224 S. A. Hageman — A Just Intonation Piano. 



Akt. XVIII. — A Just Intonation Piano; by S. A. Hageman. 



The problem of tuning and transposition and just intona- 

 tion, the practical solution of which is the subject of this 

 paper, is one which does not need to be restated for the readers 

 to whom it will chiefly come. But for their convenience, and 

 for the sake of logical completeness, the intervals of the two 

 scales true and tempered are here given. 



Taking, as is usual, the C scale for illustration, the letters 

 designating the tones of a complete octave are given with the 

 intervals between, and the fractions proportioned to the vibra- 

 tion numbers immediately below each letter. 



C|D V-EifF £ G Jf AfBifC 

 If i 4 1 I ¥ 2 



Chord lengths may be had by simply inverting the above frac- 

 tions. 



If, as is well known, we use, instead of these intervals, their 

 logarithms, we have a set of numbers that may be compared by 

 addition and subtraction and thus represent the actual magni- 

 tude of the intervals with a high degree of accuracy and con- 

 venience, especially for comparison with temperament. 



The numbers 102, 91 and 56 are modified logarithms of the 

 above fractions and may replace them — in which case 100 and 

 50 will respectively represent the two intervals of the tempered 

 diatonic scale which is here given. 



True scale C 102 D 91 E 56 F 102 G 91 A 102 B 56 C 



Tempered scale 100 100 50 100 100 100 50 



It is further desirable to append the complete duodene of C 

 as the most complete exhibition of all the tones and semitones 

 of the octave as used. 



Bb 



D 



F# 



Et? 



G 



B 



At 



C 



E 



Dfe 



F 



A 



This is taken from the English translation of Sensations of 

 Tone by Helmholtz. 



It is not considered necessary to explain these anew further 

 than to remark that, in this tabular arrangement of the com- 

 plete diatonic and chromatic scale, the intervals along the ver- 

 tical lines are pure fifths, and along horizontal lines pure 

 thirds. Thus their exact mathematical values are clearly 

 established. 





