Re: [tuning] 17-limit scale

Johnny Reinhard posted the following scale:

"1/1
18/17
17/16
34/31
19/17
17/15
20/17
17/14/
22/17
17/13
24/17
26/17
17/11
28/17
34/20
62/34
34/17"

34/20 reduces to 17/10, 62/34 to 31/17. It's a 31-limit scale, built out of a
harmonic series fragment: 17-18-19-20-22-24-26-28-31/17 and a subharmonic series
fragment: 17/17-31-16-15-14-13-11-10. I'd guess that it has a tendency to be
ambiguous about its origins, as the harmonic series is missing the one identity
(32/17) and the subharmonic series is missing the three identity (17/12).

Yep! It seems the joke was on me, and by me. Sorry Pat and Dave.

--- In tuning@y..., "Daniel Wolf" <djwolf1@m...> wrote:
[Johnny Reinhard's "harmonic 17" is] a 31-limit scale, built
out of a
> harmonic series fragment: 17-18-19-20-22-24-26-28-31/17 and a
subharmonic series
> fragment: 17/17-31-16-15-14-13-11-10.

Indeed. I was about to post a similar analysis. If we show only odd
factors, the harmonic series subset looks like 3:5:7:9:11:13:17:19:31
and the subharmonic series subset looks like 1/(31:17:15:13:11:7:5:1).
Of course we can multiply any of those numbers by 2, 4, 8 etc since
this is an octave repeating scale.

The large harmonic series subset gives ample opportunity for the 17 to
be heard as justly intoned, unlike Pat and Dave's scale where the only
harmonic series subsets containing 17 are 1:17, 3:17, 5:17, but the 17
corresponds to a different note in each case. Note that I was wrong
before when I said that there was only one 17-limit dyad in the scale.
There are 3 (out of 66), but none of them extend even to triads.

Now whether a bare 8:17, 10:17, 12:17 or 16:17 can be heard as justly
intoned, particularly when the lower note in all these is only 7 cents
away from being the 5 in a 4:5:6, may be a matter of opinion. _My_
opinion on this was of course delightfully rebutted by the description
"just the silliest SH*T I've heard in ages". I just can't help but
admire the author's tight logic and his appeal to empirically
verifiable facts, not to mention his good humour. ;-)

There is a subharmonic series segment 1/(17:5:3:1) in Pat and Dave's
scale, but my understanding is that a utonality doesn't provide any
more context for justness to become audible, than do the dyads of
which it is composed.

But there is an important point raised by both these scales that is
_not_ a matter of opinion.

--------------------------------------------------------------------
Merely looking at the ratios describing the notes in one particular
mode of a scale can be _incredibly_ misleading about what kind of
intervals are predominant.

A scale with N notes has N*(N-1)/2 intervals. So 12 notes means 66
intervals, 16 notes means 120 intervals.

When the prime factors are few (say 4 or fewer), constructing a
lattice is a good way of seeing most of the intervals of interest. But
as a more general method, looking at every mode of the scale is
guaranteed to show us every interval (in fact we get to see them
twice, the second time inverted). Also, removing all factors of two
from all numerators and denominators makes it easy to spot subsets of
harmonic and subharmonic series. (If anyone wants to see the Excel
spreadsheet just let me know).

If you look at the ratios for all intervals in Pat and Dave B's scale
you will find only those 3 ratios of 17 (and their inversions).

If you look at Johnny's scale you will find 15 ratios of 17. But,
perhaps surprisingly, this is the same or one less than the number of
ratios of 5, or 7, or 11, 13, 31. There is slightly less 17-ness in
that scale than there is 13-ness or 31-ness for example. What's
special about 17 in that scale is the that it is the identity of the
one note that is common to both the harmonic and subharmonic subsets.
---------------------------------------------------------------------

Johnny, could you please expand on "The limit in JI is based on the
powerful resonance of the first 5 harmonics."

Regards,
-- Dave Keenan

In a message dated 3/9/01 12:57:10 AM Eastern Standard Time,
D.KEENAN@UQ.NET.AU writes:

> Johnny, could you please expand on "The limit in JI is based on the
> powerful resonance of the first 5 harmonics."
>
>

As theorists, y'all are forcing a round peg into a square shape. Yes, you
could describe "harmonic 17" as 31-limit, but to me it would be like
describing 12-tET as a meantone variant.

I found lots of triad usage, and made good use of the 7 cents. Performing
accurately to the cent is doable when you hear done it right. It is
eminently repeatable. It is a red herring to think in terms of just
intonation "lock." Regardless the tuning, any interval can be played exactly
when it is internalized by the musician playing.

Specifically, Dave, by not emphasizing 1-3-5 from the overtone series, my
"harmonic 17" tuning never sounds like it is based on any limit at all. Of
course, that is due to it being used in a performed composition. On the
page, there is more room for interpretation.

Best, Johnny Reinhard