Euler in English

[Rousseau:]
>> Classical interpretation: 1/1 5/4 3/2 9/5 (36 45 54 64)
>> Euler's interpretation: 1/1 5/4 3/2 7/4 (36 45 54 63, or 4 5 6 7)

[Euler:]
>>> "the difference is so small (64/63) that it is not perceived."

[Barbaro:]
> The difference between the two interpretations above is 36/35,
> not 64/63.

Brett is correct.

9:5 = 36:20
7:4 = 35:20

The 64/63 difference is between 16/9 and 7/4:

16:9 = 64:36
7:4 = 63:36

My notes on Euler say that in _Conjecture_ [1764], he wrote:

> "The ear tends to simplify the ratio perceived, especially
> if the dissonant tones follow after a harmonious progression;
> for example, 36:45:54:65 is indistinguishable from 36:45:54:63
> [= 4:5:6:7]

I'm very surprised looking at that now to see that there's
a 13-limit ratio included! [65:36 = 2^-2 * 3^-2 * 5^1 * 13^1]

I got this citation from the 1980 Grove's Dictionary.
It's possible I copied it wrong, or that it's their typo.

In any case, The proportions Kami gave for the 'classical'
chord, 36:45:54:64, are the ratios 1/1 : 5/4 : 3/2 : 16/9.
This is almost certainly the chord Euler was using for
comparison with 4:5:6:7.

> There's a lot more on that French Euler page.
> Can we gain any insight into his GS function?

I'm halfway finished with a webpage of
an English translation of Bailhache's webpage.

In a few more days it'll be ready.

-monzo

|\=/|.-"""-. Joseph L. Monzo....................monz@juno.com
/6 6\ \ http://www.ixpres.com/interval/monzo/homepage.html
=\_Y_/= (_ ;\
_U//_/-/__/// |"...I had broken thru the lattice barrier..."|
/monz\ ((jgs; | - Erv Wilson |

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