Yahoo Tuning Groups Ultimate Backup tuning other EDOs rendered exactly in 768edo

since the cardinality of 768edo factors into
2^8 * 3^1 , it can represent every one of the
following EDOs exactly:

2 3 edo

[ 1 0 ] = 2
[ 2 0 ] = 4
[ 1 1 ] = 6
[ 3 0 ] = 8
[ 2 1 ] = 12
[ 4 0 ] = 16
[ 3 1 ] = 24
[ 5 0 ] = 32
[ 4 1 ] = 48
[ 6 0 ] = 64
[ 5 1 ] = 96
[ 7 0 ] = 128
[ 6 1 ] = 192
[ 8 0 ] = 256
[ 7 1 ] = 384
[ 8 1 ] = 768

so to whomever has typical MIDI hardware with
a resolution of 768edo: if you use any of the
above EDOs, you get exactly what you think you get.

for other EDOs with 768edo hardware, most notes
have some amount of error ... but it's never
more than 1/2 of a 768edo degree, which would be
a maximum of 25/32 (= 0.78125 = ~4/5) of a cent,
and which is a heptamu.

... which makes 768edo, at worst, a "microtemperament"
of any other tuning, by Gene's definition of
"microtemperament" as maximum error less than 1 cent.

-monz

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> http://sonic-arts.org/dict/hexamu.htm
>
>
>
> since the cardinality of 768edo factors into
> 2^8 * 3^1 , it can represent every one of the
> following EDOs exactly:
>
>
> 2 3 edo
>
> [ 1 0 ] = 2
> [ 2 0 ] = 4
> [ 1 1 ] = 6
> [ 3 0 ] = 8
> [ 2 1 ] = 12
> [ 4 0 ] = 16
> [ 3 1 ] = 24
> [ 5 0 ] = 32
> [ 4 1 ] = 48
> [ 6 0 ] = 64
> [ 5 1 ] = 96
> [ 7 0 ] = 128
> [ 6 1 ] = 192
> [ 8 0 ] = 256
> [ 7 1 ] = 384
> [ 8 1 ] = 768
>
>
> so to whomever has typical MIDI hardware with
> a resolution of 768edo: if you use any of the
> above EDOs, you get exactly what you think you get.
>
>
> for other EDOs with 768edo hardware, most notes
> have some amount of error ... but it's never
> more than 1/2 of a 768edo degree, which would be
> a maximum of 25/32 (= 0.78125 = ~4/5) of a cent,
> and which is a heptamu.
>
> ... which makes 768edo, at worst, a "microtemperament"
> of any other tuning, by Gene's definition of
> "microtemperament" as maximum error less than 1 cent.
>
>
>
> -monz

i see that you've added all (fourteen? of) these nu "mu" terms to
your dictionary as well as to your equal temperament page
(http://sonic-arts.org/dict/eqtemp.htm) . . .

speaking of 16-equal, you might want to reread

/tuning/topicId_39531.html#39572

and add victor cerullo under 16-equal . . . also 600-equal is a
notable omission of your equal temperament page, given that it was
the basis for measurement for several theorists, 1/600 octave being
the "iring" of widogast iring (1898), and the "centitone" of joseph
yasser (1932) . . . i made an effort to include historically
important divisions such as 301, 600, 1000, and 3072 (=768*4) on the
graphs i made for that page as well as the "guide" for those graphs
which is still only available at

/tuning/database?
method=reportRows&tbl=10&sortBy=10&sortDir=up

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

> and add victor cerullo under 16-equal . . . also 600-equal is a
> notable omission of your equal temperament page, given that it was
> the basis for measurement for several theorists, 1/600 octave being
> the "iring" of widogast iring (1898), and the "centitone" of joseph
> yasser (1932) . . .

I would add 224, 311 and 494 equal. Also, I have composed in 27, 41,
46, 53, 171 and 175 equal if you need more examples.

🔗monz <monz@attglobal.net>

hi paul,

> From: "Paul Erlich" <perlich@aya.yale.edu>
> To: <tuning@yahoogroups.com>
> Sent: Tuesday, July 08, 2003 3:26 PM
> Subject: [tuning] Re: other EDOs rendered exactly in 768edo
>
>
> i see that you've added all (fourteen? of) these nu "mu" terms to
> your dictionary as well as to your equal temperament page
> (http://sonic-arts.org/dict/eqtemp.htm) . . .

i figured that since these are MIDI tuning specs,
they're pretty important and should have both
names and good definitions.

i suspect that "dodekamu", "tetradekamu", and
especially "hexamu" will be the ones most frequently
consulted ... and i expect those three pages to
grow as i continue to explore these tunings.

i'm especially fascinated with 768edo right now,
since it's what i've been using all along and
didn't know it! (at least for those years that
i had the Yamaha instruments, i knew what tuning
my microtonal music used!)

> speaking of 16-equal, you might want to reread
>
> /tuning/topicId_39531.html#39572
>
> and add victor cerullo under 16-equal . . . also 600-equal is a
> notable omission of your equal temperament page, given that it was
> the basis for measurement for several theorists, 1/600 octave being
> the "iring" of widogast iring (1898), and the "centitone" of joseph
> yasser (1932) . . . i made an effort to include historically
> important divisions such as 301, 600, 1000, and 3072 (=768*4) on the
> graphs i made for that page as well as the "guide" for those graphs
> which is still only available at
>
> /tuning/database?
> method=reportRows&tbl=10&sortBy=10&sortDir=up

thanks for that. i've added Victor to 16edo,
and added 600edo with Iring and Yasser.

unfortunately, incorporating the database will
be enough work that i keep putting it off ...
eventually it will be in there.

-monz

🔗Paul Erlich <perlich@aya.yale.edu>

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

> thanks for that. i've added Victor to 16edo,
> and added 600edo with Iring and Yasser.

awesome. finally, since 4296 is the clear winner as far as the graphs
are concerned, you might want to specifically copy the annotation
from this page,

http://sonic-arts.org/dict/marc-edolist.htm

where it's listed thus:

'4296: 1992 Marc Jones (used as most convenient UHT to measure 5th
limit intervals)'

it's a very natural place to stop, as evidenced here:

http://www.kees.cc/tuning/s235.html

and it's a multiple of 12 (multimu?), due to it tempering out the
atom of kirnberger,
2923003274661805836407369665432566039311865085952/29229773394926806124
51840826835216578535400390625.
(shown as the {2,3,5} vector, [-161 84 12], in the URL above)

🔗Joseph Pehrson <jpehrson@rcn.com>

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

/tuning/topicId_45384.html#45384

>
> http://sonic-arts.org/dict/hexamu.htm
>
>
>
> since the cardinality of 768edo factors into
> 2^8 * 3^1 , it can represent every one of the
> following EDOs exactly:
>
>
> 2 3 edo
>
> [ 1 0 ] = 2
> [ 2 0 ] = 4
> [ 1 1 ] = 6
> [ 3 0 ] = 8
> [ 2 1 ] = 12
> [ 4 0 ] = 16
> [ 3 1 ] = 24
> [ 5 0 ] = 32
> [ 4 1 ] = 48
> [ 6 0 ] = 64
> [ 5 1 ] = 96
> [ 7 0 ] = 128
> [ 6 1 ] = 192
> [ 8 0 ] = 256
> [ 7 1 ] = 384
> [ 8 1 ] = 768
>
>
> so to whomever has typical MIDI hardware with
> a resolution of 768edo: if you use any of the
> above EDOs, you get exactly what you think you get.
>
>
> for other EDOs with 768edo hardware, most notes
> have some amount of error ... but it's never
> more than 1/2 of a 768edo degree, which would be
> a maximum of 25/32 (= 0.78125 = ~4/5) of a cent,
> and which is a heptamu.
>

***Why is this again? (I think it has been discussed before....) Is
it because you're considering an *interval* of two notes which
doubles the accuracy and therefore *halves* the error??

J. Pehrson

🔗Graham Breed <graham@microtonal.co.uk>

Monz:
>>for other EDOs with 768edo hardware, most notes
>>have some amount of error ... but it's never
>>more than 1/2 of a 768edo degree, which would be
>>a maximum of 25/32 (= 0.78125 = ~4/5) of a cent,
>>and which is a heptamu.

Joe P:
> ***Why is this again? (I think it has been discussed before....) Is > it because you're considering an *interval* of two notes which > doubles the accuracy and therefore *halves* the error??

If you're rounding each note to the nearest step, you can never be more than half a step out. But this is notes, not intervals. If one note is half another step too sharp, and another note is half a step too flat, the interval between them will be out by a whole step. So when considering intervals you do have to assume they could be out by one tuning step, which is exactly 1.5635 cents for 768edo.

The worst intervals in 11-limit minimax Miracle are over 3.3 cents out, so if you add an error of over 1.5 cents in the resolution, you get a total error of 4.9 cents relative to JI. Most intervals won't be this bad, and some will even be closer to JI than their ideal tempered values, but you never know when one of those bad intervals is going to crop up. If that bothers you, you could try adaptive tuning to JI, or send vibrato as pitch bends so that no intervals will always be bad. But the easiest thing would be to use a synthesizer that has a better resolution.

Graham

🔗Joseph Pehrson <jpehrson@rcn.com>

--- In tuning@yahoogroups.com, Graham Breed <graham@m...> wrote:

/tuning/topicId_45384.html#45463

> Monz:
> >>for other EDOs with 768edo hardware, most notes
> >>have some amount of error ... but it's never
> >>more than 1/2 of a 768edo degree, which would be
> >>a maximum of 25/32 (= 0.78125 = ~4/5) of a cent,
> >>and which is a heptamu.
>
> Joe P:
> > ***Why is this again? (I think it has been discussed before....)
Is
> > it because you're considering an *interval* of two notes which
> > doubles the accuracy and therefore *halves* the error??
>
> If you're rounding each note to the nearest step, you can never be
more
> than half a step out. But this is notes, not intervals. If one
note is
> half another step too sharp, and another note is half a step too
flat,
> the interval between them will be out by a whole step. So when
> considering intervals you do have to assume they could be out by
one
> tuning step, which is exactly 1.5635 cents for 768edo.
>

***Hi Graham!

Oh sure... this makes sense... I was kinda wondering about that,
since it didn't seem right that the error would only be a half step.
Thanks for clearing this up.

> The worst intervals in 11-limit minimax Miracle are over 3.3 cents
out,
> so if you add an error of over 1.5 cents in the resolution, you get
a
> total error of 4.9 cents relative to JI.

***Yes, that part I can see from Paul's chart and some of his
previous comments. Well, 11-limit starts getting a tad "dissonant"
anyway so the slight beating doesn't bother me...

J. Pehrson