Yahoo Tuning Groups Ultimate Backup tuning A weird 12-out-of-BlackJack

Paul Erlich wrote:

>BJ# cents
>20 11.667
>18 10.500
>17 9.667
>15 8.500
>13 7.333
>11 6.167
>10 5.833
>8 4.667
>6 3.500
>4 2.333
>3 1.500
>1 0.333
>
>(note that there is no note at 0 cents)

Cool. Here it is on the chain of 21 Erlich-Keenan generators (shown with
"+" signs at the notes of Paul's 12-note subset.

[Yes, I'm vain enough to like that name. Thanks Graham. :-)]

When you see it against the otonal and utonal 7-limit tetrads (and several
other chord patterns) you can see why it is a good choice.

Degrees of 72-EDO
2 9 16 23 30 37 44 51 58 65 0 7 14 21 28 35 42 49 56 63 70
+--+-----------+--+--+--+--------+--+--+--+--------------+--+
5--------------7-----1-----------------3
1/3---------------1/1---1/7------------1/5

5--------------7-----1-----------------3-----------------9-------11

Folks should note that these diagrams _must_ be viewed in a monospaced font
such as Courier.

>This has 3 7-limit otonal tetrads, 3 7-limit utonal tetrads, both of
>blackjack's 10:12:15:18s, and one 12:14:18:21.

Many chords in this subset turn out to have familiar patterns if you map it
to the 12-EDO keyboard with 51/72-EDO as C. Here's a Scala file to do that.

! 12-of-blackjack.scl
!
Paul Erlich's 12-tone proper Blackjack subset in 72-EDO
12
!
116.6666667 ! Db
200.0 ! D
316.6666667 ! Eb
383.3333333 ! E
500.0 ! F
583.3333333 ! F#
700.0 ! G
816.6666667 ! Ab
933.3333333 ! Bbb
966.6666667 ! A#
1083.333333 ! B
2/1 ! C

Step sizes in 72-EDO steps:
7 5 7 4 7 5 7 7 7 2 7 7

>Can anyone find a better 12-tone proper subset?

No. But there is of course a slightly better 12-note 7-limit subset of the
Erlich-Keenan temperament in general. Lumma's scale!

It has the same number of tetrads but more triads (I think) and is more even.

This can be aligned so it differs from the above subset in only 2 notes, or
it can be shifted so that all but one note is within Blackjack, in which
case C corresponds to 58/72-EDO.

Here's a scala file.

! Lumma_in_72.scl
!
Carl Lumma's scale in 72-EDO
12
!
116.6666667 ! Db
200.0 ! D
266.6666667 ! D#
383.3333333 ! E
500.0 ! F
583.3333333 ! F#
700.0 ! G
816.6666667 ! Ab
883.3333333 ! A
966.6666667 ! A#
1083.333333 ! B
2/1 ! C

Step sizes in 72-EDO steps:
7 5 4 7 7 5 7 7 4 5 7 7

So it's more even, and there are more familiar chord patterns (notably the
A key is A, not Bbb). What a shame it just doesn't quite fit in Blackjack.

These 12-note subsets will give maximum accessibility for keyboardists, but
only to a small (but interesting) slice of the temperament. Meantone names
work well for these.

Regards,
-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

🔗paul@stretch-music.com

--- In tuning@y..., David C Keenan <D.KEENAN@U...> wrote:
> Degrees of 72-EDO
> 2 9 16 23 30 37 44 51 58 65 0 7 14 21 28 35 42 49 56 63 70
> +--+-----------+--+--+--+--------+--+--+--+--------------+--+

Shouldn't the pattern be symmetrical about 65-0?
>
> Many chords in this subset turn out to have familiar patterns if
you map it
> to the 12-EDO keyboard with 51/72-EDO as C. Here's a Scala file to
do that.
>
> ! 12-of-blackjack.scl
> !
> Paul Erlich's 12-tone proper Blackjack subset in 72-EDO
> 12
> !
> 116.6666667 ! Db
> 200.0 ! D
> 316.6666667 ! Eb
> 383.3333333 ! E
> 500.0 ! F
> 583.3333333 ! F#
> 700.0 ! G
> 816.6666667 ! Ab
> 933.3333333 ! Bbb
> 966.6666667 ! A#
> 1083.333333 ! B
> 2/1 ! C
>
> Step sizes in 72-EDO steps:
> 7 5 7 4 7 5 7 7 7 2 7 7

Awesome! Gentlemen (and ladies), start composing!
>
> >Can anyone find a better 12-tone proper subset?
>
> No.

How about a better improper 12-tone subset?

🔗David C Keenan <D.KEENAN@UQ.NET.AU>

Paul Erlich wrote:
> Shouldn't the pattern be symmetrical about 65-0?

I messed up all right. But no, it should be symmetrical about 0. Here is
what it should have been (35 instead of 7):

Paul's proper 12 of Blackjack
2 9 16 23 30 37 44 51 58 65 0 7 14 21 28 35 42 49 56 63 70
+--+-----------+--+--+--+-----------+--+--+--+-----------+--+

I only messed up the diagram. I used the right notes for the Scala files etc.

> How about a better improper 12-tone subset?

I don't think there is a better 7-limit one at all.

But the following has 2 otonal and 2 utonal 11-limit pentads (1-3-7-9-11)
and 2 major and 2 minor 5-limit triads and 1 otonal and 1 utonal 1-5-7
triads, and probably some other useful stuff. Only otonalities are shown on
the chain below.

7-----1-----------------3-----------------9-------11
5--------------------1-----------------3
5--------------7-----1
+--+--+--+--+--------------+--+--------------+--+--+--+--+
2 9 16 23 30 37 44 51 58 65 0 7 14 21 28 35 42 49 56 63 70

! 12-of-blackjack-11.scl
!
A 12-tone 11-limit subset of Blackjack
12
!
33.33333333 ! C|
150.0 ! D;
266.6666667 ! D#
383.3333333 ! E
500.0 ! F
583.3333333 ! F#
700.0 ! G
816.6666667 ! Ab
933.3333333 ! A|
1050.0 ! B;
1083.333333 ! B
1200.0 ! C

Step sizes in 72-EDO steps:
2 7 7 7 7 5 7 7 7 7 2 7

It is not proper but it is CS.

I've used a meantone-ish notation again for this, but with half sharps "|"
and half flats ";" (as suggested by Graham, but using Manuel's
ASCIIfication). In real life the half sharp is a vertical line with two
short horizontal lines across it, and the half-flat symbol is like a flat
symbol with the top half of the loop missing.

Graham, do these correspond to your use of "^" and "v", which everyone
seems to want to use in different ways?

For those who can't make the leap to a notation with 10 (or 11!) nominals,
this seems a good approach for MIRACLE-31 and its subsets including
Blackjack (= MIRACLE-21). The full Boston 72-EDO notation seems like
overkill, except for free-pitched instruments, in which case the
appropriate symbols to indicate shifts of a single 72-EDO step up or down,
can be appended to these names. Only three pairs of modifiers are needed,
not 4. And never more than two at once.

Here's Blackjack notated in this way on the chain.

2 9 16 23 30 37 44 51 58 65 0 7 14 21 28 35 42 49 56 63 70
D| E; F; F# G Ab A| B; C; C# D Eb E| F| G; G# A Bb B| C| D;

This is centred on D (for keyboard symmetry reasons), but there are other
positions on the chain that result in 5 naturals instead of 3. e.g.

Note that this centers the D-A fifth. Another venerable institution.

Bit maybe we'd rather centre it on the C-G fifth? We still get 5 naturals.

2 9 16 23 30 37 44 51 58 65 0 7 14 21 28 35 42 49 56 63 70
E F Gb G| A; A# B C Db D| E; F; F# G Ab A| B; C; C# D Eb

Yes, I like this one best.

A 4:5:6:7:9:11 hexad on C would be spelled C E G A# D F|.
A neutral triad on C is spelled C E; G.

I've run out of time tonight. If anyone sorts that last naming scheme into
pitch order (starting from C?) please post it.

Regards,
-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

🔗David C Keenan <D.KEENAN@UQ.NET.AU>

🔗Graham Breed <graham@microtonal.co.uk>

Dave Keenan wrote:

> I've used a meantone-ish notation again for this, but with half sharps "|"
> and half flats ";" (as suggested by Graham, but using Manuel's
> ASCIIfication). In real life the half sharp is a vertical line with two
> short horizontal lines across it, and the half-flat symbol is like a flat
> symbol with the top half of the loop missing.
>
> Graham, do these correspond to your use of "^" and "v", which everyone
> seems to want to use in different ways?

Oh, hello! In meantone, yes, ^ and v would be half of # and b. This is the
same definition as Vicentino's dot. So you can use them for neutral thirds.
Also for septimal commas. In 31-equal, they come out as 1 step, and so they
can also be used for 31-note enharmonies like C#^==Db in general meantone.
Usually the context will make it obvious which is being used.

I used to use / and \ but it seems they're generally used for commas. Still,
most of my website is like this. + and - are no good because they vanish on
the lattice (/ and \ aren't much trouble) and staff. It would be nice if we
could get some kind of consensus on this.

Logically, it'd follow that # and b would refer to 2 quommas in decimal
notation, for consistency with 31-equal. But I think they're better as 3
quommas for consistency with 72-equal. ^^ and w aren't much trouble but ^^^
and vw are really worth avoiding.

I happen to have updated <http://x31eq.com/decimal_notation.htm> to
cover the meaning of / and \ to mean half of ^ and \. Also globally replaced
"Erlich-Keenan" with "Miracle".

If we want to make the decimal ^ and v distinct from Boston notation, we can
either make them triangles or not. Depending on what Boston notation actually
does. But I would prefer to keep to ASCII as I use it so much.

Graham

"I toss therefore I am" -- Sartre

🔗David C Keenan <D.KEENAN@UQ.NET.AU>

So much for sleep. Nope. It seems the best you can do is get 6 naturals in
blackjack, whichis what the one centred on C-G had (not 5 as I mistakenly
wrote). So 7 naturals notation of Lumma's scale is a mystery.

Here's are two that omit only D. The first is centred on E-B and thereby
favours G# over Ab. The second is centred on F-C and favours Ab over G#.

2 9 16 23 30 37 44 51 58 65 0 7 14 21 28 35 42 49 56 63 70
G# A Bb B| C| D; D# E F Gb G| A; A# B C Db D| E; F; F# G

2 9 16 23 30 37 44 51 58 65 0 7 14 21 28 35 42 49 56 63 70
A Bb B| C| D; D# E F Gb G| A; A# B C Db D| E; F; F# G Ab

That's 5 candidates so far. Do we vote on a canonical one? (Like the Eb to
G# standard for meantone?)

The canonical MIRACLE-10 should probably be
D# E F Gb G| A; A# B C Db
which has the maximum of 4 naturals.

This also corresponds to the obvious keyboard mapping for MIRACLE-10 too.
i.e. omit the D and G#/Ab keys.

-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

🔗paul@stretch-music.com

🔗monz <joemonz@yahoo.com>

--- In tuning@y..., paul@s... wrote:

/tuning/topicId_22532.html#22683

> --- In tuning@y..., Graham Breed <graham@m...> wrote:
>
> > If we want to make the decimal ^ and v distinct from Boston
> > notation, we can either make them triangles or not.
> > Depending on what Boston notation actually does.
>
> Boston notation doesn't use triangles.

Right.

Graham, *I'm* the one who introduced ^ and v to represent
the 1/4-tone symbols in Boston notation that couldn't be
given in ASCII.

Actually, I had modified the Boston notation before I ever
had email, to conform more to Franz Richter Herf's 72-EDO
notation. I always like the up and down arrows for
quarter-tones, so retained them here.

The < and > suggested themselves right away as analogues to
the "half-arrows" used by both the Bostonians and Herf for
1/6-tones inflections.

And the + and - came out of my experiences with Johnston
notation. I recognized long ago that 1/12-tones were an
analogy to the syntonic comma.

> See the Mook page (anyone have the link handy)?

https://www.mindeartheart.org/micro.html

> In any case, 12-tET is the norm, it's what musicians are
> trained to play, and that's why we're investigating 72-tET
> in the first place. So you had better incorporate that norm
> into your notation.

And this, from the guy who invented his *own* decatonic
notation! I very much like the simplicity of your decimal
notation to represent Miracle tunings, Graham, at least
the smaller subsets like Blackjack.

But from a practical perspective Paul does have a good point.
So I'm going along with Joe Pehrson and basing all my notation
from now on on 72-EDO. (Unless I need 144-EDO, that is...)

-monz
http://www.monz.org
"All roads lead to n^0"

🔗David C Keenan <D.KEENAN@UQ.NET.AU>

Here's one for Alison.

12-tone Blackjack subset, six 4-7-9-11 tetrads
2 9 16 23 30 37 44 51 58 65 0 7 14 21 28 35 42 49 56 63 70
1/11-----1/9---------------------------------1/1---1/7
7-----1-----------------------------------9-------11
1/11-----1/9---------------------------------1/1---1/7
7-----1-----------------------------------9-------11
1/11-----1/9---------------------------------1/1---1/7
7-----1-----------------------------------9-------11
+--+--+--+--+--+--------------------------+--+--+--+--+--+
C| D; D# E F Gb G| A; A# B C Db D| E; F; F# G Ab A| B; notation
C C# D Eb F G E F# G# A A# B keyboard key

Step sizes:
7 7 7 5 2 5 2 5 7 7 7 11

It's so uneven it isn't even CS. However, it has a strictly proper
decatonic subset (with only four tetrads). Just omit 37 and 28 (Gb and F;),
The G and E keys could then be reassigned to the notes of the same name.

! 12-of-blackjack-7-9-11.scl
!
A 12-tone subset of Blackjack with six 4-7-9-11 tetrads
12
!
116.6666667 ! D;
233.3333333 ! D#
350.0 ! E
433.3333333 ! F;
466.6666667 ! F
550.0 ! F#
583.3333333 ! Gb
666.6666667 ! G
783.3333333 ! Ab
900.0 ! A|
1016.666667 ! B;
2/1 ! C|

Joseph,

The idea with these scala definitions is to copy and paste them into a text
editor (e.g. Notepad or SimpleText), not directly into Scala, and then save
them as the filename given at the top.

Regards,
-- Dave Keenan
-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

--- In tuning@y..., paul@s... wrote:
> --- In tuning@y..., David C Keenan <D.KEENAN@U...> wrote:
>
> >
> > Paul's proper 12 of Blackjack
> > 2 9 16 23 30 37 44 51 58 65 0 7 14 21 28 35 42 49 56 63 70
> > +--+-----------+--+--+--+-----------+--+--+--+-----------+--+
>
> How about this 12-tone scale:
>
> 2 9 16 23 30 37 44 51 58 65 0 7 14 21 28 35 42 49 56 63 70
> ---+--+--+--+-----------+--+--+--+-----------+--+--+--+------
>
> Promising or unpromising?

Extremely promising harmonically! I haven't looked at melodic. Make us
a Scala file of it.

There's a sort of secondary generator happening in a lot of these
subsets isn't there. It consists of 7 miracle generators, which is the
4:5. This strongly suggests that 4:5 would be a good spacing for
guitar strings.

-- Dave Keenan

🔗Graham Breed <graham@microtonal.co.uk>

Monz:

> Graham, *I'm* the one who introduced ^ and v to represent
> the 1/4-tone symbols in Boston notation that couldn't be
> given in ASCII.

Okay. So do any existing notations actually use chevrons?
Conflicting ASCII diagrams aren't such a problem. But I wonder about
copying errors between decimal and some other 72-equal notation if the
same symbols are used for different meanings. Also the confusion of
players learning the same symbols to mean different things.

> Actually, I had modified the Boston notation before I ever
> had email, to conform more to Franz Richter Herf's 72-EDO
> notation. I always like the up and down arrows for
> quarter-tones, so retained them here.

I think "quartertones" can generally be defined as half a chromatic
semitone. Hence a neutral third is half-way between a major and
minor. So my usage in decimal notation is consistent with this.

Paul:

> > In any case, 12-tET is the norm, it's what musicians are
> > trained to play, and that's why we're investigating 72-tET
> > in the first place. So you had better incorporate that norm
> > into your notation.

Monz:

> And this, from the guy who invented his *own* decatonic
> notation! I very much like the simplicity of your decimal
> notation to represent Miracle tunings, Graham, at least
> the smaller subsets like Blackjack.

Oh, I see that is in 22ALL.pdf after all. I don't remember it, but it
may be where I got the triangles from. The two notations do look
similar, but Paul's isn't on regular manuscript paper.

12 and 7 aren't important numbers in Miracle temperament. So writing
Miracle scales in a notation invented for 7 and adapted for 12 is
always going to be complex. The only way out is to make a clean
break, which is what I've done. Anything I don't like about staff
notation can be thrown out with the 7- and 12-ness.

> But from a practical perspective Paul does have a good point.
> So I'm going along with Joe Pehrson and basing all my notation
> from now on on 72-EDO. (Unless I need 144-EDO, that is...)

Yes, from a practical perspective that's what I suggested as well.
But if a lot of music gets written in Miracle-biased scales, it's
going to look very complex in a 12*6 notation. Eventually, a point
will be reached when it's easier to make a concentrated effort to
learn a new notation than keep on making the extra effort to use a
more complex notation.

Every performer and composer will have to decide for themselves when
that point is reached. Instruments built for decimal scales will
hurry it along. For myself, as I don't think in terms of 12-equal,
and don't have to worry about performances, I made the break right
away.

The 72=12*6 coincidence is going to be important in phasing in Miracle
tuning. Music written in one of the Miracle MOS scales can be
transcribed into familiar notation for performance. But I don't make
a big thing of this as it doesn't concern me. Also, existing 11-limit
music can be transcribed into decimal notation for early adopters. It
would be convenient to have a conversion chart between decimal and
31/meantone, 12*6 and 12-equal-plus-cents notations. If nobody else
does, I'll have a look at that.

Graham

🔗paul@stretch-music.com

🔗monz <joemonz@yahoo.com>

🔗jpehrson@rcn.com

--- In tuning@y..., David C Keenan <D.KEENAN@U...> wrote:

/tuning/topicId_22532.html#22655

>
> For those who can't make the leap to a notation with 10 (or 11!)
nominals,this seems a good approach for MIRACLE-31 and its subsets
including Blackjack (= MIRACLE-21). The full Boston 72-EDO notation
seems like overkill, except for free-pitched instruments, in which
case the appropriate symbols to indicate shifts of a single 72-EDO
step up or down, can be appended to these names. Only three pairs of
modifiers are needed, not 4. And never more than two at once.

Hello Dave!

I'm so far behind now, that I'm reading the posts BACKWARDS, hoping
eventually to get to "0!"

However, I think part of the point of using 72-tET notation is the
fact the people are starting to get used to it, and there seem to be
a whole bunch of scales very closely related to it, so it becomes a
kind of "lingua franca," so to speak... so players aren't constantly
having to learn new notations for new scales.... or at least that's
how *I* am understanding this....

___________ _________ ________ ______
Joseph Pehrson