new Tuning Dictionary page: EDO prime errors

hello all,

i've just created a new webpage for the Tuning Dictionary
which shows the errors of EDO representations of prime-factors,
from 3 to 43 (an arbitrary limit), in cents and in percentages
of one EDO-step, for several of the most popular EDOs:

http://sonic-arts.org/dict/edo-prime-error.htm

it includes, at the bottom, a mouse-over applet.

-monz

>i've just created a new webpage for the Tuning Dictionary
>which shows the errors of EDO representations of prime-factors,
>from 3 to 43 (an arbitrary limit), in cents and in percentages
>of one EDO-step, for several of the most popular EDOs:

>http://sonic-arts.org/dict/edo-prime-error.htm

>-monz

Monz,

Whew! For a moment there, when I saw your headline, I thought you were saying there had been errors in your dictionary - and for a moment (but only a moment), my faith was shaken. Thank goodness the errors turn out to be in the musical phenomena themselves, not in the reference work whose infallibility I count on.

Cheers,

Kyle "Say it ain't so, Joe" Gann

hi Kyle,

--- In tuning@yahoogroups.com, Kyle Gann <kgann@e...> wrote:

> > i've just created a new webpage for the Tuning Dictionary
> > which shows the errors of EDO representations of prime-factors,
> > from 3 to 43 (an arbitrary limit), in cents and in percentages
> > of one EDO-step, for several of the most popular EDOs:
>
> > http://sonic-arts.org/dict/edo-prime-error.htm
>
> >-monz
>
> Monz,
>
> Whew! For a moment there, when I saw your headline, I thought
> you were saying there had been errors in your dictionary -
> and for a moment (but only a moment), my faith was shaken.
> Thank goodness the errors turn out to be in the musical
> phenomena themselves, not in the reference work whose
> infallibility I count on.
>
> Cheers,
>
> Kyle "Say it ain't so, Joe" Gann

thanks for the vote of confidence! :)

of course, the Dictionary was/is written (mostly) by me,
and i'm human, and therefore fallible. ... but thankfully,
paul erlich is *always* on my case correcting any errors
*he* finds! (and he's good at it).

-monz

as you know, monz, i think this is not a good way to evaluate how
well an ET represents JI sonorities, or even how the primes are best
represented in a particular ET. for instance, i find that in a major
triad, getting a good approximation of the 6:5 just minor third is
nearly, if not equally, important as getting a good approximation of
the 5:4 just major third, and when one considers the fifth too the
second-closest approximation of prime 5 is often a better choice than
the closest one.

and of course that's only considering a single chord -- many other
issues come into play when considering progressions, etc.

if there's one thing i hope to have communicated in 7 years on this
list, it's the above.

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> hello all,
>
>
> i've just created a new webpage for the Tuning Dictionary
> which shows the errors of EDO representations of prime-factors,
> from 3 to 43 (an arbitrary limit), in cents and in percentages
> of one EDO-step, for several of the most popular EDOs:
>
> http://sonic-arts.org/dict/edo-prime-error.htm
>
>
> it includes, at the bottom, a mouse-over applet.
>
>
>
> -monz

hi paul,

re:
http://sonic-arts.org/dict/edo-prime-error.htm

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

> as you know, monz, i think this is not a good way to evaluate
> how well an ET represents JI sonorities, or even how the
> primes are best represented in a particular ET. for instance,
> i find that in a major triad, getting a good approximation of
> the 6:5 just minor third is nearly, if not equally, important
> as getting a good approximation of the 5:4 just major third,
> and when one considers the fifth too the second-closest
> approximation of prime 5 is often a better choice than
> the closest one.
>
> and of course that's only considering a single chord -- many
> other issues come into play when considering progressions, etc.
>
> if there's one thing i hope to have communicated in 7 years on
> this list, it's the above.

yes, i did already know that you felt this way, and i also
knew that you were going to mention it after i announced
the new webpage.

however, i still feel that these lists and graphs give data
that is worth considering when evaluating and/or comparing
EDOs.

if it's important to consider how well an EDO represents
the 6/5 ratio, can't one see that already by looking at
how well it represents both 3 and 5?

anyway, i'll probably just add your comment to the page as
a sort of "disclaimer" ... hopefully that will make it
more acceptable to you.

-monz

Tuners,

An interesting thought occurs to me as I read this exchange below.

As I introspect on the creative process, at least for me, I realize that I,
and probably I'm not unique here, let me know, so let's say for now 'humans'
-- tend to like to simplify things into 'one size fits all this is the best
for everything' solutions. It's easy to understand the appeal. Reduced mental
clutter. Order through the chaos. Or, a quick answer to avoid the
unpredictability of changing one's mind.

I bring this up because I think that really expressive great stuff can be done
in tunings that pay no heed to the golden calf, or holy grail of 'how close
is this sound to JI'. Of course, JI is beautiful and serves a purpose. But I
also think 12-tet does. I just got through giving a 1hr 45 min recital in
12-tet, and didn't feel at all lacking for it. (is this a coming out of the
closet as a pan-acceptance of all tunings kind-of-guy?)There is plenty of
room for everyone in the world!

Of course, I've had moments of purely being inspired by the otherness that is
possible in the 'terra nova' of possibilities for a 21st century of tonality
ushered in by new tunings and tuning systems.

I think there is an almost overwhelming richness, enough for 40 lifetimes, and
it is kind of scary--fully exploring one makes one feel that one has to
forsake all the others to really accomplish something in a given tuning, like
ho a great composer like Bach accomplished something (leaving alone the
possibility that the Bach phenom is a repeatable, or quasi-repeatable thing).
So, the systematic mind needs to feel that they are making a wise
choice--hence the "this is close to this kind of JI interval " analysis....

Then, maybe monz is just doing all these different things just for the pure
fun of it. I understand that. This topic is for most of us, an obsession. And
the odd thing about obsession is how IRRATIONAL it is!!!! in a really really
frightening and fascinating way...for instance, there is not a day that goes
by where I don't repeat some of the same thoughts I've always had about
tuning in my head, over and over and over again. It almost always remains
static like that, but occassionally, there will be an epiphany. Does anyone
else know what I'm talking about? This obssession? Or does what I'm saying
remain opaque?

What are all of your thoughts on this topic?

Best,
AKJ

P.S. Every once in a while, I love doing aesthetic conversations. Don't get me
wrong, I love the technical math stuff, but I feel it has to turn into music
somewhere, and that the math is just a tool.

But I have to say, I'm so thankful to you all that you exist! I feel like this
strange need I have to think about this (not strange to me, but axiomatic) is
really quite satisfied and compelled having others who share this need.
And it's a rare need, no?

On Monday 20 October 2003 12:11 am, monz wrote:
> hi paul,
>
>
> re:
> http://sonic-arts.org/dict/edo-prime-error.htm
>
> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> > as you know, monz, i think this is not a good way to evaluate
> > how well an ET represents JI sonorities, or even how the
> > primes are best represented in a particular ET. for instance,
> > i find that in a major triad, getting a good approximation of
> > the 6:5 just minor third is nearly, if not equally, important
> > as getting a good approximation of the 5:4 just major third,
> > and when one considers the fifth too the second-closest
> > approximation of prime 5 is often a better choice than
> > the closest one.
> >
> > and of course that's only considering a single chord -- many
> > other issues come into play when considering progressions, etc.
> >
> > if there's one thing i hope to have communicated in 7 years on
> > this list, it's the above.
>
> yes, i did already know that you felt this way, and i also
> knew that you were going to mention it after i announced
> the new webpage.
>
> however, i still feel that these lists and graphs give data
> that is worth considering when evaluating and/or comparing
> EDOs.
>
> if it's important to consider how well an EDO represents
> the 6/5 ratio, can't one see that already by looking at
> how well it represents both 3 and 5?
>
> anyway, i'll probably just add your comment to the page as
> a sort of "disclaimer" ... hopefully that will make it
> more acceptable to you.
>
>
>
> -monz
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
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for man -- who has no gills. -Ambrose Bierce 'The Devils Dictionary'

On Monday 20 October 2003 12:11 am, monz wrote:
> hi paul,
>
>
> re:
> http://sonic-arts.org/dict/edo-prime-error.htm
>
> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> > as you know, monz, i think this is not a good way to evaluate
> > how well an ET represents JI sonorities, or even how the
> > primes are best represented in a particular ET. for instance,
> > i find that in a major triad, getting a good approximation of
> > the 6:5 just minor third is nearly, if not equally, important
> > as getting a good approximation of the 5:4 just major third,
> > and when one considers the fifth too the second-closest
> > approximation of prime 5 is often a better choice than
> > the closest one.
> >
> > and of course that's only considering a single chord -- many
> > other issues come into play when considering progressions, etc.
> >
> > if there's one thing i hope to have communicated in 7 years on
> > this list, it's the above.
>
> yes, i did already know that you felt this way, and i also
> knew that you were going to mention it after i announced
> the new webpage.
>
> however, i still feel that these lists and graphs give data
> that is worth considering when evaluating and/or comparing
> EDOs.
>
> if it's important to consider how well an EDO represents
> the 6/5 ratio, can't one see that already by looking at
> how well it represents both 3 and 5?

I think that's an excellent point, Monz. I don't see any reason that it
wouldn't be so, anyway.

-AKJ

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> if it's important to consider how well an EDO represents
> the 6/5 ratio, can't one see that already by looking at
> how well it represents both 3 and 5?

Nope.

There's no way to tell from the diagram whether the errors are in the
same or opposite directions, i.e. no way to tell whether to add or
subract the 3 and 5 errors to get the 3:5 (5:6) error.

And even if you showed the directions there's no way to know whether
the 3:5 error so calculated is the best 3:5 approximation in the
tuning or not. But that would be OK so long as you pointed it out to
the reader.

Still these diagrams do provide some useful information. The above
limitations just need to be explained somewhere.

hi Dave,

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

> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> > if it's important to consider how well an EDO represents
> > the 6/5 ratio, can't one see that already by looking at
> > how well it represents both 3 and 5?
>
> Nope.
>
> There's no way to tell from the diagram whether the errors
> are in the same or opposite directions, i.e. no way to tell
> whether to add or subract the 3 and 5 errors to get the 3:5
> (5:6) error.

that true, and i realized it after i sent my post.
but still ...

> And even if you showed the directions there's no way to
> know whether the 3:5 error so calculated is the best 3:5
> approximation in the tuning or not. But that would be OK
> so long as you pointed it out to the reader.
>
> Still these diagrams do provide some useful information.

thanks.

> The above limitations just need to be explained somewhere.

yes, will do.

these graphs were not intended to show how well an EDO
approximates any certain tuning ... rather, simply as a
rough guide to the accuracy and nature of prime-mapping
for different EDOs.

for example, it's easy to see that in 12edo, *all* 11-limit
ratios will be approximately midway between two adjacent
12edo degrees, and therefore, in 24edo, all 11-limit ratios
will be mapped with very little error. etc.

the idea is that this kind of data will be useful when
using my software. for example, if the user wants to
work with 13edo, he/she can see from my graph that 13edo
does a good job of representing 11 and a fair-to-mediocre
job of representing 5 and 13. so that would be a good
basis for a prime-space, from which the user would then
be able to select unison-vectors and obtain 13edo *with*
a representation of pitches in a 5,11,13 periodicity-block.
that representation in turn might assist in making harmonic
choices while composing.

-monz

For 12-eq intonation:
Error between 3rd harmonic [3/2] and nearer step in the scale:
1.9550009 cents. [A]

Error between 5th harmonic [5/4] and nearer step in the scale:
13.6862861 cents. [B]

Error between Minor 3rd [6/5] and nearer step in the scale:
15.641287 cents.

A + B = 15.641287

Feel free to Correct the maths if wrong.
Max.

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
wrote:
> On Monday 20 October 2003 12:11 am, monz wrote:
> > hi paul,
> >
> >
> > --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> > > as you know, monz, i think this is not a good way to evaluate
> > > how well an ET represents JI sonorities, or even how the
> > > primes are best represented in a particular ET. for instance,
> > > i find that in a major triad, getting a good approximation of
> > > the 6:5 just minor third is nearly, if not equally, important
> > > as getting a good approximation of the 5:4 just major third
[...]
> >
> > if it's important to consider how well an EDO represents
> > the 6/5 ratio, can't one see that already by looking at
> > how well it represents both 3 and 5?
>
> I think that's an excellent point, Monz. I don't see any reason
that it
> wouldn't be so, anyway.
>
> -AKJ

--- In tuning@yahoogroups.com, "Maximiliano G. Miranda Zanetti"
<giordanobruno76@y...> wrote:
> For 12-eq intonation:
> Error between 3rd harmonic [3/2] and nearer step in the scale:
> 1.9550009 cents. [A]
>
> Error between 5th harmonic [5/4] and nearer step in the scale:
> 13.6862861 cents. [B]
>
>
> Error between Minor 3rd [6/5] and nearer step in the scale:
> 15.641287 cents.
>
> A + B = 15.641287
>

And one has
Error between 7th harmonic [7/4] and nearer step in the scale:
-31.1740935 cents. [A]

Error between 5th harmonic [5/4] and nearer step in the scale:
13.6862861 cents. [B]

Error between septimal tritone [7/5] and nearer step in the scale:
17.4878074 cents. [B]

Again, this equals A + B.

>

Hello Maximiliano!
The problem i have with this way of thinking is that it is possible the the 19th harmonic might
be preferred to a 6/5 which for many on this list is just not minor enough. pardon the pun. So
often when one is actually confronted with the sound of a tuning, it is not unusal for me to be
surprised in either direction.

>
>
> Message: 22
> Date: Mon, 20 Oct 2003 14:23:05 -0000
> From: "Maximiliano G. Miranda Zanetti" <giordanobruno76@yahoo.com.ar>
> Subject: Re: new Tuning Dictionary page: EDO prime errors
>
> --- In tuning@yahoogroups.com, "Maximiliano G. Miranda Zanetti"
> <giordanobruno76@y...> wrote:
> > For 12-eq intonation:
> > Error between 3rd harmonic [3/2] and nearer step in the scale:
> > 1.9550009 cents. [A]
> >
> > Error between 5th harmonic [5/4] and nearer step in the scale:
> > 13.6862861 cents. [B]
> >
> >
> > Error between Minor 3rd [6/5] and nearer step in the scale:
> > 15.641287 cents.
> >
> > A + B = 15.641287
> >
>
> And one has
> Error between 7th harmonic [7/4] and nearer step in the scale:
> -31.1740935 cents. [A]
>
> Error between 5th harmonic [5/4] and nearer step in the scale:
> 13.6862861 cents. [B]
>
> Error between septimal tritone [7/5] and nearer step in the scale:
> 17.4878074 cents. [B]
>
> Again, this equals A + B.
>
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

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

> if it's important to consider how well an EDO represents
> the 6/5 ratio, can't one see that already by looking at
> how well it represents both 3 and 5?

it's not necessarily so easy, if the et is inconsistent, because then
the best approximation to 6/5 may not be simply the best
approximation of 3 minus the best approximation of 5. using the big
chart on your equal temperament page, it should be easy to find such
examples (the inconsistent ets are in the reddish hues).

> anyway, i'll probably just add your comment to the page as
> a sort of "disclaimer" ... hopefully that will make it
> more acceptable to you.

thanks!

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
wrote:
>
> topic is for most of us, an obsession. And
> the odd thing about obsession is how IRRATIONAL it is!!!! in a
really really
> frightening and fascinating way...for instance, there is not a day
that goes
> by where I don't repeat some of the same thoughts I've always had
about
> tuning in my head, over and over and over again. It almost always
remains
> static like that, but occassionally, there will be an epiphany.
Does anyone
> else know what I'm talking about? This obssession? Or does what I'm
saying
> remain opaque?
>
> What are all of your thoughts on this topic?
>
> Best,
> AKJ

welcome to the land of the obsessed. and thank goodness you're
actually making music in the midst of it all.

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
wrote:

> > if it's important to consider how well an EDO represents
> > the 6/5 ratio, can't one see that already by looking at
> > how well it represents both 3 and 5?
>
> I think that's an excellent point, Monz. I don't see any reason
that it
> wouldn't be so, anyway.
>
> -AKJ

ok, you guys asked for it . . . my favorite example of this is 64-
equal, where, like all 5-limit inconsistent ets, the best
approximation of the 6:5 minor third is not the difference between
the best approximation of the 3:2 perfect fifth and the best
approximation of the 5:4 major third . . . and in fact the major
triad that uses the best approximation of the primes seems to be only
the *third-best* major triad in 64-equal . . .

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
wrote:

> I think that's an excellent point, Monz. I don't see any reason
that it
> wouldn't be so, anyway.

It's true. However it is also true that instead of primes up to 43,
we could look instead at 11-limit ratios p/q, p,q odd<13, p>q. Here
they are:

3, 5, 7, 9, 11, 11/5, 5/3, 7/5, 11/3, 9/7, 9/5, 7/3, 11/9, 11/7

There are 14 of these, as compared to 13 odd primes <=43.

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

>
> for example, it's easy to see that in 12edo, *all* 11-limit
> ratios will be approximately midway between two adjacent
> 12edo degrees, and therefore, in 24edo, all 11-limit ratios
> will be mapped with very little error. etc.

be careful -- you really mean "ratios of 11" and not "11-limit
ratios". 3:2 is an 11-limit ratio, but it's not a ratio of 11.

> the idea is that this kind of data will be useful when
> using my software. for example, if the user wants to
> work with 13edo, he/she can see from my graph that 13edo
> does a good job of representing 11 and a fair-to-mediocre
> job of representing 5 and 13. so that would be a good
> basis for a prime-space, from which the user would then
> be able to select unison-vectors and obtain 13edo *with*
> a representation of pitches in a 5,11,13 periodicity-block.
> that representation in turn might assist in making harmonic
> choices while composing.

maybe, but you might be missing some equally valid choices, based on
other equally consonant intervals.

--- In tuning@yahoogroups.com, "Maximiliano G. Miranda Zanetti"
<giordanobruno76@y...> wrote:

> And one has
> Error between 7th harmonic [7/4] and nearer step in the scale:
> -31.1740935 cents. [A]
>
> Error between 5th harmonic [5/4] and nearer step in the scale:
> 13.6862861 cents. [B]
>
> Error between septimal tritone [7/5] and nearer step in the scale:
> 17.4878074 cents. [B]
>
> Again, this equals A + B.

excellent. now try exactly the same thing for 24-equal, and you'll
see a counterexample.

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
> >
>
> Hello Maximiliano!
> The problem i have with this way of thinking is that it is
>possible the the 19th harmonic might
> be preferred to a 6/5 which for many on this list is just not minor
>enough

that's another valid issue, but i think the basic issue of
inconsistency should be clarified first, since it can apply to higher
harmonics as well as lower ones:

http://www.sonic-arts.org/dict/consiste.htm

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
>
> excellent. now try exactly the same thing for 24-equal, and you'll
> see a counterexample.

Paul, I think there's no counterexample. It's a simple mathematical
question, so I think it will work out in any x-EQ.

However, I wrote in an unclear, mezzy way. So I will work it out
again.

First, some method. I take the initial intervals (7/4, 5/4) and find
the difference with the nearer step in the EQ scale.

Convention:
If step > rational interval, I assign a positive sign. Idem negative.
(Please don't look at my previous examples...).

12-EQ [7/4]= +31.1741 [A](CENTS)
[5/4]= +13.6863 [B]

Now the desired interval is 7/5, that implies going up a "seventh"
and stepping down a third, so I must take A - B, which equals 17.4878.
This is indeed the difference between 582.5122 and 12-EQ F#.

Now with 24-EQ.
24-EQ [7/4]= -18.8260 [A]
[5/4]= +13.6863 [B]
A - B = -32.5122.

But in 24-EQ, no note can be further from a step than 25 cents.
However, 50 - 32.5122 gives the required distance. (The sign is
reversed for we have taken the next step.)

So the rule is modified to get:

Distance = min(abs(a - b),W/2 - abs(a-b)) where W is interval width
in cents.
W = 1200/x x=steps per octave (in EDO)

And the sign is
sign = sign(a - b) if a - b < W/2
-sign(a - b) Otherwise.

Max.
PS1: I hope I'm right this time.
PS2: The rule could also be slightly modified to cope with "additive
intervals" (eg. 35/16, etc.).

Kraig, I see perfectly your point, of course I suppose anyone
here understands what you mean and partly share your position...
I just was trying to work out a simple way of handling "composed
ratios" based on Monzo's numbers.

--- In tuning@yahoogroups.com, kraig grady <kraiggrady@a...> wrote:
> >
>
> Hello Maximiliano!
> The problem i have with this way of thinking is that it is
possible the the 19th harmonic might
> be preferred to a 6/5 which for many on this list is just not minor
enough. pardon the pun. So
> often when one is actually confronted with the sound of a tuning,
it is not unusal for me to be
> surprised in either direction.
>
> >

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

> --- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
> wrote:
>
> > I think that's an excellent point, Monz. I don't see any
> > reason that it wouldn't be so, anyway.
>
> It's true. However it is also true that instead of primes up
> to 43, we could look instead at 11-limit ratios p/q, p,q odd<13,
> p>q. Here they are:
>
> 3, 5, 7, 9, 11, 11/5, 5/3, 7/5, 11/3, 9/7, 9/5, 7/3, 11/9, 11/7
>
> There are 14 of these, as compared to 13 odd primes <=43.

OK, then ... who wants to re-do my webpage?
i'm not likely to do it.

as i said, i'm interested in this primarily for use with
my software, and showing errors of the basic prime-mappings
works just fine for that purpose.

-monz

I think the point here is simply whether 6/5 error could be analysed
using the errors of the primes 3 and 5, which can be perfectly done
(see one of my previous posts):

Prime 3 error: -8.2050 a
Prime 5 error: +7.4363 b
Difference a - b = -15.6413
Step width = 18.75
Adjusted difference: +3.1087

If that is not the point, let me apologise for all this senseless
math.
--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
> wrote:
>
> > > if it's important to consider how well an EDO represents
> > > the 6/5 ratio, can't one see that already by looking at
> > > how well it represents both 3 and 5?
> >
> > I think that's an excellent point, Monz. I don't see any reason
> that it
> > wouldn't be so, anyway.
> >
> > -AKJ
>
> ok, you guys asked for it . . . my favorite example of this is 64-
> equal, where, like all 5-limit inconsistent ets, the best
> approximation of the 6:5 minor third is not the difference between
> the best approximation of the 3:2 perfect fifth and the best
> approximation of the 5:4 major third . . . and in fact the major
> triad that uses the best approximation of the primes seems to be
only
> the *third-best* major triad in 64-equal . . .

hi paul,

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> >
> > for example, it's easy to see that in 12edo, *all* 11-limit
> > ratios will be approximately midway between two adjacent
> > 12edo degrees, and therefore, in 24edo, all 11-limit ratios
> > will be mapped with very little error. etc.
>
> be careful -- you really mean "ratios of 11" and not "11-limit
> ratios". 3:2 is an 11-limit ratio, but it's not a ratio of 11.

you're absolutely correct. my bad.

i meant "ratios of 11" in regard to both 12edo and 24edo in
the above quote.

> > the idea is that this kind of data will be useful when
> > using my software. for example, if the user wants to
> > work with 13edo, he/she can see from my graph that 13edo
> > does a good job of representing 11 and a fair-to-mediocre
> > job of representing 5 and 13. so that would be a good
> > basis for a prime-space, from which the user would then
> > be able to select unison-vectors and obtain 13edo *with*
> > a representation of pitches in a 5,11,13 periodicity-block.
> > that representation in turn might assist in making harmonic
> > choices while composing.
>
> maybe, but you might be missing some equally valid choices,
> based on other equally consonant intervals.

i think we should talk more about this off-list before
posting any more here. the software doesn't miss anything.
you set up your prime-space and all the data is there.

to some extent, it may be pointless to continue this thread
until the software is available so that others can see my
position more clearly.

-monz

When I wrote

> Distance = min(abs(a - b),W/2 - abs(a-b)) where W is interval width
> in cents.
> W = 1200/x x=steps per octave (in EDO)

I of course meant

Distance = min(abs(a - b),W - abs(a-b)) where W is interval width
in cents.
W = 1200/x x=steps per octave (in EDO).
Sorry.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
> wrote:
>
> > I think that's an excellent point, Monz. I don't see any reason
> that it
> > wouldn't be so, anyway.
>
> It's true.

How can you say that Gene? There is no way to tell whether to add or
subtract the errors for p and q to get the error for p:q.

If p and q both had errors of 25% of a step then p:q might have no
error or an error of 50%! You can't tell by looking at the chart.

Monz has since retracted it himself.

I'm constantly amazed at the patience of Paul Erlich; calmly
explaining the ratio consistency issue (which is in addition to the
one I refer to above) to the latest batch of newbies. Good on you
Paul. :-)

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> > It's true.
>
> How can you say that Gene? There is no way to tell whether to add or
> subtract the errors for p and q to get the error for p:q.

The errors came with signs, so yes there is.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> > How can you say that Gene? There is no way to tell whether to add or
> > subtract the errors for p and q to get the error for p:q.
>
> The errors came with signs, so yes there is.

Ok. You're talking about the lists. I was talking about the charts, as
I assumed Monz and Aaron were. Sorry.

--- In tuning@yahoogroups.com, "Maximiliano G. Miranda Zanetti"
<giordanobruno76@y...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> >
> > excellent. now try exactly the same thing for 24-equal, and
you'll
> > see a counterexample.
>
> Paul, I think there's no counterexample. It's a simple mathematical
> question, so I think it will work out in any x-EQ.

then you haven't interpreted the question the same way i have. if
each ratio is approximated by its nearest interval in the equal
temperament, then you do indeed get a counterexample in the case of
24-equal and {5/4, 7/4, 7/5}.

> However, I wrote in an unclear, mezzy way. So I will work it out
> again.
>
> First, some method. I take the initial intervals (7/4, 5/4) and
find
> the difference with the nearer step in the EQ scale.

yes . . .

>
> Convention:
> If step > rational interval, I assign a positive sign. Idem
negative.
> (Please don't look at my previous examples...).
>
> 12-EQ [7/4]= +31.1741 [A](CENTS)
> [5/4]= +13.6863 [B]
>
> Now the desired interval is 7/5, that implies going up a "seventh"
> and stepping down a third,

aha. i, instead, view this as an autonomous 'consonant' interval,
which sounds better as a dyad when you use as close as possible an
approximation to it. that was my point _contra_ monz.

i'm not sure whay you're proposing -- how would *you* render the
major triad in 64-equal, based on the math below? note also that monz
doesn't give the *signs* of the primes' errors . . .

--- In tuning@yahoogroups.com, "Maximiliano G. Miranda Zanetti"
<giordanobruno76@y...> wrote:
> I think the point here is simply whether 6/5 error could be
analysed
> using the errors of the primes 3 and 5, which can be perfectly done
> (see one of my previous posts):
>
> Prime 3 error: -8.2050 a
> Prime 5 error: +7.4363 b
> Difference a - b = -15.6413
> Step width = 18.75
> Adjusted difference: +3.1087
>
> If that is not the point, let me apologise for all this senseless
> math.
> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
> > wrote:
> >
> > > > if it's important to consider how well an EDO represents
> > > > the 6/5 ratio, can't one see that already by looking at
> > > > how well it represents both 3 and 5?
> > >
> > > I think that's an excellent point, Monz. I don't see any reason
> > that it
> > > wouldn't be so, anyway.
> > >
> > > -AKJ
> >
> > ok, you guys asked for it . . . my favorite example of this is 64-
> > equal, where, like all 5-limit inconsistent ets, the best
> > approximation of the 6:5 minor third is not the difference
between
> > the best approximation of the 3:2 perfect fifth and the best
> > approximation of the 5:4 major third . . . and in fact the major
> > triad that uses the best approximation of the primes seems to be
> only
> > the *third-best* major triad in 64-equal . . .

hi paul, Gene, and Dave,

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> > --- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
> > wrote:
> >
> > > I think that's an excellent point, Monz. I don't see any
> > > reason that it wouldn't be so, anyway.
> >
> > It's true. However it is also true that instead of primes up
> > to 43, we could look instead at 11-limit ratios p/q, p,q odd<13,
> > p>q. Here they are:
> >
> > 3, 5, 7, 9, 11, 11/5, 5/3, 7/5, 11/3, 9/7, 9/5, 7/3, 11/9, 11/7
> >
> > There are 14 of these, as compared to 13 odd primes <=43.
>
>
>
> OK, then ... who wants to re-do my webpage?
> i'm not likely to do it.
>
> as i said, i'm interested in this primarily for use with
> my software, and showing errors of the basic prime-mappings
> works just fine for that purpose.

anyway, i've already done something like this almost
2 years ago:

http://sonic-arts.org/dict/error.htm

-monz

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

/tuning/topicId_48010.html#48020

> hi paul,
>
>
> re:
> http://sonic-arts.org/dict/edo-prime-error.htm
>
>
> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
>
> > as you know, monz, i think this is not a good way to evaluate
> > how well an ET represents JI sonorities, or even how the
> > primes are best represented in a particular ET. for instance,
> > i find that in a major triad, getting a good approximation of
> > the 6:5 just minor third is nearly, if not equally, important
> > as getting a good approximation of the 5:4 just major third,
> > and when one considers the fifth too the second-closest
> > approximation of prime 5 is often a better choice than
> > the closest one.
> >
> > and of course that's only considering a single chord -- many
> > other issues come into play when considering progressions, etc.
> >
> > if there's one thing i hope to have communicated in 7 years on
> > this list, it's the above.
>
>
> yes, i did already know that you felt this way, and i also
> knew that you were going to mention it after i announced
> the new webpage.
>
> however, i still feel that these lists and graphs give data
> that is worth considering when evaluating and/or comparing
> EDOs.
>
> if it's important to consider how well an EDO represents
> the 6/5 ratio, can't one see that already by looking at
> how well it represents both 3 and 5?
>
> anyway, i'll probably just add your comment to the page as
> a sort of "disclaimer" ... hopefully that will make it
> more acceptable to you.
>

***Could someone please explain a bit what this situation is all
about? I'm not entirely getting it...

In fact, I'm not even "partially" getting it... although I know
partials are involved... (enough!)

JP

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_48010.html#48047

> --- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
> wrote:
>
> > > if it's important to consider how well an EDO represents
> > > the 6/5 ratio, can't one see that already by looking at
> > > how well it represents both 3 and 5?
> >
> > I think that's an excellent point, Monz. I don't see any reason
> that it
> > wouldn't be so, anyway.
> >
> > -AKJ
>
> ok, you guys asked for it . . . my favorite example of this is 64-
> equal, where, like all 5-limit inconsistent ets, the best
> approximation of the 6:5 minor third is not the difference between
> the best approximation of the 3:2 perfect fifth and the best
> approximation of the 5:4 major third . . . and in fact the major
> triad that uses the best approximation of the primes seems to be
only
> the *third-best* major triad in 64-equal . . .

***Ah... so.... I'm getting a glimmer now. Actually, we went over
this material before at one point...

JP

Dear Aaron,

I looked up "holy grail" in Webster's Third International.

[ME, graal, greal, fr. MF, bowl, grail, fr. ML gradalis, perh. fr. L gradus step + -alis -al; perh. fr. its having consisted originally of a series of bowls or plates arranged one above the other--more at GRADE] 2 : an eminently desirable and ultimate object of an extended effort or quest

To me there is something musical about a series of bowls.

Coming to this topic after reading Helmholtz again, it's possible that he embodies a healthy approach to this dilemma. After placing western musical on highest pedestal, carefully showing how tonality and harmony have evolved from a modal approach, he then devotes most of his research to clarifying this modal, homophonic, unison music. He chides 12ET along the way, much like an archaeologist might sit comfortably in an air conditioned tent while digging up ancient art.

rt

Aaron K. Johnson wrote:

>Tuners,
>
>An interesting thought occurs to me as I read this exchange below.
>
>As I introspect on the creative process, at least for me, I realize that I, >and probably I'm not unique here, let me know, so let's say for now 'humans' >-- tend to like to simplify things into 'one size fits all this is the best >for everything' solutions. It's easy to understand the appeal. Reduced mental >clutter. Order through the chaos. Or, a quick answer to avoid the >unpredictability of changing one's mind.
>
>I bring this up because I think that really expressive great stuff can be done >in tunings that pay no heed to the golden calf, or holy grail of 'how close >is this sound to JI'. Of course, JI is beautiful and serves a purpose. But I >also think 12-tet does. I just got through giving a 1hr 45 min recital in >12-tet, and didn't feel at all lacking for it. (is this a coming out of the >closet as a pan-acceptance of all tunings kind-of-guy?)There is plenty of >room for everyone in the world!
>
>Of course, I've had moments of purely being inspired by the otherness that is >possible in the 'terra nova' of possibilities for a 21st century of tonality >ushered in by new tunings and tuning systems.
>
>I think there is an almost overwhelming richness, enough for 40 lifetimes, and >it is kind of scary--fully exploring one makes one feel that one has to >forsake all the others to really accomplish something in a given tuning, like >ho a great composer like Bach accomplished something (leaving alone the >possibility that the Bach phenom is a repeatable, or quasi-repeatable thing). >So, the systematic mind needs to feel that they are making a wise >choice--hence the "this is close to this kind of JI interval " analysis....
>
>Then, maybe monz is just doing all these different things just for the pure >fun of it. I understand that. This topic is for most of us, an obsession. And >the odd thing about obsession is how IRRATIONAL it is!!!! in a really really >frightening and fascinating way...for instance, there is not a day that goes >by where I don't repeat some of the same thoughts I've always had about >tuning in my head, over and over and over again. It almost always remains >static like that, but occassionally, there will be an epiphany. Does anyone >else know what I'm talking about? This obssession? Or does what I'm saying >remain opaque?
>
>What are all of your thoughts on this topic?
>
>Best,
>AKJ
>
>P.S. Every once in a while, I love doing aesthetic conversations. Don't get me >wrong, I love the technical math stuff, but I feel it has to turn into music >somewhere, and that the math is just a tool.
>
>But I have to say, I'm so thankful to you all that you exist! I feel like this >strange need I have to think about this (not strange to me, but axiomatic) is >really quite satisfied and compelled having others who share this need.
>And it's a rare need, no?
>
>
>
>On Monday 20 October 2003 12:11 am, monz wrote:
>
>>hi paul,
>>
>>
>>re:
>>http://sonic-arts.org/dict/edo-prime-error.htm
>>
>>--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
>>
>>>as you know, monz, i think this is not a good way to evaluate
>>>how well an ET represents JI sonorities, or even how the
>>>primes are best represented in a particular ET. for instance,
>>>i find that in a major triad, getting a good approximation of
>>>the 6:5 just minor third is nearly, if not equally, important
>>>as getting a good approximation of the 5:4 just major third,
>>>and when one considers the fifth too the second-closest
>>>approximation of prime 5 is often a better choice than
>>>the closest one.
>>>
>>>and of course that's only considering a single chord -- many
>>>other issues come into play when considering progressions, etc.
>>>
>>>if there's one thing i hope to have communicated in 7 years on
>>>this list, it's the above.
>>>
>>yes, i did already know that you felt this way, and i also
>>knew that you were going to mention it after i announced
>>the new webpage.
>>
>>however, i still feel that these lists and graphs give data
>>that is worth considering when evaluating and/or comparing
>>EDOs.
>>
>>if it's important to consider how well an EDO represents
>>the 6/5 ratio, can't one see that already by looking at
>>how well it represents both 3 and 5?
>>
>>anyway, i'll probably just add your comment to the page as
>>a sort of "disclaimer" ... hopefully that will make it
>>more acceptable to you.
>>
>>
>>
>>-monz
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>You do not need web access to participate. You may subscribe through
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>

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> i'm not sure whay you're proposing -- how would *you* render the
> major triad in 64-equal, based on the math below? note also that
monz
> doesn't give the *signs* of the primes' errors . . .

Well, step by step...
This is 12-eq representation analysis in Monzo's page

http://sonic-arts.org/dict/edo-prime-error.htm
[This is supposed to be an hyperlink!?]

12edo best representation of prime-factor

prime ~cents error % EDO-step error

3 -1.955000865 -2
19 +2.486983868 +2
17 -4.9554095 -5

43 -11.51770564 -12
5 +13.68628614 +14
[...]

Signs are there. Monzo follows my convention... Or better, I follow
his convention!!

Second, please don't miss the point of my message, which was to
calculate best representation of complex ratios through the sheer
numbers present in Monzo's page. If I want to play C and then E,
there should be no doubt I should play steps 0 and 21 of a 64-eq
scale to get the best approximation.

Third, I could tell you best representations in 64-eq for 1, 5/4 and
3/2: Steps 0,21,37. If you want me to express the best representation
of 4:5:6 chord in 64-eq, I must admit I may be not apt to express
anything but some ignorance. I think I have found you are somewhat
interested in harmonic entropy. It looks as possible to me you have
some reasons to conclude there's a better major "chord" than steps 0-
21-37.Let me ask you something. How do you arrive to such a result?
Is it through a perfectly objective functional, or through some
subjective measuring?

In finding the representations for the ratios, one just has to
minimize the distance between the proportion and the ideal step to
stand for it. I believe there is little objection to that procedure.

--- In tuning@yahoogroups.com, "Maximiliano G. Miranda Zanetti"
<giordanobruno76@y...> wrote:
> If I want to play C and then E,
> there should be no doubt I should play steps 0 and 21 of a 64-eq
> scale to get the best approximation.

there is plenty of doubt -- C to E can mean a lot of different
things, musically speaking. for one, it immediately indicates a
diatonic scale, which for the vast majority of recorded western
musical history would be in some kind of pythagorean or meantone
tuning.

> It looks as possible to me you have
> some reasons to conclude there's a better major "chord" than steps
0-
> 21-37.Let me ask you something. How do you arrive to such a result?
> Is it through a perfectly objective functional, or through some
> subjective measuring?

both, but the ear is the final arbiter. by experimenting with triads
with different-sized fifths, i convinced myself that getting the
minor third near just is about as important as getting the major
third near just, in order to maximize the consonance of the triad as
a whole.

> In finding the representations for the ratios, one just has to
> minimize the distance between the proportion and the ideal step to
> stand for it. I believe there is little objection to that procedure.

i object when the resulting ratios are inconsistent, when they cannot
live together in a single chord, as is the case for {3:2, 5:4, 6:5}
in 64-equal or {7:4, 5:4, 7:5} in 24-equal. in such cases you are
forced to use the second-best approximation for at least one of these
ratios in order to construct the relevant chord. this issue was
overlooked by many of the researchers who studied equal temperament,
for example wendy carlos and yunik&swift, who overestimated the
ability of certain equal temperaments to approximate just intonation
by focusing only on ratios (that is, dyads) without considering
whether these ratios would combine consistently into chords.

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

> there is plenty of doubt -- C to E can mean a lot of different
> things, musically speaking. for one, it immediately indicates a
> diatonic scale, which for the vast majority of recorded western
> musical history would be in some kind of pythagorean or meantone
> tuning.

Well, I meant 4:5, which is a ratio any meantone M3 (or even pytag.
81/64) tries to represent. In fact, one of the reasons for the
appearance of meantone tuning was the need for a more perfect 3rd
than 81/64. I think everyone in this group perceives there's some
perfection in 4:5, which is not found in any of its "proxies".
However, one has also to understand that after all this time, our
tired ears find some rationale and validity in 400-cent E or even
81/64.

>[...] By experimenting with triads with different-sized fifths, i
convinced myself that getting the
> minor third near just is about as important as getting the major
> third near just, in order to maximize the consonance of the triad
as
> a whole.
>
> > In finding the representations for the ratios, one just has to
> > minimize the distance between the proportion and the ideal step
to
> > stand for it. I believe there is little objection to that
procedure.
>
> i object when the resulting ratios are inconsistent, when they
cannot
> live together in a single chord, as is the case for {3:2, 5:4, 6:5}
> in 64-equal or {7:4, 5:4, 7:5} in 24-equal. in such cases you are
> forced to use the second-best approximation for at least one of
these
> ratios in order to construct the relevant chord. this issue was
> overlooked by many of the researchers who studied equal
temperament,
> for example wendy carlos and yunik&swift, who overestimated the
> ability of certain equal temperaments to approximate just
intonation
> by focusing only on ratios (that is, dyads) without considering
> whether these ratios would combine consistently into chords.

Quite interesting. And to be honest, I was unaware of much of this.
Now I understand better what your point was.

Max.

Hi all,
Following your discussion I recall that effort of the conservative German composer Paul Hindemith (1896 - 1963) in his
"Unterweisung im Tonsatz, theoretical part" (don't know if there is an English translation of that book) where he tried to build up a new tuning system apart from 12tet, based on derivations from the harmonic scale. His theory was to start from a given note C, set the octave frame by taking the 2nd. harmonic c, proceeded to third harmonic g which he transposed one oct. downwards (G), then he set the 3rd. harmonic as second, so the new first harmonic would be F, right, here we have clear proportions such as 2/1, 3/2, etc., and thus he went up to the
6th harmonic, and he always furnished the pitches in frequencies so as to generate a new tuning for the 12 chromatic half tones within an octave, non tet, but with well distributed differences in the size of the resulting intervals. As I critisized in my Septatonic article, he refrained to use the harmonics above the 6th one, reasoning there should be nointervall smaller than a minor second, because he was afraid of creating a sound chaos.
Try and find that Hindemith book in English, it's sometimes quite polemic against avantgarde music tendencies, but still worth reading. (Everyone of us is free to draw different conclusions, right ?)
Bye
Werner

--- In tuning@yahoogroups.com, Wernerlinden@a... wrote:

/tuning/topicId_48010.html#48147

> Hi all,
> Following your discussion I recall that effort of the conservative
German composer Paul Hindemith (1896 - 1963) in his
> "Unterweisung im Tonsatz, theoretical part" (don't know if there is
an English translation of that book) where he tried to build up a new
tuning system apart from 12tet, based on derivations from the
harmonic scale. His theory was to start from a given note C, set the
octave frame by taking the 2nd. harmonic c, proceeded to third
harmonic g which he transposed one oct. downwards (G), then he set
the 3rd. harmonic as second, so the new first harmonic would be F,
right, here we have clear proportions such as 2/1, 3/2, etc., and
thus he went up to the
> 6th harmonic, and he always furnished the pitches in frequencies so
as to generate a new tuning for the 12 chromatic half tones within an
octave, non tet, but with well distributed differences in the size of
the resulting intervals. As I critisized in my Septatonic article, he
refrained to use the harmonics above the 6th one, reasoning there
should be nointervall smaller than a minor second, because he was
afraid of creating a sound chaos.
> Try and find that Hindemith book in English, it's sometimes quite
polemic against avantgarde music tendencies, but still worth reading.
(Everyone of us is free to draw different conclusions, right ?)
> Bye
> Werner

***Ummm... this is Hindemith's most famous book, known in English as
the _Craft of Musical Composition_ which describes his basic tonal
theories:

Unterweisung im Tonsatz (The Craft of Musical Composition)
Vol. 1 Theoretischer Teil (Theoretical Part), Mainz etc.: Schott
1937, 2nd expanded edition: Mainz etc.: Schott 1940

***The English version can easily be purchased at Amazon.com:

Craft of Musical Composition: Book One, Theoretical Part (Tap/159)
by Paul Hindemith

List Price: $20.95
Price: $20.95

JP

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

> ... instead of primes up to 43, we could look instead at
> 11-limit ratios p/q, p,q odd<13, p>q. Here they are:
>
> 3, 5, 7, 9, 11, 11/5, 5/3, 7/5, 11/3, 9/7, 9/5, 7/3, 11/9, 11/7
>
> There are 14 of these, as compared to 13 odd primes <=43.

i've made graphs showing the %error of an EDO degree
from all combinations of prime-factors up to the 11-limit,
for two important EDOs:

http://sonic-arts.org\dict\12-eq.htm
http://sonic-arts.org\dict\hexamu.htm

i'm sure that more will follow ... eventually ...

-monz

oops ...

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

> i've made graphs showing the %error of an EDO degree
> from all combinations of prime-factors up to the 11-limit,
> for two important EDOs:

sorry about the wrong-facing slashes in the URLs. here:

http://sonic-arts.org/dict/12-eq.htm
http://sonic-arts.org/dict/hexamu.htm

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> oops ...
>
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > i've made graphs showing the %error of an EDO degree
> > from all combinations of prime-factors up to the 11-limit,
> > for two important EDOs:
>
>
> sorry about the wrong-facing slashes in the URLs. here:
>
> http://sonic-arts.org/dict/12-eq.htm
> http://sonic-arts.org/dict/hexamu.htm
>
>
>
> -monz

awesome work, monz!

now 'all combinations of prime-factors up to the 11-limit' would
really refer to the infinity of 11-prime-limit ratios, wouldn't it?

what would seem to make most sense, especially given our experience
with 12-equal vis-a-vis ratios like 9:7 and 9:5 and 9:4, would be to
simply use the 11-limit tonality diamond's ratios; that is, to use
the 11-odd-limit?

-paul

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> > oops ...
> >
> > --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> >
> > > i've made graphs showing the %error of an EDO degree
> > > from all combinations of prime-factors up to the 11-limit,
> > > for two important EDOs:
> >
> >
> > sorry about the wrong-facing slashes in the URLs. here:
> >
> > http://sonic-arts.org/dict/12-eq.htm
> > http://sonic-arts.org/dict/hexamu.htm
> >
> >
> >
> > -monz
>
> awesome work, monz!

thanks! i try hard.

> now 'all combinations of prime-factors up to the 11-limit'
> would really refer to the infinity of 11-prime-limit ratios,
> wouldn't it?
>
> what would seem to make most sense, especially given our
> experience with 12-equal vis-a-vis ratios like 9:7 and 9:5
> and 9:4, would be to simply use the 11-limit tonality diamond's
> ratios; that is, to use the 11-odd-limit?

yes, i have to agree with you there. i'll change the text on
the webpages. thanks.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
>
> > --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> > > oops ...
> > >
> > > --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> > >
> > > > i've made graphs showing the %error of an EDO degree
> > > > from all combinations of prime-factors up to the 11-limit,
> > > > for two important EDOs:
> > >
> > >
> > > sorry about the wrong-facing slashes in the URLs. here:
> > >
> > > http://sonic-arts.org/dict/12-eq.htm
> > > http://sonic-arts.org/dict/hexamu.htm
> > >
> > >
> > >
> > > -monz
> >
> > awesome work, monz!
>
>
> thanks! i try hard.
>
>
>
> > now 'all combinations of prime-factors up to the 11-limit'
> > would really refer to the infinity of 11-prime-limit ratios,
> > wouldn't it?
> >
> > what would seem to make most sense, especially given our
> > experience with 12-equal vis-a-vis ratios like 9:7 and 9:5
> > and 9:4, would be to simply use the 11-limit tonality diamond's
> > ratios; that is, to use the 11-odd-limit?
>
>
> yes, i have to agree with you there. i'll change the text on
> the webpages. thanks.
>
>
>
>
> -monz

hi monz,

i see you've changed the text, but not the ratios! now the text
doesn't agree with the ratios you're showing -- probably you just
haven't gotten around to completing these pages yet . . . ?

-paul

hi paul,

--- In tuning@yahoogroups.com, "asarkiss" <asarkiss@y...> wrote:

> > > > http://sonic-arts.org/dict/12-eq.htm
> > > > http://sonic-arts.org/dict/hexamu.htm
>
> hi monz,
>
> i see you've changed the text, but not the ratios! now
> the text doesn't agree with the ratios you're showing
> -- probably you just haven't gotten around to completing
> these pages yet . . . ?

arrrrrgh! i did that quickly the other day just after
chatting with you ... i thought you were suggesting a
correction only for the text.

i'm not planning to change the list of ratios, because
that means creating new graphics. maybe someday, but
not now.

so, what should the text say to agree with my graphics?

-monz

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

> i'm not planning to change the list of ratios, because
> that means creating new graphics. maybe someday, but
> not now.

That's unfortunate, especially since you already have, linked from
your equal temperament page (though the link from 12 is missing),
several ETs compared against exactly the set of ratios i proposed:

http://www.sonic-arts.org/dict/ETgraphs.htm

:(

But anyway i suppose the text for now should say "ratios where the
denominator and numerator are both prime numbers below 13" . . . It
may seem i was bugging you about particulars, in this case and so
many others, but really i'm trying to impart a better understanding
of the nature and role of musical ratios . . .

Of course, whether you do it this way or the other, you're presenting
sets of ratio-approximations which are incompatible with one another
for the purpose of approximating simple JI chords; that is, both 12-
equal and 768-equal are inconsistent with respect to the set of
approximations you imply with your errors. Personally, i'd at least
warn the reader of this.

hi paul,

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > i'm not planning to change the list of ratios, because
> > that means creating new graphics. maybe someday, but
> > not now.
>
> That's unfortunate, especially since you already have,
> linked from your equal temperament page (though the link
> from 12 is missing), several ETs compared against exactly
> the set of ratios i proposed:
>
> http://www.sonic-arts.org/dict/ETgraphs.htm
>
> :(
>
> But anyway i suppose the text for now should say "ratios
> where the denominator and numerator are both prime numbers
> below 13" . . . It may seem i was bugging you about
> particulars, in this case and so many others, but really
> i'm trying to impart a better understanding
> of the nature and role of musical ratios . . .

thanks, and i appreciate your "bugging". i simply have
too much other work to concentrate on right now, and when
i do stuff like this (heavy on graphics) it's in a moment
of inspiration when i want to "strike while the iron is hot".

for me to go back now and re-do it would take me away
from too many other important things that need attention now,
and my head is no longer there right now.

> Of course, whether you do it this way or the other,
> you're presenting sets of ratio-approximations which are
> incompatible with one another for the purpose of
> approximating simple JI chords; that is, both 12-equal
> and 768-equal are inconsistent with respect to the set
> of approximations you imply with your errors. Personally,
> i'd at least warn the reader of this.

OK, that's easy enough to add.

the reason i chose 12edo and 768edo as my two examples
for this is simply that they both have much currency
as standard tunings, and my idea was simply to show people
how well or badly those tunings are approximating JI.

i'll make the two changes you suggest here, but then
really fixing it properly will have to wait until we
can chat more in depth about it.

-monz

hi paul,

OK, you win. it wasn't so much work after all, and
now that i've redone my spreadsheet, doing all other ETs
will be easy.

http://sonic-arts.org/dict/12-eq.htm

now shows a graph and table categorizing the %EDO-step
error of 12-ET for all 11-limit ratios.

i'll fix 768edo soon, and get around to others when
i have time.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> hi paul,
>
>
> OK, you win. it wasn't so much work after all, and
> now that i've redone my spreadsheet, doing all other ETs
> will be easy.
>
>
> http://sonic-arts.org/dict/12-eq.htm
>
> now shows a graph and table categorizing the %EDO-step
> error of 12-ET for all 11-limit ratios.
>
> i'll fix 768edo soon, and get around to others when
> i have time.
>
>
>
> -monz

thanks monz, that certainly enhances the logic, i feel, of these
comparisons . . .

>> Of course, whether you do it this way or the other,
>> you're presenting sets of ratio-approximations which are
>> incompatible with one another for the purpose of
>> approximating simple JI chords; that is, both 12-equal
>> and 768-equal are inconsistent with respect to the set
>> of approximations you imply with your errors. Personally,
>> i'd at least warn the reader of this.

>OK, that's easy enough to add.

i'm not seeing anything on this on the 12-equal page -- maybe you
just haven't added it yet . . .

hi paul,

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:

> >> Of course, whether you do it this way or the other,
> >> you're presenting sets of ratio-approximations which are
> >> incompatible with one another for the purpose of
> >> approximating simple JI chords; that is, both 12-equal
> >> and 768-equal are inconsistent with respect to the set
> >> of approximations you imply with your errors. Personally,
> >> i'd at least warn the reader of this.
>
> > OK, that's easy enough to add.
>
> i'm not seeing anything on this on the 12-equal page --
> maybe you just haven't added it yet . . .

oops ... i changed the graphic and its associated text,
but didn't put in that warning. sorry.

so, in order to make these graphs deal with the consistency
issue, i'd have to use bingo-card approximations rather
than closest approximations. that maked things a lot
more complicated.

anyways, i've done 768edo as well (also without the
consistency warning):

http://sonic-arts.org/dict/hexamu.htm

those will have to do for now.

-monz

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

> anyways, i've done 768edo as well (also without the
> consistency warning):
>
> http://sonic-arts.org/dict/hexamu.htm

hmmm ... in the case of 768edo, the bingo-card approach
may not be relevant anyway. the graphic i've used here
is the one which actually matters if one's hardware simply
finds the closest approximation to 768edo for whatever
MIDI data the user inputs.

the _caveat_ here is that, in most cases (i think),
MIDI hardware simply truncates data which goes beyond
the 7 bits which determine 768edo, which means that the
closest approximation is not always the one used.

i'd have to come back to this issue when i have time
to think about devising an appropriate algorithm.

-monz

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> hi paul,
>
>
> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
>
> > >> Of course, whether you do it this way or the other,
> > >> you're presenting sets of ratio-approximations which are
> > >> incompatible with one another for the purpose of
> > >> approximating simple JI chords; that is, both 12-equal
> > >> and 768-equal are inconsistent with respect to the set
> > >> of approximations you imply with your errors. Personally,
> > >> i'd at least warn the reader of this.
> >
> > > OK, that's easy enough to add.
> >
> > i'm not seeing anything on this on the 12-equal page --
> > maybe you just haven't added it yet . . .
>
>
> oops ... i changed the graphic and its associated text,
> but didn't put in that warning. sorry.
>
> so, in order to make these graphs deal with the consistency
> issue, i'd have to use bingo-card approximations rather
> than closest approximations.

or some other set of consistent approximations -- or you could put in
the warning.

> that maked things a lot
> more complicated.

so why not just put in the warning?

on 11/3/03 9:59 AM, monz <monz@attglobal.net> wrote:

> it wasn't so much work after all, and
> now that i've redone my spreadsheet, doing all other ETs
> will be easy.
>
> http://sonic-arts.org/dict/12-eq.htm
>
> now shows a graph and table categorizing the %EDO-step
> error of 12-ET for all 11-limit ratios.

Interesting that all the ratios classified as "terrible" are handled really
well by quartertones (meaning 24edo). At another level, I suppose that
result is almost a "statistical tautology", since 50%-edo-step is the
largest possible relative error. Still the particulars are a surprise to
me, admittely a newbie at this.

-Kurt

> i'll fix 768edo soon, and get around to others when
> i have time.
>
> -monz

hi Kurt,

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> on 11/3/03 9:59 AM, monz <monz@a...> wrote:
>
> > it wasn't so much work after all, and
> > now that i've redone my spreadsheet, doing all other ETs
> > will be easy.
> >
> > http://sonic-arts.org/dict/12-eq.htm
> >
> > now shows a graph and table categorizing the %EDO-step
> > error of 12-ET for all 11-limit ratios.
>
> Interesting that all the ratios classified as "terrible"
> are handled really well by quartertones (meaning 24edo).
> At another level, I suppose that result is almost a
> "statistical tautology", since 50%-edo-step is the
> largest possible relative error. Still the particulars
> are a surprise to me, admittely a newbie at this.

at the risk of being accused of repeating what you said but
in a different way -- altho i'm not sure i am saying quite
the same thing -- i'd like to add:

for any ET which has "terrible" representations which are
at or near +/-50% edo-step relative error, those ratios
will be represented excellently in the EDO which has
twice as many steps as the original one.

so, for example, as you pointed out, those which are
near 50% error in 12-ET are excellent in 24-ET.

those near 50% error in 24-ET are in turn represented
excellently in 48-ET.

those near 50% error in 17-ET are represented excellently
in 34-ET.

etc.

when you double the cardinality of an EDO, you have
degrees which fit exactly in between those of the
half-size EDO.

-monz