April 17, 1902] 



NA TURE 



561 



Resultant-Tones and the Harmonic Series. 



The following method of recovering, by inspection of the 

 harmonic series, the position of the dift'erential resultant-tone of 

 the first order for any given musical interval has occurred tome, 

 and, as far as I have been able to learn, has not yet found a 

 place in the text-books on musical acoustics which are in most 

 common use. I therefore venture to hope that a brief state- 

 ment of it may gain admittance to your columns and be the 

 means of saving some time and trouble to students of the 

 subject. 



The harmonic series up to No. lo, with the order of each 

 number placed below it, may be written thus : 



C C G' C" E" G" {B"W C" D'" E'" 

 I 2 3 4 5 6 (7) 89 10 



the brackets of course indicating that No. 7 is not identical 

 with the seventh note of the diatonic scale. As the "vibra- 

 tion numbers," or "frequencies," of these harmonics are 

 proportional to their respective order-numbers, these latter 

 may, for the purpose in hand, be treated as if they were the 

 vibration-mimbers of the corresponding harmonics. Hence the 

 difference between any two order-numbers will give the vibra- 

 tion-number of the resultant-tone for the interval formed by the 

 two corresponding harmonics, and the position of the resultant- 

 lone in the harmonic series will thus be at once assigned. 



Successive application of the above method to a series of 

 intervals gives the following results : — 



Minor tone D'" - E'", 10-9=1. Resultant tone C, 3 octaves 

 and a major tone below the graver primary. 



Major tone C" - C", 9 — 8 = I. Resultant tone C, 3 octaves 

 below the graver primary. 



Minor third E" - G", 6-5 = 1. Resultant tone C, 2 octaves 

 and major third below the graver primary. 



Major third C"-E", 5-4 = 1. Resultant tone C, 2 octaves 

 below the graver primary. 



Fourth G'-C", 4-3 = 1. Resultant tone C, a twelfth 

 below the graver primary. 



Fifth C - G', 3-2=1. Resultant tone C, i octave. 



Minor sixth E"-C"', 8-5 = 3. Resultant tone G', major 

 sixth below the graver primary. 



Major sixth G'-E", 5-3 = 2. Resultant tone C, fifth below 

 the graver primary. 



Octave C-C, 2-1 = 1. Resultant tone C, coincident with 

 the graver primary. 



Ninth C"- D'", 9 -4 = 5. Resultant tone E", a major third 

 above the graver primary. 



In order similarly to treat semi-tones, sevenths and other dis- 

 sonant intervals, it is only necessary to include higher numbers 

 of the harmonic series, and the method is evidently as applicable 

 to summational as to differential resultant-tones. 



M.VRGARET DiCKINS. 



Tardebigge Vicarage, Bromsgrove, April 5. 



Municipal Meteorology. 



In your notes in Natire for April 3 (p. 518) you mention 

 Dr. H. R. Mill's observations as to the length of the February 

 frost, the period at Torquay being reported the shortest, viz. 

 nine days. 



I think it should be pointed out that for scientific purposes 

 the Torquay temperatures should be treated with much caution, 

 as those of an extremely sheltered spot, viz. Gary Green. Up 

 to last year Torquay had two observing stations ; but one of 

 them, viz. that at Chapel Hill, was given up. On moving the 

 adoption of the report advising, among other things, the dis- 

 continuance of the said station, a member of the Town Council 

 very honestly remarked that " the range of temperature at 

 Chapel Hill was greater than that at the Princess Gardens, and 

 it was more satisfactory for the town to have as small a range of 

 temperature as possible" ! {Western Morning News, March 6, 

 1901). According to the last meteorological report, the temper- 

 atures are still taken at one of the old stations, viz. Cary Green. 

 The differences between Chapel Hill and Cary Green were 

 often considerable, eg. on March 18, 1900, Chapel Hill 23°'9, 

 Cary Green 27°. A. R. Hunt. 



Torquay, March 10. 



NO. 1694, VOL. 65] 



A Gall-making Cynipid Fly in Jamaica. 



The Hymenopterous gall-makers of the family Cynipidce, 

 though widely distributed over the earth, have seemed io be 

 totally absent from the West Indies, somewhat to the astonish- 

 ment of entomologists. However, in February, 1892, Mr. W. 

 Harris sent me a gall on Bidens reptans, collected by Mr. 

 NichoUs at Cinchona, Jamaica. The gall was a large oblong 

 swelling attached to the stem of the plant, and contained many 

 cells. Unfortunately, I was not able to breed the flies, but I 

 secured some immature fragments, which sufficed to show that 

 they were Cynipida;. Mr. W. H. Ashmead, to whom I sub- 

 mitted my notes and sketches, is of opinion that the genus must 

 be Aulax or allied thereto. Although it is impossible to 

 determine the species, it seems desirable to call attention to the 

 occurrence of this gall-maker in Jamaica ; and perhaps one of 

 your West Indian readers will be able to supply us with complete 

 information. T. D. A. Cockerell. 



East Las Vegas, New Mexico, U.S.A., April 3. 



Criticism of "The Great Persian War." 

 May I call attention to one or two points in your critic's 



treatment of my work (p. 43;])? 



(i) Your critic says that at Thermopylse " the mound and the 



Phocian Wall are the only debatable points." 



I may mention that, among many other debatable points, 



some persons who have examined the pass have located the 



fight at the West gate. 



(2) Your critic is apparently not aware that the site of 

 Trachis is uncertain. Two sites at some distance from one 

 another have been proposed. 



(3) Of Aphette he says, " it is hard to reconcile Mr. Grundy's 

 situation with the remark of Herodotus, that the wreckage of 

 the first sea-fight drifted out (the italics are your critic's) to 

 Apheta?, which is badly misrendered in the words ' was thrust 

 in upon the Persian fleet.' " 



Either your critic or Liddell and Scott are guilty of a bad 

 mistake in scholarship (t'/'i'i; L.S. sub. eK(popea>. 4. Pass, "to 

 be cast on shore" : with a reference to this very passage Herod. 

 VIII. 12 ; if. also 4K<pepw in the same dictionary). If I sin, I 

 sin in very good company. 



(4) In reference to B.C. 479, your critic asks, "Why should 

 they (the Persians) stop at Cithjeron ?" 



I would ask in reply, (i) Why did not the Persians slop in 

 Attica? (2) Why, above all, did the Greeks follow them into 

 Boeotia ? 



(5) Your critic says, "It is absurd exaggeration to call 

 Taurus ' an all but blank impassable wall.' " 



In the part of that chain which separates Asia Minor from 

 the Euphrates region between the pass from Laranda to Kelen- 

 deris and the pass of Kiskisos, a distance of 170 miles, there is 

 only one pass, the Cilician Gates. The two first mentioned 

 afford circuitous and difficult routes to the Euphrates region. 

 The Cilician Gates are by no means an easy pass. (For Map 

 vide Ramsay, "Asia Minor," p. 330.) 



(6) Your critic says that "for downright geographical non- 

 sense" it would be hard to beat the description of Pteria as the 

 chief strategic point in the Halys region, .is commanding the 

 middle portion of the valley through which the river flows. 



He is apparently under the impression that in order to com- 

 mand a valley strategically, a town must be in it. On the im- 

 portance of the position of Pteria the evidence of Herodotus 

 is sufficiently striking (H. I. 76), " Now Pteria is the strongest 

 position in the whole of this country." 



(7) Your critic asks, " Are the Phrygians and the Bithynians 

 still to be called Thracians in the days of Darius ?" 



His answer is obviously No ! Another authority, Herodotus, 

 however, speaking of the races which Crcesus ruled, mentions 

 (I. 28) Qp-hiKes 01 &vyOi re KOl Bifli/i'oi. 



(S) Your critic says, " On p. 37S a whole paragraph is based 

 on a childish mistranslation of Herodotus." 



This charge is based on a somewhat careless mistake of his 

 own. I speak of a "decision." He refers to the "dis- 

 cussion " of chapter Ixxiv, of Herod. VIII. 



I have dealt with all the major charges brought against me. 

 I cannot expect you to allow me space to deal with the minor 

 charges, though I believe them to contain inaccuracies. 



G. B. Grundy. 



