Topic: Zen Temperament
1 scales
| File | Description | Notes | Period (ยข) |
|---|---|---|---|
| zenop | Lesfip scale derived from Zen, using J O'S "good" intervals, 256/255 | 12 | 1200.0 |
Thread (11 messages)
From: john777music (2011-02-21) Subject: 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.
From: genewardsmith (2011-02-21) Subject: Re: Zen Temperament --- In [email protected], "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
From: john777music (2011-02-21) Subject: Re: Zen Temperament 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 [email protected], "genewardsmith" <genewardsmith@...> wrote: > > > --- In [email protected], "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 >
From: john777music (2011-02-21) Subject: Re: Zen Temperament 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 [email protected], "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 [email protected], "genewardsmith" <genewardsmith@> wrote: > > > > > > --- In [email protected], "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 > > >
From: genewardsmith (2011-02-22) Subject: Re: Zen Temperament --- In [email protected], "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.
From: john777music (2011-02-22) Subject: Re: Zen Temperament 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 [email protected], "genewardsmith" <genewardsmith@...> wrote: > > > > --- In [email protected], "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. >
From: Mike Battaglia (2011-02-22) Subject: Re: [tuning] Re: Zen Temperament 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
From: john777music (2011-02-22) Subject: Re: Zen Temperament 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 [email protected], 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 >
From: Mike Battaglia (2011-02-22) Subject: Re: [tuning] Re: Zen Temperament 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
From: john777music (2011-02-22) Subject: Re: Zen Temperament 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 [email protected], 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 >
From: Mike Battaglia (2011-02-26) Subject: Re: [tuning] Re: Zen Temperament 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