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Message: 7900

Date: Sat, 01 Nov 2003 03:45:23

Subject: Re: Eponyms

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:

> I agree with Monz. There's definitely no need to include the
> 2-exponents here.

If you exlude them, you need a way of making it clear they are gone.


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Message: 7901

Date: Sat, 01 Nov 2003 22:15:49

Subject: Re: Eponyms

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> and as i've already been arguing with the ratios, forget it.
> 
> there's almost always nothing valuable about retaining the
> data for prime-factor 2, unless it need be considered for
> (to cite two examples i can think of quickly):
> 
> - actual orchestral scoring where the 8ve-register must
> be considered, or 
> 
> - analyzing ancient Greek and Roman theory, which was 
> based on 4:3 "perfect-4ths" and always specified 8ves,
> and gave different names to notes an 8ve apart.

if you don't retain data for prime-factor 2, how is your software 
able to handle torsion?


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Message: 7902

Date: Sat, 01 Nov 2003 03:47:28

Subject: Re: Eponyms

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
wrote:

> There's no need for systematic names to be so unfriendly as to call
> 81/80 the [-4, 4, -1]-comma, or even the [4, -1]-comma. The name
> "5-comma" can be generated and decoded systematically, as I've 
shown.

If 81/80 is a 5-comma, it would seem the schisma is also.


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Message: 7903

Date: Sat, 01 Nov 2003 22:28:31

Subject: Re: 'neutral' intervals

From: Paul Erlich

Aaron, before we get into some sort of personality clash, let me just 
say i was apparently misinterpreting you. It sounded like you were 
saying that, among all musicians and music listeners, intervals of 
350 cents are almost always interpreted as variants of either a minor 
third or a major third. You didn't say "I hear", you used the passive 
voice, so I thought you were generalizing about the music's affect on 
human listeners. I have intimate familiarity with both having the 
experience you describe, and then no longer having it, once the 
individuality of the intervals of a different culture have a chance 
to settle in. Therefore, I simply felt it appropriate to question 
what looked like an overgeneralization on your part. No offense 
meant, and I think we can discuss this in a sharing manner with 
everyone describing their own experience and point of view. Speaking 
of which, we should probably move this discussion to the tuning list 
in order to get more viewpoints in.

> Perhaps the Arabic musicians should speak for themselves? I 
> confess that I have not speken with any Arabic musicians on this 
> issue. I'm curious to know with whom you have discussed this 
> issue with.

Some of the finest oud players in the world (as well as middle 
eastern violinists . . .). If you're anywhere near boston, I'll 
introduce you.

> There is no need to beg in order to differ. You are free to call 
> these i intervals whatever you want. I offered my take on this 
> issue for what it's worth, and apparently it's worth little to you.

On the contrary, every viewpoint shared is very valuable to me. If it 
weren't for you, these lists would be that much more limited in 
scope. Sorry if it seemed I pounced on you but I mistook your 
statement as a pouncing on my experiences.


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Message: 7904

Date: Sat, 01 Nov 2003 03:55:44

Subject: Re: Eponyms

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> so if they're being described as monzos, just leave out
> the first exponent of the vector and the first prime-factor
> of the label.
> 
> 
> ... looks like Gene and i support each other on this method 
> of description.

I think we need a way of distinguishing 2-free monzos from complete 
information monzos. I suggest <4, -1> vs [-4, 4, -1] to distinguish 
the two ways of representing 81/80; the corresponding octave class 
could be (4, -1). The rule would be [] represents an interval, <> 
represents an interval in the standard octave 1 <= q < 2, and () 
represents the octave class whose represetative is given by the 
corresponding <>.


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Message: 7905

Date: Sat, 01 Nov 2003 22:34:27

Subject: Re: Eponyms

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> but what i forget to emphasize here again is: even in
> these cases where 8ves must be considered, it's easier
> to use the monzo including 2's exponent, instead of the 
> actual ratio.

my "heuristics" (maybe they should be called "intuitions"), allow you 
to glean essential information about the commas (their complexity or 
distance in the lattice, and the error that tempering them out is 
likely to impart) directly from the numbers in the ratios. the former 
is particularly easy, it's just the number of digits in the numerator 
and denominator.

for another example, you can compare two commas with about the same 
size numerators and denominators in them and estimate their relative 
sizes in cents, and errors of tempering, just by looking at the 
*difference* between numerator and denominator in each. The prime 
factorization doesn't help here at all.


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Message: 7906

Date: Sat, 01 Nov 2003 01:51:11

Subject: Re: Eponyms

From: Carl Lumma

>So what would your systematic names for 81/80 and 64/63 look like?

As shown.

>> With the wrong ranges, you wouldn't be able to extract the 3 exponent,
>> I assumed.
>
>That's true. But I don't understand what point you're making here.

No point.  IIRC I was just explaining something I'd said earlier.

-Carl


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Message: 7907

Date: Sat, 01 Nov 2003 14:40:14

Subject: Re: Eponyms

From: Carl Lumma

> the former 
>is particularly easy, it's just the number of digits in the numerator 
>and denominator.

It seems to be particularly difficult to pin down...

() log(d)  [this list]
() odd-limit(n/d)  [in the tuning dictionary]
() # of digits ???  [just now]

-Carl


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Message: 7908

Date: Sat, 01 Nov 2003 01:55:54

Subject: Re: Eponyms

From: Carl Lumma

>Would you care to explain what your objection's are to the proposal,

I think I've done that.

>as opposed to your objections to my online personality?

Actually I was referring to the both of us being anal there.

You mention genetic predisposition, and interestingly there's this
notion of "tasters" -- that in social animals a small part of the
population has a genetic factor that makes them simply must try
everything exactly once.  If true, I am surely one.

-Carl


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Message: 7909

Date: Sat, 01 Nov 2003 22:45:23

Subject: Re: ennealimmal

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> Am I correct that the first ennealimmal scale with an octave
> is simply 9-equal, and the next is this 17-tone one...

should be 18-tone . . . apparently you don't count 0 *or* 1200?

> 
>  50.
>  133.3
>  183.3
>  266.7
>  316.7
>  400.
>  450.
>  533.3
>  583.3
>  666.7
>  716.7
>  800.
>  850.
>  933.3
>  983.3
>  1066.7
>  1116.7
> 
> ...?
> 
> Manuel, is there a convenient way to get MOS-like scales with
> non-octave periods in Scala?
> 
> -Carl


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Message: 7910

Date: Sat, 01 Nov 2003 09:57:39

Subject: The supertemperament

From: Gene Ward Smith

For any odd prime p, there is a finite list of superparticular ratios
which belong to the p-limit. For some n, the smallest n intervals on
this list will uniquely determine a val v, which will be for an equal
temperament v(2) for which v is the standard val. This defines a
function supertemp(p) from p to the "supertemperament" for p. If I
calculated the 19-limit superparticulars correctly, we have the following:

supertemp(3) = 2
supertemp(5) = 7
supertemp(7) = 72
supertemp(11) = 72
supertemp(13) = 270
supertemp(17) = 1506
supertemp(19) = 8539

For each p, there will be a range from the first n to the first n
superparticulars which give the supertemperament. The ranges up
through 19 are as follows:

3: 1
5: 2
7: 3
11: 5-9
13: 9-12
17: 9-13
19: 14-15


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Message: 7911

Date: Sat, 01 Nov 2003 14:46:36

Subject: hey Paul

From: Carl Lumma

I'm interested in these scales...

>> [4, -3, 2, 13, 8, -14] [[1, 2, 2, 3], [0, 4, -3, 2]]
>> complexity 14.729697 rms 12.188571 badness 2644.480844
>> generators [1200., -125.4687958]
>
>25:24 chroma = -6 - 4 = -10 generators -> 10 note scale
>graham complexity = 7 -> 6 tetrads

Not sure of the significance of the - in -125.  I realize
that might have been Gene.

>> [4, 2, 2, -1, 8, -6] [[2, 0, 3, 4], [0, 2, 1, 1]]
>> complexity 10.574200 rms 23.945252 badness 2677.407574
>> generators [600.0000000, 950.9775006]
>
>//
>
>> [2, 6, 6, -3, -4, 5] [[2, 0, -5, -4], [0, 1, 3, 3]]
>> complexity 11.925109 rms 18.863889 badness 2682.600333
>> generators [600.0000000, 1928.512337]
>
>25:24 chroma = 6 - 1 = 5 generators -> 10 note scale
>graham complexity = 3*2 = 6 -> 8 tetrads

I've never noticed "generators" being expressed as larger
than "periods".  Why?  Can't we just reduce by the periods
here, getting

350.9775006

and

600., 128.512337

resp.?

Again, sorry if this is more of a question for the poster
of the >>'d text (Gene?).

>> [6, -2, -2, 1, 20, -17] [[2, 2, 5, 6], [0, 3, -1, -1]]
>> complexity 19.126831 rms 11.798337 badness 4316.252447
>> generators [600.0000000, 231.2978354]
>
>25:24 chroma = -2 - 3 = 5 generators -> 10 note scale
>graham complexity = 8 -> 4 tetrads

By the way, do these temperaments have names?

-Carl


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Message: 7912

Date: Sat, 01 Nov 2003 09:58:58

Subject: Re: Eponyms

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> 
> wrote:
> 
> > There's no need for systematic names to be so unfriendly as to call
> > 81/80 the [-4, 4, -1]-comma, or even the [4, -1]-comma. The name
> > "5-comma" can be generated and decoded systematically, as I've 
> shown.
> 
> If 81/80 is a 5-comma, it would seem the schisma is also.

Have you actually read any of the several descriptions I've given of
the proposed komma naming algorithm and its inverse? Are they all
really that unclear?

81/80 is the 5-comma. 
32805/32768 is the 5-schisma.
And if you want, you can say that they are both 5-kommas.

64/63 is the 7-comma
59049/57344 is the 7-medium-diesis or 7-M-diesis
28/27 is the 7-large-diesis or 7-L-diesis

2048/2035 is the 25-comma
6561/6400 is the 25-small-diesis or 25-S-diesis

128/125 is the 125-small-diesis
250/243 is the 125-medium-diesis
531441/512000 is the 125-large-diesis

5120/5103 is the 5:7-kleisma
3645/3584 is the 5:7-comma


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Message: 7913

Date: Sat, 01 Nov 2003 22:47:23

Subject: Re: Eponyms

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> > the former 
> >is particularly easy, it's just the number of digits in the 
numerator 
> >and denominator.
> 
> It seems to be particularly difficult to pin down...
> 
> () log(d)  [this list]
> () odd-limit(n/d)  [in the tuning dictionary]

these are virtually identical for any small comma -- the intervals in 
question here.

> () # of digits ???  [just now]

yes, if the log is to the base 10, it's just the rounded number of 
digits.


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Message: 7914

Date: Sat, 01 Nov 2003 10:00:26

Subject: Re: Eponyms

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> Have you actually read any of the several descriptions I've given of
> the proposed komma naming algorithm and its inverse? Are they all
> really that unclear?

I don't buy kommas.


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Message: 7915

Date: Sat, 01 Nov 2003 14:53:05

Subject: Re: ennealimmal

From: Carl Lumma

>> Am I correct that the first ennealimmal scale with an octave
>> is simply 9-equal, and the next is this 17-tone one...
>
>should be 18-tone . . . apparently you don't count 0 *or* 1200?

d'oh.

>>  50.
>>  133.3
>>  183.3
>>  266.7
>>  316.7
>>  400.
>>  450.
>>  533.3
>>  583.3
>>  666.7
>>  716.7
>>  800.
>>  850.
>>  933.3
>>  983.3
>>  1066.7
>>  1116.7
//  2/1
>>
>> ...?
>> 
>> Manuel, is there a convenient way to get MOS-like scales with
>> non-octave periods in Scala?

-Carl


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Message: 7916

Date: Sat, 01 Nov 2003 10:06:24

Subject: Re: Eponyms

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >So what would your systematic names for 81/80 and 64/63 look like?
> 
> As shown.

As shown where? If you mean "81/80" and "64/63", I thought we agreed
that these don't qualify as names. And if we didn't, then I have to
say I find, for example, "5-schisma" to be a serious improvement over
"32805/32768" as a name.


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Message: 7917

Date: Sat, 01 Nov 2003 14:54:33

Subject: heuristic (was Re: Re: Eponyms)

From: Carl Lumma

>> It seems to be particularly difficult to pin down...
>> 
>> () log(d)  [this list]
>> () odd-limit(n/d)  [in the tuning dictionary]
>
>these are virtually identical for any small comma -- the intervals in 
>question here.

Right, but why not pick a form and stick with it.

>> () # of digits ???  [just now]
>
>yes, if the log is to the base 10, it's just the rounded number of 
>digits.

Ah yes, of course.

-Carl


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Message: 7918

Date: Sat, 01 Nov 2003 02:08:19

Subject: Re: The supertemperament

From: Carl Lumma

>For any odd prime p, there is a finite list of superparticular ratios
>which belong to the p-limit.

Here's something I can believe but which isn't immediately obvious.
Can you prove it?

>For some n, the smallest n intervals on
>this list will uniquely determine a val v, which will be for an equal
>temperament v(2) for which v is the standard val. This defines a
>function supertemp(p) from p to the "supertemperament" for p. If I
>calculated the 19-limit superparticulars correctly, we have the following:
>
>supertemp(3) = 2
>supertemp(5) = 7
>supertemp(7) = 72
>supertemp(11) = 72
>supertemp(13) = 270
>supertemp(17) = 1506
>supertemp(19) = 8539

Cool.  Howabout moving a fixed n down the list (or n's which, for each
starting point in the list, uniquely define a val)?

-Carl


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Message: 7919

Date: Sat, 01 Nov 2003 22:57:06

Subject: heuristic (was Re: Re: Eponyms)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> It seems to be particularly difficult to pin down...
> >> 
> >> () log(d)  [this list]
> >> () odd-limit(n/d)  [in the tuning dictionary]
> >
> >these are virtually identical for any small comma -- the intervals 
in 
> >question here.
> 
> Right, but why not pick a form and stick with it.

that's not my style :) actually, i try to use the second when 
supplying exact calculations . . .

> >> () # of digits ???  [just now]
> >
> >yes, if the log is to the base 10, it's just the rounded number of 
> >digits.
> 
> Ah yes, of course.

obviously a different form, but a lot easier to use when you don't 
have a calculator handy!


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Message: 7920

Date: Sat, 01 Nov 2003 02:11:00

Subject: Re: Eponyms

From: Carl Lumma

>> >So what would your systematic names for 81/80 and 64/63 look like?
>> 
>> As shown.
>
>As shown where? If you mean "81/80" and "64/63", I thought we agreed
>that these don't qualify as names.

We most certainly didn't.

>And if we didn't, then I have to
>say I find, for example, "5-schisma" to be a serious improvement over
>"32805/32768" as a name.

I didn't say I'd use 32805/32768 as a name.

-Carl


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Message: 7921

Date: Sat, 01 Nov 2003 10:20:11

Subject: Re: Eponyms

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:
> 
> > Have you actually read any of the several descriptions I've given of
> > the proposed komma naming algorithm and its inverse? Are they all
> > really that unclear?
> 
> I don't buy kommas.

Do you mean you don't like spelling it with a "k" when it's being used
as a generic term. That's fine. That's not part of the naming
algorithm. That's just me. 

There are no "kommas" in the automatically generated names.

Are you seriously saying you haven't read any of them because of "kommas"?


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Message: 7922

Date: Sat, 01 Nov 2003 00:21:38

Subject: Re: Eponyms

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >The 5, 7 and 11 are all on the same side of the ratio, or there would
> >have been a colon ":" in there.
> 
> How do you pronounce that?

Good question. I haven't been pronouncing the colon at all. 385/384 is
the first time I've felt any desire to indicate multiplication (as
5.7.11-kleisma). I don't want to pronounce the dot either, so that's a
bit of a problem. If we call it 385-kleisma that problem doesn't
occur, but I tend to think the systematic name should give the
factorisation when it gets bigger than most people can easily
factorise mentally. I'm guessing that's around 125, but we could
simply declare it to be 385. :-) And then the need for dots would be
very rare.

Another vague idea: The order of mention of primes could be different
depending whether they are being multiplied (dot) or divided (colon).

> >They are all only to the power given, namely 1.
> 
> How do you do it with higher powers?

5, 25, 125, 5^4, 5^5, ... where the latter are pronounced "five to the
four" etc.
7, 49, 343 (or 7^3), 7^4, 7^5, ...
11, 121, 11^3, 11^4, ...
I suppose 11^3 should be pronounced "eleven cubed" rather than "eleven
to the three".

Tanaka's kleisma (_the_ kleisma) has the systematic name of
5^6-kleisma (five-to-the-six-kleisma)

> >It's a kleisma so it's in the range 4.5 c (a bit arbitrary at present)
> >to 11.7 c (actually, exactly half a pythagorean comma).
> 
> 385:383 is in that range.

Yes, but the system says that the only factors omitted from the first
part of the name are factors of 2 and 3. 383 contains other primes (in
fact _is_ a rather large prime) which would therefore have to be
upfront in the name.

> >That's how a dumb algorithm would have to do it, but you or I
> >(assuming we knew something about the system) would say: Its got 385
> >as a factor along with some powers of 2 and 3. I know roughly how big
> >it is so I wonder if it's 386/385 or 385/384.
> 
> Oh, I thought you always gave the numerator.

No. 

To convert a comma ratio to its systematic name:

1. Remove all factors of 2 and 3.
2. Replace slash with colon.
3. Swap the two sides of the ratio if necessary to put the smallest
number first.
4. If it now starts with "1:", eliminate the "1:".
5. If any side of the (2,3-reduced) ratio is bigger than 125 (or maybe
385) then give its prime factorisation in some form (details yet to be
decided). 
6. Calculate the comma size in cents and use it to look up and append
the category name, preceded by a hyphen.

This is not guaranteed to give a unique name (although clashes will be
exceedingly rare). To be certain that your comma actually deserves the
name, you have to run the process in reverse (as I've described
already) trying 3-exponents in the series 0, 1, -1, 2, -2, 3, -3, ...
and octave reducing, until you get a hit on the correct size-category.
Then see if you've got your original comma ratio back again.

> >Oh 384 has prime factors of only 2's and 3's.
> 
> How do we know it's only got 2's and 3's if we're only given
> "385-kleisma"?

Because that's the system.

Even if you didn't know the system to start with, you should soon
notice, when looking at any kind of list of systematic comma names,
that there isn't an explicit power of 2 or 3 mentioned anywhere.


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Message: 7923

Date: Sat, 01 Nov 2003 02:24:08

Subject: Re: Eponyms

From: Carl Lumma

>Are you seriously saying you haven't read any of them
>because of "kommas"?

I must admit that the first time this went around, I
stopped reading when I saw it.

-Karl


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Message: 7924

Date: Sat, 01 Nov 2003 00:27:43

Subject: Re: Eponyms

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> > >>5.7.11-kleisma has no advantages over 385/384 that I can see.
> > 
> > The latter must be factored to see what it's good for, and
> > log'ed to give an exact size.  The former gives a size range,
> > and with the addition of the 3 exponent tells you what it's
> > good for (otherwise how'reyou going to say what pythagorean
> > commas are good for?).  But with the addition of the 3 exponent,
> > we loose the ability to draft size ranges.  What say you to
> > this, Dave? 
> 
> If you want to make this systematic, why not simply monzo-size range?

I agree with this for the _really_ complex commas, but I want a
reasonably non-mathematician friendly system where for example the
systematic names for 81/80 and 64/63 are 5-comma and 7-comma
respectively. You can even pronounce the "7" as "septimal" if you
want, and then it's the same as its common name.


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