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Chad Stearns added section_Other_Literature_Edward_C__.tex
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\section{Other Literature}
Edward C. Carterette and Roger A. Kendall wrote ‘On the Tuning and Stretched Octave of Javanese Gamelans’. In their paper, they primarily focus on the phenomenon of octave stretching, and they do little to examine the exact placement of the notes in the Slendro scale. Not only does this disappoint me, but they also neglect to share the recorded frequencies for the Gamelan ensembles they looked at. They do however, share the average the intervals recorded from the Slendro scale, and list them in the paper. We should note however, that the purpose of averaging intervals isnt without large problems\footnote{ To what extent this is valuable, is dubious, for the reasons mentioned in the section of this paper ‘Uniquely tuning per Gamelan Orchestra’. If two Gamelan orchestras were tuned perfectly to two different just intonation scale, their average would be imperfectly tuned to both. Also, if two Gamelan orchestras were tuned with a random degree of imprecision to the same just intonation scale, then we could only devise the just intonation scale from a statistically sufficient number of orchestras. That hey are tuned with non random imprecision, to a variety of just intonation scales, is both more plausible, and more confusing to the purpose of averaging many Gamelan orchestras.}. We shall do a cursory analysis of those average values, which I will list below\footnote{Carterette and Kendall list the intervals in cents, which can be mathematically reformulated in the decimal expression we have been using.}.
\begin{table}
\begin{tabular}{ c c }
Note & Interval \\
1 & \\
2 & 1.142742 \\
3 & 1.312666 \\
5 & 1.507857 \\
6 & 1.732073 \\
1 & 2.005785 \\
\end{tabular}
\end{table}
Carterette and Kendalls intervals make for a peculiar comparison with the intervals recorded from Arizona State Universitys Javanese Orchestra. The notes 2 and 3 are very similar, and the notes 5 and 7 are not similar.
Carterette and Kendalls recorded interval for the note 2 and note 3, incredibly, are almost exactly equal to the just intervals 8 / 7, and 21/16..
\begin{equation}
|(8/7)/1.142742| \ - \ 1 \ = \ 0.00010076
|(21/16)/1.312666| \ - \ 1 \ = \ 0.00012646
\end{equation}
This is a level of exactness that would exceed anyones anyone standard for precision in tuning. For instruments in the middle range of human hearing, it would take a full 20 seconds for a beat duration between 8/7 and Carterette and Kendalls recorded average for Slendro note 2.
Kendall and Cartettes recorded interval for note 5 and note 6 deviates from what we recorded from Arizona State Universitys Javanese orchestra. Note 5 roughly corresponds to 3/2. To my knowledge, Kendall and Carterettes recorded value for note 6 has no strong similarity to a simple just interval, it sits roughly equally between just intervals 7/4 and 12/7, and is closer to the equal tempered note 6 (2 ^ 4 / 5 ). It could be that Slendro notes 6 in Gamelan orchestras, are always tuned in equal proportion to either 7/4 or 12/7.
A just intonation model for Kendall and Carterettes recorded intervals fit far better than an equal intonation model, especially with regards to notes 2, 3, and 5. Carterette and Kendalls average recorded Slendro note 6 is closer to the equal intonation value, but not to an exceptional extent, and like we mentioned above different just intonation values for note 6 in different orchestras could result in Carterette and Kendalls average.
