> ----- Original Message -----
> From: <jpehrson@...>
> To: <tuning-math@yahoogroups.com>
> Sent: Tuesday, June 19, 2001 8:10 PM
>Subject: [tuning-math] 7/72 generator in blackjack
>
>
> Well, this isn't very advanced... but, if not math, at least it's
> arithmetic...

That's OK, Joe... this list is for math dummies like me, too,
as well as guys like Paul, Dave, and Graham who understand the
more esoteric stuff.

>
> I still don't understand how 7 of the 72-tET scale is a generator of
> blackjack. It's a great concept (spooky!) since we have been
> finding 7's to be very peculiar in some other instances...
>
> Would someone please go over that again, gently??


Take another look at the explanation and especially the diagram
below the graph at
http://www.ixpres.com/interval/dict/miracle.htm


Dave Keenan found the MIRACLE generator (~116.7 cents) by use
of the "brute force" approach: he had his computer perform
thousands (millions?.. billions?) of calculations and analyze
the resulting scales.

The ~116.7-cent generator came out on top as implying the
largest number of 11-limit consonances. 2^(7/72) happens
to be extremely close to the calculated MIRACLE generator
(which, I should emphasize, is only *one* possible MIRACLE
generator... there can be many, depending on the error method
selected).

The diagram on my Dictionary page shows how you cycle thru
intervals of 2^(7/72) on either side of 1/1, which in this
case really should be called 2^(0/72). Upon reaching the
10th note on either side, you've got Blackjack. Extending
to the 15th note on either side gives you Canasta.

This process is exactly analagous to constructing a meantone
cycle, except that instead of a "cycle of 5ths", you're
cycling thru the generator interval, whatever it may be.


An interesting digression: some scales can be thought of as
being constructed by more than one generator simultaneously.
Naturally, our familiar old 12-EDO is one such scale. It
can be thought of as a "cycle of 5ths" where each "5th" is
7 Semitones; here's an example centered on "D" (flats and
sharps are of course equivalent to their enharmonic cousins):

Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G#
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Or it can be thought of as a cycle of Semitones:

Ab - A - Bb - B - C - C# - D - Eb - E - F - F# - G - G#
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6


In either case, the generator creates a scale of 12 distinct
pitches before producing a pitch which is an exact replica
of one already existing.



-monz
http://www.monz.org
"All roads lead to n^0"





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