Tuning-Math Digests messages 1475 - 1499

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Message: 1475

Date: Tue, 28 Aug 2001 19:18:26

Subject: Re: More microtemperaments

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., graham@m... wrote:

> The departure is from a JI odd limit.  That is, all odd numbers up 
to the 
> one you pick are involved in ratios, and then you octave reduce.  
The 
> "mapping by steps" is your homomorphism.

I think the best way to answer my questions may be to do some 
detective work and then ask about anything which remains unclear. I'm 
going therefore to take a look at the first 5-limit example, and add 
commentary.

My understanding is that we want 3, 5 and 5/3 within 2.8 cents. If a 
is how sharp our "3" is, and b is how sharp the "5" is, then we want 
a and b in the hexagonal region determined by

|a| <= 2.8, |b| <= 2.8, |a-b| <= 2.8

We need further conditions to determine the tuning, so let's look at 
what Graham does.

"3 4 5 6 7 8 9 10 12 15 16 18 19 22 23 24 25 26 27 28"

I don't know what these are. 

"5/19, 317.0 cent generator

basis: (1.0, 0.26416041678685936)"

If we set r =  0.26416041678685936, then 5/19 is a convergent for r. 
It's not clear why it is singled out; convergents for r are 1/3, 1/4, 
4/15, 5/19, 9/34,14/53, ...

"mapping by period and generator: 
[(1, 0), (0, 6), (1, 5)]"

This seems to explain where r came from: if we send (a, b) to a + 
b*r, then (1,0) goes to log_2(2) = 1, (0,6) goes to 6*r which turns 
out to be log_2(3), and (1,5) goes to 1+5*r which is the 
approximation of log_2(5) we get when both octaves and fifths are 
exact and [-6,-5,6] is in the kernel. Hence, r = 3^(1/6). Is Graham's 
basic condition that all primes up to the last will be exactly 
represented?

"mapping by steps: 
[(15, 4), (24, 6), (35, 9)]"

It seems as if this may have something to do with the convergents to 
r. We have the 4-et [4, 6, 9] from the convergent 1/4 and the 15-et 
[15, 24, 35] from the convergent 4/15. We may then proceed to the 
others:

[19, 30, 44]  = [ 4,  6,  9] + [15, 24, 35]
[34, 54, 79]  = [15, 24, 35] + [19, 30, 44]
[53, 84, 123] = [19, 30, 44] + [34, 54, 79]

after which a slew of semiconvergents come in. Graham says this 
is "my homomorphism", but I'm getting a whole collection.

"unison vectors: 
[[-6, -5, 6]]"

2^(-6)*3^(-5)*6^5 = 15625/15552 is the unison vector for anything 
using the matrix M = 

[0 1]
[6 5]

to approximate log_2(3) and log_2(5) using 1 and r' as a basis, so 
that [1,  r']M = [6r', 1+5r'] and so
6 r' approximates log_2(3) and 1 + 5 r' approximates log_2(5)--the 
column vector V = 

[  1  ]
[ 6 r']
[1+5r']

has unison vectors generated by [-6, -5, 6].

"highest interval width: 6"

How did we get to intervals and scales?

"complexity measure: 6 (7 for smallest MOS)"

How is this defined? 

"highest error: 0.001126 (1.351 cents)"

5/3 is off by this amount.

"unique"

What is unique?

This system is so close to the 53-et that it would seem to make sense 
to adjust the fifth, the octave or both and make it exactly the 53-et.


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Message: 1476

Date: Tue, 28 Aug 2001 19:44:34

Subject: Re: Now I think "the hypothesis" is true :)

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:

> Gene, was it ever decided if a kernel is equivalent to a set
> of unison vectors, as we use them?

Here are some questions:

(1) Is the unison a unison vector?

(2) If q is a unison vector, is q^2 a unison vector? 

(3) Are products of unison vectors unique--that is, if we have unison 
vectors {v1, ... vn} and v1^e1 * ... * vn^en = q, are the exponents 
ei determined?


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Message: 1477

Date: Tue, 28 Aug 2001 19:57:31

Subject: Re: Now I think "the hypothesis" is true :)

From: Carl Lumma

>>Gene, was it ever decided if a kernel is equivalent to a set
>>of unison vectors, as we use them?
> 
> Here are some questions:
> 
> (1) Is the unison a unison vector?

No.

> (2) If q is a unison vector, is q^2 a unison vector?

No, but it does point to a unison.

> (3) Are products of unison vectors unique--that is, if we have
> unison vectors {v1, ... vn} and v1^e1 * ... * vn^en = q, are the
> exponents ei determined?

I don't know.

-Carl


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Message: 1478

Date: Tue, 28 Aug 2001 20:20:42

Subject: Re: Now I think "the hypothesis" is true :)

From: Carl Lumma

Sorry Bob, I did not see your message until after I had posted
mine (we both answered in the same way).  Paul also makes a good
point -- you're assuming the 'classical' 12-tone, 5-limit scale
here (such as Ellis' "duodene"), which is common practice in
many music theory text books, but often leads to trouble here,
where we take nothing for granted when it comes to JI!

-Carl

> In Just Intonation, these pitches form intervals with C that are
> not  equal. In 12-tET, the irrational approximation of both
> intervals (sq root of 2) lies between F# and Gb. On the other hand,
> in Just Intonation F# is 45/32 of the frequency of C(1.40625*Fc)and
> Gb is 36/25 (1.44)of C. So it becomes clear that F# is lower than
> Gb, and the 12-tET interval of 1.414... is an irrational
> approximation in between them.


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Message: 1479

Date: Tue, 28 Aug 2001 20:22:52

Subject: Re: Now I think "the hypothesis" is true :)

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:
> 
> > Gene, was it ever decided if a kernel is equivalent to a set
> > of unison vectors, as we use them?
> 
> Here are some questions:
> 
> (1) Is the unison a unison vector?

Yes.
> 
> (2) If q is a unison vector, is q^2 a unison vector? 

Yes.

Don't know about (3).

"Unison vector" sometimes means any element of the kernel, but 
sometimes "the set of unison vectors of G" means "the generators of 
the kernel for G" . . . and in the case of chromatic unison vectors, 
we're pointing to an _altered_ equivalence, not a true equivalence.


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Message: 1480

Date: Tue, 28 Aug 2001 21:04:59

Subject: Re: Now I think "the hypothesis" is true :)

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:

Paul's answers were yes, yes, don't know and Carl's answers were no, 
no, don't know. I've remarked that I don't know if a unison vector is 
an element of the kernel or a generator of the kernel, and apparently 
that has not been decided. If your answer to (1) is "no", then your 
answer to (3) should probably be "yes", since otherwise a product of 
unison vectors will equal 1. Let's assume Paul's answer to (3) is 
also yes, then we have two types of definition:

(1) Carl type: Unison vectors are defined to be generators of the 
kernel of some homomorphism.

(2) Paul type: Unison vectors are defined to be members of the kernel 
of some homomorphism.

To pin this down further, here is another question:

(4) If we are considering octaves to be equivalent, is 2 a unison 
vector?


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Message: 1481

Date: Tue, 28 Aug 2001 21:06:14

Subject: Re: Now I think "the hypothesis" is true :)

From: Carl Lumma

>>> Gene, was it ever decided if a kernel is equivalent to a set
>>> of unison vectors, as we use them?
>> 
>> Here are some questions:
>> 
>> (1) Is the unison a unison vector?
> 
> Yes.

Really?  Why would it be?  And how do you define 'unison vector',
then?

>> (2) If q is a unison vector, is q^2 a unison vector? 
> 
> Yes.

I think I get a different PB if I use 5:4 instead of 25:16...

> "Unison vector" sometimes means any element of the kernel, but 
> sometimes "the set of unison vectors of G" means "the generators
> of the kernel for G" . . .

Whew.

-Carl


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Message: 1482

Date: Tue, 28 Aug 2001 22:14 +0

Subject: Re: More microtemperaments

From: graham@xxxxxxxxxx.xx.xx

genewardsmith@xxxx.xxx () wrote:

> My understanding is that we want 3, 5 and 5/3 within 2.8 cents. If a 
> is how sharp our "3" is, and b is how sharp the "5" is, then we want 
> a and b in the hexagonal region determined by
> 
> |a| <= 2.8, |b| <= 2.8, |a-b| <= 2.8

Yes.

> We need further conditions to determine the tuning, so let's look at 
> what Graham does.
> 
> "3 4 5 6 7 8 9 10 12 15 16 18 19 22 23 24 25 26 27 28"
> 
> I don't know what these are. 

They're 5-limit consistent equal temperaments, being used to calculate the 
linear temperaments.

> "5/19, 317.0 cent generator
> 
> basis: (1.0, 0.26416041678685936)"
> 
> If we set r =  0.26416041678685936, then 5/19 is a convergent for r. 
> It's not clear why it is singled out; convergents for r are 1/3, 1/4, 
> 4/15, 5/19, 9/34,14/53, ...

Because 19=4+15 is the simplest sum of numbers from the above list that 
fits this temperament.

> "mapping by period and generator: 
> [(1, 0), (0, 6), (1, 5)]"
> 
> This seems to explain where r came from: if we send (a, b) to a + 
> b*r, then (1,0) goes to log_2(2) = 1, (0,6) goes to 6*r which turns 
> out to be log_2(3), and (1,5) goes to 1+5*r which is the 
> approximation of log_2(5) we get when both octaves and fifths are 
> exact and [-6,-5,6] is in the kernel. Hence, r = 3^(1/6). Is Graham's 
> basic condition that all primes up to the last will be exactly 
> represented?

The condition is that the worst error is as low as possible.

> "mapping by steps: 
> [(15, 4), (24, 6), (35, 9)]"
> 
> It seems as if this may have something to do with the convergents to 
> r. We have the 4-et [4, 6, 9] from the convergent 1/4 and the 15-et 
> [15, 24, 35] from the convergent 4/15. We may then proceed to the 
> others:
> 
> [19, 30, 44]  = [ 4,  6,  9] + [15, 24, 35]
> [34, 54, 79]  = [15, 24, 35] + [19, 30, 44]
> [53, 84, 123] = [19, 30, 44] + [34, 54, 79]
> 
> after which a slew of semiconvergents come in. Graham says this 
> is "my homomorphism", but I'm getting a whole collection.

Then this is a subtlety of "homomorphism" I wasn't aware of.  I remember 
you showing how a linear (2-D) temperament can be described using the 
mappings of two equal temperaments.

> "unison vectors: 
> [[-6, -5, 6]]"
> 
> 2^(-6)*3^(-5)*6^5 = 15625/15552 is the unison vector for anything 
> using the matrix M = 
> 
> [0 1]
> [6 5]
> 
> to approximate log_2(3) and log_2(5) using 1 and r' as a basis, so 
> that [1,  r']M = [6r', 1+5r'] and so
> 6 r' approximates log_2(3) and 1 + 5 r' approximates log_2(5)--the 
> column vector V = 
> 
> [  1  ]
> [ 6 r']
> [1+5r']
> 
> has unison vectors generated by [-6, -5, 6].

Not sure about this bit.

> "highest interval width: 6"
> 
> How did we get to intervals and scales?

From the set of 5-limit intervals: 1:1, 5:4, 6:5, 3:2 and equivalents and 
inversions.  The highest number of generators you need to describe all 
these intervals is 6.

> "complexity measure: 6 (7 for smallest MOS)"
> 
> How is this defined? 

The number before times the number of periods to an octave.  The number of 
complete otonalities you can play is the number of notes in the generated 
scale minus this.

> "highest error: 0.001126 (1.351 cents)"
> 
> 5/3 is off by this amount.

That'll be it then.

> "unique"
> 
> What is unique?

It means each interval being approximated has a unique mapping to the 
temperament.  For example, meantone fails to be unique in the 9-limit 
because 9:8 and 10:9 map the same way.

> This system is so close to the 53-et that it would seem to make sense 
> to adjust the fifth, the octave or both and make it exactly the 53-et.

Maybe, but it could be a useful way of choosing subsets of 53-et.


               Graham


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Message: 1483

Date: Tue, 28 Aug 2001 01:22:15

Subject: Further examples of Zeta function tunings

From: genewardsmith@xxxx.xxx

This came up on the other list. A "Gram tuning" is a tuning derived 
from the Gram point associated to the large value of |Z(t)| 
assoicated to an et--in practice, this seems to mean the nearest Gram 
point. A "Z tuning" is the tuning derived from the actual value where 
|Z(t)| achieves its large local maximum near the n-et referred to 
before. Here are some further examples:

15

Gram point 45
Gram tuning = 15.052, 4.14 cents flat
Z tuning = 15.053, 4.26 cents flat

19

Gram point 63
Gram tuning = 18.954, 2.93 cents sharp
Z tuning = 18.948, 3.29 cents sharp

22

Gram point 78
Gram tuning = 22.025, 1.35 cents flat
Z tuning = 22.025, 1.37 cents flat

72

Gram point 378
Gram tuning = 71.954, .763 cents sharp
Z tuning = 71.951, .823 cents sharp


It would be interesting to compare this tunings to those arrived at 
by other methods.


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Message: 1484

Date: Tue, 28 Aug 2001 21:14:42

Subject: Re: Now I think "the hypothesis" is true :)

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:
> 
> Paul's answers were yes, yes, don't know and Carl's answers were 
no, 
> no, don't know. I've remarked that I don't know if a unison vector 
is 
> an element of the kernel or a generator of the kernel, and 
apparently 
> that has not been decided. If your answer to (1) is "no", then your 
> answer to (3) should probably be "yes", since otherwise a product 
of 
> unison vectors will equal 1. Let's assume Paul's answer to (3) is 
> also yes, then we have two types of definition:
> 
> (1) Carl type: Unison vectors are defined to be generators of the 
> kernel of some homomorphism.
> 
> (2) Paul type: Unison vectors are defined to be members of the 
kernel 
> of some homomorphism.

Gene, did you read the rest of my message???


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Message: 1485

Date: Tue, 28 Aug 2001 06:14:04

Subject: Re: Now I think "the hypothesis" is true :)

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., "Carl" <carl@l...> wrote:
> --- In tuning-math@y..., genewardsmith@j... wrote:
> > What's CS?
> 
> The property that every interval in a scale appears in only
> one interval class.  For example, 3:2 appears only as a 5th
> in the diatonic scale... but in 12-tET, the tritone appears
> as both a 4th and a 5th, so the diatonic scale in 12-tET is
> non-CS.

It seems to me that in a 12-et, a tritone would always be 6 steps. 
Can you clarify?


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Message: 1486

Date: Tue, 28 Aug 2001 21:15:56

Subject: Re: Now I think "the hypothesis" is true :)

From: genewardsmith@xxxx.xxx

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

> "Unison vector" sometimes means any element of the kernel, but 
> sometimes "the set of unison vectors of G" means "the generators of 
> the kernel for G" . . . and in the case of chromatic unison 
vectors, 
> we're pointing to an _altered_ equivalence, not a true equivalence.

It seems to me we should decide which way it's going to be. As for 
your last point, the chromatic unison vector is in the kernel of one 
homomorphism but not of another.


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Message: 1487

Date: Tue, 28 Aug 2001 10:37 +0

Subject: More microtemperaments

From: graham@xxxxxxxxxx.xx.xx

I've altered my temperament finding program to accept only temperaments 
with a worst error of less than 2.8 cents.  I think this is the cutoff for 
a microtemperament.  Results are at

<3 4 5 7 8 9 10 12 15 16 18 19 22 23 25 26 27 28 29 31 34 35 37 39 41 *>
<404 Not Found * Search for http://www.microtonal.co.uk/limit7.micro in Wayback Machine>
<5 12 19 22 26 27 29 31 41 46 50 53 58 60 68 70 72 77 80 84 87 89 91 94 99 *>
<22 26 29 31 41 46 58 72 80 87 89 94 111 113 118 121 130 145 149 152 159 166 171 176 183 *>
<26 29 41 46 58 72 80 87 94 111 113 121 130 149 159 166 171 183 190 198 212 217 224 241 253 *>
<29 41 58 72 80 87 94 111 121 130 149 159 183 190 198 212 217 224 241 253 270 282 296 301 311 *>

Some of them don't have as many as 10 results.


                    Graham


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Message: 1488

Date: Tue, 28 Aug 2001 21:17:20

Subject: Re: Now I think "the hypothesis" is true :)

From: Paul Erlich

--- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:
> >>> Gene, was it ever decided if a kernel is equivalent to a set
> >>> of unison vectors, as we use them?
> >> 
> >> Here are some questions:
> >> 
> >> (1) Is the unison a unison vector?
> > 
> > Yes.
> 
> Really?  Why would it be?

It's mapped to a unison.> >> (2) If q is a unison vector, is q^2 a 
unison vector? 
> > 
> > Yes.
> 
> I think I get a different PB if I use 5:4 instead of 25:16...

Just because it's also a unison vector, doesn't mean the resulting PB 
is the same!
> 
> > "Unison vector" sometimes means any element of the kernel, but 
> > sometimes "the set of unison vectors of G" means "the generators
> > of the kernel for G" . . .
> 
> Whew.

I hoped Gene would understand this, but apparently he skipped over 
this and came to the same conclusion independently (based on your 
answer and the first part of mine).


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Message: 1489

Date: Tue, 28 Aug 2001 10:20:02

Subject: Re: More microtemperaments

From: genewardsmith@xxxx.xxx

--- In tuning-math@y..., graham@m... wrote:

> I've altered my temperament finding program to accept only 
temperaments 
> with a worst error of less than 2.8 cents.  I think this is the 
cutoff for 
> a microtemperament.  

Is there somewhere where the meaning of this results is documented? 
It's hard to tell what to think of an error less than 2.8 cents when 
one doesn't know what the error is a departure from, for instance.


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Message: 1490

Date: Tue, 28 Aug 2001 21:20:18

Subject: Re: Now I think "the hypothesis" is true :)

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > "Unison vector" sometimes means any element of the kernel, but 
> > sometimes "the set of unison vectors of G" means "the generators 
of 
> > the kernel for G" . . . and in the case of chromatic unison 
> vectors, 
> > we're pointing to an _altered_ equivalence, not a true 
equivalence.
> 
> It seems to me we should decide which way it's going to be.

Too late -- it's already been used both ways. Why don't we just drop 
the "unison vector" terminology on this list and use "kernel" 
terminology instead, as I suggested before?


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Message: 1491

Date: Tue, 28 Aug 2001 11:29 +0

Subject: Re: More microtemperaments

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <9mfr8i+di6b@xxxxxxx.xxx>
In article <9mfr8i+di6b@xxxxxxx.xxx>, genewardsmith@xxxx.xxx () wrote:

> Is there somewhere where the meaning of this results is documented? 
> It's hard to tell what to think of an error less than 2.8 cents when 
> one doesn't know what the error is a departure from, for instance.

The departure is from a JI odd limit.  That is, all odd numbers up to the 
one you pick are involved in ratios, and then you octave reduce.  The 
"mapping by steps" is your homomorphism.


                     Graham


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Message: 1492

Date: Wed, 29 Aug 2001 07:39:32

Subject: An example

From: genewardsmith@xxxx.xxx

Let's put the definitions I just gave to use by looking at how the 
Blackjack scales might be derived. We may start from either ets or 
commas, but ets are easier to find and so it probably makes the most 
sense to began there. If we look at 7-limit ets, we find 
10,12,15,19,22,27,31,41,68,72,99 as ets h_n with n between 10 and 100 
and cons(7,n)<1. If we pick h_{31} and h_{41}, we generate a rank 2 
group M which also contains h_{10} = h_{41}-h_{31}, 
h_{72}=h_{41}+h_{31}, etc. Then K=null(M) is generated by the notes 
[-5, 2, 2, -1] and [-5,-1,-2,4], which correspond to the tones 
225/224 and 2401/2400.

If we look for ets contained in M we find h_{10}, h_{11}, h_{20}, 
h_{21}, ... and so forth. If we select h_{21}, we find ker(h_{21}) is 
generated by [2,2,-1,-1] (corresponding to 35/35) and K. If we make 
[2,2,-1,-1] a chroma and {[-5,2,2,-1], [-5,-1,-2,4]} commas then K is 
the commatic kernel and L=N_7/K is a note group of rank 2. 

If we choose a tuning for L in a reasonable way we now should have a 
good tone system for the 7-limit. "Reasonable" might for instance 
mean tuning octaves pure and picking a good value for the remaining 
generator. A particularly practical form of "reasonable" is to tune 
another et in M; thus we could have 21 notes out of 31 with 36/35 one 
step, 21 notes out of 41 with 36/35 two steps, or 21 notes out of 72 
with 36/35 three steps.


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Message: 1493

Date: Wed, 29 Aug 2001 19:26:44

Subject: Re: More microtemperaments

From: Paul Erlich

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <9mhgt6+avlv@e...>
> Dave Keenan wrote:
> 
> > Something must be wrong. How come schismic didn't make it into 
> > 5-limit? Couldn't you be missing some by not taking your 
consistent 
> > ET's out far enough. But there's definitely no need to go past 
215-tET 
> > (within 2.8 cents of anything).
> 
> Yes, schismic comes from 12 and 29.  I was taking the first 20 
consistent 
> ETs, which only got as far as 28.  So I've fudged it and am now 
taking the 
> first 21 instead.  Schismic should now be top of 
> <3 4 5 7 8 9 10 12 15 16 18 19 22 23 25 26 27 28 29 31 34 35 37 39 41 *>.  I used to take all 
consistent 
> ETs with fewer than 100 notes, but this meant a lot more were 
considered 
> for 15- than 5-limit.
> 
> 
>                       Graham

SO how do you know you're still not missing any?


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Message: 1494

Date: Wed, 29 Aug 2001 10:07 +0

Subject: Re: More microtemperaments

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <9mhgt6+avlv@xxxxxxx.xxx>
Dave Keenan wrote:

> Something must be wrong. How come schismic didn't make it into 
> 5-limit? Couldn't you be missing some by not taking your consistent 
> ET's out far enough. But there's definitely no need to go past 215-tET 
> (within 2.8 cents of anything).

Yes, schismic comes from 12 and 29.  I was taking the first 20 consistent 
ETs, which only got as far as 28.  So I've fudged it and am now taking the 
first 21 instead.  Schismic should now be top of 
<3 4 5 7 8 9 10 12 15 16 18 19 22 23 25 26 27 28 29 31 34 35 37 39 41 *>.  I used to take all consistent 
ETs with fewer than 100 notes, but this meant a lot more were considered 
for 15- than 5-limit.


                      Graham


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Message: 1495

Date: Wed, 29 Aug 2001 00:27:30

Subject: Re: Now I think "the hypothesis" is true :)

From: Carl Lumma

>>"Unison vector" sometimes means any element of the kernel, but 
>>sometimes "the set of unison vectors of G" means "the generators
>>of the kernel for G" . . . 

Ah, so my answers were based on the latter, and yours on the
former?

>>>>(2) If q is a unison vector, is q^2 a unison vector? 
>>>> 
>>>Yes.
>>
>>I think I get a different PB if I use 5:4 instead of 25:16...
>
>Just because it's also a unison vector, doesn't mean the resulting
>PB is the same!

And your reasoning here an example of the former?

>>>> (1) Is the unison a unison vector?
>>> 
>>> Yes.
>> 
>> Really?  Why would it be?
> 
> It's mapped to a unison.

Sounds like a tautology to me.

-Carl


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Message: 1496

Date: Wed, 29 Aug 2001 01:24:06

Subject: Defining CS and propriety for newbies (was: Now I think "the hypothesis" is tru)

From: Dave Keenan

--- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:
> "Interval class" is just a bad way to say "scale step".

Eek! No it isn't. I remember you had this problem before.

A scale step is the distance between two _consecutive_ scale degrees. 
For example, in western diatonic scales we have whole-tone steps and 
half-tone steps. We don't have minor third steps or perfect fifth 
steps.

>  "Every
> interval" is just a bad way to say "every acoustic interval".
> Does that help?

I don't think it helps. "Acoustic" simply means "relating to sound". 
Everything we deal with here is acoustic.

I don't think that any the above terminology is very good, when trying 
to define CS or propriety for newbies.

Instead of your "interval class" I use "number of scale steps", and 
instead of your "interval" I use "size (in cents)". In these 
definitions I use "interval" to mean "the distance between a specific 
pair of notes of the scale".

Now the definitions:

A scale is proper if all intervals spanning the same number of scale 
steps, have a range of sizes (in cents) that does not overlap but may 
meet, the range of sizes for any other number of scale steps.

A scale is strictly-proper if all intervals spanning the same number 
of scale steps, have a range of sizes (in cents) that is disjoint from 
(does not meet or overlap), the range of sizes for any other number of 
scale steps.

Examples.

1. Improper
Number of steps in interval                               4
              1                2                  3 <---------->
Ranges   <-------->        <-------->        <---------->
|        |        |        |        |        |        |        |
0       100      200      300      400      500      600      700 etc.
Interval size (cents)

2. Proper
Number of steps in interval
              1                2                  3       4
Ranges   <-------->        <-------->        <--------x-------->
|        |        |        |        |        |        |        |
0       100      200      300      400      500      600      700 etc.
Interval size (cents)

3. Strictly proper
Number of steps in interval
              1                2                  3       4
Ranges   <-------->        <-------->        <-------> <------->
|        |        |        |        |        |        |        |
0       100      200      300      400      500      600      700 etc.
Interval size (cents)


The following is supposedly Erv Wilson's definition of CS, as conveyed 
by Kraig Grady.

A scale is CS if all intervals of the same size (in cents), span the 
same number of scale steps.

CS is supposed to be a useful property for a scale to have, but notice 
that, by this definition, any random scale that has 
no-two-intervals-the-same-size is trivially CS, even if it has two 
intervals that differ by only 0.00001 cent spanning different numbers 
of scale steps!

A more meaningful definition for CS would be of the form:

A scale is CS if all intervals in the same range of sizes (in cents), 
(with all ranges defined so as to be disjoint), span the same number 
of scale steps.

Notice that this is almost equivalent to strict-propriety, written 
conversely. However a scale which is not strictly proper (i.e. it has 
number-of-step ranges that meet or overlap) might be able to have 
these ranges split into sub-ranges in such a way thay they no longer 
overlap and it is thereby CS.

4. CS?
Number of steps in interval                         4        4
              1                2               3   <->  3  <--->
Ranges   <-------->        <-------->        <--->     <->
|        |        |        |        |        |        |        |
0       100      200      300      400      500      600      700 etc.
Interval size (cents)

But clearly, this division into non-overlapping sub-ranges must be 
musically meaningful and in particular the sub-ranges must not be 
allowed to be too narrow, or else we are back to the trivial case 
where every subrange can consist of a single size.

Various ways of defining allowable ranges for CS, have been proposed, 
but none universally agreed upon. I ask their authors to explain what 
they are, should they be so inclined.

-- Dave Keenan


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Message: 1497

Date: Wed, 29 Aug 2001 01:27:25

Subject: Re: More microtemperaments

From: Dave Keenan

--- In tuning-math@y..., graham@m... wrote:
> I've altered my temperament finding program to accept only 
temperaments 
> with a worst error of less than 2.8 cents.  I think this is the 
cutoff for 
> a microtemperament.  Results are at
> 
> <3 4 5 7 8 9 10 12 15 16 18 19 22 23 25 26 27 28 29 31 34 35 37 39 41 *>
> <404 Not Found * Search for http://www.microtonal.co.uk/limit7.micro in Wayback Machine>
> <5 12 19 22 26 27 29 31 41 46 50 53 58 60 68 70 72 77 80 84 87 89 91 94 99 *>
> <22 26 29 31 41 46 58 72 80 87 89 94 111 113 118 121 130 145 149 152 159 166 171 176 183 *>
> <26 29 41 46 58 72 80 87 94 111 113 121 130 149 159 166 171 183 190 198 212 217 224 241 253 *>
> <29 41 58 72 80 87 94 111 121 130 149 159 183 190 198 212 217 224 241 253 270 282 296 301 311 *>
> 
> Some of them don't have as many as 10 results.

Oh Graham, you're wonderful!

-- Dave Keenan


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Message: 1498

Date: Wed, 29 Aug 2001 01:35:34

Subject: Re: More microtemperaments

From: Dave Keenan

--- In tuning-math@y..., graham@m... wrote:
> I've altered my temperament finding program to accept only 
temperaments 
> with a worst error of less than 2.8 cents.  I think this is the 
cutoff for 
> a microtemperament.  Results are at
> 
> <3 4 5 7 8 9 10 12 15 16 18 19 22 23 25 26 27 28 29 31 34 35 37 39 41 *>
> <404 Not Found * Search for http://www.microtonal.co.uk/limit7.micro in Wayback Machine>
> <5 12 19 22 26 27 29 31 41 46 50 53 58 60 68 70 72 77 80 84 87 89 91 94 99 *>
> <22 26 29 31 41 46 58 72 80 87 89 94 111 113 118 121 130 145 149 152 159 166 171 176 183 *>
> <26 29 41 46 58 72 80 87 94 111 113 121 130 149 159 166 171 183 190 198 212 217 224 241 253 *>
> <29 41 58 72 80 87 94 111 121 130 149 159 183 190 198 212 217 224 241 253 270 282 296 301 311 *>
> 
> Some of them don't have as many as 10 results.

Something must be wrong. How come schismic didn't make it into 
5-limit? Couldn't you be missing some by not taking your consistent 
ET's out far enough. But there's definitely no need to go past 215-tET 
(within 2.8 cents of anything).

-- Dave Keenan


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Message: 1499

Date: Wed, 29 Aug 2001 01:55:00

Subject: Re: Now I think "the hypothesis" is true :)

From: Dave Keenan

--- In tuning-math@y..., "Carl Lumma" <carl@l...> wrote:
> What recent threads have called "steps" are
> actually 2nds.

Yes. "Seconds" are the _only_ things that are called steps. Lest ye 
doubt, please see p63 of
404 Not Found * Search for http://depts.washington.edu/pnm/clampitt.pdf in Wayback Machine
"A step interval is an interval whose two boundary pitches are 
adjacent pitches of a scale."

-- Dave Keenan


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