TUNING digest 934

Manuel,

Thanks for doing the heavy lifting on figuring self-similarities in
the Partch scale. A great reminder of how powerful your SCALA program
is for PC users.

Carter

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David Malkin wrote:

> Thanks for all the help from List members. I've made it to Page 117. Harry
> talks about how notes can be taken in different senses (Otonal or Utonal).
> How does he come up with the notion that the 12TET scale can be taken in
> 144 senses?

He just means that C is the tonic of C, the leading tone of C#, the flat 7th of D, etc. The pitch C has 12
"senses" depending on which pitch you consider the tonic. Do this for all 12 notes and all 12 tonics and you
get 12 * 12 144 different "senses" of some pitch taken against some tonic in 12tet.

> Also, this statement he makes is very cryptic to me:
>
> "As for the senses in which the thirteen degrees may be taken, 1/1,
> 3/2, and 4/3 may each be taken in six; 6/5, 5/3, 5/4, and 8/5 may each be
> taken in three and the other six ratios may each be taken in two, making a
> total of 42 senses."

I'm not sure which 13 degrees he's talking about, but he's doing the same thing; seeing how many ways each of
the 13 pitches he's isolated can be interpreted musically depending on which tonic (Numerary Nexus actually,
which is almost like tonic) is assumed. I believe that not all pitch relationships are counted as valid senses
by Partch (see below) so counting senses in a Partch scale is not as simple as s n * n, where n is the
number of pitches and s is the number of senses.

I'm going on memory again, but here's how I think it works:

For instance, 5/7 is neither in the otonality nor utonality of 1,
otonality of 1 1/1 3/1 5/1 7/1, etc.
utonality of 1 1/1 1/3 1/5 1/7, etc.
so 5/7 has no valid sense when 1 is considered the Numerary Nexus.

It is however in the utonality of 5,
utonality of 5 5/1 5/3 5/5 5/7 (bingo), etc.
and it's also in the otonality of 7,
otonality of 7 1/7 3/7 5/7 (bingo) 7/7, etc.
so 5/7 has at least two valid senses.

I think he's counting senses to argue that even though you can't modulate indefinitely in a just scale with a
fixed number of tones (43 in his case) you can still play in many keys and have the tones to cover the senses
required to play in those keys. If you work with instruments which allow adjustment of pitch, you don't have
to worry about counting senses and playing in only certain keys, because you can modulate indefinitely and
never run out of notes.

Matt Nathan

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