Yahoo Tuning Groups Ultimate Backup tuning Some 12 Mappable Scales

Hi Tuners,

I would like to share a few of my favorite 12 tone mappable Just Scales.

I generally view all scales as subsets of larger systems. A scale, to me, is a set of tones that are interrelated not only to one another in some way, but to other sets as well. The scale sets here are slices of the infinity of tonal possiblities. In my view, harmonics and subharmonics are both integral to the tonal fabric and the scales here reflect that view.

I like this first one because it incorporates lots of sevens and can be full of dissonance or quite sweet, depending on the desired result. It is also, I think, a possible 7-limit interpretation/extrapolation of the melodic minor. I am showing the relative just ratios in a row of triads to highlight that interpretation.

The ascending 6th and 7th are shown in brackets (as is the ascending octave triad.) The descending octave triad is a harmonic 5-6-7.

5/4 4/3 3/2 5/3 7/4 2/1 (2/1) 21/10 (20/9) 7/3 (5/2)
1/1 10/9 5/4 4/3 3/2 5/3 (5/3) 7/4 (40/21) 2/1 (2/1)
5/6 20/21 1/1 10/9 5/4 4/3 (10/7) 3/2 100/63) 5/3 (5/3)
--------------------------------------------------------------------------------------------------

The next 12 tone mappable scale is one I originally submitted to this list as the 'intersection of sets' scale a few years back. It is the intersection of the 7-limit n:n+1 harmonic/subharmonic relationships of 1/1 and 3/2. There are two conjoined 8-tone scales sharing the bold faced intervals. It's a great scale for using a two tone drone consisting of 1/1 and 3/2.

1/1 9/8 (6/5) 5/4 4/3 3/2 (8/5) 5/3 (12/7) 7/4 (9/5) 15/8 2/1

--------------------------------------------------------------------------------------------------

The next is not exactly defined by an 'n'-limit. It comprises the harmonic and subharmonic subsets of of 2:3, 4:5, 8:9 and 16:17, plus four additional notes; 17/12 & 24/17 (which are very, very close in pitch and can be mapped to 600 cents with a +or- 3 cent error), 17/10 (which is 6/5 times 17/12), and 20/17 (which is 5/4 times 16/17). This scale can be very 'Spanish-ish' sounding...

1/1 17/16 9/8 20/17 5/4 4/3 (24/17 & 17/12) 3/2 8/5 17/10 16/9 32/17 2/1
--------------------------------------------------------------------------------------------------

The last scale is a very basic, very sweet 5-limit scale:

1/1 135/128 9/8 6/5 5/4 27/20 45/32 3/2 8/5 27/16 9/5 15/8 2/1
--------------------------------------------------------------------------------------------------
Enjoy!
Robin

🔗Robin Perry <jinto83@yahoo.com>

Well... that first scale doesn't read the way I wrote it. I will
try to re-format that to make better sense:

5/6, 1/1, 5/4
20/21, 10/9, 4/3
1/1, 5/4, 3/2
10/9, 4/3, 5/3
5/4, 3/2, 7/4
4/3, 5/3, 2/1
(10/7, 5/3, 2/1) ascending
3/2, 7/4, 21/19
(100/63, 40/21, 20/9) ascending
5/3, 2/1, 73
5/3, 2/1, 5/2

--- In tuning@yahoogroups.com, Robin Perry <jinto83@...> wrote:
>
> Hi Tuners,
>
> I would like to share a few of my favorite 12 tone mappable Just
Scales.
>
> I generally view all scales as subsets of larger systems. A
scale, to me, is a set of tones that are interrelated not only to
one another in some way, but to other sets as well. The scale sets
here are slices of the infinity of tonal possiblities. In my view,
harmonics and subharmonics are both integral to the tonal fabric and
the scales here reflect that view.
>
> I like this first one because it incorporates lots of sevens and
can be full of dissonance or quite sweet, depending on the desired
result. It is also, I think, a possible 7-limit
interpretation/extrapolation of the melodic minor. I am showing the
relative just ratios in a row of triads to highlight that
interpretation.
>
> The ascending 6th and 7th are shown in brackets (as is the
ascending octave triad.) The descending octave triad is a harmonic 5-
6-7.
>
> 5/4 4/3 3/2 5/3 7/4 2/1 (2/1) 21/10 (20/9)
7/3 (5/2)
> 1/1 10/9 5/4 4/3 3/2 5/3 (5/3) 7/4 (40/21)
2/1 (2/1)
> 5/6 20/21 1/1 10/9 5/4 4/3 (10/7) 3/2 100/63) 5/3
(5/3)
> -----------------------------------------------------------------
---------------------------------
>
> The next 12 tone mappable scale is one I originally submitted to
this list as the 'intersection of sets' scale a few years back. It
is the intersection of the 7-limit n:n+1 harmonic/subharmonic
relationships of 1/1 and 3/2. There are two conjoined 8-tone scales
sharing the bold faced intervals. It's a great scale for using a two
tone drone consisting of 1/1 and 3/2.
>
> 1/1 9/8 (6/5) 5/4 4/3 3/2 (8/5) 5/3 (12/7) 7/4 (9/5) 15/8 2/1
>
> -----------------------------------------------------------------
---------------------------------
>
> The next is not exactly defined by an 'n'-limit. It comprises
the harmonic and subharmonic subsets of of 2:3, 4:5, 8:9 and 16:17,
plus four additional notes; 17/12 & 24/17 (which are very, very
close in pitch and can be mapped to 600 cents with a +or- 3 cent
error), 17/10 (which is 6/5 times 17/12), and 20/17 (which is 5/4
times 16/17). This scale can be very 'Spanish-ish' sounding...
>
> 1/1 17/16 9/8 20/17 5/4 4/3 (24/17 & 17/12) 3/2 8/5 17/10 16/9
32/17 2/1
> -----------------------------------------------------------------
---------------------------------
>
> The last scale is a very basic, very sweet 5-limit scale:
>
> 1/1 135/128 9/8 6/5 5/4 27/20 45/32 3/2 8/5 27/16 9/5 15/8 2/1
> -----------------------------------------------------------------
---------------------------------
> Enjoy!
> Robin
>

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

--- In tuning@yahoogroups.com, Robin Perry <jinto83@...> wrote:

> The ascending 6th and 7th are shown in brackets (as is the
ascending octave triad.) The descending octave triad is a harmonic 5-
6-7.

This way of describing the scale strikes me as a bit murky.

I would say the core of this scale are the two otonal and two utonal
tetrads it contains; these form a ten-note scale, also epimorphic,
which is an example of a "line scale"; a chain of contiguous tetrads
in tetrad space along a straight line. For tetrad space:

http://www.xenharmony.com/sevlat.htm

The key paragraph being:

If [a b c] is any triple of integers, then it represents the major
tetrad with root 3^((-a+b+c)/2) 5^((a-b+c)/2) 7^((a+c-c)/2) if a+b+c
is even, and the minor tetrad with root 3^((-1-a+b+c)/2) 5^((1+a-
b+c)/2 7^((1+a+b-c)/2) if a+b+c is odd. Each unit cube corresponds to
a either a stellated hexany, also called a tetradekany, or
dekatesserany, or else a transposed 7-limit tonality diamond.

The ten-note core of your scale in my view consists of the line of
tetrads [0,-2,-1], [0,-1,-1],[0,-1,0],[0,0,0]]. This gives the scale
[10/9, 7/6, 5/4, 4/3, 10/7, 3/2, 5/3, 7/4, 40/21, 2], which like your
scale has the nice property of being epimorphic. The question then
is, how to add two more seven-limit notes to get another nice
epimorphic scale? I'm not clear why you want to add 21/20 and 100/63;
why not for instance 28/27 and 14/9?

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

--- In tuning@yahoogroups.com, Robin Perry <jinto83@...> wrote:

> The next 12 tone mappable scale is one I originally submitted to
this list as the 'intersection of sets' scale a few years back.

It's been in the Scala archives since as a result.

> The next is not exactly defined by an 'n'-limit.

It looks as if (17/12)/(24/17) = 289/288 should be tempered out. Using
200-et does this for you with the cents values coming out evenly.

> The last scale is a very basic, very sweet 5-limit scale:
>
> 1/1 135/128 9/8 6/5 5/4 27/20 45/32 3/2 8/5 27/16 9/5 15/8 2/1

This is a transposition of the famous Ellis duodene.

🔗Robin Perry <jinto83@yahoo.com>

Hi Gene,

Thank you for the examination of this one. I didn't intend to make
murk around this scale. I used the analogy of the melodic minor
because it seemed like a good way to convey the pattern inherent in
the scale. Maybe a better way of understanding it would be to
stretch it beyond the octave, which is how I first concieved it.

It follows a very predictable pattern, which is the method behind
the madness of using the 21/20 and the 100/63 as opposed to using a
28/27 and a 14/9.

I'll put the 'raised 6th' triad first and continue from there with
the pattern:

5/7 - 5/6 - 1/1 (Utonal 7-6-5)
50/63 - 20/21 - 10/9 (Otonal 5-6-7)
5/6 - 1/1 - 5/4 (Utonal 6-5-4)
20/21 - 10/9 - 4/3 (Utonal 7-6-5)
1/1 - 5/4 - 3/2 (Otonal 4-5-6)
10/9 - 4/3 - 5/3 (Utonal 6-5-4)
5/4 - 3/2 - 7/4 (Otonal 5-6-7)
4/3 - 5/3 - 2/1 (Otonal 4-5-6)
3/2 - 7/4 - 21/10 (Utonal 7-6-5)
5/3 - 2/1 - 7/3 (Otonal 5-6-7)

The pattern is U-O-U-U-O-U-O-O...

The last two tones in each triad are the first two tones in the next
to next triad.

So, if we go beyond a 12 mappable set, the next triads in the
sequence would be:

7/4 - 21/20 - 21/8 (Utonal 6-5-4)
2/1 - 7/3 - 42/15 (Utonal 7-6-5)

I do understand why you might propose using 28/27 and 14/9, but they
aren't what this scale is 'about.' It's not so much of a 'line
scale' as it is a 'zig-zag' scale which features the 21/20 interval
prominantly.

And, on the comments about the other scales:

Thanks for letting me know that the Intersection scale is in the
Scala archive. I didn't know that. I also didn't know about the
Ellis duodene. Thank you.

As for the 3,5,9,17 scale: If you do want to use a temperament, I
think that 46 Equal does the trick with this one.

Thanks again, Gene.

Regards,

Robin

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, Robin Perry <jinto83@> wrote:
>
> > The ascending 6th and 7th are shown in brackets (as is the
> ascending octave triad.) The descending octave triad is a harmonic
5-
> 6-7.
>
> This way of describing the scale strikes me as a bit murky.
>
> I would say the core of this scale are the two otonal and two
utonal
> tetrads it contains; these form a ten-note scale, also epimorphic,
> which is an example of a "line scale"; a chain of contiguous
tetrads
> in tetrad space along a straight line. For tetrad space:
>
> http://www.xenharmony.com/sevlat.htm
>
> The key paragraph being:
>
> If [a b c] is any triple of integers, then it represents the major
> tetrad with root 3^((-a+b+c)/2) 5^((a-b+c)/2) 7^((a+c-c)/2) if
a+b+c
> is even, and the minor tetrad with root 3^((-1-a+b+c)/2) 5^((1+a-
> b+c)/2 7^((1+a+b-c)/2) if a+b+c is odd. Each unit cube corresponds
to
> a either a stellated hexany, also called a tetradekany, or
> dekatesserany, or else a transposed 7-limit tonality diamond.
>
> The ten-note core of your scale in my view consists of the line of
> tetrads [0,-2,-1], [0,-1,-1],[0,-1,0],[0,0,0]]. This gives the
scale
> [10/9, 7/6, 5/4, 4/3, 10/7, 3/2, 5/3, 7/4, 40/21, 2], which like
your
> scale has the nice property of being epimorphic. The question then
> is, how to add two more seven-limit notes to get another nice
> epimorphic scale? I'm not clear why you want to add 21/20 and
100/63;
> why not for instance 28/27 and 14/9?
>

🔗Robin Perry <jinto83@yahoo.com>

Hi Tuners,

Manuel Op De Coul was kind enough to Just-ify one of my 12 Equal
compositions, "Calling You."

http://home.casema.nl/luweiwang/snd/calling_you.mp3

For those who are not familiar with Manuel, he is the creator of
Scala, a bit of software that everyone into re-tuning should have:

http://www.xs4all.nl/~huygensf/scala/

The message posted to this group concerning the scale used in the
tune is re-posted below:

Regards,

Robin

--- In tuning@yahoogroups.com, "Robin Perry" <jinto83@...> wrote:
>
> Hi Gene,
>
> Thank you for the examination of this one. I didn't intend to
make
> murk around this scale. I used the analogy of the melodic minor
> because it seemed like a good way to convey the pattern inherent
in
> the scale. Maybe a better way of understanding it would be to
> stretch it beyond the octave, which is how I first concieved it.
>
> It follows a very predictable pattern, which is the method behind
> the madness of using the 21/20 and the 100/63 as opposed to using
a
> 28/27 and a 14/9.
>
> I'll put the 'raised 6th' triad first and continue from there with
> the pattern:
>
> 5/7 - 5/6 - 1/1 (Utonal 7-6-5)
> 50/63 - 20/21 - 10/9 (Otonal 5-6-7)
> 5/6 - 1/1 - 5/4 (Utonal 6-5-4)
> 20/21 - 10/9 - 4/3 (Utonal 7-6-5)
> 1/1 - 5/4 - 3/2 (Otonal 4-5-6)
> 10/9 - 4/3 - 5/3 (Utonal 6-5-4)
> 5/4 - 3/2 - 7/4 (Otonal 5-6-7)
> 4/3 - 5/3 - 2/1 (Otonal 4-5-6)
> 3/2 - 7/4 - 21/10 (Utonal 7-6-5)
> 5/3 - 2/1 - 7/3 (Otonal 5-6-7)
>
> The pattern is U-O-U-U-O-U-O-O...
>
> The last two tones in each triad are the first two tones in the
next
> to next triad.
>
> So, if we go beyond a 12 mappable set, the next triads in the
> sequence would be:
>
> 7/4 - 21/20 - 21/8 (Utonal 6-5-4)
> 2/1 - 7/3 - 42/15 (Utonal 7-6-5)
>
> I do understand why you might propose using 28/27 and 14/9, but
they
> aren't what this scale is 'about.' It's not so much of a 'line
> scale' as it is a 'zig-zag' scale which features the 21/20
interval
> prominantly.
>
> And, on the comments about the other scales:
>
> Thanks for letting me know that the Intersection scale is in the
> Scala archive. I didn't know that. I also didn't know about the
> Ellis duodene. Thank you.
>
> As for the 3,5,9,17 scale: If you do want to use a temperament, I
> think that 46 Equal does the trick with this one.
>
> Thanks again, Gene.
>
> Regards,
>
> Robin
>
>
>
>
>
>
>
>
>
>
>
>
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@>
> wrote:
> >
> > --- In tuning@yahoogroups.com, Robin Perry <jinto83@> wrote:
> >
> > > The ascending 6th and 7th are shown in brackets (as is the
> > ascending octave triad.) The descending octave triad is a
harmonic
> 5-
> > 6-7.
> >
> > This way of describing the scale strikes me as a bit murky.
> >
> > I would say the core of this scale are the two otonal and two
> utonal
> > tetrads it contains; these form a ten-note scale, also
epimorphic,
> > which is an example of a "line scale"; a chain of contiguous
> tetrads
> > in tetrad space along a straight line. For tetrad space:
> >
> > http://www.xenharmony.com/sevlat.htm
> >
> > The key paragraph being:
> >
> > If [a b c] is any triple of integers, then it represents the
major
> > tetrad with root 3^((-a+b+c)/2) 5^((a-b+c)/2) 7^((a+c-c)/2) if
> a+b+c
> > is even, and the minor tetrad with root 3^((-1-a+b+c)/2) 5^((1+a-
> > b+c)/2 7^((1+a+b-c)/2) if a+b+c is odd. Each unit cube
corresponds
> to
> > a either a stellated hexany, also called a tetradekany, or
> > dekatesserany, or else a transposed 7-limit tonality diamond.
> >
> > The ten-note core of your scale in my view consists of the line
of
> > tetrads [0,-2,-1], [0,-1,-1],[0,-1,0],[0,0,0]]. This gives the
> scale
> > [10/9, 7/6, 5/4, 4/3, 10/7, 3/2, 5/3, 7/4, 40/21, 2], which like
> your
> > scale has the nice property of being epimorphic. The question
then
> > is, how to add two more seven-limit notes to get another nice
> > epimorphic scale? I'm not clear why you want to add 21/20 and
> 100/63;
> > why not for instance 28/27 and 14/9?
> >
>

🔗Carl Lumma <clumma@yahoo.com>

Hi Robin,

I somehow missed this thread.

> 5/4 4/3 3/2 5/3 7/4 2/1
> 1/1 10/9 5/4 4/3 3/2 5/3
> (5/3) 7/4 (40/21) 2/1 (2/1)
> 5/6 20/21 1/1 10/9 5/4 4/3
> (10/7) 3/2 100/63 5/3 (5/3)

I'm having a hard time with this chart. It was
formatted really strangely on my screen. I've tried
to piece it back together. I removed the line
beginning 2/1 as I took it to be a copy of the row
beginning 1/1. But the last row is missing an
item, I don't know why some ratios are in parenthesis,
and I don't know how the chart was constructed.

>There are two conjoined 8-tone scales sharing the bold faced
>intervals.

Please be aware that some readers, such as myself, can't
see bold-face in list messages.

-Carl

🔗Carl Lumma <clumma@yahoo.com>

Oh weird. Cool. Sorry for all the messages, I've been
a bit dense here... when I first looked at this thread (in
reverse chrono order) my eyes glazed over.

> And, on the comments about the other scales:
>
> Thanks for letting me know that the Intersection scale is in the
> Scala archive. I didn't know that. I also didn't know about the
> Ellis duodene. Thank you.
>
> As for the 3,5,9,17 scale: If you do want to use a temperament, I
> think that 46 Equal does the trick with this one.
>
> Thanks again, Gene.

Was the message you're replying to here off-list?

-Carl

🔗Carl Lumma <clumma@yahoo.com>

🔗Robin Perry <jinto83@yahoo.com>

Hi Carl,

No, the message was on list. It's one of those threads in there
somewhere...

I originally posted four scales in one message. The 'melodic minor'
one was garbled, so I reposted with corrections. Gene replied and
there it went from there.

Cheers,

Robin

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> --- In tuning@yahoogroups.com, "Robin Perry" <jinto83@> wrote:
> >
> > Hi Gene,
> >
> > Thank you for the examination of this one. I didn't intend to
make
> > murk around this scale. I used the analogy of the melodic minor
> > because it seemed like a good way to convey the pattern inherent
in
> > the scale. Maybe a better way of understanding it would be to
> > stretch it beyond the octave, which is how I first concieved it.
> >
> > It follows a very predictable pattern, which is the method
behind
> > the madness of using the 21/20 and the 100/63 as opposed to
using a
> > 28/27 and a 14/9.
> >
> > I'll put the 'raised 6th' triad first and continue from there
with
> > the pattern:
> >
> > 5/7 - 5/6 - 1/1 (Utonal 7-6-5)
> > 50/63 - 20/21 - 10/9 (Otonal 5-6-7)
> > 5/6 - 1/1 - 5/4 (Utonal 6-5-4)
> > 20/21 - 10/9 - 4/3 (Utonal 7-6-5)
> > 1/1 - 5/4 - 3/2 (Otonal 4-5-6)
> > 10/9 - 4/3 - 5/3 (Utonal 6-5-4)
> > 5/4 - 3/2 - 7/4 (Otonal 5-6-7)
> > 4/3 - 5/3 - 2/1 (Otonal 4-5-6)
> > 3/2 - 7/4 - 21/10 (Utonal 7-6-5)
> > 5/3 - 2/1 - 7/3 (Otonal 5-6-7)
> >
> > The pattern is U-O-U-U-O-U-O-O...
> > The last two tones in each triad are the first two tones
> > in the next to next triad.
>
> Oh weird. Cool. Sorry for all the messages, I've been
> a bit dense here... when I first looked at this thread (in
> reverse chrono order) my eyes glazed over.
>
> > And, on the comments about the other scales:
> >
> > Thanks for letting me know that the Intersection scale is in the
> > Scala archive. I didn't know that. I also didn't know about
the
> > Ellis duodene. Thank you.
> >
> > As for the 3,5,9,17 scale: If you do want to use a temperament,
I
> > think that 46 Equal does the trick with this one.
> >
> > Thanks again, Gene.
>
> Was the message you're replying to here off-list?
>
> -Carl
>

🔗Robin Perry <jinto83@yahoo.com>

Hi Carl,

Yes, I did write that tune with that scale in mind. It's the story
of my (now over, thank god) long distance relationship with my
wife. She was in Australia and I was in the U.S. We are now both
in Australia. I like the scale because it's got such dimension to
it. You can go happy or bluesy, depending on the emphasis.

Thanks... glad you liked it.

Cheers,

Robin

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > Hi Tuners,
> >
> > Manuel Op De Coul was kind enough to Just-ify one of my 12 Equal
> > compositions, "Calling You."
> >
> > http://home.casema.nl/luweiwang/snd/calling_you.mp3
> >
> > For those who are not familiar with Manuel, he is the creator of
> > Scala, a bit of software that everyone into re-tuning should
have:
> >
> > http://www.xs4all.nl/~huygensf/scala/
> >
> > The message posted to this group concerning the scale used in
the
> > tune is re-posted below:
> >
> > Regards,
> >
> > Robin
>
> This totally rocks, which is why I went back through this
> thread. Did you write the piece with this scale in mind?
> Because it works perfectly!
>
> -Carl
>