Zen Temperament

Jacques pointed out that my 7 limit Seventh Heaven scale (which crams in as many sevens as possible) contained a few approximate 11 limit intervals. So I decided to build a just scale, based on 3,5,7 prime limits (emphasis on the sevens) with the idea that tempering this just scale might yield all the good approximate 11 and 13 limit intervals (these are: 11/8, 11/7, 11/6 and 13/7 according to my own taste) as well as all the other lower limit good intervals. I'm working only with intervals an octave or less wide.

My Blue Temperament and Blue Just scales do not contain 11/8, 11/7 or 11/6.

Here's the just scale I worked out...

Zen Just
1/1
15/14
9/8
7/6
9/7
4/3
7/5
3/2
14/9
5/3
9/5
27/14
2/1

I chose 27/14 to preserve symmetry: the notes going up from 1/1 are the exact mirror image of the notes going down from 3/2. So either 1/1 or 3/2 can be used as the tonic. There are no just 11 or 13 limit intervals (an octave or less wide) in this scale. This scale has 71 *good* intervals (see the list of good intervals below).

Next I tempered each note by not more than 6.776 cents (256/255) to yield as many good intervals as possible . The result had 90 out of a possible 144 good intervals (an octave or less wide over 12 keys) but a few of the good intervals listed below did not occur.

So I tempered the scale another way so that all of the good intervals occurred at least once. The resulting scale has only 83 good intervals but each and every *good* interval (according to my taste and listed below) occurs at least twice. Here's the scale...

Zen Temperament
0.0
114.1
203.9
265.7
436.2
498.0
587.9
702.0
758.4
890.0
1011.9
1143.5
1200.0

Here's how often each *good* interval occurs over twelve keys in Zen Temperament (within 6.776 cents accuracy)...

9/8....2 times
8/7....2 times
7/6....4
6/5....6
5/4....5
9/7....4
4/3....5
11/8...4
7/5....4
10/7...4
3/2....5
11/7...2
8/5....5
5/3....6
12/7...4
7/4....2
9/5....2
11/6...2
13/7...3
2/1...12

So if you're looking for maximum variety, this scale should be good.

John.

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:

> Zen Temperament
> 0.0
> 114.1
> 203.9
> 265.7
> 436.2
> 498.0
> 587.9
> 702.0
> 758.4
> 890.0
> 1011.9
> 1143.5
> 1200.0

Here's something you might compare it to:

! zenop.scl
Lesfip scale derived from Zen, using J O'S "good" intervals, 256/255
12
!
116.56676
204.62247
266.88263
435.27508
497.53524
585.59096
702.15771
761.02630
885.22578
1016.93193
1141.13141
1200.00000

Gene,

I had a look at your Zenop scale. By my reckoning the Zenop scale does not contain an 11/7 (within 6.776 cents accuracy). The point of my scale was to have *all* good intervals represented.
Also it seems that my Zen Temperament has 83 *good* intervals over 12 keys and the Zenop has only 79 *good* intervals (from my list of good intervals in my last post).

What do you make of this?

Maybe there's a bug in my program. Can you find an 11/7 in the Zenop scale you gave me?

John.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
> --- In tuning@yahoogroups.com, "john777music" <jfos777@> wrote:
>
> > Zen Temperament
> > 0.0
> > 114.1
> > 203.9
> > 265.7
> > 436.2
> > 498.0
> > 587.9
> > 702.0
> > 758.4
> > 890.0
> > 1011.9
> > 1143.5
> > 1200.0
>
> Here's something you might compare it to:
>
>
> ! zenop.scl
> Lesfip scale derived from Zen, using J O'S "good" intervals, 256/255
> 12
> !
> 116.56676
> 204.62247
> 266.88263
> 435.27508
> 497.53524
> 585.59096
> 702.15771
> 761.02630
> 885.22578
> 1016.93193
> 1141.13141
> 1200.00000
>

Gene,

I made a programming error somewhere. The number of good dyads (that I posted earlier) in each scale may not be correct. I'm still sure though that Zen Temperament contains all of the good intervals I mentioned but the Zenop scale has no 11/7.

Igs and Mike, I'll be checking 1/4 Comma Meantone vs Blue Temperament again.

John.

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
>
> Gene,
>
> I had a look at your Zenop scale. By my reckoning the Zenop scale does not contain an 11/7 (within 6.776 cents accuracy). The point of my scale was to have *all* good intervals represented.
> Also it seems that my Zen Temperament has 83 *good* intervals over 12 keys and the Zenop has only 79 *good* intervals (from my list of good intervals in my last post).
>
> What do you make of this?
>
> Maybe there's a bug in my program. Can you find an 11/7 in the Zenop scale you gave me?
>
> John.
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> >
> >
> > --- In tuning@yahoogroups.com, "john777music" <jfos777@> wrote:
> >
> > > Zen Temperament
> > > 0.0
> > > 114.1
> > > 203.9
> > > 265.7
> > > 436.2
> > > 498.0
> > > 587.9
> > > 702.0
> > > 758.4
> > > 890.0
> > > 1011.9
> > > 1143.5
> > > 1200.0
> >
> > Here's something you might compare it to:
> >
> >
> > ! zenop.scl
> > Lesfip scale derived from Zen, using J O'S "good" intervals, 256/255
> > 12
> > !
> > 116.56676
> > 204.62247
> > 266.88263
> > 435.27508
> > 497.53524
> > 585.59096
> > 702.15771
> > 761.02630
> > 885.22578
> > 1016.93193
> > 1141.13141
> > 1200.00000
> >
>

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:

> Maybe there's a bug in my program. Can you find an 11/7 in the Zenop scale you gave me?

No bug, it's just that least squares isn't trying to maximize good intervals, but to minimize squared error. If it gets pulled in a certain way by the optimization process, it may drop an interval such as 11/7 below the threshold, which in this case is 256/255. It might also find other good intervals to take the place of what is dropped, or not, as it works out. A constraint requiring that all the intervals be kept would be a different optimization procedure. Doing that suggests using linear programming.

In this case, the 11/7s have migrated to being 13.833 cents off, but other things are better.

Thanks Gene,

I know nothing about least squares and I intend to post my own optimization algorithm in a few days after I have worked out the details.

I found the bug and it seems that both my Zen Temperament and your Zenop have 79 good intervals each out of a possible 144. Zenop has no 11/7 (within 6.776 cents accuracy) and my Zen Temperament also has 79 good intervals but *all* of the good intervals occur within 6.776 cents accuracy.

Here are the good intervals and how often they occur over 12 keys in Zen Temperament within 6.776 cents...

9/8....2 times
8/7....2 times
7/6....4
6/5....6
5/4....5
9/7....4
4/3....5
11/8...4
7/5....2
10/7...2
3/2....5
11/7...2
8/5....5
5/3....6
12/7...4
7/4....2
9/5....2
11/6...2
13/7...3
2/1...12

To Igs and Mike, I checked 1/4 Comma Meantone vs Blue Temperament again and my original post was good: 1/4 CMT has 90 good intervals (within 6.776 cents accuracy) out of a possible 144 and Blue Temperament has 97 good intervals out of a possible 144 (within 6.776 cents accuracy).

John.

www.johnsmusic7.com

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "john777music" <jfos777@> wrote:
>
> > Maybe there's a bug in my program. Can you find an 11/7 in the Zenop scale you gave me?
>
> No bug, it's just that least squares isn't trying to maximize good intervals, but to minimize squared error. If it gets pulled in a certain way by the optimization process, it may drop an interval such as 11/7 below the threshold, which in this case is 256/255. It might also find other good intervals to take the place of what is dropped, or not, as it works out. A constraint requiring that all the intervals be kept would be a different optimization procedure. Doing that suggests using linear programming.
>
> In this case, the 11/7s have migrated to being 13.833 cents off, but other things are better.
>

On Mon, Feb 21, 2011 at 8:21 PM, john777music <jfos777@...> wrote:
>
> To Igs and Mike, I checked 1/4 Comma Meantone vs Blue Temperament again and my original post was good: 1/4 CMT has 90 good intervals (within 6.776 cents accuracy) out of a possible 144 and Blue Temperament has 97 good intervals out of a possible 144 (within 6.776 cents accuracy).

Feel free to compare these ones as well:

120.00000
192.00000
312.00000
384.00000
504.00000
576.00000
696.00000
816.00000
888.00000
1008.00000
1080.00000
2/1

and

118.51852
192.59259
311.11111
385.18518
503.70370
577.77778
696.29630
814.81482
888.88889
1007.40741
1081.48148
2/1

-Mike

Hi Mike,

I checked out the two scales you posted.

Here's my list of good intervals and how often each occurs over 12 keys in scale A and scale B (within 6.776 cents or 256/255 accuracy)...

Ints..A...B
9/8...0...0
8/7...0...2
7/6...3...3
6/5...9...9
5/4...8...8
9/7...4...4
4/3...11..11
11/8..0...0
7/5...6...6
10/7..6...6
3/2...11..11
11/7..0...0
8/5...8...8
5/3...9...9
12/7..3...3
7/4...0...2
9/5...0...0
11/6..0...0
13/7..0...0
2/1...12..12

Scale A has 90 good intervals (within 6.776 accuracy) and scale B has 94 good intervals. Blue Temperament has 97.

I'm surprised that both scales have 11 good Fourths and 11 good Fifths. Very impressive. Blue Temperament has only 9 good Fourths and Fifths.

I have an idea that every note in a scale should "go" with the tonic (1/1) in melody. My formula for melody is

2/x + 2/y

.....if the result is 0.25 or higher then the melody interval is good. Below 0.25 is bad. Note that this formula applies to sine waves only. I intend to write a program that evaluates melodic intervals with a regular harmonic series in the next few days. The point I'm making is that some of the notes in the scales you gave me may not "go" with the tonic in melody according to a *good* melodic progression. I'll have more to say on this soon.

What are the scales you gave me called? And if it's not too complicated to explain, how were they built?

John.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Mon, Feb 21, 2011 at 8:21 PM, john777music <jfos777@...> wrote:
> >
> > To Igs and Mike, I checked 1/4 Comma Meantone vs Blue Temperament again and my original post was good: 1/4 CMT has 90 good intervals (within 6.776 cents accuracy) out of a possible 144 and Blue Temperament has 97 good intervals out of a possible 144 (within 6.776 cents accuracy).
>
> Feel free to compare these ones as well:
>
> 120.00000
> 192.00000
> 312.00000
> 384.00000
> 504.00000
> 576.00000
> 696.00000
> 816.00000
> 888.00000
> 1008.00000
> 1080.00000
> 2/1
>
> and
>
> 118.51852
> 192.59259
> 311.11111
> 385.18518
> 503.70370
> 577.77778
> 696.29630
> 814.81482
> 888.88889
> 1007.40741
> 1081.48148
> 2/1
>
> -Mike
>

On Tue, Feb 22, 2011 at 2:19 PM, john777music <jfos777@...> wrote:
>
> Scale A has 90 good intervals (within 6.776 accuracy) and scale B has 94 good intervals. Blue Temperament has 97.
>
> I'm surprised that both scales have 11 good Fourths and 11 good Fifths. Very impressive. Blue Temperament has only 9 good Fourths and Fifths.
//snip
> What are the scales you gave me called? And if it's not too complicated to explain, how were they built?

The first one is meantone taken out of 50-equal. The second one is
meantone taken out of 81-equal. Try this one, which is Erv Wilson's
golden meantone:

118.93000
192.42800
311.35800
384.85600
503.78600
577.28400
696.21400
815.14400
888.64200
1007.57200
1081.07000
2/1

And also this one, which is 12 notes of magic temperament, taken out
of 41-equal:

58.53659
321.95122
380.48781
439.02439
497.56098
702.43903
760.97561
819.51220
878.04878
1082.92683
1141.46342
2/1

-Mike

Erv Wilson's Golden Meantone has 94 good intervals (within 6.776 cents or 256/255 accuracy) out of a possible 144. This scale also has 11 good Fourths and Fifths.

As for 12 notes of magic temperament, taken out of 41-equal, this has only 92 good intervals and only seven good Fourths and Fifths. It also has 11 good 5/4's and 8/5's.

Have you (or anyone) got a scale that might beat Blue Temperament's 97 good intervals?

John.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Tue, Feb 22, 2011 at 2:19 PM, john777music <jfos777@...> wrote:
> >
> > Scale A has 90 good intervals (within 6.776 accuracy) and scale B has 94 good intervals. Blue Temperament has 97.
> >
> > I'm surprised that both scales have 11 good Fourths and 11 good Fifths. Very impressive. Blue Temperament has only 9 good Fourths and Fifths.
> //snip
> > What are the scales you gave me called? And if it's not too complicated to explain, how were they built?
>
> The first one is meantone taken out of 50-equal. The second one is
> meantone taken out of 81-equal. Try this one, which is Erv Wilson's
> golden meantone:
>
> 118.93000
> 192.42800
> 311.35800
> 384.85600
> 503.78600
> 577.28400
> 696.21400
> 815.14400
> 888.64200
> 1007.57200
> 1081.07000
> 2/1
>
> And also this one, which is 12 notes of magic temperament, taken out
> of 41-equal:
>
> 58.53659
> 321.95122
> 380.48781
> 439.02439
> 497.56098
> 702.43903
> 760.97561
> 819.51220
> 878.04878
> 1082.92683
> 1141.46342
> 2/1
>
> -Mike
>

On Tue, Feb 22, 2011 at 3:37 PM, Mike Battaglia <battaglia01@...> wrote:
>
> The first one is meantone taken out of 50-equal. The second one is
> meantone taken out of 81-equal. Try this one, which is Erv Wilson's
> golden meantone:

Sorry, to correct, that should have been Kornerup's golden meantone.
Sorry about that and thanks to Jacques Dudon for pointing this out to
me offlist.

-Mike