Yahoo Tuning Groups Ultimate Backup tuning-math Enharmonic diesis?

Monz,

In
http://sonic-arts.org/dict/19edo.htm
you refer to 625:648 (62.57 c) as the enharmonic diesis.

But the enharmonic dieses you described in
http://sonic-arts.org/monzo/aristoxenus/tutorial.htm
would seem to be in the approximate range of 45 to 56 cents (a fourth
minus either a Pythagorean or just major third and then divided by two).

And elsewhere, including Margo Schulter here
/makemicromusic/topicId_166.html#166
I find 125:128 (41.06 c) being called the enharmonic
diesis, or simply the enharmonic.

In Scala, or here
http://www.xs4all.nl/~huygensf/doc/intervals.html
we find
31:32 (54.96 c) called the Greek enharmonic, and
512:525 (43.41 c) called the Avicenna enharmonic diesis.

Apart from your 19edo page, they all seem to point away from
considering anything as large as 625:648 (62.57 c) to be an enharmonic
diesis.

Just thought you'd like to know.

🔗monz <monz@attglobal.net>

hi Dave,

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> Monz,
>
> In
> http://sonic-arts.org/dict/19edo.htm
> you refer to 625:648 (62.57 c) as the enharmonic diesis.
>
> But the enharmonic dieses you described in
> http://sonic-arts.org/monzo/aristoxenus/tutorial.htm
> would seem to be in the approximate range of 45 to 56 cents
> (a fourth minus either a Pythagorean or just major third
> and then divided by two).
>
> And elsewhere, including Margo Schulter here
> /makemicromusic/topicId_166.html#166
> I find 125:128 (41.06 c) being called the enharmonic
> diesis, or simply the enharmonic.
>
> In Scala, or here
> http://www.xs4all.nl/~huygensf/doc/intervals.html
> we find
> 31:32 (54.96 c) called the Greek enharmonic, and
> 512:525 (43.41 c) called the Avicenna enharmonic diesis.
>
> Apart from your 19edo page, they all seem to point away
> from considering anything as large as 625:648 (62.57 c) to
> be an enharmonic diesis.
>
> Just thought you'd like to know.

thanks for that.

i have to point out that the reason i called that 19edo
(or 1/3-comma meantone ... they're essentially the same size)
interval the "enharmonic diesis" is simply because that's
the way that particular interval functions in those tunings,
i.e., the difference between Ab and G#.

you're correct that theory treatises generally refer to
the "enharmonic diesis" as being somewhere in the neighborhood
of 41 cents, which is roughly the size of the JI version
125:128 = [3, 5]-monzo [0, -3].

but in 19edo, treating it as 1/3-comma meantone and mapping
the pitch-names according to generator "5ths", the difference
between G#:Ab is that between the 12th and 13th degrees of
19edo, thus one degree of 19edo which equals ~63.15789474
cents.

so in this particular case, the epithet "enharmonic" refers
not to the size-range of the interval, but to its harmonic
or scalar function.

-monz

🔗Dave Keenan <d.keenan@bigpond.net.au>

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

> > Apart from your 19edo page, they all seem to point away
> > from considering anything as large as 625:648 (62.57 c) to
> > be an enharmonic diesis.
> >
> > Just thought you'd like to know.
>
>
>
> thanks for that.
>
> i have to point out that the reason i called that 19edo
> (or 1/3-comma meantone ... they're essentially the same size)
> interval the "enharmonic diesis" is simply because that's
> the way that particular interval functions in those tunings,
> i.e., the difference between Ab and G#.
>
> you're correct that theory treatises generally refer to
> the "enharmonic diesis" as being somewhere in the neighborhood
> of 41 cents, which is roughly the size of the JI version
> 125:128 = [3, 5]-monzo [0, -3].
>
> but in 19edo, treating it as 1/3-comma meantone and mapping
> the pitch-names according to generator "5ths", the difference
> between G#:Ab is that between the 12th and 13th degrees of
> 19edo, thus one degree of 19edo which equals ~63.15789474
> cents.
>
> so in this particular case, the epithet "enharmonic" refers
> not to the size-range of the interval, but to its harmonic
> or scalar function.

Silly me. I didn't read far enough. I was just searching on
"enharmonic diesis" and saw the value of 62.565148 cents and the terms
"great" and "diminished second" associated with it. But you are quite
correct, in precise 1/3-comma-meantone (but not 19-EDO) the tempered
size of 125;128 is precisely the untempered 625:648.

So who was the turkey who called 125:128 the "great" diesis? The
sooner we lose that name the better. I notice Scala doesn't use it.

And is it correct to call it a diminished second? Isn't G# to Ab a
_doubly_ diminished second? But considered as a doubly diminished
second (i.e. a Pythagorean comma), its tempered size in 1/3-comma
meantone is _minus_ 62.565148 cents.

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> 125:128 = [3, 5]-monzo [0, -3].

I've been meaning to say:

I don't think it is a good idea to use this terminology "[3,
5]-monzo". I always read the "[3, 5]" _as_ a monzo and then do a
double-take when I find the _real_ monzo such as the "[0, -3]" above.
After all, [3, 5]-monzo translates as 3^3*5^5-monzo or 84375-monzo.

I think it would be better to write e.g.

125:128 = 3,5-monzo [0, -3].

or just

125:128 = [0, -3]

After all, if the reader doesn't already know what the square brackets
mean, they probably won't be much the wiser for being told it is a
"3,5-monzo". "3,5-exponent vector" would be more useful. But I
understand you're saying "3,5" here because there has recently been
some vehement objection by one party, to the practice of leaving out
the 2-exponent, and the recently-proposed selective use of
commas(punctuation sense) can't be considered standard yet.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗monz <monz@attglobal.net>

hi Dave,

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

> Silly me. I didn't read far enough. I was just searching
> on "enharmonic diesis" and saw the value of 62.565148 cents
> and the terms "great" and "diminished second" associated
> with it. But you are quite correct, in precise
> 1/3-comma-meantone (but not 19-EDO) the tempered
> size of 125;128 is precisely the untempered 625:648.

just for the benefit of those trying to keep track:

the 1/3-comma meantone "enharmonic diesis" is the
ratio 625:648 = ~62.565148 cents.

one degree of 19edo is 2^(1/19) = ~63.15789474 cents.

19edo and 1/3-comma meantone are close enough in pitch
that there is probably never an audible difference.

> So who was the turkey who called 125:128 the "great" diesis?
> The sooner we lose that name the better. I notice Scala doesn't
> use it.

i'm pretty sure that the first turkey who did that was
Alexander Ellis, in one of his many voluminous appendices
to his translation of Helmholtz's _On The Sensations Of Tone_.

> And is it correct to call it a diminished second? Isn't G#
> to Ab a _doubly_ diminished second? But considered as a
> doubly diminished second (i.e. a Pythagorean comma), its
> tempered size in 1/3-comma meantone is _minus_ 62.565148 cents.

G#:A is a minor-2nd, so G#:Ab is simply a diminished-2nd.

examples of a doubly-diminished-2nd would be G#:Abb or Gx:Ab .

-monz

🔗monz <monz@attglobal.net>

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> hi Dave,
>
>
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
>
> > So who was the turkey who called 125:128 the "great" diesis?
> > The sooner we lose that name the better. I notice Scala doesn't
> > use it.
>
>
> i'm pretty sure that the first turkey who did that was
> Alexander Ellis, in one of his many voluminous appendices
> to his translation of Helmholtz's _On The Sensations Of Tone_.

Ellis's use of the qualifier "great" was to distinguish
125:128 from the small interval which we now call the
"magic comma", 3,5-monzo [-1, 5] = ratio 3072:3125 =
~29.61356846 cents, which he dubbed "small diesis".

Rameau, on the other hand, called 125:128 the "minor diesis",
in contrast to his "major diesis", 3,5-monzo [-5, 3] =
ratio 250:243 = ~49.16613727 cents.

it's important to note these two historical usages, if
only because (i assume) they've been repeated in so many
other theoretical works. exactly *how* often they've been
repeated, i couldn't say.

i do agree that a newer, more systematic method of naming
is called for, and am happy to be acknowledged as having
set some sort of precedent in that endeavor.

-monz

🔗monz <monz@attglobal.net>

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

> Ellis's use of the qualifier "great" was to distinguish
> 125:128 from the small interval which we now call the
> "magic comma", 3,5-monzo [-1, 5] = ratio 3072:3125 =
> ~29.61356846 cents, which he dubbed "small diesis".
>
>
> Rameau, on the other hand, called 125:128 the "minor diesis",
> in contrast to his "major diesis", 3,5-monzo [-5, 3] =
> ratio 250:243 = ~49.16613727 cents.

i'm quite happy to simply refer to 125:128 as the
"enharmonic diesis", since that is its notational
function in a JI analysis of "common-practice"
(i.e., meantone-based) music.

-monz

🔗monz <monz@attglobal.net>

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
>
> i'm quite happy to simply refer to 125:128 as the
> "enharmonic diesis", since that is its notational
> function in a JI analysis of "common-practice"
> (i.e., meantone-based) music.

unfortunately, under the criteria of notation, the
qualifier "enharmonic" also refers to other small
intervals a syntonic-comma (or its multiples) larger
and smaller than 125:128, such as

ratio 625:648 = <2 3,5>-monzo [3 4, -4] = ~62.565148 cents, and
ratio 2025:2048 = <2 3, 5>-monzo [11 -4, -2] = ~19.55256881 cents.

i'm also trying to establish here a new standard usage
of brackets. since using commas to separate groups of
exponents in a monzo eliminates the need for contrasting
brackets to indicate the presence or absence of prime-factor 2,
i propose that we use angle-brackets for the prime-factors
themselves, and square-brackets for the actual monzo of
the exponents.

-monz

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> > --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> >
> >
> > i'm quite happy to simply refer to 125:128 as the
> > "enharmonic diesis", since that is its notational
> > function in a JI analysis of "common-practice"
> > (i.e., meantone-based) music.
>
>
>
> unfortunately, under the criteria of notation, the
> qualifier "enharmonic" also refers to other small
> intervals a syntonic-comma (or its multiples) larger
> and smaller than 125:128, such as
>
> ratio 625:648 = <2 3,5>-monzo [3 4, -4] = ~62.565148 cents, and
> ratio 2025:2048 = <2 3, 5>-monzo [11 -4, -2] = ~19.55256881 cents.
>
>
> i'm also trying to establish here a new standard usage
> of brackets. since using commas to separate groups of
> exponents in a monzo eliminates the need for contrasting
> brackets to indicate the presence or absence of prime-factor 2,
> i propose that we use angle-brackets for the prime-factors
> themselves, and square-brackets for the actual monzo of
> the exponents.

Well I suppose you could, but I don't think this is necessary, and we
could save the angle-brackets for something more important. There is
obviously no need for selective use of commas between the primes
themselves. They provide no additional information there. I would
simply call these 2,3,5-monzos. Of course the important thing to know
is whether they are 2,...-monzos (complete monzos, octave specific
monzos) or 3,..-monzos (2-free monzos, octave equivalent monzos).

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

🔗monz <monz@attglobal.net>

hi Dave,

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

> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > G#:A is a minor-2nd, so G#:Ab is simply a diminished-2nd.
> >
> > examples of a doubly-diminished-2nd would be G#:Abb or Gx:Ab .
>
> Yes, of course. Thanks for taking the time to correct me.

nothing to it. i teach my students this stuff every day.

(... the minor-2nd and diminished-2nd, that is.

only the more advanced students get into double-sharps
and double-flats ... it usually takes a year or two of
lessons before we get that far into theory.)

-monz

🔗monz <monz@attglobal.net>

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> >
> > i'm also trying to establish here a new standard usage
> > of brackets. since using commas to separate groups of
> > exponents in a monzo eliminates the need for contrasting
> > brackets to indicate the presence or absence of prime-factor 2,
> > i propose that we use angle-brackets for the prime-factors
> > themselves, and square-brackets for the actual monzo of
> > the exponents.
>
> Well I suppose you could, but I don't think this is necessary,
> and we could save the angle-brackets for something more
> important. There is obviously no need for selective use of
> commas between the primes themselves. They provide no
> additional information there.

huh? they would simply separate the groups of primes to
show how the exponents are grouped. i know that that's
"a given", so i guess maybe you're right.

> I would simply call these 2,3,5-monzos.

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

> Of course the important thing to know is whether they
> are 2,...-monzos (complete monzos, octave specific
> monzos) or 3,..-monzos (2-free monzos, octave equivalent
> monzos).

of course ... which i why i continue to like labeling the
monzo with its constituent prime-factors, altho the comma
convention makes it unnecessary.

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

great minds thinking alike! ;-)

i had actually already thought of that too ... but since my
understanding of wedgies is lagging far behind that of many
of you others, i'll refrain from commenting further.

-monz

🔗Paul Erlich <perlich@aya.yale.edu>

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

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

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

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

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

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

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

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
>
> > So who was the turkey who called 125:128 the "great" diesis?
>
> probably the same person (not bird) who called 3125:3072 the "small"
> diesis.

Yes. Monz already told me it was Ellis for both of these, and it's all
beautifully described at
http://sonic-arts.org/dict/diesis.htm
I should have looked there first.

You're right, Ellis was no turkey.

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

In George's and my comma naming system it is called the 3125-comma or
5^5-comma, but it doesn't get its own sagittal symbol because it's too
close to what we assume will be notationally-more-common; the
5:7-comma [-9 6, 1 -1] 29.22 c, whose symbol is, in ASCII longhand,
'|) which literally represents a 5-schisma plus a 7-comma.

> > And is it correct to call it a diminished second? Isn't G# to Ab a
> > _doubly_ diminished second?
>
> no. G to Ab is a minor second, and G# to Ab is a diminished second.

Yes. Thanks. Monz already corrected me on that.

> > its tempered size in 1/3-comma
> > meantone is _minus_ 62.565148 cents.
>
> no, G# is lower than Ab in 1/3-comma meantone.

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

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> > >
> > > i'm also trying to establish here a new standard usage
> > > of brackets. since using commas to separate groups of
> > > exponents in a monzo eliminates the need for contrasting
> > > brackets to indicate the presence or absence of prime-factor 2,
> > > i propose that we use angle-brackets for the prime-factors
> > > themselves, and square-brackets for the actual monzo of
> > > the exponents.
> >
> > Well I suppose you could, but I don't think this is necessary,
> > and we could save the angle-brackets for something more
> > important. There is obviously no need for selective use of
> > commas between the primes themselves. They provide no
> > additional information there.
>
>
> huh? they would simply separate the groups of primes to
> show how the exponents are grouped. i know that that's
> "a given", so i guess maybe you're right.

I assumed that's what you intended, and there's no harm in a bit of
redundancy. But I see this as _doubly_ redundant.

If you know the system with the commas in the monzos, then you know
that e.g. [1 1, 1] is a 2,3,5-monzo and not a 3,5,7-monzo, so adding
the words "2,3,5-monzo" is redundant (but still a good idea). And if
you're told it's a 2,3,5-monzo then you can ignore the commas and just
line up exponents with primes. So "2,3 5-monzo" is doubly redundant
and forces us to use something like the angle-brackets to hold it
together: <2,3 5>-monzo.

But while we intend the monzo [1, 2 3] to be quite different from the
monzo [1 2, 3], a 2,3,5-monzo is of course no different from a <2,3
5>-monzo.

Someone seeing the term "<2,3 5>-monzo" might wonder if there were
such things as <2 3,5>-monzos and <2 3 5>-monzos, as different categories.

> > I would simply call these 2,3,5-monzos.
>
>
> simply using a comma between *every* prime, and no spaces.
> i suppose i like that. (i don't sound too convinced, tho.)

I think mathematicians name things like that all the time. Don't they
Gene?

Don't forget to add the stuff about the selective comma(punctuation
sense) convention to your "monzo" dictionary entry when you get a chance.

> > Maybe a good use for angle-brackets would be for wedgies
> > since, as I understand it, these are in a _very_ different
> > domain from that of monzos, and the angle-brackets are
> > suggestive of wedges themselves.
>
> great minds thinking alike! ;-)
>
> i had actually already thought of that too ... but since my
> understanding of wedgies is lagging far behind that of many
> of you others, i'll refrain from commenting further.

I'm afraid I don't understand them either. I couldn't even tell you
the difference between a wedgie and a val. :-) I read Gene's "val"
definition in your dictionary but I'm none the wiser. I don't see an
entry for "wedgie". I guess some time I'd like to see some carefully
explained examples of how vals and wedgies are used. I've never
managed to follow it on tuning-math. I fully expect that when I
eventually put in the effort to understand these things, I'll say "Is
that all they are? Well why didn't you _say_ so?". :-)

-- Dave Keenan

🔗Gene Ward Smith <gwsmith@svpal.org>

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

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

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> > simply using a comma between *every* prime, and no spaces.
> > i suppose i like that. (i don't sound too convinced, tho.)
>
> I think mathematicians name things like that all the time. Don't
they
> Gene?

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

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

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

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > > Maybe a good use for angle-brackets would be for wedgies
> > > since, as I understand it, these are in a _very_ different
> > > domain from that of monzos, and the angle-brackets are
> > > suggestive of wedges themselves.
> >
> >
> > great minds thinking alike! ;-)
>
> I see no point in it myself; one is hardly likely to confuse a wedgie
> with an interval, though telling an 11-limit linear wedgie from an 11-
> limit planar wedgie is another matter. There are also vals to
> consider; it's too hard writing these as column vectors, which is
> really the clearest way to do it.

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

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
> > > simply using a comma between *every* prime, and no spaces.
> > > i suppose i like that. (i don't sound too convinced, tho.)
> >
> > I think mathematicians name things like that all the time. Don't
> they
> > Gene?
>
> They name things all the time, but usually abstract things, not
> notations. Those have a tendency to change by authorial whim.

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

> > I'm afraid I don't understand them either. I couldn't even tell you
> > the difference between a wedgie and a val. :-) I read Gene's "val"
> > definition in your dictionary but I'm none the wiser. I don't see an
> > entry for "wedgie". I guess some time I'd like to see some carefully
> > explained examples of how vals and wedgies are used.
>
> My web site seems to have a problem though the site itself is up, but
> I assume it will be working soon. If so, you might take a look at
> what's already there, and suggest what you think ought to be there.

The URL again?

🔗Gene Ward Smith <gwsmith@svpal.org>

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

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

Physicists sometimes use bra and ket vectors; if you did that, the
monzo for 81/80 would be a ket, [-4 4 -1> and the val for 5-limit 12-
et would be a bra, <12 19 28]. Putting them together would give the
bra-ket, angle bracket, or inner product: <12 19 28 | -4 4 -1> = 0.

See:

http://mathworld.wolfram.com/ContravariantVector.html

http://mathworld.wolfram.com/CovariantVector.html

http://mathworld.wolfram.com/Ket.html

http://mathworld.wolfram.com/Bra.html

http://mathworld.wolfram.com/One-Form.html

http://mathworld.wolfram.com/AngleBracket.html

🔗Graham Breed <graham@microtonal.co.uk>

🔗monz <monz@attglobal.net>

hello all,

did we ever reach a consensus on the use or
non-use of angle-brakets and/or pipe symbols
in expressing monzos? i have not been following
tuning-math much lately, and just want to be
sure that the Encyclopaedia entry is up-to-date.

-monz

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

/tuning-math/message/7527

> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
> wrote:
>
> > I admit to being shamefully ignorant of these things.
> > It sounds then like vals could use the angle-brackets.
>
> Physicists sometimes use bra and ket vectors; if you
> did that, the monzo for 81/80 would be a ket, [-4 4 -1>
> and the val for 5-limit 12-et would be a bra, <12 19 28].
> Putting them together would give the bra-ket, angle bracket,
> or inner product: <12 19 28 | -4 4 -1> = 0.
>
> See:
>
> http://mathworld.wolfram.com/ContravariantVector.html
>
> http://mathworld.wolfram.com/CovariantVector.html
>
> http://mathworld.wolfram.com/Ket.html
>
> http://mathworld.wolfram.com/Bra.html
>
> http://mathworld.wolfram.com/One-Form.html
>
> http://mathworld.wolfram.com/AngleBracket.html

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗monz <monz@attglobal.net>

hi paul and Gene,

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> >
> > --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> > > hello all,
> > >
> > >
> > > did we ever reach a consensus on the use or
> > > non-use of angle-brakets and/or pipe symbols
> > > in expressing monzos?
> >
> > Not really. Angle-brakets seem to be well accepted,
> > but so far as "|" versus "]" goes, some people use one,
> > and some the other.
>
> The monzo would have a "[", not a "]". I'm using these,
> not pipe symbols or even vertical lines, in my paper.

it was my understanding that the monzo by itself uses [...>
and the val uses <...] , and that putting them together
one would use the pipe symbol instead of the two square
brackets thus: <...|...> . has this become established usage?

also, what about the suggestion to use comma punctuation
after the exponents of 3, 11, 19, 31, etc.? is that
established at all?

-monz

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> hi paul and Gene,
>
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <gwsmith@s...>
> > wrote:
> > >
> > > --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> > > > hello all,
> > > >
> > > >
> > > > did we ever reach a consensus on the use or
> > > > non-use of angle-brakets and/or pipe symbols
> > > > in expressing monzos?
> > >
> > > Not really. Angle-brakets seem to be well accepted,
> > > but so far as "|" versus "]" goes, some people use one,
> > > and some the other.
> >
> > The monzo would have a "[", not a "]". I'm using these,
> > not pipe symbols or even vertical lines, in my paper.
>
>
> it was my understanding that the monzo by itself uses [...>
> and the val uses <...] , and that putting them together
> one would use the pipe symbol instead of the two square
> brackets thus: <...|...> . has this become established usage?

That's the way my paper does it. However I don't use the word "monzo"
(sorry joe), but I've suggested here in a post entitled "Who's Val?"
that if they're called monzos, their counterparts should be called
breeds.

> also, what about the suggestion to use comma punctuation
> after the exponents of 3, 11, 19, 31, etc.? is that
> established at all?

Some people liked that, but someone didn't -- was it Dave or George?

🔗Carl Lumma <ekin@lumma.org>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗monz <monz@attglobal.net>

hi paul,

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > >However I don't use the word "monzo"
> >
> > Why not?
> >
> > -Carl
>
> I want the minimum of jargon, and the maximum sense
> of "I could have thought of this myself and I probably
> did at one point" in the reader.

i guess i'm just being selfish, but i am disappointed,
and wish you'd use "monzo" in your paper ... mainly
because it's my feeling that its use in a paper by *you*
would give the term a cachet that it doesn't currently have.

as Gene pointed out when he first named the "vector of
prime-factor exponents" after me, it's useful to have
one word to replace a whole phrase, if you're going to
be referring to it often.

-monz

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> hi paul,
>
>
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >
> > > >However I don't use the word "monzo"
> > >
> > > Why not?
> > >
> > > -Carl
> >
> > I want the minimum of jargon, and the maximum sense
> > of "I could have thought of this myself and I probably
> > did at one point" in the reader.
>
>
> i guess i'm just being selfish, but i am disappointed,
> and wish you'd use "monzo" in your paper ...

Don't worry . . . you (along with Carl Lumma and Joseph Pehrson) are
credited in the paper with providing "crucial germinative interest".

I don't refer to the general item enough to warrant giving it a
name . . . i say "in vector form the syntonic comma is represented
as [-4 4 -1>" in between an arithmetic and a geometrical explanation
of this fact, and thereafter i simply give specific ones without
referring to them by a general name (rather, i use them, as well as
standard ratios, to represent commas, so i simply refer to them as
commas).

🔗Carl Lumma <ekin@lumma.org>

>> > > >However I don't use the word "monzo"
>> > >
>> > > Why not?
>> >
>> > I want the minimum of jargon, and the maximum sense
>> > of "I could have thought of this myself and I probably
>> > did at one point" in the reader.
>>
>> i guess i'm just being selfish, but i am disappointed,
>> and wish you'd use "monzo" in your paper ...
>
>Don't worry . . . you (along with Carl Lumma and Joseph Pehrson) are
>credited in the paper with providing "crucial germinative interest".
>
>I don't refer to the general item enough to warrant giving it a
>name . . . i say "in vector form the syntonic comma is represented
>as [-4 4 -1>" in between an arithmetic and a geometrical explanation
>of this fact, and thereafter i simply give specific ones without
>referring to them by a general name (rather, i use them, as well as
>standard ratios, to represent commas, so i simply refer to them as
>commas).

I'm sure you're doing the right thing for your paper, Paul, as
you did with The Forms of Tonality. Come to think of it, has Forms
ever been submitted to something like Xenharmonikon or 1/1?

-Carl

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> > > >However I don't use the word "monzo"
> >> > >
> >> > > Why not?
> >> >
> >> > I want the minimum of jargon, and the maximum sense
> >> > of "I could have thought of this myself and I probably
> >> > did at one point" in the reader.
> >>
> >> i guess i'm just being selfish, but i am disappointed,
> >> and wish you'd use "monzo" in your paper ...
> >
> >Don't worry . . . you (along with Carl Lumma and Joseph Pehrson)
are
> >credited in the paper with providing "crucial germinative
interest".
> >
> >I don't refer to the general item enough to warrant giving it a
> >name . . . i say "in vector form the syntonic comma is
represented
> >as [-4 4 -1>" in between an arithmetic and a geometrical
explanation
> >of this fact, and thereafter i simply give specific ones without
> >referring to them by a general name (rather, i use them, as well
as
> >standard ratios, to represent commas, so i simply refer to them as
> >commas).
>
> I'm sure you're doing the right thing for your paper, Paul, as
> you did with The Forms of Tonality. Come to think of it, has Forms
> ever been submitted to something like Xenharmonikon or 1/1?
>
> -Carl

XH can't do color. I don't know if it's "JI" enough for 1/1.

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> as Gene pointed out when he first named the "vector of
> prime-factor exponents" after me, it's useful to have
> one word to replace a whole phrase, if you're going to
> be referring to it often.

I actually named it after you initially in the descriptions of some
of my Maple routines written for my own personal use, without any
thought of making it public, but then I started thinking of them as
monzos and didn't have another name for them. I *did* have a name for
vals, but I suppose we could use a name for vals as explicitly
written out bra vectors, if we want to become obsessional about it.

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> >
> > it was my understanding that the monzo by itself uses [...>
> > and the val uses <...] , and that putting them together
> > one would use the pipe symbol instead of the two square
> > brackets thus: <...|...> . has this become established usage?
>
> That's the way my paper does it. However I don't use the
word "monzo"
> (sorry joe), but I've suggested here in a post entitled "Who's
Val?"
> that if they're called monzos, their counterparts should be called
> breeds.
>
> > also, what about the suggestion to use comma punctuation
> > after the exponents of 3, 11, 19, 31, etc.? is that
> > established at all?
>
> Some people liked that, but someone didn't -- was it Dave or George?

I'm the one who suggested the idea of the comma punctuation, because
it remedies the difficulty in reading simple ratios involving higher
primes (such as 23/16) -- something I perceived to be a serious
problem with Monz's notation. See:
/tuning-math/message/7435

I think there was only one person who disliked the idea (I'm pretty
sure it wasn't Dave, because he's a stickler for perceptual
improvements in notation), IIRC on the grounds that the commas are a
convenience rather than a necessity. I responded that commas are
widely accepted as serving a similar purpose as place-markers in
large decimal numbers. However, not enough people expressed any
further opinions one way or another for my suggestion to be either
adopted or rejected.

Now's the time for others to speak up so the issue can be resolved.

--George

🔗monz <monz@attglobal.net>

hi George (and everyone else),

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> > >
> > > also, what about the suggestion to use comma punctuation
> > > after the exponents of 3, 11, 19, 31, etc.? is that
> > > established at all?
> >
> > Some people liked that, but someone didn't -- was it
> > Dave or George?
>
> I'm the one who suggested the idea of the comma
> punctuation, because it remedies the difficulty in
> reading simple ratios involving higher primes
> (such as 23/16) -- something I perceived to be a
> serious problem with Monz's notation. See:
> /tuning-math/message/7435
>
> I think there was only one person who disliked the
> idea (I'm pretty sure it wasn't Dave, because he's
> a stickler for perceptual improvements in notation),
> IIRC on the grounds that the commas are a convenience
> rather than a necessity. I responded that commas are
> widely accepted as serving a similar purpose as
> place-markers in large decimal numbers. However,
> not enough people expressed any further opinions one
> way or another for my suggestion to be either
> adopted or rejected.
>
> Now's the time for others to speak up so the issue
> can be resolved.

i did agree with George that comma punctuation is good,
after the exponent of 3 and then after every third exponent
after that.

yes, let's please resolve it now so that i can put it
into the Encyclopaedia and be done with it.

-monz

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
> I think there was only one person who disliked the idea (I'm pretty
> sure it wasn't Dave, because he's a stickler for perceptual
> improvements in notation), IIRC on the grounds that the commas are a
> convenience rather than a necessity. I responded that commas are
> widely accepted as serving a similar purpose as place-markers in
> large decimal numbers. However, not enough people expressed any
> further opinions one way or another for my suggestion to be either
> adopted or rejected.
>
> Now's the time for others to speak up so the issue can be resolved.

Yes. I wholeheartedly support putting a comma after the 3-exponent and
then after every third exponent beyond that. Not only because of the
improvement in readability (not having to count all the way from the
end to figure which exponent you're looking at), but also because it
allows one to distinguish octave-equivalent vectors (and even the
2,3-reduced ones George and I sometimes use in describing notational
commas relative to a chain of fifths) from complete vectors.

[2 3, 5 7 11, 13 17 19> ordinary, complete
[3, 5 7 11, 13 17 19> octave-equivalent, 2-reduced
[, 5 7 11, 13 17 19> 2,3-reduced

And I agree that 11 and 19 are two natural stopping places, for
various reasons.

I don't remember anyone objecting to these commas, but maybe some
wouldn't bother using them that way themselves.

I don't like the pipe or vertical bar since it is too easily confused
with a digit one. You folks who still have perfect eyesight may not
think so, but wait 'til you get a bit older.

Even for the inner product I prefer
<2 3, 5][2 3, 5>
or
<2 3, 5].[2 3, 5>