Yahoo Tuning Groups Ultimate Backup tuning tina - unit of interval measurement: 8539-edo

Hello all,

I've discovered something i've been searching for for a
long time: an EDO which can be used as an logarithmic
*integer* unit of interval measurement, namely, 8539-edo.

This tuning has a very low error in JI up to the 31-limit,
and good consistency. Thus, one may use the up-to-4-digit
integer values without any decimal places for a huge number
of JI intervals.

I propose that we replace cents with tinas.

I've made webpage for it in the Encyclopedia:

http://tonalsoft.com/enc/t/tina.aspx

The 12-edo semitone is exactly 711 7/12 (= 711.58,3...) tinas.

Below is a table of tina values for common 31-limit JI
interval sizes (from my webpage, somewhat abridged). There
is no rounding error and the decimal parts are in almost
all cases negigible.

The only interval i know of which seems to have any
real importance but is too small to be represented in
tinas is the atom (of Kirnbirger).

interval name tinas ratio

octave ........................ 8539 ...... 1/1
8386 .... 160/81
8345 ..... 63/32
8247 .... 125/64
8160 ..... 64/33
31st harmonic ................. 8148 ..... 31/16
septimal major-7th ............ 8091 ..... 27/14
8068 ..... 77/40
8036 ..... 48/25
7938 ..... 40/21
pyth major-7th ................ 7897 .... 243/128
7883 .... 256/135
7814 ..... 66/35
17th subharmonic .............. 7792 ..... 32/17
just maj-7th / 15th harm ..... 7744 ..... 15/8
7730 ... 4096/2187
7689 ..... 28/15
undecimal neutral-7th ......... 7467 ..... 11/6
7435 ..... 64/35
7365 ..... 20/11
29th harmonic ................. 7326 ..... 29/16
7273 .... 231/128
pyth aug-6th .................. 7255 .. 59049/32768
just minor-7th ................ 7241 ...... 9/5
7102 ... 3645/2048
pyth min-7th / 9th subharm .... 7088 ..... 16/9
just aug-6th .................. 6949 .... 225/128
7th harmonic .................. 6894 ...... 7/4
septimal minor-7th ............ 6894 ...... 7/4
6862 ..... 96/55
6672 ..... 55/32
septimal major-6th ............ 6640 ..... 12/7
6599 ... 2187/1280
6585 .... 128/75
pyth maj-6th / 27th harm ..... 6446 ..... 27/16
19th subharmonic .............. 6422 ..... 32/19
6363 .... 176/105
just major-6th ................ 6293 ...... 5/3
6169 ..... 33/20
6099 .... 105/64
6067 ..... 18/11
13th harmonic ................. 5981 ..... 13/8
5822 ..... 77/48
pyth aug-5th .................. 5804 ... 6561/4096
just min-6th / 5th subharm .... 5790 ...... 8/5
5651 .... 405/256
pyth minor-6th ................ 5637 .... 128/81
5568 ..... 11/7
just aug-5th .................. 5498 ..... 25/16
5443 ..... 14/9
5411 .... 256/165
5374 ..... 99/64
septimal 5th / 21st subh ...... 5189 ..... 32/21
5148 .... 243/160
perfect-5th / 3rd harmonic .... 4995 ...... 3/2
just wolf-5th ................. 4842 ..... 40/27
4801 .... 189/128
4718 ..... 22/15
4703 .... 375/256
4648 ..... 35/24
11th subharmonic .............. 4616 ..... 16/11
4524 .... 231/160
large just dim-5th ............ 4492 ..... 36/25
23rd harmonic ................. 4471 ..... 23/16
septimal aug-4th .............. 4394 ..... 10/7
pyth aug-4th / tritone ........ 4353 .... 729/512
small just dim-5th ............ 4339 ..... 64/45
just aug-4th / tritone ........ 4200 ..... 45/32
pyth dim-5th .................. 4186 ... 1024/729
septimal dim-5th .............. 4145 ...... 7/5
23rd subharmonic .............. 4068 ..... 32/23
4047 ..... 25/18
11th harmonic ................. 3923 ..... 11/8
3891 ..... 48/35
3836 .... 512/375
3821 ..... 15/11
3711 . 177147/131072
3697 ..... 27/20
3558 .. 10935/8192
perfect-4th / 3rd subharm ..... 3544 ...... 4/3
3405 .... 675/512
septimal-4th / 21st harm ...... 3350 ..... 21/16
3252 .... 125/96
3165 .... 128/99
3128 .... 165/128
septimal major-3rd ............ 3096 ...... 9/7
3073 ..... 77/60
3041 ..... 32/25
pyth major-3rd / ditone ....... 2902 ..... 81/64
2888 .... 512/405
2819 ..... 44/35
just major-3rd / 5th harm ... 2749 ...... 5/4
13th subharmonic .............. 2558 ..... 16/13
undecimal neutral-3rd ......... 2472 ..... 11/9
2440 .... 128/105
2370 ..... 40/33
2278 ..... 77/64
pyth aug-2nd .................. 2260 .. 19683/16384
just minor-3rd ................ 2246 ...... 6/5
19th harmonic ................. 2117 ..... 19/16
2107 ... 1215/1024
pyth min-3rd / trihemitone .... 2093 ..... 32/27
2024 ..... 33/28
1954 ..... 75/64
septimal minor-3rd ............ 1899 ...... 7/6
1867 ..... 64/55
1677 ..... 55/48
septimal maj-2nd / 7th subh ... 1645 ...... 8/7
1604 .... 729/640
pyth maj-2nd / tone / 9th harm 1451 ...... 9/8
just minor-2nd / small tone ... 1298 ..... 10/9
29th subharmonic .............. 1213 ..... 32/29
1174 ..... 11/10
1104 ..... 35/32
1072 ..... 12/11
948 ..... 27/25
850 ..... 15/14
pyth aug-prime / apotome ...... 809 ... 2187/2048
just min-2nd / 15th subharm ... 795 ..... 16/15
17th harmonic ................. 747 ..... 17/16
large just aug-prime .......... 656 .... 135/128
pyth minor-2nd / limma ........ 642 .... 256/243
601 ..... 21/20
573 ..... 22/21
small just aug-prime .......... 503 ..... 25/24
pyth tricomma ................ 501 1.50E+17/1.44E+17
31st subharmonic .............. 391 ..... 32/31
undecimal-diesis / 33rd harm .. 379 ..... 33/32
maximal-diesis ................ 350 .... 250/243
septimal-diesis ............... 347 ..... 36/35
enharmonic diesis ............. 292 .... 128/125
large biseptimal-comma ........ 254 ..... 49/48
small biseptimal-comma ........ 249 ..... 50/49
magic-comma .................. 211 ... 3125/3072
septimal comma ................ 194 ..... 64/63
pythagorean-comma ............ 167 . 531441/524288
syntonic-comma ................ 153 ..... 81/80
diaschisma .................... 139 ... 2048/2025
semicomma .................... 72 2109375/2097152
kleisma ....................... 58 .. 15625/15552
septimal-kleisma .............. 55 .... 225/224
septimal-schisma .............. 27 33554432/33480783
mercator-comma ............... 26 1.94E+25/1.93E+25
nondecimal-schisma ............ 24 .... 513/512
skhisma ...................... 14 .. 32805/32768
monzisma ...................... 2 4.50E+17/4.50E+17
nanisma ...................... 1 6.49E+32/6.49E+32
origin / prime / unison ....... 0 ...... 1/1

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

🔗monz <monz@tonalsoft.com>

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

🔗monz <monz@tonalsoft.com>

Hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
> >
> > Hello all,
> >
> >
> > I've discovered something i've been searching for for a
> > long time: an EDO which can be used as an logarithmic
> > *integer* unit of interval measurement, namely, 8539-edo.
> >
> > This tuning has a very low error in JI up to the 31-limit,
> > and good consistency. Thus, one may use the up-to-4-digit
> > integer values without any decimal places for a huge number
> > of JI intervals.
> >
> > I propose that we replace cents with tinas.
>
> And unlike 11664, it isn't divisible by 12. In
> fact 8539 is a prime number. So for some purposes
> it is less than ideal. They are both standout
> systems, of course.

Yes, i've pointed that out myself. As i already said
in another post, it doesn't matter to me that it's
not divisible by 12, because i'm only interested in
the fact that it can be used as an integer unit of
interval measurement, so who cares if it doesn't divide
by 12?

> What's curious is that it's been around for a
> while and now suddenly is newsworthy. What was
> the tipping point? I guess one factor is that
> earlier, 8539 was discussed in the context of
> 13-21 limit systems, and would be listed with
> a bunch of others. Now we are noticing it fits
> well with Dave and George's particular requirements,
> though in fact I did mention it to them some while
> back.

Hmm, i totally missed the previous discussion of it.
(There was about a year or so that i hardly read the
tuning lists.)

This time around i found it on my own, and immediately
recognized that it filled a need that i've felt for
a long time. With all the writing and calculating i
do concerning musical intervals, it's a real pain to
always have to deal with decimal-place accuracy when
using cents (or any other not-so-accurate unit).

So it's really nice to have a unit that gives a
very good representation of a very wide variety of
intervals with only four digits, with a level of
resolution 7 times greater than cents.

Tina values of intervals can be added together using
only those integer values and the error of the result
will generally not be too bad.

The fact that one degree of 12-edo falls almost
midway between 711 and 712 tinas also doesn't bother
me much, because i've found that the error of the
12-edo 5th (4981 tinas) is only - 8 & 1/3 % of a
tina, which means that you'd have to go to 6 5ths
to reach an error of half a tina. That's accurate
enough for me.

There are some interesting discrepancies among the
meantones. The 1/4-comma meantone 5th maps to 4957
tinas, while that of 31-edo maps to 4958 tinas, and
the 1/6-comma meantone 5th maps to 4969 while that
of 55-edo maps to 4968.

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

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

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

> Yes, i've pointed that out myself. As i already said
> in another post, it doesn't matter to me that it's
> not divisible by 12, because i'm only interested in
> the fact that it can be used as an integer unit of
> interval measurement, so who cares if it doesn't divide
> by 12?

I'd like measurement systems to be divisible bu 12
in order eg to make the difference in a single midi
note number, which is 100 cents, come out evenly.
But clearly for many purposes it won't matter.

What are you looking for? I doubt it is exactly
same as what Dave and George want.

From my point of view, something like 54624 is
pretty darned good. It is under 1% error in the
5-limit, and under 5% error in the 11-limit, is
27-limit consistent, and divisible by 12. But it
probably won't interest you, Dave or George.

>
> > What's curious is that it's been around for a
> > while and now suddenly is newsworthy. What was
> > the tipping point? I guess one factor is that
> > earlier, 8539 was discussed in the context of
> > 13-21 limit systems, and would be listed with
> > a bunch of others. Now we are noticing it fits
> > well with Dave and George's particular requirements,
> > though in fact I did mention it to them some while
> > back.
>
>
> Hmm, i totally missed the previous discussion of it.
> (There was about a year or so that i hardly read the
> tuning lists.)
>
> This time around i found it on my own, and immediately
> recognized that it filled a need that i've felt for
> a long time. With all the writing and calculating i
> do concerning musical intervals, it's a real pain to
> always have to deal with decimal-place accuracy when
> using cents (or any other not-so-accurate unit).
>
> So it's really nice to have a unit that gives a
> very good representation of a very wide variety of
> intervals with only four digits, with a level of
> resolution 7 times greater than cents.
>
> Tina values of intervals can be added together using
> only those integer values and the error of the result
> will generally not be too bad.
>
>
> The fact that one degree of 12-edo falls almost
> midway between 711 and 712 tinas also doesn't bother
> me much, because i've found that the error of the
> 12-edo 5th (4981 tinas) is only - 8 & 1/3 % of a
> tina, which means that you'd have to go to 6 5ths
> to reach an error of half a tina. That's accurate
> enough for me.
>
> There are some interesting discrepancies among the
> meantones. The 1/4-comma meantone 5th maps to 4957
> tinas, while that of 31-edo maps to 4958 tinas, and
> the 1/6-comma meantone 5th maps to 4969 while that
> of 55-edo maps to 4968.
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

🔗Andreas Sparschuh <a_sparschuh@yahoo.com>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote/asked :
>
>..... why 8539 instead of 6079, 6691, or 11664, for
> instance?
>
The 3-limit quality of an n-EDO can be determinated,
simply by calculating the defect of the PC as error in Cents:

n-EDOs:

665 * ln( 3^12 / 2^19) / ln(2) = ~ 13.0007558... degree(665)
===> ~ 13.0007558... / 13 = ~ 1.00005814....
~0.1 Cents sharp, almost inperceptible.

6 079 * ln(3^12 / 2^19) / ln(2) = ~ 118.844503....deg(6079)
====> ~ 119 / 118.844503... = 1.00130841...
~2.3 Cents flat, a little more than a schisma.

6 691 * ln(3^12 / 2^19) / ln(2) = ~ 130.809108...deg(6691)
====> 130.809108... / 130 = 1.00622391...
~10.7 Cents sharp, almost 1/2 comma off, hence inacceptable.

8 539 * ln((3^12) / (2^19))) / ln(2) = ~ 166.937524...deg(8539)
====> 167 / 166.9375 = 1.00037439...
~0.65 Cents flat, but more than 6 times worser as "665"
that means also still problematic too.

11 664 * ln((3^12) / (2^19))) / ln(2) = ~ 228.031301...deg(11664)
=====> 228.031301 / 228 = ~ 1.00013729...
~0.23 Cents, why all that efford for yielding twice worser than "665"?

/tuning-math/message/16529

190 537 * ln( 3^12 / 2^19) / ln(2) = ~ 3 724.999 998 76...deg
======> 3725 / 3 724.999 998 7... = ~ 1.000 000 003 ...
less than ~ 6 * 10^-7 Cents deviation barely.

Just compare your own preferred n-EDO to that one, then decide!

A.S.

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...>
wrote:

> The 3-limit quality of an n-EDO can be determinated,
> simply by calculating the defect of the PC as error in Cents:

These aren't errors in cents at all; they are
errors in terms of what I used to call relative
cents until Paul objected. If you take an error
e in cents, then n*e/12 is a percentage error,
which we've been using a lot lately as Dave and
George seem to like it.

> n-EDOs:
>
> 665 * ln( 3^12 / 2^19) / ln(2) = ~ 13.0007558... degree(665)
> ===> ~ 13.0007558... / 13 = ~ 1.00005814....
> ~0.1 Cents sharp, almost inperceptible.

It's actually -.001363768 cents, which is completely
inperceptible. The percent error is -.0755754826328 %,
which is, of course, very small. But why do we care
in particular about the errors in the Pythagorean
comma when evaluating a notational edo? If we divide
it by 12 yet again, we get the percentage error in 3,
which seems a more reasonable thing to look at.

In fact, the eror
>
> 6 079 * ln(3^12 / 2^19) / ln(2) = ~ 118.844503....deg(6079)
> ====> ~ 119 / 118.844503... = 1.00130841...
> ~2.3 Cents flat, a little more than a schisma.

Um, no. The percent error of 3 is 1.3 %, which isn't too
bad.

>
> 6 691 * ln(3^12 / 2^19) / ln(2) = ~ 130.809108...deg(6691)
> ====> 130.809108... / 130 = 1.00622391...
> ~10.7 Cents sharp, almost 1/2 comma off, hence inacceptable.

Actually, 1.6 %, which might be acceptable to a lot
of people, though maybe not Dave and George.

>
> 8 539 * ln((3^12) / (2^19))) / ln(2) = ~ 166.937524...deg(8539)
> ====> 167 / 166.9375 = 1.00037439...
> ~0.65 Cents flat, but more than 6 times worser as "665"
> that means also still problematic too.

1/2 of 1 %, which is not bad at all; I'd not call it
problematic.

>
> 11 664 * ln((3^12) / (2^19))) / ln(2) = ~ 228.031301...deg(11664)
> =====> 228.031301 / 228 = ~ 1.00013729...
> ~0.23 Cents, why all that efford for yielding twice worser
than "665"?

Because there's more to life than the 3-limit, that's
why. Let's check some percentage errors:

For 3/2, 665 gives a 0.006 % error, and 11664 gives a
0.26 % error, so 665 is way better.

For 5/4 and 6/5, 665 gives an 8% error. 11664 gives
a 3 % error.

For 7/4 and 7/6, 665 gives an 11 % error and
11664 gives 1.2 % and 1.5 %.

For 7/5, 665 gives a 19 % error, and 11664 gives
1.85 %. Oops.

Furthermore: 665 is consistent only up to the 9-limit,
whereas 11664 is consistent to the 27 limit.

> 190 537 * ln( 3^12 / 2^19) / ln(2) = ~ 3 724.999 998 76...deg
> ======> 3725 / 3 724.999 998 7... = ~ 1.000 000 003 ...
> less than ~ 6 * 10^-7 Cents deviation barely.
>
> Just compare your own preferred n-EDO to that one, then decide!

It has a 21 % error in the 5-limit, a 23 % error in
the 9-limit, a 49 % error in the 11-limit, and past
that isn't consistent. I conclude that it's useless
for the purposes of most people looking at these things.

It is, however, a denominator for a convergent to
log2(3). Those things are trivial to find.

🔗monz <monz@tonalsoft.com>

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

> The fact that one degree of 12-edo falls almost
> midway between 711 and 712 tinas also doesn't bother
> me much, because i've found that the error of the
> 12-edo 5th (4981 tinas) is only - 8 & 1/3 % of a
> tina, which means that you'd have to go to 6 5ths
> to reach an error of half a tina. That's accurate
> enough for me.
>
> There are some interesting discrepancies among the
> meantones. The 1/4-comma meantone 5th maps to 4957
> tinas, while that of 31-edo maps to 4958 tinas, and
> the 1/6-comma meantone 5th maps to 4969 while that
> of 55-edo maps to 4968.

I've added to my tina webpage

http://tonalsoft.com/enc/t/tina.aspx

a table of the tina values for all of the commonly used
intervals, in all of the standard keys, in some of the
most important EDO meantones. The intervals are listed
as a chain-of-5ths, in decreasing generator order, with
the tonic of each key as the zeroth generator. 53-edo
is also shown for comparison, as a representation of
pythagorean tuning.

The percentage errors for 12-, 55-, and 31-edo are quite
low, those for 43-, 50-, and 19-edo not as good, and the
error for 53-edo almost as bad as it can get, at 49%
(i.e., the 5th of 53-edo is almost midway between two
tina values, at ~4994.5094) -- the values shown in the
53-edo column are actually quite accurate for real
pythagorean JI tuning.

You can easily see the divergence for 12-edo, where
the diminished-2nd (i.e., C:Dbb for example) maps to
one degree higher than zero, and the augmented-7th
(i.e., C:B# for example), which represents the
pythagorean-comma, maps to one degree lower than zero.

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

🔗Jon Szanto <jszanto@cox.net>

🔗Gene Ward Smith <genewardsmith@sbcglobal.net>

🔗monz <monz@tonalsoft.com>

Hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Jon Szanto" <jszanto@> wrote:
> >
> > --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
> > > I've added to my tina webpage
> >
> > Oh, for crap's sake, be useful and go change a diaper.
> > No one needs tinas right now...
>
> Why not? Good name for a girl baby. Can you imagine
> Eightthousandfivehundredandthirtynine Monzo?

Ha, good one! :-)

Actually, it's really ironic that you say that, because
i have an Aunt Tina who is almost 90 years old and she
never married or had any children, and the last few of
my generation who were born were all boys (including me),
so she never got to have a baby named after her.

Mama and i had already had a long debate about our
baby's name and had already picked Lelani by the time
my mother told me about this. Then i stayed up all night
Friday night making my webpage about 8539-edo, calling
the unit "hepticent", and just before going to bed
early Saturday morning i read Dave Keenan's message
here and decided i liked George and Dave's name "tina"
for it a lot better.

So when Lelani was born less than a day later, it was
already ironic that i had applied my Aunt's name to
something else that was my "baby" (the idea to use
8539-edo as a measurement unit).

Anyway guys, i'm only here typing on the tuning list
while i'm home alone and Mama and Lelani are still
resting in the hospital. Take advantage of the fact
that i have this time to be here now ... pretty soon,
when they're home, i *will* be too busy changing
diapers and stuff like that. ;-P

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

🔗Andreas Sparschuh <a_sparschuh@yahoo.com>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> These aren't errors in cents at all; they are
> errors in terms of what I used to call relative
> cents until Paul objected.
agreed, hence that kind of erroes should be called more apt
as "meta"-Cents in a metaphorically sense.

> If you take an error
> e in cents, then n*e/12 is a percentage error,
> which we've been using a lot lately as Dave and
> George seem to like it.
also right, in that case it makes also more sense to
express that deviations also in percent units too.

> But why do we care
> in particular about the errors in the Pythagorean
> comma when evaluating a notational edo? If we divide
> it by 12 yet again, we get the percentage error in 3,
> which seems a more reasonable thing to look at.

also an truism, refer it back to the fundamental root
of 3-imit: the plain 5th.
The resulting PC: 3^12/2^19 is dervied from a dozen 5ths.

>
> > 190 537 * ln( 3^12 / 2^19) / ln(2) = ~ 3 724.999 998 76...deg
> > ======> 3725 / 3 724.999 998 7... = ~ 1.000 000 003 ...
>
> It has a 21 % error in the 5-limit, a 23 % error in
> the 9-limit, a 49 % error in the 11-limit, and past
> that isn't consistent. I conclude that it's useless
> for the purposes of most people looking at these things.
hence appearenty irrelevant for the aims that group here.
May be not comletely futil:
Rather something for physicists that do want to convert:
http://en.wikipedia.org/wiki/Stoney_units#Stoney_units
into
http://en.wikipedia.org/wiki/Planck_units
by the "3-limit?":
http://en.wikipedia.org/wiki/Fine-structure_constant
rational or logarithmic(base 2) approximations:
alpha = ~ (12*40/41)^-2 = ~ ((2^(3*3*757/190537))/12)^2
>
> It is, however, a denominator for a convergent to
> log2(3). Those things are trivial to find.

Alike if one knews already the:
http://en.wikipedia.org/wiki/Bohr_model
then especially the
http://en.wikipedia.org/wiki/Balmer_series
spectra turns out to be an trivial sub-case.
>
A.S.