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Message: 8075 Date: Tue, 11 Nov 2003 09:58:54 Subject: Re: Enharmonic diesis? From: Graham Breed Dave Keenan wrote:

> But the question remains: Why did he call something as small as 29.6 c > a diesis at all. Why not a comma? Particularly given the history of > the term "diesis" as first applying to something as large as 90.2 c > and later approximate quartertones and fifthtones. 3125:3072 is more > like a seventhtone.

Presumably because it maps to 19 fifths in meantone, making it dimensionally equivalent to other dieses. A comma, by this reckoning, would be tempered out in meantone. Graham

Message: 8076 Date: Tue, 11 Nov 2003 03:44:44 Subject: Re: Enharmonic diesis? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> > simply using a comma between *every* prime, and no spaces. > > i suppose i like that. (i don't sound too convinced, tho.)

> > I think mathematicians name things like that all the time. Don't

they

> Gene?

They name things all the time, but usually abstract things, not notations. Those have a tendency to change by authorial whim.

> I'm afraid I don't understand them either. I couldn't even tell you > the difference between a wedgie and a val. :-) I read Gene's "val" > definition in your dictionary but I'm none the wiser. I don't see an > entry for "wedgie". I guess some time I'd like to see some carefully > explained examples of how vals and wedgies are used.

My web site seems to have a problem though the site itself is up, but I assume it will be working soon. If so, you might take a look at what's already there, and suggest what you think ought to be there.

Message: 8077 Date: Tue, 11 Nov 2003 10:00:24 Subject: Re: Vals? From: Graham Breed Dave Keenan wrote:

> So can that obscure definition of "val" on Monz's site now be replaced > with "ET-prime-mapping", which can in turn be defined as > "a list of the numbers of steps approximating each prime number in > some ET", and proceeding to show how to use one to calculate the > number of steps corresponding to some ratio in some ET?

AIUI, the only difference is that vals don't have to describe any temperament, so there one step of abstraction beyond an ET. Graham

Message: 8078 Date: Tue, 11 Nov 2003 04:03:07 Subject: Definition of microtemperament From: Dave Keenan Monz, I just noticed that your definition of microtemperament is quite wrong. Sorry I didn't pick this up sooner. Definitions of tuning terms: microtemperament,... * [with cont.] (Wayb.) What you've actually defined here is "planar temperament". Here's my definition of "microtemperament". A microtemperament is a temperament where the consonances sound justly intoned to most listeners in ordinary musical use. The allowed errors in the approximated ratios are therefore somewhat context-dependent but would always be less than 3 cents. A JI scale might be microtempered to increase the number of available consonances or to regularise the scale for some purpose such as allowing more full-width continuous frets on a stringed instrument. Microtemperament may also be used to introduce deliberate slight mistunings to avoid phase-locking when a JI scale is implemented on an electronic instrument. A microtemperament may be equal, linear, planar or of any dimension less than that of the JI scale being approximated.

Message: 8079 Date: Tue, 11 Nov 2003 21:41:31 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> Couldn't I (in fact didn't I) just define an (unqualified) ET-mapping > in exactly the same way?

Go ahead and do so, however a val is not necessarily an et-mapping of any kind. And then a "p-limit ET-mapping" would be a

> restricted one. > > So they're exactly the same!!!!

I havn't seen your definition.

> So why have we been calling them "vals" all this time?

Because there wasn't a good word for "finitely generated homomorphic mapping from Q+ to Z" or "Z-linear combination of padic valuations" already in existence. A mathworld

> search on the term finds nothing. Did you invent the term?

You bet. We needed a term for it, and there wasn't one. Is it

> merely an obscure synonym for "homomorphism", or "group homomorphism"? > The fact that it _is_ a group homomorphism is far from being its most > important characteristic as far as microtonality is concerned. The > fact that it maps ratios to steps of ETs is of far more interest.

A val *does not necessarily* map ratios to steps of an ET, but it *is* always a homomorphism. If you insist, you could replace the term with "finitely generated homomorphic mapping from Q+ to Z", I suppose, but I imagine "a p-limit homomorphic mapping to the integers" or something like that would suit you better.

> Even if the term is in common use in some area of mathematics, I don't > understand why you have been using it all this time when the perfectly > transparent terms ET-mapping, or ET-prime-mapping would have served us > far better.

It most certainly would not have served me better. You do what you like, but please don't expect me to follow your lead.

Message: 8080 Date: Tue, 11 Nov 2003 04:05:27 Subject: Re: Enharmonic diesis? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

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

> > > Maybe a good use for angle-brackets would be for wedgies > > > since, as I understand it, these are in a _very_ different > > > domain from that of monzos, and the angle-brackets are > > > suggestive of wedges themselves.

> > > > > > great minds thinking alike! ;-)

> > I see no point in it myself; one is hardly likely to confuse a wedgie > with an interval, though telling an 11-limit linear wedgie from an 11- > limit planar wedgie is another matter. There are also vals to > consider; it's too hard writing these as column vectors, which is > really the clearest way to do it.

I admit to being shamefully ignorant of these things. It sounds then like vals could use the angle-brackets.

Message: 8081 Date: Tue, 11 Nov 2003 22:27:30 Subject: Re: Eponyms From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>

wrote:

> > --- In tuning-math@xxxxxxxxxxx.xxxx "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.

Message: 8082 Date: Tue, 11 Nov 2003 04:11:18 Subject: Re: Enharmonic diesis? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

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

> > > simply using a comma between *every* prime, and no spaces. > > > i suppose i like that. (i don't sound too convinced, tho.)

> > > > I think mathematicians name things like that all the time. Don't

> they

> > Gene?

> > They name things all the time, but usually abstract things, not > notations. Those have a tendency to change by authorial whim.

I mean they name things n,m-whatevers where n and m are natural number s or primes or something, with a comma between them and no spaces?

> > I'm afraid I don't understand them either. I couldn't even tell you > > the difference between a wedgie and a val. :-) I read Gene's "val" > > definition in your dictionary but I'm none the wiser. I don't see an > > entry for "wedgie". I guess some time I'd like to see some carefully > > explained examples of how vals and wedgies are used.

> > My web site seems to have a problem though the site itself is up, but > I assume it will be working soon. If so, you might take a look at > what's already there, and suggest what you think ought to be there.

The URL again?

Message: 8083 Date: Tue, 11 Nov 2003 22:35:54 Subject: Re: Linear temperament names? From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...>

wrote:

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

> > > I'm suspecting that blind faith in log-flat badness measures

may

> > > have something to do with this.

> > > > blind faith in log-flat badness and believing it's a useful

measure

> > for any purpose are two completely different issues.

> > That depends how I take the word "any" here. If I interpret it as > "some" then I agree. Sure, it's useful for some purposes, and I'm

very

> grateful to Gene for providing us with that tool.

yes, it's just what was needed to complement a series of 10x-zooms on the ET chart.

> > > Don't bother with a "you just never know" type of answer,

because if

> > > you're going to clutter an already cluttered (but very clever)

> > diagram

> > > with lines corresponding to them, and give them meaningless or

> > cryptic

> > > names and fill up a huge table (43 temperaments!),

> > > > going to? i already did! or do you think i'm going to add further > > clutter (since you said already cluttered . . .)?

> > I'm sorry. "Going to" was just a manner of speaking. I was

imagining a

> time before all the more complex or inaccurate temperaments were

added. at that time, there were lines temperaments with much more badness, and those have been removed.

> > monz made tables for ETs as large as 4296 so i wanted to show how > > those ETs were related to one another. perhaps the best solution

is

> > to make different versions of the table to go with the different

zoom

> > levels, so that it's never presented as that huge.

> > I don't think that would help. The question would still remain as to > why anyone would care that, for example, 4296-ET and 3125-ET are > related by a linear temperament in which the comma [71 -99, 37] > vanishes (or has any of its other given properties), since one can > come up with a linear temperament for _any_ pair of 5-limit ETs, and > you're clearly not proposing to list them all.

i don't think it hurts, though, given that if someone cares enough about 4296-ET or 3125-ET to zoom that far into the chart, they should be equally interested in linear temperaments of that level of complexity.

> > > The same goes for 5-limit linear temperaments having any error

> > greater

> > > than the one that uses the same neutral third generator as both

> > major

> > > and minor third. Why would anyone care about such so-called 5-

limit

> > > temperaments?

> Hmm. Do you really think people will appreciate that from the mere > inclusion of such a temperament on the diagram and in the table?

no, more text would be needed.

> > > In my opinion, the diagram (and table) would be greatly

improved if

> > > these did not appear at all,

> > > > they pretty much don't appear in the diagrams you would be

looking

> > at! it's only at the zoom levels that don't interest you anyway

where

> > you would see these lines.

> > Hmm. 7 of them appear on the default zoom:10 level and 13 of them on > the zoom:100 level. These are the levels of most interest to me,

and I

> suspect of most interest to most musicians.

ok, list these 7 and 13. why is the zoom 100 level of such interest to you?

> By the way, I've said it before, but I'll say it again: Those

diagrams

> really are marvelous. So much information at a glance. The idea of > making the numerals smaller as the numbers get bigger was

particularly

> brilliant.

there's more text that needs to be added to explain the scale-tree character of what you're seeing here, and how it implies all sorts of wonderful DE scales at a glance.

Message: 8084 Date: Tue, 11 Nov 2003 04:26:12 Subject: Vals? From: Dave Keenan Here's a simple question for anyone who thinks they can answer it. How is a val different from an ET-mapping? i.e. a list of the numbers of steps approximating each prime in some ET.

Message: 8085 Date: Tue, 11 Nov 2003 22:41:09 Subject: Re: Definition of microtemperament From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> Monz, > > I just noticed that your definition of microtemperament is quite > wrong. Sorry I didn't pick this up sooner. > > Definitions of tuning terms: microtemperament,... * [with cont.] (Wayb.) > > What you've actually defined here is "planar temperament". > > Here's my definition of "microtemperament". > > A microtemperament is a temperament where the consonances sound

justly

> intoned to most listeners in ordinary musical use. The allowed

errors

> in the approximated ratios are therefore somewhat context-dependent > but would always be less than 3 cents.

I would have said "would always be less than about 3 cents" or "... less than 3.5 cents" in order to include Miracle. Or don't you consider that a microtemperament, and if not, then what should we call it? --George

Message: 8086 Date: Tue, 11 Nov 2003 05:29:42 Subject: Re: Definition of microtemperament From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> Monz, > > I just noticed that your definition of microtemperament is quite > wrong. Sorry I didn't pick this up sooner.

It's been discussed before; I liked error less than a cent as the cutoff.

Message: 8087 Date: Tue, 11 Nov 2003 22:42:03 Subject: Re: A Riemann zeta peek at 75 equal From: Paul Erlich considering that it's not so great in the 5-limit, i was surprised to see it show up as so many of your low-geometric-badness 5-limit linear temperaments: tetracot, orwell, vavoom, *and* misty! --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> I've put a jpg of Z(x) for values of x near 75 in the Photos > directory. Here Z is the Riemann zeta function along the critical > line, twisted in the usual way to get a real function of a real > variable, and then rescaled by the factor ln(2)/2 pi in the

argument,

> so as to make it correspond to divisions of the octave. > > The interest of 75 is that there is a zero of the Reimann zeta > function near 75, and two reasonably large peaks, rather than the > usual one, near to it. The highest peak is the flat octaves peak, > which for a range of values around the peak corresponds to the > 19-limit val > > [75, 119, 174, 211, 260, 278, 307, 319] > > which has 7-limit comma basis > > [225/224, 1728/1715, 15625/15309] > > The other peak is the sharp octaves peak; it has 19-limit val > > [75, 119, 174, 210, 259, 277, 306, 318] > > and comma basis > > [686/675, 875/864, 5120/5103] > > The standard val, in case anyone cares, would be > > [75, 119, 174, 211, 259, 278, 307, 319] > > The three are already distinct in the 11-limit.

Message: 8088 Date: Tue, 11 Nov 2003 05:42:22 Subject: Re: Enharmonic diesis? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> I admit to being shamefully ignorant of these things. It sounds then > like vals could use the angle-brackets.

Physicists sometimes use bra and ket vectors; if you did that, the monzo for 81/80 would be a ket, [-4 4 -1> and the val for 5-limit 12- et would be a bra, <12 19 28]. Putting them together would give the bra-ket, angle bracket, or inner product: <12 19 28 | -4 4 -1> = 0. See: Contravariant Vector -- from MathWorld * [with cont.] Covariant Vector -- from MathWorld * [with cont.] Ket -- from MathWorld * [with cont.] Bra -- from MathWorld * [with cont.] One-Form -- from MathWorld * [with cont.] Angle Bracket -- from MathWorld * [with cont.]

Message: 8089 Date: Tue, 11 Nov 2003 22:43:52 Subject: Re: Eponyms From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote:

> > --- In tuning-math@xxxxxxxxxxx.xxxx "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?

Message: 8090 Date: Tue, 11 Nov 2003 05:45:35 Subject: Re: Vals? From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote:

> Here's a simple question for anyone who thinks they can answer it. > > How is a val different from an ET-mapping? i.e. a list of the

numbers

> of steps approximating each prime in some ET.

You can identify them; however a val, as I defined it, would define a homomorphic mapping from every positive rational number to an associated integer. Usually you would restrict this to a given prime limit.

Message: 8091 Date: Tue, 11 Nov 2003 22:50:18 Subject: Re: Vals? From: Paul Erlich in fairness to gene, i believe he derived it from the term "valuation" which is an entire field of mathematics. --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

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

wrote:

>

> > Couldn't I (in fact didn't I) just define an (unqualified) ET-

mapping

> > in exactly the same way?

> > Go ahead and do so, however a val is not necessarily an et-mapping

of

> any kind. > > And then a "p-limit ET-mapping" would be a

> > restricted one. > > > > So they're exactly the same!!!!

> > I havn't seen your definition. >

> > So why have we been calling them "vals" all this time?

> > Because there wasn't a good word for "finitely generated homomorphic > mapping from Q+ to Z" or "Z-linear combination of padic valuations" > already in existence. > > A mathworld

> > search on the term finds nothing. Did you invent the term?

> > You bet. We needed a term for it, and there wasn't one. > > Is it

> > merely an obscure synonym for "homomorphism", or "group

homomorphism"?

> > The fact that it _is_ a group homomorphism is far from being its

most

> > important characteristic as far as microtonality is concerned. The > > fact that it maps ratios to steps of ETs is of far more interest.

> > A val *does not necessarily* map ratios to steps of an ET, but it

*is*

> always a homomorphism. If you insist, you could replace the term

with

> "finitely generated homomorphic mapping from Q+ to Z", I suppose,

but

> I imagine "a p-limit homomorphic mapping to the integers" or

something

> like that would suit you better. >

> > Even if the term is in common use in some area of mathematics, I

don't

> > understand why you have been using it all this time when the

perfectly

> > transparent terms ET-mapping, or ET-prime-mapping would have

served us

> > far better.

> > It most certainly would not have served me better. You do what you > like, but please don't expect me to follow your lead.

Message: 8092 Date: Tue, 11 Nov 2003 07:34:38 Subject: Re: Vals? From: Dave Keenan Dave:

> > Here's a simple question for anyone who thinks they can answer it. > > > > How is a val different from an ET-mapping? i.e. a list of the > > numbers of steps approximating each prime in some ET.

Gene:

> You can identify them; however a val, as I defined it, would define a > homomorphic mapping from every positive rational number to an > associated integer. Usually you would restrict this to a given prime > limit.

Couldn't I (in fact didn't I) just define an (unqualified) ET-mapping in exactly the same way? And then a "p-limit ET-mapping" would be a restricted one. So they're exactly the same!!!! So why have we been calling them "vals" all this time? A mathworld search on the term finds nothing. Did you invent the term? Is it merely an obscure synonym for "homomorphism", or "group homomorphism"? The fact that it _is_ a group homomorphism is far from being its most important characteristic as far as microtonality is concerned. The fact that it maps ratios to steps of ETs is of far more interest. Even if the term is in common use in some area of mathematics, I don't understand why you have been using it all this time when the perfectly transparent terms ET-mapping, or ET-prime-mapping would have served us far better. I wouldn't be the only one who has failed to understand many of your posts over the years through the lack of this simple substitution. And I don't think I'll be the only person reading this who will be saying to themselves now, "Is that all a val is?. So can that obscure definition of "val" on Monz's site now be replaced with "ET-prime-mapping", which can in turn be defined as "a list of the numbers of steps approximating each prime number in some ET", and proceeding to show how to use one to calculate the number of steps corresponding to some ratio in some ET? Somehow I don't think the wedge-invariant or wedgie will be so easily disposed of. I'll have to reread that Grassman Algebra book some time. I remember a glimmer of understanding last time I did that, but it was a long time ago. Grassmann Algebra Book * [with cont.] (Wayb.)

Message: 8093 Date: Tue, 11 Nov 2003 22:55:43 Subject: Re: Definition of microtemperament From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "George D. Secor" <gdsecor@y...> wrote:

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

> > Monz, > > > > I just noticed that your definition of microtemperament is quite > > wrong. Sorry I didn't pick this up sooner. > > > > Definitions of tuning terms: microtemperament,... * [with cont.] (Wayb.) > > > > What you've actually defined here is "planar temperament". > > > > Here's my definition of "microtemperament". > > > > A microtemperament is a temperament where the consonances sound

> justly

> > intoned to most listeners in ordinary musical use. The allowed

> errors

> > in the approximated ratios are therefore somewhat context-

dependent

> > but would always be less than 3 cents.

> > I would have said "would always be less than about 3 cents" or "... > less than 3.5 cents" in order to include Miracle. Or don't you > consider that a microtemperament, and if not, then what should we > call it? > > --George

dave has said that miracle is a microtemperament in the 7-limit but not in the 9-limit or 11-limit.

Message: 8094 Date: Tue, 11 Nov 2003 23:15:03 Subject: Re: Eponyms From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "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.

Message: 8095 Date: Tue, 11 Nov 2003 23:16:07 Subject: Re: Eponyms From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "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?

Message: 8096 Date: Tue, 11 Nov 2003 23:26:54 Subject: Re: Eponyms From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "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.

Message: 8097 Date: Tue, 11 Nov 2003 23:33:30 Subject: Re: Eponyms From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "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.

Message: 8098 Date: Tue, 11 Nov 2003 00:29:31 Subject: Re: Enharmonic diesis? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

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

> > So who was the turkey who called 125:128 the "great" diesis?

> > probably the same person (not bird) who called 3125:3072 the "small" > diesis.

Yes. Monz already told me it was Ellis for both of these, and it's all beautifully described at Definitions of tuning terms: diesis, (c) 1998 ... * [with cont.] (Wayb.) I should have looked there first. You're right, Ellis was no turkey. But the question remains: Why did he call something as small as 29.6 c a diesis at all. Why not a comma? Particularly given the history of the term "diesis" as first applying to something as large as 90.2 c and later approximate quartertones and fifthtones. 3125:3072 is more like a seventhtone. In George's and my comma naming system it is called the 3125-comma or 5^5-comma, but it doesn't get its own sagittal symbol because it's too close to what we assume will be notationally-more-common; the 5:7-comma [-9 6, 1 -1] 29.22 c, whose symbol is, in ASCII longhand, '|) which literally represents a 5-schisma plus a 7-comma.

> > And is it correct to call it a diminished second? Isn't G# to Ab a > > _doubly_ diminished second?

> > no. G to Ab is a minor second, and G# to Ab is a diminished second.

Yes. Thanks. Monz already corrected me on that.

> > its tempered size in 1/3-comma > > meantone is _minus_ 62.565148 cents.

> > no, G# is lower than Ab in 1/3-comma meantone.

Thanks. I did get awfully confused about this didn't I.

Message: 8099 Date: Tue, 11 Nov 2003 01:57:29 Subject: Re: Linear temperament names? From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

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

> > I'm suspecting that blind faith in log-flat badness measures may > > have something to do with this.

> > blind faith in log-flat badness and believing it's a useful measure > for any purpose are two completely different issues.

That depends how I take the word "any" here. If I interpret it as "some" then I agree. Sure, it's useful for some purposes, and I'm very grateful to Gene for providing us with that tool.

> > Don't bother with a "you just never know" type of answer, because if > > you're going to clutter an already cluttered (but very clever)

> diagram

> > with lines corresponding to them, and give them meaningless or

> cryptic

> > names and fill up a huge table (43 temperaments!),

> > going to? i already did! or do you think i'm going to add further > clutter (since you said already cluttered . . .)?

I'm sorry. "Going to" was just a manner of speaking. I was imagining a time before all the more complex or inaccurate temperaments were added.

> > then you'd better > > do a lot better than that.

> > monz made tables for ETs as large as 4296 so i wanted to show how > those ETs were related to one another. perhaps the best solution is > to make different versions of the table to go with the different zoom > levels, so that it's never presented as that huge.

I don't think that would help. The question would still remain as to why anyone would care that, for example, 4296-ET and 3125-ET are related by a linear temperament in which the comma [71 -99, 37] vanishes (or has any of its other given properties), since one can come up with a linear temperament for _any_ pair of 5-limit ETs, and you're clearly not proposing to list them all.

> > The same goes for 5-limit linear temperaments having any error

> greater

> > than the one that uses the same neutral third generator as both

> major

> > and minor third. Why would anyone care about such so-called 5-limit > > temperaments?

> > for example, one can regard a given temperament as 'lame' in the > sense of not fully functioning, to help understand the structures > inherent in a distributionally even scale that is not equally > tempered. if nothing else, the temperament will signify that a > certain ratio is operating as a chromatic unison vector for > distributionally even scales of interest with cardinalities that > appear as numbers along the line representing the temperament.

Hmm. Do you really think people will appreciate that from the mere inclusion of such a temperament on the diagram and in the table? I'm still having trouble figuring out what you mean, from what you've written above. I suspect this is a bit subtle or esoteric as a reason for including them here.

> > In my opinion, the diagram (and table) would be greatly improved if > > these did not appear at all,

> > they pretty much don't appear in the diagrams you would be looking > at! it's only at the zoom levels that don't interest you anyway where > you would see these lines.

Hmm. 7 of them appear on the default zoom:10 level and 13 of them on the zoom:100 level. These are the levels of most interest to me, and I suspect of most interest to most musicians.

> > But if you really think these 25 should stay, then I'll take you up

> on

> > your offer and ask that you refer to them with single letters,

> except

> > where they are 5-limit subsets

> > you don't really mean subsets, but rather 'incarnations', right?

I suppose so. I mean where their prime-mapping matrix is a sub-matrix of the higher limit one that does deserve a name.

> > of higher-limit temperaments that _do_ > > deserve a name, in which case their names should be followed by > > "(5-limit)".

> > ok, if monz is on board we'll proceed to strip the personality out of > this page :). i have no qualms about doing so.

I guess you don't have any qualms, because you know that anyone who does would hold _me_ responsible, not you. You've out-bluffed me here. :-) I really don't think you had the right to make the offer you made (regarding replacing names with letters). I'm pretty sure that was the feeling behind Carl's "Nooooooooooooooooooooooo!". So I thought I'd call your bluff. But I've lost my nerve for this game, and I couldn't bear Carl's anguish if you actually went ahead with it. So please leave it as it is. I'm happy that you've at least heard me out. Actually, I'd be _really_ happy if you'd just change the name of the one you call "minortone" to something silly. Maybe Carl can suggest one? This is because I'd like to use the name "minortone" in the naming of an actually useful higher-limit temperament whose mapping is not a superset of that one. I've come to the conclusion that if you're going to give names to meaningless temperaments, they should be meaningless names, so as to leave the meaningful names to do real work elsewhere. By the way, I've said it before, but I'll say it again: Those diagrams really are marvelous. So much information at a glance. The idea of making the numerals smaller as the numbers get bigger was particularly brilliant. Regards, -- Dave Keenan

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