changes in intervallic pattern : RE: [tuning] Some 12 Mappable Scales

Hi

About : 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

1- For 1/1 17/16 9/8 20/17 5/4 4/3 17/12 3/2 8/5 17/10 16/9 32/17 2/1

If we assume divisions as :

A= 17/16 = 104.955

B= 18/17 = 98.955

C= 160/153 = 77.448

D= 16/15 = 111.731

Then we have intervallic pattern of ABCAD AB DACBA which has 2 enantiomorph tetrachord and A+B is symmetry division.

2- For 1/1 17/16 9/8 20/17 5/4 4/3 24/17 3/2 8/5 17/10 16/9 32/17 2/1

The pattern will change to :

ABCAD BA DACBA

Now what will happen if we have 2 isomorph tetrachord : ABCAD AB ABCAD

0

104.955

203.91

281.358

386.313

498.044

602.999

701.954

806.909

905.864

983.312

1088.267

1200.000

0: -- 1/1 -- 0.000 unison, perfect prime

1: -- 17/16 -- 104.955 17th harmonic

2: -- 9/8 -- 203.910 major whole tone

3: -- 20/17 -- 281.358 septendecimal augmented second

4: -- 5/4 --386.314 major third

5: -- 4/3 -- 498.045 perfect fourth

6: -- 17/12 -- 603.000 2nd septendecimal tritone

7: -- 3/2 --701.955 perfect fifth

8: -- 51/32 --806.910

9: -- 27/16 -- 905.865 Pythagorean major sixth

10: -- 30/17 -- 983.313

11: -- 15/8 -- 1088.269 classic major seventh

12: -- 2/1 --1200.000 octave

And also : DACBA AB ABCAD

DACBA AB DACBA

And many other pattern by changing position of divisions in tetrachords.

Now , if we consider division of E by (A*B)^(1/2)=101.955 then we have ABCAD EE ABCAD instead of ABCAD AB ABCAD :

0

104.955

203.91

281.358

386.313

498.044

599.999

701.954

806.909

905.864

983.312

1088.267

1200.000

And for ABCAD EE DACBA , we have :

0

104.955

203.91

281.358

386.313

498.044

599.999 = 600.000

701.954

813.685

918.64

996.088

1095.043

1200.000

Which we have to enantiomorph package with a symmetry line between (ABCAD E) and (E DACBA)

Shaahin Mohaajeri

Tombak Player & Researcher , Microtonal Composer

My web site <http://240edo.tripod.com/>

My farsi page in Harmonytalk <http://www.harmonytalk.com/id/908>

My tombak musics in Rhythmweb <http://www.rhythmweb.com/gdg>

My article in DrumDojo <http://www.drumdojo.com/world/persia/tonbak_acoustics.htm>

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf Of Robin Perry
Sent: Sunday, October 22, 2006 11:27 AM
To: tuning@yahoogroups.com
Subject: [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

Hi Mohajeri,

Please bear with me. I'm self-taught in all of this and I don't
really understand where you're going with this analysis. Can you
break it down a bit?

Thanks,

Robin

--- In tuning@yahoogroups.com, "Mohajeri Shahin" <shahinm@...> wrote:
>
> Hi
>
> About : 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
>
>
>
> 1- For 1/1 17/16 9/8 20/17 5/4 4/3 17/12 3/2 8/5 17/10 16/9
32/17 2/1
>
> If we assume divisions as :
>
> A= 17/16 = 104.955
>
> B= 18/17 = 98.955
>
> C= 160/153 = 77.448
>
> D= 16/15 = 111.731
>
> Then we have intervallic pattern of ABCAD AB DACBA which has 2
enantiomorph tetrachord and A+B is symmetry division.
>
>
>
> 2- For 1/1 17/16 9/8 20/17 5/4 4/3 24/17 3/2 8/5 17/10 16/9
32/17 2/1
>
> The pattern will change to :
>
> ABCAD BA DACBA
>
>
>
> Now what will happen if we have 2 isomorph tetrachord : ABCAD AB
ABCAD
>
> 0
>
> 104.955
>
> 203.91
>
> 281.358
>
> 386.313
>
> 498.044
>
> 602.999
>
> 701.954
>
> 806.909
>
> 905.864
>
> 983.312
>
> 1088.267
>
> 1200.000
>
>
>
> 0: -- 1/1 -- 0.000 unison, perfect prime
>
> 1: -- 17/16 -- 104.955 17th harmonic
>
> 2: -- 9/8 -- 203.910 major whole tone
>
> 3: -- 20/17 -- 281.358 septendecimal augmented
second
>
> 4: -- 5/4 --386.314 major third
>
> 5: -- 4/3 -- 498.045 perfect fourth
>
> 6: -- 17/12 -- 603.000 2nd septendecimal tritone
>
> 7: -- 3/2 --701.955 perfect fifth
>
> 8: -- 51/32 --806.910
>
> 9: -- 27/16 -- 905.865 Pythagorean major sixth
>
> 10: -- 30/17 -- 983.313
>
> 11: -- 15/8 -- 1088.269 classic major seventh
>
> 12: -- 2/1 --1200.000 octave
>
>
>
> And also : DACBA AB ABCAD
>
> DACBA AB DACBA
>
> And many other pattern by changing position of divisions in
tetrachords.
>
> Now , if we consider division of E by (A*B)^(1/2)=101.955 then we
have ABCAD EE ABCAD instead of ABCAD AB ABCAD :
>
> 0
>
> 104.955
>
> 203.91
>
> 281.358
>
> 386.313
>
> 498.044
>
> 599.999
>
> 701.954
>
> 806.909
>
> 905.864
>
> 983.312
>
> 1088.267
>
> 1200.000
>
>
>
> And for ABCAD EE DACBA , we have :
>
>
>
> 0
>
> 104.955
>
> 203.91
>
> 281.358
>
> 386.313
>
> 498.044
>
> 599.999 = 600.000
>
> 701.954
>
> 813.685
>
> 918.64
>
> 996.088
>
> 1095.043
>
> 1200.000
>
> Which we have to enantiomorph package with a symmetry line between
(ABCAD E) and (E DACBA)
>
>
>
>
>
>
>
>
>
>
>
>
>
> Shaahin Mohaajeri
>
> Tombak Player & Researcher , Microtonal Composer
>
> My web site <http://240edo.tripod.com/>
>
> My farsi page in Harmonytalk <http://www.harmonytalk.com/id/908>
>
> My tombak musics in Rhythmweb <http://www.rhythmweb.com/gdg>
>
> My article in DrumDojo
<http://www.drumdojo.com/world/persia/tonbak_acoustics.htm>
>
> ________________________________
>
> From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On
Behalf Of Robin Perry
> Sent: Sunday, October 22, 2006 11:27 AM
> To: tuning@yahoogroups.com
> Subject: [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
>