EDO harmonic properties you've never seen before

The dominance of the "limit" idea means that, except for the efforts
of pioneers like Margo Schulter and Dan Stearns, we know very little
about, for example, EDOs that are good at 1:7:9:11 chords when they
can't do the full 1:3:5:7:9:11.

I've just put up a spreadsheet on my website that tabulates all 512
possible subsets of the 19-limit against the EDOs from 5 to 171. When
the EDO is consistent for that set of harmonics it shows the maximum
error in cents, otherwise the cell is blank.

For me, the desire to have this information accessible came from the
19-limit-notation-for-EDOs (ETs) project. It is desirable to notate an
EDO based on a set of harmonics that it renders consistently and with
low errors.

See http://dkeenan.com/Music/EDOErrorsAndConsistency.xls.zip
Zipped Excel spreadsheet 302KB

-- Dave Keenan

> From: "dkeenanuqnetau" <d.keenan@uq.net.au>
> Subject: EDO harmonic properties you've never seen before
>
> The dominance of the "limit" idea means that, except for the efforts
> of pioneers like Margo Schulter and Dan Stearns, we know very little
> about, for example, EDOs that are good at 1:7:9:11 chords when they
> can't do the full 1:3:5:7:9:11.
>

Wonderful spreadsheet, Dave.

I've been grinding a few of these out by hand and
having them nicely put together like this is fantastic. My interest
came from 34 (looking at it as 1,3,5,9,11,13) and 28 (which other than
a great 5/4 and some very nice embedded ED identities, I couldnt fully
grok. Now I realize it is a very good solution to 1,5,11,19 which is
probably still a cross between a pelican and a platypus).

thanks a lot,

Bob Valentine

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

/tuning/topicId_34763.html#34763

> See http://dkeenan.com/Music/EDOErrorsAndConsistency.xls.zip
> Zipped Excel spreadsheet 302KB
>
> -- Dave Keenan

****This is really an *amazing* sheet, Dave! I can't imagine how you
did this... It's a nice extension to some of Paul Erlich's ET
studies...

Just for review (for the "lurkers" on this list, of course, not
because I, myself, don't understand it... :) )

It seems you are considering notating ETs by using symbols for higher-
level commas and you are using an "extended" set of written and ASCII
symbols based upon the Sims notation?? (That's what I'm rather
fantasizing...)

So there will, ultimately, be only *one* general set that will
somehow notate *everything...*

Please let me know if I'm getting a glimmer of this.

Thanks!

JP

> From: dkeenanuqnetau <d.keenan@uq.net.au>
> To: <tuning@yahoogroups.com>
> Sent: Saturday, February 23, 2002 11:12 PM
> Subject: [tuning] EDO harmonic properties you've never seen before
>
>
> ...
>
> I've just put up a spreadsheet on my website that tabulates all 512
> possible subsets of the 19-limit against the EDOs from 5 to 171. When
> the EDO is consistent for that set of harmonics it shows the maximum
> error in cents, otherwise the cell is blank.
>
> ...
>
> See http://dkeenan.com/Music/EDOErrorsAndConsistency.xls.zip
> Zipped Excel spreadsheet 302KB

BRAVO, Dave! this is really great.

i've just made a new Dictionary entry for "error", in which i
chose two subsets of your spreadsheet (11-limit, 5-22edo and 5-41edo)
to make some column graphs showing the error.

http://www.ixpres.com/interval/dict/error.htm

-monz

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning@y..., "monz" <joemonz@y...> wrote:

/tuning/topicId_34763.html#34792

>
> i've just made a new Dictionary entry for "error", in which i
> chose two subsets of your spreadsheet (11-limit, 5-22edo and 5-
41edo) to make some column graphs showing the error.
>
> http://www.ixpres.com/interval/dict/error.htm
>

****These are very colorful, Monz! I'm concerned that the labeling
situation, though, makes them less optimal than they *should* be.

Why don't you take the automatic text out, and just *manually* add
(and rotate vertically) teeny-tiny text for each column.

Or choose smaller segments of columns with the text.

Just an idea... pay no attention to me... :)

JP

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote in the thread
"EDO harmonic properties you've never seen before":
> It seems you are considering notating ETs by using symbols for
higher-
> level commas and you are using an "extended" set of written and
ASCII
> symbols based upon the Sims notation?? (That's what I'm rather
> fantasizing...)
>
> So there will, ultimately, be only *one* general set that will
> somehow notate *everything...*
>
> Please let me know if I'm getting a glimmer of this.

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
in the "EDO harmonic properties you've never seen before" thread

> It seems you are considering notating ETs by using symbols for
higher-
> level commas and you are using an "extended" set of written and
ASCII
> symbols based upon the Sims notation?? (That's what I'm rather
> fantasizing...)
>
> So there will, ultimately, be only *one* general set that will
> somehow notate *everything...*
>
> Please let me know if I'm getting a glimmer of this.

Well of course the spreadsheet
http://dkeenan.com/Music/EDOErrorsAndConsistency.xls.zip
(Zipped Excel spreadsheet 302KB)
has many uses other than the one that prompted me to generate it.

It partly supercedes Paul Hahn's consistency and error tables which
have been an invaluable resource for a decade or so. Paul Hahn only
listed the strict odd-limits but he went to 31-limit and to 1200-EDO
in some tables, and 200-EDO in others.

Folks can extract their own information, e.g. search for an ET for
some specific harmonic purpose, by adding their own formulae. For
example, Margo Schulter might add a row that only shows the
1,3,7,9,11,13 (no 5) max-error when the ET is not also 1,3,5,7,9,11,13
consistent.

So your question really belongs to the "A common notation for JI and
ETs" thread which is why I've answered it here.

The common-notation project is at this stage quite independent of the
specific symbols used. For my own convenience I am currently using the
ASCII version of the Sims accidentals v^ <> [] for the 5 7 and 11
commas and some other up/down pairs of characters that no-one is
currently using as ASCIIdentals. These don't necessarily relate to the
possible extended Sims accidentals I posted earlier as ASCII-grafix.
In particular I think the 17 and 19 commas should be represented by
symbols that give some indication of their small size relative to the
others. Mind you, very few ETs below 72-tET will require the use of a
17 or 19-comma symbol and they will all be ETs that are rarely used.
Even the 13-comma will be rare below 72-tET.

Manuel Op de Coul and Daniel Wolf have gotten to me somewhat and
convinced me that there is a living European microtonal tradition that
is incompatible with the Sims notation, and in particular, that the
problem with the slashes \ / as syntonic comma symbols can be
mitigated by putting a short vertical stroke thru the up-comma slash,
thereby making it reminiscent of a "+" sign and avoiding left-right
confusability. [You'll need Message Index, Expand Messages to see this
correctly on Yahoo's dopey web interface].

/
|/
|
/|
/

I think the traditions agree reasonable well on the septimal comma
symbols (L7, or <> in ACSII). Although I'm worried that perhaps
someone writes a harmonic seventh as G:F7 rather than G:FL? Anyone
know?

And of course we know the Europeans (and some US residents) want to
use the up and down arrows (ASCII v^) for the 11-diesis because they
have been used for quartertones.

Well I may just have to find _two_ sets of 19-limit symbols, in order
to encourage acceptance of the basic idea of notating ETs using
19-limit JI, but I will never ever support the use of ordinary "-" and
"+" as accidentals on scores.

Now Joseph there's another way in which this project is probably not
the answer to all your prayers (I don't think such an answer exists),
and this relates to the recent confusion over the two different ways
of understanding and using commas. If we use v^ <> and [] to notate a
whole host of ETs, then no matter how we do it, there is no way that
they can always represent the same pitch change in cents, across all
the ETs. I don't even expect # and b to do that.

The only feasible way of notating all ETs so that the notation
indicates shifts from 12-tET, is the Johnny Reinhardt method of
writing the + or - cents next to the notes.

As I understand it, Dan Stearn's notational proposal is similar to
Johny's in that it tries to use symbols to mean specific numbers of
cents from 12-tET, instead of meaning the difference between a
particular prime ratio and a particular-length chain of fifths, but
one problem is that it forces you to round the deviations to the
nearest multiple of 8&1/3 cents. I just don't think this is good
enough. And it certainly doesn't have the same property as Gene's and
my Rapoport-style proposal, that the best approximation to a
4:5:6:7:9:11 chord (for example) is always spelled G:Bv:D:F<:A:C] no
matter what ET you're in (except when the commas, and therefore their
accidentals, vanish in some ET).

Slightly off-topic, I have a proposal for a shorthand that allows both
Blackjack and 22-tET to be notated with only one accidental per note
and without slashes thru noteheads as Alison Monteith has done.
Someone who can make or edit fonts, should make a pair of symbols that
are a standard flat symbol with an arrow head pointing upwards on the
top of the vertical stroke and a standard sharp symbol with an
arrowhead pointing down on the bottom of the leftmost vertical stroke.

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> /
> |/
> |
> /|
> /

How is this relevant to ascii symbolism?

> Well I may just have to find _two_ sets of 19-limit symbols, in order
> to encourage acceptance of the basic idea of notating ETs using
> 19-limit JI, but I will never ever support the use of ordinary "-" and
> "+" as accidentals on scores.

Let's please not confuse things further.

I don't even expect # and b to do that.

Of course they don't.

> The only feasible way of notating all ETs so that the notation
> indicates shifts from 12-tET, is the Johnny Reinhardt method of
> writing the + or - cents next to the notes.

I imagine some other unit than cents would be feasible, for starters.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > /
> > |/
> > |
> > /|
> > /
>
> How is this relevant to ascii symbolism?

Because folk want the ASCII symbols to look like (or at least suggest)
the real symbols used on scores.

I was previously able to reject the use of \ / as comma symbols and
therefore the whole (for want of a better term) "European" system of
\/ <> v^ for 5, 7 11. If they make \ / not be left-right confusable in
real life (as above), then I can't reject them. Those guys seem pretty
adamant about keeping v^ for quarter tones (and hence the 11-diesis)
and it fits with George Secor's stuff too. The fact that Manuel Op de
Coul isn't about to change Scala to correspond to the Sims notation,
is fairly serious.

> > Well I may just have to find _two_ sets of 19-limit symbols, in
order
> > to encourage acceptance of the basic idea of notating ETs using
> > 19-limit JI, but I will never ever support the use of ordinary "-"
and
> > "+" as accidentals on scores.
>
> Let's please not confuse things further.

So Monz, how about you stop supporting + and - as accidentals on
actual scores and consider them only as ASCII approximations of \ and
/-with-a short-vertical-line-thru-it?

I'd hate it if the whole idea of notating ETs the same as 19-limit JI
got rejected on either side of the Atlantic just because it chose the
wrong set of symbols.

To me, getting down to a single set of symbols is a different
struggle, feel free to work on it. But I guess we should at least aim
for 13,17,19 symbols that will fit in with both systems.

> I don't even expect # and b to do that.
>
> Of course they don't.

No. But some folks expect an ET notation system where Ab always equals
G# etc.

> > The only feasible way of notating all ETs so that the notation
> > indicates shifts from 12-tET, is the Johnny Reinhardt method of
> > writing the + or - cents next to the notes.
>
> I imagine some other unit than cents would be feasible, for
starters.

Depends what you mean by "feasible". I meant "sufficiently accurate
while being easily understood by performers with ordinary training".

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> > I imagine some other unit than cents would be feasible, for
> starters.
>
> Depends what you mean by "feasible". I meant "sufficiently accurate
> while being easily understood by performers with ordinary training".

Why not 612 instead of 1200?

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > > I imagine some other unit than cents would be feasible, for
> > starters.
> >
> > Depends what you mean by "feasible". I meant "sufficiently
accurate
> > while being easily understood by performers with ordinary
training".
>
> Why not 612 instead of 1200?

Because cents are already a standard, and even if they weren't, most
people are used to understanding and estimating percentages in daily
life. Performers will far more easily understand, estimate and produce
a certain _percent_ of the distance between 12-tET notes, compared to
a certain number of fifty-oneths of it.

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> Because cents are already a standard, and even if they weren't, most
> people are used to understanding and estimating percentages in daily
> life. Performers will far more easily understand, estimate and produce
> a certain _percent_ of the distance between 12-tET notes, compared to
> a certain number of fifty-oneths of it.

I doubt it. Do you really think someone can learn to make something 37 cents above 300 by mental math?

On 2/25/02 1:53 PM, "genewardsmith" <genewardsmith@juno.com> wrote:

> --- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
>> Because cents are already a standard, and even if they weren't, most
>> people are used to understanding and estimating percentages in daily
>> life. Performers will far more easily understand, estimate and produce
>> a certain _percent_ of the distance between 12-tET notes, compared to
>> a certain number of fifty-oneths of it.
>
> I doubt it. Do you really think someone can learn to make something 37 cents
> above 300 by mental math?

I hear some fellow named Reinhard can...

:::runs away:::

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

/tuning/topicId_34763.html#34811

****Thanks, Dave, for all your help with this. I think I pretty much
understand your post.

>
> Slightly off-topic, I have a proposal for a shorthand that allows
both Blackjack and 22-tET to be notated with only one accidental per
note and without slashes thru noteheads as Alison Monteith has done.
> Someone who can make or edit fonts, should make a pair of symbols
that are a standard flat symbol with an arrow head pointing upwards
on the top of the vertical stroke and a standard sharp symbol with an
> arrowhead pointing down on the bottom of the leftmost vertical
stroke.

***I *did* want to mention that what you are envisioning here are
actually *standard* symbols in several music notation programs. I've
seen them both in SCORE and in Sibelius...

JP

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

/tuning/topicId_34763.html#34818

> I was previously able to reject the use of \ / as comma symbols and
> therefore the whole (for want of a better term) "European" system
of \/ <> v^ for 5, 7 11. If they make \ / not be left-right
confusable in real life (as above), then I can't reject them. Those
guys seem pretty adamant about keeping v^ for quarter tones (and
hence the 11-diesis) and it fits with George Secor's stuff too. The
fact that Manuel Op de Coul isn't about to change Scala to correspond
to the Sims notation, is fairly serious.
>

****I guess I'll have to confess that it really does look like
there's no "worldwide" standard... :)

Well, we're back to making sure we have LEGENDS on our scores, like
back in the *wild* 60's... where half of the music was the invention
of a new *notation...* :)

> > > Well I may just have to find _two_ sets of 19-limit symbols, in
> order
> > > to encourage acceptance of the basic idea of notating ETs using
> > > 19-limit JI, but I will never ever support the use of
ordinary "-" and "+" as accidentals on scores.

****Your point, Dave, about these little symbols for the syntonic
comma being illegible is, I believe well taken. I agree with Monz
that they are kind of "neat" for the small alterations of that comma,
but, really they don't work on scores.

\ and / are considerably better, I believe...

>
> > > The only feasible way of notating all ETs so that the notation
> > > indicates shifts from 12-tET, is the Johnny Reinhardt method of
> > > writing the + or - cents next to the notes.
> >
> > I imagine some other unit than cents would be feasible, for
> starters.
>
> Depends what you mean by "feasible". I meant "sufficiently accurate
> while being easily understood by performers with ordinary training".

***Yes, let's please keep that in mind. Theory is *great* but we're
talking about, maybe, music that is to be *played...* (I hope some
of it, anyway...) :)

JP

--- In tuning@y..., "monz" <joemonz@y...> wrote:

> i've just made a new Dictionary entry for "error", in which i
> chose two subsets of your spreadsheet (11-limit, 5-22edo and 5-
41edo)
> to make some column graphs showing the error.
>
> http://www.ixpres.com/interval/dict/error.htm

good info, but unfortunately

·'primes' is not a good annotation, since 9 is not a prime;

·the 3d bar graph doesn't really help get the information across,
since the order of sets of odds is arbitrary.