Yahoo Tuning Groups Ultimate Backup tuning review of 53-tET

hi Joe,

> From: "Joseph Pehrson" <jpehrson@rcn.com>
> To: <tuning@yahoogroups.com>
> Sent: Friday, October 25, 2002 12:52 PM
> Subject: [tuning] review of 53-tET
>
>
> Monz' Sonic Arts site is again down. Maybe he's working on it.

hmmm ... no, i'm not. i'm surprised that it's down.

> Anyway, I'd like to review 53-tET.
>
> What's the story again?? Isn't that a "circulating pythagorean" or
> some such?
>
> So all the fifths are very close to just. I think quite a few other
> intervals are close to just as well.
>
> Please refresh my "Swiss-cheese" memory...

53edo is a really excellent approximation to Pythagorean tuning,
and because its step-size falls nearly midway between the Pythagorean
and syntonic commas, it provides a very good representation of
5-limit JI as well.

from my "bingp-card-lattice" page:

>> 53edo tempers out the skhisma [8 1], kleisma [-5 6],
>> and semicomma [3 7], and their derivatives.
>>
>> 53 edo does not temper out the syntonic comma,
>> diaschisma, diesis, minimal diesis, magic comma,
>> or ampersand's comma.
>>
>> Note that for this big section of the lattice,
>> there is no error between the 53edo bingo-card lattice
>> and that showing the nearest 53edo approximation to JI.
>> This means, therefore, that 53edo provides a superb system
>> of integer interval-measurement for most 3-limit Pythagorean
>> and 5-limit JI tunings.

from the "equal temperament" definition:
(view in "Expand Messages" mode if viewing on the Yahoo website)

53edo prime interval
degrees vector ratio name
3 5

0 [ 8 1] 32805:32768 skhisma
1 [-4 -2] 2048:2025 diaschisma
1 [ 4 -1] 81:80 syntonic comma
2 [ 0 -3] 128:125 diesis
3 [-1 2] 25:24 JI chromatic semitone
4 [ 3 1] 135:128 Ellis larger limma, Rameau mean semitone
4 [-5 0] 256:243 limma
5 [-1 -1] 16:15 JI diatonic semitone
6 [ 3 -2] 27:25
8 [-2 1] 10:9 minor tone
9 [ 2 0] 9:8 major tone
10 [-2 -2] 256:225
14 [ 1 -1] 6:5 minor 3rd
17 [ 0 1] 5:4 major 3rd
22 [-1 0] 4:3 4th
31 [ 1 0] 3:2 5th

here is my ASCII reproduction of the 5-limit 53edo
bingo-card-lattice, showing the central periodicity-block
(i.e., the 53 pitches closest to n^0). numbers are
53edo-degrees, and numbers in parentheses show the four
pairs of pitches where two rational representations are
equally far from n^0.

(view in "Expand Messages" mode if viewing on the Yahoo website)

exponent of 3
-4 -3 -2 -1 0 1 2 3 4
e
x 5 (32)
p 4 37 15 (46)
o 3 42 20 51 29 ( 7)
n 2 47 25 3 34 12 43 (21)
e 1 52 30 8 39 17 48 26 4
n 0 35 13 44 22 0 31 9 40 18
t -1 49 27 5 36 14 45 23 1
-2 (32) 10 41 19 50 28 6
o -3 (46) 24 2 33 11
f -4 ( 7) 38 16
-5 (21)
5

-monz
"all roads lead to n^0"

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "Joseph Pehrson" <jpehrson@r...> wrote:
> > Monz' Sonic Arts site is again down. Maybe he's working on it.
> > Anyway, I'd like to review 53-tET.
> >
> > What's the story again?? Isn't that a "circulating pythagorean"
or
> > some such?
>
> Characteristic 7-limit temperaments for 53-et are schismic, orwell,
>amt, and catakleismic. I think one of my series of et articles a
>while back discussed it.

let's get more basic for joseph . . .

53-equal has a maximum error of:

0.07 cents in the 3-limit
1.41 cents in the 5-limit
6.17 cents in the 7-limit
6.17 cents in the 9-limit

so yes, it's essentially a pythagorean chain that closes after 53
fifths . . .

it also essentially emulates 5-limit just intonation . . . on a 53-
tone "donut" . . .

🔗Joseph Pehrson <jpehrson@rcn.com>

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>

/tuning/topicId_40117.html#40141

>
> let's get more basic for joseph . . .
>
> 53-equal has a maximum error of:
>
> 0.07 cents in the 3-limit
> 1.41 cents in the 5-limit
> 6.17 cents in the 7-limit
> 6.17 cents in the 9-limit
>
> so yes, it's essentially a pythagorean chain that closes after 53
> fifths . . .
>

***Thanks, Paul! Yes, that's what I needed. But, when it closes
after 53, does it close *exactly??* Probably not. How far is it
off??

> it also essentially emulates 5-limit just intonation . . . on a 53-
> tone "donut" . . .

***Yes, the 5-limit emulation is really quite fine here! (Of course
Blackjack does much better in the 7-limit, and 72-tET in the even
higher ones... not too surprising...)

J. Pehrson

🔗Joseph Pehrson <jpehrson@rcn.com>

--- In tuning@y..., "Joseph Pehrson" <jpehrson@r...> wrote:

/tuning/topicId_40117.html#40159

> --- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>
>
> /tuning/topicId_40117.html#40141
>
> >
> > let's get more basic for joseph . . .
> >
> > 53-equal has a maximum error of:
> >
> > 0.07 cents in the 3-limit
> > 1.41 cents in the 5-limit
> > 6.17 cents in the 7-limit
> > 6.17 cents in the 9-limit
> >
> > so yes, it's essentially a pythagorean chain that closes after 53
> > fifths . . .
> >
>
> ***Thanks, Paul! Yes, that's what I needed. But, when it closes
> after 53, does it close *exactly??* Probably not. How far is it
> off??
>

Heh, heh, this is funny... Of course, 53-equal *closes*! I was
meaning 53-"looping" Pythagorean, and Margo just set me right on
that!....

Thanks!

🔗monz <monz@attglobal.net>

hi Joe,

> From: "Joseph Pehrson" <jpehrson@rcn.com>
> To: <tuning@yahoogroups.com>
> Sent: Friday, October 25, 2002 7:59 PM
> Subject: [tuning] Re: review of 53-tET
>
>
> --- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...>
>
> /tuning/topicId_40117.html#40141
>
> >
> > let's get more basic for joseph . . .
> >
> > 53-equal has a maximum error of:
> >
> > 0.07 cents in the 3-limit
> > 1.41 cents in the 5-limit
> > 6.17 cents in the 7-limit
> > 6.17 cents in the 9-limit
> >
> > so yes, it's essentially a pythagorean chain that closes after 53
> > fifths . . .
> >
>
> ***Thanks, Paul! Yes, that's what I needed. But, when it closes
> after 53, does it close *exactly??* Probably not. How far is it
> off??

Joe, 53edo is an equal-temperament, so yes, it *does* close exactly
after 53 tones.

the reason why 53edo is so good at approximating Pythagorean tuning
is because of Mercator's comma, which is a ratio with hideously large
numbers which i prefer to write simply as 3^53, and it's about
3.615045866 cents.

what this means is that when constructing a Pythagorean chain,
one will first encounter the Pythagorean comma (~23.46 cents)
at 3^12, then if the chain is carried out farther, the next
note you'll find which is closer to the 1/1 is 3^53, only
~3.6 cents away.

so in the same way that 12edo can be viewed as a 1/12-Pythagorean-comma
temperament, 53edo can be viewed as a 1/53-Mercator-comma temperament.
each "5th" is narrowed by 1/53 of a Mercator-comma, so that when
you reach the 53rd one, the ~3.6 cents has vanished and you're
exactly back where you started from.

now, 1/53 of a Mercator-comma is only ~0.068208413 cent
(about 1/15-cent), so a "5th" tempered by that tiny amount
really isn't all that different from a "pure" 3:2. thus,
53edo's excellence in approximating Pythagorean tuning.

as far as emulating 5-limit JI, 53edo's "major-3rd" comes
in both the Pythagorean and JI flavors. if we think of the
53edo "5th" as our generator, +4 generators gives us the
pseudo-Pythagorean "major-3rd" resembling 81:64, and
-8 generators gives us the schismic and pseudo-JI "major 3rd"
resembling 5:4.

the 53edo schismic 3rd is ~384.9056604 cents, which is
only ~1.408053487 cents narrower than the 5:4 ratio.
so you see that it also represents 5-limit extremely well,
altho not nearly as perfectly as it does Pythagorean.

> > it also essentially emulates 5-limit just intonation . . .
> > on a 53-tone "donut" . . .
>
> ***Yes, the 5-limit emulation is really quite fine here!
> (Of course Blackjack does much better in the 7-limit, and
> 72-tET in the even higher ones... not too surprising...)

right ... any of the MIRACLE tunings give a much better overall
systemic approximation to not only 7-limit but also 11-limit.
but for a relatively-low-cardinality EDO which does a great
job of approximating both Pythagorean and 5-limit JI,
53edo can't be beat!

this was all figured out centuries ago in China.
(and of course, i'd bet that the Chinese got it somehow
from the Sumerians.)

-monz
"all roads lead to n^0"

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗monz <monz@attglobal.net>

----- Original Message -----
From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: Friday, October 25, 2002 8:53 PM
Subject: [tuning] Re: review of 53-tET

> --- In tuning@y..., "monz" <monz@a...> wrote:
> >
> > this was all figured out centuries ago in China.
> > (and of course, i'd bet that the Chinese got it somehow
> > from the Sumerians.)
>
> the chinese never looked at the 5-limit or higher, as they were
> strictly . . . what we in the west would call "pythagorean".

hmmm ... really? even the theorists never even bothered
to look at 5-limit? i'm surprised if that's true.

-monz

🔗Joseph Pehrson <jpehrson@rcn.com>

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

/tuning/topicId_40117.html#40165

> > >
> >
> > ***Thanks, Paul! Yes, that's what I needed. But, when it closes
> > after 53, does it close *exactly??* Probably not. How far is it
> > off??
>
>
> Joe, 53edo is an equal-temperament, so yes, it *does* close exactly
> after 53 tones.
>

***I found this humorous after I realized my mistake. I was thinking
of the 53-note Pythagorean "loop" that Margo has been discussing. (I
happened to come up with that "loopy" term, myself, and almost forgot
about it!)

I guess I tend to think of a Pythagorean loop as something *pure*,
maybe from reading some of Margo's essays on Medieval music... so
53edo was not so much on my mind.

I'm assuming you call it an EDO rather than an ET because what
it's "tempering" is rather vague (with all the different
possibilities??)

> the reason why 53edo is so good at approximating Pythagorean tuning
> is because of Mercator's comma, which is a ratio with hideously
large numbers which i prefer to write simply as 3^53, and it's about
> 3.615045866 cents.
>

***Is this the "map cat" or somebody related to him??

> what this means is that when constructing a Pythagorean chain,
> one will first encounter the Pythagorean comma (~23.46 cents)
> at 3^12, then if the chain is carried out farther, the next
> note you'll find which is closer to the 1/1 is 3^53, only
> ~3.6 cents away.
>
> so in the same way that 12edo can be viewed as a 1/12-Pythagorean-
comma
> temperament, 53edo can be viewed as a 1/53-Mercator-comma
temperament.
> each "5th" is narrowed by 1/53 of a Mercator-comma, so that when
> you reach the 53rd one, the ~3.6 cents has vanished and you're
> exactly back where you started from.
>
> now, 1/53 of a Mercator-comma is only ~0.068208413 cent
> (about 1/15-cent), so a "5th" tempered by that tiny amount
> really isn't all that different from a "pure" 3:2. thus,
> 53edo's excellence in approximating Pythagorean tuning.
>

***I'd hate to try to tune that on my old piano.... :)

>
> as far as emulating 5-limit JI, 53edo's "major-3rd" comes
> in both the Pythagorean and JI flavors. if we think of the
> 53edo "5th" as our generator, +4 generators gives us the
> pseudo-Pythagorean "major-3rd" resembling 81:64, and
> -8 generators gives us the schismic and pseudo-JI "major 3rd"
> resembling 5:4.
>

***Oh sure... so this is the E-G# third that Lindley was discussing
and that, quite possibly, led to further 5-limit developments of the
Renaissance. (I *think* that was the spelling...)

> the 53edo schismic 3rd is ~384.9056604 cents, which is
> only ~1.408053487 cents narrower than the 5:4 ratio.
> so you see that it also represents 5-limit extremely well,
> altho not nearly as perfectly as it does Pythagorean.
>

***It's great to have a scale that does *both* of those...

Thanks, Monz!

Joe P.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@y..., "monz" <monz@a...> wrote:
>
> ----- Original Message -----
> From: "wallyesterpaulrus" <wallyesterpaulrus@y...>
> To: <tuning@y...>
> Sent: Friday, October 25, 2002 8:53 PM
> Subject: [tuning] Re: review of 53-tET
>
>
> > --- In tuning@y..., "monz" <monz@a...> wrote:
> > >
> > > this was all figured out centuries ago in China.
> > > (and of course, i'd bet that the Chinese got it somehow
> > > from the Sumerians.)
> >
> > the chinese never looked at the 5-limit or higher, as they
were
> > strictly . . . what we in the west would call "pythagorean".
>
>
>
> hmmm ... really? even the theorists never even bothered
> to look at 5-limit? i'm surprised if that's true.

it's true that 53-equal was proposed from only a 3-limit basis in
all the published references i've ever read about chinese
musical theory.

review of 53-tET

🔗Joseph Pehrson <jpehrson@rcn.com>

🔗Gene Ward Smith <genewardsmith@juno.com>

🔗monz <monz@attglobal.net>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Joseph Pehrson <jpehrson@rcn.com>

🔗Joseph Pehrson <jpehrson@rcn.com>

🔗Joseph Pehrson <jpehrson@rcn.com>

🔗monz <monz@attglobal.net>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗monz <monz@attglobal.net>

🔗Joseph Pehrson <jpehrson@rcn.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗monz <monz@attglobal.net>