The scale

Dear fellows,

There is no a twelve tone scale that can be recognized in most of the occidental countries as being the optimal solution for producing real harmonized chords and best expressions on the piano and other instruments.

Early in this month, when working with square array diagonals, the following two decimal ratios appeared:
1.1875 and 1.78125. Surprisingly, their quotient is 1.5 = 3/2. The new task was to buid a scale where these two ratios could operate as Eb and Bb.

The way was given by the interval 2C / Bb : (1.78125 / 1.1875) = 1.12280701754
The above interval is close to equal tempered D so it will tentatively work as D.
D fifth equals to 1.12280701754 x 1.5 = 1.68421052631, this value might be A. We got the contiguous frequency tones of A and Bb and therefore we can get its interval factor by dividing (Bb / A) = (1.78125 / 1.68421052631) = 1.0576171875

The (C / C#) decimal interval is supposed to be the latest one (1.0576171875); it is also C# relative frequency with respect to C. Now we have the contiguous C# - D tone frequency values and their interval factor given by:

(D / C#) = (1.12280701754 / 1.0576171875) = 1.06163839885.

Besides the interval factor (9/8)^0.5 = 1.06066017178 = q, two types
of factors m and p work in the octave, the scale is detailed here:

C----------- ( - 2.5 cents) ----------------------- 261.25 Hz
(x 1.0576171875) = m

C# ---- 1.0576171875 = (1083 / 1024) ( 94.5 cents) ---------- 276.3025 Hz
(x 1.06163839885) = p

D ----- 1.12280701754 = (64 / 57) ( 198 cents) ------------ 293.3333
x m
Eb ---- 1.1875 = (19 / 16) ( 295 cents) ----------- 310.2344
x p
E ------ 1.26069559863 = (4096 / 3249) ( 398.6 cents) ----------- 329.3567
x m
F ------ 1.33333333333 = (4/3) ( 495.6 cents) ----------- 348.3333
x q
F# ---- 1.41421356237 = 2^0.5 ( 597.5 cents) ----------- 369.4633
x q
G ------ 1.5 = (3/2) ( 699.5 cents) ----------- 391.875
x m
Ab ----- 1.58642578125 = (3249 / 2048) ( 796.5 cents) ----------- 414.4537
x p
A ------ 1.68421052631 = (32 / 19) ( 900 cents ) ------ 440 Hz
x m
Bb --- 1.78125 = (57 / 32) ( 997 cents) -------- 465.3516
x p
B ---- 1.89104339795 = (2048 / 1083) ( 1100.5 cents)--- 494.0351 Hz
x m
2C --- (1197.5 cents) ---- 522.5 Hz
-----------------------------------------------------------
Last night we succeded in the grand piano tuning with Piagui IV scale, as we
named to the new scale. After Luis Enrique Colmenares played
some pieces, the fantastic chord qualities clearly convinced us that this
scale is even better than Piagui I. After that, the piano tuned to the tempered scale was
availed to compare chords and triad components. Notes C#, Eb, F, Ab gave the
new taste to everything listened with astonishment.

Once the first pieces were listened I waited for the opinion of Luis
Enrique, the pianist, and Armando Becerra, the technician on tuning, they are
related with the piano chord sounds over the years. Both have said enthusiastically:
"Not only the harmony but everything is clearly more pleasant than what is heard on
a piano tuned to the Tempered scale, all the expressions overcome everything
we listened before."

I sent to two members of the list the complete information of this new scale; they
didn't answer my message; since I attached a table I had to send it to their
personal address, which might be cancelled; lack of politeness is hard to endure.
I know they are still busy with microtonalty, may be this contributed to cancel their
expected answers. This is not a claim but if they take the way of ignoring what is
going on outside of their country unless it is done by the same group of the list, they
could realize that not only microtonalty is an important field. The twelve tone scales still
require research as it was proved with what was achieved last night on the piano.

Thanks

Mario

June 12

On Sun, Jun 12, 2011 at 5:08 PM, Mario Pizarro <piagui@...> wrote:
>
> Dear fellows,
>
> There is no a twelve tone scale that can be recognized in most of the occidental countries as being the optimal solution for producing real harmonized chords and best expressions on the piano and other instruments.
>
> Early in this month, when working with square array diagonals, the following two decimal ratios appeared:
> 1.1875 and 1.78125. Surprisingly, their quotient is 1.5 = 3/2. The new task was to buid a scale where these two ratios could operate as Eb and Bb.

19/16 and 57/32! Not a bad choice - 12-equal actually supports them
very well. Minor triads can sound more harmonious when they're tuned
16:19:24.

Mario, have you tried moving beyond 12-tone scales? If you were to
look at some other note, you might find that there are lots of other
harmonic possibilities that only having 12-notes doesn't offer. If
harmonic purity is what you're after, you might like looking at some
19-note scales, as well as 22-note ones as well. Guitars won't have a
problem with them, since they can have as many frets as they want.
Pianos are a bit harder, but with modern MIDI controllers like the
AXiS

http://www.c-thru-music.com/cgi/

people can explore unconventional scales. Maybe you'd enjoy giving that a shot.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> 19/16 and 57/32! Not a bad choice - 12-equal actually supports them
> very well. Minor triads can sound more harmonious when they're tuned
> 16:19:24.

You could try tempering 513/512 in 2.3.19 as an alternative to 2.3.5 with 81/80.