resolution of 72-tET

Hi Paul... or whomever...

I'm forgetting what the actual resolution of 72-tET is now. Is it
within 2-3 cents of Just for most of the intervals??

Thanks!

Joseph

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> Hi Paul... or whomever...
>
> I'm forgetting what the actual resolution of 72-tET is now. Is it
> within 2-3 cents of Just for most of the intervals??
>
> Thanks!
>
> Joseph

within 2 cents for 5-limit
within 3 cents for 7-limit
within 4 cents for 9-limit
within 4 cents for 11-limit

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:

/tuning/topicId_37831.html#37838

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> > Hi Paul... or whomever...
> >
> > I'm forgetting what the actual resolution of 72-tET is now. Is
it
> > within 2-3 cents of Just for most of the intervals??
> >
> > Thanks!
> >
> > Joseph
>
> within 2 cents for 5-limit
> within 3 cents for 7-limit
> within 4 cents for 9-limit
> within 4 cents for 11-limit

***Hmmm. Well, that's what I thought. That seems pretty accurate to
*me* so I don't know what exactly all the *hoopla* is about,
particularly if one informs the performer that the *idea* is to *go*
for the Just Intonation and eliminate beats. They're almost there
with 72-tET, and they can just move a *tiny* bit and no beating.
They might even hit it by *accident!* :)

JP

In a message dated 6/16/02 6:09:41 PM Eastern Daylight Time, jpehrson@rcn.com
writes:

> > within 2 cents for 5-limit
> > within 3 cents for 7-limit
> > within 4 cents for 9-limit
> > within 4 cents for 11-limit
>
> ***Hmmm. Well, that's what I thought. That seems pretty accurate to
> *me* so I don't know what exactly all the *hoopla* is about,
> particularly if one informs the performer that the *idea* is to *go*
> for the Just Intonation and eliminate beats. They're almost there
> with 72-tET, and they can just move a *tiny* bit and no beating.
> They might even hit it by *accident!* :)
>
> JP

Joseph, sorry to upset the applecart, but is there any evidence for this?
Specifically, is there any example of a performance that had the players move
to just positions because they were asked to? There certainly is no time in
real time to listen to the sound of none beating. It's a bit too zen for me.

best, Johnny Reinhard

--- In tuning@y..., Afmmjr@a... wrote:

/tuning/topicId_37831.html#37846

> In a message dated 6/16/02 6:09:41 PM Eastern Daylight Time,
jpehrson@r...
> writes:
>
>
> > > within 2 cents for 5-limit
> > > within 3 cents for 7-limit
> > > within 4 cents for 9-limit
> > > within 4 cents for 11-limit
> >
> > ***Hmmm. Well, that's what I thought. That seems pretty
accurate to
> > *me* so I don't know what exactly all the *hoopla* is about,
> > particularly if one informs the performer that the *idea* is to
*go*
> > for the Just Intonation and eliminate beats. They're almost
there
> > with 72-tET, and they can just move a *tiny* bit and no beating.
> > They might even hit it by *accident!* :)
> >
> > JP
>
>
>
> Joseph, sorry to upset the applecart, but is there any evidence for
this?
> Specifically, is there any example of a performance that had the
players move
> to just positions because they were asked to? There certainly is
no time in
> real time to listen to the sound of none beating. It's a bit too
zen for me.
>
> best, Johnny Reinhard

***Hi Johnny!

Well, my guess *off the top* is that it would be possible in slower
passages but not necessarily in faster....

Dunno. Apparently you were hearing Chris Washburne as playing 72
rather than Just, even though I *told* him that Just was the
objective... (Not *his* fault, necessarily, since he's so trained in
72...)

Although, other people at the concert seemed to be hearing Just...at
least *sometimes!*

I guess the only way we'll find out is for me to write lots of pieces
in 72-tET and try to arrange lots of performances!!!

After talking to John Eaton and Tristan Mureil, though, it's clear
that both of them at times use only QUARTERTONES to try to get
Just... *really!* I can *prove* that! I've got a score right here
by Eaton that shows the 7th partial as a QUARTERTONE! Several
passages of that, in fact!

Surely, 72-tET is going to do a better job of Just emulation than
*quartertones!* So it seems...

best,

Joseph

In a message dated 6/16/02 10:04:42 PM Eastern Daylight Time,
jpehrson@rcn.com writes:

> Well, my guess *off the top* is that it would be possible in slower
> passages but not necessarily in faster....
>

My question is simple to you: what is just sounding, other than lack of
beats?

> Dunno. Apparently you were hearing Chris Washburne as playing 72
> rather than Just, even though I *told* him that Just was the
> objective... (Not *his* fault, necessarily, since he's so trained in
> 72...)
>

No matter what the composer tells the performer, if he is to play to a
recording, he must match the notes on the recording. An orchestra player
cannot play a just third in a final chord of a symphony. No conductor would
allow it. Nobody would except it for it is out of tune in the context. By
extension, a recording without just but only 72 division intervals does not
bend..how could a trombonist go against the recording?

> Although, other people at the concert seemed to be hearing Just...at
> least *sometimes!*
>

What does this mean, hearing just? As someone that sings just intonation all
the time, this makes little sense. I guess I'll ask each individual person
what they mean by hearing "just sometimes."

> I guess the only way we'll find out is for me to write lots of pieces
> in 72-tET and try to arrange lots of performances!!!
>

Maybe it is a good idea to write a true experimental piece, meaning a piece
where an experiment is tried and the result is actually in question. It
won't need too many performances....only a recording.

> After talking to John Eaton and Tristan Mureil, though, it's clear
> that both of them at times use only QUARTERTONES to try to get
> Just... *really!* I can *prove* that! I've got a score right here
> by Eaton that shows the 7th partial as a QUARTERTONE! Several
> passages of that, in fact!

Please recall that talked with them both, as well. Tristan approximates his
just intervals to quartertones, yes. Does that mean he is getting the best
resolutions of his ideas? I don't think so. He has given up on players.
Too bad considering I have all the players than could do his music justice.

As for John Eaton, did you hear his opera "Danton and Robespierre" which is
polymicrotonal (Danton is just and Robespierre is in eighthtones). Actually,
John has used just on and off. But never anything for bassoon. He said
bassoonists always complained and c(w)ouldn't play his microtones. When I
say John's opera "Cry of Clytemnestra" at the Brooklyn Academy of Music in
the '80s, a prominent bassoonist would spin his bocal up and down to "effect"
quartertones. Real low tech. I heard his Mass which we are planning to
produce next years with quartertones and sixthtones. Beautiful. But not
everyone liked just, like Xenakis. But John Eaton does.

> Surely, 72-tET is going to do a better job of Just emulation than
> *quartertones!* So it seems...
>

Not for the 11th limit it doesn't.

> best,
>
> Joseph

best, Johnny

--- In tuning@y..., Afmmjr@a... wrote:

/tuning/topicId_37831.html#37853

> > Surely, 72-tET is going to do a better job of Just emulation than
> > *quartertones!* So it seems...
> >
>
> Not for the 11th limit it doesn't.
>

***Hi Johnny!

Well, the bottom line is that I'm not going to quibble about a 2 or 3
cent discrepancy. For *my* practical purposes 72-tET *is* just, and
Ezra Sims seems to agree with this as well.

If that's not satisfactory to *other* people, that's their *own*
business, and they're welcome to have that opinion...

best,

Joseph

--- In tuning@y..., Afmmjr@a... wrote:
> In a message dated 6/16/02 10:04:42 PM Eastern Daylight Time,
> jpehrson@r... writes:
>
>
> > Well, my guess *off the top* is that it would be possible in
slower
> > passages but not necessarily in faster....
> >
>
> My question is simple to you: what is just sounding, other than
lack of
> beats?

if that's your criterion, then if you play in 72-equal, with
harmonies going by 4 or 5 per second, there's no time for beats to
occur, so then by your own definition it's just sounding, right?

> > Surely, 72-tET is going to do a better job of Just emulation than
> > *quartertones!* So it seems...
> >
>
> Not for the 11th limit it doesn't.

in the 11-limit:

72-equal does just as well as 24-equal for 11:1, 11:3, 11:9, 9:1, 3:1.
72-equal does far better than 24-equal for 11:5, 11:7, 9:5, 9:7, 7:5,
7:3, 7:1, 5:3, 5:1.

so i have to differ.

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:

/tuning/topicId_37831.html#37855

>
> in the 11-limit:
>
> 72-equal does just as well as 24-equal for 11:1, 11:3, 11:9, 9:1,
3:1.

**Hi Paul!

Actually, since 24-tET is imbedded *within* 72-tET, shouldn't they do
the 11th limit about the same??

Joseph

--- In tuning@y..., Afmmjr@a... wrote:

> No matter what the composer tells the performer, if he is to play
to a
> recording, he must match the notes on the recording. An orchestra
player
> cannot play a just third in a final chord of a symphony. No
conductor would
> allow it. Nobody would except it for it is out of tune in the
context. By
> extension, a recording without just but only 72 division intervals
does not
> bend..how could a trombonist go against the recording?

Bob:
Well, there seems to be a persistent implication in your posts,
Johnny, that musicians and singers who can easily and quickly spot
deviations from just in the sub-ten cent range abound, some even in
the sub-three cent range, just oodles of them everywhere. I would
like to run into a collection of such musicians in one place
sometime. The number of commercially available recordings of
musicians (other than fixed tuned instrumentalists) who play or sing
with this kind of precision is vanishingly small in my experience. I
have a few such recordings, but they are exceptional and I wish I had
more.

On the other hand, or rather, for the same reason, the statement
that "no conductor would allow" a just third at the end of a symphony
movement I find to be a bit hard to swallow. I have heard string
quartets, for example, play just triads fairly consistently in the
performance of compositions even from the romantic period, not to
mention classical.

A conductor "not allowing" a just third in the final triad of a
symphony implies that he/she cares about the difference enough to
disallow such a thing. I find this VERY hard to swallow, since I
detect a HUGE body of symphony conductors who let pitch errors on the
order of 10-14 cents go all the time, and not in the service of JI,
but simply random mistunings. (I would not complain if they were
just.) Where does this kind of assessment come from, Johnny? Do you
live on a different planet?

Respectfully puzzled,

Bob

jpehrson2 wrote:

Hi Paul... or whomever...

I'm forgetting what the actual resolution of 72-tET is now. Is it within 2-3 cents of Just for most of the intervals??

Thanks!

Joseph

Subject:
table ordered by harmonic liimit
Date:
Wed, 2 May 2001 11:56:53 -0700
From:
"Canright, David" <dcanright@nps.navy.mil>
To:
'Rick Tagawa' <ricktagawa@earthlink.net>

72-tET Approximation of Just Intervals

Below is a table of all the intervals found in the harmonic series up to
harmonic 64, and their closest approximations by 72-tET. These are ordered
by first appearance in the harmonic series (by numerator, then by
denominator; for example, if you are only interested up to harmonic 20, stop
when you first hit 21). Not all of the "octave complements" appear (if they
involve a numerator greater than 64); of course, their tempered
approximations have equal but opposite errors. Each entry includes the
ratio, cents error relative to nearest tempered (so, for example, 3/2 is
+2.0c relative to note #42), tempered note # (0-71), keyboard # (0-5), key #
(0-11), where keyboards 1-5 are assumed progressively sharper relative to
keyboard 0.

ratio error note kb:key#

1/1 +0.0c 0 0:0
3/2 +2.0c 42 0:7
4/3 -2.0c 30 0:5
5/3 +1.0c 53 5:8
5/4 +3.0c 23 5:3
6/5 -1.0c 19 1:3
7/4 +2.2c 58 4:9
7/5 -0.8c 35 5:5
7/6 +0.2c 16 4:2
8/5 -3.0c 49 1:8
8/7 -2.2c 14 2:2
9/5 +0.9c 61 1:10
9/7 +1.8c 26 2:4
9/8 +3.9c 12 0:2
10/7 +0.8c 37 1:6
10/9 -0.9c 11 5:1
11/6 -0.6c 63 3:10
11/7 -0.8c 47 5:7
11/8 +1.3c 33 3:5
11/9 -2.6c 21 3:3
11/10 -1.7c 10 4:1
12/7 -0.2c 56 2:9
12/11 +0.6c 9 3:1
13/7 +5.0c 64 4:10
13/8 +7.2c 50 2:8
13/9 +3.3c 38 2:6
13/10 +4.2c 27 3:4
13/11 +5.9c 17 5:2
13/12 +5.2c 8 2:1
14/9 -1.8c 46 4:7
14/11 +0.8c 25 1:4
14/13 -5.0c 8 2:1
15/8 +4.9c 65 5:10
15/11 +3.6c 32 2:5
15/13 -2.3c 15 3:2
15/14 +2.8c 7 1:1
16/9 -3.9c 60 0:10
16/11 -1.3c 39 3:6
16/13 -7.2c 22 4:3
16/15 -4.9c 7 1:1
17/9 +1.0c 66 0:11
17/10 +2.0c 55 1:9
17/11 +3.6c 45 3:7
17/12 +3.0c 36 0:6
17/13 -2.2c 28 4:4
17/14 +2.8c 20 2:3
17/15 +0.0c 13 1:2
17/16 +5.0c 6 0:1
18/11 +2.6c 51 3:8
18/13 -3.3c 34 4:5
18/17 -1.0c 6 0:1
19/10 -5.5c 67 1:11
19/11 -3.8c 57 3:9
19/12 -4.4c 48 0:8
19/13 +7.0c 39 3:6
19/14 -4.6c 32 2:5
19/15 -7.4c 25 1:4
19/16 -2.5c 18 0:3
19/17 -7.4c 12 0:2
19/18 -6.4c 6 0:1
20/11 +1.7c 62 2:10
20/13 -4.2c 45 3:7
20/17 -2.0c 17 5:2
20/19 +5.5c 5 5:0
21/11 +2.8c 67 1:11
21/13 -3.1c 50 2:8
21/16 +4.1c 28 4:4
21/17 -0.8c 22 4:3
21/19 +6.6c 10 4:1
21/20 +1.1c 5 5:0
22/13 -5.9c 55 1:9
22/15 -3.6c 40 4:6
22/17 -3.6c 27 3:4
22/19 +3.8c 15 3:2
22/21 -2.8c 5 5:0
23/12 -7.0c 68 2:11
23/13 +4.4c 59 5:9
23/14 -7.2c 52 4:8
23/15 +6.7c 44 2:7
23/16 -5.1c 38 2:6
23/17 +6.7c 31 1:5
23/18 +7.7c 25 1:4
23/19 -2.6c 20 2:3
23/20 -8.0c 15 3:2
23/21 +7.5c 9 3:1
23/22 -6.4c 5 5:0
24/13 -5.2c 64 4:10
24/17 -3.0c 36 0:6
24/19 +4.4c 24 0:4
24/23 +7.0c 4 4:0
25/13 -1.2c 68 2:11
25/14 +3.8c 60 0:10
25/16 +6.0c 46 4:7
25/17 +1.0c 40 4:6
25/18 +2.1c 34 4:5
25/19 -8.2c 29 5:4
25/21 +1.8c 18 0:3
25/22 +4.6c 13 1:2
25/23 -5.6c 9 3:1
25/24 +4.0c 4 4:0
26/15 +2.3c 57 3:9
26/17 +2.2c 44 2:7
26/19 -7.0c 33 3:5
26/21 +3.1c 22 4:3
26/23 -4.4c 13 1:2
26/25 +1.2c 4 4:0
27/14 +3.7c 68 2:11
27/16 +5.9c 54 0:9
27/17 +0.9c 48 0:8
27/19 -8.3c 37 1:6
27/20 +2.9c 31 1:5
27/22 +4.5c 21 3:3
27/23 -5.7c 17 5:2
27/25 -0.1c 8 2:1
27/26 -1.3c 4 4:0
28/15 -2.8c 65 5:10
28/17 -2.8c 52 4:8
28/19 +4.6c 40 4:6
28/23 +7.2c 20 2:3
28/25 -3.8c 12 0:2
28/27 -3.7c 4 4:0
29/15 +8.0c 68 2:11
29/16 -3.8c 62 2:10
29/17 +8.0c 55 1:9
29/18 -7.7c 50 2:8
29/19 -1.3c 44 2:7
29/20 -6.7c 39 3:6
29/21 -7.9c 34 4:5
29/22 -5.1c 29 5:4
29/23 +1.3c 24 0:4
29/24 -5.7c 20 2:3
29/25 +6.9c 15 3:2
29/26 +5.7c 11 5:1
29/27 +7.0c 7 1:1
29/28 -5.9c 4 4:0
30/17 +0.0c 59 5:9
30/19 +7.4c 47 5:7
30/23 -6.7c 28 4:4
30/29 -8.0c 4 4:0
31/16 -5.0c 69 3:11
31/17 +6.7c 62 2:10
31/18 +7.8c 56 2:9
31/19 -2.5c 51 3:8
31/20 -7.9c 46 4:7
31/21 +7.6c 40 4:6
31/22 -6.3c 36 0:6
31/23 +0.1c 31 1:5
31/24 -6.9c 27 3:4
31/25 +5.7c 22 4:3
31/26 +4.5c 18 0:3
31/27 +5.8c 14 2:2
31/28 -7.1c 11 5:1
31/29 -1.2c 7 1:1
31/30 +6.8c 3 3:0
32/17 -5.0c 66 0:11
32/19 +2.5c 54 0:9
32/21 -4.1c 44 2:7
32/23 +5.1c 34 4:5
32/25 -6.0c 26 2:4
32/27 -5.9c 18 0:3
32/29 +3.8c 10 4:1
32/31 +5.0c 3 3:0
33/17 -1.7c 69 3:11
33/19 +5.8c 57 3:9
33/20 +0.3c 52 4:8
33/23 +8.3c 37 1:6
33/25 -2.7c 29 5:4
33/26 -3.9c 25 1:4
33/28 +1.1c 17 5:2
33/29 +7.0c 13 1:2
33/31 +8.2c 6 0:1
33/32 +3.3c 3 3:0
34/19 +7.4c 60 0:10
34/21 +0.8c 50 2:8
34/23 -6.7c 41 5:6
34/25 -1.0c 32 2:5
34/27 -0.9c 24 0:4
34/29 -8.0c 17 5:2
34/31 -6.7c 10 4:1
34/33 +1.7c 3 3:0
35/18 +1.2c 69 3:11
35/19 +7.6c 63 3:10
35/22 +3.8c 48 0:8
35/23 -6.5c 44 2:7
35/24 +3.2c 39 3:6
35/26 -2.1c 31 1:5
35/27 -0.7c 27 3:4
35/29 -7.8c 20 2:3
35/31 -6.6c 13 1:2
35/32 +5.1c 9 3:1
35/33 +1.9c 6 0:1
35/34 +0.2c 3 3:0
36/19 +6.4c 66 0:11
36/23 -7.7c 47 5:7
36/25 -2.1c 38 2:6
36/29 +7.7c 22 4:3
36/31 -7.8c 16 4:2
36/35 -1.2c 3 3:0
37/19 +3.8c 69 3:11
37/20 -1.6c 64 4:10
37/21 -2.8c 59 5:9
37/22 +0.0c 54 0:9
37/23 +6.4c 49 1:8
37/24 -0.6c 45 3:7
37/25 -4.6c 41 5:6
37/26 -5.9c 37 1:6
37/27 -4.5c 33 3:5
37/28 -0.8c 29 5:4
37/29 +5.1c 25 1:4
37/30 -3.6c 22 4:3
37/31 +6.3c 18 0:3
37/32 +1.3c 15 3:2
37/33 -1.9c 12 0:2
37/34 -3.6c 9 3:1
37/35 -3.8c 6 0:1
37/36 -2.6c 3 3:0
38/21 -6.6c 62 2:10
38/23 +2.6c 52 4:8
38/25 +8.2c 43 1:7
38/27 +8.3c 35 5:5
38/29 +1.3c 28 4:4
38/31 +2.5c 21 3:3
38/33 -5.8c 15 3:2
38/35 -7.6c 9 3:1
38/37 -3.8c 3 3:0
39/20 +6.2c 69 3:11
39/22 +7.8c 59 5:9
39/23 -2.5c 55 1:9
39/25 +3.2c 46 4:7
39/28 +7.0c 34 4:5
39/29 -3.8c 31 1:5
39/31 -2.6c 24 0:4
39/32 -7.5c 21 3:3
39/34 +4.2c 14 2:2
39/35 +4.0c 11 5:1
39/37 +7.8c 5 5:0
39/38 -5.0c 3 3:0
40/21 -1.1c 67 1:11
40/23 +8.0c 57 3:9
40/27 -2.9c 41 5:6
40/29 +6.7c 33 3:5
40/31 +7.9c 26 2:4
40/33 -0.3c 20 2:3
40/37 +1.6c 8 2:1
40/39 -6.2c 3 3:0
41/21 +8.3c 69 3:11
41/22 -5.6c 65 5:10
41/23 +0.8c 60 0:10
41/24 -6.2c 56 2:9
41/25 +6.4c 51 3:8
41/26 +5.2c 47 5:7
41/27 +6.5c 43 1:7
41/28 -6.4c 40 4:6
41/29 -0.5c 36 0:6
41/30 +7.5c 32 2:5
41/31 +0.7c 29 5:4
41/32 -4.3c 26 2:4
41/33 -7.5c 23 5:3
41/34 +7.4c 19 1:3
41/35 +7.3c 16 4:2
41/36 -8.2c 14 2:2
41/37 -5.6c 11 5:1
41/38 -1.8c 8 2:1
41/39 +3.2c 5 5:0
41/40 -7.3c 3 3:0
42/23 -7.5c 63 3:10
42/25 -1.8c 54 0:9
42/29 +7.9c 38 2:6
42/31 -7.6c 32 2:5
42/37 +2.8c 13 1:2
42/41 -8.3c 3 3:0
43/22 -6.5c 70 4:11
43/23 -0.1c 65 5:10
43/24 -7.1c 61 1:10
43/25 +5.6c 56 2:9
43/26 +4.3c 52 4:8
43/27 +5.7c 48 0:8
43/28 -7.3c 45 3:7
43/29 -1.4c 41 5:6
43/30 +6.6c 37 1:6
43/31 -0.2c 34 4:5
43/32 -5.1c 31 1:5
43/33 +8.2c 27 3:4
43/34 +6.6c 24 0:4
43/35 +6.4c 21 3:3
43/36 +7.6c 18 0:3
43/37 -6.5c 16 4:2
43/38 -2.7c 13 1:2
43/39 +2.4c 10 4:1
43/40 -8.1c 8 2:1
43/41 -0.9c 5 5:0
43/42 +7.4c 2 2:0
44/23 +6.4c 67 1:11
44/25 -4.6c 59 5:9
44/27 -4.5c 51 3:8
44/29 +5.1c 43 1:7
44/31 +6.3c 36 0:6
44/35 -3.8c 24 0:4
44/37 +0.0c 18 0:3
44/39 -7.8c 13 1:2
44/41 +5.6c 7 1:1
44/43 +6.5c 2 2:0
45/23 -4.7c 70 4:11
45/26 -0.3c 57 3:9
45/28 +4.7c 49 1:8
45/29 -6.0c 46 4:7
45/31 -4.8c 39 3:6
45/32 +6.9c 35 5:5
45/34 +1.9c 29 5:4
45/37 +5.5c 20 2:3
45/38 -7.3c 18 0:3
45/41 -5.5c 10 4:1
45/43 -4.6c 5 5:0
45/44 +5.6c 2 2:0
46/25 +5.6c 63 3:10
46/27 +5.7c 55 1:9
46/29 -1.3c 48 0:8
46/31 -0.1c 41 5:6
46/33 -8.3c 35 5:5
46/35 +6.5c 28 4:4
46/37 -6.4c 23 5:3
46/39 +2.5c 17 5:2
46/41 -0.8c 12 0:2
46/43 +0.1c 7 1:1
46/45 +4.7c 2 2:0
47/24 -3.1c 70 4:11
47/25 -7.1c 66 0:11
47/26 +8.3c 61 1:10
47/27 -7.0c 58 4:9
47/28 -3.3c 54 0:9
47/29 +2.6c 50 2:8
47/30 -6.1c 47 5:7
47/31 +3.8c 43 1:7
47/32 -1.2c 40 4:6
47/33 -4.4c 37 1:6
47/34 -6.1c 34 4:5
47/35 -6.3c 31 1:5
47/36 -5.1c 28 4:4
47/37 -2.5c 25 1:4
47/38 +1.3c 22 4:3
47/39 +6.4c 19 1:3
47/40 -4.1c 17 5:2
47/41 +3.1c 14 2:2
47/42 -5.3c 12 0:2
47/43 +4.0c 9 3:1
47/44 -2.5c 7 1:1
47/45 -8.1c 5 5:0
47/46 +3.9c 2 2:0
48/25 -4.0c 68 2:11
48/29 +5.7c 52 4:8
48/31 +6.9c 45 3:7
48/35 -3.2c 33 3:5
48/37 +0.6c 27 3:4
48/41 +6.2c 16 4:2
48/43 +7.1c 11 5:1
48/47 +3.1c 2 2:0
49/25 -1.6c 70 4:11
49/26 -2.9c 66 0:11
49/27 -1.5c 62 2:10
49/29 +8.1c 54 0:9
49/30 -0.6c 51 3:8
49/31 -7.4c 48 0:8
49/32 +4.3c 44 2:7
49/33 +1.0c 41 5:6
49/34 -0.6c 38 2:6
49/36 +0.4c 32 2:5
49/37 +3.0c 29 5:4
49/38 +6.8c 26 2:4
49/39 -4.8c 24 0:4
49/40 +1.3c 21 3:3
49/41 -8.1c 19 1:3
49/43 -7.2c 14 2:2
49/44 +3.0c 11 5:1
49/45 -2.6c 9 3:1
49/46 -7.3c 7 1:1
49/47 +5.5c 4 4:0
49/48 +2.4c 2 2:0
50/27 +0.1c 64 4:10
50/29 -6.9c 57 3:9
50/31 -5.7c 50 2:8
50/33 +2.7c 43 1:7
50/37 +4.6c 31 1:5
50/39 -3.2c 26 2:4
50/41 -6.4c 21 3:3
50/43 -5.6c 16 4:2
50/47 +7.1c 6 0:1
50/49 +1.6c 2 2:0
51/26 -0.3c 70 4:11
51/28 +4.8c 62 2:10
51/29 -6.0c 59 5:9
51/31 -4.8c 52 4:8
51/32 +6.9c 48 0:8
51/35 +1.8c 39 3:6
51/37 +5.6c 33 3:5
51/38 -7.3c 31 1:5
51/40 +3.9c 25 1:4
51/41 -5.5c 23 5:3
51/43 -4.6c 18 0:3
51/44 +5.6c 15 3:2
51/46 -4.7c 11 5:1
51/47 +8.1c 8 2:1
51/49 +2.6c 4 4:0
51/50 +1.0c 2 2:0
52/27 +1.3c 68 2:11
52/29 -5.7c 61 1:10
52/31 -4.5c 54 0:9
52/33 +3.9c 47 5:7
52/35 +2.1c 41 5:6
52/37 +5.9c 35 5:5
52/41 -5.2c 25 1:4
52/43 -4.3c 20 2:3
52/45 +0.3c 15 3:2
52/47 -8.3c 11 5:1
52/49 +2.9c 6 0:1
52/51 +0.3c 2 2:0
53/27 +1.0c 70 4:11
53/28 +4.7c 66 0:11
53/29 -6.1c 63 3:10
53/30 +1.9c 59 5:9
53/31 -4.9c 56 2:9
53/32 +6.8c 52 4:8
53/33 +3.6c 49 1:8
53/34 +1.9c 46 4:7
53/35 +1.7c 43 1:7
53/36 +2.9c 40 4:6
53/37 +5.5c 37 1:6
53/38 -7.3c 35 5:5
53/39 -2.3c 32 2:5
53/40 +3.9c 29 5:4
53/41 -5.6c 27 3:4
53/42 +2.7c 24 0:4
53/43 -4.7c 22 4:3
53/44 +5.5c 19 1:3
53/45 -0.1c 17 5:2
53/46 -4.8c 15 3:2
53/47 +8.0c 12 0:2
53/48 +4.9c 10 4:1
53/49 +2.5c 8 2:1
53/50 +0.9c 6 0:1
53/51 -0.1c 4 4:0
53/52 -0.4c 2 2:0
54/29 -7.0c 65 5:10
54/31 -5.8c 58 4:9
54/35 +0.7c 45 3:7
54/37 +4.5c 39 3:6
54/41 -6.5c 29 5:4
54/43 -5.7c 24 0:4
54/47 +7.0c 14 2:2
54/49 +1.5c 10 4:1
54/53 -1.0c 2 2:0
55/28 +2.1c 70 4:11
55/29 +8.1c 66 0:11
55/31 -7.4c 60 0:10
55/32 +4.3c 56 2:9
55/34 -0.7c 50 2:8
55/36 +0.4c 44 2:7
55/37 +3.0c 41 5:6
55/38 +6.8c 38 2:6
55/39 -4.9c 36 0:6
55/41 -8.1c 31 1:5
55/42 +0.2c 28 4:4
55/43 -7.2c 26 2:4
55/46 -7.3c 19 1:3
55/47 +5.5c 16 4:2
55/48 +2.3c 14 2:2
55/49 +0.0c 12 0:2
55/51 -2.6c 8 2:1
55/52 -2.9c 6 0:1
55/53 -2.5c 4 4:0
55/54 -1.6c 2 2:0
56/29 +5.9c 68 2:11
56/31 +7.1c 61 1:10
56/33 -1.1c 55 1:9
56/37 +0.8c 43 1:7
56/39 -7.0c 38 2:6
56/41 +6.4c 32 2:5
56/43 +7.3c 27 3:4
56/45 -4.7c 23 5:3
56/47 +3.3c 18 0:3
56/51 -4.8c 10 4:1
56/53 -4.7c 6 0:1
56/55 -2.1c 2 2:0
57/29 +3.2c 70 4:11
57/31 +4.4c 63 3:10
57/32 -0.5c 60 0:10
57/34 -5.5c 54 0:9
57/35 -5.7c 51 3:8
57/37 -1.9c 45 3:7
57/40 -3.5c 37 1:6
57/41 +3.7c 34 4:5
57/43 +4.6c 29 5:4
57/44 -1.8c 27 3:4
57/46 +4.5c 22 4:3
57/47 +0.6c 20 2:3
57/49 -4.9c 16 4:2
57/50 -6.5c 14 2:2
57/52 -7.7c 10 4:1
57/53 -7.4c 8 2:1
57/55 -4.8c 4 4:0
57/56 -2.7c 2 2:0
58/31 +1.2c 65 5:10
58/33 -7.0c 59 5:9
58/35 +7.8c 52 4:8
58/37 -5.1c 47 5:7
58/39 +3.8c 41 5:6
58/41 +0.5c 36 0:6
58/43 +1.4c 31 1:5
58/45 +6.0c 26 2:4
58/47 -2.6c 22 4:3
58/49 -8.1c 18 0:3
58/51 +6.0c 13 1:2
58/53 +6.1c 9 3:1
58/55 -8.1c 6 0:1
58/57 -3.2c 2 2:0
59/30 +4.2c 70 4:11
59/31 -2.5c 67 1:11
59/32 -7.5c 64 4:10
59/33 +5.9c 60 0:10
59/34 +4.2c 57 3:9
59/35 +4.0c 54 0:9
59/36 +5.3c 51 3:8
59/37 +7.8c 48 0:8
59/38 -5.0c 46 4:7
59/39 +0.0c 43 1:7
59/40 +6.2c 40 4:6
59/41 -3.2c 38 2:6
59/42 +5.1c 35 5:5
59/43 -2.3c 33 3:5
59/44 +7.9c 30 0:5
59/45 +2.3c 28 4:4
59/46 -2.4c 26 2:4
59/47 -6.3c 24 0:4
59/48 +7.2c 21 3:3
59/49 +4.9c 19 1:3
59/50 +3.2c 17 5:2
59/51 +2.3c 15 3:2
59/52 +2.0c 13 1:2
59/53 +2.3c 11 5:1
59/54 +3.3c 9 3:1
59/55 +4.9c 7 1:1
59/56 +7.0c 5 5:0
59/57 -7.0c 4 4:0
59/58 -3.7c 2 2:0
60/31 -6.8c 69 3:11
60/37 +3.6c 50 2:8
60/41 -7.5c 40 4:6
60/43 -6.6c 35 5:5
60/47 +6.1c 25 1:4
60/49 +0.6c 21 3:3
60/53 -1.9c 13 1:2
60/59 -4.2c 2 2:0
61/31 +5.2c 70 4:11
61/32 +0.2c 67 1:11
61/33 -3.1c 64 4:10
61/34 -4.7c 61 1:10
61/35 -4.9c 58 4:9
61/36 -3.7c 55 1:9
61/37 -1.1c 52 4:8
61/38 +2.7c 49 1:8
61/39 +7.7c 46 4:7
61/40 -2.8c 44 2:7
61/41 +4.5c 41 5:6
61/42 -3.9c 39 3:6
61/43 +5.4c 36 0:6
61/44 -1.1c 34 4:5
61/45 -6.7c 32 2:5
61/46 +5.3c 29 5:4
61/47 +1.4c 27 3:4
61/48 -1.7c 25 1:4
61/49 -4.1c 23 5:3
61/50 -5.7c 21 3:3
61/51 -6.7c 19 1:3
61/52 -7.0c 17 5:2
61/53 -6.6c 15 3:2
61/54 -5.6c 13 1:2
61/55 -4.1c 11 5:1
61/56 -1.9c 9 3:1
61/57 +0.8c 7 1:1
61/58 +4.0c 5 5:0
61/59 +7.7c 3 3:0
61/60 -4.7c 2 2:0
62/33 -8.2c 66 0:11
62/35 +6.6c 59 5:9
62/37 -6.3c 54 0:9
62/39 +2.6c 48 0:8
62/41 -0.7c 43 1:7
62/43 +0.2c 38 2:6
62/45 +4.8c 33 3:5
62/47 -3.8c 29 5:4
62/49 +7.4c 24 0:4
62/51 +4.8c 20 2:3
62/53 +4.9c 16 4:2
62/55 +7.4c 12 0:2
62/57 -4.4c 9 3:1
62/59 +2.5c 5 5:0
62/61 -5.2c 2 2:0
63/32 +6.1c 70 4:11
63/34 +1.1c 64 4:10
63/37 +4.7c 55 1:9
63/38 -8.1c 53 5:8
63/40 +3.1c 47 5:7
63/41 -6.3c 45 3:7
63/43 -5.4c 40 4:6
63/44 +4.8c 37 1:6
63/46 -5.5c 33 3:5
63/47 +7.2c 30 0:5
63/50 +0.1c 24 0:4
63/52 -1.1c 20 2:3
63/53 -0.8c 18 0:3
63/55 +1.8c 14 2:2
63/58 -6.8c 9 3:1
63/59 -3.1c 7 1:1
63/61 +5.9c 3 3:0
63/62 -5.6c 2 2:0
64/33 -3.3c 69 3:11
64/35 -5.1c 63 3:10
64/37 -1.3c 57 3:9
64/39 +7.5c 51 3:8
64/41 +4.3c 46 4:7
64/43 +5.1c 41 5:6
64/45 -6.9c 37 1:6
64/47 +1.2c 32 2:5
64/49 -4.3c 28 4:4
64/51 -6.9c 24 0:4
64/53 -6.8c 20 2:3
64/55 -4.3c 16 4:2
64/57 +0.5c 12 0:2
64/59 +7.5c 8 2:1
64/61 -0.2c 5 5:0
64/63 -6.1c 2 2:0

>
>
>
>
>You do not need web access to participate. You may subscribe through
>email. Send an empty email to one of these addresses:
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - unsubscribe from the tuning group.
> tuning-nomail@yahoogroups.com - put your email message delivery on hold for the tuning group.
> tuning-digest@yahoogroups.com - change your subscription to daily digest mode.
> tuning-normal@yahoogroups.com - change your subscription to individual emails.
> tuning-help@yahoogroups.com - receive general help information.
> >
>Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/ >
>
>

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> **Hi Paul!
>
> Actually, since 24-tET is imbedded *within* 72-tET, shouldn't they
do
> the 11th limit about the same??
>
> Joseph

Hi, Joseph! How so?...since 72 yields three divisions for every one
division of 24?! 24 works for the 11th harmonic so well only because
the prime 11 happens to fall almost exactly on the quarter tone as
defined in 12t-ET. 7 is not so lucky in this context, despite its
reputation for luck in others.

Cheers,

Bob

--- In tuning@y..., Rick Tagawa <ricktagawa@e...> wrote:

/tuning/topicId_37831.html#37858

>
> 72-tET Approximation of Just Intervals
>

***Thanks, Rick, for sending along this incredibly useful chart!

J. Pehrson

--- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:

/tuning/topicId_37831.html#37859

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> > **Hi Paul!
> >
> > Actually, since 24-tET is imbedded *within* 72-tET, shouldn't
they
> do
> > the 11th limit about the same??
> >
> > Joseph
>
> Hi, Joseph! How so?...since 72 yields three divisions for every one
> division of 24?! 24 works for the 11th harmonic so well only
because
> the prime 11 happens to fall almost exactly on the quarter tone as
> defined in 12t-ET. 7 is not so lucky in this context, despite its
> reputation for luck in others.
>
> Cheers,
>
> Bob

Hi Bob!

Oh... so you mean the 11-limit emulations in 72 are *not* coming from
the quartertone sets in that scale...

JP

In a message dated 6/17/02 12:12:06 PM Eastern Daylight Time,
rwendell@cangelic.org writes:

> Bob:
> Well, there seems to be a persistent implication in your posts,
> Johnny, that musicians and singers who can easily and quickly spot
> deviations from just in the sub-ten cent range abound, some even in
> the sub-three cent range, just oodles of them everywhere. I would
> like to run into a collection of such musicians in one place
> sometime.

Please come visit us in New York. Here you can meet someone for each and
every instrument who can play with the precision I demand.

The number of commercially available recordings of
musicians (other than fixed tuned instrumentalists) who play or sing
> with this kind of precision is vanishingly small in my experience. I
> have a few such recordings, but they are exceptional and I wish I had
> more.

Actually, I do have many, many such recordings in the AFMM archive. The
business of releasing CDs is a terrible lot and I have been working to have
greater repertoire so that when they do come out there will be the right
apparatus and the necessary demand. Frankly, demand is still low. The world
is a bit distracted right now.

> On the other hand, or rather, for the same reason, the statement
> that "no conductor would allow" a just third at the end of a symphony
> movement I find to be a bit hard to swallow.

As a bassoonist in many orchestras, I know what I can get away with. This
playing of a Picardy third in a final cadence would never balance the rest of
the piece correctly for a conductor for the Romantic period for certain. Now
playing bassoon to the piano in Otto Luening's Sonata for Bassoon was to be
planting just relationships around piano anchors. This is what Otto wanted.
A bassoonist could never supplant the harmony of an entire piece of music by
injecting a just interval. An exception I recently heard was a bassist
singing an ascending 7/4 in a Henze opera.

I have heard string > quartets, for example, play just triads fairly
> consistently in the
> performance of compositions even from the romantic period, not to
> mention classical.
>

String Quartets are a different matter. Old studies have demonstrated that
they are more likely to play sharper thirds than flat (e.g., famous Boston
study). Sure, they can do anything and any leader can ask any which way.
And any composer can have a different preference (e.g., Ives preferred sharp
thirds as a norm, Luening wanted flat thirds).

> A conductor "not allowing" a just third in the final triad of a
> symphony implies that he/she cares about the difference enough to
> disallow such a thing. I find this VERY hard to swallow, since I
> detect a HUGE body of symphony conductors who let pitch errors on the
> order of 10-14 cents go all the time, and not in the service of JI,
> but simply random mistunings. (I would not complain if they were
> just.)

Conductors that employ huge vibrato will always camouflage intent. But do
you think a conductor can have no exact crystal clear concept of where the
intonation should be? Doesn't Boulez know? Simply because it is
expressionistly employed, without apparent rhyme or reason to you and others,
it means there isn't any? Nope. I don't buy that. I'm finding connections
between Bach's choice of keys in Werckmeister and the way Casals plays the
cello suites, even though Casals never consulting the Werckmeister tunings
(to my knowledge).

Where does this kind of assessment come from, Johnny? Do you
> live on a different planet?
> Respectfully puzzled,
> Bob

My assessment comes from a singular career in performing microtonal music of
all styles in New York.

best, Johnny

--- In tuning@y..., Rick Tagawa <ricktagawa@e...> wrote:

> 13/8 +7.2c 50 2:8

> 19/13 +7.0c 39 3:6

> 19/16 -2.5c 18 0:3

i wouldn't put too much importance on this table, since, as you can
see from the values above, the numbers don't even agree with one
another for a simple harmonic series chord like 8:13:19 --

13/8 is 50 steps;

19/13 is 39 steps;

so 19/8 should be 89 steps;

19/16 should be 89 - 72 = 17 steps;

but the table says 19/16 is 18 steps!

so the quality of approximation that the table suggests in the 19-
limit and above is often not acheivable in practice, where chords
(rather than mere dyads) are typically used.

In a message dated 6/17/02 11:26:58 AM Eastern Daylight Time,
paul@stretch-music.com writes:

> In tuning@y..., Afmmjr@a... wrote:
> > In a message dated 6/16/02 10:04:42 PM Eastern Daylight Time,
> > jpehrson@r... writes:
> >
> >
> > > Well, my guess *off the top* is that it would be possible in
> slower
> > > passages but not necessarily in faster....
> > >
> >
> > My question is simple to you: what is just sounding, other than
> lack of
> > beats?
>
> if that's your criterion, then if you play in 72-equal, with
> harmonies going by 4 or 5 per second, there's no time for beats to
> occur, so then by your own definition it's just sounding, right?
>

I never said that is my criterion: I posed a question. You misinterpret me.

> > > Surely, 72-tET is going to do a better job of Just emulation than
> > > *quartertones!* So it seems...
> > >
> >
> > Not for the 11th limit it doesn't.
>
> in the 11-limit:
>
> 72-equal does just as well as 24-equal for 11:1, 11:3, 11:9, 9:1, 3:1.
> 72-equal does far better than 24-equal for 11:5, 11:7, 9:5, 9:7, 7:5,
> 7:3, 7:1, 5:3, 5:1.
>
> so i have to differ.
>

While this may have cleared up a bit regarding 72 and 11 limit being better
than 24: but who ever demanded better 11 limit intervals? No one. If no one
wants to volunteer what makes for the virtue of being just in intervals, I'll
certainly try. It seems most of the JI composers are off list right now.

best, Johnny Reinhard

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:

> Oh... so you mean the 11-limit emulations in 72 are *not* coming
from
> the quartertone sets in that scale...

some of them are -- namely 11:1, 11:3, and 11:9. however, 11:5 and
11:7 are 11-limit intervals that 72-equal approximates very well, but
24-equal can't.

--- In tuning@y..., Afmmjr@a... wrote:

> I never said that is my criterion: I posed a question. You
>misinterpret me.

oops -- sorry, johnny! so how would you answer the question you posed?

> > > > Surely, 72-tET is going to do a better job of Just emulation
than
> > > > *quartertones!* So it seems...
> > > >
> > >
> > > Not for the 11th limit it doesn't.
> >
> > in the 11-limit:
> >
> > 72-equal does just as well as 24-equal for 11:1, 11:3, 11:9, 9:1,
3:1.
> > 72-equal does far better than 24-equal for 11:5, 11:7, 9:5, 9:7,
7:5,
> > 7:3, 7:1, 5:3, 5:1.
> >
> > so i have to differ.
> >
>
> While this may have cleared up a bit regarding 72 and 11 limit
being better
> than 24: but who ever demanded better 11 limit intervals? No one.

who ever demanded better 11-limit intervals than what? i'm missing
your point here.

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:

/tuning/topicId_37831.html#37863

> --- In tuning@y..., Rick Tagawa <ricktagawa@e...> wrote:
>
> > 13/8 +7.2c 50 2:8
>
> > 19/13 +7.0c 39 3:6
>
> > 19/16 -2.5c 18 0:3
>
> i wouldn't put too much importance on this table, since, as you can
> see from the values above, the numbers don't even agree with one
> another for a simple harmonic series chord like 8:13:19 --
>
> 13/8 is 50 steps;
>
> 19/13 is 39 steps;
>
> so 19/8 should be 89 steps;
>
> 19/16 should be 89 - 72 = 17 steps;
>
> but the table says 19/16 is 18 steps!
>
> so the quality of approximation that the table suggests in the 19-
> limit and above is often not acheivable in practice, where chords
> (rather than mere dyads) are typically used.

***Thanks, Paul! I actually found this slightly humorous, apologies
to Rick...

JP

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:

/tuning/topicId_37831.html#37865

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
>
> > Oh... so you mean the 11-limit emulations in 72 are *not* coming
> from
> > the quartertone sets in that scale...
>
> some of them are -- namely 11:1, 11:3, and 11:9. however, 11:5 and
> 11:7 are 11-limit intervals that 72-equal approximates very well,
but
> 24-equal can't.

***Got it! Thanks!

JP

In a message dated 6/17/02 2:24:19 PM Eastern Daylight Time,
paul@stretch-music.com writes:

> who ever demanded better 11-limit intervals than what? i'm missing
> your point here.

Paul, I am suggesting that 11-limit closeness to just is a red herring. No
composer to my knowledge has ever urged a temperament for better 11-limit
intervals. Do you know any? There are none. I would further suggest that
pure 11-limit will make its most musical sense with a real just 7-limit, etc.

On a related point, I've been trying to raise the issue about defining the
justness we are trying to attain in a temperament like 72. I have been
discussing the products produced by 72-tone composers that do not match up
with the stated theories. At your encouragement I will take a stab at this.
But since I was just interrupted by a long distance call....I'll try a bit
later. best, Johnny Reinhard

Isn't part of what we're about here the creation of new possibilities
for composition? Why should we care about how many composers have
asked for better 11-limit approximations? How many composers write
for anything but common practice performance in the first place?...a
pretty small minority, I would venture.

I would like to explicitly illustrate why I, for one, would be
interested in good 11- and 13-limit approximations. I'm personally
angling (slowly) toward the following 13-limit just blues scale
(using C as an example to make it more concrete) that I have
tentatively defined, where the second number in the ratios refers to
the tonic at 1/1 as is conventional in JI:

C=1, Eb=7/6, neutral 3rd=11/9, E nat=5/4, F=4/3, F#- = 11/8, F#+ =
13/9, G=3/2, Bb=7/4

In case the F#+ at 13/9 raises eyebrows, I find the typical F#
leading up to a G in C blues a much higher value than the more
familiar F#- leading down to the F natural. 13/9 puts the perfect
fifth on G in a 27/26 relationship with this F#+, even skinnier than
the "normal" 5-limit chromatic half-step. It also puts the F#+ in a
clean 13/11 relationship with the neutral third.

In this scale, the famous "flat third" chord spelled in common
practice as C-E-G-Bb-Eb would put the lower E natural and the higher
(2*7)/6 Eb in a 28/15 (4*7/3*5) relationship aligned right along the
harmonic series. I'm interested in playing with polyphonic
counterpoint "a la renaissance" in the context of a generally more
contemporary rhythmic style in this kind of bluesy harmonic context
with all its interesting potential for unique modulation schemes,
etc. and see what falls out, where it leads me esthetically.

Why use a temperament instead of straight 13-limit JI? ...to liberate
my creative attention, pure and simple. Hope this clarifies mine and
others' motives for ET approximations of JI.

Cheers,

Bob

--- In tuning@y..., Afmmjr@a... wrote:
> In a message dated 6/17/02 2:24:19 PM Eastern Daylight Time,
> paul@s... writes:
>
>
> > who ever demanded better 11-limit intervals than what? i'm
missing
> > your point here.
>
> Paul, I am suggesting that 11-limit closeness to just is a red
herring. No
> composer to my knowledge has ever urged a temperament for better 11-
limit
> intervals. Do you know any? There are none. I would further
suggest that
> pure 11-limit will make its most musical sense with a real just 7-
limit, etc.
>
> On a related point, I've been trying to raise the issue about
defining the
> justness we are trying to attain in a temperament like 72. I have
been
> discussing the products produced by 72-tone composers that do not
match up
> with the stated theories. At your encouragement I will take a stab
at this.
> But since I was just interrupted by a long distance call....I'll
try a bit
> later. best, Johnny Reinhard

>
> On a related point, I've been trying to raise the issue about
defining the justness we are trying to attain in a temperament like
72. I have been discussing the products produced by 72-tone
composers that do not match up with the stated theories.

***This has already been well defined. We know what the tolerances
are of 72-tET regarding Just. As far as *I'm* concerned 72-tET *IS*
Just, and I'm willing to tolerate the 3 or 4 cents discrepancies.

I intend to continue using 72-tET with that objective in mind.

I can certainly understand composers who also have *theories* about
Just not liking 72-tET.

For instance: it would *obviously* not work for La Monte Young. A
discrepancy of 3 or 4 cents would cause a beat about every *minute*...

[is that right, somebody??, I need help with the math...]

In any case, it would be opposed to La Monte Young's *theories* and
ideas, so *of course* he isn't going to be able to use just.

Likewise, Partch, Kraig Grady, Pat Pagano... a whole bunch of other
people who like the *purity* of just ratios are not going to want to
use 72...

So??

I happen to find 72-tET practical... and I'm glad people pointed it
out to me!

Joseph

--- In tuning@y..., Afmmjr@a... wrote:
> In a message dated 6/17/02 2:24:19 PM Eastern Daylight Time,
> paul@s... writes:
>
>
> > who ever demanded better 11-limit intervals than what? i'm
missing
> > your point here.
>
> Paul, I am suggesting that 11-limit closeness to just is a red
herring. No
> composer to my knowledge has ever urged a temperament for better 11-
limit
> intervals. Do you know any?

erv wilson
james tenney
wendy carlos
herman, gene, etc.

> There are none.

right . . .

> I would further suggest that
> pure 11-limit will make its most musical sense with a real just 7-
>limit, etc.

meaning?

> I have been
> discussing the products produced by 72-tone composers that do not
match up
> with the stated theories.

according to jay williams and other listeners, there's more of a
match than you're giving the composers credit for. but why make such
pronouncements in advance? surely the bulk of this work is yet to
come . . . we may even be seeing some from eaton, if joseph is to be
believed . . .

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:

> For instance: it would *obviously* not work for La Monte Young. A
> discrepancy of 3 or 4 cents would cause a beat about every
>*minute*...
>
> [is that right, somebody??, I need help with the math...]

it's more like a beat every second, though it depends on which
interval, at what absolute pitch level, you're talking about.

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:

/tuning/topicId_37831.html#37882

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
>
> > For instance: it would *obviously* not work for La Monte Young.
A
> > discrepancy of 3 or 4 cents would cause a beat about every
> >*minute*...
> >
> > [is that right, somebody??, I need help with the math...]
>
> it's more like a beat every second, though it depends on which
> interval, at what absolute pitch level, you're talking about.

***Whoops. Actually, I *knew* that from my piano tuning
experiences... I don't know what I was thinking... not much.

JP

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
>
> /tuning/topicId_37831.html#37882
>
> > --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> >
> > > For instance: it would *obviously* not work for La Monte Young.
> A
> > > discrepancy of 3 or 4 cents would cause a beat about every
> > >*minute*...
> > >
> > > [is that right, somebody??, I need help with the math...]
> >
> > it's more like a beat every second, though it depends on which
> > interval, at what absolute pitch level, you're talking about.
>
>
> ***Whoops. Actually, I *knew* that from my piano tuning
> experiences... I don't know what I was thinking... not much.

As far as I'm concerned, whether there are _no_ beats, or if the beats
are slow enough that you can actually _count_ them, then it's just.

For me personally, in most cases, slow beating is much more
interesting (and human-like) than no beating.

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> ... As far as I'm concerned, whether there are _no_ beats, or if
the beats
> are slow enough that you can actually _count_ them, then it's just.

For me, if you can count them, it's tempered. (If you're doing it by
ear, then you set a temperament by counting beats, right?) However,
if they're so slow that it's difficult to count them, then it's just.

> For me personally, in most cases, slow beating is much more
> interesting (and human-like) than no beating.

I think that no beating sounds stagnant or dead, but very slow
beating is like letting in a bit of fresh air. Microtempering a JI
set can accomplish this, while making consonances out of some near-
misses. (My upper limit is about 3.25 cents error for most
intervals, but half of that for 2:3's. And 217-ET does considerably
better than that.)

--George

--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
> --- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > ... As far as I'm concerned, whether there are _no_ beats, or if
> the beats
> > are slow enough that you can actually _count_ them, then it's
just.
>
> For me, if you can count them, it's tempered.

Might it not simply be less-than-perfectly-tuned just, due to a myriad
of physical and human factors?

> (If you're doing it by
> ear, then you set a temperament by counting beats, right?)

Right, but you're counting beats in those intervals which are
essentially just. You can't tune 12-equal by counting beats in the
thirds, nor meantone by counting beats in the fifths, right?

> I think that no beating sounds stagnant or dead, but very slow
> beating is like letting in a bit of fresh air. Microtempering a JI
> set can accomplish this, while making consonances out of some near-
> misses. (My upper limit is about 3.25 cents error for most
> intervals, but half of that for 2:3's. And 217-ET does considerably
> better than that.)

Then we are in substantial agreement. I think you will find that beats
between harmonics in an interval that deviates from a small
whole-number ratio by 3.25 cents will be too fast to count over much
of the piano compass.

Here's a easily remembered rule of thumb for mentally calculating
beats, given the frequency of the relevant harmonics and the error in
cents. At 880 Hz (e.g. 2nd harmonic of concert A) the beat rate in Hz
is approximately equal to the error in cents (provided the error is
less than 20 cents). I just worked this out, so someone had better
check it. :-)

In a message dated 6/19/02 2:49:08 PM Eastern Daylight Time,
gdsecor@yahoo.com writes:

> I think that no beating sounds stagnant or dead, but very slow
> beating is like letting in a bit of fresh air.

This is completely tone based. A rich musical tone sounds gorgeously in an
acoustically resonant space. But tinny electronics or flat, lifeless
playing, dynamically flawed, would implode without some beats.

Johnny

In a message dated 6/19/02 7:45:35 PM Eastern Daylight Time,
d.keenan@uq.net.au writes:

> Then we are in substantial agreement. I think you will find that beats
> between harmonics in an interval that deviates from a small
> whole-number ratio by 3.25 cents will be too fast to count over much
> of the piano compass.
>
>

Obviously, you mean out of time. A piano piece will not allow one to count
beats within the confines of the piece. It's like someone screaming to a
friend, "Listen!...do you hear the flute that is missing. Ah it's really
beautiful." best, Johnny

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> Here's a easily remembered rule of thumb for mentally calculating
> beats, given the frequency of the relevant harmonics and the error
in
> cents. At 880 Hz (e.g. 2nd harmonic of concert A) the beat rate in
Hz
> is approximately equal to the error in cents (provided the error is
> less than 20 cents). I just worked this out, so someone had better
> check it. :-)

It is correct with 2%.
when deltaF/F << 1, the approximation

ln(1 + (deltaF/F)) ~= deltaF/F

is fairly good (try it for values less than 0.01 !!!)
So, stating that:

beatF = 2 * deltaF

(because when counting, we disregard phase change of the beat
enveloppe)

The classic formula for calculating cents approximates to

delta¢ ~= (deltaF/F) * (1200/ln(2))

thus:

beatF ~= delta¢ * F * ln(2)/600

That lead to a "generalized rule of thumb"

beatF = delta¢ * F *.001155

for F = 880 Hz:

beatF ~= 1.017 * delta¢

close within 2% to

beatF ~= delta¢

yours truly

François Laferriere

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
> > --- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> >
> > > ... As far as I'm concerned, whether there are _no_ beats, or
if
> > the beats
> > > are slow enough that you can actually _count_ them, then it's
> just.
> >
> > For me, if you can count them, it's tempered.
>
> Might it not simply be less-than-perfectly-tuned just, due to a
myriad
> of physical and human factors?
>
> > (If you're doing it by
> > ear, then you set a temperament by counting beats, right?)
>
> Right, but you're counting beats in those intervals which are
> essentially just. You can't tune 12-equal by counting beats in the
> thirds,

True, you can't do it quantitatively. But you can do a qualitative
check once you have set the temperament by playing the chromatically
ascending thirds to ensure that the beat rates gradually increase.

> nor meantone by counting beats in the fifths, right?

The D to A in the middle octave is about 2.5 times per second, which
is easy enough to count. How else would you temper the fifths by ear
in meantone?

> > I think that no beating sounds stagnant or dead, but very slow
> > beating is like letting in a bit of fresh air. Microtempering a
JI
> > set can accomplish this, while making consonances out of some
near-
> > misses. (My upper limit is about 3.25 cents error for most
> > intervals, but half of that for 2:3's. And 217-ET does
considerably
> > better than that.)
>
> Then we are in substantial agreement.

Yes.

--George

--- In tuning@y..., Afmmjr@a... wrote:
> In a message dated 6/19/02 2:49:08 PM Eastern Daylight Time,
> gdsecor@y... writes:
>
> > I think that no beating sounds stagnant or dead, but very slow
> > beating is like letting in a bit of fresh air.
>
> This is completely tone based. A rich musical tone sounds
gorgeously in an
> acoustically resonant space. But tinny electronics or flat,
lifeless
> playing, dynamically flawed, would implode without some beats.
>
> Johnny

Your point is well taken. My conclusion was reached many years ago
using a completely electronic instrument, the Motorola Scalatron.

It just occurred to me that you and others might be interested in an
experience I had with that instrument some years ago. The following
text (slightly edited) is from a book I am writing.

<< In the spring of 1975, when the instrument with the Bosanquet
generalized keyboard was under development, Erv Wilson made a trip to
the factory in Arlington Heights, Illinois to see it.

While he was there we tried out a tuning in just intonation that he
had just developed (a 20-tone Eikosany, I believe). The digitally
tuned Scalatron had available 1024 different pitches in the octave in
a subharmonic series, and the actual pitches employed for any tonal
system, whether tempered or just, would usually be an approximation
of the theoretical pitches, with a maximum error ranging from 0.4 to
0.8 cents, depending on which part of the series was required for
each particular tone. The exception to this is that just intervals
can be made exact if each tone in the system can be matched to one of
the subharmonics in a series from 1025 to 2048, which turned out to
be the case with Erv's tuning. The entire tuning could even be put
in the proper "key" by adjusting the frequency of the master
oscillator, the "fundamental" of the series. Once all of this was
done, all of the pitches were phase-locked into exact ratios.

When Erv played his phase-locked tuning, it was just intonation such
as I had never heard before, and I would have to describe its effect
as nothing short of amazing. While his 13-limit tuning in itself was
remarkable, some of the chords had a resonance that surpassed
anything I had ever heard before (or since). I did not even find any
problem with any stagnant effect. In one respect the experience
could be compared to viewing an immense crystal from various
directions, with the direction of view changing with each chord. (It
was not until I subsequently saw Erv's diagram of this tuning that I
realized how appropriate this description is.) Each chord seemed to
be a facet and each tone of a chord the corner of a facet, with each
facet having the appearance of the surface of a still pond. The
slightest movement in the air between the sound source and my ears
had the effect of temporarily disturbing the pond, and even turning
my head produced the slight ripple of a momentary phase shift. When
one of the workers in the facility walked through the room while Erv
was playing, the effect of the disturbance of the air on the sound
was evident to everyone in the room.

With the possible exception of the experience produced by the phase-
locked sine waves of the Scalatron synthesizer built for La Monte
Young, just intonation under laboratory conditions such as these is
virtually unattainable in an actual musical performance in what is
ordinarily thought to be just intonation. The latter is only an
approximation of rational intervals, a _de facto_ temperament, with
the accuracy dependent upon the pitch stability of the instrument
(for instruments of fixed pitch) or the ability of the performer (for
voices, unfretted strings, winds, etc.). >>

--George

George,

--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
> It just occurred to me that you and others might be interested in
> an experience I had with that instrument some years ago.

[snip]

That was a vivid and fascinating anecdote. It makes me think, especially with your coda about the ability (or inability) to recreate this kind of purely tuned experience with live performers: even if the incredible accuracy needed for this effect to take place, is it not a valid 'goal' to shoot for? If we *know* this kind of effect can take place (as *you* certainly know), does it not follow that one might want to compose with this in mind, knowing full well that the accuracy might not produce it but in coming close would 'approach' it?

It seems to me that when phenomena of this manner are known, there is nothing wrong with conceptualising along those lines, and working toward achieving it. If starting from the get-go with a compromised tuning - even if the compromise fits in with what we think/know is an 'accepted' error value - we are crippling at least one kind of vision.

Well, whatever. I just really like your experiential exposition, for a *lot* of reasons!

Cheers,
Jon

--- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:

> While he was there we tried out a tuning in just intonation that he
> had just developed (a 20-tone Eikosany, I believe). The digitally
> tuned Scalatron had available 1024 different pitches in the octave in
> a subharmonic series, and the actual pitches employed for any tonal
> system, whether tempered or just, would usually be an approximation
> of the theoretical pitches, with a maximum error ranging from 0.4 to
> 0.8 cents, depending on which part of the series was required for
> each particular tone.

When I first had a chance to play with electronic microtonality, it was with a box my brother built which also used undertone tuning; a frequency counter would count the vibrations of a crystal in the megahertz range, and then play a square wave in at an undertone of this ultrahigh frequency. Since the crystal was quite stable, the result was very precise tuning, quite precise even for high notes. It sounds as if this idea can be adapted to something other than square waves, and it seems to me this is an idea which still might be worth adopting.

--- In tuning@y..., "jonszanto" <JSZANTO@A...> wrote:
> George,
>
> --- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
> > It just occurred to me that you and others might be interested in
> > an experience I had with that instrument some years ago.
>
> [snip]
>
> That was a vivid and fascinating anecdote. It makes me think,
>especially with your coda about the ability (or inability) to
>recreate this kind of purely tuned experience with live performers:
>even if the incredible accuracy needed for this effect to take
>place, is it not a valid 'goal' to shoot for? If we *know* this kind
>of effect can take place (as *you* certainly know), does it not
>follow that one might want to compose with this in mind, knowing
>full well that the accuracy might not produce it but in coming close
>would 'approach' it?

if i may jump in, i would answer yes, wholeheartedly, but this still
doesn't go any way toward arguing for a strict-ji approach over an
adaptive-ji approach.

> It seems to me that when phenomena of this manner are known, there
>is nothing wrong with conceptualising along those lines, and working
>toward achieving it. If starting from the get-go with a compromised
>tuning - even if the compromise fits in with what we think/know is
>an 'accepted' error value - we are crippling at least one kind of
>vision.

--- In tuning@y..., "francois_laferriere" <francois.laferriere@o...>
wrote:
> --- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > Here's a easily remembered rule of thumb for mentally calculating
> > beats, given the frequency of the relevant harmonics and the error
> in
> > cents. At 880 Hz (e.g. 2nd harmonic of concert A) the beat rate in
> Hz
> > is approximately equal to the error in cents (provided the error
is
> > less than 20 cents). I just worked this out, so someone had better
> > check it. :-)
>
> It is correct with 2%.
...
> François Laferriere

Thanks François.

--- In tuning@y..., "jonszanto" <JSZANTO@A...> wrote [#38017]:
> George,
>
> --- In tuning@y..., "gdsecor" <gdsecor@y...> wrote [#38014]:
> > It just occurred to me that you and others might be interested in
> > an experience I had with that instrument some years ago.
>
> [snip - phase-locked tuning experience]
>
> That was a vivid and fascinating anecdote. It makes me think,
especially with your coda about the ability (or inability) to
recreate this kind of purely tuned experience with live performers:
even if the incredible accuracy needed for this effect to take place,
is it not a valid 'goal' to shoot for? If we *know* this kind of
effect can take place (as *you* certainly know), does it not follow
that one might want to compose with this in mind, knowing full well
that the accuracy might not produce it but in coming close
would 'approach' it?
>
> It seems to me that when phenomena of this manner are known, there
is nothing wrong with conceptualising along those lines, and working
toward achieving it. If starting from the get-go with a compromised
tuning - even if the compromise fits in with what we think/know is
an 'accepted' error value - we are crippling at least one kind of
vision.
>
> Well, whatever. I just really like your experiential exposition,
for a *lot* of reasons!
>
> Cheers,
> Jon

Try as you wish and try all you want, but I don't it's possible to
achieve the phenomenon I described with acoustical (i.e., non-
electronic) instruments. There are too many variables (or transient
elements) in a performance besides pitch that must also be kept under
control so as not to disturb the effect, and these involve the
audience as well as the performers. You might be able to come close
for a few fleeting moments here and there, but they would pass by so
quickly that I doubt anyone would notice.

Perhaps this would be best left to the realm of electronic music, and
let live musicians do what they do best -- playing with expressive
spontaneity.

But if my experience encourages others to strive to improve their
intonation, then by all means, go for it! There is at least as much
to be appreciated in the striving for perfection as in perfection
itself.

--George

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "gdsecor" <gdsecor@y...> wrote:
>
> > While he was there we tried out a tuning in just intonation that
he
> > had just developed (a 20-tone Eikosany, I believe). The
digitally
> > tuned Scalatron had available 1024 different pitches in the
octave in
> > a subharmonic series, and the actual pitches employed for any
tonal
> > system, whether tempered or just, would usually be an
approximation
> > of the theoretical pitches, with a maximum error ranging from 0.4
to
> > 0.8 cents, depending on which part of the series was required for
> > each particular tone.
>
> When I first had a chance to play with electronic microtonality, it
was with a box my brother built which also used undertone tuning; a
frequency counter would count the vibrations of a crystal in the
megahertz range, and then play a square wave in at an undertone of
this ultrahigh frequency. Since the crystal was quite stable, the
result was very precise tuning, quite precise even for high notes. It
sounds as if this idea can be adapted to something other than square
waves, and it seems to me this is an idea which still might be worth
adopting.

That's exactly the way it was done on the Motorola Scalatron to
achieve the pitches in the top octave; these were subsequently
divided repeatedly by 2 to get the rest of the octaves.

Some even-numbered partials were introduced by adding in some square
wave input 1, 2, and 3 octaves higher at lower amplitudes. In this
way a stairstep waveshape was successfully used to simulate a
sawtooth wave. The output was then filtered in various ways.

--George