John's Temperament v1

Here's my best shot (so far) at a twelve tone tempered scale. I started with the following symmetric/mirrored "just" (except for the sqrt2 tritone) scale...

1/1
16/15
9/8
6/5
5/4
4/3
sqrt2
3/2
8/5
5/3
16/9
15/8
2/1

Michael suggested using 10/9 instead of 9/8 and 9/5 instead of 16/9 which is clearly better so I used these notes instead. I tempered the scale so as to get as many good dyads/intervals as possible from the following list of "good" (to me at least) intervals:

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

I set the maximum deviation from a 'true' interval at 6.775877 cents (256/255). To preserve symmetry if I raised the nth note *up* from 1/1 by 'x' cents then I lowered the nth note *down* from 2/1 by 'x' cents and I left the tritone alone.

Here's what I got...

0.0 cents
115.119
188.333
315.641
388.008
498.045
600.0
701.955
811.992
884.359
1011.67
1084.88
1200.0

If you consider the 13 possible intervals over an octave (the unison and octave and eleven intervals in between) in 12 keys you have 156 (13*12) possible intervals and of these, with my scale, 94 of these are good. That's a hit rate of just over 60%, not too bad.

I haven't checked yet to see if some of the more exotic intervals (e.g. 11/8, 11/7, 11/6, 13/7) occur, more on this tomorrow.

John.

Interesting...the only dyads I don't agree on as "best" from your list are
9/7
11/8
13/7
To figure out why take 2 over those fractions and listen the the result of those dyads. For example 2 / (9/7) = 14/9 (and 14/9 IMVHO sounds very un-confident).

>"I set the maximum deviation from a 'true' interval at 6.775877 cents
(256/255). To preserve symmetry if I raised the nth note *up* from 1/1
by 'x' cents then I lowered the nth note *down* from 2/1 by 'x' cents
and I left the tritone alone."

Wow that's one strict standard...my scale allows about 9 cents from "true" intervals and 12TET allows more like a maximum of 13 cents off (or 15 cents off if you count the 1.6875-ish interval as a really bad 5/3).

>"If you consider the 13 possible intervals over an octave (the unison
and octave and eleven intervals in between) in 12 keys you have 156
(13*12) possible intervals and of these, with my scale, 94 of these are
good. That's a hit rate of just over 60%, not too bad."

I get the feeling 12TET gets about 30% of the intervals leaning on "wrong"...namely the sour major 3rd, skewed 6th at about 1.68, and sour minor second. So if you're getting 60% I'd say you are at least fairly competitive with the 70% guess-timate I'd give 12TET. It still amazes me you are trying to hit "% of correct interval" measures with all 12 notes, though, as 12TET already does that really well (and I personally feel optimizing subsets of 7-9 would yield much more substantial results vs. the diatonic scale than an alternative 12-tone scale would to 12TET).

I also hope you hit some of the more exotic intervals...because IMVHO that's a huge part of why people may like your scale over 12TET.

Best of luck!

As far as 12-tone well temperaments go it isn't the end of the world,
although the fifth from G-D is pretty out.

-Mike

On Mon, May 17, 2010 at 5:51 PM, john777music <jfos777@...> wrote:
>
>
>
> Here's my best shot (so far) at a twelve tone tempered scale. I started with the following symmetric/mirrored "just" (except for the sqrt2 tritone) scale...
>
> 1/1
> 16/15
> 9/8
> 6/5
> 5/4
> 4/3
> sqrt2
> 3/2
> 8/5
> 5/3
> 16/9
> 15/8
> 2/1
>
> Michael suggested using 10/9 instead of 9/8 and 9/5 instead of 16/9 which is clearly better so I used these notes instead. I tempered the scale so as to get as many good dyads/intervals as possible from the following list of "good" (to me at least) intervals:
>
> 1/1
> 9/8
> 8/7
> 7/6
> 6/5
> 5/4
> 9/7
> 4/3
> 11/8
> 7/5
> 10/7
> 3/2
> 11/7
> 8/5
> 5/3
> 12/7
> 7/4
> 9/5
> 11/6
> 13/7
> 2/1
>
> I set the maximum deviation from a 'true' interval at 6.775877 cents (256/255). To preserve symmetry if I raised the nth note *up* from 1/1 by 'x' cents then I lowered the nth note *down* from 2/1 by 'x' cents and I left the tritone alone.
>
> Here's what I got...
>
> 0.0 cents
> 115.119
> 188.333
> 315.641
> 388.008
> 498.045
> 600.0
> 701.955
> 811.992
> 884.359
> 1011.67
> 1084.88
> 1200.0
>
> If you consider the 13 possible intervals over an octave (the unison and octave and eleven intervals in between) in 12 keys you have 156 (13*12) possible intervals and of these, with my scale, 94 of these are good. That's a hit rate of just over 60%, not too bad.
>
> I haven't checked yet to see if some of the more exotic intervals (e.g. 11/8, 11/7, 11/6, 13/7) occur, more on this tomorrow.
>
> John.
>
>

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

> 0.0 cents
> 115.119
> 188.333
> 315.641
> 388.008
> 498.045
> 600.0
> 701.955
> 811.992
> 884.359
> 1011.67
> 1084.88
> 1200.0

If you get rid of that Godawful sqrt(2) tritone this begins to look a lot like 12 notes of meantone temperament. A standard gamut from
Db to F# will do ya, or if you must, from Db to B, and stick in a sqrt(2) in place of F#.

Michael,

I've been testing my scale (John's Temperament v1) and was pleased and surprised to find a few of the more exotic intervals occurring. If you take the eleven intervals (in a 12 tone system) bewteen the unison and octave exclusively, over 12 keys you have 132 intervals (11*12) and with my scale 70 of these are good.

Here is my list of "good" (to me at least) intervals and how often they occur among the 132 intervals mentioned above. The exotic intervals are markes with ***...

9/8....5 times
8/7....1 time
7/6....2 times
6/5....6
5/4....6
9/7....2 ***
4/3....8
11/8...0 ***
7/5....4
10/7...4
3/2....8
11/7...3 ***
8/5....6
5/3....6
12/7...2
7/4....1
9/5....4
11/6...0 ***
13/7...2 ***

Of the 19 intervals listed here, my scale only misses two of them: the 11/8 and 11/6. The bonuses are the 9/7, 11/7 and 13/7. All of these are within 6.775877 cents accuracy.

What do you make of this?

John.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Interesting...the only dyads I don't agree on as "best" from your list are
> 9/7
> 11/8
> 13/7
> To figure out why take 2 over those fractions and listen the the result of those dyads. For example 2 / (9/7) = 14/9 (and 14/9 IMVHO sounds very un-confident).
>
> >"I set the maximum deviation from a 'true' interval at 6.775877 cents
> (256/255). To preserve symmetry if I raised the nth note *up* from 1/1
> by 'x' cents then I lowered the nth note *down* from 2/1 by 'x' cents
> and I left the tritone alone."
>
> Wow that's one strict standard...my scale allows about 9 cents from "true" intervals and 12TET allows more like a maximum of 13 cents off (or 15 cents off if you count the 1.6875-ish interval as a really bad 5/3).
>
> >"If you consider the 13 possible intervals over an octave (the unison
> and octave and eleven intervals in between) in 12 keys you have 156
> (13*12) possible intervals and of these, with my scale, 94 of these are
> good. That's a hit rate of just over 60%, not too bad."
>
> I get the feeling 12TET gets about 30% of the intervals leaning on "wrong"...namely the sour major 3rd, skewed 6th at about 1.68, and sour minor second. So if you're getting 60% I'd say you are at least fairly competitive with the 70% guess-timate I'd give 12TET. It still amazes me you are trying to hit "% of correct interval" measures with all 12 notes, though, as 12TET already does that really well (and I personally feel optimizing subsets of 7-9 would yield much more substantial results vs. the diatonic scale than an alternative 12-tone scale would to 12TET).
>
> I also hope you hit some of the more exotic intervals...because IMVHO that's a huge part of why people may like your scale over 12TET.
>
> Best of luck!
>

>"Of the 19 intervals listed here, my scale only misses two of them: the
11/8 and 11/6...
What do you make of this?"
For one, I don't consider 11/8 good in the first place. I figured this out as 2/1 over 11/8 = 16/11, a terrible tritone-ish fifth-like interval. And since you have an octave-based scale, having a 11/8 would mean also having a 16/11. So be glad you didn't get that interval. :-)

Missing 11/6 and hitting 13/7 twice though, I feel is a mistake because I feel 11/6 is easily the superior 7th interval (though the 9/5 interval you ran into isn't bad either and I'd even say 18/11 is pretty good as a 7th).
.....
Note 2/1 over 13/7 = about 14/13...which seems too close to the maximum critical band dissonance point around 17/16 to my ears. That's the real thing I keep running into with your scales...there are few real obvious neutral second options...and thus many of your semi-tones that result from your "stretched 7ths" become too close to be used in many chords (minus things like inverted 7th chords).

I realize getting the 11/6 and often the 9/5 requires the use of neutral seconds (that's how my latest scales get the 11/6)...but this comes back to my point about 12+ tone somewhat equally spaced tunings being somewhat unrealistic. And, again, I keep on thinking 7-9 tones or so would enable much better dyadic ratios.

For an example of neutral second chords, look at the attached mp3. If that doesn't change you mind on whether neutral seconds can be at least semi-consonant I don't know what can. :-)
****************
I think that's the most major downside to your tuning...though it has a few new good intervals it by and large stays on the same "mean-tone" type constructs that ultimately help form 12TET. So it's purity is good but it's new-ness...maybe not so much.
****************

Meanwhile the good news...you hit the more exotic 9/7 and 11/7 ratios very well and managed to hit 5/3 instead of 12TET's 37/22-ish interval for the 6th (5/3 sounds significantly better IMVHO...it's almost as if 12TET sounds 15 cents or so off).

I'll also take a shot at composing with your new scale as well when I get the time.

Some questions:
1) If those are your good intervals...what to your "bad" intervals look like and what notes are they formed between?
2) How come you are still using 9/8 instead of the 10/9 swap I suggested to straighten out your sour 5th? (again I keep thinking 10/9 = nearly neutral second...and how much more flexibility you'd have if you allowed 12/11, 11/10, and 10/9 to be considered on the "good list" of intervals)

Michael,

you said:"For one, I don't consider 11/8 good in the first place. I figured this out as 2/1 over 11/8 = 16/11, a terrible tritone-ish fifth-like interval. And since you have an octave-based scale, having a 11/8 would mean also having a 16/11. So be glad you didn't get that interval."

I know very little about conventional music theory, for example I don't know what a neutral second is, all I know is numbers. I've heard "octave equivalence" mentioned a few times and I'm not sure what it means but it seems to me that it means that 2/1 over 11/8 (16/11) should sound as good as 11/8. To me this is not the case and just because 16/11 sounds bad it shouldn't mean that 11/8 also sounds bad.

For example, to me the 15/8 is bad but the 15/4 is good.

You asked me (1) what are my bad intervals? Over a one octave range the bad intervals are any and all intervals that are not part of my list of 21 good intervals. This is based on my calculator (have you tried the latest version 7.2 yet?). I have tested all of the 21 "good" intervals I listed before and to my ear they all sound acceptable.

(2) I *did* use 10/9 instead of 9/8 in my scale before I tempered it. So the 1st note was 1/1, the 2nd was 16/15 and the 3rd was 10/9. BTW I have tested 10/9 a few times and I just don't like it. For me the narrowest good interval (in harmony) is 9/8.

I tried to play your mp3 attachment but nothing happened when I clicked on it. I'd still like to hear it, maybe you could uploed it to the "Files" section.

I also tried your Infinity scale and I like it. To me it sounds like the first 4 notes and the last four notes sound like two complete and independent groups.

John.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Some questions:
> 1) If those are your good intervals...what to your "bad" intervals look like and what notes are they formed between?
> 2) How come you are still using 9/8 instead of the 10/9 swap I suggested to straighten out your sour 5th? (again I keep thinking 10/9 = nearly neutral second...and how much more flexibility you'd have if you allowed 12/11, 11/10, and 10/9 to be considered on the "good list" of intervals)
>

>"I know very little about conventional music theory, for example I don't
know what a neutral second is"
I'd say the general range of from 12/11 to 10/9 works well as a neutral second range.

>"For example, to me the 15/8 is bad but the 15/4 is good."
However to get the 15/4 you need to have a 15/8 in the octave below (at least in an octave-based scale). Hence you can't have the good without the bad...in that case and many others. Same goes in that if you have 15/8 (even if you did like it), you also get 16/15 by virtue of the octave period (2/1 over 15/8...the "octave inversion"). So again, before you consider an interval "good", I'd highly recommend checking what the above mentioned "octave inversion" is. Otherwise you may be unknowingly ensuring you have bad intervals elsewhere in your scale.

>"You asked me (1) what are my bad intervals? Over a one octave range the
bad intervals are any and all intervals that are not part of my list of
21 good intervals."
Right but I meant...which starting notes do those bad intervals occur on?

>"(2) I *did* use 10/9 instead of 9/8 in my scale before I tempered it. So the 1st note was 1/1, the 2nd was 16/15 and the 3rd was 10/9. "
Ah, ok...sorry I'm not good at quickly analyzing cent values and you gave 9/8 in the fraction-based listing of your scale so it confused me.

>"BTW I have tested 10/9 a few times and I just don't like it. For me the narrowest good interval (in harmony) is 9/8."
Ok but, as long as you stick with such a limitation, I question how much more pure your scale is going to sound than things like 1/4 comma mean-tone and other scales which, in general, do a great job of purifying intervals of 9/8 and higher. Unless you are simply going for different good intervals rather than more "pure" ones, in which case I'd suggest going for more exotic intervals.

>"I also tried your Infinity scale and I like it. To me it sounds like the first 4 notes and the last four notes sound like two complete and
independent groups."

Glad you enjoyed it but, dare I ask...which version of it did you try? I keep updating it slightly so I have about 6 versions floating around...the latest of which (when tempered) is
----latest version of "Infinity" scale
1
1.09359
1.2261
1.34095
1.463
1.63492
1.83316
2

Far as the mp3, yes I'll try and upload it to the files section under my folder as of now...and I'm still fairly convinced it will make you think twice about your "smaller than 9/8 doesn't usually work well in harmony" as it uses "illegal" 3+ note chains of intervals under 9/8 as chords! ;-)
The latest version achieves 100% of notes within 10 cents of my list of favored dyads and a good 80% or so within 7 cents of them...then again, I consider 12/11 through 10/9 as good (unlike yourself)...so I have a bit more slack on "purity".

The Infinity scale I tried was: 1/1, 11/10, 5/4, 11/8, 3/2, 5/3, 11/6, 2/1.

I've thought about it and I think that, using my approach, I don't need to consider octave inversions when I'm working out a scale.

"which starting notes do those bad intervals occur on?" That would take some time to figure out.

John.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> >"I know very little about conventional music theory, for example I don't
> know what a neutral second is"
> I'd say the general range of from 12/11 to 10/9 works well as a neutral second range.
>
> >"For example, to me the 15/8 is bad but the 15/4 is good."
> However to get the 15/4 you need to have a 15/8 in the octave below (at least in an octave-based scale). Hence you can't have the good without the bad...in that case and many others. Same goes in that if you have 15/8 (even if you did like it), you also get 16/15 by virtue of the octave period (2/1 over 15/8...the "octave inversion"). So again, before you consider an interval "good", I'd highly recommend checking what the above mentioned "octave inversion" is. Otherwise you may be unknowingly ensuring you have bad intervals elsewhere in your scale.
>
> >"You asked me (1) what are my bad intervals? Over a one octave range the
> bad intervals are any and all intervals that are not part of my list of
> 21 good intervals."
> Right but I meant...which starting notes do those bad intervals occur on?
>
> >"(2) I *did* use 10/9 instead of 9/8 in my scale before I tempered it. So the 1st note was 1/1, the 2nd was 16/15 and the 3rd was 10/9. "
> Ah, ok...sorry I'm not good at quickly analyzing cent values and you gave 9/8 in the fraction-based listing of your scale so it confused me.
>
> >"BTW I have tested 10/9 a few times and I just don't like it. For me the narrowest good interval (in harmony) is 9/8."
> Ok but, as long as you stick with such a limitation, I question how much more pure your scale is going to sound than things like 1/4 comma mean-tone and other scales which, in general, do a great job of purifying intervals of 9/8 and higher. Unless you are simply going for different good intervals rather than more "pure" ones, in which case I'd suggest going for more exotic intervals.
>
> >"I also tried your Infinity scale and I like it. To me it sounds like the first 4 notes and the last four notes sound like two complete and
> independent groups."
>
> Glad you enjoyed it but, dare I ask...which version of it did you try? I keep updating it slightly so I have about 6 versions floating around...the latest of which (when tempered) is
> ----latest version of "Infinity" scale
> 1
> 1.09359
> 1.2261
> 1.34095
> 1.463
> 1.63492
> 1.83316
> 2
>
> Far as the mp3, yes I'll try and upload it to the files section under my folder as of now...and I'm still fairly convinced it will make you think twice about your "smaller than 9/8 doesn't usually work well in harmony" as it uses "illegal" 3+ note chains of intervals under 9/8 as chords! ;-)
> The latest version achieves 100% of notes within 10 cents of my list of favored dyads and a good 80% or so within 7 cents of them...then again, I consider 12/11 through 10/9 as good (unlike yourself)...so I have a bit more slack on "purity".
>

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

I tried another brower and still didn't get the attachment. I suspect you can't get it except via email. If so, would people please quit using friggin attachments, and if not, would someone explain how to get them?

Michael,

when building a scale I consider all intervals less than an octave wide that occur over a *TWO* octave range (1/1 to 4/1) so I think this covers all octave inversions, am I right?

John.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> >"I know very little about conventional music theory, for example I don't
> know what a neutral second is"
> I'd say the general range of from 12/11 to 10/9 works well as a neutral second range.
>
> >"For example, to me the 15/8 is bad but the 15/4 is good."
> However to get the 15/4 you need to have a 15/8 in the octave below (at least in an octave-based scale). Hence you can't have the good without the bad...in that case and many others. Same goes in that if you have 15/8 (even if you did like it), you also get 16/15 by virtue of the octave period (2/1 over 15/8...the "octave inversion"). So again, before you consider an interval "good", I'd highly recommend checking what the above mentioned "octave inversion" is. Otherwise you may be unknowingly ensuring you have bad intervals elsewhere in your scale.
>
> >"You asked me (1) what are my bad intervals? Over a one octave range the
> bad intervals are any and all intervals that are not part of my list of
> 21 good intervals."
> Right but I meant...which starting notes do those bad intervals occur on?
>
> >"(2) I *did* use 10/9 instead of 9/8 in my scale before I tempered it. So the 1st note was 1/1, the 2nd was 16/15 and the 3rd was 10/9. "
> Ah, ok...sorry I'm not good at quickly analyzing cent values and you gave 9/8 in the fraction-based listing of your scale so it confused me.
>
> >"BTW I have tested 10/9 a few times and I just don't like it. For me the narrowest good interval (in harmony) is 9/8."
> Ok but, as long as you stick with such a limitation, I question how much more pure your scale is going to sound than things like 1/4 comma mean-tone and other scales which, in general, do a great job of purifying intervals of 9/8 and higher. Unless you are simply going for different good intervals rather than more "pure" ones, in which case I'd suggest going for more exotic intervals.
>
> >"I also tried your Infinity scale and I like it. To me it sounds like the first 4 notes and the last four notes sound like two complete and
> independent groups."
>
> Glad you enjoyed it but, dare I ask...which version of it did you try? I keep updating it slightly so I have about 6 versions floating around...the latest of which (when tempered) is
> ----latest version of "Infinity" scale
> 1
> 1.09359
> 1.2261
> 1.34095
> 1.463
> 1.63492
> 1.83316
> 2
>
> Far as the mp3, yes I'll try and upload it to the files section under my folder as of now...and I'm still fairly convinced it will make you think twice about your "smaller than 9/8 doesn't usually work well in harmony" as it uses "illegal" 3+ note chains of intervals under 9/8 as chords! ;-)
> The latest version achieves 100% of notes within 10 cents of my list of favored dyads and a good 80% or so within 7 cents of them...then again, I consider 12/11 through 10/9 as good (unlike yourself)...so I have a bit more slack on "purity".
>

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
>
> The Infinity scale I tried was: 1/1, 11/10, 5/4, 11/8, 3/2, >5/3, 11/6, 2/1.

That's a very nice scale. If you forget about trying to make the tuning a wind-chime tuning and accept that different intervals are going to be used in different ways in different contexts, you can greatly enrichen this scale by doing things like dropping a 22/21 above the 1/1 and the 3/2 (like a Db and an Ab, at 22/21 and 11/7, but your "Db" is 8/7 above your "Bb", trippy but sweet).

Then you'll have a strong and lovely alternative to a leading tone, tonicizing Do or Sol. Like this: 1/1, 11/12 (the 11/6 below the 1/1), 22/21, 1/1. C, Bb, Db, C. Kind of wraps around the tonic rather than simply rising to it.

The 11/7 will work this way tonicizing/emphasizing Sol, you can add a 33/28 to the scale, very nice dark minor third, to get the same action on your 11/10 "re", and so on and so forth.

This kind of wrap-around is common in modal musics, but it sounds great and it works, and it will let you anchor things at those times you wish them to be anchored, without a leading tone or V-I. You've got to have compositional tools that aren't specific to 12-tET or meantone or major/minor triadic thinking, otherwise you'll always be falling into wonky and wobbly sounding tunes, or be limited to a few crude sequences, with these xenharmonic tunings.

-Cameron Bobro

John>"when building a scale I consider all intervals less than an octave wide
that occur over a *TWO* octave range (1/1 to 4/1) so I think this
covers all octave inversions, am I right?"
Yes. However doing that check in limiting your original set of desirable intervals saves you the hassle of checking the 2 over (original interval) dyads for ANY scale you use which uses that dyad list. Think of it as writing a re-usable function instead of redundant code in a computer program.

John>"I've thought about it and I think that, using my approach, I don't need
to consider octave inversions when I'm working out a scale."
Well, doing so would likely cut the number of bad intervals possible almost in half...considering that for every interval there is another interval equal to 2 over that interval. Like it or not, every time you have an 11/8, you also have a 16/11 between the top of the 11/8 dyad and an octave over the root tone of the 11/8. This is perhaps the main reason the newer versions of the Infinity scale are different than the old ones, the other being the use of alternative 5ths to gain a higher percentage of 5ths that are "within 10 cents of perfect intervals".

>"which starting notes do those bad intervals occur on?" That would take some time to figure out."
Right, but if you worked out that info gave me it I could easily use if to help you optimize the scale.