Meantones in EDOs (was Re: More on shruti-s)

Hi Ozan.

I wrote:

> > Actually amazingly close to 119-EDO, don't you think? :-)
> >
>
>
> 1.4 cent highest absolute difference. I wouldn't say amazingly, but yes,
> pretty much.

Don't forget that in order you got the 1.4 cent difference, you have to
"travel" way up to the 119th fifth. Do you know about any other EDO which
approximates some of the "famous" meantones in that closely, or even more
closely?

Petr

Hi Ozan.

I wrote:

> Don't forget that in order you got the 1.4 cent difference, you have to
> "travel" way up to the 119th fifth. Do you know about any other EDO which
> approximates some of the "famous" meantones in that closely, or even more
> closely?

OK, you've successfully persuaded me to find one. It's a chain of 81 fifths
of 5/19-comma meantone. There you get a difference of just ~0.068 cents --
you see, after stacking 81 fifths!
Well, I think this is the best approximation to one of the "known" meantones
that I have ever found.
Or do you think there are even better ones in the class of EDOs not
exceeding 200?

Petr

Hi again, Ozan.

I wrote:

> Well, I think this is the best approximation to one of the "known"
meantones
> that I have ever found.
> Or do you think there are even better ones in the class of EDOs not
> exceeding 200?

Wow! Hell, I must have got mad. Of course, there are. Have you tried
comparing 1/11-comma meantone and 12-EDO?

Petr

Petr,

----- Original Message -----
From: "Petr Par�zek" <p.parizek@chello.cz>
To: <tuning@yahoogroups.com>
Sent: 24 �ubat 2006 Cuma 11:30
Subject: [tuning] Meantones in EDOs (was Re: More on shruti-s)

> >
> > 1.4 cent highest absolute difference. I wouldn't say amazingly, but yes,
> > pretty much.
>
> Don't forget that in order you got the 1.4 cent difference, you have to
> "travel" way up to the 119th fifth. Do you know about any other EDO which
> approximates some of the "famous" meantones in that closely, or even more
> closely?
>
> Petr
>
>

You admitted yourself to 12-edo!

Petr,

>
> Wow! Hell, I must have got mad. Of course, there are. Have you tried
> comparing 1/11-comma meantone and 12-EDO?
>
> Petr
>
>

The calculation is fairly simple:

log (3/2) times 1200 divided by log 2 makes:
701.95500086538741774448673273738 cents

1/11th of log (81/80) times 1200 divided by log 2 makes:
1.9551172360649866848694195965683 cents

Subtracting this amount from the pure fifth gives:
699.99988362932243105961731314081 cents

(12/7) times this value yields a very close octave:
1199.9998005074098818164868225271 cents

Where the half-tone is 1/12th of this octave:
99.999983375617490151373901877257 cents

Compared to 12-edo:

Step size is 100.0000 cents
1: 99.999: 1: 100.0000 cents, diff. 0.000008 steps, 0.0008 cents
2: 200.000: 2: 200.0000 cents, diff. 0.000002 steps, 0.0002 cents
3: 300.000: 3: 300.0000 cents, diff. -0.000003 steps, -0.0003 cents
4: 400.000: 4: 400.0000 cents, diff. 0.000004 steps, 0.0005 cents
5: 500.000: 5: 500.0000 cents, diff. -0.000001 steps, -0.0001 cents
6: 599.999: 6: 600.0000 cents, diff. 0.000007 steps, 0.0007 cents
7: 700.000: 7: 700.0000 cents, diff. 0.000001 steps, 0.0001 cents
8: 799.999: 8: 800.0000 cents, diff. 0.000009 steps, 0.0009 cents
9: 900.000: 9: 900.0000 cents, diff. 0.000003 steps, 0.0003 cents
10: 1000.000: 10: 1000.0000 cents, diff. -0.000002 steps, -0.0002 cents
11: 1099.999: 11: 1100.0000 cents, diff. 0.000005 steps, 0.0006 cents
12: 1200.000: 12: 1200.0000 cents, diff. -0.000000 steps, -0.0000 cents
Total absolute difference : 0.000048 steps, 0.0049 cents
Average absolute difference: 0.000004 steps, 0.0004 cents
Root mean square difference: 0.000005 steps, 0.0005 cents
Highest absolute difference: 0.000009 steps, 0.0009 cents

It is practically identical!

Cordially,
Ozan

Petr,

>
> OK, you've successfully persuaded me to find one. It's a chain of 81
fifths
> of 5/19-comma meantone. There you get a difference of just ~0.068 cents --
> you see, after stacking 81 fifths!
> Well, I think this is the best approximation to one of the "known"
meantones
> that I have ever found.
> Or do you think there are even better ones in the class of EDOs not
> exceeding 200?

88 equal divisions of an octave stretched by 0.0663 cents seems to represent
the Harrison/Lucy meantone perfectly:

(300/Pi)+600 cents divided by 51 times 88.

>
> Petr
>
>
>

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:

> 88 equal divisions of an octave stretched by 0.0663 cents
> seems to represent the Harrison/Lucy meantone perfectly:
>
> (300/Pi)+600 cents divided by 51 times 88.

88-edo represents LucyTuning *very well*, but *not* perfectly!
If Charles Lucy sees what you wrote, he's going to give you
a hard time!

... I'm simply pointing out that the two tunings, while
extremely similar, are not exactly the same, that's all.
Your use of the word "perfectly" implies that there's
no difference.

-monz
http://tonalsoft.com
Tonescape microtonal music software

Did you notice that dividing (300/Pi)+600 cents by 51 and multiplying it by
88 yields an octave of 1200.0663 cents which, when divided by 88/51 results
in the Lucy/Harrison fifth again?

Of course, what was I thinking?

It is not the first time I mentioned this to Charles. He seems to have
brushed it aside. 88-edso (0.0663 cents stretched octave) represents his
system only too `perfectly`.

Yet, nobody is perfect!

Oz.

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 01 Mart 2006 �ar�amba 23:53
Subject: [tuning] Meantones in EDOs (was Re: More on shruti-s)

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> > 88 equal divisions of an octave stretched by 0.0663 cents
> > seems to represent the Harrison/Lucy meantone perfectly:
> >
> > (300/Pi)+600 cents divided by 51 times 88.
>
>
> 88-edo represents LucyTuning *very well*, but *not* perfectly!
> If Charles Lucy sees what you wrote, he's going to give you
> a hard time!
>
> ... I'm simply pointing out that the two tunings, while
> extremely similar, are not exactly the same, that's all.
> Your use of the word "perfectly" implies that there's
> no difference.
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>
>
>

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@> wrote:
>
> > 88 equal divisions of an octave stretched by 0.0663 cents
> > seems to represent the Harrison/Lucy meantone perfectly:
> >
> > (300/Pi)+600 cents divided by 51 times 88.
>
>
> 88-edo represents LucyTuning *very well*, but *not* perfectly!
> If Charles Lucy sees what you wrote, he's going to give you
> a hard time!

Yes, but since there's no audible difference, the hard times he gives
seems pretty dubious. "Perfectly" seems apt particularly since there
is no consonant interval which is exactly approximated.