stretched-octave eq temps

Gregg Gibson wrote

>} I agree also that the fifths major thirds of 31-tone equal are slightly
>} but noticeably smoother and more pleasing than those of the 19-tone
>} equal. However, if the 19-tone octave is stretched by 2-3 cents, its
>} fifths become quite as good as those of the 31-tone.
>
>My calculations indicate that 2-3 cents is indeed the optimal amount of
>stretching for the 19-tone octave in order to maximize the overall consonance
>of intervals which are approximations of ratios using numbers no higher than
>6 (including the octave and double octave). This stretching results in what
>can be called 18.96TET. The consonance, relative to JI, of 19TET is 94.53%,
>while that of 18.96TET is 96.87%. However, 31TET is better at 97.30%, and
>30.98TET (31-tone octave streched by just under a cent) is still better at
>97.48%.
>
>By way of comparison, I get 85.07% for 12TET, and 86.70% for 12.02TET
>(12-tone octave reduced by 2 cents). It appears that the ear prefers
>stretched octaves as an equivalence interval anyway, especially for tones
>with weak harmonic content, or piano tones where the harmonics are slighly
>stretched, so adjusting the octave to increase consonance seems more viable
>in cases such as 19TET, where the octave needs to be increased, than 12TET,
>where the octave needs to be decreased.
>
>22TET gets a 92.21% score, and 22.01TET a 92.24% score, by this measure.
>However, increasing the highest number allowed in the ratios from 6 to 8, we
>get the following scores:
>
>12.00TET -- 54.48%
>12.06TET -- 61.56%
>19.00TET -- 79.28%
>18.94TET -- 83.84%
>22.00TET -- 87.68%
>22.03TET -- 88.77%
>31.00TET -- 98.037%
>30.99TET -- 98.043%
>
>(15 and 27 could also fit in this sequence of increasing consonance).
>
>The above assumes that the ear assesses intervals with a resolution of 1%.
>Changing this assumption does not change the rank-order of the results; only
>the absolute consonance levels depend on this assumption.


SMTPOriginator: tuning@eartha.mills.edu
From: "Paul H. Erlich"
Subject: series of improving equal temperaments
PostedDate: 16-12-97 18:45:20
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