ii6 



NATURE 



[Dec. 9, 1875 



THE GRAPHIC METHOD OF REPRESENTING 

 MUSICAL INTERVALS* 



THE object of the paper was to explain a method of repre- 

 senting musical intervals, which was very useful in giving a 

 clear idea to the mind of relations often complex and obscure. 



The author pointed out that there was a natural tendency to 

 refer the positions of musical notes to positions in space. It was 

 by no means clear that there was any real physical or physiolo- 

 gical relation between the two things, but somehow or other the 

 idea had become so firmly rooted in the mind that it had deve- 

 loped itself in expressions of every-day use. For example, it 

 was customary to call a note with rapid vibrations a hi^h note, 

 and one with slow vibrations a low note. Few people considered 

 whether there was any natural justification for these terms ; 

 probably there was none, but they had existed almost ever since 

 music had taken a definite form, and bad given rise to the form 

 of notation employed to express the positions of musical sounds. 



It followed from this that the musical idea of distance between 

 two notes, which was technically called a musical interval, 

 might be considered as having an analogy between the high and 

 low positions of the two notes respectively, a greater interval 

 being represented by a greater space, and vice versA ; and carry- 

 ing this idea out to its full extent it became possible to represent 

 musical intervals to the eye in such a way as to convey ideas of 

 comparative magnitude precisely analogous to the impressions 

 which these intervals would make on the ear. This the author 

 called the graphic method of representing intervals. 



The idea of such a method had been embodied from early 

 times in the word scaU., which was derived from the Latin scala, 

 a ladder, thereby clearly implying an analogy between the 

 spaces of the steps and the intervals of the notes. Mr. Hullah, 

 in some of his elementary books, had actually made use of a 

 diagram of a ladder for this purpose, and he had introduced the 

 improvement of representing the intervals between the third and 

 fourth and between the seventh and eighth steps (of the diatonic 

 major scale) as only half the length of the other degrees, 

 thereby embodying, in a graphic mode, the distinction in magni- 

 tude between the whole tones and the semitones. What the 

 author proposed to do in this paper was merely to establish this 

 mode on definite principles, and; to give it more capability and 

 more accuracy. 



It was well known that the scientific definition of a musical 

 interval was expressed by the ratio which the vibration-number 

 of the higher sound bore to that of the lower one, and it had 

 been shown that the idea of the magnitude of the interval in a 

 musical sense might be expressed by the logarithm of this ratio. 

 Hence, by plotting down this logarithm with a scale of equal 

 parts, and drawing a line of that length, such a line would be a 

 correct graphic representation of the magnitude of the interval. 



The author explained the mode of doing this in a simple prac- 

 tical way, which might be put in practice by anyone, with the 

 aid of a small table of logarithms, as easily as working a simple 

 sum in arithmetic ; and he calculated and laid down several 

 examples in the presence of the audience. It would be, he said, 

 sufficiently accurate to express the distances in three places of 

 figures, as, for example : — 



The interval of an octave would be expressed by a line whose 

 length was — 



= log. 2 = 301 



That of a major — 



Sixth = log. f =222 



That of a minor — 



Sixth = log. I =204 



That of a fifth— 



= log. I = 176 



That of a fourth— 



= log.|... =125 



That of a major — 



Third - log. \ = 97 



That of a minor — 



Third = log. f = 79 



And so on for any others. 



It would be seen how truly'these numbers corresponded to the 

 ideas of the intervals existing in musical practice, for, according 

 to the usual musical rules — 



• Abstract of a paper read by W. Pole, F.R.S., Mus. Doc, Oxon., at the 

 second meeting of the Musical Association for the Advancement of the Art 

 and Science of Music on Dec. 6, at the Beethoven Rooms, Harley Street, 

 Mr. Bosanquet in the chair. 



Fifth + Fourth = Octave. 



Major Sixth + Minor Third = Octave. 



Minor Sixth + Major Third = Octave. 



Major Third + Minor Third = Fifth. 



Fourth + Minor Third = Minor Sixth. 



And so on. 



The author then, as a more extended illustration of the prin- 

 ciple, showed the process of determination of the exact positions 

 of the various notes of the modern musical scale, including all 

 the accidental sharps and flats necessary for chromatic purposes 

 and for modulation ; and he proceeded to draw the sanje on a 

 large diagram, making the octave 3 feet long. This enabled the 

 audience to appreciate clearly many delicate points of intona- 

 tion, which were difficult to be conveyed to the mind by any 

 process of verbal description, and which the author explained 

 and commented on in their theoretical and practical bearings. 

 He also drew a corresponding scale on the plan of equal tem- 

 perament, and pointed out the more imporcant differences between 

 this and the true scale, concluding with some remarks on the 

 subject of intonation generally. 



THE SWEDISH ARCTIC EXPEDITION 



THE following extracts are taken from a letter addressed to 

 Mr. Oscar Dickson, of Gothenburg by|Dr. F. R. Kjellman, 

 who (and not his brother Dr. Theel Kjellman, as was stated 

 by mistake in Nature, vol. xiii. p. 75) was in command of the 

 Proven, the vessel of the Swedish Arctic Expedition during the 

 return voyage from the mouth of the Jenesei to Norway. The 

 Proven left the mouth of the Jenesei on the 19th August, fell in 

 with ice on the 23rd in 75° 22' N. lat. and 66° 30 E. long, from 

 Greenwich ; sailed along the edge of the ice until, a little south of 

 Cape Middendorff, it was found to connect itself wi'.h the land 

 so as to bar all passage northwards. The Proven then turned 

 south and was carried by a current twelve miles south of Ma- 

 totschkin Scharr. 



•' Before going farther I may perhaps be permitted to make 

 some remarks on the higher vertebrate animals which we found 

 to inhabit or visit the Kara Sea. The walrus occurs here plenti- 

 fully, and has of late years been the object of exterminating pur- 

 suit on the part of the Norwegians. At many places on the 

 Samoyede peninsula and White Island we saw great herds of 

 these beautiful animals. The Kara Sea has three species of seals, 

 Phoca barbata, hispida, and Gnrnlandica. The last-named was 

 that which we saw most frequently and in greatest numbers. Off 

 Obi and Jenesei white fish (? dolphins) were very common, and 

 on the east coast of Novaya Zemlya we saw a large fin- whale 

 (fenhval). If I add that one day, as we lay becalmed between 

 Udde Bay and Matotschkin Scharr, an ice- bear quite unexpectedly 

 came swimming out to our vessel, where he, of course, soon met 

 his death, I have named all the mammalia we saw during our 

 navigation of the Kara Sea. The bird world was exceedingly 

 poor. I may almost say that it was a great raiity to see a tern 

 or a mew. The alka {Uria Briinnic/m), which, occars in such 

 immense numbers on the west coast of Novaya Zemlya, is 

 believed to be absent on the east coast. We saw here only one, 

 and it appeared to have gone astray. Only some few species of 

 fish were observed." 



The Prlrjeii passed through Matotschkin Scharr on the loth 

 and nth September, arriving at Hammerfest on the 26th of the 

 same month, and at Tromso on the 3rd October. Dr. Kjellman 

 sums up the scientific results of the expedition as follows :— 



"We botanists have endeavoured not only to ascertain what 

 species of plants Novaya Zemlya possesses, but also to get an 

 insight into the varying distribution of the different species, the 

 nature of the vegetation at different localities, in different lati- 

 ■tudes, at varying heights above the sea, at varying dis- 

 tances from the seashore, &c. We have made a great 

 number of such observations, and thereby will, I believe, be 

 in a position to give such an account of the vegetation of 

 Novaya Zemlya as will satisfy the requirements of science. Of 

 flowering plants we have rich collections from Matotschkin 

 Scharr, from many places on the west coast of Southern Novaya 

 Zemlya, from Waigats Island and the mainland lying opposite to 

 it, from the Samoyede peninsula and the region lymg round 

 Dickson's Harbour, and these collections contain a considerable 

 number of species new to those localities. The phanerogamic 

 egetation of Novaya Zemlya has a strong resemblance to that 

 of Spitzbergen, but at the same time, as might be expected from 



