Tuning-Math Digests messages 6475 - 6499

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Message: 6475

Date: Thu, 13 Feb 2003 15:48:38

Subject: Re: "Ultimate" 5-limit again

From: Gene W Smith

On Thu, 13 Feb 2003 22:45:32 -0000 "wallyesterpaulrus
<wallyesterpaulrus@xxxxx.xxx>" <wallyesterpaulrus@xxxxx.xxx> writes:
>
> probably i goofed. can you help fill in all the data for the row 
> that 
> i would need to add to this table:
> 
> Yahoo groups: /tuning/database? *
> method=reportRows&tbl=10&sortBy=5&sortDir=up
> 
> i'll then add the corresponding green line in the relevant et 
> graphs . . .

Unless you want to send it to me, you'll need to wait until my modem
problem is fixed. I'm going to try to install another driver now.


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Message: 6477

Date: Thu, 13 Feb 2003 01:14:55

Subject: Re: vanishing diatonic semitone

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" 
> <d.keenan@u...> wrote:
> 
> > It's been a stretch for me to accept neutral thirds and pelogic as
> > 5-limit temperaments.
> 
> pelogic is among the most useful 5-limit temperaments ever 
> discovered. just try a few different inharmonic timbres and you'll 
> quickly find one that works. be sure to listen to some gamelan music 
> first :)

Pelog may be among the most useful MOS scales ever discovered, but if
it _requires_ inharmonic timbres, in what sense is it an approximation
of 5-limit JI?


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Message: 6479

Date: Thu, 13 Feb 2003 01:52:17

Subject: Re: vanishing diatonic semitone

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> if we had listened to you about badness, you wouldn't be seeing any 
> green lines on the graph with 494 and 612 on it.

I was about to write "You've got me there", however I realised that it
is still the case that I have no interest in 494 or 612
_as_5_limit_temperaments_. Presumably they would appear in my kind of
list at much higher limits, although 624 looks like it might replace
612 beyond 17-limit.

> things like this are useful to know.

Maybe so.

> > Next you'll be wanting the one where 9:10 vanishes. ;-)
> 
> not really -- this one just seems like a severe omission at the 
> moment, if you look at the graphs and charts.

And it probably sounds just great with inharmonic timbres. ;-)


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Message: 6480

Date: Thu, 13 Feb 2003 02:03:52

Subject: Re: 5LimitTemp.xls

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> Dave,
> 
> The degeneracy column seems broken.  I've got Excel 2000.
> 
> -Carl

I've only got Excel 97. Do you have the Analysis Toolpack (or
whatever) installed so the GCD function works? Look it up in Excel Help.


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Message: 6481

Date: Thu, 13 Feb 2003 05:52:15

Subject: Re: 5LimitTemp.xls

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >> The degeneracy column seems broken.  I've got Excel 2000.
> >> 
> >> -Carl
> >
> >I've only got Excel 97. Do you have the Analysis Toolpack (or
> >whatever) installed so the GCD function works? Look it up in Excel
> >Help.
> 
> Ah, now it works.  That is, if only rows 17-19, 27-29, 32-34,
> 44-46 are supposed to be degenerate, and the rest blank.

That's correct.


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Message: 6482

Date: Thu, 13 Feb 2003 09:31:47

Subject: Re: vanishing diatonic semitone

From: Graham Breed

wallyesterpaulrus  wrote:

> the timbres that people like sethares talk about, even if they don't 
> always say so, start as harmonic and then each harmonic (up to 6, 8, 
> 12, whatever) is "tweaked" toward the nearest et (or whatever) 
> position. therefore, it's an approximation of an approximation of 5-
> limit JI :)

That's not true.  Many of Sethares' timbres are measured from physical 
objects.  It is true that the ones that start as harmonic timbres start 
as harmonic timbres.


                        Graham


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Message: 6483

Date: Fri, 14 Feb 2003 13:28:47

Subject: Re: That poor overloaded word "comma"

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Dave wrote:
>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?

If I remember correctly either by Jon Fonville or Ben Johnston.
In an article in Perspectives of New Music vol. 29 no. 2, 1991.

Manuel


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Message: 6484

Date: Fri, 14 Feb 2003 13:32:43

Subject: Re: That poor overloaded word "comma"

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Carl wrote:
>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... 

Why, letters are cheap.

>And periodicity blocks aren't always block-shaped. etc.

Periodicity blobs then? :-)
A more literal translation of the Dutch name is repetition blocks,
but that isn't shorter.

Manuel


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Message: 6485

Date: Fri, 14 Feb 2003 13:58:29

Subject: Re: scala show data

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Thanks for the bug report Carl, it will be fixed in the
next version. Only ETs are affected.

>I also notice that "Lumma stability" is the title of the value
>in the show data output, but "Lumma instability" is the title
>in the help for show data.

Oops.

>Also in the help, the return type is given as n>1.  But if it
>really is stability you're returning, it would be 0 <= n <= 1,
>right?

No, that's the condition for the value to be given, i.e. only
if the number of tones is greater than 1.

Manuel


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Message: 6487

Date: Fri, 14 Feb 2003 19:15:50

Subject: Re: A common notation for JI and ETs

From: Gene W Smith

On Fri, 14 Feb 2003 12:28:10 +0100 manuel.op.de.coul@xxxxxxxxxxx.xxx
writes:

> I haven't followed the thread (300 messages in my inbox), but
> unison vector isn't the only term that Fokker used, also
> homophonic interval and defining interval. I like the latter
> better than unison vector. 

I agree.

One has the constructing intervals
> and defining intervals which form a periodicity block.
> Defining commas would be an alternative.

I think this is what I was calling the commas and chroma, or what people
having been calling the "commatic" and "chromatic" unison vectors.


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Message: 6488

Date: Fri, 14 Feb 2003 23:51:36

Subject: Re: A common notation for JI and ETs

From: monz

hi George,


> From: <gdsecor@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Friday, February 14, 2003 11:33 AM
> Subject: [tuning-math] Re: A common notation for JI and ETs
>
>
> It's my understanding that the ancient Greek "diesis" was an
> interval in the enharmonic genus (varying considerably in size,
> but smaller than a limma) and that Italian theorists of the
> later Middle Ages or Renaissance first used the term "diesis"
> to include the limma (and it wouldn't surprise me a bit if
> *that* resulted from a misunderstanding of Greek usage, just
> as the Greek modes were misinterpreted in a prior century). 
> I did a little checking last night in both the Oxford and
> Harvard Dictionaries, and they state that the use of the 
> term "diesis" for the limma dates from the Italian Renaissance
> at the earliest.  This is in contrast to Marchettus of Padua
> (14th century), who used the term "diesis" for a fifth-tone
> (of varying size) in a 29-tone octave (divided rationally,
> not an ET); his limma was 2/5 of a tone (two dieses) and
> apotome 3/5 of a tone.  (See Margo Schulter's paper
> _Enharmonic Excursion to Padua, 1318: Marchettus, the
> cadential diesis, and neo-Gothic tunings_, 
<http://value.net/~mschulter/marchetmf.txt *> (ASCII text) or 
<http://value.net/~mschulter/marchetmf.zip - Ok *> (text and PostScript)


"diesis" is a musical term which indeed has a long and
confusing history.  you might profit from reading this:
Definitions of tuning terms: diesis, (c) 1998 by Joe Monzo *


and i think you missed this -- which you'd also probably
be interested in at least in passing:
Onelist Tuning Digest # 483 message 26, (c)2000 by Joe Monzo *



-monz


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Message: 6491

Date: Fri, 14 Feb 2003 12:28:10

Subject: Re: A common notation for JI and ETs

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Paul wrote:
>> 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?

I haven't followed the thread (300 messages in my inbox), but
unison vector isn't the only term that Fokker used, also
homophonic interval and defining interval. I like the latter
better than unison vector. One has the constructing intervals
and defining intervals which form a periodicity block.
Defining commas would be an alternative.

Manuel


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Message: 6492

Date: Sat, 15 Feb 2003 17:08:57

Subject: Re: A common notation for JI and ETs

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

>>One has the constructing intervals
>> and defining intervals which form a periodicity block.
>> Defining commas would be an alternative.

Gene wrote:

>I think this is what I was calling the commas and chroma, or what people
>having been calling the "commatic" and "chromatic" unison vectors.

Not quite, the defining intervals comprise both commatic
and chromatic ones. The (same number of) constructing intervals are the
"prime" intervals, 3/1, 5/1, etc.
So "defining commas" isn't a good term, they aren't necessarily commas.

Manuel


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Message: 6497

Date: Sun, 16 Feb 2003 23:55:45

Subject: Re: A common notation for JI and ETs

From: monz

hi paul,


> From: <wallyesterpaulrus@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Sunday, February 16, 2003 6:25 PM
> Subject: [tuning-math] Re: A common notation for JI and ETs
>
>
> --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> > hi paul,
> > 
> > 
> > 
> > > From: <wallyesterpaulrus@y...>
> > > To: <tuning-math@xxxxxxxxxxx.xxx>
> > > Sent: Saturday, February 15, 2003 1:02 PM
> > > Subject: [tuning-math] Re: A common notation for JI and ETs
> > >
> > >
> > > --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> 
> > > wrote:
> > > 
> > > > and i think you missed this -- which you'd also probably
> > > > be interested in at least in passing:
> > > > Onelist Tuning Digest # 483 message 26, (c)2000 by Joe Monzo *
>
> <etc., snip>


thanks!



> your faithful scrutinizer,
> paul


just don't become "The Central Scrutinizer"!! ;-)

[reference to Frank Zappa's _Joe's Garage_ for those
who don't get it ... ]



-monz


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Message: 6498

Date: Sun, 16 Feb 2003 10:02:05

Subject: Re: A common notation for JI and ETs

From: David C Keenan

At 10:06 PM 14/02/2003 +0000, you wrote:
>It's my understanding that the ancient Greek "diesis" was an interval
>in the enharmonic genus (varying considerably in size, but smaller
>than a limma) and that Italian theorists of the later Middle Ages or
>Renaissance first used the term "diesis" to include the limma (and it
>wouldn't surprise me a bit if *that* resulted from a misunderstanding
>of Greek usage, just as the Greek modes were misinterpreted in a
>prior century).  I did a little checking last night in both the
>Oxford and Harvard Dictionaries, and they state that the use of the
>term "diesis" for the limma dates from the Italian Renaissance at the
>earliest.  This is in contrast to Marchettus of Padua (14th century),
>who used the term "diesis" for a fifth-tone (of varying size) in a 29-
>tone octave (divided rationally, not an ET); his limma was 2/5 of a
>tone (two dieses) and apotome 3/5 of a tone.  (See Margo Schulter's
>paper _Enharmonic Excursion to Padua, 1318: Marchettus, the cadential
>diesis, and neo-Gothic tunings_,
><http://value.net/~mschulter/marchetmf.txt *> (ASCII text) or
><http://value.net/~mschulter/marchetmf.zip - Ok *> (text and PostScript)

I'll go along with that. It seems the 1959 Oxford was wrong about the 
ancient Greeks using "diesis" for the limma, but they've fixed it now. It 
seems that in fact they used it to mean quarter-tone. Or half of what's 
left when you take a "ditone" out of a perfect fourth (as per the 
enharmonic genus). You say "varying considerably in size" but just how much 
is it likely to have varied in ancient times? I can't imagine the perfect 
fourth varying (intentionally) much outside of Pythagorean 3:4 to 
Aristoxenean (12-ET), 498 to 500 cents. And I can't see the ditone varying 
much outside Pythagorean 64:81 to Didymus' 4:5, 408 TO 386 cents. See

Tutorial on ancient Greek Tetrachord-theory *

So that puts the ancient greek enharmonic diesis at 45 to 57 cents, or 
almost exactly our middle range of dieses, which would seem to make the 
prefix "neo-" quite inappropriate for this group.

>By the way, I also read that "diesis" is also an Italian word
>(pronounced dee-EH-sis), so it might not be inappropriate to use
>prefixes of Latin origin to modify it.
>
> > >... I tried to
> > >identify what I would call the prototypical diesis in each group:
> >
> > Shouldn't you instead be looking at the primary interpretation of
>the most
> > commonly ocurring sagittal symbol in each group?
>
>The most popular that could be claimed in each group are:
>
>small:  25/1 - 256400:6561, ~43.0c
>    and  125/1 ? 125:128, ~41.1c
>    and  11/5 ? 44:45, ~38.9c
>medium: 11/1 ? 32:33, ~53.3c
>    and  35/1 ? 35:36, ~48.8c
>large:  7/1 ? 27:28, ~63.0c
>         13/1 ? 26:27, ~65.3c
>
>I'll discuss the small group below.
>
>The 11 comma, 32:33, definitely takes the prize in the medium group,
>so neo- can't be dismissed as a group prefix on the basis of
>popularity.
>
>For the large group, you might think of skipping over 7/1 (since it
>is much more likely that the 7-comma, 63:64, would be used to notate
>this,

Yes. We only have the popularity of the ratios (sans 2's and 3's) and not 
the different commas that may be used for a given ratio, and I would  not 
think 27:28 would be very popular, particularly since it requires a schisma 
accent.

Although it isn't relevant to this discussion, I take this opportunity to 
remind us that we only have the popularity of the ratios in the scale 
archive, and not their popularity in actual use. This could be calculated 
if we knew the popularity of each scale in actual use, but it seems likely 
that scales which contain ratios which occur rarely in the archive, would 
be unpopular scales. Thus our numbers are likely to be exaggerating the 
popularity (in use) of the less popular (in the archive) ratios.

>  but I see this as an opportunity to point out that it is not a
>stretch of the imagination that a large diesis *does* function and
>*sound* like a small semitone, e.g., in the resolution of 21/16 to
>5/4.

Sure. But that doen't have any bearing on the naming of these categories of 
diesis, does it?

> > But the minor diesis 125:128 is rarely used in the sagittal
>notation,
> > having symbol .//|.
>
>Even though it's the second most popular small diesis.  But I would
>attribute the popularity of both 25/1 and 125/1 as being due to
>transposition of 5-limit chords within a scale rather than use of a
>5^2 or 5^3 harmony in a chord.

Certainly!

>   Now if popularity were restricted to
>actual use of the diesis as determining an interval that is most
>likely to occur in a chord, then I think that 11/5, or the 5:11
>diesis, would probably be considered the representative interval.

Yes. But popularity isn't, and shouldn't be, restricted in that way.

>  But it's third place in overall popularity, so it's not clear-cut.
>(Hence I don't think that the small group has a good chance of
>disqualifying the use of the neo- prefix for the middle group.)

Agreed.

> > By far the most common in this range will be the 25-
><small>diesis //|. I
> > can't find anywhere this has been previously named, presumably
>because it
> > is simply a double syntonic comma. So, considered as a "comma" in
>its own
> > right it is almost as "neo-" as the 11 and 13 commas below.
>
>Two intervals it produces are 16:25 and 24:25, an augmented fifth and
>augmented prime, and these are both part of traditional harmony,
>hence not new.  And ratios of 7 are not completely new -- even Partch
>admitted that the 7th harmonic is implied in 12-ET.  It is only with
>11 and 13 that radically new intervals are introduced, and I think
>that 11 is the more radical of the two.  So I would stand by the
>label neo- as characterizing the middle group of dieses.

I retract my argument that the small group might be considered new.

> > And there are
> > other commas in this group which are probably newer.
>
>You're always going to find new theoretical intervals, but I maintain
>that 11 has the *newest* and *most exotic* sound once you go beyond
>the bounds of 12-ET or traditional harmony.  13 comes close, but with
>17 and 19 you get intervals that are very similar to what you have in
>12-ET (except when you relate them to 11 and 13).  This is the sense
>in which I am using neo- to describe the middle diesis group, and the
>sound will be characteristic of *any* of the dieses in the group,
>because they all approximate the 11 and 13 dieses in *size.*

However it would seem that they would not have been new to the ancient Greeks.

> > >45-57 cents -- 32:33, the unidecimal diesis (or quartertone),
> > >introduces some of the *strangest new* harmonies encountered in
> > >alternative tunings.  I thought neo- might be more descriptive of
>an
> > >interval such as this, rather than some nondescript label (such as
> > >meso-) that suggests that it might be average or middlin'.
> >
> > But it _is_ average as far as size goes, and that's what these
>prefixes are
> > supposed to be about.
>
>While average in size, it's very un-average, i.e., out-of-the-
>ordinary, in the characteristic *sound* of the intervals (ratios of
>11) that result from its use.  I think that there is an advantage in
>having a label that is musically (rather than strictly
>mathematically) descriptive.

I wouldn't call the size of an interval in cents a strictly mathematical 
description. It certainly realates to how it sounds.

> > And anyway, how long does something remain "new"?
>
>As I said above, it's new relative to 12-ET and traditional harmony,
>and it will continue to be new to anyone unfamiliar with alternative
>tunings.  At present this includes most of the musical world, and it
>will always be radically new to anyone who is beginning the study of
>alternative tunings.  I don't expect traditional harmonic values to
>disappear from the musical scene any time soon.

This just seems like too specific a viewpoint in time and space. It seems 
likely that other cultures now, and our culture at other times 
did-not/do-not/will-not see these as new.

> > Also, I should think that if 125:128 is prototypical of the small
>group
> > then 243:250 would be that for the medium group. But again this is
>not a
> > common comma to want to notate. It might be notated as /|) or (|~ .
>
>I don't see how you drew that conclusion -- 125:128 has a special
>place in history as *the meantone diesis* that 243:250 does not
>share.  Besides that, you have 11/1 higher on the popularity list
>than 125/1.

I took "prototypical" to mean "coming first in time" and assumed that a 
5-prime-limit diesis would have been noticed in this category before 11.

>So then we might want something else besides meta-.

Yes.

 > My main objection is that neo- tells one nothing about the size.

>But it does tell one about the main characteristic of intervals
>altered by dieses in this size range: they are the ones that *sound*
>most foreign to anyone who is twelve-oriented, i.e., like
>quartertones.  The over-57-cent group, on the other hand, sound like
>small semitones (though they could be most accurately described as
>third-tones), while the under-45-cent group could be described as
>fifth-tones.

Yes. It's uncanny how well these boundaries correspond to the mid points 
between 1/3, 1/4 and 1/5 of a Pythagorean tone.

But even if we had prefixes that meant 1/3, 1/4 and 1/5 I think this would 
be a mistake since the symbols don't always come out as these in various ETs.

>So if you don't like meta-, then what?
>
> > In regard to having the right _meaning_, the best Greek set I can
>find are
> >   hypo-
> >   meso-
> >   hyper-
>
>Hypo- and hyper- do not offer the opportunity to distinguish by
>abbreviation.

I already suggested the abbreviations "o" and "er" for these, which to me 
seem reasonably distinct vowel sounds. But I assume you do not find them so.

>   For example, I might want to use 7-c and 7-h to stand
>for 7-comma and 7-hyperdiesis in a diagram showing the actual symbols
>(where words would take up too much space), but I don't have that
>option if two ranges begin with the same letter.  Besides, they sound
>too much alike when spoken, so one might be misunderstood for the
>other.

I agree that hypo and hyper are too similar, and few people know which has 
which meaning or even that they have different meanings.

> > If we were to depart from the Greek
> >   minor
> >   neutral
> >   major
> > would be obvious enough, and so would
> >   small
> >   medium (or mean)
> >   large
>
>You've made a better case against meta- than neo-, and I suppose that
>would also eliminate proto-.  I would like to see two short prefixes
>for large and small that begin with different letters.  In fact, I
>would like to see separate letters for each of the following:
>
>Schisma - s
>Kleisma - k
>Comma - c
>[Small] diesis - x
>[Medium] diesis - y
>[Large] diesis - z
>where x, y, and z are letters different from s, k, and c.  So we
>could abbreviate intervals as 5c, 7c, 5s, 7z, 11y, 11z, 13y, 13z,
>etc.  (We don't need a separate letter for schismina, because it
>isn't a symbol in the notation.)

This would be nice, but I wouldn't let it stop us from using something if 
there was sufficient other reason.

> > It is unfortunate that the word "diesis" already has two more
>syllables
> > than we'd like it to have. This is presumably why we feel compelled
>to
> > shorten any prefix we might add to it, down to a single syllable.
>
>Or at least a single consonant.

But not if that consonant is "w" (whose name has 3 syllables).

>So if you don't like proto-, neo-, and meta-, then what would you
>suggest?

I think we should use "m" for the middle group, whether it stands for 
"mid", "middle", medium", "mean", or "meso".

I think english speaking folks are used to seeing the letters L M H for low 
medium, high, on appliances etc. and we could claim lepto- meso- hadro-. So 
that's a possibility but I can't help thinking there may yet be a better 
solution. Any other suggestions?
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 6499

Date: Sun, 16 Feb 2003 16:25:58

Subject: Re: A common notation for JI and ETs

From: David C Keenan

I wrote
"I think english speaking folks are used to seeing the letters L M H for 
low medium, high, on appliances etc. and we could claim lepto- meso- 
hadro-. So that's a possibility but I can't help thinking there may yet be 
a better solution. Any other suggestions?"

We're also used to seeing S M L on t-shirts etc. It's a pity L can stand 
for either large or low, and S will clash with schisma. We could always use 
sd md ld for the dieses and plain s for the schisma.

We alreadty seem to have both been using small medium large as our default 
terms.


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