48 REPOET— 1876. 



digital to the end of the coloured immediately above it in direct line there is 

 always the chromatic semitone of 128 : 135, and from the end of each coloured 

 digital to the end of the white immediately above the comma of 80 : 81 always 

 appears. 



Between each white or flat note, as E|; , and each coloured or sharp note, as DJf, 

 at the distance of six removes looking across the diagram or finger-board, the schisma 

 of the scale is always found, 32,768 : 32,805. 



The only other relation of the scale is represented by a round digital on the 

 finger-board and by 7 minor on the diagram ; it is tuned as 15 : 16, to the 6th of 

 the major scale, and supplies the sharpened 7th and 6th tones of the modern minor 

 scale ; "it also gives the imperfect chromatic semitone of 24 : 25 in relation to the 

 5th of the major scale. 



This finger-board is termed "natural " because no extra digitals lite the five black 

 digitals of the ordinary keyboard are required to produce the chromatic tones. 

 Eveiy coloured digital is sharp in relation to the white below it ; and every white 

 digital is flat in relation to the coloured above it, the relation being always 128 : 135. 



On this finger-board only 4 musical relations, viz. 8 : 9, 9 : 10, 15:6, and 128 : 135, 

 are found, and 3 musical difierences, viz. 24 : 25, 80 : 81, and 32,768 : 32,805. 



All the larger intervals of the scale are formed by adding 8:9, 9:10, and 15 : 16 

 together, and all the smaller intervals are produced by subtracting or dividing 

 these, thus : — 



8 : 9 less 15 : 16 = 128 : 135, the chromatic semitone ; 

 9:10 „ 15:16= 24 : 25, the imperfect chromatic semitone ; 

 8:9 „ 9 : 10 = 80 : 81, the comma ; and 

 8 : 9 „ 9 : 10 = 128 : 135, which being 



squared and divided by 9 : 10, gives the schisma 32,768 ; 32,805. Thus all the in- 

 tervals and relations of the musical scale proceed from these three simple elements, 

 8 : 9, 9 : 10, and 15 : 16. 



By adding a comma and a schisma together, the comma of Pythagoras is produced. 

 This is always found between keys changed enharmonically, as from C [^ to B. 



The three series of digitals upon this finger-board, white, coloured, and roimd, are 

 very easily tuned by perfect fifths throughout, and connected together by major 

 thirds. The tuning is diagonal, producing every interval perfectly in its proper place. 



Tuned in this way, this instrument (within its range or compass) is mathemati- 

 cally and musically perfect, without compromise or approximation of any kind, and 

 requiring neither equations, decimals, nor logarithms to explain it. It is very easily 

 played upon*. 



Oil a Practical Method of Tuninq a Major Third. 

 By Sir W. Thomson, D. G\L., F.B.S. 



Instkuments, &c. 

 On a Form of Gasholder giving a uniform Flow of Gas. By Prof. Barkett. 



Diagrams and Description of the new Lecture-Tahle for Physical Demonstration 

 in the Itoyal College of Science for Ireland. By Prof. Barrett, 



Two neiv Forms of Apparatus for the Experimental Illustration of the Expan- 

 sion of Solids by Heat. By Prof. Barrett, 



* The principles of construction, tuning, &,c. of tliR voioe-harmonium ■will bo found 

 fully explained in ' Mueic in Common Things,' part ii. Collin* and Co. : London, Edin- 

 burgh, and GlRggoTf. 



