho ..ring being «toppe,l -cess'vely at 363, 34, . . -^^^-^ 
 
 „,„n,ling .litterences in column 4 „f the 'f '^ f, "■:^^-f^-t''. To complet; the 
 
 1 • I QC V c! ns n'duired bv Tafiore s rule ^p. -i^o auuve;. -• t 
 
 :;; r.lfetp!c.f #;o 48 i'"!. .x, .hviaed into 13 equal part,; substitute or the raUo 
 :;:;ror3:^;3.:2e.anausethe«.,eagain;the<,^^^^^^^^^^^ 
 
 .66. the sum being J-^'f ^;^;^:^::^'::^ :^rref:l>; the .'able is to be us«. in 
 If the law be that i.f thf ^(iiuirc kxit. , as w ^.^i^ied bv 2: for example, in 
 
 "'■'■'■'^^"- ""' T';: !.:,>;i:r.h %ta::: -^us^ thtl^^ '•"■ *"" - *; 
 
 ,1„- type resonator, <alln,g the eqmvai therefore be 1.0, 
 
 huger hole, is 0.6 (more -"■-«>•• "f/^.^Zrespon.ling nun,l«,rs from column 
 l'::r£^'^:^t i:^:^:t:L ... th- quotients to the fundamental 
 pitch. The resulta are as follows: 
 
 10 
 16 
 22 
 28 
 34 
 
 E. S. 
 
 39.86 
 
 48.00 
 
 53.51 
 
 57.69 
 
 61.05 
 
 8.14 4.07 A + .07 E. S. 
 
 13.65 6.83 C-.17. 
 
 17.83 8.92 D - .08. • 
 
 21.19 10.60 E - .40. 
 
 line corresponds to the type resonator differences are to 
 
 The table may a so be used ^^^J^^J^^^ll,^'^^^^^ l.ars whose lengths are 
 be doubled instead of halved, ^hus with a seiies o^ ^ ^ ^^^^.^^^ .^ 
 
 24, 23, etc., to 17, the compass will be 2 X [i>o.^^ -i^-"'^^ 
 practically an octave, as stated on page 436. 
 
 Table for computing musical intervals. 
 
 N. 
 
 Log. N. E. S. 
 
 t> 
 
 7 .... 
 
 H 
 
 9.... 
 10... 
 
 n 
 ij ... 
 
 13... 
 II ... 
 
 ir. ... 
 If. . . . 
 
 17 ... 
 Its . . . 
 
 li> . . . 
 
 •JO . . 
 
 0.0000 
 .3010 
 .4771 
 .0021 
 .fi990 
 . 77.H2 
 .8451 
 .9031 
 
 . gw-i 
 
 1.0000 
 .0414 
 . 0792 
 1139 
 .1461 
 . 1701 
 .2041 
 
 .2:w 
 
 . 2553 
 . 27S.S 
 .3010 
 
 Dif. 
 
 12.00 
 19.02 
 24.00 
 27. 86 
 31.02 
 33.69 
 36.00 
 
 :w.oi 
 39.8:: 
 
 41.51 
 
 43.02 
 
 44.41 
 
 45. 69 
 
 46.88 
 
 48.00 
 
 49.05 
 
 .50.04 
 
 50.98 
 
 61. hd 
 
 N. 
 
 12. 00 
 7.02 
 4.98 
 3.86 
 3.16 
 2.67 
 2.31 
 2.04 
 1.82 
 1.65 
 1..51 
 1.39 
 1.28 
 1.19 
 1.12 
 1.05 
 0.99 
 0.94 
 0.88 
 
 21.. 
 22.. 
 23.. 
 24.. 
 26. 
 26. 
 27. 
 28. 
 
 29. 
 
 30. 
 
 31. 
 
 33.. 
 34.. 
 35.. 
 36. 
 37. 
 38. 
 39. 
 40. 
 
 Log. K. E. S 
 
 1. 3222 
 .3424 
 .3617 
 .3802 
 .3979 
 .4150 
 .4314 
 . 4472 
 .4624 
 .4771 
 . 4914 
 . 5051 
 . 5185 
 . 5315 
 .5441 
 .5563 
 . 5082 
 . 5798 
 .5911 
 . 6021 
 
 Dif. 
 
 52.70 
 
 0.84 
 
 53.51 
 
 .81 
 
 54.28 
 
 .77 
 
 55.02 
 
 .74 
 
 56.73 
 
 .71 
 
 56.41 
 
 .68 
 
 57.06 
 
 .65 
 
 57.69 
 
 .63 
 
 58.30 
 
 .61 
 
 58.88 
 
 .58 
 
 .59.45 
 
 .57 
 
 60.00 
 
 .55 
 
 60.53 
 
 .53 
 
 61.05 
 
 .52 
 
 61.55 
 
 .50 
 
 62.04 
 
 .49 
 
 62.52 
 
 .48 
 
 C2.98 
 
 .46 
 
 63.43 
 
 .45 
 
 63. 86 
 
 .43
 
 EXPLANATION OF P.LATE 1. 
 STRINGEU INSTRUMENTS. 
 
 Fig. 1. Sm.\ll Turkish Tamboura. 
 
 (Cat. No. 95312, U. S. N. M.) 
 
 Fig. 2. Medium Colascioni (Italian). 
 (Cat. No. 95307, U. S. N. M.) 
 
 XoTE.— The scale shown on this and most of the following plates is 20 centimeters 
 long. 
 
 444 
 
 -u\
 
 Report of U. S. National Museum, 1900.— Wead. 
 
 Plate 1, 
 
 Stringed Instruments.
 
 EXPLANATION OF PLATE 2. 
 FLUTES WITH EQUAL-SPACED HOLES, TYPE A. 
 
 Fig. 1. Pipe from Susa. Engel, 'Music of the INIost Ancient Nations, p. 77. 
 
 Fig. 2. Bone Flute, about 6 inrlies long, disinterred at Truxillo, Peru. British 
 
 iMuseuni. Engel, Musical Instruments, p. 64. 
 Figs. 3, 4. Aztec Pipes, called by Mexicans pUo; usual form; scale, a, b, c#, e, f#. 
 
 P^ngel, Musical Instruments, j). 62. 
 Fig. 5. Aztec Pipe; unusual form. P^ngel, Musical Instruments, p. 62. 
 
 446
 
 Report of U. S. National Museum, 1900.— Wead. 
 
 Plate 2. 
 
 i'Oi 
 
 m 
 
 roi 
 
 Flutes with equal-spaced Holes.
 
 EXPLANATION OF PLATE 3. 
 
 FLUTES WITH KQUAL-SPACED HOLES, TYPE A. 
 
 Kijr. 1. Double Flageolet. Mexico. 
 
 (Cat. No. 19717.3, V. S. N. M. Report 1890, fig. 2.50b.) 
 
 Fi^'. ~. Aztec Flageolet {pito). Mexico. 
 
 (Cat. No. 172819, IT. S. N. M. Report 18%, fig. 252. ) 
 
 Fig. H. Stone Flageolet. Mexico. 
 (Cat. No. 98948, U. S. N. M.) 
 
 V\<i. 4. ]>oxE Flageolet. Costa Rica. 
 
 (Cut. No. 18108, U. S. N. M. Report 1896, fig. 27H.) 
 
 Fig. 5. BoxE Flageolet. Amazon. 
 
 (Cat. No. 5719, V. S. N. M.) 
 
 Fig. (>. Bamboo Whistle. Tliibet. 
 
 (Cut. No. lf.716.5a, U. .S. N. M. Report 1S9G, plate 69.) 
 
 Fig. 7. B.v.Miioo Whistle. Thibet. 
 
 (Cut. No. l(i71Gob, U. .S. N. M. Reporl l,s%, pluto 09.) 
 
 Fig. H. Shepherd's Pipe, with ueed. Arabia. 
 
 (Cut. No. 936.5.5, U. S. N. M.) 
 Fit'. '•'. lIoK.v {Sdillolorri). Finland. 
 
 (Cut. No. 9.5080, r. S. N. M.) 
 44S 
 
 I
 
 Report of U. S. National Museum, 1900.— Wead. 
 
 Plate 3. 
 
 Flutes with equal-spaced Holes.
 
 EXPLANATION OF PLATE 4. 
 
 FLUTES WITH KQUAL-SPACEI) HOLES, TYPE A. 
 
 Fijr. 1. "Dtkect Flute. Peru. 
 
 (Cat. No. 95901, U. S. N. M.) 
 
 Fig. 2. Flute or Flageolet. Kiowa Indians. 
 (Cat. No. 1535*4, U. S. N. M.) 
 
 Fig. 3. Flute or Flageolet. Mohave Indians. 
 (Cat. No. J07535, U. S. N. M.) 
 
 Fig. 4. Flute or Flacjeolet. Dakota Indians. 
 (Cat. No. 23724, U. S. N. M.) 
 
 Fig. 5. Transverse Flute (Ti-tzu). China. 
 (Cat. No. 13044C, U. S. N. M.) 
 
 Fig. <>. Transverse Flute {Koina Fuye). Japan. 
 (Cat. No. 93205, U. S. N. M.) 
 
 Fig. 7. Oboe {Pec Chmvar). Siani. 
 (Cat. No. 27313. U. S. N. M.) 
 
 Fig. 8. Flageolet {Sopilka). Little Russia. 
 (Cat. No. 96466, U. S. N. M.) 
 
 Fig. 9. Double Flageolet. Thibet. 
 (Cat. No. 95816, U. S. N. M.) 
 
 450
 
 Report of U. S. National Museum, 1900.— Wead. 
 
 Plate 4. 
 
 Flutes with equal-spaced Holes.
 
 EXPLANATIONOFPLATE5. 
 
 FLUTES WITH HOLES IX TWO GROUPS, TYPE B. 
 
 Fn nil Pnetorius's Syntagma Musicum of 1618, to show tinger-holes grouped in two sets. 
 452
 
 Report of U. S. National Museum, 1900 — Wead. 
 
 Plate 5. 
 
 
 Flutes with Holes in Two Groups.
 
 EXPLANATION OF PLATE 6. 
 
 FLUTES WITH IIOLKS IN TWO GROUPS, TYPE B. 
 
 Fig. 1. Flageolet {Sonling). Java. 
 (Cat. No. 95G69, U. S. N. M.) 
 
 Fig. 2. DiKEcr Flute. Ceylon. 
 
 (Cat. No. 95727, U. S. N. M.) 
 
 Fig. 3. Direct Flute {Manjairah). Syria. 
 (Cat. No. 95150, U. S. N. M.) 
 
 Fig. 4. German D Flute. New York. 
 (Cat. No. 55624, U. S. N. M.) 
 
 Fig. 5. Flageolet {Soulinf/). Java. 
 (Cat. No. 95666, U. S. N. M.) 
 
 Fig. iS. Transverse Flute {Murali). BengaL 
 (Cat. No. 92707, U. S. N. M.) 
 
 Fig. 7. Transverse Flute. Manila. 
 
 (Cat. No. 95061, U. S. N. M.) 
 
 Fig. 8. Transverse Flute. Manila. 
 (Cat. No. 95060, U. S. N. M.) 
 
 454
 
 Report of U. S. National Museum, 1900.— Wead. 
 
 Plate 6. 
 
 Flutes with Holes in Two Groups.
 
 \.
 
 EXPLANATION OF PLATE 7. 
 
 CENTRAL AMEKICAN KESONATOltS OK WHISTLES. 
 
 Fig. 1. Costa Kka. 
 
 (Report U. S. Nat. Mils., 1«9(;, p. til7. Scale: f, a, e, i1, c Cat. No. 59970, V. S. N. M.) 
 
 Fig. 2. Costa Rka. 
 
 (Report U. S. Nut. Mux., 1«9(;, fiK- 'ifW. locale: d, e, f}I, k, a. Cat. No.599G9, U. S. N. M.) 
 
 F]g. 8. Costa Rica. 
 
 (Report U. S. Nat. Mus., 189(5, fl;,'. 262. Scale: k\>,M, b, c, dl», d, eb, e, f. Cat. No. 28952, c . 
 N.M.) 
 
 Fig. 4. CJosta Rica. 
 
 (Report U.S. Nat. Mua., 1896, fig. 269. Scale: f,g,a,Ijl', u. Cat. No. 28956, U.S. N. M.) 
 
 Fig. 5. C'osta Rica. 
 
 (Report U.S. Nat. Mus., 1896, p. 61 7. Scale: db,i,f^,>i\>,ht>. ('at. No. t;(HH5, U. S. N.M.) 
 
 Fig. 6. Pana.via, Chikiqui. 
 
 (Report U. S. Nat. Mus., 1896, figs. 304-6. Holmes, Report Bureau Ethnol., 1884-5, figs. 
 245-246. Scale: end closed, f, g, ab, bb; open, f#, gift, a#, b. Cat. No. 109682, U. S. N. M.) 
 
 Fig. 7. C'osTA Rica. 
 
 ( Report U. S. Nat. Mus., 1896, p. 614. Scale: gb, bb, cb, db, eb. Cat. No. 28954, U. S. N. M.) 
 
 Fig. s. Costa Rica. 
 
 (Report U.S. Nat. Mus., 1896, fig. 270. Scale: ab,bb,b,c,db,d, eb. Cat. No.6123, U. S.N. M.) 
 456
 
 Report of U. S. National Museum, 1900,— Wead, 
 
 Plate 7. 
 
 Central American Resonators, or Whistles.
 
 NAT MUS 1900 32
 
 EXPLANATION OF PLATE 8. 
 
 COMPOSITE INSTRUMENTS. 
 
 Fi-'. L Pan's Pipes. Cairo, Egypt. 
 
 (Crtt.No.94()53,ir:S.N.M.) 
 
 Vhj. -. K.v.NTELE. Finland. 
 
 (Cat. No. 95691, U. S. N. M.) 
 
 Fig.;-!. :\I(>KKiN. Japan. Two bars turned edgewise to show their form. 
 
 (Cat. No. 9(5841, U. S. N. M.) 
 
 Tlic i)aper scale is 20 centimeters long. 
 458
 
 Report of U. S. National Museum, 1900.— Wead. 
 
 Plate 8. 
 
 Composite Instruments.
 
 EXPLANATION OF PLATE 9. 
 pan's pipes. 
 Fijr. 1. Pan's Pipes (/S(f/a^/-)- Egypt. 
 
 ((•at.N().94653,U.S.N.M.) 
 
 Fig. 2. Pan's Pipes. Fiji Archipelago. 
 
 (Report r. S. Nat. Mus., 1896, p. 5.59; Cat. No. 23942, U. 8. N. M. ) 
 
 Fig. 8. Pan's Pipes {Huayra Puhura) . Peru, from an ancient grave. 
 (Cat. No. 136869, U. S. N. M.) 
 
 4(50
 
 Report of U. S. National Museum, 1900.— Wead. 
 
 Plate 9. 
 
 PAN'S Pipes.
 
 t
 
 EXPLANATION OF PLATE 10. 
 
 SCALES GIVEN IJY RESONATORS. 
 
 The construction of this chart has been explained in the appendix. To use it, find 
 in the l)ase line the number which expresses the radius of the finger holes, that 
 of the mouth hole being considered 1.0, and erect a perpendicular therefrom; 
 the heights of the points of intersection with the successive curves, measured on 
 the left-hand scale, give the pitch of the successive notes produced as the holes 
 1, 2, 3, et(^, are opened, expressed in equally tempered semitones, E. R. The 
 dotted line corresponds to the position on the chart of the type resonator. The 
 chart shows clearly how the successive intervals become smaller as the number of 
 open holes increases, and how the total compass is small if tlie finger holes are 
 relatively small. 
 
 Use may be made of the chart for many ready calculations of intervals other than 
 those due to equal differences, and by doubling the readings in K. S. the result 
 may be applied to string ratios; e. g., find the interval corresjionding to the ratio 
 5 : 4, or 1+0.25; the chart gives directly 1.9; the double of wliich is 3.8 E. S. The 
 table in the appendix gives more accurately 3.86 E. S., showing that the just 
 Third is 0.14 E. S. flatter than the piano Third. 
 
 462
 
 Report of U. S. National Museum, 1900.— Wead. 
 
 Plate 10. 
 
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 Scales given by Resonators. 
 
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