Mailing list post
From: genewardsmith (2010-06-25)
Subject: Highschool scales
--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
>
> I left out a note, the NPJ scale should be:
> 1/1, 15/14, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 9/5, 15/8, 2/1.
And if you leave it in, NPJ is strictly proper, epimorphic, and other good things it wouldn't be if you left it out.
Your scale reminds me somewhat of two twelve-note scales (others in the series having different numbers of notes) I constructed long ago, two of the very first scales I ever considered in my long history of considering scales, which to my surprise have seemingly not been proposed by anyone else and stuck into the Scala directory, so I give them below.
Long, long ago and not every far away when I was in high school, I took the famous Ptolemy-Zarlino JI diatonic
9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2
with intervals 9/8 10/9 16/15 9/8 10/9 9/8 16/15, and split both the 9/8 and 10/9 in half by
9/8 = 15/14 * 21/20 (square denominator to two triangular)
10/9 = 16/15 * 25/24 (triangular denominator to two square)
and used them to expand the scale. The choices involving 10/9 are virtally automatic: between 9/8 and 5/4 we choose between 6/5 and 75/64, so we choose 6/5; between 3/2 and 5/3 we choose between 8/5 and 25/16, and so select 8/5. This gives us a natural nine-note extension which is more or less self-recommending, and which showed up in my survey of 9-note, 5-limit Fokker blacks as "mavlim7", one of the 27/25&135/128 blocks.
Splitting 9/8 and introducing the 7-limit is where it gets interesting. Between 5/3 and 15/8 I have a choice between 7/4 and 25/14, and so of course choose 7/4. I don't have any clear reason to choose either 15/14 or 21/20 between 1 and 9/8, so I decide to construct two scales. Now between 4/3 and 3/2, I must decide between 7/5 and 10/7. But clearly 7/5 goes with 21/20, and 10/7 with 15/14, and so I am done, having constructed the two 12-note "Highschool" scales shown below.
! 09highschool.scl
Nine note Highschool scale
9
!
9/8
6/5
5/4
4/3
3/2
8/5
5/3
15/8
2
! 10highschool1.scl
First 10-note Highschool scale
10
!
21/20
9/8
5/4
4/3
7/5
3/2
5/3
7/4
15/8
2
! 10highschool2.scl
Second 10-note Highschool scale
10
!
15/14
9/8
5/4
4/3
10/7
3/2
5/3
7/4
15/8
2
! 12highschool1.scl
First 12-note Highschool scale
12
!
21/20
9/8
6/5
5/4
4/3
7/5
3/2
8/5
5/3
7/4
15/8
2
! 12highschool2.scl
Second 12-note Highschool scale
12
!
15/14
9/8
6/5
5/4
4/3
10/7
3/2
8/5
5/3
7/4
15/8
2
Full thread (3 messages)
From: john777music (2010-06-25)
Subject: NPJ scale correction
I left out a note, the NPJ scale should be:
1/1, 15/14, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 9/5, 15/8, 2/1.
From: genewardsmith (2010-06-25)
Subject: Highschool scales
--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
>
> I left out a note, the NPJ scale should be:
> 1/1, 15/14, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 9/5, 15/8, 2/1.
And if you leave it in, NPJ is strictly proper, epimorphic, and other good things it wouldn't be if you left it out.
Your scale reminds me somewhat of two twelve-note scales (others in the series having different numbers of notes) I constructed long ago, two of the very first scales I ever considered in my long history of considering scales, which to my surprise have seemingly not been proposed by anyone else and stuck into the Scala directory, so I give them below.
Long, long ago and not every far away when I was in high school, I took the famous Ptolemy-Zarlino JI diatonic
9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2
with intervals 9/8 10/9 16/15 9/8 10/9 9/8 16/15, and split both the 9/8 and 10/9 in half by
9/8 = 15/14 * 21/20 (square denominator to two triangular)
10/9 = 16/15 * 25/24 (triangular denominator to two square)
and used them to expand the scale. The choices involving 10/9 are virtally automatic: between 9/8 and 5/4 we choose between 6/5 and 75/64, so we choose 6/5; between 3/2 and 5/3 we choose between 8/5 and 25/16, and so select 8/5. This gives us a natural nine-note extension which is more or less self-recommending, and which showed up in my survey of 9-note, 5-limit Fokker blacks as "mavlim7", one of the 27/25&135/128 blocks.
Splitting 9/8 and introducing the 7-limit is where it gets interesting. Between 5/3 and 15/8 I have a choice between 7/4 and 25/14, and so of course choose 7/4. I don't have any clear reason to choose either 15/14 or 21/20 between 1 and 9/8, so I decide to construct two scales. Now between 4/3 and 3/2, I must decide between 7/5 and 10/7. But clearly 7/5 goes with 21/20, and 10/7 with 15/14, and so I am done, having constructed the two 12-note "Highschool" scales shown below.
! 09highschool.scl
Nine note Highschool scale
9
!
9/8
6/5
5/4
4/3
3/2
8/5
5/3
15/8
2
! 10highschool1.scl
First 10-note Highschool scale
10
!
21/20
9/8
5/4
4/3
7/5
3/2
5/3
7/4
15/8
2
! 10highschool2.scl
Second 10-note Highschool scale
10
!
15/14
9/8
5/4
4/3
10/7
3/2
5/3
7/4
15/8
2
! 12highschool1.scl
First 12-note Highschool scale
12
!
21/20
9/8
6/5
5/4
4/3
7/5
3/2
8/5
5/3
7/4
15/8
2
! 12highschool2.scl
Second 12-note Highschool scale
12
!
15/14
9/8
6/5
5/4
4/3
10/7
3/2
8/5
5/3
7/4
15/8
2
From: john777music (2010-06-26)
Subject: Re: Highschool scales
Thanks Gene,
nice to get a bit of positive feedback,
John.
--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
>
>
>
> --- In tuning@yahoogroups.com, "john777music" <jfos777@> wrote:
> >
> > I left out a note, the NPJ scale should be:
> > 1/1, 15/14, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 9/5, 15/8, 2/1.
>
> And if you leave it in, NPJ is strictly proper, epimorphic, and other good things it wouldn't be if you left it out.
>
> Your scale reminds me somewhat of two twelve-note scales (others in the series having different numbers of notes) I constructed long ago, two of the very first scales I ever considered in my long history of considering scales, which to my surprise have seemingly not been proposed by anyone else and stuck into the Scala directory, so I give them below.
>
> Long, long ago and not every far away when I was in high school, I took the famous Ptolemy-Zarlino JI diatonic
>
> 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2
>
> with intervals 9/8 10/9 16/15 9/8 10/9 9/8 16/15, and split both the 9/8 and 10/9 in half by
>
> 9/8 = 15/14 * 21/20 (square denominator to two triangular)
>
> 10/9 = 16/15 * 25/24 (triangular denominator to two square)
>
> and used them to expand the scale. The choices involving 10/9 are virtally automatic: between 9/8 and 5/4 we choose between 6/5 and 75/64, so we choose 6/5; between 3/2 and 5/3 we choose between 8/5 and 25/16, and so select 8/5. This gives us a natural nine-note extension which is more or less self-recommending, and which showed up in my survey of 9-note, 5-limit Fokker blacks as "mavlim7", one of the 27/25&135/128 blocks.
>
> Splitting 9/8 and introducing the 7-limit is where it gets interesting. Between 5/3 and 15/8 I have a choice between 7/4 and 25/14, and so of course choose 7/4. I don't have any clear reason to choose either 15/14 or 21/20 between 1 and 9/8, so I decide to construct two scales. Now between 4/3 and 3/2, I must decide between 7/5 and 10/7. But clearly 7/5 goes with 21/20, and 10/7 with 15/14, and so I am done, having constructed the two 12-note "Highschool" scales shown below.
>
> ! 09highschool.scl
> Nine note Highschool scale
> 9
> !
> 9/8
> 6/5
> 5/4
> 4/3
> 3/2
> 8/5
> 5/3
> 15/8
> 2
>
> ! 10highschool1.scl
> First 10-note Highschool scale
> 10
> !
> 21/20
> 9/8
> 5/4
> 4/3
> 7/5
> 3/2
> 5/3
> 7/4
> 15/8
> 2
>
> ! 10highschool2.scl
> Second 10-note Highschool scale
> 10
> !
> 15/14
> 9/8
> 5/4
> 4/3
> 10/7
> 3/2
> 5/3
> 7/4
> 15/8
> 2
>
> ! 12highschool1.scl
> First 12-note Highschool scale
> 12
> !
> 21/20
> 9/8
> 6/5
> 5/4
> 4/3
> 7/5
> 3/2
> 8/5
> 5/3
> 7/4
> 15/8
> 2
>
> ! 12highschool2.scl
> Second 12-note Highschool scale
> 12
> !
> 15/14
> 9/8
> 6/5
> 5/4
> 4/3
> 10/7
> 3/2
> 8/5
> 5/3
> 7/4
> 15/8
> 2
>