Blackjack and 23-Skidoo -- for Joe Pehrson

Hello, there, Joe Pehrson, and please let me thank you and Joe Monzo for
educating me a bit on Blackjack, which doesn't include any 400-cent
thirds, a consideration which might make my comments not so relevant.

In a regular tuning, or a similar system like a well-temperament, the
assumption is that "G-B" represents a major third formed from a chain
of four fifths (G-D-A-E-B). However, in Blackjack, the chains are
built up not from fifths, but from secors, of course.

The result is that we have a different kind of system, and my wisest
response, now that I've been clued in, might be to let you and others
ad-JUST the terminology however it best suits your musical purposes.

Incidentally, a minor correction: I referred to "just" thirds in
Zarlino's 2/7-comma meantone, but meant "near-just," since the
approximations of 5:4 and 6:5, and also 9:7 and 7:6, are all
tempered.

Anyway, while my remarks might relate to some dialogues that come up
on the List, I'd say that yours are much more to the point concerning
Blackjack or any other tuning: the main thing is to make music, as
Ivor Darreg has also urged, and not let notational niceties stop one.

When dealing with 13-EDO and 20-EDO, tunings where the usual thirds
are not formed from regular chains of "fifths" at 8/13 octave and
12/20 octave, I simply adopt a spelling that tells me where things are
located on the keyboard. That means that "D-A" in 20-EDO is 11/20
octave or 660 cents rather than the usual "fifth" of 720 cents; while
"G-D" or "A-E" in 13-EDO is 7/13 octave or about 646 cents rather than
the "fifth" of about 738 cents (calling for the right timbres to have
a consonant effect like that of a 3:2).

Maybe a system of special interest here, Joe, is the Wonder Tuning,
with a "bisecor" generator equal to precisely a third of a 3:2 fifth,
or (3:2)^1/3, about 233.985 cents. Manuel Op de Coul, as I noted last
year, gives a version of this in the Scala archive, which in turn he
associates with some ideas of Adriaan Fokker.

The version of this tuning I've come recently to call "23-Skidoo" has
a chain of 23 bisecors, or 24 notes. I'm tempted to conclude that a
quintal notation might be best, grouping the notes into sets of five,
maybe something like this, with the first note of the tuning placed at
0 cents:

1 2 3 4 5

I 234 468 702 936 1170

II 204 438 672 906 1140

III 174 408 642 876 1110

IV 144 378 612 846 1080

V 114 348 582

Anyway, with Wonder in its "23-Skidoo" set, unlike Blackjack, we have
a bit of a quandary as to how to decide which thirds and sixths, if
any, set a notational "standard."

I call Wonder a "partially just" tuning, because we have pure 3:2
fifths and 4:3 fourths -- but also an irrational element of
temperament in dividing a pure fifth into three equal parts.

Here we do have some "regular" 408-cent major thirds at 81:64, as in
Pythagorean: 3 bisecors yield a 3:2, and 12 bisecors this third. Thus we
get Pythagorean chains by taking a set of notes at intervals of every
three bisecors.

However, as you point out with Blackjack, the main point of tuning
23-Skidoo is to get other kinds of thirds -- here, more specifically,
the near-6:7:9 variety. Four bisecors less an octave give us a
264-cent minor third only about 2.81 cents smaller than 7:6, while
seven give us a 438-cent major third only the same amount wider than
9:7. The bisecor itself gives us a near-8:7 or near-7:4 at this same
accuracy. The result is that 12:14:18:21 or 14:18:21:24 is available in an
approximate version at 17 locations, with the fifths pure and the other
intervals off by only this 2.81 cents.

Thus the near-just major and minor thirds at 438 cents and 264 cents
set the norm, although we have some Pythagorean thirds and other sizes
as well.

When a tuning isn't based on a regular chain of fifths, the usual
conventions I discussed in my previous post may not apply -- and if
someone chooses to follow them, the motivations might be somewhat
different than in a tuning where they fit the expected keyboard
layout, for example.

Anyway, thank you for a reply which not only educated me about
Blackjack, but has gotten me interested again in the Wonder Tuning and
the 23-Skidoo set.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:

> Maybe a system of special interest here, Joe, is the Wonder Tuning,
> with a "bisecor" generator equal to precisely a third of a 3:2 fifth,
> or (3:2)^1/3, about 233.985 cents. Manuel Op de Coul, as I noted last
> year, gives a version of this in the Scala archive, which in turn he
> associates with some ideas of Adriaan Fokker.

It seems reasonable to associate this with the 7-limit linear temperament with map [[0,3,-24,-1],[1,1,7,3]] and wedgie
[3,-24,-1,65,10,-45]. Do you suppose anyone would object if I started calling this temperament Wonder?

This is done well by the generator 23/118, incidentally.

> The version of this tuning I've come recently to call "23-Skidoo" has
> a chain of 23 bisecors, or 24 notes.

Can you explain why 24 notes, instead of what seem like the more obvious choices of 21 or 26?

> I call Wonder a "partially just" tuning, because we have pure 3:2
> fifths and 4:3 fourths -- but also an irrational element of
> temperament in dividing a pure fifth into three equal parts.

The deviation from justness of the rms optimized tuning of the fifth of "Wonder", or of the 118-et tuning of the fifth for that matter, is very slight, and many people would happily call them just also.

--- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:

/tuning/topicId_33913.html#33913

> Maybe a system of special interest here, Joe, is the Wonder Tuning,
> with a "bisecor" generator equal to precisely a third of a 3:2
fifth, or (3:2)^1/3, about 233.985 cents. Manuel Op de Coul, as I
noted last year, gives a version of this in the Scala archive, which
in turn he associates with some ideas of Adriaan Fokker.
>

***Thank you so much, Margo, for your contribution with illustration
of these other scales based upon other than "conventional" 3:2
generating chains. There are obviously lots of possibilities to
explore!

best,

Joe Pehrson

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:

> Can you explain why 24 notes, instead of what seem like the more
>obvious choices of 21 or 26?

if margo will permit me to interject, i believe it's because margo
likes to use two keyboards, each of which has 12 notes per octave.