Fwd: Two 21-tone JI scales: detemperings of blackjack

Paul

I would believe naïvely that an interval like 64/63 is too
small to be seen as a step but I would never use that as an
argument since it concerns musical judgment rather than maths
and I am searcher in music-math relations, not musician.

However I mentionned (post 1000) a problem concerning linear
generator picking up unison vectors using convexity. That
suggests 64/63 could be a unison vector misinterpreted as a
step.

Besides you wrote precedently:

<<
Personally, I think Blackjack, let along Canasta, have
too many notes to be heard and conceptualized in their
entirety, the way diatonic scales and their Middle-Eastern
cousins are, and perhaps my decatonic scales can be.
>>

So rather than comment on receivability in gammier theory of
your detempered values, I would suggest a reinterpretation
of the Blackjack set using incidentally a decatonic gammier.

You have to judge if it has interest from musical viewpoint
or even may concern your decatonic scales. If not, it won't be
bad for it's not strictly derived from gammier theory. I had
to use ad hoc maths to insert anew consistency into a reduced
(so desorganized) set of 21 intervals out of a gammier having
41 intervals originally.

-----

As alternative to the Blackjack set seen as the scale 0 2
7 9 14 16 21 23 28 30 35 37 42 44 49 51 56 58 63 65 70 72
having alternatively steps 2 and 5, let us work on this
approach where the same set would be seen as

9 16 23 30 37 44 51 58 65
0 72
7 14 21 28 35 42 49 56 63

where 2 and 70 are missing for being considered as unison
vectors. The values 2 and 70 would keep sense only for the
transposition and would be stranger in the modes begining
with 0.

-----

In appearance, there is no problem to describe the melodic
relations here since the steps seem to be simply 7 and 9.
>From the tempered viewpoint it would be hard to go over
the following solution.

9---16---23---30---37---44---51---58---65
/ / / / / / / / / \
0 / / / / / / / / 72
\ / / / / / / / / /
7---14---21---28---35---42---49---56---63

However, I recall that the Blackjack set is supposed to
be very closed from just values in 11-limit and it's a
reduced set (after Canasta 31 and Miracle 41), so the
steps are not forcely all retained.

So I want to refine the problem seeking to detemper in
such way that the intervals would be both of minimal
sonance and melodically well-organized.

-----

I have a link to images (post 1000) showing that the gammier
ib1215 would be the simplest solution using 41 intervals and
where it is easy to see that the step relations would be
completely desorganized with the Blackjack set.

Starting from that it seemed clear that odd 11 couldn't be
used to reorganize the melodic relations (even if it keeps
sense vertically). Thus, seeking to reorganize the just
relations in the 7-limit I used the following relation

12/11 = 35/32 modulo 385/384

to find an optimal solution which requires to use also the
step 5 with 7 and 9. I don't give more details since it's not
a systematic approach.

-----

In the following graph the edges (--,\,---) correspond
to (5,7,9).

(2) seen as unison vector
\
0 --- 9
\ \
7 --- 16
\ \
14 --- 23 -- 28
\ \ \
21 --- 30 -- 35
\ \
37 -- 42 --- 51
\ \ \
44 -- 49 --- 58
\ \
56 --- 65
\ \
63 --- 72
\
(70) seen as unison vector

which is a regular lattice (in sense of "treillis") if
the unison vectors are removed. So, detempering as

(5,7,9) == (21/20, 16/15 or 15/14, 35/32)

we obtain this JI lattice having all low sonance excepted
the two using odd 105.

| 35/32 | 21/20 | 35/32 |

--- ( 1 )----35/32
| |
16/15 | |
| |
--- 16/15-----7/6
| |
15/14 | |
| |
--- 8/7------5/4-----21/16
| | |
16/15 | | |
| | |
--- 128/105----4/3------7/5
| |
15/14 | |
| |
--- 10/7------3/2----105/64
| | |
16/15 | | |
| | |
--- 32/21-----8/5------7/4
| |
15/14 | |
| |
--- 12/7-----15/8
| |
16/15 | |
| |
--- 64/35----( 2 )

So, rather than that

9---16---23---30---37---44---51---58---65
/ / / / / / / / / \
0 / / / / / / / / 72
\ / / / / / / / / /
7---14---21---28---35---42---49---56---63

we have that

9---16---23---30---37---44 51---58---65
/ / / \ / \ \ \ / / / \
0 / / \ \ \ \ / / 72
\ / / / \ \ \ / \ / / /
7---14---21 28---35---42---49---56---63

-----

I add simply that 9 as a step has to correspond to 35/32 here
to give consistency while it may appear as 12/11 in chord.

Finally, I could draw images later if this JI interpretation
has sense at musical viewpoint.

Pierre

I replied at

tuning-math@yahoogroups.com