Re: [tuning] Re: Schismas, tuning systems, and tastes

Grahms's suggestion that all commas are alike rings true for Charles Ives's
finding commonality between the Pythagorean difference between the C and B#
and the 25-cent eighthtone. This appears the basis of Ives's theoretical
thinking.

Grahms's perspective is also why I consider the enharmonic of the ancient
Greeks to be more about the 5/4 major third and less about the 2 different
quartertones which make up the tetrachord.

Johnny Reinhard

Graham Breed wrote,

> I find it hard to see why, with 17 notes available, you wouldn't be
pulled away from the Pythagorean intervals, and towards the 5- or
7-limit.

On a different but still somewhat related note... I've experimented a
bit with taking sequences of a common note name -- say Fbb, Fb, F, F#,
Fx for example -- as a series segment; say the F example above as a
9:10:11:12:13:14. Now if you were to map this to a 7 by 5 grid you
would have:

Ebbb---Bbbb---Fbb----Cbb----Gbb----Dbb----Abb
\ \ \ \ \ \ \
\ \ \ \ \ \ \
Ebb----Bbb----Fb-----Cb-----Gb-----Db-----Ab
\ \ \ \ \ \ \
\ \ \ \ \ \ \
Eb-----Bb-----F------C------G------D------A
\ \ \ \ \ \ \
\ \ \ \ \ \ \
E------B------F#-----C#-----G#-----D#-----A#
\ \ \ \ \ \ \
\ \ \ \ \ \ \
E#-----B#-----Fx-----Cx-----Gx-----Dx-----Ax

This in turn could be seen as a sort of 13-limit,

Gbb
\
\
Gb
\
\
C------G------D
\
\
G#
\
\
Gx

pseudo 17 as it would sit on a circle of 17-tET (or near 17-tET)
fifths:

1/1
4/3 3/2
16/9 9/8
32/27 27/16
99/64 104/81
33/32 52/27
11/8 13/9
11/6 13/12
11/9 13/8
44/27 39/32
88/81 117/64
45/32 112/81
15/8 28/27
5/4 14/9
5/3 7/6
10/9 7/4
40/27 21/16
160/81 63/32

Interestingly, if you were to take the 11/9 slightly short of pure it
would make each superparticular ratio in this sequence equivalent -
i.e., all the best 11/10, 12/11, 13/12, and 14/13 approximations would
equal 2 if 17 were stretched up to somewhere around 17.282... in a
straight, unadulterated 17-tET I've also experiment a bit with some
scales based on differing combinations of a 1:3:7:9:11:13 consistency,
though this really is a separate approach from the one I just outlined
here.

Dan

Johnny!
The 5/4 does appear to be the more stable element of the enharmonic. But I believe there
is great significance n the case of Ptolemy who divided the remaining 16/15 into 3 parts
constructing a larger and a smaller interval. 45-46-47-48=46/45+ 48/46=24/23. The nature of
this choice could be cosmological thus extramusical in its choice. My intuition on this is
that he is not just being clever. As code has not been broken on this one, i see no reason to
dismiss these differences in size. The difference tones generated by this compared to a
32-31-30 division is important. One implies a tonic the other the dominant.

Afmmjr@aol.com wrote:

> Grahms's perspective is also why I consider the enharmonic of the ancient
> Greeks to be more about the 5/4 major third and less about the 2 different
> quartertones which make up the tetrachord.
>
> Johnny Reinhard

-- Kraig Grady
North American Embassy of Anaphoria island
www.anaphoria.com

Kraig!

Ptolemy was a c. 200 scientist from Alexandria, living long after and far
away from when and where the Enharmonic genus was in practice. As much as we
love Ptolemy's tuning specificity, it doesn't lead me to believe that there
was any special recognition for the distinctions of these sizes (except maybe
extra-musical as you suggest). Speaking of a "dominant" in tetrachordal
music may be out of place in terms of the time line.

Johnny Reinhard

Johnny!
thanks for the corrections on the dates. One could speculate that he was well aware of
older sources though and would think this is more than possible. Do we really know that the
enharmonic was no longer being used at this time. Dominant was not a good choice but the
difference in difference tones generated of one set would imply tones a fifth above the other.

Afmmjr@aol.com wrote:

Speaking of a "dominant" in tetrachordal

> music may be out of place in terms of the time line.
>
> Johnny Reinhard
>
>

-- Kraig Grady
North American Embassy of Anaphoria island
www.anaphoria.com