3-6-9 temperaments

Drawing on my experience listening to and tuning
meantone 5ths, I decided that 699 cents is about the
limit for 5ths of a certain class. Flatter fifths
are certainly useable, but seem to leave a realm
cast by fifths in the 699-702 range.

I looked up the corresponding change in harmonic
entropy, applied it sharp of 3:2, and lo and behold
I arrived at 705 cents. In other words, you can
inflict about the same amount of confusion by
mistuning a 5th either 3 cents sharp or 3 cents flat.

I applied the harmonic entropy delta to 5/4, and
wound up with +/- 6 cents. And *this* seemed to
correspond to a realm I'd noticed with major 3rds.

I then did the same thing for 7/4, and this time
found an assymetry. The range is about 962-978, or
about 7 cents flat / 9 cents sharp. I don't have
enough hands-on experience with minor 7ths to talk
about realms, but 3-6-9 seems like a nice progression.

So here is a list of ETs < 100 that have 3/2, 5/4,
and 7/4 approximations <= 3, 6, and 9 cents
respectively...

41 46 53 63 65 72 75 77 80 82 84 87 89 92 94 96 99

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:

> I looked up the corresponding change in harmonic
> entropy, applied it sharp of 3:2, and lo and behold
> I arrived at 705 cents.

What, specifically, are you doing to compute harmonic
entropy? Paul always blew me off when I asked that.

> > I looked up the corresponding change in harmonic
> > entropy, applied it sharp of 3:2, and lo and behold
> > I arrived at 705 cents.
>
> What, specifically, are you doing to compute harmonic
> entropy? Paul always blew me off when I asked that.

I'm looking up values in a list he gave me. :)

But the algorithm is explained many times in the archives
here, as well as on the harmonic_entropy list.

I have half of the thing done in scheme. I think
John deLaubenfels coded it up completely.

I don't recall Paul blowing you off about it... I recall
him begging you to take an interest in it.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:

> But the algorithm is explained many times in the archives
> here, as well as on the harmonic_entropy list.

That's how Paul blew me off. The algorithm has
free parameters, so this doesn't answer my
question.

> I don't recall Paul blowing you off about it... I recall
> him begging you to take an interest in it.

I did, but no one thought my ideas (trying to
use the ? function, etc) were likely to get
anywhere, and that's probably true. If there
was a quick and easy way to do this stuff, I
didn't find it. But of course, since people
kept blowing me off, I also didn't have much
traction to try.

Carl Lumma wrote:

> So here is a list of ETs < 100 that have 3/2, 5/4,
> and 7/4 approximations <= 3, 6, and 9 cents
> respectively...
> > 41 46 53 63 65 72 75 77 80 82 84 87 89 92 94 96 99
> I wonder what you'd get if you try the equivalent change in harmonic entropy around 2/1, and include ETs with tempered octaves?

> > But the algorithm is explained many times in the archives
> > here, as well as on the harmonic_entropy list.
>
> That's how Paul blew me off. The algorithm has
> free parameters, so this doesn't answer my
> question.

What's your question? There only free parameter is s.
There is also the matter of how to generate the ratios,
(i.e. order of Farey series used), but the calculation
converges pretty rapidly. Also the choice of wether
to use a Farey series to start with or a Mann series
doesn't seem to matter much. s can be anything you
want it to be, but 1% seems to be a pretty good value.

> > I don't recall Paul blowing you off about it... I recall
> > him begging you to take an interest in it.
>
> I did, but no one thought my ideas (trying to
> use the ? function, etc) were likely to get
> anywhere,

I never understood what you were trying to do with ?.
In particular, the problem we wanted you to tackle was
the n-adic cases (Paul's algorithm is dyadic).

I'm afraid I don't have the algorithm to hand, so I'd
have to do what you'd have to do: search some list
archives or use google.

-Carl

> > So here is a list of ETs < 100 that have 3/2, 5/4,
> > and 7/4 approximations <= 3, 6, and 9 cents
> > respectively...
> >
> > 41 46 53 63 65 72 75 77 80 82 84 87 89 92 94 96 99
>
> I wonder what you'd get if you try the equivalent change in harmonic
> entropy around 2/1, and include ETs with tempered octaves?

You get 2 cents +/-. I doubt it's enough to change
any of the above numbers (per approx. octave), but I
could be wrong.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:

> What's your question? There only free parameter is s.
> There is also the matter of how to generate the ratios,
> (i.e. order of Farey series used), but the calculation
> converges pretty rapidly.

I need to know

(1) The standard deviation
(2) The list of ratios being used
(3) The weighting factor for a given ratio

Then I have a specific sum I could evaluate. What
I *don't* need is a demand I read the harmomic
entropy list instead.

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> Drawing on my experience listening to and tuning
> meantone 5ths, I decided that 699 cents is about the
> limit for 5ths of a certain class. Flatter fifths
> are certainly useable, but seem to leave a realm
> cast by fifths in the 699-702 range.
>
> I looked up the corresponding change in harmonic
> entropy, applied it sharp of 3:2, and lo and behold
> I arrived at 705 cents. In other words, you can
> inflict about the same amount of confusion by
> mistuning a 5th either 3 cents sharp or 3 cents flat.
>
> I applied the harmonic entropy delta to 5/4, and
> wound up with +/- 6 cents. And *this* seemed to
> correspond to a realm I'd noticed with major 3rds.
>
> I then did the same thing for 7/4, and this time
> found an asymmetry. The range is about 962-978, or
> about 7 cents flat / 9 cents sharp. I don't have
> enough hands-on experience with minor 7ths to talk
> about realms, but 3-6-9 seems like a nice progression.
>
> So here is a list of ETs < 100 that have 3/2, 5/4,
> and 7/4 approximations <= 3, 6, and 9 cents
> respectively...
>
> 41 46 53 63 65 72 75 77 80 82 84 87 89 92 94 96 99
>
> -Carl
>

Hmm .. meantones never quite make it, then! 3+3+3+3+6 = 18, not
enough to bridge the gap to the syntonic comma. Probably not even
with tempered octaves.

Since the harmonic entropy curve seems to be parabolic about its
deep minima, it's not so surprising that the fifth and third can be
about equally flat or sharp of pure. The asymmetry of 7/4 suggests
that there may be a slight sort of interference from other minor
sevenths. That is, a somewhat sharp harmonic seventh could be in
danger of being 'recognised' as a somewhat flat 5-limit m7.

What was the goal of tuning the meantone fifths you mentioned? Piano
work?

~~~T~~~

> > What's your question? There only free parameter is s.
> > There is also the matter of how to generate the ratios,
> > (i.e. order of Farey series used), but the calculation
> > converges pretty rapidly.
>
> I need to know
>
> (1) The standard deviation
> (2) The list of ratios being used
> (3) The weighting factor for a given ratio
>
> Then I have a specific sum I could evaluate. What
> I *don't* need is a demand I read the harmomic
> entropy list instead.

If you point me to a textbook on Grassman algebra, I don't
acuse you of demanding I do something I do not need.
As I explained, I'm not witholding anything from you. To
answer your questions I'd have to consult the archives.
Do you want me to do that for you?

I don't recall any weighting factor. You are perhaps
referring to the distance between mediants on either side
of ratios in a Farey series. Typically a Farey series
is used between 1/1 and 4/1. Paul has also just used
sqrt(n*d) I believe, because it's equivalent. The
standard distribution of the curve over the true ratio
is I believe s. Or at least, s somehow gives the width
of the gaussian over the true ratio. Again I'd have to
look to remember how it's done.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:

> If you point me to a textbook on Grassman algebra, I don't
> acuse you of demanding I do something I do not need.
> As I explained, I'm not witholding anything from you.

I'm not accusing you of doing so or demanding
anything, since you explained you were using
a table. You might put the table up somewhere
though.

> Hmm .. meantones never quite make it, then! 3+3+3+3+6 = 18, not
> enough to bridge the gap to the syntonic comma. Probably not even
> with tempered octaves.

Seems that way. Maybe Herman will find otherwise.

> Since the harmonic entropy curve seems to be parabolic about its
> deep minima, it's not so surprising that the fifth and third can be
> about equally flat or sharp of pure.

The third especially can't. I think if you look at the numbers,
the third tolerates sharpness better than flatness beyond
6 cents. For example, 372cents is 4.5448256, while 400cents
is 4.5369463.

> What was the goal of tuning the meantone fifths you mentioned? Piano
> work?

Yeah. And just experimenting with meantones, which for some
reason I've done a bit of.

-Carl

> The third especially can't. I think if you look at the numbers,
> the third tolerates sharpness better than flatness beyond
> 6 cents. For example, 372cents is 4.5448256, while 400cents
> is 4.5369463.

This actually doesn't quite seem as big a difference as I was
expecting. Then again, the scaling of harmonic entropy always
seemed a bit unintuitive to me.

-Carl

What was the reason you expected a flat M3 to be more entropic than an
equal-and-oppositely sharp one? Intuitively I haven't a clue.

~~~T~~~

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > The third especially can't. I think if you look at the numbers,
> > the third tolerates sharpness better than flatness beyond
> > 6 cents. For example, 372cents is 4.5448256, while 400cents
> > is 4.5369463.
>
> This actually doesn't quite seem as big a difference as I was
> expecting. Then again, the scaling of harmonic entropy always
> seemed a bit unintuitive to me.
>
> -Carl
>

Many years of listening to them.

-Carl

> What was the reason you expected a flat M3 to be more entropic
> than an equal-and-oppositely sharp one? Intuitively I haven't
> a clue.
>
> ~~~T~~~
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@> wrote:
> > > The third especially can't. I think if you look at the numbers,
> > > the third tolerates sharpness better than flatness beyond
> > > 6 cents. For example, 372cents is 4.5448256, while 400cents
> > > is 4.5369463.
> >
> > This actually doesn't quite seem as big a difference as I was
> > expecting. Then again, the scaling of harmonic entropy always
> > seemed a bit unintuitive to me.
> >
> > -Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> Many years of listening to them.
>
> -Carl
>
> > What was the reason you expected a flat M3 to be more entropic
> > than an equal-and-oppositely sharp one? Intuitively I haven't
> > a clue.

I think it makes sense if you realize 6/5 is only
25/24 from 5/4 whereas 4/3 is 16/15 away.

This is off topic, but an interesting test of your ability to discern fine
differences in musical passages. The developer is in med school but formerly
worked for NI on Reaktor and Absynth.

Its actually a pretty difficult quiz, and experienced musicians might only
get about 80% correct. It takes about 5 minutes, but if you know the sounds are
different you can immediately hit the button and go on to the next one.

http://www.jakemandell.com/tonedeaf/

Charles Lucy lucy@lucytune.com

----- Promoting global harmony through LucyTuning -----

For information on LucyTuning go to: http://www.lucytune.com

LucyTuned Lullabies (from around the world):
http://www.lullabies.co.uk

Skype user = lucytune

I scored 76 percent on the rhythym test, 89 percent on the tonedeaf test, reliably differentiated tones 1.2 hz apart at 500 hz, on my first try. This is fun.

Oz.

----- Original Message -----
From: Charles Lucy
To: tuning@yahoogroups.com
Sent: 23 Nisan 2007 Pazartesi 13:13
Subject: [tuning] Hearing test - taken from Logic user group list

This is off topic, but an interesting test of your ability to discern fine
differences in musical passages. The developer is in med school but formerly
worked for NI on Reaktor and Absynth.

Its actually a pretty difficult quiz, and experienced musicians might only
get about 80% correct. It takes about 5 minutes, but if you know the sounds are
different you can immediately hit the button and go on to the next one.

http://www.jakemandell.com/tonedeaf/

Charles Lucy lucy@lucytune.com

----- Promoting global harmony through LucyTuning -----

For information on LucyTuning go to: http://www.lucytune.com

LucyTuned Lullabies (from around the world):
http://www.lullabies.co.uk

Skype user = lucytune