High thirds and categorical perceptions

"'--Where am I? Why do I do this?--'these are natural inquires. They
have assailed thousands before our day; they will afflict thousands in
years to come and perhaps there is no form of interrogation so loaded
with subtle torture,--unless it is to be asked for a light in a
strange depot by a man you've just selected out of seventeen thousand
as the one man most likely to have a match." (Charles E. Ives,
postlude, _114 SONGS_)

The following scale is one that I use in the second verse of an
arrangement I did of Lowell Mason's hymn "WORK FOR THE NIGHT IS
COMING":

1/1, 8/7, 9/7, 7/5, 32/21, 12/7, 27/14, 2/1

It's a peculiar scale that I found by ear, and it can be analytically
looked at in a number of different ways... for instance, I find this
scale to most sound like a collection of 3s 7s and 11s, and as such it
could also be looked at as say the IV of a:

11/6
\ ,21/16
\ ,
1/1---------3/2---------9/8--------27/16-------81/64

where the 7/5 is flattened by a 441/440:

88/63
\ ,1/1
\ ,
32/21--------8/7--------12/7---------9/7--------27/14

But really this scale seems to be just the type of JI scale that could
benefit from tempering; say 17-tET where the scale is still strictly
proper, the 243/242 disappears, and a series of apparently beneficial
compromises are struck round the 64/63:

0 212 424 494 706 918 1059 1200
0 212 282 494 706 847 988 1200
0 71 282 494 635 776 988 1200
0 212 424 565 706 918 1129 1200
0 212 353 494 706 918 988 1200
0 141 282 494 706 776 988 1200
0 141 353 565 635 847 1059 1200

Only problem is, no matter what I seemed to try for a more
analytically manageable compromise, none of them ever seemed to sound
nearly as good as the original scale of 1/1, 8/7, 9/7, 7/5, 32/21,
12/7, 27/14, 2/1, to me. And while it has been my experience that once
one has gotten used and attached to a music and it's tuning -- and if
nothing is aurally manifesting itself as a tuning related problem --
anything else you try is bound to be up against it... However, I
really don't think that was the whole issue here.

Earlier I said that I find this scale to most sound like a collection
of 3s 7s and 11s, and if you look at the 1/1, 9/8, 81/64, 21/16, 3/2,
27/16, 11/6, 2/1 as a sort of pseudo V where the neutralish 11/6 is
seen as a b7th (note that this would then make the original scale the
pseudo I), then you could look at this as a Pythagorean chain of 3/2s
offset by 64/63s and 33/32s:

S = add a 64/63
s = subtract a 64/63
U = add a 33/32
u = subtract a 33/32
US = add a 22/21
su = subtract a 22/21

1 1 9S8 81S64 4US3 3S2 27S16 243S128 2 1
1 1 9 8 32U27 4 3 3 2 27 16 16s9 2 1
1 1 256U243 32 27 4 3 3 2 128s81 16 9 2 1
1 1 9u8 81u64 729u512 3su2 27u16 243u128 2 1
1 1 9 8 81 64 4s3 3 2 27 16 16U9 2 1
1 1 9 8 32s27 4 3 3 2 128U81 16 9 2 1
1 1 256s243 32 27 4 3 1024U729 128 81 16 9 2 1
1 1 9S8 81S64 4US3 3S2 27S16 243S128 2 1

If you were to then ignore the 441/440 that separates the offset
combined commas from a 7/5 and a 10/7, and call the 88/63 as a 7/5,
and the 63/44 as a 10/7, you would then have:

1/1 8/7 9/7 7/5 32/21 12/7 27/14 2/1
1/1 9/8 11/9 4/3 3/2 27/16 7/4 2/1
1/1 12/11 32/27 4/3 3/2 14/9 16/9 2/1
1/1 12/11 11/9 11/8 10/7 18/11 11/6 2/1
1/1 9/8 81/64 21/16 3/2 27/16 11/6 2/1
1/1 9/8 7/6 4/3 3/2 18/11 16/9 2/1
1/1 28/27 32/27 4/3 16/11 128/81 16/9 2/1
1/1 8/7 9/7 7/5 32/21 12/7 27/14 2/1

This offers a much better view of the array of 3s 7s and 11s that I
most hear this scale to be.

Dan

[Dan Stearns, TD 519.11]
>The following scale is one that I use in the second verse of an
>arrangement I did of Lowell Mason's hymn "WORK FOR THE NIGHT IS
>COMING":
>
>1/1, 8/7, 9/7, 7/5, 32/21, 12/7, 27/14, 2/1

Fascinating.

>It's a peculiar scale that I found by ear, and it can be analytically
>looked at in a number of different ways... for instance, I find this
>scale to most sound like a collection of 3s 7s and 11s, and as such it
>could also be looked at as say the IV of a:
>
>11/6
> \ ,21/16
> \ ,
> 1/1---------3/2---------9/8--------27/16-------81/64
...
>If you were to then ignore the 441/440 that separates the offset
>combined commas from a 7/5 and a 10/7, and call the 88/63 as a 7/5,
>and the 63/44 as a 10/7, you would then have:
>
>1/1 8/7 9/7 7/5 32/21 12/7 27/14 2/1
>1/1 9/8 11/9 4/3 3/2 27/16 7/4 2/1
>1/1 12/11 32/27 4/3 3/2 14/9 16/9 2/1
>1/1 12/11 11/9 11/8 10/7 18/11 11/6 2/1
>1/1 9/8 81/64 21/16 3/2 27/16 11/6 2/1
>1/1 9/8 7/6 4/3 3/2 18/11 16/9 2/1
>1/1 28/27 32/27 4/3 16/11 128/81 16/9 2/1
>1/1 8/7 9/7 7/5 32/21 12/7 27/14 2/1
>
>This offers a much better view of the array of 3s 7s and 11s that I
>most hear this scale to be.
...

You could use 31-tET, with steps of 6, 5, 4, 4, 5, 5, 2 (still strictly proper). Call it A#, C, D, Dx(Fbb), F, G, A. Of course all those Pythagorean thirds and sixths will become excellent ratios of 5 (while the 2:3's and 8:9's are not so good).

Assuming that's unacceptable, we can do something that isn't an ET (EDO) and doesn't go all the way to meantone. We can _microtemper_ it. i.e. Distribute the 440:441 (3.93c) and 242:243 (7.14c) in such a way that we get Just 5:7's and we don't care about 4:5, 5:6, 10:11, 7:11.

To optimally (minimax) distribute the 242:243, as per my web page
http://dkeenan.com/Music/DistributingCommas.htm
narrow the 2:3's by 1.43c (1/5 of this comma) and leave the 8:11's just. (So 6:11's will be 1.43c wide, 9:11's will be 2.86c wide, 8:9's will be 2.86c narrow).

Now this takes care of 2.86c of the 440:441 leaving 1.07c. So we narrow the 4:7's by this amount and make the 5:7's just. (7:9's will be 1.79c narrow, 6:7's will be 0.36c wide).

To summarise:
3's 1.43c narrow
7's 1.07c narrow
11's just

With rounding to the nearest cent, this means:

Bb\ C D E\ F G A Bb\
0 232 433 582 732 932 1133 1200

(Major thirds happen to be 401 cents, minor thirds 299 cents)

Compare this to your original
1/1, 8/7, 9/7, 7/5, 32/21, 12/7, 27/14, 2/1
0 231 435 583 729 933 1137 1200

Nothing has moved very far, but I think you will notice an improvement.

Regards,

-- Dave Keenan
http://dkeenan.com

[David C Keenan:]
> You could use 31-tET, with steps of 6, 5, 4, 4, 5, 5, 2 (still
strictly proper).

Yes, and awhile after I finished the piece I thought that 31e might be
a good compromise if I ever wanted to have it recorded with real
instruments (and by "real instruments" I just mean the actual intended
instruments and not the usual take what you can get MIDI demo). But it
didn't work out... Part of the problem was the tuning of the first
verse (which is an even less analytically cooperative JI scale than
that of the second verse):

1/1, 10/9, 7/6, 21/16, 3/2, 14/9, 7/4, 2/1

You could look this as the VI of a:

1/1, 9/8, 9/7, 4/3, 3/2, 12/7, 40/21, 2/1

Which probably 'wants' to be something more like:

4/3---------1/1--------,3/2--------,9/8
` , ` , `
`12/7-------`9/7--------`27/14

And while I seem to remember being almost mildly enthusiastic about a
21/11 7th in the I, it just seemed that I never could swallow any
substitute for the 10/9, and that I wanted to (without really wanting
to!) retain the 81/80's (note that 31e would not distinguish between a
27/14, a 40/21, or a 21/11 version):

,G.-----,D.
A'-/+\-`E'-/+\-`B-------F#
| / | \ | / | \ | |
|/ ,Db.\+/-,Ab,\+--,Eb,-+--,Bb,----,F,
Eb'----`Bb'----,F'-----,C'-----,G'-----,D.-----,A
`G'-----`D'--+--`A'--+\-`E'-/+\-`B' /|
| | \ | / | \ | / |
Gb------Db.\+/-,Ab.\+/-,Eb
`Bb'----`F'

>Call it A#, C, D, Dx(Fbb), F, G, A.

This sort of QCM spelling starts to get a bit harrowing right there at
the ("11/9") Fbb = Dx enharmonic turnaround... How a spelling of say
Bbb, C, D, E, F, G, A, Bbb derived from a 31e circle of fifths with a
fifth size at 17.87, i.e., (LOG(11)-LOG(1))*(31/LOG(2^6))?

>Of course all those Pythagorean thirds and sixths will become
excellent ratios of 5 (while the 2:3's and 8:9's are not so good).
Assuming that's unacceptable,

Yeah, in the case of the 1/1, 8/7, 9/7, 7/5, 32/21, 12/7, 27/14,
2/1 -- and especially that tuning in that particular standing musical
context -- 31e wasn't quite right.

>we can do something that isn't an ET (EDO) and doesn't go all the way
to meantone.

I've been trying to just use "e" as a sort of truce between an ET and
an ED... but if I was going to use one or the other in this context,
it would certainly be ET as everything is quite clearly and
consistently dealing with tempering.

>We can _microtemper_ it. . . With rounding to the nearest cent, this
means:

Bb\ C D E\ F G A Bb\
0 232 433 582 732 932 1133 1200

Thanks Dave, this looks good, and I'll definitely try it out when I
get a hunk of time to do so. Any thoughts on "microtempering" the 1/1,
10/9, 7/6, 21/16, 3/2, 14/9, 7/4, 2/1?

Dan

[D.Stearns, TD 522.10]
>Yes, and a while after I finished the piece I thought that 31e might be
>a good compromise if I ever wanted to have it recorded with real
>instruments (and by "real instruments" I just mean the actual intended
>instruments and not the usual take what you can get MIDI demo). But it
>didn't work out... Part of the problem was the tuning of the first
>verse (which is an even less analytically cooperative JI scale than
>that of the second verse):
>
>1/1, 10/9, 7/6, 21/16, 3/2, 14/9, 7/4, 2/1
>
>You could look this as the VI of a:
>
>1/1, 9/8, 9/7, 4/3, 3/2, 12/7, 40/21, 2/1
>
>Which probably 'wants' to be something more like:
>
> 4/3---------1/1--------,3/2--------,9/8
> ` , ` , `
> `12/7-------`9/7--------`27/14
>
>And while I seem to remember being almost mildly enthusiastic about a
>21/11 7th in the I, it just seemed that I never could swallow any
>substitute for the 10/9, and that I wanted to (without really wanting
>to!) retain the 81/80's (note that 31e would not distinguish between a
>27/14, a 40/21, or a 21/11 version):

In other words "between a 9/8, a 10/9 or a 49/44 version".

...
>Thanks Dave, this looks good, and I'll definitely try it out when I
>get a hunk of time to do so. Any thoughts on "microtempering" the 1/1,
>10/9, 7/6, 21/16, 3/2, 14/9, 7/4, 2/1?

Nope. What intervals or chords does the 10/9 take part in? If you want it to be both the root of a 5:7:9 and (the third of a supermajor triad on the 7/4 or the fifth of a subminor triad on the 3/2) then it's gotta be in some meantone.

C D D# E# G G# A# (C)

Although the sharpened notes are independent of the natural ones. i.e. you can juggle the errors in the 7's somewhat independently.

It's hard to imagine a 5:7 or 5:9 from the 10/9 being so important that it couldn't be closer to a 9/8. Or is it for melodic reasons that it needs to be 10/9? Perhaps to even out the first two steps. That second step (D to D#) is pretty small at 63 cents. But the 5th step (G to G#) is the same.

Regards,

-- Dave Keenan
http://dkeenan.com