Eponyms

On the subject of eponyms:

Manuel, I'd prefer it if Scala did not refer to 384:385 as Keenan's
kleisma, although I thank Paul for his sentiments in proposing it.

Now that I've found what I think is a good system for naming kommas,
I'd prefer it to be called "385-kleisma" or "5.7.11-kleisma". I think
I prefer the latter, and would pronounce it "five seven eleven kleisma".

Does anyone have any objection to this, or want to propose another name?

Regards,
-- Dave Keenan

>Now that I've found what I think is a good system for naming kommas,
>I'd prefer it to be called "385-kleisma"

In your scheme, the term kleisma tells us that the denominator must
be 384, and not 383?

>or "5.7.11-kleisma".

...tells us how to combine factors of 5, 7, and 11 to get the
right ratio?

Just asking.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >Now that I've found what I think is a good system for naming kommas,
> >I'd prefer it to be called "385-kleisma"
>
> In your scheme, the term kleisma tells us that the denominator must
> be 384, and not 383?
>
> >or "5.7.11-kleisma".
>
> ...tells us how to combine factors of 5, 7, and 11 to get the
> right ratio?

I already did. Sorry I didn't give examples.

The 5, 7 and 11 are all on the same side of the ratio, or there would
have been a colon ":" in there. They are all only to the power given,
namely 1. So we can immediately fill in the monzo for all the primes
greater than two [? ? 1 1 1].

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).

Try successive exponents of 3 in this sequence 0, 1, -1, 2, -2, 3, -3
... and with each of those, whatever power of 2 that octave-reduces it
to lowest terms, i.e. puts it in the range -600 to +600 cents. As soon
as you hit one whose absolute value in cents is actually in the
kleisma range, you've found it. If it is a negative number of cents,
negate all the exponents in the monzo.

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 384 has prime factors
of only 2's and 3's. Calculate size in cents. Yep that's it, 385/384.

The careful choice of the range boundaries for schismina, schisma,
kleisma, comma, small diesis, medium diesis, large diesis, etc. (at
square roots of various 3-kommas) is what makes the powers of 3 and 2
unambiguous.

But even before you've done any such processing, you immediately know
roughly how big it is and what its good for, namely turning 7-limit
into an approximation of 11-limit).

If you want to see potential values for komma-size-category
boundaries, put the integers from 1 to 53 in spreadsheet column A.
These represent exponents of 3. In column B calculate the associated
exponent of 2 as
=ROUND(A1 * LN(3)/LN(2), 0)
and in column C calculate the square root of the implied 3-komma in
cents as
=ABS(A1 * LN(3)/LN(2) - B1)*600

But we have to reject any where both the 2 and the 3 exponents are
even. This is because the square root would then be rational, and we
would have a 3-comma with an ambiguous category because it would be
right on the boundary. So change that to
=IF(ISEVEN(A1)*ISEVEN(B1), 0, ABS(A1 * LN(3)/LN(2) - B1)*600)

Then sort the list on column C, the size in cents.

It is more important to have category boundaries at square roots of
kommas with smaller exponents of 3, but not if they are too close to a
boundary for one with an even-lower 3-exponent.

The result follows, showing my proposal.

I note that we have to go out to 3^200 before we find a good place for
the schisma/kleisma boundary, at 4.499913461 cents = sqrt(3^200/2^317).

53 84 1.807522933 schismina/schisma
41 65 9.92248226
12 19 11.73000519 kleisma/comma
29 46 21.65248745
17 27 33.38249264 comma/small diesis
36 57 35.19001558
5 8 45.11249784 small diesis/medium diesis
46 73 55.0349801
7 11 56.84250303 medium diesis/large diesis
19 30 68.57250822 large diesis/?
22 35 78.49499048 ?
31 49 80.30251341
43 68 92.03251861
51 81 100.1474779
2 3 101.9550009 ?
39 62 111.8774831
27 43 123.6074883
26 41 125.4150113
15 24 135.3374935
3 5 147.0674987 ?
50 79 148.8750216
9 14 158.7975039
32 51 168.7199862
21 33 170.5275091
33 52 182.2575143
8 13 192.1799965
45 71 193.9875195
49 78 202.1024788
37 59 213.832484
16 25 215.6400069
25 40 225.5624892
13 21 237.2924944
40 63 239.1000173
1 2 249.0224996 ?
42 67 258.9449818
11 17 260.7525048
23 36 272.48251
18 29 282.4049922
35 55 284.2125151
47 74 295.9425203

Again, the point of these boundaries is that they let you extract the
powers of 2 and 3 from the comma name, if it includes enough
information about the exponents of the primes higher than 3.

-- Dave Keenan

I realise I missed this the first time:

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> In your scheme, the term kleisma tells us that the denominator must
> be 384, and not 383?

Yes! But not directly of course. All it tells us directly is the size
range. From that, and the 385, we can get the factors of 2 and 3, as I
explained in the previous message. It's such a neat trick, I guess
it's hard to believe it works.

But note that the number given in the name is not necessarily the
numerator or demominator of the comma ratio, it's the comma ratio with
the 2's and 3's removed, (and inverted if it's less than one).

> >or "5.7.11-kleisma".
>
> ...tells us how to combine factors of 5, 7, and 11 to get the
> right ratio?

Yes. When the dots are read as multiplication, this contains no more
and no less information than "385-kleisma" since prime factorisations
are unique.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> On the subject of eponyms:
>
> Manuel, I'd prefer it if Scala did not refer to 384:385 as Keenan's
> kleisma, although I thank Paul for his sentiments in proposing it.
>
> Now that I've found what I think is a good system for naming kommas,
> I'd prefer it to be called "385-kleisma" or "5.7.11-kleisma". I
think
> I prefer the latter, and would pronounce it "five seven eleven
kleisma".
>
> Does anyone have any objection to this, or want to propose another
name?

There isn't really much point in naming commas like 385/384 in the
first place, but if you do, the name should be easier than the thing
it is naming. 5.7.11-kleisma has no advantages over 385/384 that I
can see.

hi Gene,

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
>
> > On the subject of eponyms:
> >
> > Manuel, I'd prefer it if Scala did not refer to 384:385
> > as Keenan's kleisma, although I thank Paul for his sentiments
> > in proposing it.
> >
> > Now that I've found what I think is a good system for
> > naming kommas, I'd prefer it to be called "385-kleisma"
> > or "5.7.11-kleisma". I think I prefer the latter, and would
> > pronounce it "five seven eleven kleisma".
> >
> > Does anyone have any objection to this, or want to propose
> > another name?
>
> There isn't really much point in naming commas like 385/384
> in the first place, but if you do, the name should be easier
> than the thing it is naming. 5.7.11-kleisma has no advantages
> over 385/384 that I can see.

i disagree quite strongly.

to me, the only thing the ratio shows is that it's
superparticular (or epimoric, if you prefer Greek over Latin).

i think 5.7.11-kleisma is a much better name ... altho i
think my first choice would be to use the monzo and call it
the [-7 -1 1 1 1]-kleisma, or if you can do without the 2
(which i also always prefer if possible), [-1 1 1 1]-kleisma.

:)

-monz

Gene wrote:
>5.7.11-kleisma has no advantages over 385/384 that I can see.

I think so too. It looks like it's the simplest undecimal
kleisma and there isn't another one called that so I'll change
the name in "undecimal kleisma".

Manuel

--- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:
>
> Gene wrote:
> >5.7.11-kleisma has no advantages over 385/384 that I can see.

So call it the 385-comma. I agree that in this case the systematic
name is not much better than the ratio. But it does at least tell you
its approximate size in cents.

Manuel:
> I think so too. It looks like it's the simplest undecimal
> kleisma and there isn't another one called that so I'll change
> the name in "undecimal kleisma".

No it's not the simplest. What about 243/242 (7.1 c) which you call
the neutral third comma. I call it the 121-kleisma. Calling it a comma
implies that it is bigger than it really is.

Although 896/891 (9.7 c) is more complex than either 243/242 (7.1 c)
or 385/384 (4.5 c), it occurs more commonly as a notational comma
relative to Pythagorean, since it serves to notate simpler ratios.

I notice you call 896/891 (9.7 c) the undecimal semicomma. I call it
the 7:11-kleisma. Is there a general rule as to how a semicomma
differs from a kleisma? This seems unlikely since you list "semicomma
majeur" as an alternate name for _the_ kleisma (the 5^6-kleisma,
15625/15552, 8.1 c). And 2109375/2097152 (10.1 c) you call _the_
semicomma or Fokker's comma.

What will you call 2835/2816 (11.6 c). I call it the 11:35-kleisma.
There's also 2893401/2883584 (5.9 c). I call it the 11:49-kleisma.
Admittedly these last two are not very much in need of a name.

Dave wrote:
>Calling it a comma implies that it is bigger than it really is.

The list isn't systematic in that sense. It's more a list of
historical names. So "comma" is used in the general sense and can
be a comma of any size.
I think "neutral third comma" was a name proposed by Brian.

>Is there a general rule as to how a semicomma
>differs from a kleisma?

What you thought, no. "Semicomma majeur" comes from Rameau.

>And 2109375/2097152 (10.1 c) you call _the_
>semicomma or Fokker's comma.

Fokker called it the semicomma and I added "Fokker's comma"
for he mentioned it in his books, noticed its vanishing in
31-tET, etc. Whether he borrowed the name from Rameau I don't
know but it's quite possible.

>What will you call 2835/2816 (11.6 c). I call it the 11:35-kleisma.
>There's also 2893401/2883584 (5.9 c). I call it the 11:49-kleisma.
>Admittedly these last two are not very much in need of a name.

No, I haven't thought about it.

Manuel

Pertaining to the discussion of interval names, I believe that the
term 'neutral', although it seems sensible enough, is
problematic in practice and should be reconsidered.

The concern arises because a so-called neutral third sounds
'major' in one context and 'minor' in another. Use of these
'neutral' intervals in a melodic context almost never results in a
musical functionality which could be described as 'neutral'. In my
experience, the same ambiguity presents itself when these
intervals are used harmonically.

This has led me to call these intervals 'narrow major' or 'wide
minor' depending on context. This dual perspective is
advantageous for descriptions of musical function.

I apply the umbrella term 'ambiguous' rather than 'neutral' to
these intervals, referring to an 'ambisecond' or 'ambithird', etc.
'Ambi' also brings to mind 'ambu' which suggests that these
intervals can move about in terms of function.

I have not heard of others using the terms or prefixes that I use,
but it seems likely that someone else would easily come up with
something similar.

Regards,
Aaron Hunt

At 06:30 PM 10/30/2003, you wrote:
>--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>> >Now that I've found what I think is a good system for naming kommas,
>> >I'd prefer it to be called "385-kleisma"
>>
>> In your scheme, the term kleisma tells us that the denominator must
>> be 384, and not 383?
>>
>> >or "5.7.11-kleisma".
>>
>> ...tells us how to combine factors of 5, 7, and 11 to get the
>> right ratio?
>
>I already did. Sorry I didn't give examples.
>
>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?

>They are all only to the power given, namely 1.

How do you do it with higher powers?

>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.

>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.

>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"?

-Carl

>>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?

-Carl

--- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:
>
> Gene wrote:
> >5.7.11-kleisma has no advantages over 385/384 that I can see.
>
> I think so too. It looks like it's the simplest undecimal
> kleisma and there isn't another one called that so I'll change
> the name in "undecimal kleisma".

A fine name. Go for it.

--- In tuning-math@yahoogroups.com, 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?

>> 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?

Example?

-Carl

--- In tuning-math@yahoogroups.com, "gooseplex" <cfaah@e...> wrote:

> Pertaining to the discussion of interval names, I believe that the
> term 'neutral', although it seems sensible enough, is
> problematic in practice and should be reconsidered.
>
> The concern arises because a so-called neutral third sounds
> 'major' in one context and 'minor' in another.

That's usually only a result of insufficient exposure to the music in
question. Arabic musicians don't hear their neutral thirds as 'major'
in one context and 'minor' in another, same for neutral seconds.

> Use of these
> 'neutral' intervals in a melodic context almost never results in a
> musical functionality which could be described as 'neutral'.

Almost never?

> In my
> experience, the same ambiguity presents itself when these
> intervals are used harmonically.

So you mean almost never *in your experience*?

> This has led me to call these intervals 'narrow major' or 'wide
> minor' depending on context. This dual perspective is
> advantageous for descriptions of musical function.

i beg to differ.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "gooseplex" <cfaah@e...> wrote:
>
> > Pertaining to the discussion of interval names, I believe that
the
> > term 'neutral', although it seems sensible enough, is
> > problematic in practice and should be reconsidered.
> >
> > The concern arises because a so-called neutral third sounds
> > 'major' in one context and 'minor' in another.
>
> That's usually only a result of insufficient exposure to the music
in
> question. Arabic musicians don't hear their neutral thirds
as 'major'
> in one context and 'minor' in another, same for neutral seconds.
>
> > Use of these
> > 'neutral' intervals in a melodic context almost never results in
a
> > musical functionality which could be described as 'neutral'.
>
> Almost never?
>
> > In my
> > experience, the same ambiguity presents itself when these
> > intervals are used harmonically.
>
> So you mean almost never *in your experience*?
>
> > This has led me to call these intervals 'narrow major' or 'wide
> > minor' depending on context. This dual perspective is
> > advantageous for descriptions of musical function.
>
> i beg to differ.

I can't imagine how anyone (such as Margo Schulter or myself) who has
spent any significant amount of time using a 17-tone temperament,
either equal or well-tempered (to improve non-5 ratios of 7), could
avoid hearing certain 2nds, 3rds, 6th, and 7ths as distinctly
neutral. A prime example of this is the two-voice progression
consisting of the interval of an augmented 6th, Eb3-C#4, resolved to
an octave, D3-D4, by melodic intervals of 2 degrees of 17, which are
unmistakably neutral 2nds.

--George

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> I can't imagine how anyone (such as Margo Schulter or myself) who
has
> spent any significant amount of time using a 17-tone temperament,
> either equal or well-tempered (to improve non-5 ratios of 7), could
> avoid hearing certain 2nds, 3rds, 6th, and 7ths as distinctly
> neutral. A prime example of this is the two-voice progression
> consisting of the interval of an augmented 6th, Eb3-C#4, resolved
to
> an octave, D3-D4, by melodic intervals of 2 degrees of 17, which
are
> unmistakably neutral 2nds.
>
> --George

George,

I'm confused. Wouldn't the resolving intervals be 1 degree of 17 each?

-Paul

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...>
> wrote:
>
> > I can't imagine how anyone (such as Margo Schulter or myself) who
> has
> > spent any significant amount of time using a 17-tone temperament,
> > either equal or well-tempered (to improve non-5 ratios of 7),
could
> > avoid hearing certain 2nds, 3rds, 6th, and 7ths as distinctly
> > neutral. A prime example of this is the two-voice progression
> > consisting of the interval of an augmented 6th, Eb3-C#4, resolved
> to
> > an octave, D3-D4, by melodic intervals of 2 degrees of 17, which
> are
> > unmistakably neutral 2nds.
> >
> > --George
>
> George,
>
> I'm confused. Wouldn't the resolving intervals be 1 degree of 17
each?
>
> -Paul

Oops, sorry -- I was in too much of a hurry! The first interval
should be a major 6th with tones E-semiflat and C-semisharp.

--George

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >If you want to make this systematic, why not simply monzo-size range?
>
> Example?

225/224 becomes the [-5,2,2,-1]-kleisma, whereas 385/384 is the
[-7,-1,1,1,1]-kleisma.

--- In tuning-math@yahoogroups.com, 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

So these are two advantages of "5.7.11-kleisma" over "385/384".

> (otherwise how'reyou going to say what pythagorean
> commas are good for?).

There aren't too many of them that have come to my attention so far.
These are well known and have common names: Pythagorean-limma,
apotome, Pythagorean-comma. If the previously described naming system
was simply applied they would all be 1-<whatevers> where "<whatever>"
stands for the correct size category. But I would change this to
3-<whatevers> or even better, Pythagorean-<whatevers>. So the limma
and comma would be the same as their common names and the apotome
would probably have the systematic name: Pythagorean-semitone. The
3-exponent can be extracted from the size category if necessary.

I think that in any list of common commas you will find that more than
90% of them have a 3-exponent other than zero. So it is fairly
unnecessary to include, up front, the fact that they have 3's in them;
and it would be technically redundant since the 3-exponent can be
extracted from the size category in conjunction with the numbers that
are supplied.

> But with the addition of the 3 exponent,
> we loose the ability to draft size ranges. What say you to
> this, Dave?

I'm not sure what you mean by "to draft size ranges"? If you mean "to
decide the boundaries of size ranges", I don't understand why you
would lose that ability. You could keep the same ranges and the up
front information about the 3-exponent would simply be redundant.

I don't want to call 81/80 the 5:3^4-comma, but just the 5-comma. And
64/63 would be simply the 7-comma, not the 7.9-comma. That's the whole
point of setting the size-category boundaries so carefully, to
eliminate the need to have the power of 3 explicit in the systematic name.

--- In tuning-math@yahoogroups.com, 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.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, 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.

>> >If you want to make this systematic, why not simply monzo-size range?
>>
>> Example?
>
>225/224 becomes the [-5,2,2,-1]-kleisma, whereas 385/384 is the
>[-7,-1,1,1,1]-kleisma.

That works.

-Carl

>> >>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
>
>So these are two advantages of "5.7.11-kleisma" over "385/384".

Since factoring is as hard as extracting the 3 exponent given
the size range, the only difference is whether you prefer to
know a range by memorizing a few words, or guestimate a range
by doing some quick division.

>> (otherwise how'reyou going to say what pythagorean
>> commas are good for?).
>
>There aren't too many of them that have come to my attention so far.
>These are well known and have common names: Pythagorean-limma,
>apotome, Pythagorean-comma. If the previously described naming system
>was simply applied they would all be 1-<whatevers> where "<whatever>"
>stands for the correct size category. But I would change this to
>3-<whatevers> or even better, Pythagorean-<whatevers>. So the limma
>and comma would be the same as their common names and the apotome
>would probably have the systematic name: Pythagorean-semitone. The
>3-exponent can be extracted from the size category if necessary.

The idea of leaving out the 3's is clever but not beneficial, in
my opinion.

>> But with the addition of the 3 exponent,
>> we loose the ability to draft size ranges. What say you to
>> this, Dave?
>
>I'm not sure what you mean by "to draft size ranges"? If you mean "to
>decide the boundaries of size ranges", I don't understand why you
>would lose that ability. You could keep the same ranges and the up
>front information about the 3-exponent would simply be redundant.

With the wrong ranges, you wouldn't be able to extract the 3 exponent,
I assumed.

-Carl

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > >If you want to make this systematic, why not simply monzo-size range?
> >
> > Example?
>
> 225/224 becomes the [-5,2,2,-1]-kleisma, whereas 385/384 is the
> [-7,-1,1,1,1]-kleisma.

I agree with Monz. There's definitely no need to include the
2-exponents here. They're musically irrelevant in most cases, and if
you do need them, the mere fact that these are commas of some kind,
and hence smaller than 600 cents, is enough to give you their
2-exponents.

I wouldn't want to start using monzos in this role until things got
really complex, like if you would otherwise have more than say 12
characters in the numeric part.

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.

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

Again a cool idea, and I find these sort of inquiries fascinating, but
I try to avoid them when I can't see them being very useful. YMMV.

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

I've so far tried my best not to mention the term "anal retentive".
:)

>> 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.

Yeah, sorry I didn't catch that until later.

>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.

Again, nice touch but...

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> The idea of leaving out the 3's is clever but not beneficial, in
> my opinion.

I'm a little confused here. On the one hand you'd apparently be quite
happy to call something the "fartisma", because all you need is a
"hook" to hang the meaning on, and names with numbers in them are
boring, but when I show you a way to eliminate some of the numbers
while still remaining systematic and unambiguous, you want to keep all
the numbers, even redundant ones.

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

> 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.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:

> I think that in any list of common commas you will find
> that more than 90% of them have a 3-exponent other than
> zero. So it is fairly unnecessary to include, up front,
> the fact that they have 3's in them;

i can pretty much agree with that, with one very important
comma residing in that other 10%: the enharmonic diesis,
ratio 128/125, [3,5]-monzo version: the [ 0 -3]-diesis.

> I don't want to call 81/80 the 5:3^4-comma, but just
>the 5-comma.

[3,5]-monzo version: the [4 -1]-comma.

> And 64/63 would be simply the 7-comma, not the 7.9-comma.

[3,5,7]-monzo version: the [-2 0 -1]-comma.

> That's the whole point of setting the size-category
> boundaries so carefully, to eliminate the need to have
> the power of 3 explicit in the systematic name.

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.

-monz

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >Another vague idea: The order of mention of primes could be different
> >depending whether they are being multiplied (dot) or divided (colon).
>
> Again a cool idea, and I find these sort of inquiries fascinating, but
> I try to avoid them when I can't see them being very useful. YMMV.

Seems useful to me.

I think we're now past the point where all the common comma's and
temperaments have been named.

My purpose in proposing systematic naming methods for both linear
temperaments and commas is to avoid drowning in lots of meaningless
names where, if you haven't been following the tuning-math list
religiously for the past x years the only way you have of figuring out
what someone's talking about is to look the names up in a database
somewhere.

> >Tanaka's kleisma (_the_ kleisma) has the systematic name of
> >5^6-kleisma (five-to-the-six-kleisma)
>
> I've so far tried my best not to mention the term "anal retentive".
> :)

It takes all kinds. That's funny. Gene says I'm sloppy, and you say
I'm anal-retentive. A contradiction, wot?

Actually, I think I'm just your classic engineer/architect type. I
design systems. I'm good at it. I make my living designing systems of
several different kinds. I'm apparently genetically predisposed to it.
You, presumably, have other wonderful and complementary qualities.

Would you care to explain what your objection's are to the proposal,
as opposed to your objections to my online personality?

--- In tuning-math@yahoogroups.com, "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.

i'm sorry to respectfully disagree with you, Dave, but i
don't see anything unfriendly about "[4 -1]-comma".

(note that i don't consider the comma punctuation necessary.)

admittedly, "5-comma" is a whole lot easier and, yes,
i'll admit, friendlier.

but for me, so used to visualizing tunings on a lattice,
"[4 -1]-comma" tells me exactly what i need to know.

if i'm picturing the whole prime-space on a lattice,
[4 -1] helps me to *immediately* set a boundary in my
mind which filters out a large number of redundant
lattice-points.

in fact, when i hear or read the word "syntonic" the
first thing i think about is the vector on the lattice
which would describe it in a prime-space ... and then
the second thing i think about very quickly after that
is 3^4 / 5^1 .

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.

-monz

> > term 'neutral', although it seems sensible enough, is
> > problematic in practice and should be reconsidered.
> >
> > The concern arises because a so-called neutral third
sounds
> > 'major' in one context and 'minor' in another.
>
> That's usually only a result of insufficient exposure to the music
in
> question. Arabic musicians don't hear their neutral thirds as
'major'
> in one context and 'minor' in another, same for neutral
seconds.

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. I never suggested that Arabic musicans hear these
intervals the way I hear them.

> > Use of these
> > 'neutral' intervals in a melodic context almost never results in
a
> > musical functionality which could be described as 'neutral'.
>
> Almost never?

Yes, that is what I said. Almost never.

> > In my
> > experience, the same ambiguity presents itself when these
> > intervals are used harmonically.
>
> So you mean almost never *in your experience*?

That's right. How could I mean anything else?

> > This has led me to call these intervals 'narrow major' or 'wide
> > minor' depending on context. This dual perspective is
> > advantageous for descriptions of musical function.
>
> i beg to differ.

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. I
happen to think that 'ambi' makes more sense than 'neutral' as a
descriptor for these intervals. 'Ambi' literally means 'on both
sides of' and that is exactly what these intervals represent for
me. You disagree. That's fine.

Regards,
Aaron Hunt

>
> I can't imagine how anyone (such as Margo Schulter or myself)
who has
> spent any significant amount of time using a 17-tone
temperament,
> either equal or well-tempered (to improve non-5 ratios of 7),
could
> avoid hearing certain 2nds, 3rds, 6th, and 7ths as distinctly
> neutral. A prime example of this is the two-voice progression
> consisting of the interval of an augmented 6th, Eb3-C#4,
resolved to
> an octave, D3-D4, by melodic intervals of 2 degrees of 17,
which are
> unmistakably neutral 2nds.
>
> --George

No, it's not for lack of exposure that I say what I say. On the
contrary, I believe that exposure to these intervals in a variety of
contexts supports exactly what I am saying. You may be
misunderstanding me. I am by no means saying that these
seconds are indistinguishable from larger or smaller intervals
around them. If a person *confuses* these intervals with
so-called major or minor intervals, that is simply a case of bad
ears.

This is not what I'm saying at all. Sure, each interval has a
sonance all its own, and can be identified unmistakably as what
it is, once it becomes familiar.

I'm saying that the character of 'neutral' intervals is one of
ambiguity, or 'on both sides of ' the majoir / minor category, and
in various contexts these intervals tend to create phantom
impressions of being other than they are in terms of function,
even though their size and quality is unmistakably 'in between'.

So you don't buy my take on this. OK. I find it amazing that it
appears difficult for you to make the smallest admission that a
so-called neutral interval can give the impression of a 'major' or
'minor' sonority in some contexts.

Regards,
AH

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> in fact, when i hear or read the word "syntonic" the
> first thing i think about is the vector on the lattice
> which would describe it in a prime-space ... and then
> the second thing i think about very quickly after that
> is 3^4 / 5^1 .

actually that's not true. i don't visualize the numbers as
3^4 / 5^1 , but rather as 3^4 * 5^-1 , since that's exactly
how the lattice works. and that visualization agrees
exactly with the monzo of the syntonic comma.

> 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.

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.

-monz

--- In tuning-math@yahoogroups.com, "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.

--- In tuning-math@yahoogroups.com, "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.

--- In tuning-math@yahoogroups.com, "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 <>.

>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

>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

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "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

--- In tuning-math@yahoogroups.com, "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.

--- In tuning-math@yahoogroups.com, 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.

>> >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

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "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"?

>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

hi Gene,

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> --- In tuning-math@yahoogroups.com, "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 <>.

i like that idea a lot !!!!! :) :)

awesome!

-monz

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:

> 81/80 is the 5-comma.

using the convention Gene just proposed which i accept
(OK, i'll keep the comma punctuation) :

<3, 5>-monzo: <4, -1>-comma.

> 32805/32768 is the 5-schisma.

<3, 5>-monzo: <8, 1>-schisma.

> 64/63 is the 7-comma

<3, 5, 7>-monzo: <-2, 0, -1>-comma.

> 59049/57344 is the 7-medium-diesis or 7-M-diesis

<3, 5, 7>-monzo: <10, 0, 1>-diesis.

> 28/27 is the 7-large-diesis or 7-L-diesis

<-3, 0, 1>-diesis.

> 2048/2035 is the 25-comma

Dave, here i'm not sure if your ratio is correct,
because 2035 = 5 * 11 * 37 .

> 6561/6400 is the 25-small-diesis or 25-S-diesis

<8, -2>-diesis.

> 128/125 is the 125-small-diesis

<0, -3>-diesis

> 250/243 is the 125-medium-diesis

<-5, 3>-diesis.

> 531441/512000 is the 125-large-diesis

<12, -3>-diesis.

> 5120/5103 is the 5:7-kleisma

<-6, 1, -1>-kleisma.

> 3645/3584 is the 5:7-comma

<6, 1, -1>-comma.

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
>
> <snip>
>
> > 59049/57344 is the 7-medium-diesis or 7-M-diesis
>
> <3, 5, 7>-monzo: <10, 0, 1>-diesis.

oops ... my bad. missed a sign.

that should be <10, 0, -1>-diesis.

> > 2048/2035 is the 25-comma
>
> Dave, here i'm not sure if your ratio is correct,
> because 2035 = 5 * 11 * 37 .

Dave made a typo here and the ratio should be 2048/2025,
in monzo form: the <-4, -2>-comma.

here's the whole list of Dave's "kommas", from largest
to smallest, with the [2, 3, 5, 7]-monzos and cents:

(if viewing on the Yahoo web interface, you'll have
to forward it to your email account to see it properly.)

2 3 5 7 cents ratio

[-12, 12, -3, 0]-diesis 64.51886879 531441 : 512000
[ 2, -3, 0, 1]-diesis 62.96090387 28 : 27
[-13, 10, 0, -1]-diesis 50.72410218 59049 : 57344
[ 1, -5, 3, 0]-diesis 49.16613727 250 : 243
[ -8, 8, -2, 0]-diesis 43.01257919 6561 : 6400
[ 7, 0, -3, 0]-diesis 41.05885841 128 : 125
[ -9, 6, 1, -1]-comma 29.21781259 3645 : 3584
[ 6, -2, 0, -1]-comma 27.2640918 64 : 63
[ -4, 4, -1, 0]-comma 21.5062896 81 : 80
[ 11, -4, -2, 0]-comma 19.55256881 2048 : 2025
[ 10, -6, 1, -1]-kleisma 5.757802203 5120 : 5103
[-15, 8, 1, 0]-schisma 1.953720788 32805 : 32768

or if you prefer <3, 5, 7>-monzos:

3 5 7 cents ratio

< 12, -3, 0>-diesis 64.51886879 531441 : 512000
< -3, 0, 1>-diesis 62.96090387 28 : 27
< 10, 0, -1>-diesis 50.72410218 59049 : 57344
< -5, 3, 0>-diesis 49.16613727 250 : 243
< 8, -2, 0>-diesis 43.01257919 6561 : 6400
< 0, -3, 0>-diesis 41.05885841 128 : 125
< 6, 1, -1>-comma 29.21781259 3645 : 3584
< -2, 0, -1>-comma 27.2640918 64 : 63
< 4, -1, 0>-comma 21.5062896 81 : 80
< -4, -2, 0>-comma 19.55256881 2048 : 2025
< -6, 1, -1>-kleisma 5.757802203 5120 : 5103
< 8, 1, 0>-schisma 1.953720788 32805 : 32768

-monz

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "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.

(1) I don't like having two words which sound the same and with
related meanings

(2) I think the goofy spelling, if you do this, should be for your
new meaning, and not imposed on an established one. "Comma in a
particular range of cents"="komma", in other words.

--- In tuning-math@yahoogroups.com, "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?

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.

--- In tuning-math@yahoogroups.com, "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.

> 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

--- In tuning-math@yahoogroups.com, 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.

>> 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

--- In tuning-math@yahoogroups.com, 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!

hi paul,

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> --- In tuning-math@yahoogroups.com, "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?

ah, thanks for mentioning a third important example!

the software retains the data for 2, but the default
in *displaying* the data to the user is to start the
prime-series with 3.

if the user wants to see the data for 2, that option
is available.

for that matter, if the user wants to see ratios,
that's available too.

the default notation right now is what we're calling
"prime ratios". for example, the 15/8 ratio is displayed
by default as 3^1 5^1 / 2^3 ... well, that's the best
i can do in ASCII equivalent, but the software uses
actual superscripts instead of carets, and puts the
numerator and denominator on separate lines with the
separator line running horizontally.

i hope to be able to get regular monzo notation, i.e.
where 15/8 would be [-3 1 1], as soon as possible ...
but it will entail a lot of work with the graphics,
so the first release will probably not have it. i'm
disappointed about that, but my partner does the
coding and he has a much larger role than i in
determining the priorities on the coding work.

-monz

--- In tuning-math@yahoogroups.com, "Paul Erlich"
<perlich@a...> wrote:
> Aaron, before we get into some sort of personality clash, let me
just
> say i was apparently misinterpreting you.... No offense
> meant,

Nobody's fault and no harm done. My original post was unclear
and sounded overly emphatic as usual, and I took your response
the wrong way. We're probably both much less severe in person
anyway. I'd much rather be friends and have fun discussing
mutual interests ...

> > 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.

Thanks for the offer; I'm in Illinois but if I'm ever out your way I'll
be sure to bother you about this!

As for neutral intervals, I might try sending a better post to the
main tuning list about them, maybe with some examples... like,
the following conventional four voice progression using 9:11

first chord
1:1
6:7
2:3
1:2
second chord
1:1
9:11 *
2:3
4:7
third chord
1:1
3:4
3:4
9:14

Because the 6:7 'minor' thirds are so small, and the small
seventh 4:7 pulls downward so strongly, the 'neutral' third can
even be slightly more narrow than 9:11 and I believe that the
effect of 'leading up' is still there. This is a case where the term
'neutral' bothers me, and I would call it instead a 'narrow major'
third. A similar example can be given for 9:11 which seems to
pull in the opposite direction, making it a 'wide minor' third (for
my ears).

best,
Aaroneous

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...> wrote:
> > > --- In tuning-math@yahoogroups.com, "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?

Hey, what happened to this thread, anyway? If things start getting
taken a little too personally, do we just pack up our stuff and go
home?

The comma-naming system that Dave was presenting is something that he
and I have been using for quite some time. It evolved over the
course of more than a year during our Sagittal development project,
and we found these names very useful in discussing commas that we
were trying to symbolize, so the names are something that both Dave
and I can take credit for. The specific boundaries between
categories were worked out by Dave, and I agree with them.

The monzo comma-naming system is so cumbersome (i.e., unfriendly),
that I can't imagine how anyone could follow a name above the 11-
limit unless it's written down. Imagine trying to mention to someone
in spoken conversation that you're trying to decide whether to use
the symbol for "<-6, 0, 0, 0, 0, 0, 0, 1> or <6, 0, 0, 0, 0, 0, 0,
1>" in a composition -- are you expecting me to be mentally prepared
to count all those zeros so I know what prime number you mean when
you finally get to the "1" that matters? The problem is, the "name"
(if you can call it that) emphasizes *powers* rather than *primes*,
so it tends to get rather cryptic.

Now if Dave says, "Should I use a 23-comma or 23-small-diesis (23S)
symbol?", I immediately know what he's talking about: these are
commas that alter tones in a 1/1 pythagorean chain to arrive at tones
in a 23/16 and/or a 32/23 pythagorean chain. The difference in using
a 23C vs. a 23S (how's that for brevity?) amounts to nothing more
than how you wish to spell the pitch, i.e., which nominal you wish to
modify. Likewise, a 7:11-comma vs. a 7:11 kleisma (or 7:11C vs.
7:11k) notates tones in 11/7 and 14/11 pythagorean chains with
alternate spellings. Discarding powers of 2 and 3 from the *names*,
therefore, is in keeping with the fact that Sagittal symbols may (in
a heptatonic notation) modify *many* (and *only*) tones in a
pythagorean chain containing 1/1. If you want to notate a certain
ratio in JI, then its comma-symbol is determined *only* by the primes
above 3 in the ratio, and the nominal being modified is determined by
the powers of 2 and 3. And there's also a clue to help you remember
which symbol represents which comma, because the comma name also
tells you the size range.

> > > 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.
>
> (1) I don't like having two words which sound the same and with
> related meanings
>
> (2) I think the goofy spelling, if you do this, should be for your
> new meaning, and not imposed on an established one. "Comma in a
> particular range of cents"="komma", in other words.

And please, let's not get hung up on "comma" vs. "komma". In the
first place, we don't want "komma" to specify a particular size,
because the "k" won't distinguish it from "kleisma" in the
abbreviations that we're using. In the second place, "comma" should
be okay for both the generic and specific-size-range terms, because
the context should make it obvious how the term is being used; as
evidence of this, read my preceding paragraph again.

But don't get me wrong -- I'm not objecting to monz's HEWM factored-
ratio notation, which is very useful in a lot of instances (as should
be evident by just reading a few other current messages). But I
would call this a (theoretical) *notation* for ratios rather than
(systematic) *names* for commas. A name should be something that is
simple enough to be easily recognized.

--George

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> Now if Dave says, "Should I use a 23-comma or 23-small-diesis (23S)
> symbol?", I immediately know what he's talking about: these are
> commas that alter tones in a 1/1 pythagorean chain to arrive at
tones
> in a 23/16 and/or a 32/23 pythagorean chain.

This is fine for your very specialized purposes, but what about the
rest of us? Do you have a name for every superparticular ratio up to
the 23-limit?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...>
> wrote:
>
> > Now if Dave says, "Should I use a 23-comma or 23-small-diesis
(23S)
> > symbol?", I immediately know what he's talking about: these are
> > commas that alter tones in a 1/1 pythagorean chain to arrive at
tones
> > in a 23/16 and/or a 32/23 pythagorean chain.
>
> This is fine for your very specialized purposes, but what about the
> rest of us? Do you have a name for every superparticular ratio up
to
> the 23-limit?

I find it very amusing that you would place me, the
*composer/theorist*, as the one with "very specialized purposes"
while you, the *mathematician*, would group yourself with the "rest
of us". It would seem to me that only those with very specialized
purposes would actually *need* to have a name for *every*
superparticular ratio up to the 23-limit.

A systematic naming system should not be something that would make
things more complicated for the rest of us. I could hardly imagine a
professor in a microtonal music course using "minus three, zero,
zero, zero, one" as a name for 26:27 when "13L-diesis" (which can
even be shortened to "13L") is so much simpler and clearer. Monzos
have their place as a specialized *notation* (which would also be of
benefit in explaining the names), but not as *names* themselves.

As for the answer to your second question: I don't know if there will
be unique names for *all* of them (probably not), but I believe that
there are enough names to cover all of those that would be of use to
most theorists. Those with very specialized purposes (such as
yourself) could devise a modification to Dave's naming system (such
as appending letters a, b, c, etc. in order of ratio complexity) to
distinguish equivocal names -- I think you're creative enough to come
up with something that would be meaningful (or perhaps Dave might
have some ideas).

In summary, let's keep the simpler things simple.

--George

hi George (and Gene)

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> A systematic naming system should not be something that
> would make things more complicated for the rest of us.
> I could hardly imagine a professor in a microtonal music
> course using "minus three, zero, zero, zero, one" as a name
> for 26:27 when "13L-diesis" (which can even be shortened
> to "13L") is so much simpler and clearer. Monzos have
> their place as a specialized *notation* (which would also
> be of benefit in explaining the names), but not as *names*
> themselves.

i see your point, and can agree with that.

... even tho *i* will always think of any rational
interval as its monzo.

(i guess that's self-evident, given the name of the term?)

;-)

-monz

hi George,

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> The monzo comma-naming system is so cumbersome
> (i.e., unfriendly), that I can't imagine how anyone
> could follow a name above the 11-limit unless it's
> written down. Imagine trying to mention to someone
> in spoken conversation that you're trying to decide
> whether to use the symbol for "<-6, 0, 0, 0, 0, 0, 0, 1>
> or <6, 0, 0, 0, 0, 0, 0, 1>" in a composition -- are
> you expecting me to be mentally prepared to count all
> those zeros so I know what prime number you mean when
> you finally get to the "1" that matters?

this is an interesting and good point.

in fact, over the years of working in extended-JI,
i've pretty much come to the conclusion that there are
so many notes available in even a rather compact 11-limit
euler-genus, that via xenharmonic bridging, that system
can imply many higher-prime ratios.

in particular, i've noticed that lots of 11-limit ratios
sound very similar to nearby 13-limit ratios.

if one accepts this aspect of my theory, then monzos
can easily be used to name all the relevant 11-limit
kommas.

on the other hand, the main reason i came up with the
idea of using monzos to represent prime-factored ratios
was that i wanted to avoid both having to always specify
the primes, and also to avoid superscripts.

at the time i originally thought of the monzo idea,
i was working in 19-limit, and it seemed a easier to me
to specify, to pick a totally random example, 133:72 as
[-3, -2, 0, 1, 0, 0, 0, 1] (with the prime-factors
2, 3, 5, 7, 11, 13, 17, 19 understood) than to write
is out as 2^-3 * 3^-2 * 7^1 * 19^1.

> The problem is, the "name" (if you can call it that)
> emphasizes *powers* rather than *primes*, so it tends
> to get rather cryptic.

when one works with the same set of prime-factors
over and over again, one very easily gets used to
remembering the primes which underly the monzo.

and as i've pointed out before, the monzo allows direct
visualization of the lattice, which in turn helps in
comprehension of the structure of the entire tuning system.

i came up with the monzo idea based on analogy with
our regular numbering system.

it doesn't take too long for one to learn, whether in school
or in everyday life, that, for example, the number 133 is a
monzo-like representation of (10^2)*1 + (10^1)*3 + (10^0)*3.

the regular arabic numeral is a nice compact way of expressing
what would look like a rather complicated mathematical expression
if it were written out in full. but once one learns how
it works, the long version is never need anymore.

anyway, i'll get off my soapbox now. i've already agreed
that for purposes of naming beyond 11-limit, the prime system
is better than the monzo system.

but i will maintain that for 3-, 5-, 7-, and 11-limit,
the monzo system works just fine.

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> ... even tho *i* will always think of any rational
> interval as its monzo.
>
> (i guess that's self-evident, given the name of the term?)
>
> ;-)

Oh, dear! Now I fear that if someone brings up the word "secor" in a
tuning context, they'll too readily associate that with the
term "irrational". ;-)

--George

hi George,

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> >
> > ... even tho *i* will always think of any rational
> > interval as its monzo.
> >
> > (i guess that's self-evident, given the name of the term?)
> >
> > ;-)
>
> Oh, dear! Now I fear that if someone brings up the
> word "secor" in a tuning context, they'll too readily
> associate that with the term "irrational". ;-)
>
> --George

oh, not necessarily!

in a series of old posts, most informatively this one:

/tuning/topicId_22793.html#23195

i presented a "rational canasta" tuning. the express
purpose of this was to be able to map the canasta scale
to the computer keyboard in the old (JustMusic) version
of my software, which was not able to accept irrational
pitches.

in my "rational canasta" tuning, a secor is:

<3,5,7,11,13>-monzo = <-4, -1, 0, 1, -1>
ratio = 5632:5265
= ~116.657 cents

this is only ~0.0101 cent less than 2^(7/72), the
"standard" secor.

and of course many other rational secors could be found.

i mentioned in one of those old posts the irony of
having to find a rational tuning which approximated
the subset of the irrational 72edo MIRACLE, which in
turn provides manifold approximations of rational
JI intervals ... and even mentioned how it conjured
up Escher images in my mind.

and in fact, with regard to my original comment you
quoted, i also think of many irrational intervals
in terms of their monzos ... even tho many folks here
find that to be pointless since irrational intervals
can be prime-factored in an infinite number of ways.

but i find it useful, for example, to see the generator
"5th" of 1/4-comma meantone as the [2,3,5]-monzo [0, 0, 1/4],
or that of 2/7-comma meantone as [1/7, -1/7, 2/7].

i just wrote "rational" to avoid having to go into
details like this ... but now you've gone and forced
to do it anyway! :)

-monz

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> As for the answer to your second question: I don't know if there
will
> be unique names for *all* of them (probably not), but I believe
that
> there are enough names to cover all of those that would be of use
to
> most theorists.

Could you list what you think of as the important commas?

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> in my "rational canasta" tuning, a secor is:
>
> <3,5,7,11,13>-monzo = <-4, -1, 0, 1, -1>
> ratio = 5632:5265
> = ~116.657 cents
>
> this is only ~0.0101 cent less than 2^(7/72), the
> "standard" secor.

though i know you wanted to use the 72-equal secor, the "standard"
secor is (as you correctly state on http://www.sonic-
arts.org/dict/secor.htm) (18/5)^(1/19) = ~116.7156 cents.

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> hi George,
>
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...> wrote:
>
> > --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> > >
> > > ... even tho *i* will always think of any rational
> > > interval as its monzo.
> > >
> > > (i guess that's self-evident, given the name of the term?)
> > >
> > > ;-)
> >
> > Oh, dear! Now I fear that if someone brings up the
> > word "secor" in a tuning context, they'll too readily
> > associate that with the term "irrational". ;-)
> >
> > --George
>
> oh, not necessarily!
>
> in a series of old posts, most informatively this one:
>
> /tuning/topicId_22793.html#23195
>
> i presented a "rational canasta" tuning. ...
>
> and of course many other rational secors could be found.

Probably distant relatives of mine. ;-)

> ...
> i just wrote "rational" to avoid having to go into
> details like this ... but now you've gone and forced
> to do it anyway! :)

Hey, you're taking my reply much too seriously.

--George

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...>
> wrote:
>
> > As for the answer to your second question: I don't know if there
will
> > be unique names for *all* of them (probably not), but I believe
that
> > there are enough names to cover all of those that would be of use
to
> > most theorists.
>
> Could you list what you think of as the important commas?

I don't have the resources to do that very readily. Wouldn't putting
them in order of product complexity (discarding any with factors
above some particular prime limit) accomplish this? One would only
need to determine at what point two commas were forced to share the
same name and then decide if they were both "important". Dave
already discussed this:

/tuning-math/messages/7320

--George

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> hi George,
>
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...> wrote:
>
> > The monzo comma-naming system is so cumbersome
> > (i.e., unfriendly), that I can't imagine how anyone
> > could follow a name above the 11-limit unless it's
> > written down. Imagine trying to mention to someone
> > in spoken conversation that you're trying to decide
> > whether to use the symbol for "<-6, 0, 0, 0, 0, 0, 0, 1>
> > or <6, 0, 0, 0, 0, 0, 0, 1>" in a composition -- are
> > you expecting me to be mentally prepared to count all
> > those zeros so I know what prime number you mean when
> > you finally get to the "1" that matters?
>
> this is an interesting and good point.
>
> in fact, over the years of working in extended-JI,
> i've pretty much come to the conclusion that there are
> so many notes available in even a rather compact 11-limit
> euler-genus, that via xenharmonic bridging, that system
> can imply many higher-prime ratios.
>
> in particular, i've noticed that lots of 11-limit ratios
> sound very similar to nearby 13-limit ratios.

No question about it! Many years ago I couldn't help noticing that
the ratio 351:352 vanishes in most of the good edos below 100.

More recently I also found that complex 7-limit ratios tend to *very
closely* approximate ratios of 13. The 13-schismina, 4095:4096 or <-
2, -1, -1, 0, -1>, vanishes in most of the best edos between 200 and
800. This was the first step in establishing an economy of symbol-
elements (flags) in Sagittal notation development, since it allows
one to notate a 13M-diesis (1024:1053) as the (approximate) sum
(product) of a 5-comma (80:81) and 7-comma (63:64) in both medium and
high-precision JI. (Only extreme-precision Sagittal JI has separate
symbols for 13-limit consonances.)

> if one accepts this aspect of my theory, then monzos
> can easily be used to name all the relevant 11-limit
> kommas.

Loocs like you kaught something from Dave Ceenan. ;-)

> on the other hand, the main reason i came up with the
> idea of using monzos to represent prime-factored ratios
> was that i wanted to avoid both having to always specify
> the primes, and also to avoid superscripts.
>
> at the time i originally thought of the monzo idea,
> i was working in 19-limit, and it seemed a easier to me
> to specify, to pick a totally random example, 133:72 as
> [-3, -2, 0, 1, 0, 0, 0, 1] (with the prime-factors
> 2, 3, 5, 7, 11, 13, 17, 19 understood) than to write
> is out as 2^-3 * 3^-2 * 7^1 * 19^1.

Sure.

> > The problem is, the "name" (if you can call it that)
> > emphasizes *powers* rather than *primes*, so it tends
> > to get rather cryptic.
>
> when one works with the same set of prime-factors
> over and over again, one very easily gets used to
> remembering the primes which underly the monzo.
>
> and as i've pointed out before, the monzo allows direct
> visualization of the lattice, which in turn helps in
> comprehension of the structure of the entire tuning system.

It looks as if Dave's and your method of designating ratios are
complementary. Dave's (which is at the moment restricted to ratios
no larger than ~69 cents) helps more in comprehending the size of the
comma and especially in identifying the Sagittal symbol used to
notate ratios having the same combination of primes >3.

BTW, have you ever tried collapsing an 11-limit lattice into 2
dimensions by mapping 11/8 to <10, 5>?

> i came up with the monzo idea based on analogy with
> our regular numbering system.
>
> it doesn't take too long for one to learn, whether in school
> or in everyday life, that, for example, the number 133 is a
> monzo-like representation of (10^2)*1 + (10^1)*3 + (10^0)*3.
>
> the regular arabic numeral is a nice compact way of expressing
> what would look like a rather complicated mathematical expression
> if it were written out in full. but once one learns how
> it works, the long version is never need anymore.

Yes, I can readily appreciate this sort of shorthand. However, its
weakness as a naming system lies in the fact that you need a good way
to verbalize what you're seeing. Decimal numbers work as names
because, in saying a number such as 2,010,030 (as "two million ten
thousand thirty"), we include words to indicate the placement of the
numerals. But this is not as easy for monzos, since primes are not
related to one another as are powers of ten, since exponents may be
negative numbers, and since numbers may not be rounded off by
dropping the lower prime exponents (as with 2.01 million).

Something that you might want to consider is replacement of the comma
following the powers of 3, 11, and 19 (and every 3rd prime
thereafter) by a semicolon (to serve as a place marker, similar in
function to a decimal point and commas in decimal numbers), so that
133:72 could be written as either [-3, -2; 0, 1, 0; 0, 0, 1] or [-2;
0, 1, 0; 0, 0, 1].

--George

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> BTW, have you ever tried collapsing an 11-limit lattice into 2
> dimensions by mapping 11/8 to <10, 5>?

you're probably referring to the 3-5-11 lattice?

the full 11-limit lattice is at least 3 dimensional if you use
one "xenharmonic bridge" as above. for the 3-5-11 case, this choice
(184528125:184549376) is probably a very good one. for the full 11-
limit case, 9800:9801 is probably better for most purposes, since
it's both a little smaller (in cents) and much simpler (i.e., shorter
in the lattice).

sorry if this duplicates a previous message; that one didn't seem to
show up . . .

hi paul,

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > in my "rational canasta" tuning, a secor is:
> >
> > <3,5,7,11,13>-monzo = <-4, -1, 0, 1, -1>
> > ratio = 5632:5265
> > = ~116.657 cents
> >
> > this is only ~0.0101 cent less than 2^(7/72), the
> > "standard" secor.
>
> though i know you wanted to use the 72-equal secor,
> the "standard" secor is (as you correctly state on
> http://www.sonic-arts.org/dict/secor.htm)
> (18/5)^(1/19) = ~116.7156 cents.

OK, thanks for pointing that out. so my "rational secor"
is only a little more than half a cent smaller than that.

-monz

hi George,

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> Loocs like you kaught something from Dave Ceenan. ;-)

i usually try to follow suggestions for standardization
... unless i very strongly disagree, as i did with Sims
72edo notation.

> BTW, have you ever tried collapsing an 11-limit lattice into 2
> dimensions by mapping 11/8 to <10, 5>?

no, i never did that. but i did do this:

/tuning/topicId_1372.html#1372
/tuning/topicId_1372.html#1380

(if you're viewing on the stupid Yahoo web interface,
you'll have to forward them to your email account to
view the lattices properly.)

> Yes, I can readily appreciate this sort of shorthand.
> However, its weakness as a naming system lies in the
> fact that you need a good way to verbalize what you're
> seeing.

which is the main reason why i agree with you about names
in general.

but i do think that for 11-limit, using only 4 exponents,
it's not so bad.

> Something that you might want to consider is replacement
> of the comma following the powers of 3, 11, and 19 (and
> every 3rd prime thereafter) by a semicolon (to serve as
> a place marker, similar in function to a decimal point
> and commas in decimal numbers), so that 133:72 could be
> written as either [-3, -2; 0, 1, 0; 0, 0, 1] or
> [-2; 0, 1, 0; 0, 0, 1].

well, since that last monzo doesn't use 2, it should be
written (following the convention proposed by Gene and
accepted by me) with angle brackets instead of square:
<-2; 0, 1, 0; 0, 0, 1>.

anyway, yes, that's an excellent idea ... except that
i was never crazy about adding the commas in the first place.
i still think i prefer the nice clean look of a single space
and nothing else, separating the numbers in the monzo.

-monz

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> > i just wrote "rational" to avoid having to go into
> > details like this ... but now you've gone and forced
> > to do it anyway! :)
>
> Hey, you're taking my reply much too seriously.

i know ... but still, i thought of it before i wrote
my original post, so i figured that since you brought
it up (albeit as a joke) i might as well give a nice
fat response.

:)

... isn't it so much nicer when communication here
is this pleasant?

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> hi George,
>
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...> wrote:
>
> > Loocs like you kaught something from Dave Ceenan. ;-)
>
> i usually try to follow suggestions for standardization
> ... unless i very strongly disagree, as i did with Sims
> 72edo notation.

I agree with Gene that using the conventional spelling for the new
(defined-range) meaning of comma and the unconventional spelling for
the commonly accepted (i.e., generic) meaning for a comma would not
be a good idea, so I advise that the new spelling be dropped. (It's
not required for a comma-naming system to work, anyway.)

> > ...
> > Something that you might want to consider is replacement
> > of the comma following the powers of 3, 11, and 19 (and
> > every 3rd prime thereafter) by a semicolon (to serve as
> > a place marker, similar in function to a decimal point
> > and commas in decimal numbers), so that 133:72 could be
> > written as either [-3, -2; 0, 1, 0; 0, 0, 1] or
> > [-2; 0, 1, 0; 0, 0, 1].
>
> well, since that last monzo doesn't use 2, it should be
> written (following the convention proposed by Gene and
> accepted by me) with angle brackets instead of square:
> <-2; 0, 1, 0; 0, 0, 1>.

I used the square brackets to make the point that you would no longer
need the angle brackets in order to show that the exponent of 2 is
omitted. But perhaps the semicolon looks too much like a comma, so
that the angle brackets would be more distinctive in indicating this
(however, see the following).

> anyway, yes, that's an excellent idea ... except that
> i was never crazy about adding the commas in the first place.
> i still think i prefer the nice clean look of a single space
> and nothing else, separating the numbers in the monzo.

It's very nice that you brought this up, because that's the next
thing I was going to suggest. Why not modify my suggestion above by
dropping the commas entirely, then changing the semicolons that
remain back to commas, so that the above example (133:72) would be
done this way:
[-3 -2, 0 1 0, 0 0 1] or
[-2, 0 1 0, 0 0 1].
This makes the grouping by threes more obvious (and the higher primes
much easier to locate), and angle brackets would no longer be
necessary.

--George

hi George,

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> I agree with Gene that using the conventional spelling for
> the new (defined-range) meaning of comma and the unconventional
> spelling for the commonly accepted (i.e., generic) meaning for
> a comma would not be a good idea, so I advise that the new
> spelling be dropped. (It's not required for a comma-naming
> system to work, anyway.)

OK.

>
> <snip>
>
> [regarding the format of monzos:]
>
> ... Why not modify my suggestion above by dropping the
> commas entirely, then changing the semicolons that
> remain back to commas, so that the above example (133:72)
> would be done this way:
> [-3 -2, 0 1 0, 0 0 1] or
> [-2, 0 1 0, 0 0 1].
> This makes the grouping by threes more obvious (and the
> higher primes much easier to locate), and angle brackets
> would no longer be necessary.

i like that a lot!

in fact, i find it very interesting that group the primes
by threes like this also keeps them in bunches that make
sense to me in terms of how i've used them and theorized
about them myself!

i.e., 3 is obviously extremely important both historically
and theoretically, and thus deserves to be isolated by itself
(or grouped with 2, if 2 is included).

the next comma appears after 11, and earlier in this thread
i discussed the idea that 11-limit can be a kind of boundary.
Partch thought so too. (but please, don't anyone make too
much of this comment.)

the next comma appears after 19, which i myself used as
a limit from about 1988-98.

the next comma appears after 31, which is the highest limit
Ben Johnston has used in his music.

interesting.

-monz

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> i.e., 3 is obviously extremely important both historically
> and theoretically, and thus deserves to be isolated by itself
> (or grouped with 2, if 2 is included).
>
> the next comma appears after 11, and earlier in this thread
> i discussed the idea that 11-limit can be a kind of boundary.
> Partch thought so too. (but please, don't anyone make too
> much of this comment.)
>
> the next comma appears after 19, which i myself used as
> a limit from about 1988-98.
>
> the next comma appears after 31, which is the highest limit
> Ben Johnston has used in his music.
>
> interesting.

so the primes are arranged as
2 3 , 5 7 11 , 13 17 19 , 23 29 31 , 37 41 43 , 47 53 59 , 61 67 71

looks like the next comma after 31 makes sense too -- isn't 43 the
highest limit used by george secor at least in some context?

no reply, george?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...>
> wrote:
>
> > BTW, have you ever tried collapsing an 11-limit lattice into 2
> > dimensions by mapping 11/8 to <10, 5>?
>
> you're probably referring to the 3-5-11 lattice?
>
> the full 11-limit lattice is at least 3 dimensional if you use
> one "xenharmonic bridge" as above. for the 3-5-11 case, this choice
> (184528125:184549376) is probably a very good one. for the full 11-
> limit case, 9800:9801 is probably better for most purposes, since
> it's both a little smaller (in cents) and much simpler (i.e.,
shorter
> in the lattice).
>
> sorry if this duplicates a previous message; that one didn't seem
to
> show up . . .

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> I don't have the resources to do that very readily. Wouldn't
putting
> them in order of product complexity (discarding any with factors
> above some particular prime limit) accomplish this?

Not really; however we can simply take everything below a certain
prime limit and below a limit for what I call "epipermicity", which
is, for p/q>1 in reduced form, log(p-q)/log(q). It can be shown this
gives a finite list of commas if the epimermicity limit is less than
one.

One would only
> need to determine at what point two commas were forced to share the
> same name and then decide if they were both "important".

NEVER! Two commas with the same name makes no sense at all.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...>
> wrote:
>
> > I don't have the resources to do that very readily. Wouldn't
> putting
> > them in order of product complexity (discarding any with factors
> > above some particular prime limit) accomplish this?
>
> Not really; however we can simply take everything below a certain
> prime limit and below a limit for what I call "epipermicity",

i think you mean "epimericity"

> which
> is, for p/q>1 in reduced form, log(p-q)/log(q). It can be shown
this
> gives a finite list of commas if the epimermicity limit is less
than
> one.

i'll try 7-limit some other time, but since i still have my 5-limit
list in matlab's memory, here's the top rankings (for intervals < 600
cents) by epimericity -- 1/1 shows up as best but actually its
epimericity is 0/0 so is undefined:

numerator denominator
1 1
16 15
6 5
81 80
4 3
9 8
10 9
5 4
25 24
27 25
128 125
32805 32768
250 243
135 128
2048 2025
15625 15552
256 243
648 625
32 27
3125 3072
75 64
78732 78125
6561 6400
20000 19683
125 108
27 20
32 25
25 18
625 576
1600000 1594323
144 125
393216 390625
256 225
16875 16384
2187 2048
81 64
2109375 2097152
800 729
6561 6250
1125 1024
3125 2916
100 81
531441 524288
45 32
20480 19683
2187 2000
729 640
2048 1875
243 200
16384 15625
125 96
729 625
1076168025 1073741824
3456 3125
6115295232 6103515625
1224440064 1220703125
1594323 1562500
1990656 1953125
274877906944 274658203125
262144 253125
625 512
10485760000 10460353203
1215 1024
62500 59049
7629394531250 7625597484987
78125 73728
273375 262144
4096 3645
67108864 66430125
129140163 128000000
2500 2187
162 125
1638400 1594323
531441 512000
4194304 4100625
82944 78125
32768 30375
390625000 387420489
244140625 241864704
390625 373248
9765625 9565938
1953125 1889568
4294967296 4271484375
18225 16384
625 486
34171875 33554432
2560 2187
768 625
140625 131072
31381059609 31250000000
9375 8192
etc.

considering that i had numerators and denominators well in excess of
10^50 in the list, i'm inclined to believe gene that a given
epimericity cutoff will yield a finite list. and it's a good list
too -- i'm kind of pleased with this as a temperament ranking, with
meantone very near the top, augmented, schismic, pelogic,
diaschismic, blackwood, kleismic, and diminished forming a
consecutive block of interesting and eminently useful systems (given
their characteristic DE scales), while more unlikely choices for
human music making, like semisuper, parakleismic, and ennealimmal, as
well as many simpler systems with high error, fall further down --
and of course monstrosities like atomic don't appear at all. i wonder
if even dave could stomach such a ranking -- the very simple
temperaments with high error are easy enough to mentally toss out for
the user seeking a certain goodness of approximation . . .

i actually used log(p-q)/log(odd limit), which makes the rankings a
little different for the big (in cents) intervals but does not affect
the ranking of the small commas.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "George D. Secor"
> <gdsecor@y...>
> > wrote:
> >
> > > I don't have the resources to do that very readily. Wouldn't
> > putting
> > > them in order of product complexity (discarding any with
factors
> > > above some particular prime limit) accomplish this?
> >
> > Not really; however we can simply take everything below a certain
> > prime limit and below a limit for what I call "epipermicity",
>
> i think you mean "epimericity"
>
> > which
> > is, for p/q>1 in reduced form, log(p-q)/log(q). It can be shown
> this
> > gives a finite list of commas if the epimermicity limit is less
> than
> > one.
>
> i'll try 7-limit some other time, but since i still have my 5-limit
> list in matlab's memory, here's the top rankings (for intervals <
600
> cents) by epimericity -- 1/1 shows up as best but actually its
> epimericity is 0/0 so is undefined:
>
> numerator denominator
> 1 1
> 16 15
> 6 5
> 81 80
> 4 3
> 9 8
> 10 9
> 5 4
> 25 24
> 27 25
> 128 125
> 32805 32768
> 250 243
> 135 128
> 2048 2025
> 15625 15552
> 256 243
> 648 625
> 32 27
> 3125 3072
> 75 64
> 78732 78125
> 6561 6400
> 20000 19683
> 125 108
> 27 20
> 32 25
> 25 18
> 625 576
> 1600000 1594323
> 144 125
> 393216 390625
> 256 225
> 16875 16384
> 2187 2048
> 81 64
> 2109375 2097152
> 800 729
> 6561 6250
> 1125 1024
> 3125 2916
> 100 81
> 531441 524288
> 45 32
> 20480 19683
> 2187 2000
> 729 640
> 2048 1875
> 243 200
> 16384 15625
> 125 96
> 729 625
> 1076168025 1073741824
> 3456 3125
> 6115295232 6103515625
> 1224440064 1220703125
> 1594323 1562500
> 1990656 1953125
> 274877906944 274658203125
> 262144 253125
> 625 512
> 10485760000 10460353203
> 1215 1024
> 62500 59049
> 7629394531250 7625597484987
> 78125 73728
> 273375 262144
> 4096 3645
> 67108864 66430125
> 129140163 128000000
> 2500 2187
> 162 125
> 1638400 1594323
> 531441 512000
> 4194304 4100625
> 82944 78125
> 32768 30375
> 390625000 387420489
> 244140625 241864704
> 390625 373248
> 9765625 9565938
> 1953125 1889568
> 4294967296 4271484375
> 18225 16384
> 625 486
> 34171875 33554432
> 2560 2187
> 768 625
> 140625 131072
> 31381059609 31250000000
> 9375 8192
> etc.
>
> considering that i had numerators and denominators well in excess
of
> 10^50 in the list, i'm inclined to believe gene that a given
> epimericity cutoff will yield a finite list. and it's a good list
> too -- i'm kind of pleased with this as a temperament ranking, with
> meantone very near the top, augmented, schismic, pelogic,
> diaschismic, blackwood, kleismic, and diminished forming a
> consecutive block of interesting and eminently useful systems
(given
> their characteristic DE scales), while more unlikely choices for
> human music making, like semisuper, parakleismic, and ennealimmal,
as
> well as many simpler systems with high error, fall further down --
> and of course monstrosities like atomic don't appear at all. i
wonder
> if even dave could stomach such a ranking -- the very simple
> temperaments with high error are easy enough to mentally toss out
for
> the user seeking a certain goodness of approximation . . .

there were actually a log of ties in the rankings -- for example log
(1)=0 so all superparticulars got a zero score. here's a list again
with the scores shown in the third column:

1 1 -Inf
16 15 0
6 5 0
81 80 0
4 3 0
9 8 0
10 9 0
5 4 0
25 24 0
27 25 0.210309918
128 125 0.227535398
32805 32768 0.3472592
250 243 0.35424875
135 128 0.396697481
2048 2025 0.41184295
15625 15552 0.444302056
256 243 0.466943504
648 625 0.487048023
32 27 0.488324507
3125 3072 0.493376213
75 64 0.555391285
78732 78125 0.568834688
6561 6400 0.578161697
20000 19683 0.582442033
125 108 0.586791476
27 20 0.590414583
32 25 0.604530978
25 18 0.604530978
625 576 0.604530978
1600000 1594323 0.605251545
144 125 0.6098276
393216 390625 0.610445976
256 225 0.634033151
16875 16384 0.636604282
2187 2048 0.641650248
81 64 0.644725481
2109375 2097152 0.646280585
800 729 0.646676406
6561 6250 0.653073083
1125 1024 0.65690632
3125 2916 0.663875781
100 81 0.670035965
531441 524288 0.673219541
45 32 0.673805299
20480 19683 0.675686222
2187 2000 0.680222895
729 640 0.680955483
2048 1875 0.683790172
243 200 0.684718377
16384 15625 0.686782398
125 96 0.697406178
729 625 0.704584463
1076168025 1073741824 0.706932191
3456 3125 0.721011768
6115295232 6103515625 0.72260722
1224440064 1220703125 0.723318814
1594323 1562500 0.725946912
1990656 1953125 0.727163637
2.74878E+11 2.74658E+11 0.729258578
262144 253125 0.731984665
625 512 0.7343228
10485760000 10460353203 0.739050418
1215 1024 0.739496502
62500 59049 0.741519042
7.62939E+12 7.6256E+12 0.743614519
78125 73728 0.744596936
273375 262144 0.745006093
4096 3645 0.745199871
67108864 66430125 0.745516574
129140163 128000000 0.746753932
2500 2187 0.747202793
162 125 0.747863148
1638400 1594323 0.748755323
531441 512000 0.749061613
4194304 4100625 0.75181536
82944 78125 0.752731448
32768 30375 0.753804891
390625000 387420489 0.757524855
244140625 241864704 0.757919646
390625 373248 0.758254069
9765625 9565938 0.75830862
1953125 1889568 0.763530363
4294967296 4271484375 0.765348818
18225 16384 0.766324469
625 486 0.76649026
34171875 33554432 0.768629073
2560 2187 0.770007566
768 625 0.770897186
140625 131072 0.773133533
31381059609 31250000000 0.773337708
9375 8192 0.773667414

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> i actually used log(p-q)/log(odd limit), which makes the rankings a
> little different for the big (in cents) intervals but does not
affect
> the ranking of the small commas.
>
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <gwsmith@s...>
> > wrote:
> > > --- In tuning-math@yahoogroups.com, "George D. Secor"
> > <gdsecor@y...>
> > > wrote:
> > >
> > > > I don't have the resources to do that very readily. Wouldn't
> > > putting
> > > > them in order of product complexity (discarding any with
> factors
> > > > above some particular prime limit) accomplish this?
> > >
> > > Not really; however we can simply take everything below a
certain
> > > prime limit and below a limit for what I call "epipermicity",
> >
> > i think you mean "epimericity"
> >
> > > which
> > > is, for p/q>1 in reduced form, log(p-q)/log(q). It can be shown
> > this
> > > gives a finite list of commas if the epimermicity limit is less
> > than
> > > one.
> >
> > i'll try 7-limit some other time, but since i still have my 5-
limit
> > list in matlab's memory, here's the top rankings (for intervals <
> 600
> > cents) by epimericity -- 1/1 shows up as best but actually its
> > epimericity is 0/0 so is undefined:
> >
> > numerator denominator
> > 1 1
> > 16 15
> > 6 5
> > 81 80
> > 4 3
> > 9 8
> > 10 9
> > 5 4
> > 25 24
> > 27 25
> > 128 125
> > 32805 32768
> > 250 243
> > 135 128
> > 2048 2025
> > 15625 15552
> > 256 243
> > 648 625
> > 32 27
> > 3125 3072
> > 75 64
> > 78732 78125
> > 6561 6400
> > 20000 19683
> > 125 108
> > 27 20
> > 32 25
> > 25 18
> > 625 576
> > 1600000 1594323
> > 144 125
> > 393216 390625
> > 256 225
> > 16875 16384
> > 2187 2048
> > 81 64
> > 2109375 2097152
> > 800 729
> > 6561 6250
> > 1125 1024
> > 3125 2916
> > 100 81
> > 531441 524288
> > 45 32
> > 20480 19683
> > 2187 2000
> > 729 640
> > 2048 1875
> > 243 200
> > 16384 15625
> > 125 96
> > 729 625
> > 1076168025 1073741824
> > 3456 3125
> > 6115295232 6103515625
> > 1224440064 1220703125
> > 1594323 1562500
> > 1990656 1953125
> > 274877906944 274658203125
> > 262144 253125
> > 625 512
> > 10485760000 10460353203
> > 1215 1024
> > 62500 59049
> > 7629394531250 7625597484987
> > 78125 73728
> > 273375 262144
> > 4096 3645
> > 67108864 66430125
> > 129140163 128000000
> > 2500 2187
> > 162 125
> > 1638400 1594323
> > 531441 512000
> > 4194304 4100625
> > 82944 78125
> > 32768 30375
> > 390625000 387420489
> > 244140625 241864704
> > 390625 373248
> > 9765625 9565938
> > 1953125 1889568
> > 4294967296 4271484375
> > 18225 16384
> > 625 486
> > 34171875 33554432
> > 2560 2187
> > 768 625
> > 140625 131072
> > 31381059609 31250000000
> > 9375 8192
> > etc.
> >
> > considering that i had numerators and denominators well in excess
> of
> > 10^50 in the list, i'm inclined to believe gene that a given
> > epimericity cutoff will yield a finite list. and it's a good list
> > too -- i'm kind of pleased with this as a temperament ranking,
with
> > meantone very near the top, augmented, schismic, pelogic,
> > diaschismic, blackwood, kleismic, and diminished forming a
> > consecutive block of interesting and eminently useful systems
> (given
> > their characteristic DE scales), while more unlikely choices for
> > human music making, like semisuper, parakleismic, and
ennealimmal,
> as
> > well as many simpler systems with high error, fall further down --

> > and of course monstrosities like atomic don't appear at all. i
> wonder
> > if even dave could stomach such a ranking -- the very simple
> > temperaments with high error are easy enough to mentally toss out
> for
> > the user seeking a certain goodness of approximation . . .

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
...
>
> considering that i had numerators and denominators well in excess of
> 10^50 in the list, i'm inclined to believe gene that a given
> epimericity cutoff will yield a finite list. and it's a good list
> too -- i'm kind of pleased with this as a temperament ranking, with
> meantone very near the top, augmented, schismic, pelogic,
> diaschismic, blackwood, kleismic, and diminished forming a
> consecutive block of interesting and eminently useful systems (given
> their characteristic DE scales), while more unlikely choices for
> human music making, like semisuper, parakleismic, and ennealimmal, as
> well as many simpler systems with high error, fall further down --
> and of course monstrosities like atomic don't appear at all. i wonder
> if even dave could stomach such a ranking -- the very simple
> temperaments with high error are easy enough to mentally toss out for
> the user seeking a certain goodness of approximation . . .

It's not bad. It seems to sufficiently penalise excessive complexity,
but as you observe, it does not sufficiently penalise excessive error.

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> > hi George,
> >
> > --- In tuning-math@yahoogroups.com, "George D. Secor"
> <gdsecor@y...> wrote:
> >
> > > Loocs like you kaught something from Dave Ceenan. ;-)
> >
> > i usually try to follow suggestions for standardization
> > ... unless i very strongly disagree, as i did with Sims
> > 72edo notation.

Thanks Monz, but I actually never proposed it as any kind of standard.
I first just did it as an expedient in one particular post, where I
thought there was potential for confusion, and I explained the usage
at the start of that post.
/tuning-math/message/6875
But apparently a few people failed to interpret this sentence
correctly, and instead assumed I was inventing a new term, and they
didn't like it so they didn't read any further.

But I didn't learn this until much later, so I carried on using it. It
seemed to me to be a convenient way to keep the two meanings distinct,
at least in writing. Had I known that this was actually a _barrier_ to
understanding the proposed naming system, I would have stopped using
it sooner.

> It's very nice that you brought this up, because that's the next
> thing I was going to suggest. Why not modify my suggestion above by
> dropping the commas entirely, then changing the semicolons that
> remain back to commas, so that the above example (133:72) would be
> done this way:
> [-3 -2, 0 1 0, 0 0 1] or
> [-2, 0 1 0, 0 0 1].
> This makes the grouping by threes more obvious (and the higher primes
> much easier to locate), and angle brackets would no longer be
> necessary.

I like this very much. It is also MATLAB/Octave compatible. Since
commas are optional, but semicolons indicate the end of a row (and so
would make a matrix, not a vector). Isn't that right Paul?

The angle-bracket thing was OK too. But I was going to suggest that
the octave-specific vectors should use the angle-brackets, since I
think octave-equivalent ones using square brackets have been around a
lot longer. But with this scheme they can all use square brackets.

And as pointed out by Monz, it will be easy to remember that the
commas come after the exponents of 3, 11, 19 and 31 since these have
indeed seemed to be natural stopping (resting?) places.

So a Pythagorean comma(generic sense) could be given in
octave-specific form as [x y] or [x y,] without ambiguity, but in
octave equivalent form it would have to be [y,] although I don't
recall ever having seen a vextor with only octaves in it, so if you
saw [y] you'd be pretty sure it was meant to be [y,].

Then there's the possibility of 2-and-3-free monzos being used to name
very complex commas in George's and my system. These would have to
_start_ with a comma. For example, the atom of Kirnberger is also the
[,12]-schismina. Although I don't know how you pronounce that so it's
clear you're giving a prime exponent, and not a factor.
"five-to-the-twelve-schismina" works just as well for me in this case.

And indeed, how should we pronounce the commas(punctuation sense) so
they don't get confused with commas(generic sense) or commas(specific
size range sense)?

I'm guessing these won't be very comma-n in spoken conversation
between musicians, so we can ignore this problem. But I think we've
comma long way.

George made a good point about who "the rest of us" might be.

I then realised that mathematical types searching for temperaments
want names for the commas(generic sense) that vanish, while musicians
using those temperaments will need names for the commas(generic sense)
that _don't_ vanish. Why would they need a name for something that
isn't there?

This dichotomy will also lead to quite different rankings of "the most
important commas(generic sense)".

>I then realised that mathematical types searching for temperaments
>want names for the commas(generic sense) that vanish, while musicians
>using those temperaments will need names for the commas(generic
sense)
>that _don't_ vanish. Why would they need a name for something that
>isn't there?

The names can be the same.

>This dichotomy will also lead to quite different rankings of "the
most
>important commas(generic sense)".

Will it? I recently argued no; Paul seemed to argue yes.

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
> >I then realised that mathematical types searching for temperaments
> >want names for the commas(generic sense) that vanish, while musicians
> >using those temperaments will need names for the commas(generic
> sense)
> >that _don't_ vanish. Why would they need a name for something that
> >isn't there?
>
> The names can be the same.

Certainly. But these two different purposes may lead to diffferent
ideas about what would constitute a good name. George and I think that
a good name will indicate the simplest ratios that can be notated with
that comma, relative to a chain of fifths.

>
> >This dichotomy will also lead to quite different rankings of "the
> most
> >important commas(generic sense)".
>
> Will it? I recently argued no; Paul seemed to argue yes.

I expect some overlap certainly, but also large regions that are of
interest to one and not the other. Anything with an absolute
3-exponent greater than 12 (or at most 18) is not going to be of much
interest for notational purposes. Are any of these of great interest
as vanishing in a useful linear temperament?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
> wrote:
> > One would only
> > need to determine at what point two commas were forced to share the
> > same name and then decide if they were both "important".
>
> NEVER! Two commas with the same name makes no sense at all.

Gene, perhaps before you assume you are the last remaining defender of
good sense, and leap, with battle-cry, down the throat of the imagined
enemy, you might check to see if they really meant what you assumed
they meant.

In the very next sentence George wrote:

> > Dave already discussed this:
> >
> > /tuning-math/messages/7320

In that, you will find:
"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."

So only the comma with the lowest absolute 3-exponent gets the simple
systematic name.

I think what George intended was, if only one is "important" then it
gets the name and the other little piggy has none. But if both are
important we have to figure a way to give them different names.

I suggest simply adding the adjective "complex" to the start of the
one with the second lowest 3-exponent, then if you need to go beyond
that, which seems very unliklely, then "hypercomplex" or some such.

For example, we have
[12 19] as the Pythagorean-comma 23.5 c
and so
[41 65] might be called the complex-Pythagorean-comma 19.8 c.

> relative to a chain of fifths.

Sacre bleu.

>> Will it? I recently argued no; Paul seemed to argue yes.
>
>I expect some overlap certainly, but also large regions that are of
>interest to one and not the other. Anything with an absolute
>3-exponent greater than 12 (or at most 18) is not going to be of much
>interest for notational purposes. Are any of these of great interest
>as vanishing in a useful linear temperament?

What I need from you/Paul is a general principle of good chromatic
vectors that differs from the general principle of good commatic
vectors we already have (namely "epimericity", etc.).

-Carl

Oops!
I serously screwed up the monzos for those commas(specific size-range
sense). I should have written:

For example, we have
[-19 12] as the Pythagorean-comma 23.5 c
and so
[65 -41] might be called the complex-Pythagorean-comma 19.8 c.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> What I need from you/Paul is a general principle of good chromatic
> vectors that differs from the general principle of good commatic
> vectors we already have (namely "epimericity", etc.).

I'll leave that to Paul, since that is slightly different again from
good commas for a general-purpose notation system.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:

> > [-3 -2, 0 1 0, 0 0 1] or
> > [-2, 0 1 0, 0 0 1].
> > This makes the grouping by threes more obvious (and the higher
primes
> > much easier to locate), and angle brackets would no longer be
> > necessary.
>
> I like this very much. It is also MATLAB/Octave compatible. Since
> commas are optional, but semicolons indicate the end of a row (and
so
> would make a matrix, not a vector). Isn't that right Paul?

i checked and you're right!

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> George made a good point about who "the rest of us" might be.
>
> I then realised that mathematical types searching for temperaments
> want names for the commas(generic sense) that vanish, while
musicians
> using those temperaments will need names for the commas(generic
sense)
> that _don't_ vanish. Why would they need a name for something that
> isn't there?

because it rules harmonic scale construction.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
> > >I then realised that mathematical types searching for
temperaments
> > >want names for the commas(generic sense) that vanish, while
musicians
> > >using those temperaments will need names for the commas(generic
> > sense)
> > >that _don't_ vanish. Why would they need a name for something
that
> > >isn't there?
> >
> > The names can be the same.
>
> Certainly. But these two different purposes may lead to diffferent
> ideas about what would constitute a good name. George and I think
that
> a good name will indicate the simplest ratios that can be notated
with
> that comma, relative to a chain of fifths.
>
> >
> > >This dichotomy will also lead to quite different rankings
of "the
> > most
> > >important commas(generic sense)".
> >
> > Will it? I recently argued no; Paul seemed to argue yes.
>
> I expect some overlap certainly, but also large regions that are of
> interest to one and not the other. Anything with an absolute
> 3-exponent greater than 12 (or at most 18) is not going to be of
much
> interest for notational purposes. Are any of these of great interest
> as vanishing in a useful linear temperament?

depends what you mean by useful, but i'd say no.

>What I need from you/Paul is a general principle of good chromatic
>vectors that differs from the general principle of good commatic
>vectors we already have (namely "epimericity", etc.).

I could see just 1/complexity, or somehow weakening the size term,
as size doesn't seem important beyond preserving propriety.

-Carl

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > i.e., 3 is obviously extremely important both historically
> > and theoretically, and thus deserves to be isolated by itself
> > (or grouped with 2, if 2 is included).

All of the new symbols in a JI heptatonic notation will be modifying
nominals in a pythagorean sequence (if we include sharped/flatted
tones under a broader concept of the term nominals), so this is a
natural place to make a separation.

> > the next comma appears after 11, and earlier in this thread
> > i discussed the idea that 11-limit can be a kind of boundary.
> > Partch thought so too. (but please, don't anyone make too
> > much of this comment.)

Partch thought that 7 was implied in a 12-tone octave in that 5:7 or
7:10 suggest consonant tritones, so 11 is the first prime that is
completely foreign to conventional harmony. Since ratios of 13 are
similar in effect to ratios of 11 (and since 11 harmonically bridges
to 13 rather easily), this makes a good case for stopping with 11.
Also, those who advocate achieving new harmonic resources by
extending the meantone temperament until it forms a closed system of
31 tones find that they have ended up with an 11-limit (tempered)
system.

So I think that there are many that would agree with 11 as a boundary.

> > the next comma appears after 19, which i myself used as
> > a limit from about 1988-98.

Mapping a harmonic series consistently into a 12-tone octave (without
skipping over any primes to reach other primes, and without skipping
over any odd harmonics to reach other odd harmonics) yields a 19-
limit set that's a favorite of mine:

16:17:18:19:20:21:22:24:25:26:28:30:32

So I also like 19 as a boundary.

> > the next comma appears after 31, which is the highest limit
> > Ben Johnston has used in his music.
> >
> > interesting.

This complete 5 octaves of harmonics, and there's a big gap between
31 and 37, the next prime.

> so the primes are arranged as
> 2 3 , 5 7 11 , 13 17 19 , 23 29 31 , 37 41 43 , 47 53 59 , 61 67 71
>
> looks like the next comma after 31 makes sense too -- isn't 43 the
> highest limit used by george secor at least in some context?

If you're referring to something that I posted some time ago, I think
that was 41. (But I did happen to come across a use for 43 when I
was subsequently rummaging through some of my old papers.) Trouble
with 43 is that 32:43 readily invites confusion with 3:4.

--George

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> no reply, george?
>
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "George D. Secor"
gdsecor@y...>
> > wrote:
> >
> > > BTW, have you ever tried collapsing an 11-limit lattice into 2
> > > dimensions by mapping 11/8 to <10, 5>?
> >
> > you're probably referring to the 3-5-11 lattice?
> >
> > the full 11-limit lattice is at least 3 dimensional if you use
> > one "xenharmonic bridge" as above. for the 3-5-11 case, this
choice
> > (184528125:184549376) is probably a very good one. for the full
11-
> > limit case, 9800:9801 is probably better for most purposes, since
> > it's both a little smaller (in cents) and much simpler (i.e.,
shorter
> > in the lattice).

Sorry, I started something that sudden time constraints didn't permit
me to finish. Also, in my haste I also skipped over 7 -- 7/4 would
also be mapped to the [7, -5] position.

--George

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>>
> considering that i had numerators and denominators well in excess
of
> 10^50 in the list, i'm inclined to believe gene that a given
> epimericity cutoff will yield a finite list.

I've shown that it does, for cutoffs less than 1, using Baker's
theorem.

i wonder
> if even dave could stomach such a ranking -- the very simple
> temperaments with high error are easy enough to mentally toss out
for
> the user seeking a certain goodness of approximation . . .

Tossing out powers such as 6561/6400 is what I'd recommend also,
though Dave might not go for it.

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> So I think that there are many that would agree with 11 as a
boundary.

I've recently done some pieces with full 13-limit chords as harmony;
the results may convince people that stopping earlier is a good plan.
So far as complete otonal and utonal chords go, however, 5-limit has
a natural 3-et triadic nature, 7-limit 4-et tetradic, 9-limit 5-et
quintadic, and 13-limit 7-et septadic. For 11-limit complete harmony,
we are stuck with a 6 val, which is a little ungainly.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...> wrote:
>
> > I don't have the resources to do that very readily. Wouldn't
putting
> > them in order of product complexity (discarding any with factors
> > above some particular prime limit) accomplish this?
>
> Not really; however we can simply take everything below a certain
> prime limit and below a limit for what I call "epipermicity", which
> is, for p/q>1 in reduced form, log(p-q)/log(q). It can be shown
this
> gives a finite list of commas if the epimermicity limit is less
than
> one.
>
> > One would only
> > need to determine at what point two commas were forced to share
the
> > same name and then decide if they were both "important".
>
> NEVER! Two commas with the same name makes no sense at all.

I wasn't suggesting giving them the same name, as I said in msg.
#7420:

<< As for the answer to your second question: I don't know if there
will be unique names for *all* of them (probably not), but I believe
that there are enough names to cover all of those that would be of
use to most theorists. Those with very specialized purposes (such as
yourself) could devise a modification to Dave's naming system (such
as appending letters a, b, c, etc. in order of ratio complexity) to
distinguish equivocal names -- I think you're creative enough to come
up with something that would be meaningful (or perhaps Dave might
have some ideas). >>

I really haven't had very much time lately to participate in this
sort of discussion, but when it seemed that Dave had dropped out I
saw that there were things that hadn't been resolved, I felt that I
had to jump in and say something. If you want a couple of specific
examples to pursue this further, I can mention a couple of instances
from which we were trying to set the kleisma-comma boundary:

152:153 (~11.4c) is definitely a 17:19-kleisma, but 1114112:1121931
(~12.1c) will either be either a (subordinate) 17:19-kleisma or the
17:19-comma, depending on where the boundary is set.

135:136 (~12.8c) will either be the 5:17-kleisma or the 5:17-comma,
but 327680:334611 (~36.2c) also claims the name 5:17-comma with our
present comma/S-diesis boundary.

These are issues that we still need to work out.

--George

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> Mapping a harmonic series consistently into a 12-tone octave (without
> skipping over any primes to reach other primes, and without skipping
> over any odd harmonics to reach other odd harmonics) yields a 19-
> limit set that's a favorite of mine:
>
> 16:17:18:19:20:21:22:24:25:26:28:30:32

As I remarked, the 3, 5, 7, 9 and 13 limits map respectivly to the
2, 3, 4, 5, and 7 standard vals. However, I had forgotten until I
checked it again that the 11-limit doesn't map to anything; in fact
the 13-limit seems to be the last odd limit whose complete otonal
chord defines a val. However, we can proceed as George does above; he
gives an incomplete 25-odd-limit chord, which defines the following
19-limit val: g12 = [12, 19, 28, 34, 42, 45, 49, 51]. While g12 isn't
what I've called the "standard" val h12, it in fact is preferable,
since it has a lower consistent badness score.

>As I remarked, the 3, 5, 7, 9 and 13 limits map respectivly to the
>2, 3, 4, 5, and 7 standard vals. However, I had forgotten until I
>checked it again that the 11-limit doesn't map to anything; in fact
>the 13-limit seems to be the last odd limit whose complete otonal
>chord defines a val.

You lost me.

And you're site's apparently down.

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:

> >As I remarked, the 3, 5, 7, 9 and 13 limits map respectivly to the
> >2, 3, 4, 5, and 7 standard vals. However, I had forgotten until I
> >checked it again that the 11-limit doesn't map to anything; in fact
> >the 13-limit seems to be the last odd limit whose complete otonal
> >chord defines a val.
>
> You lost me.

Consider the otonal chord of the n-odd-limit. This has (n+1)/2 octave
reduced elements, 1 < q[i] <= 2, where the q[i], i from 1 to (n+1)/2,
are arranged in increasing size. The n-odd-limit has pi(n) primes; we
may solve the (n+1)/2 linear equations for the val which sends q[1]
to 1, q[2] to 2, up to q[(n+1)/2]=2 to (n+1)/2. These linear
equations have a unique solution in the 3, 5, 7, 9, and 13 odd limits.
For 3 we get [2, 3], for 5 [3, 5, 7] and so forth--the standard vals
in the respective prime limits 3, 5, 7, 7, 11 for 2, 3, 4, 5, and 7.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> For 3 we get [2, 3], for 5 [3, 5, 7] and so forth--the standard
vals
> in the respective prime limits 3, 5, 7, 7, 11 for 2, 3, 4, 5, and 7.

Should be prime limits 3, 5, 7, 7 and 13.

To give a simple example, in the 5-limit, (5+1)/2 = 3, and we may
start from the 3-chord [5/4, 3/2, 2]. If we solve for a val [a, b, c]
such that 5/4, or [-2, 0, 1] is mapped to 1, 3/2 is mapped to 2, and
2 is mapped to 3 we get the equations a5 - 2 a2 = 1, a3 - a2 = 2, and
a2 = 3, the solution of which is a2 = 3, a3 = 5, and a5 = 7, so the
val in question is uniquely determined to be [3, 5, 7], the standard
3-val for the 5-limit.

hi George and paul,

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:

> > so the primes are arranged as
> > 2 3 , 5 7 11 , 13 17 19 , 23 29 31 , 37 41 43 , 47 53 59 ,
> > 61 67 71
> >
> > looks like the next comma after 31 makes sense too --
> > isn't 43 the highest limit used by george secor at least
> > in some context?
>
> If you're referring to something that I posted some time
> ago, I think that was 41. (But I did happen to come across
> a use for 43 when I was subsequently rummaging through some
> of my old papers.) Trouble with 43 is that 32:43 readily
> invites confusion with 3:4.

hmmm... come to think of it, i wrote a piece a few years back
called _Alternative Rock Chord_, which in fact was inspired
by a discussion i was having with paul on the tuning list.

http://sonic-arts.org/monzo/altrock/altrock.mid

the piece is minimalist, featuring two repeating phrases
which have only one chord each. the first chord is
tuned to 12edo, then the rest are JI. all voices but
one stay on the same pitch for the rest of the pattern,
but one voice moves 4/3 - 21/16, 43/32, 11/8, 43/32 -
21/16 - 4/3 then back to 12edo again, then the whole
pattern repeats with new parts added each time.

the frequencies of the center section of the moving part
have the proportions 42:43:44:43:42, heard against
the 4:5:6:7 drone of the other voices.

it was my specific intention in this piece to tune up
a variety of "4ths" and play them together with the
4:5:6:7 chord. traditional music-theory prohibits
sounding the "4th" along with the "major-3rd", but
that's exactly what a lot of alternative-rockers did
in the 1990s.

so the idea arose first in my imagination by wondering
what the differently-tuned "4th"s would sound like
and thinking about the numbers.

since the piece is minimalist, it was easy to use
copy-and-paste to create many similar sections, and
then just add new instrumental parts and pitch-bend data.

when i finished putting all the pitch-bend data into
the MIDI file, there was still a big chunk of music
at the end that was still in 12edo.

strangely to my ears, and finally heard it, when the
piece came back to 12edo at the end, it sounded like
a resolution of some sort.

so i decided to begin the piece in 12edo, and to
frame each repetition by constantly returning back
to 12edo.

just some observations about this piece that i thought
interesting.

the point of posting it here is that i used 43 as
part of my experiment in tuning the notes of that
moving part, and after hearing 43, deliberately decided
to keep it and not retune it to something simpler
(i.e., lower prime factor).

- monz

>To give a simple example, in the 5-limit, (5+1)/2 = 3, and we may
>start from the 3-chord [5/4, 3/2, 2]. If we solve for a val [a, b, c]
>such that 5/4, or [-2, 0, 1] is mapped to 1, 3/2 is mapped to 2, and
>2 is mapped to 3 we get the equations a5 - 2 a2 = 1, a3 - a2 = 2, and
>a2 = 3, the solution of which is a2 = 3, a3 = 5, and a5 = 7, so the
>val in question is uniquely determined to be [3, 5, 7], the standard
>3-val for the 5-limit.

What's the definition of "standard val"?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> What's the definition of "standard val"?

The vector consisting of round(n log2(p)) for primes p in ascending
order up to the chosen prime limit, considered as defining a val.

>> What's the definition of "standard val"?
>
>The vector consisting of round(n log2(p)) for primes p in ascending
>order up to the chosen prime limit, considered as defining a val.

What's n?

-C.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> What's the definition of "standard val"?
> >
> >The vector consisting of round(n log2(p)) for primes p in
ascending
> >order up to the chosen prime limit, considered as defining a val.
>
> What's n?

The number you are finding a standard val for. We could adjust this
number to the Zeta tuning or the Gram tuning, and have a Zeta val or
a Gram val, by the way; in that case n would no longer be an integer.

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> I really haven't had very much time lately to participate in this
> sort of discussion, but when it seemed that Dave had dropped out I
> saw that there were things that hadn't been resolved, I felt that I
> had to jump in and say something. If you want a couple of specific
> examples to pursue this further, I can mention a couple of instances
> from which we were trying to set the kleisma-comma boundary:
>
> 152:153 (~11.4c) is definitely a 17:19-kleisma, but 1114112:1121931
> (~12.1c) will either be either a (subordinate) 17:19-kleisma or the
> 17:19-comma, depending on where the boundary is set.
>
> 135:136 (~12.8c) will either be the 5:17-kleisma or the 5:17-comma,
> but 327680:334611 (~36.2c) also claims the name 5:17-comma with our
> present comma/S-diesis boundary.
>
> These are issues that we still need to work out.

George,

I hope you've seen my latest proposals by now in:
/tuning-math/message/7458

It resolves all the above issues. Previously we allowed our desire for
the boundaries to fall between sagittal symbols, to distort our
choices. The optimum size-category boundaries, for maximising the
number of unique simple names, are always at the irrational square
roots of 3-commas.

And I now have a more complete proposal for naming commas when they do
not have a unique name by simply using these size categories preceded
by the ratio with powers of 2 and 3 removed. We simply prepend the
following words in order of increasing absolute 3-exponent.

complex
supercomplex
hypercomplex
ultracomplex
5-complex
6-complex
7-complex
...

My 10 year old son, Hunter, confirms that super, hyper, ultra is the
correct progression, as this is used for the strengths of magic
potions in Pokemon. :-)

I don't expect these will ever really be needed. But here's a
ridiculous example to make my intentions clear.

Prime
exponent
vector Cents Name
-----------------------------------------------
[-19 12] 23.46 Pythagorean comma
[65 -41] 19.84 complex Pythagorean comma
[-103 65] 27.08 supercomplex Pythagorean comma
[149 -94] 16.23 hypercomplex Pythagorean comma
[-187 118] 30.69 ultracomplex Pythagorean comma
[233 -147] 12.61 5-complex Pythagorean comma
[-271 171] 34.31 6-complex Pythagorean comma
[298 -188] 32.46 7-complex Pythagorean comma
[-336 212] 14.46 8-complex Pythagorean comma
[382 -241] 28.84 9-complex Pythagorean comma
[-420 265] 18.08 10-complex Pythagorean comma
[466 -294] 25.23 11-complex Pythagorean comma

>> >> What's the definition of "standard val"?
>> >
>> >The vector consisting of round(n log2(p)) for primes p in
>> >ascending order up to the chosen prime limit, considered
>> >as defining a val.
>>
>> What's n?
>
>The number you are finding a standard val for.

Then what's p!?

Besides using two vals to find a lt, can you give an example
of what a single standard val would be good for?

-Carl

> traditional music-theory prohibits
> sounding the "4th" along with the "major-3rd", but
> that's exactly what a lot of alternative-rockers did
> in the 1990s.

... and jazz pianists have been voicing 'sus' chords with both the
third and the fourth for quite a long time, where the third is
usually placed in a higher octave above the fourth. The inverse
voicing is also used, but not as often.

AH

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> >> What's the definition of "standard val"?
> >> >
> >> >The vector consisting of round(n log2(p)) for primes p in
> >> >ascending order up to the chosen prime limit, considered
> >> >as defining a val.
> >>
> >> What's n?
> >
> >The number you are finding a standard val for.
>
> Then what's p!?

A prime number <= n.

> Besides using two vals to find a lt, can you give an example
> of what a single standard val would be good for?

It's easy to calculate and most of the time gives you the val you are
most interested in. However, the gram val isn't hard to compute
either, and is more likely to do that, so "standard" is really just a
connvenience. Gram vals would be harder to explain--look at the
trouble I am in now with the standard val.

>> >> >The vector consisting of round(n log2(p)) for primes p in
>> >> >ascending order up to the chosen prime limit, considered
>> >> >as defining a val.
>> >>
>> >> What's n?
>> >
>> >The number you are finding a standard val for.
>>
>> Then what's p!?
>
>A prime number <= n.
>
>> Besides using two vals to find a lt, can you give an example
>> of what a single standard val would be good for?
>
>It's easy to calculate and most of the time gives you the val you are
>most interested in. However, the gram val isn't hard to compute
>either, and is more likely to do that, so "standard" is really just a
>connvenience. Gram vals would be harder to explain--look at the
>trouble I am in now with the standard val.

It's just val I still don't understand. If a val is a homomorphism
from the rationals to the integers, I can't fathom how a "number I'm
finding a val for" could come into play. If the standard 5-limit val
for 12-equal is [12 19 28] or something, how does it come from
round(n log2(p))? Oh n is 12, eh? So vals are uniquely identified
by this n?

So how does one find a standard val for an odd limit (the start of
this thread, which perhaps George is still following)? Where do you
get your n?

-Carl

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
> wrote:
>
> > Mapping a harmonic series consistently into a 12-tone octave
(without
> > skipping over any primes to reach other primes, and without
skipping
> > over any odd harmonics to reach other odd harmonics)

I meant to say, "and without skipping over any *non-prime* odd
harmonics to reach other odd harmonics)"

> > yields a 19-
> > limit set that's a favorite of mine:
> >
> > 16:17:18:19:20:21:22:24:25:26:28:30:32
>
> As I remarked, the 3, 5, 7, 9 and 13 limits map respectivly to the
> 2, 3, 4, 5, and 7 standard vals. However, I had forgotten until I
> checked it again that the 11-limit doesn't map to anything;

This is why I personally desire to use JI (or a microtemperament)
that is at least 13-limit.

> in fact
> the 13-limit seems to be the last odd limit whose complete otonal
> chord defines a val. However, we can proceed as George does above;
he
> gives an incomplete 25-odd-limit chord, which defines the following
> 19-limit val: g12 = [12, 19, 28, 34, 42, 45, 49, 51]. While g12
isn't
> what I've called the "standard" val h12, it in fact is preferable,
> since it has a lower consistent badness score.

It's also possible to make a 17-limit decatonic set (or incomplete 21-
odd-limit chord) by the same method:

16:17:18:20:21:22:24:26:28:30:32

Taking every 3rd tone of the above scale results in a chord that
returns to the starting tone at the triple-octave. These
decatonic "4ths" are the same general size as heptatonic "3rds", and
there are 4 triads in the scale that have perfect fifths (hence four
useful modes for this scale).

4:5:6:15/2:9:11:14:17:21:26:32

--George

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> Tossing out powers such as 6561/6400 is what I'd recommend also,

i thought that went without saying.

> though Dave might not go for it.

why not? dave has expressed interest in tuning systems where a single
just lattice does not suffice as a derivation for all the pitches,
but tempering out 6561/6400 simply leads to torsion and not to such a
tuning system.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...> wrote:
> > I really haven't had very much time lately to participate in this
> > sort of discussion, but when it seemed that Dave had dropped out
I
> > saw that there were things that hadn't been resolved, ...
> > ...
> > These are issues that we still need to work out.
>
> George,
>
> I hope you've seen my latest proposals by now in:
> /tuning-math/message/7458
>
> It resolves all the above issues. Previously we allowed our desire
for
> the boundaries to fall between sagittal symbols, to distort our
> choices. The optimum size-category boundaries, for maximising the
> number of unique simple names, are always at the irrational square
> roots of 3-commas.

Good, I'm glad to see that you have been able to finalize this.

> And I now have a more complete proposal for naming commas when they
do
> not have a unique name by simply using these size categories
preceded
> by the ratio with powers of 2 and 3 removed. We simply prepend the
> following words in order of increasing absolute 3-exponent.
>
> complex
> supercomplex
> hypercomplex
> ultracomplex
> 5-complex
> 6-complex
> 7-complex
> ...
>
> My 10 year old son, Hunter, confirms that super, hyper, ultra is the
> correct progression, as this is used for the strengths of magic
> potions in Pokemon. :-)
>
> I don't expect these will ever really be needed. But here's a
> ridiculous example to make my intentions clear.
>
> Prime
> exponent
> vector Cents Name
> -----------------------------------------------
> [-19 12] 23.46 Pythagorean comma
> [65 -41] 19.84 complex Pythagorean comma
> [-103 65] 27.08 supercomplex Pythagorean comma
> [149 -94] 16.23 hypercomplex Pythagorean comma
> [-187 118] 30.69 ultracomplex Pythagorean comma
> [233 -147] 12.61 5-complex Pythagorean comma
> [-271 171] 34.31 6-complex Pythagorean comma
> [298 -188] 32.46 7-complex Pythagorean comma
> [-336 212] 14.46 8-complex Pythagorean comma
> [382 -241] 28.84 9-complex Pythagorean comma
> [-420 265] 18.08 10-complex Pythagorean comma
> [466 -294] 25.23 11-complex Pythagorean comma^

These could be even be abbreviated 3C, 3C-x, 3C-2x, 3C-3x, etc. (or
something similar), which goes to show that you can have comma names
with abbreviations that are short enough to label positions on a
lattice! How 'bout that!

--George

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...>
> wrote:
> > So I think that there are many that would agree with 11 as a
> boundary.
>
> I've recently done some pieces with full 13-limit chords as
harmony;
> the results may convince people that stopping earlier is a good
plan.
> So far as complete otonal and utonal chords go, however, 5-limit
has
> a natural 3-et triadic nature, 7-limit 4-et tetradic, 9-limit 5-et
> quintadic, and 13-limit 7-et septadic. For 11-limit complete
harmony,
> we are stuck with a 6 val, which is a little ungainly.

i think george secor was trying to make a similar point in explaining
why the 11-odd-limit was not of much use for him for compositional
purposes. on the other hand, if you listen to prent rodgers' music,
it becomes really hard to deny as a harmonic possibility.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> However, I had forgotten until I
> checked it again that the 11-limit doesn't map to anything;

because 3:1 and 9:3 map to different numbers of "steps" in a chord? i
think that was similar to george secor's reasoning . . .

> While g12 isn't
> what I've called the "standard" val h12, it in fact is preferable,
> since it has a lower consistent badness score.

why don't we switch from standard vals to "lowest consistent badness"
vals around here as a general rule? i think that would be a positive
development . . .

--- In tuning-math@yahoogroups.com, "gooseplex" <cfaah@e...> wrote:
> > traditional music-theory prohibits
> > sounding the "4th" along with the "major-3rd", but
> > that's exactly what a lot of alternative-rockers did
> > in the 1990s.
>
> ... and jazz pianists have been voicing 'sus' chords with both the
> third and the fourth for quite a long time, where the third is
> usually placed in a higher octave above the fourth. The inverse
> voicing is also used, but not as often.
>
> AH

i don't know of instances of the third being voiced below the fourth
in jazz, it seems to create a completely different chord which i've
heard in rock (and various tunings of which are wonderfully explored
in monz's piece) but not in jazz.

--- In tuning-math@yahoogroups.com, "Paul Erlich"
<perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "gooseplex"
<cfaah@e...> wrote:
> > > traditional music-theory prohibits
> > > sounding the "4th" along with the "major-3rd", but
> > > that's exactly what a lot of alternative-rockers did
> > > in the 1990s.
> >
> > ... and jazz pianists have been voicing 'sus' chords with both
the
> > third and the fourth for quite a long time, where the third is
> > usually placed in a higher octave above the fourth. The
inverse
> > voicing is also used, but not as often.
> >
> > AH
>
> i don't know of instances of the third being voiced below the
fourth
> in jazz, it seems to create a completely different chord which
i've
> heard in rock (and various tunings of which are wonderfully
explored
> in monz's piece) but not in jazz.

The interpretation of the chord as 'sus' may be debated, but
without question, the voicing happens quite a bit in jazz. An
example would be, spelled up from the bottom, G2, G3, C4, F4,
A4, B4. The question is: is this a sus chord or merely a dominant
11? Mark Levine writes in The Jazz Piano Book, "A persistent
myth about sus chords is that 'the fourth takes the place of the
third.' Jazz pianists, however, often voice the third with a sus
chord ... the third is voiced above the fourth" p. 24 Levine also
made a similar statement in his Jazz Theory book. I like Levine's
description, because these chords are in fact often used in
context as sus chords. Many people take issue with Levine's
interpretation. For example, Robert Rawlins writes in his Music
theory Online review of Levine's theory book:

"The problem that now arises--if the Mixolydian mode is
considered equivalent to the Vsus chord--is that the mode
contains an unwanted third. Levine circumvents this difficulty by
arguing that the third is not an undesirable note in sus chords: "A
persistent myth is that 'the 4th takes the place of the 3rd in a sus
chord.' This was true at one time, but in the 1960s, a growing
acceptance of dissonance led pianists and guitarists to play sus
voicings with both the 3rd and the 4th" (p. 46). Undeniably, jazz
musicians have explored this possibility. The question is how to
interpret the resulting chord. If a sus chord is to retain anything of
its presumed historical origin, then the absence of the leading
tone would seem to be requisite. If jazz theory, in practice, has
dispensed with the preparation and resolution of this
suspension, what must remain is at least the displacement of
the third of the chord. If the third is present, and we indeed have
a dominant triad with upper extensions, then it is not clear what
justifies pulling the 11th of the chord into the basic structure and
calling it a sus chord. If one were to argue that the voicings
generally employed in contexts where both the third and fourth
are present seem to suggest the sus chord, then it will have to
be attributed to intended ambiguity, much as a twentieth-century
composer might flirt with the ambiguity between major and
minor tonality. There is little to justify the conclusion that sus
chords implicitly contain the third, which is available anytime one
wishes to include it in the harmonic structure. Again, Levine is
avoiding the obvious: The Mixolydian mode is indeed roughly
equivalent with a Vsus chord, with the exception of the third,
which is completely foreign to theharmony."

http://www.societymusictheory.org/mto/issues/mto.00.6.1/mto.00
.6.1.rawlins.html

AH

sorry, that example should be:
G2, G3, B3, F4, C5
the one in the other message is the more common 'third above
the fourth'
AH

--- In tuning-math@yahoogroups.com, "gooseplex" <cfaah@e...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich"
> <perlich@a...> wrote:
> > --- In tuning-math@yahoogroups.com, "gooseplex"
> <cfaah@e...> wrote:
> > > > traditional music-theory prohibits
> > > > sounding the "4th" along with the "major-3rd", but
> > > > that's exactly what a lot of alternative-rockers did
> > > > in the 1990s.
> > >
> > > ... and jazz pianists have been voicing 'sus' chords with both
> the
> > > third and the fourth for quite a long time, where the third is
> > > usually placed in a higher octave above the fourth. The
> inverse
> > > voicing is also used, but not as often.
> > >
> > > AH
> >
> > i don't know of instances of the third being voiced below the
> fourth
> > in jazz, it seems to create a completely different chord which
> i've
> > heard in rock (and various tunings of which are wonderfully
> explored
> > in monz's piece) but not in jazz.
>
>
> The interpretation of the chord as 'sus' may be debated, but
> without question, the voicing happens quite a bit in jazz. An
> example would be, spelled up from the bottom, G2, G3, C4, F4,
> A4, B4.

no, there the third is *above* the fourth, not below it.

> The question is: is this a sus chord or merely a dominant
> 11? Mark Levine writes in The Jazz Piano Book, "A persistent
> myth about sus chords is that 'the fourth takes the place of the
> third.' Jazz pianists, however, often voice the third with a sus
> chord ... the third is voiced above the fourth" p. 24 Levine also
> made a similar statement in his Jazz Theory book. I like Levine's
> description, because these chords are in fact often used in
> context as sus chords.

you're avoiding my question, which was about the third being voiced
*below* the fourth. re-read my comments above again.

> Many people take issue with Levine's
> interpretation. For example, Robert Rawlins writes in his Music
> theory Online review of Levine's theory book:
>
> "The problem that now arises--if the Mixolydian mode is
> considered equivalent to the Vsus chord--is that the mode
> contains an unwanted third. Levine circumvents this difficulty by
> arguing that the third is not an undesirable note in sus chords: "A
> persistent myth is that 'the 4th takes the place of the 3rd in a
sus
> chord.' This was true at one time, but in the 1960s, a growing
> acceptance of dissonance led pianists and guitarists to play sus
> voicings with both the 3rd and the 4th" (p. 46). Undeniably, jazz
> musicians have explored this possibility. The question is how to
> interpret the resulting chord. If a sus chord is to retain anything
of
> its presumed historical origin, then the absence of the leading
> tone would seem to be requisite. If jazz theory, in practice, has
> dispensed with the preparation and resolution of this
> suspension, what must remain is at least the displacement of
> the third of the chord. If the third is present, and we indeed have
> a dominant triad with upper extensions, then it is not clear what
> justifies pulling the 11th of the chord into the basic structure
and
> calling it a sus chord. If one were to argue that the voicings
> generally employed in contexts where both the third and fourth
> are present seem to suggest the sus chord, then it will have to
> be attributed to intended ambiguity, much as a twentieth-century
> composer might flirt with the ambiguity between major and
> minor tonality. There is little to justify the conclusion that sus
> chords implicitly contain the third, which is available anytime one
> wishes to include it in the harmonic structure. Again, Levine is
> avoiding the obvious: The Mixolydian mode is indeed roughly
> equivalent with a Vsus chord, with the exception of the third,
> which is completely foreign to theharmony."
>
> http://www.societymusictheory.org/mto/issues/mto.00.6.1/mto.00
> .6.1.rawlins.html
>
> AH

still nothing relevant, i'm afraid.

--- In tuning-math@yahoogroups.com, "gooseplex" <cfaah@e...> wrote:

> sorry, that example should be:
> G2, G3, B3, F4, C5

where did you find this? on page 24 of the jazz piano book, you say?
i'll check when i get home . . .

--- In tuning-math@yahoogroups.com, "Paul Erlich"
<perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "gooseplex"
<cfaah@e...> wrote:
>
> > sorry, that example should be:
> > G2, G3, B3, F4, C5
>
> where did you find this? on page 24 of the jazz piano book, you
say?
> i'll check when i get home . . .

No, you'll find Levine's example on p. 25 is (G root assumed in
bass) F3, B3. E4, A4, C5.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> > Tossing out powers such as 6561/6400 is what I'd recommend also,
>
> i thought that went without saying.
>
> > though Dave might not go for it.
>
> why not? dave has expressed interest in tuning systems where a single
> just lattice does not suffice as a derivation for all the pitches,
> but tempering out 6561/6400 simply leads to torsion and not to such a
> tuning system.

I call this "twin meantone" and consider it to have the same errors
but twice the complexity of meantone, and so I rank it above some
things like pelogic and Blackwood (quintuple thirds). It seems to me
that it would have two unconnected just lattices.

But don't take any notice of me, I can't even tell my torsions from my
contorsions. :-)

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
>
> > However, I had forgotten until I
> > checked it again that the 11-limit doesn't map to anything;
>
> because 3:1 and 9:3 map to different numbers of "steps" in a chord?
i
> think that was similar to george secor's reasoning . . .

My reasoning is simply that trying to define one leads to an
inconsistent system of linear equaltions.

> why don't we switch from standard vals to "lowest consistent
badness"
> vals around here as a general rule? i think that would be a
positive
> development . . .

Do you have a name?

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > wrote:
> >
> > > Tossing out powers such as 6561/6400 is what I'd recommend
also,
> >
> > i thought that went without saying.
> >
> > > though Dave might not go for it.
> >
> > why not? dave has expressed interest in tuning systems where a
single
> > just lattice does not suffice as a derivation for all the
pitches,
> > but tempering out 6561/6400 simply leads to torsion and not to
such a
> > tuning system.
>
> I call this "twin meantone" and consider it to have the same errors
> but twice the complexity of meantone, and so I rank it above some
> things like pelogic and Blackwood (quintuple thirds). It seems to me
> that it would have two unconnected just lattices.
>
> But don't take any notice of me, I can't even tell my torsions from
my
> contorsions. :-)

"twin meantone" is not what results from tempering out 6561/6400 at
all, but it is an example of what i was invoking your name in
describing above. i beg you to think about it on your own and try to
come up with an answer, especially if it contradicts mine. working
through this intuitively doesn't require any tuning-math knowledge to
understand, it seems like a pretty practical question.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <gwsmith@s...>
> > wrote:
> >
> > > However, I had forgotten until I
> > > checked it again that the 11-limit doesn't map to anything;
> >
> > because 3:1 and 9:3 map to different numbers of "steps" in a
chord?
> i
> > think that was similar to george secor's reasoning . . .
>
> My reasoning is simply that trying to define one leads to an
> inconsistent system of linear equaltions.

are the two facts related?

> > why don't we switch from standard vals to "lowest consistent
> badness"
> > vals around here as a general rule? i think that would be a
> positive
> > development . . .
>
> Do you have a name?

eh?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> > > because 3:1 and 9:3 map to different numbers of "steps" in a
> chord?
> > i
> > > think that was similar to george secor's reasoning . . .
> >
> > My reasoning is simply that trying to define one leads to an
> > inconsistent system of linear equaltions.
>
> are the two facts related?

The first fact implies the second fact.

> > > why don't we switch from standard vals to "lowest consistent
> > badness"
> > > vals around here as a general rule? i think that would be a
> > positive
> > > development . . .
> >
> > Do you have a name?
>
> eh?

I was thinking of something like "optimal (p-limit) val", but was
wondering if you wanted to name it. It's harder to compute, but
obviously a good deal more significant.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> > > > why don't we switch from standard vals to "lowest consistent
> > > badness"
> > > > vals around here as a general rule? i think that would be a
> > > positive
> > > > development . . .
> > >
> > > Do you have a name?
> >
> > eh?
>
> I was thinking of something like "optimal (p-limit) val", but was
> wondering if you wanted to name it.

we *are* talking about ETs at this point, aren't we?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> > I was thinking of something like "optimal (p-limit) val", but was
> > wondering if you wanted to name it.
>
> we *are* talking about ETs at this point, aren't we?

Indeed we are.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > > I was thinking of something like "optimal (p-limit) val", but
was
> > > wondering if you wanted to name it.
> >
> > we *are* talking about ETs at this point, aren't we?
>
> Indeed we are.

so how about "mapping" instead of "val" with the implication
(preferably stated along with n) that we are talking about ET.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> so how about "mapping" instead of "val" with the implication
> (preferably stated along with n) that we are talking about ET.

I don't see the point. What about optimal et?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > so how about "mapping" instead of "val" with the implication
> > (preferably stated along with n) that we are talking about ET.
>
> I don't see the point. What about optimal et?

we're talking about how to optimally map primes to a given et, right?

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> > I don't see the point. What about optimal et?
>
> we're talking about how to optimally map primes to a given et,
right?

I don't count it as an et unless it has a mapping; anyway "optimal
val" is shorter and sweeter.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > > I don't see the point. What about optimal et?
> >
> > we're talking about how to optimally map primes to a given et,
> right?
>
> I don't count it as an et unless it has a mapping; anyway "optimal
> val" is shorter and sweeter.

ok.