Limitations on Fixed n-Tone Equal Tempered Divisions
Aline Honingh, E. Bilotta, G. Buzzanca, V. Cafagna, G. Di Maio, M. Francaviglia, G. Nottoli, M. Olivetti Belardinelli, Pietro Pantano, Tarabella L. · 2004
To construct an-tone equal tempered division other than, one usually uses a goodness-of-fit approach to find the-tone system that best approximates a number of intervals from just intonation. This method leads to equal tempered divisions of size, and. However, to be able to use an-tone equal tempered system for a keyboard application, there are some restrictions to be taken into consideration. We demand a surjective mapping from the notenames to the units of the-tone equal division, in order to have a suitable mapping from a score to e.g. the keys of a piano. We will show that this demand translates into two mathematical conditions which lead to the following values for: 5,7,12,19,26,31,43,45,50,55,69,74,81,88. Combining this result with the general results for a goodness-of-fit approach, we conclude that good divisions of the octave are and, which have indeed been used in musical practice. Since there is a limit to the closeness just intonation can be approximated by equal temperament in the method used, we expand our note-name system so as to differentiate between different frequency ratios having the same note-name. Certain conditions apply to this new system as well and the resulting values for can serve as an explanation as to why certain-tone temperaments have been used in the past. 1