Re: [tuning] Re: For Dan Stearns -- neo-Gothic

Margo Schulter wrote,

> Sometimes I may find it easy to write about theory or mathematics,
harder to communicate _enthusiasm_ for the music itself which provides
an occasion for the diversions of theory and mathematics.

Yes, I agree with this; the format of this forum just seems to be
better suited to sustaining threads that pertain to the various
theoretical or various mathematical angles. This understandably drives
some to distraction. After recieving the TD for almost a year and a
half now I personally remain pretty enthusiastic about the various
diversions of theory and mathematics here.

> You remarks lend me encouragement in focusing on the musical
experience itself.

I try to point out actual microtonal music that I'm personally
enthusiastic about here at the TD whenever possible, and when you
earlier commented on 17-tET's extra "unusual" or "neutral" intervals,
and mixing them with the usual medieval ones, you had my ears watering
(so to speak), and hoping to get an earful. Maybe you could eventually
post some of your improvisations to something like John Starrett's
TuningPunk site...

> Here your "17.095-tet" is a neat description of Pythagorean tuning
with pure fifths, and 17-note tunings (Gb-A#) are described and
advocated by the early 15th-century theorists Prosdocimus of
Beldemandis (1413) and Ugolino of Orvieto (c. 1430-1440).

Well, if the actual ~17.095-tET, or:

(LOG(2)-LOG(1))*(10/LOG(3/2))

were mapped to a circle of fifths from Dbb - B# you'd have:

F 491 702 G
Bb 983 211 D
Eb 281 913 A
Ab 772 421 E
Db 70 1123 B
Gb 562 632 F#
Cb 1053 140 C#
Fb 351 842 G#
Bbb 913 281 D#
Ebb 211 983 A#
Abb 702 491 E#
Dbb 0 1193 B#

This would give five sets of enharmoniclly distinct spellings at Db=1
C#=2, Fb=5 E=6, Gb=8 F#=9, Ab=11 G#=12, and Cb=15 B=16.

Now if (LOG(2)-LOG(1))*(10/LOG(3/2)), or ~17.095-tET is considered
consistent thru the 4:5:6:7, a lattice representation would always use
the best rounded (LOG(N)-LOG(D))*(T/LOG(2)) representations (were "N"
and "D" are the numerator and denominator of the relevant consonant
ratios, and "T" is the temperament) of the primary consonances, i.e:

5/3 5/4
\ /
\ 10/7 /
\ | /
7/6. \ | / .7/4
'.\|/.'
4/3---------1/1---------3/2
.'/|\'.
8/7' / | \ 12/7
/ | \
/ 7/5 \
/ \
8/5 6/5

as say:
16
/|\
/ | \
/ | \
/ .14 . \
/.' '.\
7-----------0----------10
\'. .'/
\ '3' /
\ | /
\ | /
\|/
11

And a byproduct of consistency -- which is only really concerned with
a consistent representation of the relevant consonant intervals -- is
that it occasionally gives rise to some fairly counterintuitive
scenarios. For example, say you take a modulation from C# minor to Db
major as

25/24-----25/16
\ /
\ /
\ /
\ /
5/4
|
|
7/4------21/16
\ /
\ /
\ /
\ /
21/20

In ~17.095-tET the 21/20 = 1 and the 25/24 = 2, and a modulation that
should be sharpening ~13.8� (a 126/125 commatic shift) is actually
flattening ~70.2�. A 225/224 commatic shift from a 15/14, 9/7, 45/28
minor triad to 16/15, 32/25, 8/5 minor triad would similarly result in
a perhaps disconcerting 8� = 70� enharmonically distinct shift from 2,
6, 12 to 1, 5, 11.

Dan