Yahoo Tuning Groups Ultimate Backup makemicromusic 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

🔗Rozencrantz the Sane <rozencrantz@...>

Wow. I have no idea what that means. I mean, I know all of the words,
but what is the rationale behind all this? I've used 1200 and even 300
as perfectly adequate integer units. Is the 31st harmonic even audible
as a source of consonance? Is one step of, say, 8538 audible as a unit
of dissonance?

On 4/28/07, 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.
>
>
> 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).

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

--Tristan
http://dolor-sit-amet.deviantart.com

🔗Jon Szanto <jszanto@...>

🔗Dave Keenan <d.keenan@...>

🔗Kraig Grady <kraiggrady@...>

🔗monz <monz@...>

Hi Tristan,

--- In MakeMicroMusic@yahoogroups.com, "Rozencrantz the Sane"
<rozencrantz@...> wrote:
>
> Wow. I have no idea what that means. I mean, I know
> all of the words, but what is the rationale behind all
> this? I've used 1200 and even 300 as perfectly adequate
> integer units. Is the 31st harmonic even audible
> as a source of consonance? Is one step of, say,
> 8538 audible as a unit of dissonance?
>
> On 4/28/07, 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.
> >
> >
> > I've made webpage for it in the Encyclopedia:
> >
> > http://tonalsoft.com/enc/t/tina.aspx
> >
> > <snip>

I re-quoted what i wrote because it lays out the rationale:
i've been wanting a unit of interval measure which has low
error for a wide variety of intervals, mainly so that i
can just use the integer value without having to bother
with decimal places and the accumulation of error from
rounding. 8539-edo accomplishes this remarkably well,
especially for JI up to 31-limit.

The next prime-factor, 37, doesn't fare so well in this
division, which is too bad, because Ezra Sims uses that
as part of the harmonic basis of his music and i'd like
to be able to quantify what he does by using tinas, but
they won't work. But Ben Johnston has composed in JI
up to the 31-limit, and so his work can be analyzed well
using tinas.

I can't make any claims as to the audibility of one
degree of 8539-edo ... in fact, most of the time
311-edo would probably work just fine as an integer
unit of measurement, and even one degree of that is
probably pretty hard to recognize audibly.

But the point is that using 8539-edo gives almost the
same level of resolution as cents with one decimal place
(i.e., the resolution of 12000-edo), but the level of
error is *much* lower up to the 31-limit, and it's
still one less digit to remember ... and 8539 is also
good for prime-factor 41. It's just too bad that
its approximation of 37 isn't so good.

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

🔗Rozencrantz the Sane <rozencrantz@...>

I guess I'm having a hard time seeing how errors that small are
relevant, or what was wrong with the larger units I've seen used. I
understand the desire to eliminate rounding-errors, but for that it
seems like it would be easier to use prime notation. Especially since
that seems closer to how the numbers actually behave. You don't write
a step of 153 Tinas because that's a nicer melodic interval than 154,
you write a step of [-4, 4, -1> because you're moving from F 5/4 above
D to F 16/9 above G. (If my math is wrong, forgive me, I don't have a
lattice in front of me.)

On 4/28/07, monz <monz@...> wrote:
> Hi Tristan,
>
> I re-quoted what i wrote because it lays out the rationale:
> i've been wanting a unit of interval measure which has low
> error for a wide variety of intervals, mainly so that i
> can just use the integer value without having to bother
> with decimal places and the accumulation of error from
> rounding. 8539-edo accomplishes this remarkably well,
> especially for JI up to 31-limit.
>
> The next prime-factor, 37, doesn't fare so well in this
> division, which is too bad, because Ezra Sims uses that
> as part of the harmonic basis of his music and i'd like
> to be able to quantify what he does by using tinas, but
> they won't work. But Ben Johnston has composed in JI
> up to the 31-limit, and so his work can be analyzed well
> using tinas.
>
> I can't make any claims as to the audibility of one
> degree of 8539-edo ... in fact, most of the time
> 311-edo would probably work just fine as an integer
> unit of measurement, and even one degree of that is
> probably pretty hard to recognize audibly.
>
> But the point is that using 8539-edo gives almost the
> same level of resolution as cents with one decimal place
> (i.e., the resolution of 12000-edo), but the level of
> error is *much* lower up to the 31-limit, and it's
> still one less digit to remember ... and 8539 is also
> good for prime-factor 41. It's just too bad that
> its approximation of 37 isn't so good.
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software

--Tristan
http://dolor-sit-amet.deviantart.com

🔗Igliashon Jones <igliashon@...>

🔗Gene Ward Smith <genewardsmith@...>

🔗Kraig Grady <kraiggrady@...>

🔗monz <monz@...>

Hi Tristan,

--- In MakeMicroMusic@yahoogroups.com, "Rozencrantz the Sane"
<rozencrantz@...> wrote:
>
> I guess I'm having a hard time seeing how errors that
> small are relevant, or what was wrong with the larger
> units I've seen used. I understand the desire to
> eliminate rounding-errors, but for that it seems like
> it would be easier to use prime notation. Especially
> since that seems closer to how the numbers actually
> behave. You don't write a step of 153 Tinas because
> that's a nicer melodic interval than 154, you write
> a step of [-4, 4, -1> because you're moving from F 5/4
> above D to F 16/9 above G. (If my math is wrong, forgive
> me, I don't have a lattice in front of me.)

Again, the point is not to think of 8539-edo as a tuning,
but rather simply to make use of it as an interval
measurement unit because you don't need decimal places.

There's nothing wrong with using cents, savarts, or
any other unit as long as you either:

a) use only the integer values and don't worry about the
error amounts or the rounding errors which accumulate; or

b) use the decimal places and put up with the hassle
of having to use them, and having to decide to how many
places of accuracy you want to perform your calculations.

8539-edo does away with both problems because:

a) the error is so small that there essentially won't be
any if you use only the integer values; and

b) you always have the same 4-digit level of resolution,
without needing a decimal point.

Yes, it's true that if you're working in JI, the only
absolutely accurate notations are either ratios, or
the prime-factor-and-exponent notation. The latter is
simpler than the former and also easier to calculate,
which are the reasons why i proposed it in the first place.
But the drawback is that it still doesn't give a logarithmic
picture of pitch, which 8539-edo does, and that's one
important aspect of how we hear pitch. The harmonic lattice
is the other, and so that's why Tonescape uses both together.

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

🔗monz <monz@...>

Hi Iglashion,

--- In MakeMicroMusic@yahoogroups.com, "Igliashon Jones"
<igliashon@...> wrote:
>
> --- In MakeMicroMusic@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.
>
> Monz, I'm sure you're very excited about this discovery,
> but isn't this more appropriate for the tuning list, or
> tuning-math even? I thought the whole point of MMM was
> to avoid the theoretical discussions and just focus on
> *making microtonal music*. Not that your discovery is
> not important, but given that there exist forums of
> which you are also a member, and in which discussions
> of this sort are much more appropriate, I (presumably
> among others) would appreciate if this discussion were
> moved elsewhere.

Actually, the discussion of small units of interval
measurement, out of which the discussion of 8539-edo grew,
is on tuning-math.

I only posted it here precisely because i believe that
this is a very useful *practical* discovery. The act of
creating microtonal music frequently involves dealing
with cents values or some other logarithmic interval
measurement, and my position is that tinas are far better
than cents.

I won't be talking about it much more here ... but i
*will* be adding tina values alongside all the cents
values on my webpages, over time.

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