> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>"
> <d.keenan@u...> wrote:
>
>> The prime-exponent vector is only one specific mathematical
>> representation of these things. One could also argue that there is
>> really only one "unison vector" and that's [0 0 0 ...].
>
> in conventional theory, there's not only a "perfect unison" but also
> an "augmented unison", etc.
>
>> Now with Paul's term "commatic unison vector", which is contrasted
>> with "chromatic unison vector", we have a third sense of "comma".
>> Meaning that which vanishes (or is distributed so you don't notice
>> it). Could that be the original meaning of "comma"? No, it seems
> that
>> they were so named purely because of their small size (but not
>> undetectability).
>
> this assumes that just intonation was actually used in practice. i
> disagree. the commas were found in JI theory, not in musical practice.
>
That's interesting because it makes them kind of the opposite of an
"anomaly". I assume we're talking ancient Greeks here. But I don't see
that this affects my point, which is that commas are near-misses
(between ratios), but not necessarily near-misses that vanish. e.g.
the Pythagorean comma does not vanish in a Pythagorean scale.
>> "Commatic unison vector" translates to "commatic comma", which looks
>> like a redundancy.
>
>> What's wrong with "vanishing comma" vs. "chromatic comma" or
>> "distributed comma" versus "chromatic comma"?
>
> however, i have no
> problem with "vanishing comma" or "distributed comma".
Message: 6351 - Contents - Hide Contents
Date: Fri, 07 Feb 2003 00:05:34
Subject: Re: A common notation for JI and ETs
From: Dave Keenan
--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> actually, fokker only considered just intonated periodicity blocks,
> where none of the unison vectors vanish.
Then wouldn't it have been better if he had called them "period
vectors" or "periodicity vectors". Could a better translation do so
even now?
Message: 6352 - Contents - Hide Contents
Date: Fri, 07 Feb 2003 02:04:12
Subject: Re: A common notation for JI and ETs
From: Dave Keenan
--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>"
<gdsecor@y...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>"
> <d.keenan@u...> wrote:
>> Here's another data point relevant to the comma-name boundaries
>> discussion.
>>
>> 49:125 E-13.469 Fb-36.929
>>
>> 36.929 c must be notated as ~|) which should make it a comma.
>> Therefore 13.469 c ought to be a kleisma, as it would be with a
> 13.47
>> c boundary.
>
> I haven't had a chance to keep up with all of your latest boundary
> changes, but your method makes sense, and assuming you haven't make
> any miscalculations, what you have should be okay. (I'm dropping my
> proposal for a 120:121 boundary, because it serves no useful purpose,
> except to suggest a ballpark value.)
>
>> As for the names of the categories - how about
>>
>> hypodiesis
>> diesis
>> hyperdiesis
>>
>> Which can be abbreviated to
>>
>> odiesis
>> diesis
>> ediesis
>
> I think that the idea is good, but I have two suggestions.
>
> 1) I was thinking of having three different prefixes (such as mini,
> midi, and maxi, although probably not those three) and
> allowing "diesis" to remain a more general term that would cover all
> three.
I was thinking that would be good too. emdiasis?
Mini- midi- maxi- are not Greek, but then we've already use -ina which
is not Greek. Incidentally -ina really means female rather than small,
but it still has the correct implication.
Sub-, super-, infra-, ultra- are not Greek either.
> 2) The three prefixes should be more than just a single letter --
> odiesis and ediesis sound too much alike.
Could be odiesis (pron. oadiesis) and erdiesis.
> I was about to suggest three prefixes used in organic chemistry:
> ortho, meta, and para. The original 125:128 diesis would then
> appropriately be termed an orthodiesis. The abbreviations o-diesis,
> m-diesis, and p-diesis would even work.
But I thought you objected to single letters?
Unfortunately these three prefixes do not generally represent 3
degrees of a single property and I fear that few musicians would be
familiar with the naming of benzene derivatives. Meta- would generally
be more appropriate for the largest, as you suggest below.
> Unfortunately, the
> particular dieses that we're using the o and m characters for are
> both in the para category.
I wouldn't place too much importance on this. But I note that in the
three categories we have these symbols.
small dq /|~ (|( ~|\ //| |~)
middle unv /|) (|~ /|\ |)) (/|
large owm |\) (|) (|\
But I find there is not much hope of making our prefixes match up with
any of these, except possibly in the large category.
> Perhaps we could use meta for the largest
> group (the meaning, "beyond," would still apply) and find a couple of
> other prefixes that wouldn't conflict with (and might even tie in
> with) the letters q and n for the small and middle ranges.
Good luck!
There aren't very many prefixes starting with q. The only one that is
even slightly appropriate is "quasi-" but that means "almost but not
quite" and would be better used for those things that have
historically been called dieses but are smaller than 36.93 cents.
"meso-" is _the_ Greek prefix meaning middle. It is used with various
other Greek pairs such as:
hypo- under
meso- middle
hyper- over
endo- inside
meso- middle
ecto- outside
proto- (or pro-) earlier or to the front
meso- middle
meta- later or to the rear
lepto- fine small thin delicate
meso- middle
hadro- thick stout
Message: 6353 - Contents - Hide Contents
Date: Fri, 07 Feb 2003 02:28:02
Subject: Re: A common notation for JI and ETs
From: Dave Keenan
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
>
>> But you can't force all commas to vanish. So shall we count
>> you as vote #2 for "vanishing comma"?
>
> What I say is "commas of temperament T", meaning those which belong
to the kernal of T, or vanish.
You can get away with this if you put it in a context where you've
previously explained what you mean. But generally I find it
problematic. It would seem more correct to say that T is a temperament
of certain commas.
In a sense, the 5-comma 80:81 is not a comma _of_ meantone because you
don't notice it's existence when composing or playing in meantone, but
you _do_ notice it in something like 22-ET because it becomes a step.
If I were to ask George in our notation discussion [hey I managed to
make this message have something to do with its subject heading] what
are the commas of some ET, he might well assume I mean, "Which ones
can we use to notate it?", and therefore "Which ones _don't_ vanish?".
Message: 6354 - Contents - Hide Contents
Date: Fri, 07 Feb 2003 14:19:03
Subject: Re: A common notation for JI and ETs
From: David C Keenan
A somewhat belated response.
At 10:13 PM 24/01/2003 +0000, you wrote:
>--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>"
><gdsecor@y...> wrote:
>--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...>
>wrote [#5565]:
>> George,
>>
>> To help with the development of the outline font, could you put up
>a bitmap
>> or gif with all the _up_ symbols we've found a need for so far,
>showing (in
>> the single-shaft case) the keyboard character that you want them
>mapped to.
>> This would be an update of your Symbols3.bmp which is the latest I
>can find
>> on the tuning-math files area. Don't include any combinations with
>5'
>> accents, just one with the acute accent beside a plain shaft.
>
>Here's the new version of Symbols:
>
>
Yahoo groups: /tuning- * [with cont.]
>math/files/secor/notation/Symbols6.gif
Thanks.
>As you can see by the filename, I've done quite a bit of work with
>symbols since the last one was posted.
Yes!
>I have 30 different single-shaft symbols there in the main portion of
>the figure -- no three-flaggers and no 5' comma alterations. (Some 5'
>comma symbols are at the very bottom.) As you noted, below, we don't
>need /|(, but I already have it in the graphic, and if I remove it
>from there now, then we'll probably change our minds for some reason
>and will want it back. :-) So I'm leaving it there for the time
>being, even though I don't think we'll ever use it.
OK.
> However, recall
>that /||( was the apotome complement of ~|~, and ~||~ the complement
>of /|(. So it looks as if eliminating /|( would also require
>eliminating ~|~.
Good point.
>Are we using ~|~ for anything?
>It was formerly the 5:19 comma,
>but )/|, which is exact, has replaced it for that purpose.
Has it. I don't remember agreeing to )/|. Does it have any other uses or
reasons for existence?
> We also
>agreed to use ~|~ for 7deg342 and 8deg388 (which decision still
>stands), but we could also replace it with )/| for both of those.
Which ever symbol we use, ~|~ or )/|, it should probably represent both
5:19 and one of 5:77 or 77. I guess no symbol can represent both 24.88 and
26.01 since they are more than a half 5-schisma apart. If it were to
represent 5:19 and 5:77 then it couldn't be ~|~ because 25.43 (the sum of
flag values) is more than half a 5-schisma from 24.06. I guess that means
it should be )/| but I really don't like these two-flags-on-same-side
symbols, except //|
>This would then eliminate both /|( and ~|~ from the notation. So
>that brings me down to 28 symbols.
But what would be the complement of )/| ? Wouldn't it still need to be /|( ?
>There's one that I tried changing from what you had: ~)| -- 5th from
>left at the very top. Near the bottom right I have an area
>labeled "experimental", where I have three versions of this symbol,
>and I chose the one in which I thought that the separate flags could
>most easily be identified
I don't think it is necessary for the separate flags to be easily
identified if it impacts on a consistent style across all symbols.
> (if you agree; however you propose to
>replace this, which I'll answer below).
>
>Here is what I now have for a keyboard layout (just tentative, easily
>subject to change, except for the top row). The most common symbols
>are at the top (standard 217 symbols in top row, plus 5' comma at far
>left), less common going downward; nothing is assigned to the bottom
>row, so there's plenty of room left, should we need it. Degrees of
>217 and 494 are given to help in establishing a reasonable
>progression by size:
>
> ` 1 2 3 4 5 6 7 8 9 0 - = key
>'| |( ~| ~|( /| |) |\ (|( //| /|) /|\ (|) (|\ symbol
> 1 1 2 3 4 5 6 7 8 9 10 11 12 deg217
> 1 2 4 6 9 11 13 16 18 20 22 25 27 deg494
>
> Q W E R T Y U I O P [ ] \
> )| )|( |~ )|~ )|) (| ~|) |~) (|~ (/| |\)
> 1 2 3 4 6 6 7 8 9 10 11
> 1 3 7 9 12 14 15 19 21 23 24
>
> A S D F G H J K L ; '
> ~)| )/| ~|~ )|\ ~|\ |))
> 2 5 5 7 8 10
> 5 10 11 14 17 22
>
> Z X C V B N M , . /
>
>
>> I'm thinking that we no longer need any 3-flag symbols (other than
>those
>> where the third "flag" is a 5' accent)
>
>The only two that I ever seriously considered are ~|() as the 7:17
>comma (see discussion below) and )|)) as ~1/2-apotome for 15deg311
>and 19deg400 (which would be its own complement), but the latter may
>be replaced with '|)), 392:405, which is also self-complementing.
'|)) certainly has a more popular rational meaning [5:49] than )|))
[19*49]. But why not use '(/| ?
>> and the only
>> 2-flags-on-the-same-side symbols that we need are //| and |)) and
>|\),
>> although I'm still wondering whether you think we'd be better off
>keeping
>> (/| instead of |)), since ||\) seems to be the only choice for its
>complement.
>
>I am very impressed by how well (/| and |\) work as complements, so I
>would hesitate to dump one of them. One problem is that (/| doesn't
>work in very many ETs as the 7^2 diesis (not valid in 270, 306, 311,
>342, 364, or 494) where you might want to map this for JI; for all of
>these /|\ has the right number of degrees, but is almost 1.3 cents
>off. (/| works in 388, but 388 isn't 1,7,49-consistent. So (/| is
>fine for as the 49 diesis for JI, but it wouldn't be usable in most
>ETs.
I see the problem.
>You would still need |)) in the font for '|)) for 19deg400, and if
>you have it there you might as well use it for 15deg311 and 29deg612
>(see below). It also occurs in the hemififth notation I gave in msg.
>#5387.
OK. I guess we need (/| _and_ |)). Sigh.
>> We could replace
>>
>> ~)| 17:19 comma
>> and
>> |~) 13:19 comma
>>
>> which are way down the popularity list anyway (Nos 71 and 45,
>Ocurrences
>> 0.08% and 0.15%),
>>
>> with
>>
>> ')|( 17:19 comma
>
>I think not. The ~)| symbol is exact for the 17:19 comma and will
>therefore be valid in any ET that's 19-limit consistent. I think you
>need a really good reason to prefer something having a 5' comma over
>something without it that's exact. Besides, you would be removing a
>flag combination for apotome-complement pairs, ~)|| with ~|\ and ~)|
>with ~||\.
OK. ~)| stays.
>> and
>> '//| 13:19 comma
>
>This one you can make a better case for, because it works in most of
>the good divisions above 270 and the size is almost right on. It
>also solves the problem of bad symbol arithmetic using |~) for
>19deg494. So I'll agree with '//| for the 13:19 diesis.
OK. Well at least I got rid of one. :-)
>But this doesn't necessarily eliminate the |~) symbol -- it presently
>has )||( as its apotome complement, and conversely )|( has ||~) as
>its apotome complement. It has also been proposed for use as
>11deg306, 13deg342, and 13deg364 (although accented symbols could now
>be used for these). It would not be good to have a double-shaft
>symbol in the notation without a single-shaft version of it.
Oh dear. So I didn't get rid of any yet! :-(
>> Whaddya think?
>
>I done thunk!
>
>> I understand we will never have a use for /|( since it is a synonym
>for |)
>
>I would tend to agree.
>
>> and we don't yet have a use for )|\ which is very close to ~|) and
>could be
>> replaced by '(|. '(| also has no known use so far, but we get it
>for free.
>
>I was using )|\ for 10deg364 (there was no other option) and also for
>the 31 comma, 243:243, but '(| will do very nicely for the 31 comma,
>while '|\ will take care of 10deg364. Since )|\ has no rational
>complement assigned, there is no problem with eliminating this
>symbol. So I agree.
Hoorah.
>> So I count 26 single-shaft up symbols in all.
>> I'm thinking we may need to
>> revisit the apotome-complement issue again, with this symbol
>> reorganisation.
>
>Which I addressed as I went along. If we keep ~)| and also have |))
>around for use as '|)), then I think that's 27 symbols, but you'd
>better check that.
Still gotta decide re )/| and /|( complements.
>> I don't think that an un-accented symbol should ever have a
>> complement that is accented or vice versa. Is this possible?
>
>Yes, I think that will work. An easy-to-use rule for 5' symbol
>complements that would follow logically from this is: if a|b and
>c||d are are apotome complements, then 'a|b and .c||d should also be,
>and also .a|b and 'c||d. The principle that applies here is that
>flags in the second half-apotome (i.e., those used for double-shaft
>symbols) are arrived at according to the definition of apotome
>complements, such that:
>a|b equals /||\ minus c||d, where /||\ is the apotome,
>and it does not necessarily follow in the notation for any particular
>ET that
>a||b equals (|) plus a|b,
>only that this is highly desirable, so that symbol arithmetic in the
>second half-apotome is usually consistent (and never very obvious if
>it isn't). To ensure that this is generally true, we have defined
>apotome complements with very small offsets, i.e., a|b ~ /||\ minus
>c||d.
>I don't know whether the foregoing has been explicity stated in our
>previous discussions, but I thought that this would be a good time to
>do it.
Sounds good. Do you have a spreadsheet that shows the complements and their
offsets?
>I don't think we need to be overly concerned about whether
>interlacing of complements is strictly maintained with the addition
>of the 5' rule,
No.
>although it would be interesting to see just how many
>exceptions there are.
Yes.
>> It might be a
>> good idea to try notating 612-ET and 624-ET before settling this.
>
>Okay. Here's a go at it:
>
>612: '| )| |( '|( )|( ~)| .~|( ~|( '~|( ./| /| '/| .|)
>|) '|) |\ (| '(| .(|( (|
>( .//| //| '//| ./|) /|) ./|\ /|\ (/| '|)) |\) (|) '(|)
>(|\
>
>624: '| )| |( '|( )|( ~)| .~|( ~|( '~|( ./| /| '/| .|)
>|) '|) .|\ |\ '|\ .(|( (|
>( .//| //| '//| ./|) /|) '/|) ./|\ /|\ '/|\ .(|)
>(|) '(|) .(|\ (|\
>
>And maybe we should do a few others besides.
Eek! I just can't face these at present. I was hoping there'd be some
memorable pattern like every third degree is notated without 5-schisma
accent and then there's one up and one down from that.
>> By the way, in your otherwise excellent quick reference, I must
>object to
>> the line
>> 7:17 diesis 448:459 ~41.995c (for 217 mapping)
>>
>> A correct symbol for this diesis would be either .~|\ or (not
>quite) '//|
>
>This one continues to be a problem (also, this was your turn to get
>the 5' symbols mixed up).
So I did. :-)
> '~|\ is just the right size, but it's not
>valid in most of the best 17-limit consistent ETs: 217, 311, 494; the
>only important one that handles it is 388.
But if it isn't _needed_ for notating any of these then who cares?
> .//| is around a cent
>off, but it works in 311 and 494, but not in 217 or 388. I hesitate
>to use a symbol containing 5' that isn't even valid in half of the
>ETs into you might want to map JI.
Forget .//|
> With ~|() we have no problem,
>because it's exact and therefore valid everywhere. And I think the
>symbol looks pretty good -- see the experimental section of my latest
>file: third column of symbols from the right. And if this is the ony
>non-5'-three-flagger that we allow, then nobody will be able confuse
>it with anything else.
>
>However, if we adopt ~|(), then there is the problem of an apotome
>complement for it, and I really can't see a good choice for that -- )
>||( is the only thing that's valid in 494, and it's not valid in 217,
>270, 311, or 388 -- plus the fact that we would then need to find a
>complement for ~|\. So this just opens up a can of worms, in
>addition to the issue of a 3-flagger.
>
>Okay, I'm convinced that we should eliminate ~|() from consideration,
>so there will be no unaccented 3-flag symbols in the notation.
Phew!
> For
>the 7:17 diesis symbol we must then make a choice between '~|\
>and .//|.
>
>Points in favor of '~|\:
>1) It's almost exactly the right size (as the 23' comma +5' comma)
>2) It's less than 1/2 cent from the right size (as the sum of the 3
>flags)
>3) It contains a 17-comma flag (as a memory aid)
>Points against:
>1) It's not valid in 217, 311, or 494 (but is in 388)
>
>Points in favor of .//|:
>1) It's valid in both 311 and 494
>2) It's less than a cent from the proper size
>3) It uses a more familiar symbol (same as for the 125-diesis)
>Points against:
>1) It's not valid 217 or 388
>2) It's nearly a cent from the proper size
>
>Almost looks like a tossup, so I think we should look at this purely
>from a JI perspective and use the one that's closest in size --
>therefore '~|\ gets my vote for the 7:17 diesis.
Agreed!
Please let me know when you've update the quick reference with the new
single-ASCII symbols as well as this.
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page * [with cont.] (Wayb.)
Message: 6355 - Contents - Hide Contents
Date: Sat, 08 Feb 2003 20:31:35
Subject: Re: A common notation for JI and ETs
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
>
>> how about "commatic unison"?
>
> Ick. If threatened, I'll pull inside my shell and start
saying "kernel elements".
"kernel elements" gets my vote whole-heartedly!!
(just get monz to fix his "metric" discussion . . .)
Message: 6356 - Contents - Hide Contents
Date: Sat, 08 Feb 2003 20:42:11
Subject: Re: A common notation for JI and ETs
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>"
<d.keenan@u...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
> <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
>> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>"
>> >
>>> The prime-exponent vector is only one specific mathematical
>>> representation of these things. One could also argue that there
is
>>> really only one "unison vector" and that's [0 0 0 ...].
>>
>> in conventional theory, there's not only a "perfect unison" but
also
>> an "augmented unison", etc.
>>
>>> Now with Paul's term "commatic unison vector", which is
contrasted
>>> with "chromatic unison vector", we have a third sense
of "comma".
>>> Meaning that which vanishes (or is distributed so you don't
notice
>>> it). Could that be the original meaning of "comma"? No, it
seems
>> that
>>> they were so named purely because of their small size (but not
>>> undetectability).
>>
>> this assumes that just intonation was actually used in practice.
i
>> disagree. the commas were found in JI theory, not in musical
practice.
>>
>
> That's interesting because it makes them kind of the opposite of an
> "anomaly". I assume we're talking ancient Greeks here.
really? what ancient greek source(s) did you have in mind?
> But I don't see
> that this affects my point, which is that commas are near-misses
> (between ratios), but not necessarily near-misses that vanish. e.g.
> the Pythagorean comma does not vanish in a Pythagorean scale.
right . . . but the words "commatic" and "chromatic" (paul hahn's
terms), when used in conjunction with "unison vector", conjure up an
analogy with the diatonic case -- their function there is explored in
my pamphlet _the forms of tonality_ . . .
>
>>> "Commatic unison vector" translates to "commatic comma", which
looks
>>> like a redundancy.
>>
>> how do you make that translation?
>
> unison vector = small interval = comma
unison vector means something different than "small interval".
> But now that I think about it it seems to me that the "unison" in
> "unison vector" could imply that it vanishes in the temperament
under
> discussion, so "chromatic unison vector" would be a contradiction.
nope -- see "augmented unison_ above.
> Sure there are such things as augmented unisons, but when
unqualified,
> unison is usually taken to mean 1:1. After all, "augmented unison"
> means "a unison with something added".
but it's still a unison.
>
>>> What's wrong with "vanishing comma" vs. "chromatic comma" or
>>> "distributed comma" versus "chromatic comma"?
>>
>> "chromatic comma" seems like a contradiction.
>
> I agree that those used for chromatic purposes are usually larger
and
> tend to be called "limma", "apotome" or "semitone" rather than
comma.
> Is that the basis of the contradiction you see, or is it something
> else? If so, what? I don't see the above preventing us from using
> "comma" as a generic term that covers these too.
ok -- too few words, too many meanings. sorry paul hahn.
> Does anyone see a problem with using "chroma" as a synonym for
> . . . "chromatic unison vector".
i've used it that way . . . but then of course we have the problem
that "chroma" is also used for a certain small set of fixed ratios.
> For a given a linear temperament let N be the cardinality of the
> proper MOS whose cardinality is closest to 8.25 (personal
> rule-of-thumb). I'd like to call any small interval that corresponds
> to a chain of N generators (according to the temperaments prime
> mapping), a chroma. Does this seem OK?
wow . . . sounds compelling . . . you'd exclude the pentatonic
meantone and the blackjack miracle . . . ?
Message: 6357 - Contents - Hide Contents
Date: Sat, 08 Feb 2003 20:43:05
Subject: Re: A common notation for JI and ETs
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>"
<d.keenan@u...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
> <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
>> actually, fokker only considered just intonated periodicity
blocks,
>> where none of the unison vectors vanish.
>
> Then wouldn't it have been better if he had called them "period
> vectors" or "periodicity vectors". Could a better translation do so
> even now?
maybe so. manuel?
Message: 6358 - Contents - Hide Contents
Date: Sat, 08 Feb 2003 20:46:06
Subject: Re: A common notation for JI and ETs
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>"
<d.keenan@u...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith
> <genewardsmith@j...>" <genewardsmith@j...> wrote:
>> --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <clumma@y...>"
> <clumma@y...> wrote:
>>
>>> But you can't force all commas to vanish. So shall we count
>>> you as vote #2 for "vanishing comma"?
>>
>> What I say is "commas of temperament T", meaning those which
belong
> to the kernal of T, or vanish.
>
> You can get away with this if you put it in a context where you've
> previously explained what you mean. But generally I find it
> problematic. It would seem more correct to say that T is a
temperament
> of certain commas.
>
> In a sense, the 5-comma 80:81 is not a comma _of_ meantone because
you
> don't notice it's existence when composing or playing in meantone,
but
> you _do_ notice it in something like 22-ET because it becomes a
step.
in a sense, 80:81, as a vector in the just lattice, is one of the
most noticeable things in meantone, since it takes you back to where
you started!
> If I were to ask George in our notation discussion [hey I managed to
> make this message have something to do with its subject heading]
what
> are the commas of some ET, he might well assume I mean, "Which ones
> can we use to notate it?", and therefore "Which ones _don't_
>vanish?".
2401:2400 doesn't vanish in 12-equal . . .
Message: 6359 - Contents - Hide Contents
Date: Sat, 08 Feb 2003 09:37:17
Subject: Re: partial-specific theory
From: Graham Breed
Carl:
> Wow, Graham, what have you got up your sleeve? Do I
> understand correctly that you're doing away with the
> 'extraction of fundamental' abstraction that we've been
> relying on here since the dawn of time? Do we really
> have the tools to, and would there be any benefit from,
> consider all the partials all of the time?
Me:
> I'm not sure what you're talking about there, but I don't
> think it applies to me.
Carl:
> What were you writing about ratios being insufficient?
Yes, ratios can be done away with, but what were you writing about
'extraction of the fundamental'?
Me:
> "Intervals are defined as vectors in terms of a minimal subset
> of the partials relative to the fundamental (which, for inharmonic
> timbres, will probably be the whole set)."
Oh, I can see that might cause confusion. I didn't mean the whole set
of partials in the timbre will be used in calculations. But each
partial you do consider (except the fundamental) will be a dimension of
the vector space. That isn't the case with harmonic timbres -- the 9th
partial is represented as twice the 3rd partial, for example.
Graham
Message: 6360 - Contents - Hide Contents
Date: Sat, 08 Feb 2003 10:30:03
Subject: Re: partial-specific theory
From: Carl Lumma
>> >Intervals are defined as vectors in terms of a minimal subset
>> of the partials relative to the fundamental (which, for
>> inharmonic timbres, will probably be the whole set)."
>
>Oh, I can see that might cause confusion. I didn't mean the
>whole set of partials in the timbre will be used in calculations.
>But each partial you do consider (except the fundamental) will
>be a dimension of the vector space. That isn't the case with
>harmonic timbres -- the 9th partial is represented as twice the
>3rd partial, for example.
Aha!
-Carl
Message: 6361 - Contents - Hide Contents
Date: Sun, 09 Feb 2003 12:18:27
Subject: Re: Janata paper
From: Graham Breed
Carl Lumma wrote:
I've been hit with a lot of reading the past few days. Has anybody else
been looking at this:
Pitch Schemata * [with cont.] (Wayb.)
> This paper has been the subject of some debate recently,
> on the SpecMus and PsyMus lists.
What goes on on SpecMus? It's members only and closed archives. The
nearest I can find to PsyMus is psycho_musicians, which is much quieter.
Also closed archives.
> Also, 'quasi-tempered meantone' is ambiguous. What the
> paper found is: each triad in 12-tET is associated with
> a region of maximum activity in the brain. The particular
> area for each triad differs between listeners and between
> sessions for a given listener, but the distances between
> the areas is always related to the number of common pitches
> between the diatonic keys rooted on them. For example,
> A major and F# minor have the same key signature, and AMaj
> and F#min chords activate the same regions in the brain of
> a given person on a given day. Maybe one of our temperament
> gurus can tell us what temperaments(s) this 7-of-12 setup
> represents...where the pythagorean, syntonic, and augmented
> commas vanish and the 25:24 is chromatic?
12-tET
Graham
Message: 6362 - Contents - Hide Contents
Date: Sun, 09 Feb 2003 03:42:44
Subject: That poor overloaded word "comma"
From: Dave Keenan
--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith
> <genewardsmith@j...>" <genewardsmith@j...> wrote:
>> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
> <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
>>
>>> how about "commatic unison"?
>>
>> Ick. If threatened, I'll pull inside my shell and start
> saying "kernel elements".
>
> "kernel elements" gets my vote whole-heartedly!!
>
> (just get monz to fix his "metric" discussion . . .)
Please, no. Not "kernel elements". I can take anything but that.
Bamboo under the fingernails, "commatic unison vectors", anything.
I've seen grown men bite their own heads off rather than use "kernel
elements".
Seriously.
I think we're mostly agreed that we ought not use "comma" or
"commatic" to mean "vanishing" if we can help it. Sorry Paul Hahn. But
the concepts described in Paul Erlich's 'The Forms of Tonality' are
very important ones, and there is a certain poetry in "commatic"
versus "chromatic" "unison vectors" that I am loath to lose. However I
think we can have our cake and eat it too.
Firstly it seems like "period vectors" would be a more obvious term
for the vectors bounding a periodicity block, and there would be no
need to appeal to augmented unisons in justifying its use. Are there
cases where such a bounding vector corresponds to a half-octave (or
other period which is an integral fraction of the interval of
equivalence)? If so, this would also resolve the problem of using
"unison vector" to refer to such a thing.
Now we have the problem of preserving the poetry of "commatic" versus
"chromatic".
"vanishing" certainly says what we mean quite clearly, and
"evanescent" is a nice sounding synonym, but we really would like a
word ending in "-ic". And here it is. "achromatic".
So we would have chromatic and achromatic period vectors, and we would
also be free to speak of the corresponding musical objects as
chromatic and achromatic commas. But if you're not concerned with
contrasting it against chromatic, "vanishing" would still be clearer.
This also solves the problem of "commatic" and "chromatic" sounding so
alike when spoken.
At least now the word "comma" only has to carry two meanings, where
one is a generalisation of the other. Here's my suggestion.
Comma
1. A difference between pitch ratios that is typically smaller than a
scale step. May include schismas, kleismas, commas (2), dieses, limmas
and small semitones.
2. A difference between pitch ratios that is typically in the range 10
to 40 cents, as opposed to schismas and kleismas being smaller, and
dieses, limmas and small semitones being larger.
My sentence "I assume we're talking ancient Greeks" should have ended
in a question mark. I see that the earliest known use of "comma" in
English in relation to pitch was in 1597 (Shorter OED) "A minute
interval or difference of pitch". Does anyone know if the ancient
Greeks used it in relation to pitch?
If one is willing to shorten "chromatic period vector" or "chromatic
comma" to simply "chroma" (as introduced by Paul Erlich), then one
could refer to a vanishing comma as an "achroma".
As Paul pointed out, there is the problem that certain specific ratios
are called chromas. Can anyone explain the rationale behind these? See
Scala's intnam.par. Most are called "Pth-partial chroma" where P is a
prime not smaller than 13.
Pth-partial chromas
P ratio cents
----------------------
13 64:65 26.84c
17 50:51 34.28c
19 95:96 18.13c
23 45:46 35.05c
29 144:145 11.98c
31 30:31 56.77c
Some of these are P/5 commas and some are P*5 commas. They are all
superparticular, and size apparently has little to do with it. Can you
explain this, Manuel? Who named these? In what publication? How long ago?
The only other chroma in intnam.par is:
large chroma (major limma)
128:135 92.18c
There doesn't seem to be any corresponding "small chroma" (or minor
limma for that matter).
Given that those "prime partial chromas" are somewhat obscure and the
"large chroma" has another name, I wouldn't be as worried about using
"chroma" and "chromatic" to mean "non-vanishing" as I was about using
"comma" or "commatic" to mean vanishing. Particularly since the large
chroma _is_ non-vanishing in diatonic scales.
I understand now that while it is fine to speak of "the achromas (or
vanishing commas) _of_ a temperament", it would be a bad idea to speak
of the chroma _of_ a temperament since the chroma is relative to a
particular MOS or periodicity block in the temperament. However I note
that the rule of "proper with cardinality nearest 8.25" isn't too bad
at giving the most popular MOS in a temperament. It gives the diatonic
for meantone and the pentatonic for Pythagorean. Sure it predicts the
decatonic in miracle, which is wrong, but the population of miracle
users _is_ rather small ... so far. :-) And even though the miracle
decatonic may never be popular, at least one person has advocated a
_notation_ for it where the chroma corresponds to 10 generators.
Message: 6363 - Contents - Hide Contents
Date: Sun, 09 Feb 2003 01:31:30
Subject: Re: That poor overloaded word "comma"
From: Carl Lumma
>I think we're mostly agreed that we ought not use "comma" or
>"commatic" to mean "vanishing" if we can help it. Sorry Paul Hahn.
Why?
"Commatic" seems to make sense. And "vanishing comma", if the
context doesn't already make the vanishing clear. "Kernel elements"
sounds fine to me, too.
>But the concepts described in Paul Erlich's 'The Forms of Tonality'
>are very important ones, and there is a certain poetry in "commatic"
>versus "chromatic" "unison vectors" that I am loath to lose. However
>I think we can have our cake and eat it too.
Aw, heck. Looks like we'll just have to create a term out of
one of our last names...
>"vanishing" certainly says what we mean quite clearly, and
>"evanescent" is a nice sounding synonym,
I can't parody this. Ok, I'll indulge you. . .
>So we would have chromatic and achromatic period vectors, and we would
>also be free to speak of the corresponding musical objects as
>chromatic and achromatic commas. But if you're not concerned with
>contrasting it against chromatic, "vanishing" would still be clearer.
"Achromatic"? I don't know, man. I think "vanishing" is better.
>Comma
>1. A difference between pitch ratios that is typically smaller than a
>scale step. May include schismas, kleismas, commas (2), dieses, limmas
>and small semitones.
>2. A difference between pitch ratios that is typically in the range 10
>to 40 cents, as opposed to schismas and kleismas being smaller, and
>dieses, limmas and small semitones being larger.
This looks good.
>If one is willing to shorten "chromatic period vector" or "chromatic
>comma" to simply "chroma" (as introduced by Paul Erlich),
"Chroma" is bad because this term is used to mean "pitch classes" in a
growing amount of literature. Meanwhile, "chromatic period vector"
is perfect, as far as a neuroimaging study by researchers at Dartmouth
College is concerned. They basically found, if I understand correctly,
a map of meantone, as a quasi-tempered block, in patterns of brain
activity of experienced listeners, when they hear chord progressions
whose roots span the tuning.
Another question is 'what do we call blocks?'. While nice, "blocks"
alone is too general. But can an acronym as short as "PBs" really be
taken seriously? Certainly "periodicity blocks" is too long... And
periodicity blocks aren't always block-shaped. etc.
-Carl
Message: 6364 - Contents - Hide Contents
Date: Sun, 9 Feb 2003 01:48:44
Subject: Re: That poor overloaded word "comma"
From: monz
hi Dave and Carl,
> From: "Carl Lumma" <ekin@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Sunday, February 09, 2003 1:31 AM
> Subject: Re: [tuning-math] That poor overloaded word "comma"
>
>
>> But the concepts described in Paul Erlich's
>> 'The Forms of Tonality' are very important ones,
>> and there is a certain poetry in "commatic"
>> versus "chromatic" "unison vectors" that I am
>> loath to lose. However I think we can have our
>> cake and eat it too.
Dave, i'd like you to go into more detail about
Paul's concepts and your feelings about the use
of "comma", if you don't mind.
>> Comma
>> 1. A difference between pitch ratios that is typically
>> smaller than a scale step. May include schismas, kleismas,
>> commas (2), dieses, limmas and small semitones.
>> 2. A difference between pitch ratios that is typically
>> in the range 10 to 40 cents, as opposed to schismas and
>> kleismas being smaller, and dieses, limmas and small
>> semitones being larger.
>
> This looks good.
yeah, inasmuch as certain other words in tuning
terminology are also overloaded ("tone" and "diesis"
both come to mind), i guess i can accept this easily
enough too.
... actually, i admire its conciseness so much i've
added it to the Tuning Dictionary.
>> If one is willing to shorten "chromatic period vector"
>> or "chromatic comma" to simply "chroma" (as introduced
>> by Paul Erlich),
>
> "Chroma" is bad because this term is used to mean
> "pitch classes" in a growing amount of literature.
yes, Carl's right, so i have that objection to it too.
as well, "chroma" also meant something else (i'm still
not entirely sure exactly what) to the ancient Greeks
who coined the term.
> Meanwhile, "chromatic period vector" is perfect,
> as far as a neuroimaging study by researchers at
> Dartmouth College is concerned. They basically found,
> if I understand correctly, a map of meantone, as
> a quasi-tempered block, in patterns of brain activity
> of experienced listeners, when they hear chord
> progressions whose roots span the tuning.
Carl, *PLEASE* give us a link or some reference
to this interesting study!!!
> Another question is 'what do we call blocks?'.
> While nice, "blocks" alone is too general. But can
> an acronym as short as "PBs" really be taken seriously?
> Certainly "periodicity blocks" is too long... And
> periodicity blocks aren't always block-shaped. etc.
yes, good point here too.
"cell" gets my vote. if that's not descriptive enough,
how about "harmonic cell"? or perhaps "periodicity cell"
is good?
-monz
Message: 6365 - Contents - Hide Contents
Date: Sun, 09 Feb 2003 19:44:16
Subject: Re: Janata paper
From: Carl Lumma
Hadn't seen it.
>> This paper has been the subject of some debate recently,
>> on the SpecMus and PsyMus lists.
>
>What goes on on SpecMus?
You'll have to join and find out.
>nearest I can find to PsyMus is psycho_musicians, which is
>much quieter. Also closed archives.
PsyMus isn't on Yahoo, and it costs money.
-Carl
Message: 6366 - Contents - Hide Contents
Date: Sun, 09 Feb 2003 02:43:12
Subject: Janata paper
From: Carl Lumma
>> >eanwhile, "chromatic period vector" is perfect,
>> as far as a neuroimaging study by researchers at
>> Dartmouth College is concerned. They basically found,
>> if I understand correctly, a map of meantone, as
>> a quasi-tempered block, in patterns of brain activity
>> of experienced listeners, when they hear chord
>> progressions whose roots span the tuning.
>
This paper has been the subject of some debate recently,
on the SpecMus and PsyMus lists.
Also, 'quasi-tempered meantone' is ambiguous. What the
paper found is: each triad in 12-tET is associated with
a region of maximum activity in the brain. The particular
area for each triad differs between listeners and between
sessions for a given listener, but the distances between
the areas is always related to the number of common pitches
between the diatonic keys rooted on them. For example,
A major and F# minor have the same key signature, and AMaj
and F#min chords activate the same regions in the brain of
a given person on a given day. Maybe one of our temperament
gurus can tell us what temperaments(s) this 7-of-12 setup
represents...where the pythagorean, syntonic, and augmented
commas vanish and the 25:24 is chromatic?
Hopefully that's more accurate. Maybe Paul has some
comments.
-Carl
Message: 6367 - Contents - Hide Contents
Date: Sun, 09 Feb 2003 19:46:05
Subject: Re: Janata paper
From: Carl Lumma
>i don't know where you're getting 7. they just used the good old
>12-equal torus.
How would you instruct someone to build such a torus? Janata uses
key signatures, which is from where I get 7.
-Carl
Message: 6368 - Contents - Hide Contents
Date: Sun, 09 Feb 2003 19:55:10
Subject: Re: That poor overloaded word "comma"
From: Carl Lumma
Dave Keenan wrote...
>I suspect the existing use of "chroma" that Carl is referring to is
>practically that, a synonym for "pitch". In this sense it is used to
>refer to a quality of a sound and as such will only appear as "the
>chroma of <something>" and not as "a chroma".
The use I was referring to is as a synonym for "pitch class", not
a pitch, and the form "a chroma" definitely comes up!
-Carl
Message: 6369 - Contents - Hide Contents
Date: Sun, 09 Feb 2003 20:09:33
Subject: Re: Janata paper
From: Carl Lumma
>> >syMus isn't on Yahoo, and it costs money.
>>
>> -Carl
>
>?? i didn't pay for my membership . . .
I just looked at the "membership form" this morning,
and it was basically a credit-card form. How did
you join?
-C.
Message: 6370 - Contents - Hide Contents
Date: Sun, 09 Feb 2003 20:19:37
Subject: Re: Janata paper
From: Carl Lumma
>> >ow would you instruct someone to build such a torus?
>
>you can look at hall's paper where a plastic inflatable one is
>depicted, or take schoenberg's or krumhansl's diagrams and
>connect the opposite pairs of edges . . .
I mean, what leads you to the torus?
>> Janata uses key signatures, which is from where I get 7.
>
>is the 7 just a result of the conventional naming, or is it more
>than that?
As I wrote earlier, I think Janata is saying the brain allocates
groups of neurons to each of the 24 diatonic keys, in such a way
that the distance between any pair of groups is related to the
number of pitches shared by their associated diatonic keys. Was
that your understanding? If true, it's a fantastic justification
for using partially-tempered periodicity blocks in music theory.
-Carl
Message: 6371 - Contents - Hide Contents
Date: Sun, 09 Feb 2003 23:52:30
Subject: Re: That poor overloaded word "comma"
From: Carl Lumma
>You adequately explain what you mean by "unison vector" and
>"chromatic", but for "commatic" we could be forgiven for thinking you
>were referring only to its small size.
>
>Had you written, "Notationally it is evident that 25:24 or 128:135
>serves as a /chromatic/ unison vector while 80:81 serves as a
>/achromatic/ unison vector." there would be no problem since most
>people would take achromatic to be the opposite of chromatic.
I don't know about that. I took the sentence to be invoking a
definition for those terms. Though I admit I also already knew
what he was talking about, I doubt any different choice of
emphasized words would have mattered if I didn't. Maybe there
just need to be some bandwidth devoted to the definitions here.
-Carl
Message: 6372 - Contents - Hide Contents
Date: Mon, 10 Feb 2003 16:32:53
Subject: Re: Janata paper
From: Carl Lumma
>dicot is a temperament, generated by neutral thirds. in what sense
>does it make sense to speak of an untempered temperament?
Well, that has to do with not having names for blocks.
>no, janata's torus has 12 keys, i don't see why you need 7
Ok. That's different, then.
>except to get the conventional letter-naming -- and *that* derives
>from the chromatic unison vector being 25:24, 135:128, or etc.
I choose 25:24. "Untempered dicot"!
-Carl
Message: 6373 - Contents - Hide Contents
Date: Mon, 10 Feb 2003 18:29:12
Subject: Re: Janata paper
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...>
wrote:
>>> PsyMus isn't on Yahoo, and it costs money.
>>>
>>> -Carl
>>
>> ?? i didn't pay for my membership . . .
>
> I just looked at the "membership form" this morning,
> and it was basically a credit-card form. How did
> you join?
>
> -C.
i just sent an e-mail to the person who approved memberships,
and was approved. there's virtually nothing of interest on that list,
anyway.
Message: 6374 - Contents - Hide Contents
Date: Mon, 10 Feb 2003 00:46:06
Subject: Re: That poor overloaded word "comma"
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan
<d.keenan@u...>" <d.keenan@u...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
> <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith
>>
>>> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
>> >>
>>>> how about "commatic unison"?
>>>
>>> Ick. If threatened, I'll pull inside my shell and start
>> saying "kernel elements".
>>
>> "kernel elements" gets my vote whole-heartedly!!
>>
>> (just get monz to fix his "metric" discussion . . .)
>
> Please, no. Not "kernel elements". I can take anything but that.
> Bamboo under the fingernails, "commatic unison vectors",
anything.
> I've seen grown men bite their own heads off rather than use
"kernel
> elements".
>
> Seriously.
>
> I think we're mostly agreed that we ought not use "comma" or
> "commatic" to mean "vanishing" if we can help it. Sorry Paul
Hahn. But
> the concepts described in Paul Erlich's 'The Forms of Tonality'
are
> very important ones, and there is a certain poetry in "commatic"
> versus "chromatic" "unison vectors" that I am loath to lose.
However I
> think we can have our cake and eat it too.
>
> Firstly it seems like "period vectors" would be a more obvious
term
> for the vectors bounding a periodicity block, and there would be
no
> need to appeal to augmented unisons in justifying its use.
well, some of these period vector might end up being used as
augmented unisons . . .
> Are there
> cases where such a bounding vector corresponds to a
half-octave (or
> other period which is an integral fraction of the interval of
> equivalence)?
no, at least not the way gene and i have been doing things.
>
> As Paul pointed out, there is the problem that certain specific
ratios
> are called chromas. Can anyone explain the rationale behind
these?
well, the "traditional" ones, 24:25 and 128:135, should be
obvious -- they are the chromatic unison vectors of the diatonic
scale (see _forms of tonality_).
>
> The only other chroma in intnam.par is:
>
> large chroma (major limma)
> 128:135 92.18c
>
> There doesn't seem to be any corresponding "small chroma"
(or minor
> limma for that matter).
look up 24:25 in *my* interval list.
achromas? hmm . . . i don't know if i like that . . .
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