A cure for 15 equal?

Here is a scale which might help cure the strange and horrible fascination exerted by 15et, which sorry, folks, isn't that well in tune. This is distinctly better.

! augene15br1.scl
Augene[15] with a brat of 1
15
!
93.470871322074944904
186.94174264414988983
213.05825735585011027
306.52912867792505515
400.00000000000000000
493.47087132207494484
586.94174264414988984
613.05825735585011039
706.52912867792505511
800.00000000000000000
893.47087132207494492
986.94174264414988972
1013.0582573558501103
1106.5291286779250552
1200.0000000000000000
! period=2^(1/3), generator=(2/5) 2^(1/3) + 1

>"Here is a scale which might help cure the strange and horrible
fascination exerted by 15et, which sorry, folks, isn't that well in
tune. This is distinctly better.

! augene15br1. scl
Augene[15] with a brat of 1
15
!
93.4708713220749449 04
186.941742644149889 83"

While there's a lot of 15TET which is far from in tune (IE the 5/4 is fairly far off), it does have good matches for 11/10, 6/5, 11/8, 50/33 (alternative 3/2), 5/3, and 11/6. That seems to be the sweet spot in it by far to my ears...the rest seems pretty bastardized (either the ratios from sour dyads relative to the root or form many sour dyads relative to many other notes in the tuning).

When you say "this is distinctly better" my question becomes "better for what dyads or what chords"? There seems to be an overflow of people saying things like "this is excellent for 11,9,7,5...limit" or "this is more in tune" without saying what they are trying to tune to...and I'm fairly sure "anything in the x-limit tonality diamond" does not count as specific as people can like certain, say, 9-limit intervals and hate others. It all seems to kind of leave me with the long "homework assignment" of analyzing all the dyads for, say, "Augene[15]", myself.

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

> When you say "this is distinctly better" my question becomes "better for what dyads or what chords"?

Well, for starters it has a fifth, so you can play major and minor triads, chords which may not interest you but which are important to some people. Its 7/4 isn't as good as what 15et gives you, but it's still much superior to the best 12et can scrape up for it. Same comment for 11/8.

My problem with 15et is that I keep hearing out of tune fifths in it. If people will use it as a strictly no-threes system, I think I would like it much better. Treat 15et xenharmonically, or use something like the scale I gave if you want a serious try at the 5-limit in the mix. 2^(3/5) is one of those frigging intervals which isn't a real fifth but which is close enough to one that there's not much else it can be either, and centering your music around it as people tend to do is going to be a problem for some people.

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

> When you say "this is distinctly better" my question becomes "better for what dyads or what chords"?

By the way, four sharp fifths of 704.377 cents and an exact 11/7 could be used instead as an augene tuning. I think the added bonus of three exact 11/7s in the scale would be a great addition.

Gene>"Well, for starters it has a fifth, so you can play major and minor
triads, chords which may not interest you but which are important to some
people."
Pardon my questioning of common knowledge but, to my ears 1.51515 IE 50/33 makes a pretty solid substitute 5th (unlike, say, 16/11) for things like triads...it's not as relaxed by itself as a dyad but, as Igs showed me, when combined with other tones in chords it quickly begins to sound like a smooth/relaxed 5th.

If you or other absolutely need something within a few cents of 3/2 though...you're right 15TET doesn't quite cut it and your suggestion is probably a better option.

>"2^(3/5) is one of those frigging intervals which isn't a real fifth but which is close enough to one that there's not much else it can be
either, and centering your music around it as people tend to do is going to be a problem for some people."
One thing I really wonder about the other scales you posted...are all the 5ths in those scales also within a few cents of the 3/2 fifth (or only some 5ths but not others)?

_,_._,___

I made a new (but rather, different) version of my scale. This time around I focused on making the best intervals "except" the minor second type intervals...the exact opposite approach I used for the original Infinity scale.

To do this I

A) Took all my favorite larger (IE 6/5 and up) intervals I learned about from making the first Infinity scale (note some of these are not across-the-board agreed upon as consonant). These include
1.2 6/5
1.2222 11/9
1.25 5/4
1.3 13/10
1.35 27/20
1.375 11/8
1.46666 22/15
1.5 3/2
1.515 50/33
1.6 8/5
1.625 13/8
1.66666666 5/3
1.75 7/4
1.8 9/5
1.833333333 11/6
1.875 15/8

B) Made a program which generates random scales and only accepts/prints scales having all possible dyads within 2 octaves with have value within 10 cents of at least one of the above "favorite dyads".

The resulting scales (two out of about 4 or so scale results given by my program, all of which had very similar values) was as follows:
1
1.093593 (about 12/11)
1.191945 (about 6/5)
1.3035 (about 13/10)
1.495 (about 3/2)
1.629453 (about 13/8)
1.743815 (about 7/4)
2

..........and................
1
1.1212 (about 9/8)
1.2487 (about 5/4)
1.336438 (about 4/3)
1.49 (about 3/2)
1.6762 (about 5/3)
1.8269 (about 11/6)
2

****************************************
This program also opens up the possibility of people handing me lists of dyads/intervals they like and my throwing them into the program to get a scale with all dyads within a certain number of cents of those desired dyads.
*****************************

John...you might specifically want to check this out as it to a large extent tries to find the best scale possible disregarding intervals under 6/5...a similar goal (as I understand it) to your NPT scale.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Gene>"Well, for starters it has a fifth, so you can play major and minor
> triads, chords which may not interest you but which are important to some
> people."

> Pardon my questioning of common knowledge but, to my ears 1.51515 IE 50/33 makes a pretty solid substitute 5th (unlike, say, 16/11) for things like triads...it's not as relaxed by itself as a dyad but, as Igs showed me, when combined with other tones in chords it quickly begins to sound like a smooth/relaxed 5th.

It's OK in the relaxed department, it's just not a fifth.

> One thing I really wonder about the other scales you posted...are all the 5ths in those scales also within a few cents of the 3/2 fifth (or only some 5ths but not others)?

Yes.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> I made a new (but rather, different) version of my scale. This time around I focused on making the best intervals "except" the minor second type intervals...the exact opposite approach I used for the original Infinity scale.

Both of the scales you obtained are strictly proper and quite close to scales which turned up in my survey of strictly proper seven note scales in 31 equal. The first turns up in the mothra family, and I gave it the name "inverse quahog". The second was one of the ones I never got around to putting in a fsmily, and I wonder if finishing that project would be a good idea, as finally there seems to be some interest.

Here are the 31 equal versions:

! ra1.scl
random 1 scale, "Inverse quahog"
7
!
154.83870967741935484
309.67741935483870968
464.51612903225806452
696.77419354838709677
851.61290322580645161
967.74193548387096774
1200.0000000000000000

! ra2.scl
random 2
7
!
193.54838709677419355
387.09677419354838710
503.22580645161290323
696.77419354838709677
890.32258064516129032
1045.1612903225806452
1200.0000000000000000

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

>
> My problem with 15et is that I keep hearing out of tune fifths in it. If people will use it as a strictly no-threes system, I think I would like it much better. Treat 15et xenharmonically, or use something like the scale I gave if you want a serious try at the 5-limit in the mix. 2^(3/5) is one of those frigging intervals which isn't a real fifth but which is close enough to one that there's not much else it can be either, and centering your music around it as people tend to do is going to be a problem for some people.
>

Easley Blackwood didn't seem to mind the rough fifth of 15-EDO, and it really seems like a LOT of people on this list find it perfectly agreeable in triads. I think you're the first person I've heard describe it as unpleasant. I have played 15-EDO myself on guitar, WITH distortion, and I definitely did not find it harmonically objectionable. I think the average guitarist might find it a tad rough but still perfectly useable...especially considering that many guitarists out there can't even tune by ear (i.e. they don't notice beating until it's really bad). 15-EDO is a rough-and-rowdy tuning perfect for playing blues or anything based on simple chord progressions.

Where 15-EDO really flops, at least in my experience, is in the melody department...it just seems tough to write a good melody in any of the <10-note MOS scales in 15-EDO over a chord progression, because so many notes in the melodic scales seem to clash with each other. Just about every 15-EDO song I've ever heard sounds good until the melody comes in. I think the familiarity of the chords really underscores the weirdness of the melodic structures. But in some ways, the utter simplicity of it (only 3 more notes than 12, the symmetric Blackwood scale, the "all-fourths" guitar tuning it makes possible) makes up for its melodic weakness. This is why I consider it "commercially-viable".

Igs,

Did you see my 15TETfirstmelody.mp3 example sent on Fri, May 7, 2010 9:35:25 PM?
I found a subset of 15TET very close in many ways to my Infinity scale (the one I think you and Gene said sounded like Porcupine)...and found loads of melodic potential with it.
Again the scale I used is 1 (root) 3 5 8 10 12 14....16 (next octave) in 15TET.

And, of course, I used the 1.515 "substitute 5th" and actually love the brightness of that tone...in some cases I find it better to use than the usual 3/2 5th.

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> Easley Blackwood didn't seem to mind the rough fifth of 15-EDO,

He talks about acoustic guitar helping to hide it.

>and it really seems like a LOT of people on this list find it
>perfectly agreeable in triads.

WHO?

>I think you're the first person I've heard describe it as
>unpleasant.

It's grating. On the edge of recognizability for 3:2.

> Where 15-EDO really flops, at least in my experience, is in
> the melody department...

Wow, I couldn't disagree more. Some of my favorite microtonal
melodies are in 15, including some written by you! The blackwood
comparisons you posted lately... I noted that 15 was best
melodically.

-Carl