Some scales on the Blackjack guitar

BTW, anyone converting a 12-tET guitar to blackjack should note that you
don't need to remove the 5th, 10th, 12th, 17th, or 22nd frets.

I give a fingerboard diagram below with all the Blackjack notes marked in
Graham Breed's decimal notation. Read > and < simply as "up" and "down".

The notes marked "x" are extra to the reference blackjack scale, the one
that occurs on the highest string. They can nevertheless be very useful in
making other scales available (such as a JI major and minor, Arabic,
bagpipe, and possibly even a 2 of {1,3,5,7,9} dekany). But at first we'll
only consider scales available within a single blackjack scale.

Any scale available in blackjack is playable between the nut and 7th fret
or between the octave (21st) and 28th fret (and all the way up the high
string).

You might want to save this message somewhere, because when I get time I'll
post some recognised scales (from the Scala archives) in the decimal notation.

Blackjack guitar
fingerboard Approx 12-tET fret position
5 8 1< 4 7 0< open open
5> 8> 1 4> 7> 0 1fr

6 9 2< 5 8 1< 2fr 1fr
6> 9> 2 5> 8> 1 3fr
2fr
7 0< 3< 6 9 2< 4fr
7> 0 3 6> 9> 2 5fr
3fr
8 1< 4< 7 0< 3< 6fr
8> 1 4 7> 0 3 7fr 4fr

9 2< x 8 1< 4< 8fr
9> 2 5 8> 1 4 9fr 5fr
x x 5> x x 4> 10fr

0 3 6 9> 2 5 11fr 6fr
x x 6> x x 5> 12fr
7fr
1 4 7 0 3 6 13fr
x 4> 7> x x 6> 14fr
8fr
2 5 8 1 4 7 15fr
x 5> 8> x 4> 7> 16fr 9fr

3 6 9 2 5 8 17fr
x 6> 9> x 5> 8> 18fr 10fr

4 7 0< 3 6 9 19fr 11fr
4> 7> 0 x 6> 9> 20fr

5 8 1< 4 7 0< 21fr 12 fr
5> 8> 1 4> 7> 0 22fr

6 9 2< 5 8 1< 23fr 13fr
6> 9> 2 5> 8> 1 24fr
14fr
7 0< 3< 6 9 2< 25fr
7> 0 3 6> 9> 2 26fr
15fr
8 1< 4< 7 0< 3< 27fr
8> 1 4 7> 0 3 28fr 16fr

Mohajira

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

--- In tuning@y..., David C Keenan <D.KEENAN@U...> wrote:
> BTW, anyone converting a 12-tET guitar to blackjack should note that
> you
> don't need to remove the 5th, 10th, 12th, 17th, or 22nd frets.
<snip>
True, but sanding the fingerboard is made more difficult by leaving
these frets in.

John Starrett

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

[...]
> Mohajira

And that was it. No scales made it into this message. Dave, if you
did a ton of work mapping out various scales and it all got
truncated, I'm very very sorry.

----- Original Message -----
From: David C Keenan <D.KEENAN@UQ.NET.AU>
Subject: [tuning] Some scales on the Blackjack guitar

I can't heads nor tails out of your notation. I suggest
taking the graphic of your fret board and adding dots
to where the notes of the scale are.

* David Beardsley
* http://biink.com
* http://mp3.com/davidbeardsley

Sorry I haven't posted the scales yet. That "Mohajira" was left on the end
of the previous message by mistake.

"David Beardsley" <davidbeardsley@b... <mailto:davidbeardsley@b...>> wrote:

> A 3 octave major scale with 4 notes on a string, slide on the half steps.

I think you may have the wrong idea about the Blackjack scale. It gives you
an amazingly big slice of the 11-limit for just 21 notes, but it does not
give you a 5-limit JI major or minor.

However, the Blackjack _guitar_ has some additional notes available on some
strings that just happen to give you two JI majors (a secor apart) and a JI
minor. I'll show below, on the fingerboard, C major and E minor as in this
5-limit lattice.

D#
A E B F# C# (wrong C#?)
F C G D

Blackjack guitar
fingerboard Approx 12-tET fret position
. G . . F# . open open
E . B D# . . 1fr

. . . . G . 2fr 1fr
F . C E . B 3fr
2fr
F# . C# . . . 4fr
. . . F . C 5fr
3fr
G . D F# . C# 6fr
. B . . . . 7fr 4fr

. . . G . D 8fr
. C . . B . 9fr 5fr
A . E . . D# 10fr

. . . . C . 11fr 6fr
. . F A . E 12fr
7fr
B . F# . . . 13fr
. D# . . . F 14fr
8fr
C . G B . F# 15fr
. E . . D# . 16fr 9fr

. . . C . G 17fr
. F . . E . 18fr 10fr

. F# . . . . 19fr 11fr
D# . . . F . 20fr

. G . . F# . 21fr 12fr 8ve
E . B D# . . 22fr

. . . . G . 23fr 13fr
F . C E . B 24fr
14fr
F# . C# . . . 25fr
. . . F . C 26fr
15fr
G . D F# . C# 27fr
. B . . . . 28fr 16fr

You will notice that there are only two and a bit complete octaves of these
scales available (from E to G). From C to C there's only one complete
octave (because A is missing from the top string). You will also see that
he best place to play these scales (no missing notes) is between the 6th
and 12th frets.

Dave Beardsley, If you want to tell me more about what you want from a JI
guitar, like give me the complete gamut with some indication as to which
notes are essential and which aren't, I'll have a go at designing one for
you with all full width frets. But you may still have to learn a new open
string tuning because my method's sucess is dependent on the synergy
possible between open string tuning and scale-rotation for fretting.

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

I decided it was too daunting to map out all the archived subsets of
Blackjack on the fingerboard. So anyone can email me with one or two from
the lists below, that particularly interest you, and I'll see what I can do.

Here are the 35 scales Manuel found that were subsets of Blackjack to
within 5 cents in any interval (including dissonances).

Blackjack is a:
superset of ARIST_CHROM3.SCL size: 7
superset of ARIST_DIATDOR.SCL size: 7
superset of ARIST_DIATRED.SCL size: 7
superset of MOHAJIRA.SCL size: 7
superset of XENAKIS_SCHROM.SCL size: 7
approximate superset of AL-FARABI.SCL size: 7, diff. 4.9354 cents
approximate superset of AL-FARABI_CHROM2.SCL size: 7, diff. 4.9354 cents
approximate superset of AL-FARABI_DIAT.SCL size: 7, diff. 4.3185 cents
approximate superset of AL-FARABI_DOR2.SCL size: 7, diff. 4.3185 cents
approximate superset of CROSS_7.SCL size: 7, diff. 2.9804 cents
approximate superset of DIAPHONIC_5.SCL size: 5, diff. 2.1592 cents
approximate superset of DIV_FIFTH2.SCL size: 5, diff. 4.9354 cents
approximate superset of EFG3377.SCL size: 9, diff. 4.1142 cents
approximate superset of EFG33777A.SCL size: 10, diff. 4.1142 cents
approximate superset of HARRISON_MIX4.SCL size: 5, diff. 2.9804 cents
approximate superset of HELMHOLTZ.SCL size: 7, diff. 4.9354 cents
approximate superset of HEXANY15.SCL size: 5, diff. 2.9804 cents
approximate superset of HEXANY18.SCL size: 5, diff. 4.3185 cents
approximate superset of HEXANY19.SCL size: 5, diff. 2.9804 cents
approximate superset of HEXANY49.SCL size: 6, diff. 4.3185 cents
approximate superset of KRING2.SCL size: 7, diff. 2.1592 cents
approximate superset of KRING4.SCL size: 7, diff. 2.9804 cents
approximate superset of LUMMADEC1.SCL size: 10, diff. 4.8387 cents
approximate superset of PIPEDUM7_9.SCL size: 9, diff. 4.9354 cents
approximate superset of PS-ENH.SCL size: 7, diff. 4.9354 cents
approximate superset of PYGMIE.SCL size: 5, diff. 4.1142 cents
approximate superset of SLENDRO5_2.SCL size: 5, diff. 2.1592 cents
approximate superset of SLENDRO_7_1.SCL size: 5, diff. 4.3185 cents
approximate superset of SLENDRO_A1.SCL size: 5, diff. 2.1592 cents
approximate superset of SLENDRO_A2.SCL size: 5, diff. 4.3185 cents
approximate superset of SLENDRO_M.SCL size: 5, diff. 2.1592 cents
approximate superset of SLENDRO_PB.SCL size: 5, diff. 2.6666 cents
approximate superset of SLENDRO_PC.SCL size: 5, diff. 2.6666 cents
approximate superset of SLENDRO_S1.SCL size: 5, diff. 4.1142 cents
approximate superset of SPON_TERP.SCL size: 5, diff. 2.5921 cents

Here are the descriptions from the Scala files, in the same order.

PsAristo 3 Chromatic, 7 + 7 + 16 parts

PsAristo Redup. Diatonic, 14 + 2 + 14 parts

Aristo Redup. Diatonic, Dorian Mode, 14 + 14 + 2 parts

Mohajira (Dudon) Two 3 + 4 + 3 Mohajira tetrachords, neutral diatonic
Xenakis's Byzantine Liturgical mode, 7 + 16 + 7 parts
Al-Farabi Syn Chrom

Al-Farabi's Chromatic permuted

Al-Farabi's Diatonic
Dorian mode of Al-Farabi's Diatonic

3-5-7 cross reduced by 2/1, quasi diatonic, similar to Zalzal's, Flynn
Cohen
D5-tone Diaphonic Cycle

Divided Fifth #2, From Schlesinger, see Chapter 8, p. 160

Genus [3377]

Genus [33777] with comma discarded which disappears in 31-tET

A "mixed type" pentatonic, Lou Harrison

Helmholtz's Chromatic scale and Gipsy major from Slovakia

1.3.5.15 2)4 hexany (1.15 tonic) degenerate, symmetrical pentatonic

1.7.49.343 Hexany, a degenerate pentatonic form

1.5.7.35 Hexany, a degenerate pentatonic form

1.3.21.49 2)4 hexany (1.21 tonic)

Double-tie circular mirroring of 6:7:8

Double-tie circular mirroring of 4:5:7

Carl Lumma, two 5-tone 7/4-chains, 5/4 apart in 31-tET, TL 9-2-2000

225/224, 49/48 and 36/35 are homophonic intervals

Dorian mode of an Enharmonic genus found in Ptolemy's Harmonics

Pygmie scale

A slendro type pentatonic which is based on intervals of 7, no. 2

Septimal Slendro 1, From HMSL Manual, also Lou Harrison, Jacques Dudon

Dudon's Slendro A1, "Seven-Limit Slendro Mutations", 1/1 8:2'94 hexany
1.3.7.21
Dudon's Slendro A2 from "Seven-Limit Slendro Mutations", 1/1 8:2 Jan
1994
Dudon's Slendro M from "Seven-Limit Slendro Mutations", 1/1 8:2 Jan
1994
"Blown fifth" medium slendro, von Hornbostel

"Blown fifth" modern slendro, von Hornbostel

Dudon's Slendro S1 from "Seven-Limit Slendro Mutations", 1/1 8:2 Jan
1994
Subharm. 6-tone series, guess at Greek poet Terpander's, 6th c. BC &
Spondeion, Winnington-Ingram (1928)

Here are a few more approximate subsets I found where no just interval is
off by more than 3.0 c and no pitch is off by more than 7.1c.

Filename: Notes:
cluster6g.scl 6
cluster6h.scl 6
efg357.scl 8
hexany1.scl 6
serre_enh.scl 7
barbour_chrom3.scl 7

Descriptions:
Six-Tone Triadic Cluster 4:5:7

Six-Tone Triadic Cluster 4:7:5

Genus sextum [357] & 7-limit Octony, see ch.6 p.118
Two out of 1 3 5 7 hexany

Dorian mode of the Serre's Enharmonic

Barbour's #3 Chromatic

That's 41 subsets found so far. Undoubtedly some of these will map to the
same scale in Blackjack, but it still seems an impressively large list.

These last 6 were found by looking for exact subsets of this 7-limit
rationalisation of Blackjack. Note that there are many other
rationalisations of Blackjack which would uncover other subsets (e.g. shift
almost any note below by a 224:225 or a 384:385).

! temp.scl
!
7-limit RI blackjack

21
!
64/63
16/15
35/32
8/7
7/6
128/105
5/4
21/16
4/3
7/5
10/7
3/2
32/21
8/5
105/64
12/7
7/4
64/35
15/8
63/32
2/1

And don't forget, there are additional notes on the Blackjack guitar that
make other scales possible. If you have a particular scale of 5 to 12 notes
that isn't mentioned above, and you want to know if it's playable on the
Blackjack guitar, send me email.

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

Sorry about all the blank lines in my last post to this thread.

Here are another 32 Blackjack-subset scales from the scala archive. This is
getting ridiculous. :-) They are all 5 or 7-limit rational.

Filename: Notes:
cluster6a 6
cons8 4
cons9 5
efg335 6
exp2 7
kayolonian_p 9
olympos 5
pipedum_12a 12
turkish 7
diat_smal 7
efg557 6
hexany20 6
minor_5 5
al-farabi_g10 7
awraamoff 12
cluster6b 6
efg337 6
efg355 6
farey3 5
farey4 9
harrison_5 5
harrison_5_4 5
harrison_mix2 5
hirajoshi2 5
indian-raja 6
kayolonian_z 9
octony_min 8
pelog_alv 7
slendro5_4 5
slendro_7_2 5
slendro_7_4 5
slovak_min 7

Description:
Six-Tone Triadic Cluster 4:5:6
Set of intervals with num + den <= 8 not exceeding 2/1
Set of intervals with num + den <= 9 not exceeding 2/1
Genus secundum [335]
Two times expanded major triad
Kayolonian scale P
Scale of ancient Greek flutist Olympos, 6th century BC as reported by Partch
128/125 and 2048/2025 are homophonic intervals
Turkish, 5-limit from Palmer on a Turkish music record, harmonic minor
inverse
"Smallest number" diatonic scale
Genus septimum [557]
3.5.7.105 Hexany
A minor pentatonic
Al-Farabi's Greek genus chromaticum forte
Awraamoff Septimal Just
Six-Tone Triadic Cluster 4:6:5
Genus quintum [337]
Genus tertium [355]
Farey fractions between 0 and 1 until 3rd level, normalised by 2/1
Farey fractions between 0 and 1 until 4th level, normalised by 2/1
From Lou Harrison, a pelog style pentatonic
From Lou Harrison, a pelog style pentatonic
A "mixed type" pentatonic, Lou Harrison
Another Japanese pentatonic koto scale
A folk scale from Rajasthan, India
Kayolonian scale Z
Octony on Harmonic Minor, from Palmer on an album of Turkish music
Bill Alves JI Pelog, 1/1 vol. 9 no. 4, 1997. 1/1=293.33
A slendro type pentatonic which is based on intervals of 7, no. 4
Septimal Slendro 2, From Lou Harrison, Jacques Dudon's APTOS
Septimal Slendro 4, from Lou Harrison, Jacques Dudon, called "NAT"
Gipsy minor scale from Slovakia

The offer still stands, for anyone to email me one or two scales and have
me show how they map onto the Blackjack guitar. If they are in the Scala
archive, just the name will do, otherwise cents and/or ratios.

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

--- In tuning@y..., David C Keenan <D.KEENAN@U...> wrote:
>
> 225/224, 49/48 and 36/35 are homophonic intervals

This looks interesting.

Do we have the regular Arabic diatonic?

I guess you could make one post with all the Slendro scales.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., David C Keenan <D.KEENAN@U...> wrote:
> >
> > 225/224, 49/48 and 36/35 are homophonic intervals
>
> This looks interesting.
>
> Do we have the regular Arabic diatonic?

It doesn't fit in blackjack since it spans 24 secors where blackjack
only has 20. But, like the JI major and minor and the pythagorean
pentatonic, it is available on the Blackjack guitar (possibly in 3
positions).

> I guess you could make one post with all the Slendro scales.

Ok. So you're asking me to map those 3 scales (incl. one
representative slendro?) on the fingerboard?

-- Dave Keenan

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@y..., David C Keenan <D.KEENAN@U...> wrote:
> > >
> > > 225/224, 49/48 and 36/35 are homophonic intervals
> >
> > This looks interesting.

You didn't comment -- can you post the structure of this scale?
> >
> > Do we have the regular Arabic diatonic?
>
> It doesn't fit in blackjack since it spans 24 secors where
blackjack
> only has 20. But, like the JI major and minor and the pythagorean
> pentatonic, it is available on the Blackjack guitar (possibly in 3
> positions).

Cool.
>
> > I guess you could make one post with all the Slendro scales.
>
> Ok. So you're asking me to map those 3 scales (incl. one
> representative slendro?) on the fingerboard?

Only if you feel like it. I won't have much practical use for this.
I'm more interested in what these slendro scales actually are. I
presume I'll be able to try them with not too much error on my 31-tET
guitar.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > > --- In tuning@y..., David C Keenan <D.KEENAN@U...> wrote:
> > > >
> > > > 225/224, 49/48 and 36/35 are homophonic intervals
> > >
> > > This looks interesting.
>
> You didn't comment -- can you post the structure of this scale?

Oh. You can look up any of the scales I posted, in the Scala archive.
You only had to match up the description with the filename (listed in
the same order). This one is "pipedum7_9.scl"
15/14 7/6 5/4 4/3 3/2 45/28 12/7 15/8 2/1
What do you suppose "pipedum" stands for? There are a lot of them.
Manuel?

> > Ok. So you're asking me to map those 3 scales (incl. one
> > representative slendro?) on the fingerboard?
>
> Only if you feel like it. I won't have much practical use for this.

In that case, I don't feel like it. :-)

> I'm more interested in what these slendro scales actually are. I
> presume I'll be able to try them with not too much error on my
> 31-tET guitar.

The following eleven Blackjack-subset slendro scales are in the Scala
archive. Can you tell me why that wasn't clear from my earlier posts,
so I don't make the same mistake again. Thanks.

SLENDRO5_2.SCL
SLENDRO5_4.SCL
SLENDRO_7_1.SCL
SLENDRO_7_2.SCL
SLENDRO_7_4.SCL
SLENDRO_A1.SCL
SLENDRO_A2.SCL
SLENDRO_M.SCL
SLENDRO_PB.SCL
SLENDRO_PC.SCL
SLENDRO_S1.SCL

The Scala archive can be downloaded from
http://www.xs4all.nl/~huygensf/scala/downloads.html

So far we've found 73 scales in the Scala archive that can be
considered subsets of Blackjack (to better accuracy than that with
which a JI major or minor could be considered a subset of 1/4 comma
meantone).

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

> Oh. You can look up any of the scales I posted, in the Scala
archive.
> You only had to match up the description with the filename (listed
in
> the same order). This one is "pipedum7_9.scl"
> 15/14 7/6 5/4 4/3 3/2 45/28 12/7 15/8 2/1
> What do you suppose "pipedum" stands for? There are a lot of them.
> Manuel?

Parellelepipedum.

>
> The following eleven Blackjack-subset slendro scales are in the
Scala
> archive. Can you tell me why that wasn't clear from my earlier
posts,

It was.

Paul erlich wrote:

> I'm more interested in what these slendro scales actually are.

I've compared those 11 and about the only thing they have in common is that
they are octave based scales of 5 notes where the steps are all between 204
and 294 cents. So I'm guessing that's a definition of a Slendro scale.

So that's why so many are possible in Blackjack. It's easy to find a string
of 5 steps in an octave chosen from {200c, 233c, 267c, 300c}.

Those 11 slendro scales map down to 7 distinct scales in blackjack. The
most popular one is the most even one, with steps of 233 233 233 233 267
cents. This one is available in 13 positions (keys) within Blackjack.

That there might be any significance to the idea of a _justly_intoned_
slendro scale, looks like nonsense to me. There are too many claimants. To
put it another way: It seems that a justly intoned scale may happen to be
Slendro, but I suspect that a slendro scale has no requirement to have any
of its intervals even approximately justly intoned (at least not for any
harmonic timbre).

> I presume I'll be able to try them with not too much error on my
> 31-tET guitar.

In 31-tET you would be choosing slendro steps from {194c, 232c, 271c,
310c}. 194 and 310 are getting rather extreme, so you should probably stick
to the maximally even one, with steps of 6 6 6 6 7 steps of 31-tET.

But lets look at how much you are presuming here, in the case of other
scales that are _intended_ to be just. This is also for Seth Austen on his
zither.

Intvl Error in miracle temperament
as subset of
72-tET 31-tET
--------------------
2:3 -2.0 -5.2
4:5 -3.0 .8
5:6 1.0 -6.0
4:7 -2.2 -1.1
5:7 .8 -1.9
6:7 -0.2 4.1
4:9 -3.9 -10.4
5:9 - .9 -11.1
7:9 -1.8 -9.3
4:11 -1.3 -9.4
5:11 1.7 -10.2
6:11 .6 -4.2
7:11 .8 -8.3
9:11 2.6 1.0

max abs 3.9 11.1

So you can see that 31-tET is not too bad at the 7-limit (max 6c error,
twice that of the 72-tET version) but its miracle-style approximations to
ratios of 9 and 11 (except 9:11) are _dreadful_.

So you can only get an approximation to the 7-limit (and neutral thirds)
behaviour of Blackjack, in 31-tET, not its 9 or 11-limit behaviour. e.g.
none of those interesting 7:9:11 supermajor augmented triads, or 6:7:9
subminors or 1/(9:7:6) supermajors, or any of the ASSs.

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

>> What do you suppose "pipedum" stands for? There are a lot of them.
>> Manuel?

>Parellelepipedum.

Parallellepipedum more precisely. The files are named after the
Scala command with which they were created. The whole word was too
long for a command name, so I abbreviated it to "pipedum", which I
found easy to remember. The alternatives "periodicity block" or
the Dutch "herhalingsblok" were too long too, and I don't like
acronymic command names.

Manuel

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

/tuning/topicId_27122.html#27162

> I decided it was too daunting to map out all the archived subsets of
> Blackjack on the fingerboard. So anyone can email me with one or
two from the lists below, that particularly interest you, and I'll
see what I can do.

It sure looks like there is plenty to do here! Maybe if
the "Blackjack Guitar" were marketed properly, it could be a "big
seller" in the era of world musics.... Well, just a thought.

____________ _________ _____
Joseph Pehrson

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

/tuning/topicId_27122.html#27192

>
> Intvl Error in miracle temperament
> as subset of
> 72-tET 31-tET
> --------------------
> 2:3 -2.0 -5.2
> 4:5 -3.0 .8
> 5:6 1.0 -6.0
> 4:7 -2.2 -1.1
> 5:7 .8 -1.9
> 6:7 -0.2 4.1
> 4:9 -3.9 -10.4
> 5:9 - .9 -11.1
> 7:9 -1.8 -9.3
> 4:11 -1.3 -9.4
> 5:11 1.7 -10.2
> 6:11 .6 -4.2
> 7:11 .8 -8.3
> 9:11 2.6 1.0
>
> max abs 3.9 11.1
>

Thanks so much, Dave, for this interesting chart! I printed and
saved it right away...

________ ___________ _______
Joseph Pehrson