Yahoo Tuning Groups Ultimate Backup tuning How to get the Miracle "white notes" on a standard keyboard

Here is some practical advice for those wanting to get a taste of Miracle
on a standard (Halberstadt) keyboard, with an octave still being an octave,
and preserving as many familiar chord patterns as possible.

You can consider the tuning I am about to give to be either the 10 "white
notes" of miracle, unavoidable mapped to a combination of black and white
keys; or a septimal pentatonic on the black keys and another on 5 of the
white keys. Two of the white keys are redundant and are tuned the same as
adjacent black keys.

In the columns below I show the keyboard key, the cents deviation from
12-tET, and four different notations:
Graham Breed's decimal numbers, decatonic letters, Tartini/Fokker 31 note,
Tartini/Fokker 72 note.

Kbd Cent Dec Dec T/F T/F
Key dev num let 31 72
--------------------------
A +33 0 A A| A|\
Bb +50 1 B B; B;
B -50 1 B B; B;
C -33 2 C C; C;/
C# -17 3 L C# C#\
D 0 4 D D D
Eb +17 5 N Eb Eb/
E +33 6 E E| E|\
F +50 7 F F| F|
F# -50 7 F F| F|
G -33 8 G G; G;/
G# -16 9 H G; G#\

The decimal numerals have been realigned relative to Grahams earlier
conversion chart.

The decatonic letters are slightly different from my earlier proposal. I
switched from K and L to L and N so that the numbers agree with the letters
modulo 10. e.g. ABC could also be IJK and D could also be M.

The ASCIIfied Tartini/Fokker notation is Manuel's and uses
| half sharp
; half-flat
/ comma up
\ comma down

In real life Tartini's half sharp looks more like

|
-|-
|
-|-
|

and Tartini's half flat looks more like

|
|
|
| /
|/

Fokker's comma strokes are just as they are in ASCII.

The pentatonics are generated by Manuel Op de Coul's generator (~14/72 oct,
every second miracle generator). They are a quasi-equal pentatonic or
slendro scale (Dudon's slendro_s2.scl). They are also a degenerate
1.7.49.343 Hexany (hexany18.scl).

The decatonic is similar to a half Euler-Fokker Genus 357777
(halfefg357777.scl) (9.3 c error).

I would have given a Scala file for this 12-tET to Miracle-10 mapping but I
assume the 2/1 must be C, and if C is 0/1200 cents then some other notes
will need 83 cent deviations and I know some devices' tuning tables only
allow -64 to +63 cents deviation. Can someone help me out here?

If we need to make it clear that a letter is a decimal letter and not a
diatonic letter, I propose we use a subscript "o", as in "Ao" (read A
decimal). This is unnecessary for chromatic alterations if we use "^" and
"v" for these as Graham Breed proposes, and Monz stops using them for ASCII
72-EDO 1/4-tones (possibly using | and ; instead).

So you can think of "o" as the natural of "^" and "v".

I'm going to give some chords and things using decimal letter notation, so
you might want to put some little stickers on your G#, C# and Eb keys
reading respectively "H", "L" and "N". And remember that Bo can be played
on either the B or Bb key, similarly Fo can be played on either the F or F#
key.

The two Slendro pentatonics are ACDEG and BLNFH.

Here are some triads in root position to get you started:

Neutral : ALE, BDF, CNG, LEH
4:5:7 : FAN, GBE, HCF
4:6:7 : CGA, LHB
6:9:11 : AEH

If you find other chords, please post them. There are the utonal versions
of the non-neutral ones above. There are also some ASSes in there
somewhere. I'm sure you will find other interesting ones by listening, that
I haven't found with the numbers. And I'm dying to hear some more music in
this scale. :-)

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

🔗David C Keenan <D.KEENAN@UQ.NET.AU>

🔗manuel.op.de.coul@eon-benelux.com

🔗jpehrson@rcn.com

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

/tuning/topicId_23187.html#23523

> As requested by Joseph Pehrson via email.
>
> ! Miracle-10.scl
> !
> The 10 "white notes" of Miracle mapped to 12 keys
> 12
> !
> 116.6666667 ! 3 L C#\
> 233.3333333 ! 4 D D
> 350.0 ! 5 N Eb/
> 466.6666667 ! 6 E E|\
> 583.3333333 ! 7 F F|
> 583.3333333 ! 7 F F|
> 700.0 ! 8 G G;/
> 816.6666667 ! 9 H G#\
> 966.6666667 ! 0 A A|\
> 1083.333333 ! 1 B B;
> 1083.333333 ! 1 B B;
> 2/1 ! 2 C C;/
>

Hello Dave and list!

I thank you so much for this! However, I have some questions and
problems. In the first place, I am not so certain that I just want
Blackjack "white" notes.... Part of the "charm" of the scale was in
the different alterations of step sizes... It was MOS and kind of
like 12-tET in that way... similar, but different. I'm not seeing
this in your new "white note" version.

ALSO, as a musician I see particular problems in the REPETITION of
frequencies in the mapping. That really seems to be a waste. I
would rather not use a scale that does that, sorry. I would,
frankly, even rather use an INFERIOR scale that doesn't repeat like
that...

Now, additionally, I really need 72-note degree values and Monz
note values. I understand, Dave, that you take objection to those
and the fact that they don't show the theoretical commas, or
something else that you're getting at, but the point is that I have
the same kind of "stupidity" as Monz and Ezra Sims.... I was able to
take in THAT notation RIGHT AWAY and YOURS is still a puzzle for me.

I like puzzles, but my intent is to WRITE MUSIC and as soon as
possible with this stuff. I hope I'm not being offensive in any way,
since I appreciate your genius with this stuff and you are the
originator of much of it on the highest level.

Maybe Monz is really the person to help me "translate" Dave Keenanese!

Anyway, I tried to do it myself and this is what I get:

> ! Miracle-10.scl
> !
> The 10 "white notes" of Miracle mapped to 12 keys
> 12

MONZ 72

> !
> 116.6666667 ! 3 L C#\ C#- 7
> 233.3333333 ! 4 D D D> 14
> 350.0 ! 5 N Eb/ Ev 21
> 466.6666667 ! 6 E E|\ F< 28
> 583.3333333 ! 7 F F| F#- 35
> 583.3333333 ! 7 F F| F#- 35
> 700.0 ! 8 G G;/ G 42
> 816.6666667 ! 9 H G#\ G#+ 49
> 966.6666667 ! 0 A A|\ Bb< 58
> 1083.333333 ! 1 B B; B- 65
> 1083.333333 ! 1 B B; B- 65
> 2/1 ! 2 C C;/ C 72

I hope this is right. However, I'm seeing some peculiar "anomalies"
right away... for one thing, Monz has a D> for 233.333333 cents, and
the Keenan is just a "D" with no alteration...

There are other things that seem kind of weird about it. Monz has a
G at 700.0 cents (sounds reasonable!), but the Keenan has significant
departure...

It must be that we are starting from a different point of reference
or some such.

The Keenen notation is just to complicated for me. I need
a "translator" into Monz ascii, if I didn't do it correctly.

Could somebody PLEASE help me?

________ _______ _____
Joseph Pehrson

🔗monz <joemonz@yahoo.com>

--- In tuning@y..., jpehrson@r... wrote:

/tuning/topicId_23187.html#23541

> Now, additionally, I really need 72-note degree values and
> Monz note values. I understand, Dave, that you take objection
> to those and the fact that they don't show the theoretical
> commas, or something else that you're getting at, but the
> point is that I have the same kind of "stupidity" as Monz
> and Ezra Sims.... I was able to take in THAT notation
> RIGHT AWAY and YOURS is still a puzzle for me.

Hi Joe,

I want to state first of all that I agree with you that
Dave's notation has its uses, and I am in no way trying to
disparage it.

But I understand your need for a notation that you can
put to practical use right away, and I agree with that too.

Here's the Miracle-10 scale with my ASCII 72-EDO note names:

! Miracle-10.scl
!
The 10 "white notes" of Miracle mapped to 12 keys
12
!
116.6666667 ! 3 L C#\ C#+ 7
233.3333333 ! 4 D D D> 14
350.0 ! 5 N Eb/ Ev 21
466.6666667 ! 6 E E|\ F< 28
583.3333333 ! 7 F F| F#- 35
583.3333333 ! 7 F F| F#- 35
700.0 ! 8 G G;/ G 42
816.6666667 ! 9 H G#\ G#+ 49
966.6666667 ! 0 A A|\ Bb< 58
1083.333333 ! 1 B B; B- 65
1083.333333 ! 1 B B; B- 65
2/1 ! 2 C C;/ C 72

Joe, your version was 90% correct. The only ASCII 72-EDO
notation you had wrong was the first one. You had C#-
instead of C#+.

Here's a formula that you might find helpful: Notice that
in ASCII 72-EDO, as you cycle thru a MIRACLE scale by
means of the 7/72-"octave" generator, the accidental
for each successive 12-EDO nominal bumps up to the next higher
symbol. So if you start on C, there is no accidental. Then,

C# gets a +
D gets a >
Eb gets a ^ ...or the simpler enharmonic Ev, which starts
a new pattern:
F gets a <
F# gets a -
G has no accidental
G# gets a +

etc.

You can see this process vividly by looking at my Ztar
blackjack mapping, shown in the first mapping example at
<http://www.ixpres.com/interval/monzo/blackjack/blackjack.htm>.

The break that Dave made in the cycle in this particular
scale is that after G#+, he used Bb< [= 2^(58/72)] instead
of A> [= 2^(56/72)]. If you look again at my blackjack mapping
you'll see that up to that point, the 10-note scale uses the
lower of each pair of blackjack notes, then at that point
switches to the higher of the last two pairs.

This is probably actually shown more clearly on Dave's own
chart <http://dkeenan.com/Music/MiraclePitchChart.htm>,
but the lack of ASCII 72-EDO notation there makes it hard
for me to see it too. I know you're philosophically opposed
to ASCII 72-EDO for MIRACLE, Dave, but how about adding it
anyway, to help those of us who like it? It would probably
lead to greater understanding of your own notational preferences.

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

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., jpehrson@r... wrote:

> Hello Dave and list!
>
> I thank you so much for this! However, I have some questions and
> problems. In the first place, I am not so certain that I just want
> Blackjack "white" notes.... Part of the "charm" of the scale was
in
> the different alterations of step sizes... It was MOS and kind of
> like 12-tET in that way... similar, but different. I'm not seeing
> this in your new "white note" version.

Well, the 10-tone "white note" scale is MOS, but I'm afraid it
doesn't function as any kind of "natural" subset of blackjack. You'll
find very few consonant chords therein.

Trust me, I just spent hours playing with blackjack on my keyboard.
>
> Now, additionally, I really need 72-note degree values and Monz
> note values. I understand, Dave, that you take objection to those
> and the fact that they don't show the theoretical commas, or
> something else that you're getting at, but the point is that I have
> the same kind of "stupidity" as Monz and Ezra Sims.... I was able
to
> take in THAT notation RIGHT AWAY and YOURS is still a puzzle for me.

You're absolutely right, Joseph. Tell us what notation you want to
use, and we'll use it. In fact, if you want to use "keyboard"
notation, with D4 and B5 forming an acoustical "octave" (2:1), we'll
be happy to use that too.
>
> I like puzzles, but my intent is to WRITE MUSIC and as soon as
> possible with this stuff.

Good for you! So, do you have blackjack up on your keyboard? I put
the "0" note at D4, for symmetry. Found all kind of great chord
progressions (hope to post about these chords soon, once you pick a
notation).

> I hope I'm not being offensive in any way,
> since I appreciate your genius with this stuff and you are the
> originator of much of it on the highest level.

I think part of what Dave Keenan may be missing is how you, a
keyboardist from (mainly) the 20th century, is able to see the
keyboard as a featureless chain of identical intervals, without being
distracted by all the black/white and up/down stuff. This is not an
easy talent to acquire (it's why I'm a guitarist instead), but once
you have it, you have it, and you can put it to use in your
microtonal explorations.

🔗Paul Erlich <paul@stretch-music.com>

🔗jpehrson@rcn.com

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_23187.html#23560

>
> You're absolutely right, Joseph. Tell us what notation you want to
> use, and we'll use it. In fact, if you want to use "keyboard"
> notation, with D4 and B5 forming an acoustical "octave" (2:1),
we'll
> be happy to use that too.
> >
> > I like puzzles, but my intent is to WRITE MUSIC and as soon as
> > possible with this stuff.
>
> Good for you! So, do you have blackjack up on your keyboard? I put
> the "0" note at D4, for symmetry. Found all kind of great chord
> progressions (hope to post about these chords soon, once you pick a
> notation).
>

Yes, I do... and I've already started working on the new piece.
HOWEVER, I am going with the ENTIRE set of blackjack 21...

Remember, I wanted to write a piece with 19-tones per octave, so 21
isn't that much different.

I am using the Sims/Maneri notation, and have it already in my
computer...

It will be a piece for trombone and electronics, and the "centered"
pitch on my keyboard is actually an octave BELOW C4 the way I have it
set up... since that is closer to the center of the trombone range.

I get at least a couple of full trombone octaves with this method.

I REALLY like the Maneri/Sims notation. I admit there could be
improvements to it, but it seems even MORE intuitive than the "cents
notation" method I was using previously. The smallest unit, after
all, is only 1/12 of a whole tone!!!

As far as the MIDI setup is concerned, I am rather limited, since I
only have 8 voices to use with my present setup. (Limitations are
good, though... I really don't feel I've totally exhausted everything
there yet... I WILL move on to more complex gear at some point!)

Generally speaking, I have set each channel to a different timbre, so
I get eight, of course, in all. It's great to have that multi-
timbral possibility.

HOWEVER, with "blackjack," which, actually, I prefer
calling "Miracle21," I am using one channel with THREE MIDI voices on
it. The reason, obviously, is that HARMONY is SO important in the
MIRACLE scales, and I want to make the most of this aspect...

Independent lines are fine, and I certainly can and have created
harmony this way... but I needed a "chordal" channel, so to speak...

So that's what's happening right now... and the piece is already
started, so too late for me to change the tuning now right in the
middle of it!

I *do* intend, however, to try other variants of the MIRACLE family,
but, for now, 21-note "blackjack" is what I'm working with...

________ _______ _____
Joseph Pehrson

🔗jpehrson@rcn.com

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., jpehrson@r... wrote:

> >
> > Good for you! So, do you have blackjack up on your keyboard? I
put
> > the "0" note at D4, for symmetry. Found all kind of great chord
> > progressions (hope to post about these chords soon, once you pick
a
> > notation).
> >
>
> Yes, I do... and I've already started working on the new piece.
> HOWEVER, I am going with the ENTIRE set of blackjack 21...

That's what I was doing too.
>
> It will be a piece for trombone and electronics, and the "centered"
> pitch on my keyboard is actually an octave BELOW C4 the way I have
it
> set up... since that is closer to the center of the trombone range.

OK . . . so C3 will be note 0/72?

If so, can I post a bunch of tetrads for you in "keyboard" notation?

For example, C3-G3-C4-F4 is a 4:5:6:7 chord . . . soooo concordant!

I'll be happy to use the Monzo 72-tET notation instead, if you
prefer . . . or both . . .

🔗D.Stearns <STEARNS@CAPECOD.NET>

🔗jpehrson@rcn.com

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_23187.html#23567

> --- In tuning@y..., jpehrson@r... wrote:
>
> > >
> > > Good for you! So, do you have blackjack up on your keyboard? I
> put the "0" note at D4, for symmetry. Found all kind of great chord
> > > progressions (hope to post about these chords soon, once you
pick a notation).
> > >
> >
> > Yes, I do... and I've already started working on the new piece.
> > HOWEVER, I am going with the ENTIRE set of blackjack 21...
>
> That's what I was doing too.
> >
> > It will be a piece for trombone and electronics, and
the "centered" pitch on my keyboard is actually an octave BELOW C4
the way I have it set up... since that is closer to the center of the
trombone range.
>
> OK . . . so C3 will be note 0/72?

This is correct... or at least what *I'm* doing...

>
> If so, can I post a bunch of tetrads for you in "keyboard" notation?
>
> For example, C3-G3-C4-F4 is a 4:5:6:7 chord . . . soooo concordant!
>

Well, that would be TERRIFIC! Actually the "keyboard" notation is
how I am going to initially write the piece... Later, the trombone
part gets "translated" to 72-tET notation...

It might be nice to accompany the "keyboard" notation with numbers
from 0-21, just so I keep aware of what I'm doing...

However if you are willing to post the "keyboard template" notation,
the 21-tone note numbers and the 72-tET Monzo/Sims pitches, that
might be optimal... I guess we could even add 72-tET numbers, if
that's not too much work...

THANKS!!!

_________ _______ _____
Joseph Pehrson

🔗Paul Erlich <paul@stretch-music.com>

🔗David C Keenan <D.KEENAN@UQ.NET.AU>

Manuel Op de Coul wrote:

>Dave Keenan wrote:
>>The 10 "white notes" of Miracle mapped to 12 keys
>[snip]
>> 966.6666667 ! 0 A A|\
>> 1083.333333 ! 1 B B;
>> 1083.333333 ! 1 B B;
>> 2/1 ! 2 C C;/
>
>Why not put the 11th "white note" 1166.667 instead on the B?
>Then you have a nicely high leading tone more.
>Dunno, I just see an easy pattern here: 11-21-31-41.

1166.667 would be Cv. It would not be the 11th note in a chain of Miracle
generators. The chain I gave starts on A, not C (as per the numbers in the
comments). So instead we'd need Av at 933.333 cents. Putting this in will
distort the keyboard mapping and destroy some familiar patterns. However,
here it is. I extended the chain by one in the other direction as well.

! Miracle-12.scl
!
A chain of 12 Miracle generators for mapping to standard keyboard
12
!
116.6666667 ! 3 L C#\
233.3333333 ! 4 D D
350.0 ! 5 N Eb/
466.6666667 ! 6 E E|\
583.3333333 ! 7 F F|
700.0 ! 8 G G;/
816.6666667 ! 9 H G#\
850.0 ! 9^ H^ Ab/
933.3 ! 0v Av A
966.6666667 ! 0 A A|\
1083.333333 ! 1 B B;
2/1 ! 2 C C;/

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

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

--- In tuning@y..., jpehrson@r... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> /tuning/topicId_23187.html#23560
>
> > I think part of what Dave Keenan may be missing is how you, a
> > keyboardist from (mainly) the 20th century, is able to see the
> > keyboard as a featureless chain of identical intervals, without
> being
> > distracted by all the black/white and up/down stuff. This is not
an
> > easy talent to acquire (it's why I'm a guitarist instead), but
once
> > you have it, you have it, and you can put it to use in your
> > microtonal explorations.
>
> Right.... It also comes from playing a lot of Schoenberg's piano
> music as a teenager...

Aha! Paul you're absolutely right. I've apparently rejected the 20th
century in that regard. I see a Halberstadt keyboard as a slice of
meantone. I'm often offended by incorrect spelling. Someone will say
"and then we play an Ab minor seventh" and I'll say "don't you mean a
G# minor seventh" (based on what preceded it) and they will say, "same
thing". Yuck! 12-tET really has won hasn't it.

Sorry Joseph. I said I'd do anything to help and then renegged, didn't
I. I hope you can understand why now, from my previous post
"ASCII-dental mess".

I had no idea you were so advanced with Miracle-21. Forget 10 or 12,
go for 21. It certainly shows much more of what Miracle is capable of.

Regards,
-- Dave Keenan

🔗jpehrson@rcn.com

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

/tuning/topicId_23187.html#23584

>
> Sorry Joseph. I said I'd do anything to help and then renegged,
didn't I. I hope you can understand why now, from my previous post
> "ASCII-dental mess".
>

That was great... I've added a few comments, myself..

> I had no idea you were so advanced with Miracle-21. Forget 10 or
12, go for 21. It certainly shows much more of what Miracle is
capable of.
>

I've already started working with it. I decided to devote one MIDI
channel to 4 voice harmony now... since HARMONY is obviously so
important in this scale...

Cuts down on my number of voices until I get more equipment, but at
least I can incorporate some of Paul Erlich's harmonic explorations.

My assumption is that many of these will be in four parts... (??)

[May use "weirder" ones, though, too... but should know the basics]

Thanks, everybody! This is terrific!

________ ______ ____
Joseph Pehrson

🔗Paul Erlich <paul@stretch-music.com>

🔗jpehrson@rcn.com

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_23187.html#23623

> --- In tuning@y..., jpehrson@r... wrote:
>
> > Cuts down on my number of voices until I get more equipment, but
at
> > least I can incorporate some of Paul Erlich's harmonic
explorations.
> >
> > My assumption is that many of these will be in four parts... (??)
>
> Hi Joseph!
>
> I was planning to post a big list of tetrads in the Blackjack
scale.
> However, I would recommend allowing five-part harmony. For example,
> some of the tetrads are 1/(5:6:7:9), so being able to add the 1/4
> (common overtone) would be a valuable asset. In fact, I was
planning
> to include, in the list of tetrads, a recommended fifth note for
some
> of them.

Hi Paul!

Well, I'll certainly consider that... however I'm running out of MIDI
voices. This is the typical MIDI scourge... "need more equipment,
need more equipment..."

I'll be interested in hearing what you come up with!!!

_________ ______ _______
Joseph Pehrson

🔗D.Stearns <STEARNS@CAPECOD.NET>

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., jpehrson@r... wrote:

> Well, I'll certainly consider that... however I'm running out of
MIDI
> voices. This is the typical MIDI scourge... "need more equipment,
> need more equipment..."
>
Well, should I stick with just plain tetrads, then?

> I'll be interested in hearing what you come up with!!!

It's going to be a rather comprehensive list of the tetrads in
Blackjack tuning, organized by generic interval pattern (i.e.,
keyboard interval pattern). For example, the first list will have the
keyboard chords C3-G3-C4-F4, C#3-G#3-C#4-F#4, D3-A3-D4-G4, etc. Most
of the chords are "consonant" and will be indicated in JI form. A few
will simply be marked as "dissonant". In any case, with many generic
interval patterns, many of which produce "consonant" chords in most
positions, the possibilities for voice leading are immense -- but at
least the tables I'll produce will help you navigate them.

So, let's decide on notation.

I take it you're mapping C3 on your keyboard to the "center" of the
blackjack tuning, as well as to the real pitch C3? Please clarify if
this is not correct. Then, I'd like to use keyboard notation and only
one 72-tET notation, if at all possible. It's up to you, Joseph.

🔗Paul Erlich <paul@stretch-music.com>

🔗monz <joemonz@yahoo.com>

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_23187.html#23632

> Personally, I don't see much logic to thinking of ten
> "white notes" in this scale. The ten "white notes" give
> almost none of the consonant chords for which the Miracle
> scales were designed. I think the logic of how consonant
> chords are constructed is absolutely clear in 72-tET notation
> based on 12-tET "naturals". The Miracle scales are, in my
> view, mainly acting as a way of getting access to a great
> number of these chords with a limited number of keys on
> a keyboard.

AHA!... a very good reason to choose some form of 72-EDO
over one of the other (decimal; meantone-based) notations
for this particular family of tunings.

72-EDO is a very clear and simple representation of the
virtual pitch continuum, and this is exactly what Maneri
refers to in his teachings.

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

🔗D.Stearns <STEARNS@CAPECOD.NET>

Paul Erlich wrote,

<<Personally, I don't see much logic to thinking of ten "white notes"
in this scale. The ten "white notes" give almost none of the consonant
chords for which the Miracle scales were designed. I think the logic
of how consonant chords are constructed is absolutely clear in 72-tET
notation based on 12-tET "naturals". The Miracle scales
are, in my view, mainly acting as a way of getting access to a great
number of these chords with a limited number of keys on a keyboard.>>

Yes I agree with all of this, excellent points! But I think this
underscores some of the problems as well. The "miracle scale" is not a
scale paradigm at all in the diatonic sense where there's a symbiotic
correspondence between a horizontal and a vertical structure.

There's a lot of unanswered (but fascinating) questions beyond the
5-limit. One of which is what kind of paradigm shift might be
necessary to accommodate a horizontal concept that'll accompany higher
harmonics in their readymade horizontal sense... whether by the weight
of acculturation or some innate psychoacoustic threshold, clearly a
"white key" concept collapses pretty quickly once the scale expands
very much beyond 7.

What could a non generalized diatonic diatonic be... any ideas?

--Dan Stearns

🔗monz <joemonz@yahoo.com>

🔗jpehrson@rcn.com

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_23187.html#23631

> It's going to be a rather comprehensive list of the tetrads in
> Blackjack tuning, organized by generic interval pattern (i.e.,
> keyboard interval pattern). For example, the first list will have
the keyboard chords C3-G3-C4-F4, C#3-G#3-C#4-F#4, D3-A3-D4-G4, etc.
Most of the chords are "consonant" and will be indicated in JI form.

That's GREAT.... probably that notation is all I need... I probably
don't even need 21-note, note numbers, although that helps remind me
that I am working in something, er, "different..."

A few
> will simply be marked as "dissonant". In any case, with many
generic interval patterns, many of which produce "consonant" chords
in most positions, the possibilities for voice leading are immense --
but at least the tables I'll produce will help you navigate them.
>

Well, a lot of this will be discovered in the process of composition
and listening... but these basics are crucial...

I'll probably have impetus to add to my MIDI equipment, too, since
HARMONY is so especially important for such MIRACLES...

> So, let's decide on notation.
>
> I take it you're mapping C3 on your keyboard to the "center" of the
> blackjack tuning, as well as to the real pitch C3? Please clarify
if this is not correct. Then, I'd like to use keyboard notation and
only one 72-tET notation, if at all possible. It's up to you, Joseph.

This is correct. C3 on my synth (octave below middle C) is C3 on my
12-tET piano. The 21-tone scale proceeds upward and downward from
there with the SCALA "map C-linear..." mapping...

________ ______ ______ ___
Joseph Pehrson

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:

> Yes I agree with all of this, excellent points! But I think this
> underscores some of the problems as well. The "miracle scale" is
not a
> scale paradigm at all in the diatonic sense where there's a
symbiotic
> correspondence between a horizontal and a vertical structure.

Well . . . as I hope my tetrad lists for Joseph Pehrson will make
clear . . . there _is_ a lot of potential for this kind of logic in
the Blackjack scale.

But ignoring that, think of Partch's _Genesis of a Music_. Does he,
anywhere in that book, put forward what a 7-limit or 11-limit "white-
key scale" would look like? No! A lot of his music is "Tonal Flux",
that is, just voice-leading between one hexad (or subset thereof) and
another, with lots of small melodic intervals, and no restriction on
which pitches from the master set are being used. Certainly
the "modernism" that Joseph is going for shouldn't exclude this kind
of approach, should it?

🔗Paul Erlich <paul@stretch-music.com>

🔗jpehrson@rcn.com

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_23187.html#23651

> --- In tuning@y..., jpehrson@r... wrote:
> > --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> >
> >>Then, I'd like to use keyboard notation and
> >> only one 72-tET notation, if at all possible. It's up to you,
> >>Joseph.
> >
> > This is correct. C3 on my synth (octave below middle C) is C3 on
> my
> > 12-tET piano. The 21-tone scale proceeds upward and downward
from
> > there with the SCALA "map C-linear..." mapping...
> >
> I still would like to include a 72-tET notation, since it will make
> it much easier to see what the chords should actually "sound like"
if
> someone is unfamiliar with JI theory. So which 72-tET notation
should
> I use?

Well, frankly, I'm already familiar with the Monzo... but what's
Keenan going to do to us?? :)

_________ ______ ____
Joseph Pehrson

🔗Paul Erlich <paul@stretch-music.com>

🔗monz <joemonz@yahoo.com>

--- In tuning@y..., jpehrson@r... wrote:

/tuning/topicId_23187.html#23654

> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> /tuning/topicId_23187.html#23651
>
> > --- In tuning@y..., jpehrson@r... wrote:
> > > --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > >
> > >>Then, I'd like to use keyboard notation and
> > >> only one 72-tET notation, if at all possible. It's up to you,
> > >>Joseph.
> > >
> > > This is correct. C3 on my synth (octave below middle C) is C3
on
> > my
> > > 12-tET piano. The 21-tone scale proceeds upward and downward
> from
> > > there with the SCALA "map C-linear..." mapping...
> > >
> > I still would like to include a 72-tET notation, since it will
make
> > it much easier to see what the chords should actually "sound
like"
> if
> > someone is unfamiliar with JI theory. So which 72-tET notation
> should
> > I use?
>
> Well, frankly, I'm already familiar with the Monzo... but what's
> Keenan going to do to us?? :)
>
> _________ ______ ____
> Joseph Pehrson

Paul, please use mine, at least in addition to the other one
you chose. I'd like to be able to understand it quickly
without having to translate. And it seems like I might be
able to drag Joe by the arm after all. Thanks.

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

🔗D.Stearns <STEARNS@CAPECOD.NET>

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., jpehrson@r... wrote:
> --- In tuning@y..., jpehrson@r... wrote:
>
> /tuning/topicId_23187.html#23647
> >
> > This is correct. C3 on my synth (octave below middle C) is C3 on
> my 12-tET piano. The 21-tone scale proceeds upward and downward
from
> > there with the SCALA "map C-linear..." mapping...
> >
>
> Oh... just to clarify... it's actually the C4 on my synth (position
> of "middle C" on it) that is tuned to C3 on the regular piano...
>
> Center of the keyboard... for the trombone range.
>
> The 21-tone scale proceeds upward and downward from that pitch.
>
> ("map C-linear")

OK, Joseph. I show below that in a certain range on the keyboard (and
of course a "major thirteenth" [acoustical 2:1 octave] above or below
that range), most of the tetrads you get by stacking one "perfect
fifth" and two "perfect fourths", in any order, are 9-limit
consonant. The keyboard range is F#3 to D#6 -- corresponding to a
pitch range of Ab|2 to Eb|4. Of course, any other range would have as
many consonant tetrads as well -- you'd just have to use inversions
of these chords, which will show up with a "major third" on the
keyboard (since "perfect fifth" + "perfect fourth" + "perfect fourth"
+ "major third" = "major thirteenth" = acoustical 2:1 octave).

Let me know if you find any errors, or if the table doesn't seem to
correspond with what you're hearing from your keyboard.

Legend for 72-tET notation:

A,B,C,D,E,F,G,b,# as in 12-tET
| = 1/4 tone up
> = 1/6 tone up
^ = 1/12 tone up
v = 1/12 tone down
< = 1/6 tone down

Blackjack scale in 72-tET notation and keyboard notation

72-tET notation keyboard notation
lower middle upper

A#< G#3 F5
Bb| C2 A3 F#5
Bv C#2 A#3 G5
C< D2 B3 G#5
C D#2 C4 A5
C> E2 C#4 A#5
C#^ F2 D4 B5
Db| F#2 D#4 C6
D> G2 E4 C#6
D#< G#2 F4 D6
Eb| A2 F#4 D#6
Ev A#2 G4 E6
F< B2 G#4 F6
F C3 A4 F#6
F#v C#3 A#4 G6
Gb^ D3 B4 G#6
G D#3 C5 A6
G> E3 C#5 A#6
G#^ F3 D5 B6
Ab| F#3 D#5 C7
A> G3 E5

Keyboard Pattern: P5-P4-P4 (inversions include P4-P4-M3, P4-M3-P5, M3-
P5-P4)

keyboard notation 72-tET notation quality
E3 B3 E4 A4 G> C< D> F dissonant
F3 C4 F4 A#4 G#^ C#^ D#< F#v dissonant
F#3 C#4 F#4 B4 Ab| C> Eb| Gb^ 4:5:6:7
G3 D4 G4 C5 A> C#^ Ev G dissonant
G#3 D#4 G#4 C#5 A#< Db| F< G> 4:5:6:7
A3 E4 A4 D5 Bb| D> F G#^ dissonant
A#3 F4 A#4 D#5 Bv D#< F#v Ab| 4:5:6:7
B3 F#4 B4 E5 C< Eb| Gb^ A> dissonant
C4 G4 C5 F5 C Ev G A#< 4:5:6:7
C#4 G#4 C#5 F#5 C> F< G> Bb| 1/(9:7:6:5)
D4 A4 D5 G5 C#^ F G#^ Bv 4:5:6:7
D#4 A#4 D#5 G#5 Db| F#v Ab| C< 1/(9:7:6:5)
E4 B4 E5 A5 D> Gb^ A> C 4:5:6:7
F4 C5 F5 A#5 D#< G A#< C> dissonant
F#4 C#5 F#5 B5 Eb| G> Bb| C#^ 4:5:6:7
G4 D5 G5 C6 Ev G#^ Bv Db| dissonant
G#4 D#5 G#5 C#6 F< Ab| C< D> 4:5:6:7
A4 E5 A5 D6 F A> C D#< dissonant
A#4 F5 A#5 D#6 F#v A#< C> Eb| dissonant
B4 F#5 B5 E6 Gb^ Bb| C#^ Ev dissonant
C5 G5 C6 F6 G Bv Db| F< dissonant

Keyboard Pattern: P4-P5-P4 (inversions include P5-P4-M3, P4-M3-P4, M3-
P5-P4)

keyboard notation 72-tET notation quality
E3 A3 E4 A4 G> Bb| D> F dissonant
F3 A#3 F4 A#4 G#^ Bv D#< F#v dissonant
F#3 B3 F#4 B4 Ab| C< Eb| Gb^ dissonant
G3 C3 G4 C5 A> C Ev G dissonant
G#3 C#4 G#4 C#5 A#< C> F< G> 2:7/3:3:7/2
A3 D4 A4 D5 Bb| C#^ F G#^ dissonant
A#3 D#4 A#4 D#5 Bv Db| F#v Ab| 2:7/3:3:7/2
B3 E4 B4 E5 C< D> Gb^ A> dissonant
C4 F4 C5 F5 C D#< G A#< 2:7/3:3:7/2
C#4 F#4 C#5 F#5 C> Eb| G> Bb| 1/3:2/5:1/2:3/5
D4 G4 D5 G5 C#^ Ev G#^ Bv 2:7/3:3:7/2
D#4 F#4 D#5 G#5 Db| F< Ab| C< 1/3:2/5:1/2:3/5
E4 A4 E5 A5 D> F A> C 2:7/3:3:7/2
F4 A#4 F5 A#5 D#< F#v A#< C> dissonant
F#4 B4 F#5 B5 Eb| Gb^ Bb| C#^ 2:7/3:3:7/2
G4 C5 G5 C6 Ev G Bv Db| dissonant
G#4 C#5 G#5 C#6 F< G> C< D> 2:7/3:3:7/2
A4 D5 A5 D6 F G#^ C D#< dissonant
A#4 D#5 A#5 D#6 F#v Ab| C> Eb| dissonant
B4 E5 B5 E6 Gb^ A> C#^ Ev dissonant
C5 F5 C6 F6 G A#< Db| F< dissonant

Keyboard Pattern: P4-P4-P5 (inversions include P4-P5-M3, P5-M3-P4, M3-
P4-P4)

keyboard notation 72-tET notation quality
E3 A3 D4 A4 G> Bb| C#^ F dissonant
F3 A#3 D#4 A#4 G#^ Bv Db| F#v dissonant
F#3 B3 E4 B4 Ab| C< D> Gb^ dissonant
G3 C3 F4 C5 A> C D#< G dissonant
G#3 C#4 F#4 C#5 A#< C> Eb| G> 1/(7:6:5:4)
A3 D4 G4 D5 Bb| C#^ Ev G#^ dissonant
A#3 D#4 G#4 D#5 Bv Db| F< Ab| 1/(7:6:5:4)
B3 E4 A4 E5 C< D> F A> dissonant
C4 F4 A#4 F5 C D#< F#v A#< 1/(7:6:5:4)
C#4 F#4 B4 F#5 C> Eb| Gb^ Bb| 5:6:7:9
D4 G4 C5 G5 C#^ Ev G Bv 1/(7:6:5:4)
D#4 F#4 C#5 G#5 Db| F< G> C< 5:6:7:9
E4 A4 D5 A5 D> F G#^ C 1/(7:6:5:4)
F4 A#4 D#5 A#5 D#< F#v Ab| C> dissonant
F#4 B4 E5 B5 Eb| Gb^ A> C#^ 1/(7:6:5:4)
G4 C5 F5 C6 Ev G A#< Db| dissonant
G#4 C#5 F#5 C#6 F< G> Bb| D> 1/(7:6:5:4)
A4 D5 G5 D6 F G#^ Bv D#< dissonant
A#4 D#5 G#5 D#6 F#v Ab| C< Eb| 1/(7:6:5:4)
B4 E5 A5 E6 Gb^ A> C Ev dissonant
C5 F5 A#5 F6 G A#< C> F< dissonant

I'll post other patterns later.

🔗Paul Erlich <paul@stretch-music.com>

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

I agree. The 10 naturals of the decimal notation for Miracle do not
constitute a usable melodic entity. So in that sense they do not
constitute a white note scale.

Clearly, there is no single such "white note scale" in Miracle. There
are many. The only one you get in Miracle-10 is a Slendro-type
pentatonic. Pretty slim pickings. You get way more in Miracle-21.

But I'm pretty sure that if you could make the break from the 12-tET
based notation to a decimal one you would find that the decimal
notation, while still requiring accidentals in almost all consonant
chords, would automatically tell you where the wolves are. Or from the
other point of view, would automatically tell you which 72-EDO notes
are actually available to you in a particular subset such as
Miracle-21.

I'm committed to providing both 10-naturals and 7-naturals notations
in anything I write or diagram. Based on what folks have written so
far in the ASCIIdental thread, the 7-naturals one, if it uses arrows
at all, will use them consistently with the Sims notation, not the
Richter-Herf.

-- Dave Keenan

🔗jpehrson@rcn.com

🔗Paul Erlich <paul@stretch-music.com>

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> There are almost certainly no generalised diatonics (by Paul's
> criteria, and probably many looser ones) in Miracle, except THE
> diatonic (in 12-tET) which doesn't appear until a chain of 37 notes.

Graham Breed might argue that the neutral-thirds heptatonic scale is a generalized diatonic, but I
find those neutral triads painfully dissonant.

> It's a whole other ball-game, in which perhaps harmony and melody are
> not so tightly related. Maybe that's just the extended-JI ball-game in
> general.

Maybe.
>
> I don't think an 11-limit generalised diatonic can exist. We see that
> Paul had to accept 17.5 c errors to get a 7-limit gen diatonic,
> considerably more than the errors of the 5-limit diatonic in meantone.
> I think the errors would have to go up to considerably more than 17.5
> c, in which case it couldn't really be called 11-limit.
>
But if you loosen the definition of generalized diatonic a bit . . . for example, if you allow chords
to be formed from different permutations of a set of two generic chord-constructing intervals . . .
and you allow a small number of unsaturated chords . . . you see that Blackjack itself doesn't look
so bad as a generalized diatonic! Certainly, the 33-cent steps make it very unlike any
"white-note scale" heretofore considered . . . but who knows how this ball-game will evolve!

🔗monz <joemonz@yahoo.com>

🔗jpehrson@rcn.com

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_23187.html#23674

>
> OK, Joseph. I show below that in a certain range on the keyboard
(and of course a "major thirteenth" [acoustical 2:1 octave] above or
below that range), most of the tetrads you get by stacking
one "perfect fifth" and two "perfect fourths", in any order, are 9-
limit
> consonant. The keyboard range is F#3 to D#6 -- corresponding to a
> pitch range of Ab|2 to Eb|4. Of course, any other range would have
as
> many consonant tetrads as well -- you'd just have to use inversions
> of these chords, which will show up with a "major third" on the
> keyboard (since "perfect fifth" + "perfect fourth" + "perfect
fourth"
> + "major third" = "major thirteenth" = acoustical 2:1 octave).
>
> Let me know if you find any errors, or if the table doesn't seem to
> correspond with what you're hearing from your keyboard.
>

Thank you so VERY much for this, Paul! This is PRACTICAL
(microtonality). I REALLY appreciate it!

Actually, it looks as though there wasn't a SINGLE error on the scale
posting... I was confused at first, since you used more FLATS with
quartertone sharps than you did in your post #22400 on blackjack...

Was there any reason for that?? Are these "new" ones "better...?"

In any case, the chord postings are VERY interesting as well. It's
funny how the same chords seem to alternate between being dissonant
and consonant. I assume that has something to do with the
alternating step sizes in this scale, correct??

Oh... I should mention that for the purposes of HARMONY, I am
using "middle C" as C4... only for the trombone part is it C3.

That shouldn't make any difference, though, from the point of view of
the chords, since everything is shifted down an octave with C4
becoming C3 as the "center" of the scale.

Thanks so VERY much for your help! Please send along other such
studies with this scale, if you can!

__________ _______ ________
Joseph Pehrson

🔗jpehrson@rcn.com

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_23187.html#23675

> By the way, Joseph, you may notice that in the keyboard range C4 to
> A5 (pitch range C3 to C4), _any_ of the 15 possible ways of
stacking
> one keyboard "perfect fifth" and two keyboard "perfect fourths"
will
> lead to a 9-limit consonant tetrad. So this is kind of a "sweet
spot"
> of this tuning (other sweet spots, of course, corresponding to
> the "inversions" involving keyboard "major thirds"). If you were
> planning to voice a lot of tetrads within the pitch range C3 to C4
> and avoid intervals smaller than a subminor third in your voicings,
> you made an amazingly good choice of how to pitch the "center" of
the blackjack scale!

Well, for "blackjack" it was only "luck of the draw..." since middle
C, of course, would come immediately to mind as a starting point...

________ _______ _____
Joseph Pehrson

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., jpehrson@r... wrote:
>
> Actually, it looks as though there wasn't a SINGLE error on the scale
> posting... I was confused at first, since you used more FLATS with
> quartertone sharps than you did in your post #22400 on blackjack...

Really?
>
> Was there any reason for that?? Are these "new" ones "better...?"

I doubt it. Can you show me what you mean?
>
> In any case, the chord postings are VERY interesting as well. It's
> funny how the same chords

By "same", you just mean "same keyboard intervals", right?

And you did try these out and listen to them, yes?

> seem to alternate between being dissonant
> and consonant.

Yes, that's the pattern through much of the scale.

> I assume that has something to do with the
> alternating step sizes in this scale, correct??

You bet! In the non-octave version of blackjack, in which the step sizes strictly alternate, each
pattern gives you a strictly alternating sequence of two kinds of consonant chord. The only
reason you get dissonant chords at all with these patterns is because of the two consecutive
small steps around each occurence of the pitch "C" (not "C" on the keyboard, but the actual pitch
"C").

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > I agree. The 10 naturals of the decimal notation for Miracle do
not
> > constitute a usable melodic entity.
>
> You mean harmonic, not melodic, right? Melodically, it's extremely
"efficient", by Rothenberg's
> definition.

Er, no. I meant melodic. It's not all _that_ bad harmonically, is it?
It has more triads than notes. So how is its stability? The answers
you get for these Rothenberg quantities depends entirely on whether
you consider the 1 step of 150c to be quite distinct from the 9 of
117c. Otherwise it might as well be 10-EDO with extremely low
efficiency.

>
> > Clearly, there is no single such "white note scale" in Miracle.
There
> > are many. The only one you get in Miracle-10 is a Slendro-type
> > pentatonic.
>
> Why is that more of a "white note scale" than Miracle-10?

I guess I was thinking mostly of Miller's magic number 7 +-2.

-- Dave Keenan

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > > I agree. The 10 naturals of the decimal notation for Miracle do
> not
> > > constitute a usable melodic entity.
> >
> > You mean harmonic, not melodic, right? Melodically, it's extremely
> "efficient", by Rothenberg's
> > definition.
>
> Er, no. I meant melodic. It's not all _that_ bad harmonically, is it?
> It has more triads than notes.

What kind of triads? I didn't find many useful triads when I was playing around with it the other
night. Intervals like 11:9 don't really work as consonances for me except in large otonal chords.

> So how is its stability?

Maybe I meant stability -- I just remember that scales with only one of one step size looked
best on one of these measures.

> The answers
> you get for these Rothenberg quantities depends entirely on whether
> you consider the 1 step of 150c to be quite distinct from the 9 of
> 117c.

Well, I was assuming that you would.

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> What kind of triads? I didn't find many useful triads when I was
playing around with it the other
> night. Intervals like 11:9 don't really work as consonances for me
except in large otonal chords.

Fair enough. I've included some below anyway. And some stacked 7:8
chords that give rise to 2:3s but probably sound like mud. Maybe
there's really only 5 consonant chords altogether.

Degrees of 72-EDO arranged as a chain of Miracle generators (7/72
oct).

44 51 58 65 0 7 14 21 28 35
+--------Miracle-10--------+
5--------------7-----1 3 of
1/1---1/7------------1/5 3 of
7-----1-----------------3 2 of
1/3---------------1/1---1/7 2 of
3-----------------9-------11 1 of
1/11-----1/9---------------1/3 1 of
1------11/9-------3 (neutral triad) 4 of
+-----+-----+-----+ stacked 7:8s 4 of
+-----+-----+-----+-----+ " " 2 of
44 51 58 65 0 7 14 21 28 35

> Maybe I meant stability -- I just remember that scales with only one
of one step size looked
> best on one of these measures.

You were right the first time. They have maximum efficiency.

> > The answers
> > you get for these Rothenberg quantities depends entirely on
whether
> > you consider the 1 step of 150c to be quite distinct from the 9 of
> > 117c.
>
> Well, I was assuming that you would.

How realistic is that?

-- Dave Keenan

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

🔗graham@microtonal.co.uk

In-Reply-To: <9ehvvs+dkoi@eGroups.com>
Paul Erlich wrote:

> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> > There are almost certainly no generalised diatonics (by Paul's
> > criteria, and probably many looser ones) in Miracle, except THE
> > diatonic (in 12-tET) which doesn't appear until a chain of 37 notes.
>
> Graham Breed might argue that the neutral-thirds heptatonic scale is a
> generalized diatonic, but I find those neutral triads painfully
> dissonant.

Yes, I was going to do that. They fulfil most of the criteria so long as
you bend the rules on harmony a bit.

I find I can get acculturated to neutral triads fairly quickly. If I hear
enough of them in a row, the 31-equal minor third even works as a
dissonance. Melodically, the MOS at least is fairly bland. But I find
this ideal for a bleached-out, apathetic kind of feel. It would suit a
lot of Stina Nordenstam's music, which is why I had a go at converting
that one piece. If I get into the right mood again, I'll try and finish
that, if summoning up the motivation doesn't itself destroy the apathy ;)

There are some cool pentatonics within both neutral third heptatonics.
They work well melodically, but I don't think would fulfil the harmonic
criteria for a generalised diatonic.

But taking either a neutral-third or fifth-based MOS as a Miracle diatonic
is somewhat bizarre. The meantone diatonic doesn't have the 5-limit
intervals spelt right. And if you're tuned to 72-equal, it means you're
only taking 1/6 of the scale. There are easier ways of getting the same
result!

I think it makes more sense to use the JI diatonic, and allow for some
"enharmonicism" (see my other post). But that drags you into 12-note
thinking, so you're not really exploiting the miracle-ness.

Similarly for the neutral-third scales. The "wolf" minor third isn't
anything special in Miracle. Taking a neutral third MOS from Blackjack
means one third of the gamut. And no modulation by notes within the scale
can cover any more of the gamut. So the ideas I suggested about
modulating between the MOS and Rast scales won't work at all -- or won't
make sense in context.

Also, when you get to the Rast it is important the 5-limit intervals
should be in tune. To get all the 11-limit intervals of these scales to
work means they become really fluid concepts. The distinction between
major and neutral seconds is blurred, but perhaps an approach like this
based on the neutral-third MOS would work.

> But if you loosen the definition of generalized diatonic a bit . . .
> for example, if you allow chords to be formed from different
> permutations of a set of two generic chord-constructing intervals . . .
> and you allow a small number of unsaturated chords . . . you see that
> Blackjack itself doesn't look so bad as a generalized diatonic!
> Certainly, the 33-cent steps make it very unlike any "white-note scale"
> heretofore considered . . . but who knows how this ball-game will
> evolve!

Indeed, Blackjack sort of works, but so many notes! I think it's good
that Miracle thinking doesn't guide you into a simple, "white note"
diatonic. It leaves to composer free to define whatever key centers and
modes they like, and also "encourages" them to think about how some tuning
details should be resolved. It's a post modern temperament, gaining value
from the multitude of scales it contains, rather than it's rendition of
one particular scale.

Graham

🔗graham@microtonal.co.uk

In-Reply-To: <9ehra9+r6j6@eGroups.com>
Dave Keenan wrote:

> But I'm pretty sure that if you could make the break from the 12-tET
> based notation to a decimal one you would find that the decimal
> notation, while still requiring accidentals in almost all consonant
> chords, would automatically tell you where the wolves are. Or from the
> other point of view, would automatically tell you which 72-EDO notes
> are actually available to you in a particular subset such as
> Miracle-21.

This is how I feel. 7, 10 and 12 interval classes all slice up 11-limit
pitch space in different ways. Only 7 gives you neutral seconds, but it
doesn't have a tritones. 12 doesn't have neutral thirds, and hence does
place major and minor thirds in different interval classes. 10 lumps 7:6
in with 8:7 rather than 6:5, which does make some sense but means
diminished triads don't look symmetric.

11:10 and 10:9 are relatively complex intervals in Miracle temperament, so
you probably wouldn't use them as much as in free JI. And if you want
them to be the same as each other or nearby intervals, you're in the wrong
temperament. This is an offshoot of the not-having-neutral-seconds
problem.

When I did the original 31-equal Blackjack lattice, I found it was
impossible for all 5-limit triads to be spelt correctly at the same time.
I think this will cause trouble if you expect 12- or 7-based notation to
make consistent sense for miracle-based progressions.

The lattice is here if anybody thinks they can improve on it:

</tuning/topicId_21940.html#21945>

Hmm, I think it would work if you used standard notation, without the ^
and v. And I bet it'll look a right mess in Boston notation if you try to
preserve those double sharps and flats.

Graham

🔗graham@microtonal.co.uk

In-Reply-To: <9ei4ag+bg82@eGroups.com>
Dave Keenan wrote:

> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > > I agree. The 10 naturals of the decimal notation for Miracle do
> not
> > > constitute a usable melodic entity.
> >
> > You mean harmonic, not melodic, right? Melodically, it's extremely
> "efficient", by Rothenberg's
> > definition.
>
> Er, no. I meant melodic. It's not all _that_ bad harmonically, is it?
> It has more triads than notes. So how is its stability? The answers
> you get for these Rothenberg quantities depends entirely on whether
> you consider the 1 step of 150c to be quite distinct from the 9 of
> 117c. Otherwise it might as well be 10-EDO with extremely low
> efficiency.

On the nominals, only the "minor" 4-step (and complement) is not an
11-limit consonance. Even that isn't too bad in 7-equal, approximating
21:16(?). So it depends on how bad you think the 11-limit is
harmonically.

Melodically, it's dull as ditchwater. 10 almost equal steps? <yawn>

I do find, with the 10 nominals as black notes, those 10 interval classes
do seem to "mean" something. So you could maybe take a 7-note subset and
call it the "mode", but allow diesis shifts relative to that. I'll call
this "enharmonicism" whereas taking notes outside of the 7 interval
classes would be "chromaticism". So you're taking 7 out of 10 interval
classes rather than 7 pitch classes from the 10 nominals.

This is taking "interval class" to mean intervals we consciously think of
as different, but group together for convenience. "Pitch classes" are
notes that may or may not have the same pitch or be written the same, but
are considered equivalent in the relevant temperament.

If you're really lucky, I might observe this distinction myself ;)

So another approach to 9-limit harmony would be to define a regular
diatonic mode in terms of 12 interval classes. You can then use
enharmonic shifts to make a minor third 7:6 or a tone 8:7, and also choose
which kind of tritone you want.

The problem I have with 12 interval classes for 11-limit enharmonicism is
that there's no concept of "neutral third". So you have to set the
submajor and superminor thirds to be equivalent. This is a problem for
Miracle temperament where that equivalence is important. So we go back to
our 7 nominals, which work fine defined on 31, but lose the "7 from
12-ness".

The Boston 72-equal notation does still have the 7 nominals behind it, so
neutral thirds are still there. But it's still going to be cumbersome for
Miracle progressions. I don't have examples to hand, but hopefully when
they come in they'll demonstrate this. I think the example on my website
would make a lot of sense with 12 interval classes, and so isn't good for
what I'm explaining here.

> > > Clearly, there is no single such "white note scale" in Miracle.
> There
> > > are many. The only one you get in Miracle-10 is a Slendro-type
> > > pentatonic.
> >
> > Why is that more of a "white note scale" than Miracle-10?
>
> I guess I was thinking mostly of Miller's magic number 7 +-2.

It also has a higher density of 7-limit intervals, doesn't it?

Graham

🔗jpehrson@rcn.com

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., jpehrson@r... wrote:
> >
> > Actually, it looks as though there wasn't a SINGLE error on the
scale
> > posting... I was confused at first, since you used more FLATS
with
> > quartertone sharps than you did in your post #22400 on
blackjack...
>
> Really?
> >
> > Was there any reason for that?? Are these "new" ones "better...?"
>
> I doubt it. Can you show me what you mean?
> >

FROM THE ORIGINAL POST #22400:

v = 1/4 tone flat
< = 1/6 tone flat
{ = 1/12 tone flat
} = 1/12 tone sharp
> = 1/6 tone sharp
^ = 1/4 tone sharp

So the blackjack scale is

note degree Semitones
C< 70 11&2/3
B{ 65 10&5/6
Bv 63 10&1/2
Bb< 58 9&2/3
A> 56 9&1/3
Av 51 8&1/2
Ab} 49 8&1/6
G> 44 7&1/3
G 42 7
Gb} 37 6&1/6
F#{ 35 5&5/6
F 30 5
F< 28 4&2/3
E{ 23 3&5/6
Ev 21 3&1/2
Eb< 16 2&2/3
D> 14 2&1/3
Dv 9 1&1/2
Db} 7 1&1/6
C> 2 0&1/3
C 0 0

FROM THE NEW POST #23674. NATURALLY, the notations are different as
well! :)

A,B,C,D,E,F,G,b,# as in 12-tET
| = 1/4 tone up
> = 1/6 tone up
^ = 1/12 tone up
v = 1/12 tone down
< = 1/6 tone down

Blackjack scale in 72-tET notation and keyboard notation

72-tET notation keyboard notation
lower middle upper

As you can see there are several time when you use a quarter-tone
sharp from a "flat" pitch in the NEW version below, which was the 12-
tET pitch ABOVE with a quarter-tone FLAT in the version above.

I see, for example a b quartertone flat in the previous notation, and
now it's a Bb quartertone sharp. Also I see a D quartertone flat
which is NOW a Db quartertone sharp.

It really doesn't make much difference, but I suppose a note WITHOUT
a "b" or "#" is a bit simpler.

My assumption was that maybe you did this so you only needed to use
ONE quartertone notation, the one UPWARD "|"...

For now, I guess I will go with the "new" way, since you are already
writing chords with it...

> > In any case, the chord postings are VERY interesting as well.
It's funny how the same chords
>
> By "same", you just mean "same keyboard intervals", right?

Yes, indeed, that's what I meant.

>
> And you did try these out and listen to them, yes?
>

Why, of course... That's what it's all about!

> > seem to alternate between being dissonant
> > and consonant.
>
> Yes, that's the pattern through much of the scale.
>
> > I assume that has something to do with the
> > alternating step sizes in this scale, correct??
>
> You bet! In the non-octave version of blackjack, in which the step
sizes strictly alternate, each pattern gives you a strictly
alternating sequence of two kinds of consonant chord. The only
> reason you get dissonant chords at all with these patterns is
because of the two consecutive small steps around each occurence of
the pitch "C" (not "C" on the keyboard, but the actual pitch
> "C").

I "guess" I understand this... not fully though. I can see that in
almost every instance of the "dissonant" case, one of these 72-tone
notation "C's" is present... There were a few other cases, though...

I'm also hoping you will come up with some "chord progressions" not
just parallel chords, although those are surely interesting from the
standpoint of the comparison of concordance...

I'm working on some myself... Actually, using a perfect fifth and
then the "third" of a triad in the octave BELOW and then running up
and down in "parallel" on the keyboard produces some amazing
results... Some INCREDIBLY concordant sonorities....

Thanks so much for your help with this scale!

___________ ________ ____
Joseph Pehrson

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., jpehrson@r... wrote:

> My assumption was that maybe you did this so you only needed to use
> ONE quartertone notation, the one UPWARD "|"...

I hate that semicolon.
>
> I "guess" I understand this... not fully though. I can see that in
> almost every instance of the "dissonant" case, one of these 72-tone
> notation "C's" is present... There were a few other cases,
though...

The C-pitch doesn't have to be present for a dissonance to
occur . . . the chord (in the primary voicing) has to _straddle_ a C-
pitch in order for a dissonance to occur. U C?

>
> I'm also hoping you will come up with some "chord progressions" not
> just parallel chords, although those are surely interesting from
the
> standpoint of the comparison of concordance...

Well, those weren't meant to be chord progressions -- just chord
charts, if you will. I'll have a couple more chord charts . . .
anyway, you can get some amazing chord progressions from the voice
leading . . . for example, allow the upper "fifth" to contract inward
to a "fourth" while the lower "fourth" expands outward to a "fifth" --
aaaahh!!! Of course, chains of chords with common tones are always
nice, too.
>
> I'm working on some myself... Actually, using a perfect fifth and
> then the "third" of a triad in the octave BELOW and then running up
> and down in "parallel" on the keyboard produces some amazing
> results... Some INCREDIBLY concordant sonorities....

You mean something like E3-C4-G4 on the keyboard, or do you mean
something else?

🔗jpehrson@rcn.com

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_23187.html#23731

> --- In tuning@y..., jpehrson@r... wrote:
>
> > My assumption was that maybe you did this so you only needed to
use ONE quartertone notation, the one UPWARD "|"...
>
> I hate that semicolon.
> >

We have to do something about this mess... or revert "back" to
Monzonote...

> > I "guess" I understand this... not fully though. I can see that
in almost every instance of the "dissonant" case, one of these 72-
tone notation "C's" is present... There were a few other cases,
> though...
>
> The C-pitch doesn't have to be present for a dissonance to
> occur . . . the chord (in the primary voicing) has to _straddle_ a
C- pitch in order for a dissonance to occur. U C?
>

I'm going to "mess around" with this a little more... with, probably,
some more questions after I do the "physical" testing..

> >
> > I'm working on some myself... Actually, using a perfect fifth
and then the "third" of a triad in the octave BELOW and then running
up and down in "parallel" on the keyboard produces some amazing
> > results... Some INCREDIBLY concordant sonorities....
>
> You mean something like E3-C4-G4 on the keyboard, or do you mean
> something else?

Yes I mean that. And it seems when you play them running down or up
in "parallel," two or three are "moderately" consonant and then,
suddenly, an AMAZING one occurs... so good it's like from outer
space... (trying to be "technical" here... :) )

I'm sure this one will have a ready explanation.... (??)

____________ _______ ______
Joseph Pehrson

🔗Paul Erlich <paul@stretch-music.com>

🔗jpehrson@rcn.com

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., jpehrson@r... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@y..., jpehrson@r... wrote:
> > >
> > > Actually, it looks as though there wasn't a SINGLE error on the
> scale
> > > posting...

I did make an error, one "C3" should have been "C4". Anyway, here's
some more 9-limit consonant tetrads:

Legend for 72-tET notation:

A,B,C,D,E,F,G,b,# as in 12-tET
| = 1/4 tone up
> = 1/6 tone up
^ = 1/12 tone up
v = 1/12 tone down
< = 1/6 tone down

Blackjack scale in 72-tET notation and keyboard notation

72-tET notation keyboard notation
lower middle upper

Keyboard Pattern: P5-m3-P5 (inversions include m3-P5-M3, P5-M3-P5, M3-
P5-m3)

keyboard notation 72-tET notation quality
E3 B3 D4 A4 G> C< C#^ F dissonant
F3 C4 D#4 A#4 G#^ C#^ Db| F#v dissonant
F#3 C#4 E4 B4 Ab| C> D> Gb^ magic
G3 D4 F4 C5 A> C#^ D#< G dissonant
G#3 D#4 F#4 C#5 A#< Db| Eb| G> magic
A3 E4 G4 D5 Bb| D> Ev G#^ dissonant
A#3 F4 G#4 D#5 Bv D#< F< Ab| magic
B3 F#4 A4 E5 C< Eb| F A> dissonant
C4 G4 A#4 F5 C Ev F#v A#< magic
C#4 G#4 B4 F#5 C> F< Gb^ Bb| dissonant
D4 A4 C5 G5 C#^ F G Bv magic
D#4 A#4 C#5 G#5 Db| F#v G> C< dissonant
E4 B4 D5 A5 D> Gb^ G#^ C magic
F4 C5 D#5 A#5 D#< G Ab| C> dissonant
F#4 C#5 E5 B5 Eb| G> A> C#^ magic
G4 D5 F5 C6 Ev G#^ A#< Db| dissonant
G#4 D#5 F#5 C#6 F< Ab| Bb| D> magic
A4 E5 G5 D6 F A> Bv D#< dissonant
A#4 F5 G#5 D#6 F#v A#< C< Eb| magic
B4 F#5 A5 E6 Gb^ Bb| C Ev dissonant
C5 G5 A#5 F6 G Bv C> F< dissonant

Keyboard Pattern: m3-M6-P4 (inversions include M6-P4-M3, P4-M3-m3, M3-
m3-M6)

keyboard notation 72-tET notation quality
E3 G3 E4 A4 G> A> D> F 4:9/2:6:7
F3 G#3 F4 A#4 G#^ A#< D#< F#v dissonant
F#3 A3 F#4 B4 Ab| Bb| Eb| Gb^ 4:9/2:6:7
G3 A#3 G4 C5 A> Bv Ev G dissonant
G#3 B3 G#4 C#5 A#< C< F< G> 4:9/2:6:7
A3 C4 A4 D5 Bb| C F G#^ dissonant
A#3 C#4 A#4 D#5 Bv C> F#v Ab| dissonant
B3 D4 B4 E5 C< C#^ Gb^ A> dissonant
C4 D#4 C5 F5 C Db| G A#< dissonant
C#4 E4 C#5 F#5 C> D> G> Bb| 1/(9:8:6:5)
D4 F4 D5 G5 C#^ D#< G#^ Bv dissonant
D#4 F#4 D#5 G#5 Db| Eb| Ab| C< 1/(9:8:6:5)
E4 G4 E5 A5 D> Ev A> C dissonant
F4 G#4 F5 A#5 D#< F< A#< C> 4:9/2:6:7
F#4 A4 F#5 B5 Eb| F Bb| C#^ dissonant
G4 A#4 G5 C6 Ev F#v Bv Db| 4:9/2:6:7
G#4 B4 G#5 C#6 F< Gb^ C< D> dissonant
A4 C5 A5 D6 F G C D#< 4:9/2:6:7
A#4 C#5 A#5 D#6 F#v G> C> Eb| dissonant
B4 D5 B5 E6 Gb^ G#^ C#^ Ev 4:9/2:6:7
C5 D#5 C6 F6 G Ab| Db| F< dissonant

Keyboard Pattern: P4-M6-m3 (inversions include M6-m3-M3, m3-M3-P4, M3-
P4-M6)

keyboard notation 72-tET notation quality
E3 A3 F#4 A4 G> Bb| Eb| F dissonant
F3 A#3 G4 A#4 G#^ Bv Ev F#v 1/7:1/6:2/9:1/4
F#3 B3 G#4 B4 Ab| C< F< Gb^ dissonant
G3 C3 A4 C5 A> C F G 1/7:1/6:2/9:1/4
G#3 C#4 A#4 C#5 A#< C> F#v G> dissonant
A3 D4 B4 D5 Bb| C#^ Gb^ G#^ 1/7:1/6:2/9:1/4
A#3 D#4 C5 D#5 Bv Db| G Ab| dissonant
B3 E4 C#5 E5 C< D> G> A> 1/7:1/6:2/9:1/4
C4 F4 D5 F5 C D#< G#^ A#< dissonant
C#4 F#4 D#5 F#5 C> Eb| Ab| Bb| 5:6:8:9
D4 G4 E5 G5 C#^ Ev A> Bv dissonant
D#4 F#4 F5 G#5 Db| F< A#< C< 5:6:8:9
E4 A4 F#5 A5 D> F Bb| C dissonant
F4 A#4 G5 A#5 D#< F#v Bv C> dissonant
F#4 B4 G#5 B5 Eb| Gb^ C< C#^ dissonant
G4 C5 A5 C6 Ev G C Db| dissonant
G#4 C#5 A#5 C#6 F< G> C> D> 1/7:1/6:2/9:1/4
A4 D5 B5 D6 F G#^ C#^ D#< dissonant
A#4 D#5 C6 D#6 F#v Ab| Db| Eb| 1/7:1/6:2/9:1/4
B4 E5 C#6 E6 Gb^ A> D> Ev dissonant
C5 F5 D6 F6 G A#< D#< F< 1/7:1/6:2/9:1/4