Yahoo Tuning Groups Ultimate Backup tuning-math My first post: sine-based well-temperament

Hello, and happy Easter if you're celebrating.

I'm already a member of tuning and MMM, and I'm more interested in the practical than the theoretical - but I've been experimenting with extended Pythagorean so much lately. I've decided on a 41-tone "well-temperament" for my own music, balancing out the need for smooth transposition and accurate 11-limit JI approximation (Partch and Wilson being the two most important models). I posted the pitches in tuning, and I hate to cross-post, so I'll just explain it in brief. It's 9 pure fifths up and down and 23 wolves, easily generated by Scala.

That led me to another experiment, and this may have been done before: well-temperaments with the amount of tempering of each fifth derived from the sine function. This example is a "smoothed out" variation on Kirnberger III all pitches in cents:

0.00000 (1/1)
101.55531
198.04500
302.42936
397.66152
501.08095
600.00000
698.91905
802.33848
897.57064
1001.95500
1098.44469
1200.00000 (2/1)

(For a Kirnberger similarity, 1/1 should be assigned to the note D.)

The formula for the temperament (negative in this case):

T = sin (n*pi/6) * log_2 (531441/524288)
(n = number of fifths away from the central note; if n is negative, |n| is the number of fourths away)

Then subtract T from the untempered pitch.

I compared my temperament with the Scala archive, and the files with the closest pitches are "sorge1.scl" (Georg Andreas Sorge, 1744) and "scottd4.scl" (Dale Scott's temperament #4, 1999).

I've done something similar with 41-tone, but put the steeper temperings at the most remote ends of the chain of fifths, and used the 41-tone comma. I don't know how practical such a complicated temperament really is, especially considering 1/23 of a 41-tone comma is only 0.86282 cents, but it's fun anyways.

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Danny Wier" <dawiertx@s...> wrote:

> I'm already a member of tuning and MMM, and I'm more interested in the
> practical than the theoretical - but I've been experimenting with
extended
> Pythagorean so much lately. I've decided on a 41-tone
"well-temperament" for
> my own music, balancing out the need for smooth transposition and
accurate
> 11-limit JI approximation (Partch and Wilson being the two most
important
> models).

It seemed to me it didn't deliver that much accurate 11-limit tuning.
The 11-limit schismatic temperament which seems to be relevant here is
the 53&118 temperament, and flattening the fifths about a quarter of a
cent might work better as a starting point for a well-temperament
based around this.

Another alternative would be Miracle[41], which I think has been named
Studloco, which has (in the 72-equal tuning) 35 fifths of 700 cents
each, and six wolf fifths of 716.667 cents each. While that does not
close the circle of fifths to the accuracy you seem to want, it has
the advantage giving you a lot of 11-limit harmony. If you try it out
you could be the first ever to actually write anything in Studloco.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Danny Wier <dawiertx@sbcglobal.net>

Gene Ward Smith wrote:

> It seemed to me it didn't deliver that much accurate 11-limit tuning.
> The 11-limit schismatic temperament which seems to be relevant here is
> the 53&118 temperament, and flattening the fifths about a quarter of a
> cent might work better as a starting point for a well-temperament
> based around this.

If you mean 53- and 118-EDO, then I compared my tuning with the former, and my temperament has a better approximation of Partch's just scale, minus 11/10 and 20/11, and Wilson's 41-tone pure scale. I haven't considered 118 enough however; I only recently discovered it while playing with a circle of thirds.

I don't like 41-EDO because of its relative weakness in 5-limit, though it is strong on 7- and 11-limit. My goal was to get the best features of both 41- and 53-tone, and still have a smoother trip around the circle of fifths. I keep the fifths true for a while then start widening them ever-so-slightly to better reach the septimals and undecimals.

This temperament is by no means a final product; I'm still experimenting.

> Another alternative would be Miracle[41], which I think has been named
> Studloco, which has (in the 72-equal tuning) 35 fifths of 700 cents
> each, and six wolf fifths of 716.667 cents each. While that does not
> close the circle of fifths to the accuracy you seem to want, it has
> the advantage giving you a lot of 11-limit harmony. If you try it out
> you could be the first ever to actually write anything in Studloco.

I already use MIRACLE when improvising on fretless bass, since my instrument is defretted and still has lines in 12-tone, and it makes it easier when playing with guitarists. And yes, 41-out-of-72-ET is indeed called Studloco.

The kind of music I want to write would be compatible with multiple tunings, including MIRACLE, Pythagorean and schismic. But I might run into a problem with 72-tone, since my old 12-tone compositions got really adventurous in modulation and polyphony.

Whatever the case, meantone is out, because I like having two different sizes for a major second, and I intend to exploit them plenty.

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Danny Wier" <dawiertx@s...> wrote:

> If you mean 53- and 118-EDO, then I compared my tuning with the
former, and
> my temperament has a better approximation of Partch's just scale, minus
> 11/10 and 20/11, and Wilson's 41-tone pure scale. I haven't
considered 118
> enough however; I only recently discovered it while playing with a
circle of
> thirds.

118 shows up as a good system particularly in the 5 and 11 limits. By
53&118 I mean the (in this case, 11-limit) temperament supported by
both; with generator a slightly flattened fifth/sharpened fourth, and
a mapping to primes given by

[<1 2 -1 19 -9|, <0 -1 8 -39 30|]

In order for this to come out exactly to 11, the fourth should be
5632^(1/30), about a third of a cent sharp. The 7 is 19 octaves up and
39 fourths down, and for that to be exact, the fourth should be
(524288/7)^(1/39), about 0.19 cents sharp. A good compromise value is
118 equal, which has a fourth about 0.26 cents sharp. A fifth this
flat, or fourth this sharp, works well for this temperament and you at
least want it in a range near to the 118 value. If you move it more in
the direction of making the 7 better, you get 171-equal, which is very
accurate in the 7-limit, but at the expense of 11. This system only
barely gives you any 7s within the scope of 41 notes, and
really it would make more sense to go up to 53 notes, or use another
system, as below.

Another schismatic system is 41&53, which probably makes more aense
for what you are doing. It involves slightly sharp fifths rather than
slightly flat ones. This is a less complex system, but also a less
accurate one. The MOS for this in 94-equal is 40 fifths of 55/94, and
one of 56/94, 12.9 cents sharp. Having more than one "wolf" fifth
would turn it into a well-temperament. The mapping in this case is

[<1 2 -1 -3 13|, <0 -1 8 14 -23|]

where the notation means, for instance, that an 11 is given by 30
generators of a fourth, down nine octaves.

That does sound like you were aiming at 41&53, then. You could try
starting out with fifths about 0.23 cents sharp instead of pure
fifths, and then widen a touch at the end to complete the circle of
fifths.

🔗Graham Breed <gbreed@gmail.com>

Danny Wier wrote:

> If you mean 53- and 118-EDO, then I compared my tuning with the former, and > my temperament has a better approximation of Partch's just scale, minus > 11/10 and 20/11, and Wilson's 41-tone pure scale. I haven't considered 118 > enough however; I only recently discovered it while playing with a circle of > thirds.

I take it "Partch's just scale" is the 43 note one from Genesis. What's "Wilson's 41-tone pure scale"?

I'm not sure how we're getting from regular to well temperaments, but the problem you're likely to face is that no 41 note constant structure can uniquely represent the 11-limit tonality diamond. That is, 11:10 and 10:9 map to the same number of steps in 41-equal, and so either they have to be compromised or one has to be omitted.

You probably know that Wilson fitted Partch's 43 note scale to a 41 note keyboard with split keys. He used the 41&29 mapping of

<<1 2 -1 -3 -4], <0 -1 8 14 18]]

(Gene, why do you mix [] and <| for these things?)

so that looks like one place for you to start. The accuracy of the linear temperament isn't good, but I don't think you can do better while keeping the diamond to within 41 notes. And all scales using the logic Wilson inferred from Partch require the 11-limit diamond.

The simplest constant structure that does uniquely map the 11-diamond is the consistent mapping to 58 notes. So, as you're interested in 41 notes, let's look at 41&58. The mapping is

<<1 1 -5 -1 2], <0 2 25 13 5]]

It has a complexity of 25 and uniquely represents the 11-limit, and so looks like a good answer. Unfortunately, not to the question you asked. Whatever that might have been. Still, if it means anything to you, the optimal generator is a neutral third of around 351.5 cents. The linear temperament is accurate to within 6 cents.

> I don't like 41-EDO because of its relative weakness in 5-limit, though it > is strong on 7- and 11-limit. My goal was to get the best features of both > 41- and 53-tone, and still have a smoother trip around the circle of fifths. > I keep the fifths true for a while then start widening them ever-so-slightly > to better reach the septimals and undecimals.

If you want to compromise 41 and 53, 41&53 is an obvious choice, so see Gene's post. This temperament uniquely represents the 11-limit diamond, but not within 41 notes -- not too much of a surprise, because that's impossible, see above. The complexity is 37 generators, and so you need at least 37*2+1=75 notes for the diamond. The 94 note MOS can handle it, but not the 53 note one. Again, not much of a surprise given what I said above and the fact that 53<58.

> I already use MIRACLE when improvising on fretless bass, since my instrument > is defretted and still has lines in 12-tone, and it makes it easier when > playing with guitarists. And yes, 41-out-of-72-ET is indeed called Studloco.

The point of miracle is that it uniquely represents the 11-limit, is consistent with 41-equal, and has the lowest possible complexity given these other constraints. The 11-limit diamond minus 11/10 and 20/11 fits within the 41 notes. Although the Genesis scale doesn't, one of Partch's older 43 note scales does, minus 11/10 and 20/11 which are impossible to include and require 45 notes of miracle.

(I thought studloco was 41 of miracle, not 41 from 72.)

> The kind of music I want to write would be compatible with multiple tunings, > including MIRACLE, Pythagorean and schismic. But I might run into a problem > with 72-tone, since my old 12-tone compositions got really adventurous in > modulation and polyphony.

In which case miracle and 41&53 both look relevant, because both include pythagorean within the "good" region. Perhaps average the two? I'm not sure why 72-equal, or some nearby tuning, is a particular problem for modulation, assuming you're going to round it off for a well temperament. The same problems must surely exist for any linear temperament???

> Whatever the case, meantone is out, because I like having two different > sizes for a major second, and I intend to exploit them plenty. Meantone is out anyway, because it isn't consistent with 41. Magic is, but I don't think it's relevant for your purposes as it has no advantages over 41&29 that I can see.

Graham

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
> Danny Wier wrote:

> <<1 2 -1 -3 -4], <0 -1 8 14 18]]
>
> (Gene, why do you mix [] and <| for these things?)

Since it can be regarded as a row of column vectors, it might make
sense to use |> on the outside; I don't see a mathematical
justification for <| on the outside. I also don't see that it really
matters what we use. Is your notation simply based on aesthetics, or a
means of underlining the point?

> (I thought studloco was 41 of miracle, not 41 from 72.)

I was using it for 41 of miracle, but also using the 72-et tuning,
which is a good 11-limit miracle tuning.

> Meantone is out anyway, because it isn't consistent with 41. Magic is,
> but I don't think it's relevant for your purposes as it has no
> advantages over 41&29 that I can see.

Of course, one can ask why we are stuck with 41--is that some physical
constraint of Danny's setup? Otherwise, for instance, unidec, the
46&72 system with mapping

[<2 5 8 5 6|, <0 -6 -11 2 3|]

might be considered as a way to tune 46 notes. Unidec has a period of
600 cents and generator a slightly sharp 14/11, and distinguishes
every 11-limit consonance. Another interesting choice would be wizard,
the 22&72 system, where 44 or 50 notes would be reasonable sizes to
use; it has a half-octave period and a generator a flat major third.
Wizard has a mapping

[<2 1 5 2 8|, <0 6 -1 10 -3|]

🔗Danny Wier <dawiertx@sbcglobal.net>

From: "Graham Breed":

> I take it "Partch's just scale" is the 43 note one from Genesis. What's
> "Wilson's 41-tone pure scale"?

Wilson had a 41-tone 11-limit scale in Xenharmonicon 3, 1975.

Lower tetrachord
1/1, 64/63, 28/27, 256/243, 16/15, 12/11, 10/9,
9/8, 8/7, 7/6, 32/27, 6/5, 27/22, 5/4,
81/64, 9/7, 21/16,
Middle dichord
4/3, 256/189, 112/81, 1024/729, 64/45, 16/11, 40/27,
Upper tetrachord
3/2, 32/21, 14/9, 128/81, 8/5, 18/11, 5/3,
27/16, 12/7, 7/4, 16/9, 9/5, 81/44, 15/8,
243/128, 27/14, 63/32, 2/1

It's not mirror-image symmetrical like Partch, since it's made out of two disjunct tetrachords. The "mother scale" is Pythagorean, pitches I call "black". Between limmas, two septimal "blue" pitches are placed; between apotomes, two quintal "red" pitches are added next to the black pitches, and one undecimal "green" pitch is placed between the red ones.

> I'm not sure how we're getting from regular to well temperaments, but
> the problem you're likely to face is that no 41 note constant structure
> can uniquely represent the 11-limit tonality diamond. That is, 11:10
> and 10:9 map to the same number of steps in 41-equal, and so either they
> have to be compromised or one has to be omitted.

About my use of Partch-43... I removed 11/10 and 20/11 so I can get a 41/53 pattern, but of course they map to distinct pitches in MIRACLE, which I use when it's more convenient.

And I'm just now reading up on schismic temperaments, so I'll have to get back to you and Gene later on. But what's wrong with microtonal.co.uk? It has a case of the 401s apparently.

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Danny Wier" <dawiertx@s...> wrote:

> Wilson had a 41-tone 11-limit scale in Xenharmonicon 3, 1975.

This is a detempered MOS for 41&53, with fifths ranging from -23 to
17. Is anyone familiar with this artice? Is this scale a Fokker block?

The TM basis for 41&53 schismatic is {225/224, 385/384, 2200/2187},
and Tenney reducing Wilson's scale with respect to this gives the
following:

! schis41.scl
Tenney reduced version of Wilson_41
! 41&53 <<1 -8 -14 23 -15 -25 33 -10 81 113||
41
!
50/49
25/24
81/77
15/14
12/11
10/9
9/8
8/7
7/6
25/21
6/5
27/22
5/4
63/50
9/7
21/16
4/3
200/147
25/18
7/5
10/7
16/11
40/27
3/2
32/21
14/9
100/63
8/5
18/11
5/3
27/16
12/7
7/4
16/9
9/5
50/27
15/8
154/81
27/14
49/25
2

Scala shows it as being similar to Wilson_41, as expected, but also
wants to compare it to the scale below, which claims to be meantone
but which when you analyze it turns out to be a 41 note MOS in 41&53
temperament. The particular tuning used is the one which makes 7/6
come out pure.

! MEANSEPT3.SCL
!
Mean-tone scale with septimal minor third

41
!
35.950 cents
62.453 cents
88.957 cents
124.907 cents
151.410 cents
177.914 cents
213.864 cents
240.367 cents
7/6
293.374 cents
329.324 cents
355.828 cents
382.331 cents
418.281 cents
444.785 cents
471.288 cents
497.791 cents
49/36
560.245 cents
586.748 cents
622.698 cents
649.202 cents
675.705 cents
711.655 cents
738.159 cents
764.662 cents
791.166 cents
827.116 cents
853.619 cents
880.122 cents
916.073 cents
942.576 cents
969.079 cents
995.583 cents
1031.533 cents
1058.036 cents
1084.540 cents
1120.490 cents
1146.993 cents
1173.497 cents
2/1

🔗Graham Breed <gbreed@gmail.com>

Gene Ward Smith wrote:

> This is a detempered MOS for 41&53, with fifths ranging from -23 to
> 17. Is anyone familiar with this artice? Is this scale a Fokker block?

http://www.anaphoria.com/xen3b.PDF

It's diagram 8 on page 10, which strangely enough follows diagram 4 on page 9. Shows a keyboard with a 41&29 mapping, and these ratios on the keys. Works with 41&53 if you move the ll-limit keys from the top to the bottom.

A few pages later is where he shows the Partch scale on the same keyboard.

> The TM basis for 41&53 schismatic is {225/224, 385/384, 2200/2187},
> and Tenney reducing Wilson's scale with respect to this gives the
> following:

Okay, I think I see how you could Tenney reduce a scale. Would any 41 note constant structure give the same result? What happens if you reduce wrt 41&29 instead?

> Scala shows it as being similar to Wilson_41, as expected, but also
> wants to compare it to the scale below, which claims to be meantone
> but which when you analyze it turns out to be a 41 note MOS in 41&53
> temperament. The particular tuning used is the one which makes 7/6
> come out pure.
> > ! MEANSEPT3.SCL
> !
> Mean-tone scale with septimal minor third No reason to assume the 11-limit is there? In which case it's a 12&29 "macro" schismic. Could be 41&53 or 41&29.

Graham

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

> It's diagram 8 on page 10, which strangely enough follows diagram 4 on
> page 9. Shows a keyboard with a 41&29 mapping, and these ratios on the
> keys. Works with 41&53 if you move the ll-limit keys from the top to
> the bottom.

As usual, I can't figure out where his numbers come from or sometimes
even what the diagrams are diagrams of.

> Okay, I think I see how you could Tenney reduce a scale. Would any 41
> note constant structure give the same result?

It depends on what the range of generators is; -20 to 20, for
instance, would give a different result.

What happens if you
> reduce wrt 41&29 instead?

Here are three different schismatic detemperings for fifths from -20
to 20:

Commas 100/99, 225/224, 245/242. Circle of fifths consists of 24 pure
3/2s, two flat by 2835/2816 (11.6 cents), four flat by 225/224 (7.7
cents), one flat by 243/242 (7.1 cents), three flat by 441/400 (3.9
cents), one sharp by 441/440, two sharp by 896/891 (9.7 cents), four
sharp by 100/99 (17.4 cents.)

! tenn41a.scl
29&41 Tenney reduced fifths from -20 to 20
41
!
45/44
25/24
22/21
15/14
12/11
10/9
9/8
63/55
7/6
25/21
6/5
11/9
5/4
14/11
9/7
55/42
4/3
15/11
11/8
88/63
63/44
16/11
22/15
3/2
32/21
14/9
11/7
8/5
18/11
5/3
27/16
189/110
110/63
16/9
9/5
11/6
15/8
21/11
27/14
55/28
2

Commas 225/224, 385/384, 2200/2187. Circle of fifths consists of 24
pure 3/2s, seven flat by 225/224, one flat by 385/384 (4.5 cents), two
sharp by 540/539 (3.2 cents), one sharp by 243/242, five sharp by
2200/2187 (10.3 cents), and one sharp by 4000/3969 (13.4 cents.)

! tenn41b.scl
41&53 Tenney reduced fifths from -20 to 20
41
!
55/54
25/24
81/77
15/14
27/25
10/9
9/8
8/7
7/6
25/21
6/5
27/22
5/4
63/50
9/7
21/16
4/3
27/20
25/18
7/5
10/7
36/25
40/27
3/2
32/21
14/9
100/63
8/5
44/27
5/3
27/16
12/7
7/4
16/9
9/5
50/27
15/8
154/81
27/14
49/25
2

Commas 385/384, 3388/3375, 4375/4374. Circle of fifths consists of 34
pure 3/2s, two flat by 8019/8000 (4.1 cents), one flat by 32805/32768
(2 cents), two sharp by 5632/5625 (2.2 cents), and the wolf, sharp by
20000/19683 (27.7 cents.)

! tenn41c.scl
53&118 Tenney reduced fifths from -20 to 20
41
!
81/80
25/24
256/243
2187/2048
27/25
10/9
9/8
729/640
88/75
32/27
19683/16384
100/81
8192/6561
81/64
32/25
33/25
4/3
27/20
25/18
1024/729
729/512
36/25
40/27
3/2
243/160
25/16
128/81
6561/4096
81/50
32768/19683
27/16
75/44
44/25
16/9
9/5
50/27
4096/2187
243/128
48/25
99/50
2

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

Danny,

What made you think of the sine function?

I'm guessing you wanted an adjustment that was zero for
the unison and octave, but positive and continuous everywhere
in between.

Just curious!

Regards,
Yahya

-----Original Message-----
________________________________________________________________________
Date: Sun, 27 Mar 2005 21:33:48 -0600
From: "Danny Wier"
Subject: My first post: sine-based well-temperament

Hello, and happy Easter if you're celebrating.

I'm already a member of tuning and MMM, and I'm more interested in the
practical than the theoretical - but I've been experimenting with extended
Pythagorean so much lately. I've decided on a 41-tone "well-temperament" for
my own music, balancing out the need for smooth transposition and accurate
11-limit JI approximation (Partch and Wilson being the two most important
models). I posted the pitches in tuning, and I hate to cross-post, so I'll
just explain it in brief. It's 9 pure fifths up and down and 23 wolves,
easily generated by Scala.

That led me to another experiment, and this may have been done before:
well-temperaments with the amount of tempering of each fifth derived from
the sine function. This example is a "smoothed out" variation on Kirnberger
III all pitches in cents:

0.00000 (1/1)
101.55531
198.04500
302.42936
397.66152
501.08095
600.00000
698.91905
802.33848
897.57064
1001.95500
1098.44469
1200.00000 (2/1)

(For a Kirnberger similarity, 1/1 should be assigned to the note D.)

The formula for the temperament (negative in this case):

T = sin (n*pi/6) * log_2 (531441/524288)
(n = number of fifths away from the central note; if n is negative, |n| is
the number of fourths away)

Then subtract T from the untempered pitch.

I compared my temperament with the Scala archive, and the files with the
closest pitches are "sorge1.scl" (Georg Andreas Sorge, 1744) and
"scottd4.scl" (Dale Scott's temperament #4, 1999).

I've done something similar with 41-tone, but put the steeper temperings at
the most remote ends of the chain of fifths, and used the 41-tone comma. I
don't know how practical such a complicated temperament really is,
especially considering 1/23 of a 41-tone comma is only 0.86282 cents, but
it's fun anyways.

________________________________________________________________________

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🔗Graham Breed <gbreed@gmail.com>

Me:
>>It's diagram 8 on page 10, which strangely enough follows diagram 4 on >>page 9. Shows a keyboard with a 41&29 mapping, and these ratios on the >>keys. Works with 41&53 if you move the ll-limit keys from the top to >>the bottom.

Gene:
> As usual, I can't figure out where his numbers come from or sometimes
> even what the diagrams are diagrams of.

The scale structure is shown on the right. 6 different chains of pure fifths, with high primes moving from the numerator to the denominator as you go up the page. Smaller keyboards use simpler prime limits. It happens that no ratio has more than one instance of a prime over 3. The ratios are naturally more complex than yours because they have more threes.

It looks like you can generate these scales pretty easily, so how about some with 58 notes? Say based around 58&46 and 58&41. The more 11-limit diamonds the better, so they're natural extensions of Partch's idea of filling out the diamond.

Graham

🔗Danny Wier <dawiertx@sbcglobal.net>

Yahya Abdal-Aziz escriba:

> Danny,
>
> What made you think of the sine function?
>
> I'm guessing you wanted an adjustment that was zero for
> the unison and octave, but positive and continuous everywhere
> in between.

Just some crazy idea I came up with the other day.

If you were to make a graph of the amount of tempering of each fifth, with the number of fifths as the X axis and the temperament as the Y axis, you would get a straight line in an equal temperament, and some type of crooked line in well-temperament. Kirnberger III slopes downward from C to E, then the slope is completely flat outside of that portion of the circle of fifths. If your temperament is circular, then a point opposite of the central tone involves tempering a fifth the full amount of a comma (Pythagorean in 12-tone, 2^65/3^41 in 41-tone, etc.): F-sharp tempered a full comma downward produces G-flat, and you come full circle.

I thought of making the slope as smooth as possible (minimizing instantaneous changes in slope as much as possible), and the sine function came to mind. Well-temperaments are circular by nature, so I wanted a circular function. There are two types of "sine temperaments" (or "trigonometric temperaments"): ones with infinite slope at the central note, which I recommend for 12-tone WT, and the other kind with zero slope at the central note, which I'm investigating for my 41-tone WT.

There are other well temperaments that smooth out the slopes in different ways, some of them pretty old. Where's that list again....

~Danny~

🔗Carl Lumma <ekin@lumma.org>

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, "Danny Wier" <dawiertx@s...>
wrote:
>
> If you were to make a graph of the amount of tempering of each
fifth, with
> the number of fifths as the X axis and the temperament as the Y
axis, you
> would get a straight line in an equal temperament, and some type of
crooked
> line in well-temperament. Kirnberger III slopes downward from C to
E, then
> the slope is completely flat outside of that portion of the circle
of
> fifths. If your temperament is circular, then a point opposite of
the
> central tone involves tempering a fifth the full amount of a comma
> (Pythagorean in 12-tone, 2^65/3^41 in 41-tone, etc.): F-sharp
tempered a
> full comma downward produces G-flat, and you come full circle.
>
> I thought of making the slope as smooth as possible (minimizing
> instantaneous changes in slope as much as possible), and the sine
function
> came to mind. Well-temperaments are circular by nature, so I wanted
a
> circular function. There are two types of "sine temperaments" (or
> "trigonometric temperaments"): ones with infinite slope at the
central note,
> which I recommend for 12-tone WT, and the other kind with zero
slope at the
> central note, which I'm investigating for my 41-tone WT.
>
> There are other well temperaments that smooth out the slopes in
different
> ways, some of them pretty old. Where's that list again....
>
> ~Danny~

Danny, may I congratulate you for being one of the very, very few
courageous adventurers into the world of non-12 irregular
temperaments. Because the possibilities (and variations thereon) are
almost endless, I found that it can take a lot of time and patience
to arrive at something that I would want to call "highly
satisfactory." I can think of at least a couple of instances where I
have returned to a particular problem after several years with a
fresh perspective that ended up producing a significantly better
result than before. So I'm offering the following as food for
thought.

In devising well-temperaments of 17 and 19 tones, I found that a
smooth progression of "moods" around the circle of fifths can be
obtained in each with only two different sizes of fifths, and in
addition, there are more keys in which the chords have the minimum
amount of error than if the size of the fifths had been gradually
varied.

My 17-tone well-temperaments has 9 fifths ~707.2205c (Ab to B, 6:11
almost exact) and 8 fifths ~704.3770 (Cv to G^, 7:11 exact), while my
19-WT has 9 fifths ~695.6296c (F to E#, 5/17-comma equal-beating with
7:9 almost exact) and 8 fifths ~693.2064c (Fb to F). If you want to
try either of these, you'll find .scl files here:
/tuning-math/files/secor/scl/

There's also an alternate version of my 19-WT, secor19p3.scl, with 3
auxiliary tones that permit some 13-limit harmony. To play it on
Scala's chromatic clavier, first enter the commands "set notation
sa38" and "set sagittal mixed". The three auxiliary tones not in the
circle of 19 are easily identified by arrow-symbols, and their
purpose is explained here (in great detail):
/tuning/topicId_38076.html#38287

With a circle of 41 fifths you wouldn't have to alter the fifths very
much to make a dramatic difference in the tones 15 to 20 fifths down
the chain.

On the other hand, my own attempts at altered-fifth (15-limit)
temperaments of 29 and 41 tones have resulted my "high-tolerance
temperament", in which most of the fifths are the same size. See:
/makemicromusic/topicId_6820.html#6847
which message also references:
/tuning-math/message/7574
Files for the .scl listings shown are also in the link to the files
section that I gave above.

As I observed in one of those messages, the amount of error in the
best keys is so small that it's almost impossible to tell that it's a
temperament. However, the number of "good" keys in each is rather
small (11 out of 41, 6 out of 29, and 3 out of 17), so I don't think
that this is what you're after. But you might want to play around
with a couple of these in Scala, just to see if you get any new ideas.

--George

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

> It looks like you can generate these scales pretty easily, so how about
> some with 58 notes? Say based around 58&46 and 58&41.

58&41 has TM basis {243/242, 441/440, 896/891}, and generator 11/9. I
reduced from -28 to 29 generators, and got this:

! tenn58.scl
Chain of 11/9s -28 to 29 Tenney reduced by {243/242,441/440,896/891}
58
!
56/55
45/44
28/27
21/20
16/15
15/14
12/11
11/10
10/9
9/8
8/7
55/48
7/6
32/27
6/5
40/33
11/9
56/45
5/4
14/11
9/7
64/49
21/16
4/3
27/20
15/11
11/8
7/5
45/32
10/7
16/11
22/15
40/27
3/2
32/21
49/32
14/9
11/7
8/5
45/28
18/11
33/20
5/3
27/16
12/7
96/55
7/4
16/9
9/5
20/11
11/6
28/15
15/8
21/11
27/14
49/25
55/28
2

Adding in 126/125 to the mix gives the whole 58-et 11-limit, and this:

! ten58.scl
58 Tenny reduced via 11-limit commas {126/125,243/242,441/440,896/891}
58
!
56/55
36/35
28/27
21/20
16/15
15/14
12/11
11/10
10/9
9/8
8/7
55/48
7/6
32/27
6/5
40/33
11/9
56/45
5/4
14/11
9/7
35/27
21/16
4/3
27/20
15/11
11/8
7/5
45/32
10/7
16/11
22/15
40/27
3/2
32/21
49/32
14/9
11/7
8/5
45/28
18/11
33/20
5/3
27/16
12/7
55/32
7/4
16/9
9/5
20/11
11/6
28/15
15/8
21/11
27/14
35/18
55/28
2