>>> 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.
Message: 6482 - Contents - Hide Contents
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
Message: 6483 - Contents - Hide Contents
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
Message: 6484 - Contents - Hide Contents
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
Message: 6485 - Contents - Hide Contents
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
Message: 6486 - Contents - Hide Contents
Date: Fri, 14 Feb 2003 19:33:46
Subject: Re: A common notation for JI and ETs
From: gdsecor
--- In tuning-math@xxxxxxxxxxx.xxxx David C Keenan <d.keenan@u...>
wrote:
>
>> --- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>"
>> > --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>"
>> > I think that the term "comma" has been used in a broad sense to
>> denote smaller intervals (which we now call kleisma and schisma)
more
>> often than larger ones,
>
> Possibly more often. But I expect it _has_ been used to cover
larger ones
> often enough.
>
>> inasmuch as the term "diesis" has been used
>> for the latter since at least the 14th century. So I would be
>> inclined not to use the term "comma" for anything above ~37 cents,
>> even in a broader sense.
>
> What term do you suggest we use for all these intervals typically
less than
> a scale step, from schisminas to small semitones?
What about "microtone." As a subdivision of that I would just have
two broad terms -- comma and diesis, with the ~37-cent boundary. The
former would be divided into comma, kleisma, schisma, and schismina,
and the latter into 3 groups (which we have yet to name).
> Here's what my Shorter Oxford (1959) has to say:
>
> Comma ...
> 3. Mus. A minute interval or difference of pitch 1597.
> ...
>
> Diesis ...
> 1. Mus. a. In ancient Gr. music, the pythagorean semitone (ratio
> 243:256). b. Now, the interval equal to the difference between
three major
> thirds and an octave (ratio 125:128); usually called enharmonic
diesis.
> ...
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_,
<Xenharmonic Excursion to Padua, 1318: * [with cont.] (Wayb.)> (ASCII text) or
<http://value.net/~mschulter/marchetmf.zip - Type Ok * [with cont.] (Wayb.)> (text and PostScript)
That's just for your information -- we're not debating boundaries,
anyway.
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, 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. I will have something to say about 26:27 below, in response to
your mention of complements.
>> 37-45 cents -- 125:128, the meantone diesis, is not only in the
group
>> with the *smallest size*, but is also the diesis by which three
4:5s
>> *fall short* of (i.e., on the near side of) an octave. So I
thought
>> that peri- or intra- might be appropriate. Of these two I prefer
>> peri-. But proto- is also good, for a couple of reasons: it is
>> similar in meaning to peri-, and it is the opposite of meta-
(should
>> we use that term for the large group). Besides, 125:128, which is
>> probably the best-known of any diesis in any group (and thus, on
>> account of its prominence, the one with the strongest claim to the
>> label proto-diesis), would validate an additional shade of meaning
by
>> which the term could be applied to this group.
>
> 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. 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.
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.)
> 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.
> 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.*
>> 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.
>> Even the
>> 13 diesis (1024:1053, the second most prominent member of the
group,
>> and the one that's actually symbolized by an "n") is new and
strange.
>
> But their complements in the large-diesis group are just as new and
> strange.
I disagree. The middle group sound like quartertones, but the large
group (third-tones) sound like the small semitones of 17 or 19-ET.
Why would the 11 and 13 complementary dieses sound less strange?
Take a look at how the symbols are used: 11/8 of C is F/|\, but 11/8
of F is B(!) -- that's a Pythagorean B, rather high (or strange) for
a tritone (relative to F) inasmuch as you're not alternating a 9:10
with the 8:9's (as would be done in JI) -- so you need a larger
alteration to lower the pitch to make it equivalent to Bb/|\.
Likewise, 13/8 of C is A(!/ -- that's a Pythagorean A, 27/16, rather
high for a major sixth, which requires a large (65-cent) 13 diesis to
bring it down to 13/8 -- but if you had a melody that moved from 5/3
to 13/8, the interval would be around 44 cents -- much closer to a 48-
cent 13 diesis.
> 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.
> 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.
> The 11-<medium>diesis /|\ will certainly be the most common in this
group.
>
>> 57-69 cents -- 625:648, besides being a *large* diesis (~27:28, or
>> 1deg19) is also the amount by which four 5:6s *exceed* (i.e., go
>> beyond) an octave.
>
> I agree that the prototypical diesis in the large group is the
major diesis
> 625:648, again not something we'd commonly use since it is '(|) .
>
> Clearly the 11-<large>diesis (|) will be the most common here.
>
>> I believe that we agree that meta- is a good
>> prefix for this group.
>
> Well, no. Only that it applies to this group better than it does to
the
> medium group.
>
> The use of "meta-" to mean "beyond" is a recent departure from the
Greek
> usage. As the Shorter Oxford puts it:
>
> "In supposed analogy to 'Metaphysics' (misaprehended as
meaning 'the
> science of that which transcends the physical'), meta- has been
prefixed to
> the name of a science, to form a designation of a higher science of
the
> same nature but dealing with ulterior problems."
So then we might want something else besides meta-.
> But why not use prefixes that are a valid description of _all_ the
commas
> in the group, rather than just ones that may be typical in any
sense? i.e.
> ones that relate to size.
>
>> Need I say more?
>
> I'm afraid so. :-)
So I've said more.
> 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.
> And if one adopts the modern sense of meta- one might take a meta-
diesis to
> be a difference between dieses, in the same way that a diesis is a
> difference between other intervals. For example, we might well have
used
> the term meta-comma to describe the differences between commas that
we
> instead called schismas and now schisminas.
>
> If one takes the biological meanings of proto- front and meta- rear
(of
> organisms) it is unclear that there is any correspondence with
small and
> large. If one takes the temporal meaning of proto- before or early
or
> primitive and meta- after or late or advanced, then it is only
slightly
> more clear.
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. 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.
> 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.)
So if you don't like proto-, neo-, and meta-, then what would you
suggest?
> 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.
> We might
> instead shorten "diesis" to "di" for convenience when spoken (say
in
> rehearsals) and then not need to shorten the prefixes.
Yes, that's possible.
> Boundaries good. Labels still need work.
Yes, still.
--George
Message: 6487 - Contents - Hide Contents
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.
Message: 6488 - Contents - Hide Contents
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_,
<Xenharmonic Excursion to Padua, 1318: * [with cont.] (Wayb.)> (ASCII text) or
<http://value.net/~mschulter/marchetmf.zip - Type Ok * [with cont.] (Wayb.)> (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 ... * [with cont.] (Wayb.)
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)200... * [with cont.] (Wayb.)
-monz
Message: 6489 - Contents - Hide Contents
Date: Fri, 14 Feb 2003 02:08:47
Subject: Re: "Ultimate" 5-limit again
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
The comma is 2^(-69) 3^45 5^(-1). Is that all you need?
Message: 6490 - Contents - Hide Contents
Date: Fri, 14 Feb 2003 03:13:55
Subject: Re: "Ultimate" 5-limit again
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:
>
>> probably i goofed. can you help fill in all the data for the row
> that
> The comma is 2^(-69) 3^45 5^(-1). Is that all you need?
here's the contents of the database (in very non-database form --
hopefully your modem will heal soon). note that this is intended to
go along with the charts on monz's equal temperament dictionary entry:
"comma name(s)", "as 2^x*3^y*5^z, [x y z]
is . . .", "numerator", "denominator", "cents", "temperament name
(s)", "sample ETs/MOSs (notes per octave)", "[period, optimal
generator] (cents)", "[2 3 5] map in terms of [per., gen.]", "optimum
RMS error (cents)"
" large limma", "0 3 -2", "27", "25", "133.237575", "beep", "4, 5,
9", "[1200, 268.056439]", "[[1,0] [2,-2] [3,-3]]", "35.609240"
" small diesis", "-10 -1
5", "3125", "3072", "29.613568", "magic", "3, 16, 19, 22, 25, 35, 41,
60, 63, 79", "[1200, 379.967949]", "[[1,0] [0,5] [2,1]]", "4.569472"
" minimal diesis", "5 -9
4", "20000", "19683", "27.659848", "tetracot", "7, 27, 34, 41, 48,
61, 75", "[1200, 176.282270]", "[[1,0] [1,4] [1,9]]", "2.504205"
" diaschisma", "11 -4 -2", "2048", "2025", "19.552569", "diaschismic,
5-limit pajara", "12, 22, 34, 46, 56, 58, 70, 78, 80, 90", "[600,
105.446531]", "[[2,0] [3,1] [5,-2]]", "2.612822"
" pythagorean comma", "-19 12
0", "531441", "524288", "23.460010", "aristoxenean", "12, 48, 60, 72,
84, 96", "[100, 14.663787]", "[[12,0] [19,0] [28,-1]]", "1.382394"
"-", "2 9 -7", "78732", "78125", "13.399010", "semisixths", "19, 27,
46, 65, 73, 84", "[1200, 442.979297]", "[[1,0] [-1,7] [-
1,9]]", "1.157498"
"-", "26 -12 -3", "67108864", "66430125", "17.598848", "misty", "12,
63, 75, 87, 99, 205", "[400, 96.787939]", "[[3,0] [5,-1]
[6,4]]", "0.905187"
" würschmidt's comma", "17 1 -
8", "393216", "390625", "11.445290", "würschmidt", "28, 31, 34, 37,
65, 71, 96, 99", "[1200, 387.819673]", "[[1,0] [-1,8]
[2,1]]", "1.071950"
" kleisma", "-6 -5 6", "15625", "15552", "8.107279", "kleismic,
hanson", "15, 19, 23, 34, 53, 72, 83, 87, 91, 125", "[1200,
317.079675]", "[[1,0] [0,6] [1,5]]", "1.029625"
" semicomma", "-21 3
7", "2109375", "2097152", "10.061000", "orwell", "9, 22, 31, 53, 75,
84, 97, 128, 243", "[1200, 271.589600]", "[[1,0] [0,7] [3,-
3]]", "0.800410"
"-", "9 -13 5", "1600000", "1594323", "6.1535581", "amity", "39, 46,
53, 60, 99, 152, 205, 311", "[1200, 339.508826]", "[[1,0] [3,-5] [6,-
13]]", "0.383104"
" minor chroma, classic chromatic semitone", "-3 -1
2", "25", "24", "70.672427", "dicot", "3, 4, 7, 10", "[1200.
350.977500]", "[[1,0] [1,2] [2,1]]", "28.851897"
"-", "32 -7 -
9", "4294967296", "4271484375", "9.491569", "escapade", "22, 43, 65,
87, 152, 217", "[1200, 55.275493]", "[[1,0] [2,-9] [2,7]]", "0.483108"
" parakleisma", "8 14 -
13", "1224440064", "1220703125", "5.291732", "parakleismic", "19, 42,
61, 80, 99, 118, 217", "[1200, 315.250913]", "[[1,0] [5,13] [6,-
14]]", "0.276603"
" schisma", "-15 8 1", "32805", "32768", "1.953721", "schismic,
helmholtz/groven", "29, 41, 53, 65, 77, 89, 118, 171, 200", "[1200,
498.272487]", "[[1,0] [2,-1] [-1,8]]", "0.161693"
"-", "24 -21
4", "10485760000", "10460353203", "4.199837", "vulture", "48, 53, 58,
217, 270, 323", "[1200, 475.542233]", "[[1,0] [0,4] [-
6,21]]", "0.153767"
"-", "23 6 -
14", "6115295232", "6103515625", "3.338011", "semisuper", "16, 18,
34, 50, 84, 118, 152, 270, 388", "[600, 71.146064]", "[[2,0] [4,-7]
[5,-3]]", "0.194018"
"'19-tone comma'", "-14 -19
19", "19073486328125", "19042491875328", "2.815547", "enneadecal", "19
, 152, 171, 323, 494, 665", "[63.157894, 7.292252]", "[[19,0] [30,1]
[44,1]]", "0.104784"
"-", "39 -29
3", "68719476736000", "68630377364883", "2.246116", "tricot", "53,
388, 441, 494, 547, 600", "[1200, 565.988015]", "[[1,0] [3,-3] [16,-
29]]", "0.057500"
"-", "-68 18
17", "295578376007080078125", "295147905179352825856", "2.523151", "va
voom", "118, 311, 547, 665", "[1200, 111.875426]", "[[1,0] [0,17] [4,-
18]]", "0.058853"
"-", "38 -2 -
15", "274877906944", "274658203125", "1.384290", "semithirds", "118,
205, 323, 441, 559, 1000", "[1200, 193.199615]", "[[1,0] [4,-15] [2,
2]]", "0.060822"
" ennealimma", "1 -27
18", "7629394531250", "7625597484987", "0.861826", "ennealimmal", "72,
99, 171, 243, 270, 441, 612", "[133.333333, 49,008820]", "[[9,0]
[15,-2] [22,-3]]", "0.025593"
" major chroma, major limma, limma ascendant", "-7 3
1", "135", "128", "92.178716", "pelogic", "7, 9, 16, 23", "[1200,
522.862346]", "[[1,0] [2,-1] [1,3]]", "18.077734"
"-", "-16 35 -
17", "50031545098999707", "50000000000000000", "1.091895", "minortone"
, "125, 171, 217, 388, 559, 730, 901", "[1200, 182.466089]", "[[1,0]
[-1,17] [-3,35]]", "0.025466"
"-", "-53 10
16", "9010162353515625", "9007199254740992", "0.569430", "kwazy", "118
, 494, 612, 730, 1342", "[600, 162.741892]", "[[2,0] [1,8] [6,-
5]]", "0.017725"
"-", "91 -12 -
31", "2475880078570760549798248448", "2474715001881122589111328125", "
0.814860", "astro", "118, 1171, 2224", "[1200, 132.194511]", "[[1,0]
[5,-31] [1,12]]", "0.014993"
"-", "37 25 -
33", "116450459770592056836096", "116415321826934814453125", "0.522464
", "whoosh", "441, 730, 1171, 3072", "[1200, 560.546970]", "[[1,0]
[17,-33] [14,-25]]", "0.012388"
" monzisma", "54 -37
2", "450359962737049600", "450283905890997363", "0.292396", "monzismic
", "53, 559, 612, 665, 1171, 1783", "[1200, 249.018448]", "[[1,0] [2,-
2] [10,-37]]", "0.005738"
"-", "-36 -52
51", "444089209850062616169452667236328125", "444002166576103304796646
509039845376", "0.339362", "egads", "441, 1342, 1783, 3125", "[1200,
315.647874]", "[[1,0] [15,-51] [16,-52]]", "0.004660"
"-", "-107 47
14", "162285243890121480027996826171875", "162259276829213363391578010
288128", "0.277035", "fortune", "612, 1901, 2513, 3125", "[1200,
221.567865]", "[[1,0] [-1,14] [11,-47]]", "0.003542"
"-", "-17 62 -
35", "381520424476945831628649898809", "381469726562500000000000000000
", "0.230068", "senior", "171, 1000, 1171, 1342, 2513, 3684", "[1200,
322.801387]", "[[1,0] [11,-35] [19,-62]]", "0.003022"
"-", "144 -22 -
47", "22300745198530623141535718272648361505980416", "2229758394562963
9856633730232715606689453125", "0.245429", "gross", "118, 1783, 1901,
3684", "[1200, 91.531021]", "[[1,0] [-2,47] [4,-22]]", "0.002842"
"-", "-90 -15
49", "17763568394002504646778106689453125", "1776308649528226802416196
7871623168", "0.046966", "pirate", "730, 1783, 2513, 4296", "[1200,
185.754179]", "[[1,0] [-6,49] [0,15]]", "0.000761"
" limma, pythagorean minor second", "8 -5
0", "256", "243", "90.224996", "blackwood", "5, 10, 15, 25", "[240,
84.663787]", "[[5,0] [8,0] [12.-1]]", "12.759741"
"-", "71 -99
37", "171798691840000000000000000000000000000000000000", "171792506910
670443678820376588540424234035840667", "0.062327", "raider", "1171,
3125, 4296", "[1200, 343.296099]", "[[1,0] [-9,37] [-
26,99]]", "0.000511"
" atom of kirnberger", "161 -84 -
12", "2923003274661805836407369665432566039311865085952", "29229773394
92680612451840826835216578535400390625", "0.015361", "atomic", "(12,)
600, 612, 3072, 3684, 4296", "[100, 1.955169]", "[[12,0] [19,1] [28,-
7]]", "0.000120"
" diatonic semitone", "4 -1 -
1", "16", "15", "111.731285", "father", "3, 5, 8", "[1200,
442.179356]", "[[1,0] [2,-1] [2,1]]", "45.614107"
" major diesis", "3 4 -
4", "648", "625", "62.565148", "diminished, 'octatonic'", "4, 8, 12,
16, 28", "[300, 94.134357]", "[[4,0] [6,1] [9,1]]", "11.060060"
" maximal diesis", "1 -5
3", "250", "243", "49.166137", "porcupine", "7, 8, 15, 22, 29, 37,
59", "[1200, 162.996026]", "[[1,0] [2,-3], [3,-5]]", "7.975801"
" diesis, great diesis, minor diesis", "7 0 -
3", "128", "125", "41.058858", "augmented, diesic", "3, 9, 12, 15,
18, 27, 39, 42", "[400, 91.201856]", "[[3,0] [5,-1]
[7,0]]", "9.677666"
"-", "-14 3 4", "16875", "16384", "51.119858", "negri", "9, 10, 19,
28, 29", "[1200, 126.238272]", "[[1,0] [2,-4] [2,3]]", "5.942563"
" comma, syntonic comma, comma of didymus", "-4 4 -
1", "81", "80", "21.506290", "meantone, 'diatonic'", "5, 7, 12, 19,
26, 31, 43, 45, 50, 55, 69, 74, 81, 88", "[1200,
503.835154]", "[[1,0] [2,-1] [4,-4]]", "4.217731"
Message: 6491 - Contents - Hide Contents
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
Message: 6492 - Contents - Hide Contents
Date: Sat, 15 Feb 2003 17:08:57
Subject: Re: A common notation for JI and ETs
From: manuel.op.de.coul@xxxxxxxxxxx.xxx
>> >ne 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
Message: 6494 - Contents - Hide Contents
Date: Sat, 15 Feb 2003 20:58:29
Subject: Re: A common notation for JI and ETs
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx Gene W Smith
<genewardsmith@j...> wrote:
>
>
> On Fri, 14 Feb 2003 12:28:10 +0100 manuel.op.de.coul@e...
> 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.
no. the distinction between commatic and chromatic unison
vectors falls completely outside of fokker's conception, or that of
wuerschmidt from whom the "defining" and "constructing" labels
were, i believe, derived. if i recall correctly, the "constructing
intervals" are simply the 1-step intervals in the just intonation
periodicity block -- for example, different instances of 12-tone in
the 5-limit can have constructing interval sets
{243:256, 2187:2048}
{243:256, 135:128, 16:15}
{25:24, 135:128, 16:15}
etc.
Message: 6495 - Contents - Hide Contents
Date: Sat, 15 Feb 2003 21:02:40
Subject: Re: A common notation for JI and ETs
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...>
wrote:
i brought this page up to dave and george very recently (last
week) here on this list, and they indeed found it very useful for
their discussion. unfortunately, several erroneous statements
persist on this page, most notably:
"It can be seen easily from the lattice that all the intervals are
made up of various combinations of the ones described by
Paul."
of course, we all know you're very busy right now, and i at least
appreciate your brief and all too infrequent visits to this list.
-paul
Message: 6496 - Contents - Hide Contents
Date: Sat, 15 Feb 2003 21:03:52
Subject: Re: A common notation for JI and ETs
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx manuel.op.de.coul@e...
wrote:
> The (same number of) constructing intervals are the
> "prime" intervals, 3/1, 5/1, etc.
oh . . . i must have been thinking of the wrong term, or perhaps of
wuerschmidt's usage instead of fokker's . . .
Message: 6497 - Contents - Hide Contents
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:
>>>
> 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
Message: 6498 - Contents - Hide Contents
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_,
><
Xenharmonic Excursion to Padua, 1318: * [with cont.] (Wayb.)> (ASCII text) or
><
http://value.net/~mschulter/marchetmf.zip - Type Ok * [with cont.] (Wayb.)> (text and PostScript)
>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 * [with cont.] (Wayb.)
Message: 6499 - Contents - Hide Contents
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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