Tuning-Math Digests messages 1700 - 1724

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Message: 1700

Date: Mon, 01 Oct 2001 04:18:40

Subject: catching up

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., jon wild <wild@f...> wrote:

I understand from the rest of what you
> wrote that the requirement is stronger still than connected: that 
the span
> of the set is also limited.

It is at least for classic blocks, which is what I just defined. The 
question is what to do about what I was calling "semiblocks"; simply 
requiring them to come from a convex region (which is in effect what 
Paul suggests in his Gentle Introduction) is pretty weak, since it 
allows for more extreme examples than he gave, but my definition may 
still be too restrictive--it doesn't include his example of the 
Indian diatonic, with sixth degree raised a comma, for instance. That 
makes for a distance of 8/7 to 4/3, which is a little far. Possibly a 
semiblock should just be convex and the diameter serves as a measure 
of how extreme it is, so the above example would be an 
8/7-semiblock.


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Message: 1701

Date: Mon, 01 Oct 2001 22:38:13

Subject: Re: 34-tone ET scale

From: Paul Erlich

34 is the next entry after 12 in this accounting of periodicity 
blocks:

S235 *

What that says to me is that, if one is going to use strict 5-limit 
JI, and one is seeking an "even" system with more than 12 notes but 
fewer than 53, one must go to 34. Not 34-tET, but 34-tJI.


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Message: 1702

Date: Mon, 01 Oct 2001 23:14:27

Subject: 46 (was Re: Pauls fingerboard kit)

From: Paul Erlich

--- In tuning-math@y..., graham@m... wrote:
> Bob Valentine wrote:
> 
> > What does 46 have that everyone seems gaga about? Is 46
> > better at 13

Yes, but I'm not planning on using it for 13-limit. More important is 
that 46 is better than 41 in 5-limit (which is the main interval 
flavor for Indian music). Though 34 is better still in the 5-limit, 
46 allows me to access 7- and 11-flavors with much higher accuracy.


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Message: 1703

Date: Mon, 01 Oct 2001 04:37:22

Subject: catching up

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., jon wild <wild@f...> wrote:

> I understand from the rest of what you
> wrote that the requirement is stronger still than connected: that 
the span
> of the set is also limited. Thanks --Jon

It is at least for classic blocks, which is what I just defined. 

The question is what to do about what I was calling "semiblocks"; 
simply requiring them to come from a convex region with a span of 
less than 1 on the first coordinate (which I think is in effect what 
Paul suggests in his Gentle Introduction) is pretty weak, and it 
allows for more extreme examples than he gave, but my definition may 
still be too restrictive--it doesn't include his example of the 
Indian diatonic, with sixth degree raised a comma, for instance. That 
makes for a distance of 8/7 to 4/3, which is a little far. Possibly a 
semiblock should just be convex and the diameter would serve as a 
measure of how extreme it is, so the above example would be an 8/7-
semiblock.


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Message: 1704

Date: Tue, 2 Oct 2001 16:08 +01

Subject: Re: 41, 46 and 58

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <9pbgq1+86qa@xxxxxxx.xxx>
Gene wrote:

> Here are relativized n-consistent goodness measures for odd n to 25, 
> for 41, 46 and 58:

How are these calculated?  It looks like lower numbers are better, so it's 
really a badness measure.


                      Graham


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Message: 1705

Date: Tue, 02 Oct 2001 16:19:03

Subject: Re: 41, 46 and 58

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <9pbgq1+86qa@e...>
> Gene wrote:

> > Here are relativized n-consistent goodness measures for odd n to 
25, 
> > for 41, 46 and 58:

> How are these calculated?  It looks like lower numbers are better, 
so it's 
> really a badness measure.

This is the same w-consistent measure cons(w, n) introduced in
Yahoo groups: /tuning-math/message/860 *
And yes, it is a badness measure, but you could always take the 
reciprocal. :)


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Message: 1706

Date: Tue, 02 Oct 2001 16:22:07

Subject: How consistent are cents?

From: genewardsmith@xxxx.xxx

If you want an idea of what a randomly chosen consistent badness 
measure looks like, chew on this:

Cons(w, 1200) for w from 2 to 49

3    53.998800
5    12.42619668
7    5.183728199
9    5.183728199
11   2.895954014
13   3.263002903
15   3.263002903
17   2.576191158
19   2.216354953
21   2.216354953
23   1.952782773
25   2.703605969
27   2.703605969
29   2.450061526
31   2.264455891
33   2.264455891
35   2.264455891
37   2.123104197
39   2.180931116
41   2.066877882
43   1.975042642
45   1.975042642
47   1.899581355
49   1.899581355


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Message: 1707

Date: Tue, 2 Oct 2001 17:45 +01

Subject: Re: 41, 46 and 58

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <9pcpdn+pinr@xxxxxxx.xxx>
> Gene wrote:

> This is the same w-consistent measure cons(w, n) introduced in
> Yahoo groups: /tuning-math/message/860 *
> And yes, it is a badness measure, but you could always take the 
> reciprocal. :)

And "less than or equal to l" should be "less than or equal to w".  I was 
wondering how a prime could be less than 1.

It looks like h(q_i) is the number of steps in the ET for the ratio q_i.  
So h(2) is the number of steps to the octave.  So cons(w, n) assumes 
consistency which -- warning! -- doesn't hold for 46-equal in the 
15-limit, because there's one ambiguous interval.

This formula:

n^(1/d) * max(abs(n*log_2(q_i) - h(q_i))

has a parenthesis unclosed.  From the one lower down, I think it should be

n^(1/d) * max(abs(n*log_2(q_i)) - h(q_i))


The n^(1/d) means you're scaling according to the size of the octave and 
the number of prime dimensions.

Then you're taking the largest deviation for any interval within the 
limit, right?  In which case, the parenthesising should be

n^(1/d) * max(abs(n*log_2(q_i) - h(q_i)))

and the other formula must be wrong.

That could be re-written

n^(1/d) * n * max(|tempered_pitch - just_pitch|)

where pitches are in octaves.  Which can be simplified to

n^(1+1/d) * max(|tempered_pitch - just_pitch|)

Is that right?  I think I almost understand it!  So why the 1+1/d?


                    Graham


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Message: 1708

Date: Tue, 02 Oct 2001 17:40:39

Subject: Re: Paul blocks

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> I see I was misreading Paul's construction, which actually only 
> allows for an extra comma which is either the sum or difference of 
> the other commas. This is a very nice idea, and can be formalized 
in 
> the same way as the Fokker block, via a norm.
> 
> If we have a Fokker-type norm in the 5-limit, which is 
> 1/n [hn, v1, v2], where hn is an n-et and the other are defined on 
> octave equivalence classes, then instead of taking the maximum of 
the 
> absolute values of these valuations, we can take instead the 
maximum 
> of the absolute value of |hn(q)| together with the median of 
> {|v1(q)|, |v2(q)|, |v1(q)-v2(q)|}, or else 
> {|v1(q), |v2(q)|, |v1(q)+v2(q)|}--in other words, we sort three 
> absolute values, and take the one in the middle. This also gives us 
a 
> norm, and we can then define Paul blocks in the 5-limit in the same 
> way as Fokker blocks. To generalize to higher dimensions, it seems 
we 
> would need to take combinations of n vals k at a time, for k from 1 
> to n, giving us the verticies of the n-measure polytope 
(=hypercube.)
> We then would take a maximum over the n smallest. At least, that 
> seems right, but I haven't really thought it through carefully.

I wish I could understand this. Can I ask you, what were you 
misreading, and what led you to a correct reading?


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Message: 1709

Date: Tue, 02 Oct 2001 17:43:25

Subject: Re: 41, 46 and 58

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:

> 41 is clearly an 
> excellent system for 7, 9 or 11, and deserves some respect!

Oh yes . . . but it's mentioned far more than 46 in the literature, 
which is why I suspect Robert Valentine was puzzled about the 46 
hoopla.


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Message: 1710

Date: Tue, 02 Oct 2001 17:45:55

Subject: Re: 41, 46 and 58

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning-math@y..., graham@m... wrote:
> > > Bob Valentine wrote:
> 
> > > > What does 46 have that everyone seems gaga about? Is 46
> > > > better at 13
> 
> > Yes, but I'm not planning on using it for 13-limit. More 
important 
> is 
> > that 46 is better than 41 in 5-limit (which is the main interval 
> > flavor for Indian music). Though 34 is better still in the 5-
limit, 
> > 46 allows me to access 7- and 11-flavors with much higher 
accuracy.
> 
> Here are relativized n-consistent goodness measures for odd n to 
25, 
> for 41, 46 and 58:
> 
> 41:
> 
> 3     .67803586
> 5    1.380445520
> 7     .7433989824
> 9     .8004237371
> 11    .9169187332
> 13   1.010934526
> 15   1.010934526
> 17   1.138442704
> 19   1.042105199
> 21   1.042105199
> 23   1.399948567
> 25   1.399948567
> 
> 46:
> 
> 3    4.21934770
> 5    1.297511950
> 7    1.181094581
> 9    1.181094581
> 11    .8584628865
> 13    .8850299838
> 15   1.082290028
> 17    .9526171496
> 19   1.188323665
> 21   1.188323665
> 23   1.109794688
> 25   1.270499309
> 
> 58:
> 
> 3    4.18614652
> 5    2.499272068
> 7    1.270307523
> 9    1.270307523
> 11    .9740696116
> 13    .8435935409
> 15    .9018214272
> 17    .9309801788
> 19   1.393450457
> 21   1.393450457
> 23   1.295990287
> 25   1.721256452
> 
> We see that 41 has values less than 1 in the 3, 7, 9 and 11 limits; 
> 46 in 11, 13 and 17; and 58 in 11, 13, 15, and 17. 41 is clearly an 
> excellent system for 7, 9 or 11, and deserves some respect!

Gene, I don't know what you mean by n-consistent here. 46 is only 
consistent through the 13-limit, so it's inconsistent in the 17-
limit. So how can 46 have a finite value for "17-consistent goodness"?


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Message: 1711

Date: Tue, 02 Oct 2001 17:47:26

Subject: Re: How consistent are cents?

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> If you want an idea of what a randomly chosen consistent badness 
> measure looks like, chew on this:
> 
> Cons(w, 1200) for w from 2 to 49
> 
> 3    53.998800
> 5    12.42619668
> 7    5.183728199
> 9    5.183728199
> 11   2.895954014
> 13   3.263002903
> 15   3.263002903
> 17   2.576191158
> 19   2.216354953
> 21   2.216354953
> 23   1.952782773
> 25   2.703605969
> 27   2.703605969
> 29   2.450061526
> 31   2.264455891
> 33   2.264455891
> 35   2.264455891
> 37   2.123104197
> 39   2.180931116
> 41   2.066877882
> 43   1.975042642
> 45   1.975042642
> 47   1.899581355
> 49   1.899581355

Again, I don't know what you mean by consistent here. 1200-tET is 
consistent through the 9-limit, but not the 11-limit.


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Message: 1712

Date: Tue, 02 Oct 2001 17:49:10

Subject: Re: Digest Number 124

From: Paul Erlich

Gene -- do you have any response here? Have you been misunderstanding 
something all along?

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., genewardsmith@j... wrote:
> 
> > The first system has jargon wherein 81/80 and 25/24 are "commatic 
> > unison vectors" and 16/15 is a "chromatic unison vector" in a 
> > situation where we are seeking a 7-note "periodiity block" scale;
> 
> Something's wrong here . . . in a 7-tone PB, specificially the 
> diatonic scale, 81:80 is the commatic unison vector, 25:24 is a 
> chromatic unison vector, and 16:15 is not a unison vector at all, 
but 
> a "step vector".
> 
> > No; to say the scale is "defined" by the fact that 16/15 is 
> > a "chromatic unison vector" (something of a misnomer, I fear) and 
> > 81/80 and 25/24 are "commatic unison vectors"
> 
> Misnomer because it's incorrect!


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Message: 1713

Date: Tue, 02 Oct 2001 03:32:53

Subject: Paul blocks

From: genewardsmith@xxxx.xxx

I see I was misreading Paul's construction, which actually only 
allows for an extra comma which is either the sum or difference of 
the other commas. This is a very nice idea, and can be formalized in 
the same way as the Fokker block, via a norm.

If we have a Fokker-type norm in the 5-limit, which is 
1/n [hn, v1, v2], where hn is an n-et and the other are defined on 
octave equivalence classes, then instead of taking the maximum of the 
absolute values of these valuations, we can take instead the maximum 
of the absolute value of |hn(q)| together with the median of 
{|v1(q)|, |v2(q)|, |v1(q)-v2(q)|}, or else 
{|v1(q), |v2(q)|, |v1(q)+v2(q)|}--in other words, we sort three 
absolute values, and take the one in the middle. This also gives us a 
norm, and we can then define Paul blocks in the 5-limit in the same 
way as Fokker blocks. To generalize to higher dimensions, it seems we 
would need to take combinations of n vals k at a time, for k from 1 
to n, giving us the verticies of the n-measure polytope (=hypercube.)
We then would take a maximum over the n smallest. At least, that 
seems right, but I haven't really thought it through carefully.


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Message: 1714

Date: Tue, 02 Oct 2001 18:10:24

Subject: Re: Paul blocks

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> I wish I could understand this. Can I ask you, what were you 
> misreading, and what led you to a correct reading?

It needs fixing anyway, so I will try again later. What I had thought 
you were saying would have been equivalent to saying that after a 
linear transformation of the parallepiped into a hypercube of measure 
1, we allow ourselves to chop up the hypercube and reassemble it into 
a convex body, also of measure 1 (no Banach-Tarski, please!) This 
tiles the n-space, but it allows us too much latitude.

Instead, you were putting restrictions on how the square could be 
chopped up and reassembled, and we should presumably have some 
restrictions also if we generalize this. I'll try to work the mess I 
posted out in a way which makes more sense, but I need to define the 
norm by first creating a region, and then defining the norm of a 
point by scaling the region and finding what scale factor makes the 
point lie on the boundry.

We could try transforming back to a regular hexagon instead, but then 
I still want to know how to generalize it. Would the 3-D version be 
the bee-type honeycomb of rhombic dodecahedra?


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Message: 1715

Date: Tue, 02 Oct 2001 03:54:46

Subject: Re: Some "ABC good" intervals

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> --- In tuning-math@y..., genewardsmith@j... wrote:
> > I found a web page, ABC-Ratios >1.4 *, 
> with 
> > a list of all 148 known "good" ABCs according to the definition 
you 
> > will find there of "good" (there are others.) 

> I don't see a definition there of "good", and most of these are not 
> superparticular ratios at all . . . so what, really, is the ABC 
> conjecture?

If we have three relatively prime positive integers A, B, C such that 
A + B = C, and if we define a "radical" function rad(N) as the 
product of the primes dividing N, we can look at C/rad(ABC); it turns 
out that this can be arbitarily large. The ABC conjecture says that 
C/rad(ABC)^e, for any e>1, cannot become arbitrarily large. It's a 
conjecture in elementary number theory with a large number of very 
powerful and not always elementary consequences, and since it is the 
latest fad I thought of it when I saw that the list of m/n in the p-
limit, with |m-n|<d, should be finite. It seems it is also an easy 
consequence of Baker's theorem, and hence is true--a useful thing to 
know.

Any ABC triple such that ln(C)/ln(rad(C)) > 1.4 is rather arbitarily 
termed "good"; they are pretty rare and some of them turn up in music 
theory, it seems.


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Message: 1716

Date: Tue, 02 Oct 2001 18:27:33

Subject: Re: Paul blocks

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > I wish I could understand this. Can I ask you, what were you 
> > misreading, and what led you to a correct reading?
> 
> It needs fixing anyway, so I will try again later. What I had 
thought 
> you were saying would have been equivalent to saying that after a 
> linear transformation of the parallepiped into a hypercube of 
measure 
> 1, we allow ourselves to chop up the hypercube and reassemble it 
into 
> a convex body, also of measure 1 (no Banach-Tarski, please!) This 
> tiles the n-space, but it allows us too much latitude.

Too much latitude? Why?
> 
> Instead, you were putting restrictions on how the square could be 
> chopped up and reassembled,

I was?

> We could try transforming back to a regular hexagon instead, but 
then 
> I still want to know how to generalize it.

Why a regular hexagon? And in what version of the lattice? No, I see 
a desirable class of 5-limit periodicity blocks defined as a hexagon 
of any shape, as long as opposite sides are parallel and congruent.

> Would the 3-D version be 
> the bee-type honeycomb of rhombic dodecahedra?

Yes, but they certainly don't have to be regular. Also, they can 
be "degenerate", with certain faces vanishing or combining with other 
faces, so that one ends up with hexagonal prisms or parallelepipeds.


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Message: 1717

Date: Tue, 02 Oct 2001 04:06:18

Subject: Re: Some "ABC good" intervals

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> I don't see a definition there of "good", and most of these are not 
> superparticular ratios at all . . . so what, really, is the ABC 
> conjecture?

If we have three relatively prime positive integers A, B, C such that 
A + B = C, and if we define a "radical" function rad(N) as the 
product of the primes dividing N, we can look at C/rad(ABC); it turns 
out that this can be arbitarily large. The ABC conjecture says that 
C/rad(ABC)^e, for any e>1, cannot become arbitrarily large. It's a 
conjecture in elementary number theory with a large number of very 
powerful and not always elementary consequences, and since it is the 
latest fad I thought of it when I saw that the list of m/n in the 
p-limit, with |m-n|<d, should be finite. It seems it is also an easy 
consequence of Baker's theorem, and hence is true--a useful thing to 
know. Any ABC triple such that ln(C)/ln(rad(ABC)) > 1.4 is rather 
arbitarily termed "good"; they are pretty rare and some of them turn 
up in music theory, it seems. For superparticular ratios this measure 
becomes ln(C)/ln(rad(BC)), and looking at the "goodness" of our 
favorite commas might be interesting, I suppose.


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Message: 1718

Date: Tue, 02 Oct 2001 19:18:18

Subject: Re: Digest Number 124

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> Gene -- do you have any response here? Have you been 
misunderstanding 
> something all along?

It's the same thing which made it so hard for me to get the 
terminology in the first place--in the tempered situation, we have 
*two* scale-step vals. For instance, we could have h12 and h7, and 
then we would have h12(16/15) = h12(25/24) = 1, but h7(16/15) = 1 and 
h7(25/24) = 0. So 25/24 is a "unison" according to h7 but it is also 
a "step" on the piano keyboard--according to h12.

We have (16/15, 25/24, 81/80)^(-1) = [h7, h5, h3]. For the purpose of 
constructing a JI scale approximating to h7, 16/15 is a step and 
25/24 is a comma. If we temper out 81/80, we can pick an et of the 
form n h7 + m h5, and then in that et 16/15 will be n intervals and 
25/24 will be m intervals, but in the scale 16/15 is still one step 
and 25/24 not an allowed step; it is still a unison so far as h7 is 
concerned.


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Message: 1719

Date: Tue, 02 Oct 2001 04:45:53

Subject: 41, 46 and 58

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., graham@m... wrote:
> > Bob Valentine wrote:

> > > What does 46 have that everyone seems gaga about? Is 46
> > > better at 13

> Yes, but I'm not planning on using it for 13-limit. More important 
is 
> that 46 is better than 41 in 5-limit (which is the main interval 
> flavor for Indian music). Though 34 is better still in the 5-limit, 
> 46 allows me to access 7- and 11-flavors with much higher accuracy.

Here are relativized n-consistent goodness measures for odd n to 25, 
for 41, 46 and 58:

41:

3     .67803586
5    1.380445520
7     .7433989824
9     .8004237371
11    .9169187332
13   1.010934526
15   1.010934526
17   1.138442704
19   1.042105199
21   1.042105199
23   1.399948567
25   1.399948567

46:

3    4.21934770
5    1.297511950
7    1.181094581
9    1.181094581
11    .8584628865
13    .8850299838
15   1.082290028
17    .9526171496
19   1.188323665
21   1.188323665
23   1.109794688
25   1.270499309

58:

3    4.18614652
5    2.499272068
7    1.270307523
9    1.270307523
11    .9740696116
13    .8435935409
15    .9018214272
17    .9309801788
19   1.393450457
21   1.393450457
23   1.295990287
25   1.721256452

We see that 41 has values less than 1 in the 3, 7, 9 and 11 limits; 
46 in 11, 13 and 17; and 58 in 11, 13, 15, and 17. 41 is clearly an 
excellent system for 7, 9 or 11, and deserves some respect!


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Message: 1720

Date: Tue, 02 Oct 2001 19:35:36

Subject: Re: Digest Number 124

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > Gene -- do you have any response here? Have you been 
> misunderstanding 
> > something all along?
> 
> It's the same thing which made it so hard for me to get the 
> terminology in the first place--in the tempered situation, we have 
> *two* scale-step vals. For instance, we could have h12 and h7, and 
> then we would have h12(16/15) = h12(25/24) = 1, but h7(16/15) = 1 
and 
> h7(25/24) = 0. So 25/24 is a "unison" according to h7 but it is 
also 
> a "step" on the piano keyboard--according to h12.

What if we substituted the word "second" instead of "step"?

Better yet, can we avoid bringing 12 into this at all? I see no 
reason it should be brought in.

> We have (16/15, 25/24, 81/80)^(-1) = [h7, h5, h3]. For the purpose 
of 
> constructing a JI scale approximating to h7, 16/15 is a step and 
> 25/24 is a comma. If we temper out 81/80, we can pick an et of the 
> form n h7 + m h5, and then in that et 16/15 will be n intervals and 
> 25/24 will be m intervals, but in the scale 16/15 is still one step 
> and 25/24 not an allowed step; it is still a unison so far as h7 is 
> concerned.

Right . . .


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Message: 1721

Date: Tue, 2 Oct 2001 21:48:50

Subject: Miracle web page

From: Graham Breed

I've added a page to my website on Miracle temperament.
<Miracle temperaments *>.
It's not connected up yet, I'll try and sort that out tomorrow.


             Graham

"I toss therefore I am" -- Sartre


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Message: 1722

Date: Tue, 02 Oct 2001 20:54:07

Subject: Re: Miracle web page

From: Paul Erlich

--- In tuning-math@y..., Graham Breed <graham@m...> wrote:

> I've added a page to my website on Miracle temperament.

Coincidentally, I just posted (to the tuning list) the design of a 
MIRACLE guitar fingerboard. It's actually the Canasta scale in 72-
tET, centered on A, with the open strings tuned conventionally in 12-
tET. Feel free to reproduce this on your website . . .


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Message: 1723

Date: Tue, 02 Oct 2001 21:29:04

Subject: Re: Miracle web page

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., Graham Breed <graham@m...> wrote:

> "I toss therefore I am" -- Sartre

Are you sure that isn't "I am therefore I toss"?

My favorite is "Slime is the agony of water", from Being and 
Nothingness.


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Message: 1724

Date: Tue, 02 Oct 2001 21:26:55

Subject: Re: Digest Number 124

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> Better yet, can we avoid bringing 12 into this at all? I see no 
> reason it should be brought in.

Since we aren't tempering, leaving it out is the thing to do.


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