Is it 17?

Since I don't have any electronic tuning equipment, I can't actually
measure what intervals I am producing on the harpsichord. So when I
found a diminished 5th b-f' (or rather two of them, g#-d' being tuned
in parallel, with g#-b and d-f both pure 6/5) that was quite a lot
narrower than 7-10, but still sounded non-beating and could be tuned
just by ear, I wondered for some time what it could have been.

I started from d and tuned f# pure: then it was clear by playing
around with c#, that g# was actually somewhat sharper even than 45/64.
By splitting the difference into two tempered fourths I could estimate
that g#-d was about 604-605 cents. Then by surveying possible dim5's
with possibly audible coincident upper partials I found 17/12 at 603
cents.

This made a lot of sense to me, as the interval sounded suspiciously
'normal'. By retuning b up to 7/10 below f' I could create a
diminished chord g#-b-d' extremely close to ET (1 - 119/100 - 17/12).
The giveaway was the sound of g#-b-f' when b was put back to 6/5 above
g#. I had initially noticed this had an unusual ringing to it, which
then made sense as the result of periodicity in 10-12-17.

Why the tritone f-b sounded (to me) rather less consonant than the
dim5 b-f' might be explained by the fact that 24/17 goes quite a bit
higher in the harmonic series than 17/12.

Sceptical or interested parties are invited to listen in:
launch.groups.yahoo.com/group/tuning_files/files/sphaerenklang/sweet_17.mp3

One answer I could give to Paul's ironic lament that he no longer
recognizes ET intervals: Just think of it as 17-limit!
~~~T~~~

Tom Dent wrote:

> Then by surveying possible dim5's with
> possibly audible coincident upper partials I found 17/12 at
> 603 cents.

It's an interesting point. The difference tone has a relative frequency of 5 and the nearest surrounding difference tone is 7. If one can pick up these by ear, then it's possible to tune the interval nicely by listening for the synchronicity regarding the common fundamental of 1. Another similarity is the minor third of 19:16 which has a difference tone of 3.

Petr