Re: searching for some small interval measurement units

The message quoted below comes from the tuning list:
/tuning/topicId_70649.html#71261

Since George wrote that response to Gene's suggestions I have
suggested to George that in extreme sagittal we might in fact have a
use for a unit smaller than the mina (approx 2460-EDO). Call it the
"tina". Incidentally these are pronounced "meena" and "teena".

It's not that we would attempt to notate every tina, but they could be
useful in setting the boundaries between the "capture zones" of the
various symbols that we do have.

It would be useful if the tina was somewhere between about half to one
tenth of a mina. But please note that the mina used in sagittal design
work is exactly 1/233 of an apotome [-11 7> rather than 1/2460 of an
octave and we would likewise need the tina to be an integral division
of the apotome, not the octave.

Any suggestions?

-- Dave Keenan

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
>
> > In evaluating all of those divisions that have been popping up, I
> > agree that the lowest primes should have very low degree-%
> > deviations. For 3, this should be significantly lower than 5%,
since
> > at that point a chain of 10 fifths (i.e., 3^10) results in a 50%
> > deviation, which is midway between degrees. Likewise, it's
highly
> > desirable that 5 and 7 each be below 10%, since it's not unusual
for
> > those primes to be taken to the 4th or 5th power in notational
commas.
> >
> > You'll see that 2460 and 11664 fare quite well under these
> > circumstances. (I also see that 324296 is exceptional.)
>
> 324296 would be just about perfect if only it was
> divisible by 12. A 12-divisible system which also
> does well on 3 and 5 is 31920--have you given any
> thought to the potential uses of that?

No, because:

1) It isn't good enough for 7, failing at 7^3 and 5*7^2;
2) 27-limit consistency is adequate for most purposes;
3) 11664 has 0.10-cent resolution, which is around the limit of
audibity, accurate enough, I would think, for even the most demanding
JI fanatic.

Would we really need more precision than that for an alternative MIDI
standard?

--George

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:
>... unit smaller than the mina (approx 2460-EDO). Call it the
> "tina". Incidentally these are pronounced "meena" and "teena".

> It would be useful if the tina was somewhere between about half to one
> tenth of a mina. But please note that the mina used in sagittal design
> work is exactly 1/233 of an apotome [-11 7> rather than 1/2460 of an
> octave and we would likewise need the tina to be an integral division
> of the apotome, not the octave.
>
> Any suggestions?
>
>
The 3-limit intervals alike the
apotome 3^7/2^11 = 2187/2048 are
usually approximated by the:

http://www.research.att.com/~njas/sequences/A060528
http://www.research.att.com/~njas/sequences/A005664
" Denominators of convergents to log_2 3."

Lectures and discussions:
1. in german:
http://delphi.zsg-rottenburg.de/muslekt3.html
http://groups.google.de/group/de.sci.mathematik/browse_thread/thread/c8396c0fc076056c/f14514b715caec87%23f14514b715caec87
http://groups.google.de/group/de.sci.mathematik/browse_thread/thread/c718426b3e4b30f6/d98482d149f630c0%23d98482d149f630c0
http://groups.google.de/group/de.sci.mathematik/browse_thread/thread/4c47329adaa1e3af/61995a6522fa7640%2361995a6522fa7640

2. in french/spanish:
http://www3.udg.edu/csocial/Premis_recerca/premis2006/treballs%20premiats/88_Les%20escales%20musicals.pdf
-
that also appear in 3x+1 cycles, the Collatz sequences:
http://www.ericr.nl/wondrous/cycles.html
http://mathforum.org/kb/message.jspa?messageID=3396858&tstart=0
and Waring's problem:
http://groups.google.de/group/sci.math/browse_thread/thread/66a86644532e81d6/cc31b3ad77a60a42?lnk=st&q=12+53+306+665+15601+31867+111202&rnum=4&hl=de#cc31b3ad77a60a42

Here comes Collatz's original instruction as poem:

"Geht es nicht durch zwei
erhöhe nach Mal-Drei.
Andernfalls bei glatter Zahl
fahr halbierend Du zu Tal."

attempting to translate that intranslatable spoonerism:

"If you can't divde by two
do raise triplewise.
Otherwise when odd number
halve it for behalve asunder."

Is there in that group any native english-speaker that has a better
apt translation in order to explain Collatz's algorithm?

http://mathforum.org/kb/plaintext.jspa?messageID=5316965

For proper reading of the original-spacings in that ASCII
decoded letter email:
<click on: "Show Meassage Option", then "Use Fixed Width Font">

> Nr
> 1 1.33333333333333333333333333 0.66666666666666666666666667
> 2^2 2^1
> --- ---
> 3^1 3^1
>
that are musically the intervals 5th: 3/2 and the 4th 4/3

> 2 1.18518518518518518518518519 0.88888888888888888888888889
> 2^5 2^3
> --- ---
> 3^3 3^2
next are the pythagorean minor 3rd: 32/27 and the major tone: 9/8

>
> 3 1.05349794238683127572016461 0.93644261545496113397347965
> 2^8 2^11
> --- ----
> 3^5 3^7
then the limma: 256/243 and apotome 2187/2048

>
> 4 1.03931824834385566014811364 0.98654036854514423990621725
> 2^27 2^19
> ---- ----
> 3^17 3^12
Al-Farabi's subsemitone 2^27/3^17 and the Pyth.-Comma 531441/528244
http://en.wikipedia.org/wiki/Al-Farabi

>
> 5 1.01152885180860850383729052 0.99791404625731122600598255
> 2^65 2^84
> ---- ----
> 3^41 3^53
the minor-PC 2^65/3^41 and Jing-Fang's or Mercator's Comma
http://en.wikipedia.org/wiki/Jing_Fang
http://en.wikipedia.org/wiki/53_equal_temperament#History

>
> 6 1.00102276179641176722083140 0.99893467461992588900707938
> 2^485 2^569
> ----- -----
> 3^306 3^359
Iassac Newton's hypothetical 612=2*306 ET and Chien-lo-Chics's comma
Lit. entry: Encyclopedia old MGG(1) Supplement Vol.

>
> 7 1.00097906399186788426531760 0.99995634684223816858146752
> 2^1539 2^1054
> ------ ------
> 3^971 3^665

971: ???-still nonamed???-
(1 200 * ln(1.0009790639...)) / ln(2) = ~1.694...Cents
with
1023/1022 < 2^1539/3^871 < 1022/1021 less than 0.1%

665:
(1 200 * ln(1 / 0.99995634...)) / ln(2) = ~0.07557...Cents
or ~(1/13.23...)Cents
with
1 / 0.999956346... = 1.00004366... ~4.366...ppm
http://en.wikipedia.org/wiki/Parts-per_notation

That's the first 3-limt approx. error in that series below the
http://en.wikipedia.org/wiki/Just_noticeable_difference
for small intervals or pitch-differences relative in frequencies:

22908/22907 < 3^665/2^1054 < 22909/22908

Who can discriminate that?
A
http://en.wikipedia.org/wiki/Microbat
has a
"range in frequency from 14,000 to over 100,000 hertz, well beyond the
range of the human ear (typical human hearing range is considered to
be from 20Hz to 20000 Hz)"
but no human can distinguish the incerment from 23kHz to 23001Hz?

But back to history:
http://www.xs4all.nl/~huygensf/doc/measures.html
Î" "Delfi unit: 1/665 part of an octave
Used in Byzantine music theory? Approximately 1/12 part of the
syntonic comma and 1/13 part of the Pythagorean comma. "
Here some links about that in modern greek:
http://www.google.de/search?hl=de&q=delfi-unit+665&btnG=Suche&meta=
There it is regulary labeled as:
uppercase Î",
http://en.wikipedia.org/wiki/Delta_(letter)

That ratio
3^665/2^1054 was also independently found again
back in the 19.th century by:
http://de.wikipedia.org/wiki/Moritz_Wilhelm_Drobisch
Some call it modern:
http://www.google.de/search?hl=de&q=satanic-comma+665&btnG=Suche&meta=
http://www.google.de/search?hl=de&q=satanic-comma&btnG=Suche&meta=
It deserves i.m.o. a more friendly name: simple "δ"-unit

>
> 8 1.00001819475389302953363369 0.99997454080187273598498660
> 2^24727 2^25781
> ------- -------
> 3^15601 3^16266

Remark: 16266 = 2711 * 3 * 2
Hence, i do suggest that both 15601-ET or 16266-ET as
apt candidates for the desired new 'alternatve' MIDI standard
in that chain of historically 3-limit refinements.

3^15601:
(1200 * ln(1.000018194753...)) / ln(2) = ~0.031499...Cents
or ~1/23 Cents
One step in 15601 ET amouts:
1200C / 15 601 = ~0.0769...C = (1/13.0008333...)C
because 12*13=156.

3^16266:
(1200 * ln(1 / 0.99997454...)) / ln(2) = ~0.044076...Cents

16266 ET
in δ-units of 2^(1/16266)cents
1 200C / 16 266 = (1/13.555)C exact
or
1 Cent = 13.555 δ (lowercase δ)
http://en.wikipedia.org/wiki/Delta_(letter)

One Delfi unit Î" := 2^(1/665) are
16 266 / 665 = ~ 24.46..δ in
compare:
http://arbuz.uz/t_octava.html
in russian:
"Периоды более высоких порядков раскладываются на нижележащие как на
составляющие модули‚ связующим звеном выступает цикл 665:
16266 = 665 х 24 + 359 â"€ 53‚..."

"δ"-units steps accordingly are defined as:
δ := 2^(1/16266) = 1/13.555 Cents

hence that approximate
Drobisch's Comma 3^665/2^1054 in ~ one single δ step
Chien lo Chic's Comma: 3^359/2^569 in ~25δ
Newton's Commma (from 612ET) 2^485/3^306 in ~24δ
Jing-Fang/Mercator = Cien * Newton = 3^53/2^84 in 49δ := 25δ+24δ
The 2^65/3^41 minor pyth. comma: 267δ
The 3^12/2^19 major PC: 318δ with 318 = 11*7*2^2
as alternative to TUs:
http://www.xs4all.nl/~huygensf/doc/measures.html

"Temperament Unit: 1/720 part of a Pythagorean comma

This measure was developed by organ builder John Brombaugh to describe
very small intervals as integer values. In this measure, the syntonic
comma is almost exactly 660 Temperament Units and the schisma 60.
Because 720 is divisible by all numbers from 2 to 6 and more, most
temperaments can be described by only integer values. In a
well-temperament, -720 TU must be distributed over the cycle of
fifths. One Grad is 60 TU. There are 36828.6282 Temperament Units in
an octave."
hence 77δ ~= 180 TUs

The schisma 32805/32768 = 5*3^8/2^15 has 26δ

>
> 9 1.00001092971425174755742972 0.99999273509254192127917314
> 2^75235 2^50508
> ------- -------
> 3^47468 3^31867
>
Hmm, here i see no need for human beeings,
hence no need to baptize 3-limit intervals
smaller than δ.

Conclusion:
The "δ"-unit, defined as δ := 2^(1/16226) ist just fine
enough in order to distinguish Drobisch-ian Comma steps:

24δ ~ Newton's Commma (from 612ET) 2^485/3^306 in ~24δ
24.5δ ~ Î" Delfi unit 2^(1/665) part of an octave (out od use)
25δ Chien lo Chic's Comma: 3^359/2^569
26δ schisma 32805/32768 = 5*3^8/2^15 has 26δ
27δ ~ 2Cents 16266/600 = 27.11

have a lot of fun with δ

A.S.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:
>
> ... But please note that the mina used in sagittal design
> work is exactly 1/233 of an apotome [-11 7> rather than 1/2460 of an
> octave and we would likewise need the tina to be an integral division
> of the apotome, not the octave.
>
> Any suggestions?

Since the 3 error in 2460-ET is extremely low (less than 0.78% of
1deg2460, or 0.00378 cents narrow), I'd say that the two are virtually
identical.

For 11664-ET (for which a single degree corresponds to 1/1105 apotome)
it's even lower: less than 0.261% of 1deg11664, or 0.00027 cents narrow.

--George

--- In tuning-math@yahoogroups.com, "Andreas Sparschuh"
<a_sparschuh@...> wrote:
> Remark: 16266 = 2711 * 3 * 2
> Hence, i do suggest that both 15601-ET or 16266-ET as
> apt candidates for the desired new 'alternatve' MIDI standard
> in that chain of historically 3-limit refinements.

Hi Andreas,

Your delta-prime which is one degree of 16266-EDO or 1541-EDA
(apotome) is certainly in the right size range and is brilliant for
approximating 3-limit ratios, but we need to consistently approximate
ratios between primes up to at least 23 (31 would be nice) and higher
powers of the lower primes up to something like
3^12
5^7
7^5
11^2.

--- In tuning-math@yahoogroups.com, "Andreas Sparschuh"
<a_sparschuh@...> up and done did write:

> Here comes Collatz's original instruction as poem:
>
> "Geht es nicht durch zwei
> erhöhe nach Mal-Drei.
> Andernfalls bei glatter Zahl
> fahr halbierend Du zu Tal."
>
> attempting to translate that intranslatable spoonerism:
>
> "If you can't divde by two
> do raise triplewise.
> Otherwise when odd number
> halve it for behalve asunder."
>
> Is there in that group any native english-speaker that has a better
> apt translation in order to explain Collatz's algorithm?

--- In tuning-math@yahoogroups.com, "Andreas Sparschuh"
<a_sparschuh@...> up and done did write:

>
> "Geht es nicht durch zwei
> erhöhe nach Mal-Drei.
> Andernfalls bei glatter Zahl
> fahr halbierend Du zu Tal."
>
> attempting to translate that intranslatable spoonerism:
>
> "If you can't divde by two
> do raise triplewise.
> Otherwise when odd number
> halve it for behalve asunder."
>
> Is there in that group any native english-speaker that has a better
> apt translation in order to explain Collatz's algorithm?
>

When division by two cannot be
multiply the thing, by three.
When even numbers are in play
halve it down all the way.

is how I'd translate it, but shouldn't it be something like

Geht es nicht durch zwei
erhöhe nach Mal-Drei,
und leg noch eins dabei.
Andernfalls bei glatter Zahl
fahr halbierend Du zu Tal.

to be the Collatz algorithm, correct me if I'm wrong.

-Cameron Bobro

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
>
> For 11664-ET (for which a single degree corresponds to 1/1105 apotome)
> it's even lower: less than 0.261% of 1deg11664, or 0.00027 cents
> narrow.
>
Dear George,

http://mathworld.wolfram.com/PythagoreanTriple.html
"The first few primes of the form 4x+1 are 5, 13, 17, 29, 37, 41, 53,
61, 73, 89, 97, 101, 109, 113, 137, ... (Sloane's A002144), so the
smallest side lengths which are the hypotenuses of 1, 2, 4, 8, 16, ...
primitive right triangles are 5, 65, 1105, 32045, 1185665, 48612265,
... (Sloane's A006278)."

http://www.research.att.com/~njas/sequences/A002144
1105 = 17 * 13 *5
http://www.research.att.com/~njas/sequences/A006278
hence
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Pythag/pythag.html
delivers:

Triples with hypotenuse=1105:
1: 663,884,1105 =221x[3,4,5] P=2652 A=293046 r=221 m=. n=.
2: 425,1020,1105 =85x[5,12,13] P=2550 A=216750 r=170 m=. n=.
3: 169,1092,1105 =13x[13,84,85] P=2366 A=92274 r=78 m=. n=.
4: 105,1100,1105 =5x[21,220,221] P=2310 A=57750 r=50 m=. n=.
5: 47,1104,1105 primitive P=2256 A=25944 r=23 m=24 n=23
6: 700,855,1105 =5x[171,140,221] P=2660 A=299250 r=225 m=. n=.
7: 744,817,1105 primitive P=2666 A=303924 r=228 m=31 n=12
8: 520,975,1105 =65x[15,8,17] P=2600 A=253500 r=195 m=. n=.
9: 561,952,1105 =17x[33,56,65] P=2618 A=267036 r=204 m=. n=.
10: 272,1071,1105 =17x[63,16,65] P=2448 A=145656 r=119 m=. n=.
11: 576,943,1105 primitive P=2624 A=271584 r=207 m=32 n=9
12: 264,1073,1105 primitive P=2442 A=141636 r=116 m=33 n=4
13: 468,1001,1105 =13x[77,36,85] P=2574 A=234234 r=182 m=. n=.

my 13 preferred subdivisions of the 1105 deg apotome 2187/2048
in 11664 EDO.

A. S.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:
>
> > Remark: 16266 = 2711 * 3 * 2
>
> Your delta-prime which is one degree of 16266-EDO or 1541-EDA
> (apotome) is certainly in the right size range and is brilliant for
> approximating 3-limit ratios, but we need to consistently approximate
> ratios between primes up to at least 23 (31 would be nice) and higher
> powers of the lower primes up to something like

Pschoacoustic experiments yield the result that
our brains represent neurologically
5-limit internally as correlate of 2^15/3^8 = 32768/6561 = ~4.99436...
7-limit respectively in 3-limit approx: 2^25/3^14 = ~7.015398...
11-limit accordingly in 3-limit approx: 3^23/2^33 = ~10.9597...
13-lim...
...&ct also for other higer primes in the overtone series too.

> 3^12:
3^13/2^19 = 2^(318.00044.../16266) = ~2^(318/16266) the PC

> 5^7 = 78125
about an ("kleisma=5^6/3^5/2^6")*5*243*64
http://en.wikipedia.org/wiki/53_equal_temperament
"and 15625/15552, known as the kleisma."

Internally 7 schismata off, neurophysiologically representation:
2^15/3^8)^7 = ~77510.275...

> 7^5 = 16807
respctively (2^27/3^14)^5 = ~16992.67016...

> 11^2 = 121
(3^23/2^33)^2 = ~120.1152...
>
because human brains turend out to be electro-physiologically
3-limit restricted by crucial experiments,
as far is i do already know from preliminary research at:
http://www.klinikum.uni-heidelberg.de/Musikalische-Verarbeitung-und-der-auditorische-Kortex.5503.0.html

At the moment i try to find out where exactly and how inbetween
http://en.wikipedia.org/wiki/Basilar_membrane
and
http://en.wikipedia.org/wiki/Heschl%27s_gyrus
above schismatic transitions from 5, 7, 11, ...& higer limits
back to simple 3-limit reduction exactly happens.
That's an experimental challenge!

A.S.

--- In tuning-math@yahoogroups.com, "Cameron Bobro" <misterbobro@...>
wrote:
> > Collatz's original instruction as poem:
> >
> > "Geht es nicht durch zwei
> > erhöhe nach Mal-Drei.
> > Andernfalls bei glatter Zahl
> > fahr halbierend Du zu Tal."
> >
>
> When division by two cannot be
> multiply the thing, by three.
> When even numbers are in play
> halve it down all the way.

much better than my one - congratulation!

>
> is how I'd translate it, but shouldn't it be something like
>
> Geht es nicht durch zwei
> erhöhe nach Mal-Drei,
> und leg noch eins dabei.
> Andernfalls bei glatter Zahl
> fahr halbierend Du zu Tal.
>
> to be the Collatz algorithm, correct me if I'm wrong.
that's even more precise, how to procede in:
http://en.wikipedia.org/wiki/Collatz_conjecture
http://mathworld.wolfram.com/CollatzProblem.html
http://www.google.de/search?hl=de&q=collatz+sequence&btnG=Google-Suche&meta=

application in

http://www.strukturbildung.de/Andreas.Sparschuh/

>Pschoacoustic experiments yield the result that
>our brains represent neurologically
>5-limit internally as correlate of 2^15/3^8 = 32768/6561 = ~4.99436...
>7-limit respectively in 3-limit approx: 2^25/3^14 = ~7.015398...
>11-limit accordingly in 3-limit approx: 3^23/2^33 = ~10.9597...
>13-lim...
>...&ct also for other higer primes in the overtone series too.

Hi Andreas,

Do you have a reference for this? Preferrably in English?

-Carl

Hi Dave (and George),

Actually, using the criteria of an accurate small division
of the apotome, Gene's suggestion of 31920-edo does quite
well: 31920 divides the apotome into 3024 logarithmically
equal parts with very little error. This division is
looking quite good to me as a unit of measurement.

Hmm ... this also points out one shortcoming of the 311-edo
that i've been so amazed by lately: the biggest shortcoming
of 311 is its approximation of prime-factor 3. So the
apotome (3^7) is represented by 30 degrees of 311 (i.e.,
divided into 30 logarithmically equal parts), whereas
the actual size of the apotome is almost exactly halfway
between 29 and 30 degrees of 311-edo.

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

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:

> The message quoted below comes from the tuning list:
> /tuning/topicId_70649.html#71261
>
> Since George wrote that response to Gene's suggestions
> I have suggested to George that in extreme sagittal we might
> in fact have a use for a unit smaller than the mina (approx
> 2460-EDO). Call it the "tina". Incidentally these are
> pronounced "meena" and "teena".
>
> It's not that we would attempt to notate every tina, but
> they could be useful in setting the boundaries between the
> "capture zones" of the various symbols that we do have.
>
> It would be useful if the tina was somewhere between about
> half to one tenth of a mina. But please note that the mina
> used in sagittal design work is exactly 1/233 of an apotome
> [-11 7> rather than 1/2460 of an octave and we would likewise
> need the tina to be an integral division of the apotome,
> not the octave.
>
> Any suggestions?
>
> -- Dave Keenan
>
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@>
> wrote:
> >
> > 324296 would be just about perfect if only it was
> > divisible by 12. A 12-divisible system which also
> > does well on 3 and 5 is 31920--have you given any
> > thought to the potential uses of that?
>
> No, because:
>
> 1) It isn't good enough for 7, failing at 7^3 and 5*7^2;
> 2) 27-limit consistency is adequate for most purposes;
> 3) 11664 has 0.10-cent resolution, which is around the limit
> of audibity, accurate enough, I would think, for even the
> most demanding JI fanatic.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:

> It would be useful if the tina was somewhere between about half to one
> tenth of a mina. But please note that the mina used in sagittal design
> work is exactly 1/233 of an apotome [-11 7> rather than 1/2460 of an
> octave and we would likewise need the tina to be an integral division
> of the apotome, not the octave.

Why do you prefer to divide an exact apotome?

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

> But back to history:
> http://www.xs4all.nl/~huygensf/doc/measures.html
> Î" "Delfi unit: 1/665 part of an octave
> Used in Byzantine music theory?

Was it or wasn't it used in Byzantine theory? Where,
if anywhere, are citations?

> Remark: 16266 = 2711 * 3 * 2
> Hence, i do suggest that both 15601-ET or 16266-ET as
> apt candidates for the desired new 'alternatve' MIDI standard
> in that chain of historically 3-limit refinements.

15601 is of course excellent in the 3-limit. It is
consistent but no longer very good in the 5-limit.
It is inconsisent in higher limits. I think it's
basically useless.

16266 is even worse. It at least isn't a prime number,
but it's not divisible by 3. It's consistent but bad
through the 5-limit, and inconsistent afterwards. It
seems to have no redeeming features at all.

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

> Pschoacoustic experiments yield the result that
> our brains represent neurologically
> 5-limit internally as correlate of 2^15/3^8 = 32768/6561 = ~4.99436...
> 7-limit respectively in 3-limit approx: 2^25/3^14 = ~7.015398...
> 11-limit accordingly in 3-limit approx: 3^23/2^33 = ~10.9597...
> 13-lim...
> ...&ct also for other higer primes in the overtone series too.

What experiment shows our brains are wired using garibaldi
temperament?

--- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:
>
> Hi Dave (and George),
>
>
> Actually, using the criteria of an accurate small division
> of the apotome, Gene's suggestion of 31920-edo does quite
> well: 31920 divides the apotome into 3024 logarithmically
> equal parts with very little error. This division is
> looking quite good to me as a unit of measurement.

In the 3-limit, we have the following errors in
percentage terms:

612: 0.295 %
2460: -0.775 %
11664: -0.261 %
31920: -0.302 %

However, George's complaint was about the
7-limit.

Hi Gene, George, and Dave,

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "monz" <monz@> wrote:
> >
> > Hi Dave (and George),
> >
> >
> > Actually, using the criteria of an accurate small division
> > of the apotome, Gene's suggestion of 31920-edo does quite
> > well: 31920 divides the apotome into 3024 logarithmically
> > equal parts with very little error. This division is
> > looking quite good to me as a unit of measurement.
>
> In the 3-limit, we have the following errors in
> percentage terms:
>
> 612: 0.295 %
> 2460: -0.775 %
> 11664: -0.261 %
> 31920: -0.302 %
>
> However, George's complaint was about the
> 7-limit.

The one that looks really good to me in 7-limit
(so far i've only searched up to 5400-edo) is 3125-edo.

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

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@> wrote:
>
> > It would be useful if the tina was somewhere between about half to one
> > tenth of a mina. But please note that the mina used in sagittal design
> > work is exactly 1/233 of an apotome [-11 7> rather than 1/2460 of an
> > octave and we would likewise need the tina to be an integral division
> > of the apotome, not the octave.
>
> Why do you prefer to divide an exact apotome?

Gene,

George pointed out to me that since we're asking for at least 3^12
consistency the difference between the EDO and the EDA will be
insignificant and so I now realise we're not going to miss any good
EDAs by looking for good EDOs.

But the reason is that apotomes are represented by sharps and flats
(or two extra arrow shafts in pure sagittal) so it's the space of an
apotome that is subdivided by the various combinations of sagittal
flags and accents.

We don't require the EDO to be divisible by 12, but of course we'd be
very happy if the best EDA for our purposes was also an EDO divisible
bt 12 and therefore usable as a MIDI tuning unit, as is the case for
the 2460-EDO/233-EDA mina.

Monz,

As you probably know we already have levels of sagittal JI notation
in the works with resolutions corresponding to 612-EDO and 2460-EDO.
3025-EDO is too close to 2460-EDO to be used for the next level, and
31920-EDO is too far away.

11664-EDO/1105-EDA is definitely in the right ball park. Are there any
other contenders between say 5000-EDO and 25000-EDO?

It would be good to see their odd consistency limits and their
percentage errors for primes up to 11.

-- Dave Keenan

While I personally have trouble believing in the audibility of
23-limit JI in anything but Dream House type situations, I can't help
remembering Ben Johnston's use of the 31-limit and thinking that we
should try to honor that with this extreme log pitch unit.

-- Dave Keenan

Heya Dave-

A little off-topic, but what's the queue like for
Keenan-built choobs, and what's the pricing like?

Tx,

-Carl

Hi Dave,

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:

> As you probably know we already have levels of sagittal
> JI notation in the works with resolutions corresponding
> to 612-EDO and 2460-EDO. 3025-EDO is too close to 2460-EDO
> to be used for the next level, and 31920-EDO is too far away.

That's a typo: it's 3125-edo.

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

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:

> 11664-EDO/1105-EDA is definitely in the right ball park. Are there any
> other contenders between say 5000-EDO and 25000-EDO?

What prime limits are you looking for? A high-limit
system, which may not have the accuracy in the 3-limit
you want, is 16808. Good in the 3-limit, but I don't
know how good you need otherwise, are 7927, 8539, and
20203. I suppose the thing to do would be to give the
conditions you require.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@> wrote:
>
> > 11664-EDO/1105-EDA is definitely in the right ball park. Are there any
> > other contenders between say 5000-EDO and 25000-EDO?
>
> What prime limits are you looking for? A high-limit
> system, which may not have the accuracy in the 3-limit
> you want, is 16808. Good in the 3-limit, but I don't
> know how good you need otherwise, are 7927, 8539, and
> 20203. I suppose the thing to do would be to give the
> conditions you require.
>

Our requirements are still a bit fuzzy, but here's an attempt at
firming them up. In odd-limit terms, we _must_ have 27-limit
consistency, but we'd really like 31-limit if possible. In addition to
that we'd like to have errors in the lower primes less than the
following (as a percentage of a degree).

3 2%
5 7%
7 10%
11 25%

-- Dave K

I just noticed that the following percentages are approximately
proportional to the square of the prime number, in case that's of any
use. Approx p^2/500. They were based on considering the lowest
consistent power we could tolerate for each prime.

> 3 2%
> 5 7%
> 7 10%
> 11 25%

-- Dave K

Hi Dave,

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:
>
> I just noticed that the following percentages are approximately
> proportional to the square of the prime number, in case that's of any
> use. Approx p^2/500. They were based on considering the lowest
> consistent power we could tolerate for each prime.

That's similar to the method i used. I wanted one number which
shows me the "score" of an EDO in the approximation-to-JI game.
So i created a function which divides the percentage error
for each prime by that prime, then adds the results:

c = cardinality of EDO
p = prime-factor
f = floating-point approximation of EDO degrees to prime-factor

m = map of prime-factor in EDO degrees
= round(f,0)

e = error of EDO for that prime-factor, as fraction of EDO degree
= m - f

"score" = (e_3/3) + (e_5/5) + (e_7/7) + (e_11/11)

> > 3 2%
> > 5 7%
> > 7 10%
> > 11 25%

Since the mina (2460-edo) already achieves an approximation
with errors lower than those values, and we're looking for
higher cardinalities than that, it makes sense to either
decrease the allowable error, or add more primes. Any ideas
on values for lower error?

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

--- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:
>
> Hi Dave,
>
>
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@> wrote:
> >
> > I just noticed that the following percentages are approximately
> > proportional to the square of the prime number, in case that's of any
> > use. Approx p^2/500. They were based on considering the lowest
> > consistent power we could tolerate for each prime.
>
>
> That's similar to the method i used. I wanted one number which
> shows me the "score" of an EDO in the approximation-to-JI game.
> So i created a function which divides the percentage error
> for each prime by that prime, then adds the results:
>
> c = cardinality of EDO
> p = prime-factor
> f = floating-point approximation of EDO degrees to prime-factor
>
> m = map of prime-factor in EDO degrees
> = round(f,0)
>
> e = error of EDO for that prime-factor, as fraction of EDO degree
> = m - f
>
> "score" = (e_3/3) + (e_5/5) + (e_7/7) + (e_11/11)

Oops, my bad ... i neglected to mention that a lower score
is better.

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

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> Heya Dave-
>
> A little off-topic, but what's the queue like for
> Keenan-built choobs, and what's the pricing like?

Gidday Carl,

The last time someone asked that, I think I put them off with a "Lets
wait and see how the first two choobs go".

Robin seems happy with his choob so far. He's named it "Roz" after
Roswell, New Mexico. A refresher: It's an 8 string in microtempered
7-limit JI with an otonal tetrad on the low 4 open strings and utonal
on the high 4.

When he came to take delivery it was awesome to see and hear him play
it so well so quickly. He had clearly been thinking about the tuning
and the guitar design deeply and for a long time.

At the moment he's unhappy about being unable to play it. He badly cut
his left thumb at work (4 stitches). :-(

Incidentally, I described it earlier as a baritone guitar. This is no
longer the case. Robin initially wanted the tuning to go from a low G#
(below the standard guitar low E) to a high B (a standard guitar 2nd
string). I was surprised, but said it might be on the limits of
do-ability. However, when I installed the appropriate string gauges
and tuned it up, the low G# was just mud. i.e. low in volume and so
inharmonic that its pitch was poorly defined. It became clear that
either we'd need to use a longer scale like a bass guitar, or an even
heavier string with more tension than I was willing to give it on PVC
pipe. I have been keeping the total tension below about 400 newtons
(~= 40 kg weight on planet earth).

So Robin opted to take Roz up a fourth (low C#, high E) and I restrung
it with the new gauges and lowered the action accordingly and the bass
was nice and clean. Then however the top string was noticeably lower
in volume and brightness than the others and I realised that with 8
strings and standard string spacing at the saddle, the outer strings
were a lot further from the piezo than they had been on the 6-string
narrow-spaced prototype.

A second piezo installed under the treble side of the bridge and
connected in parallel with the first, solved that problem beautifully.
So from now on I will always use 2 piezo discs. This should also
prevent dropout when bowing, as you can never be bowing at right
angles to both discs. I need to borrow a bow from my sister the
violinist some time, to test this.

Mad Max 3 had to wait until I caught up on a few other commitments,
but he's now fretted up for a Bb to D# 12-of-meantone (40 fretlets
over 2 octaves) with standard EADGBE open tuning. This is the first
Choob to use clear fretline and I'm currently installing a red neon
tube inside it, to be powered via the second channel of a stereo cable.

But you're right, there'd be no harm in having a queue, in which case
you'll be the first on it, after me. I've been thinking about pricing.
Here's a formula I could live with. This is Australian dollars.

$200 + $30 per string + $5 per fret or fretlet

So a 6-string with 24 frets comes to AU$500.

But right now (I can't say for how long) I'll give a 20% discount to
any brave soul willing to take a risk on something new.

These prices include a strap and sock and hard case. The sock is the
black cloth tube that you slip over the choob before you slide it
inside it's case. Internal lighting will be extra (I haven't worked
that out yet). You can pay by electronic transfer or PayPal. I may ask
for payment in advance if I don't know you well. (Not you Carl)

When shipping outside Australia I figure I should ship it with the
ballast chamber empty and open. I'll assemble it to test it, and then
take it apart enough to give access. I wouldn't want customs to have
to take it apart just to prove there's nothing in there but a little
bit of Aussie beach. So I'm afraid you'll have to source your own wet
sand and complete the assembly. Instructions will of course be provided.

Assistance with designing a guitar or bass for your desired microtonal
tuning will be provided free of charge, as always, since that's pure
fun. :-) The design discussion can be carried out on one of the tuning
lists if you think others would be interested, or by email.

Regards
-- Dave Keenan

--- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@> wrote:
> > c = cardinality of EDO
> > p = prime-factor
> > f = floating-point approximation of EDO degrees to prime-factor
> >
> > m = map of prime-factor in EDO degrees
> > = round(f,0)
> >
> > e = error of EDO for that prime-factor, as fraction of EDO degree
> > = m - f
> >
> > "score" = (e_3/3) + (e_5/5) + (e_7/7) + (e_11/11)
>
> Oops, my bad ... i neglected to mention that a lower score
> is better.

That was obvious enough. Also you would be using the absolute values
of the errors.

So if you changed that to divide by the square of the primes you'd
make us happy. And if you also raised the resulting weighted errors to
some power (a parameter) before adding them, and then took the
corresponding root of their sum, you could use a power of 2 to give
RMS or approach minimax by using a larger power.

Yes, minas already acheive this. But there is no need to do
significantly better in _relative_ terms at higher cardinalities as we
will still benefit from lower _absolute_ errors. However, as I
mentioned, we would like to go to 31 limit consistency if possible.

-- Dave

Oops! I wrote:
A refresher: It's an 8 string in microtempered
7-limit JI with an otonal tetrad on the low 4 open strings and utonal
on the high 4.

It actually has a 3:4:5:6:7 chord on the low 5 strings and the inverse
on the high five (the middle 6:7 is common to both).

-- Dave Keenan

At 07:39 PM 4/22/2007, you wrote:
>--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>>
>> Heya Dave-
>>
>> A little off-topic, but what's the queue like for
>> Keenan-built choobs, and what's the pricing like?
>
>Gidday Carl,
>
>The last time someone asked that, I think I put them off with a "Lets
>wait and see how the first two choobs go".
>
>Robin seems happy with his choob so far. He's named it "Roz" after
>Roswell, New Mexico. A refresher: It's an 8 string in microtempered
>7-limit JI with an otonal tetrad on the low 4 open strings and utonal
>on the high 4.
>
>When he came to take delivery it was awesome to see and hear him play
>it so well so quickly. He had clearly been thinking about the tuning
>and the guitar design deeply and for a long time.
>
>At the moment he's unhappy about being unable to play it. He badly cut
>his left thumb at work (4 stitches). :-(

I think I heard something about that on MMM. :(

>This is the first
>Choob to use clear fretline and I'm currently installing a red neon
>tube inside it, to be powered via the second channel of a stereo cable.

Awesome!

>But you're right, there'd be no harm in having a queue, in which case
>you'll be the first on it, after me. I've been thinking about pricing.
>Here's a formula I could live with. This is Australian dollars.
>
>$200 + $30 per string + $5 per fret or fretlet
>
>So a 6-string with 24 frets comes to AU$500.
>
>But right now (I can't say for how long) I'll give a 20% discount to
>any brave soul willing to take a risk on something new.

Wow! I should just write you a check. I'd like to consult with
you a bit on tuning at some point, maybe off-list. I'm thinking
of 22-ET, but the Roz tuning sounds interesting.

>These prices include a strap and sock and hard case. The sock is the
>black cloth tube that you slip over the choob before you slide it
>inside it's case. Internal lighting will be extra (I haven't worked
>that out yet). You can pay by electronic transfer or PayPal. I may ask
>for payment in advance if I don't know you well. (Not you Carl)

I'll pay in advance anyway. What's your e-mail address for
purposes of PayPal?

>When shipping outside Australia I figure I should ship it with the
>ballast chamber empty and open. I'll assemble it to test it, and then
>take it apart enough to give access. I wouldn't want customs to have
>to take it apart just to prove there's nothing in there but a little
>bit of Aussie beach. So I'm afraid you'll have to source your own wet
>sand and complete the assembly. Instructions will of course be provided.
>
>Assistance with designing a guitar or bass for your desired microtonal
>tuning will be provided free of charge, as always, since that's pure
>fun. :-) The design discussion can be carried out on one of the tuning
>lists if you think others would be interested, or by email.

Well this is just great news.

I was thinking about buying one of these
http://blackbirdguitars.com
and maybe I still will, but I think a Choob has a lot of things
going for it as a first step into guitarland.

By the way, I asked my friend who's a guitarist 'seduced into being
a guitar tech' about piezo vs. magnetic, and he says piezos:
* tend to sound brittle
* don't overdrive as well
I'm not worried about either of these, and at any rate observations
made on a normal guitar may not apply to choobs.

Oh: string material. Are these steel? I have soft hands, owing
to never having done an honest day's work in my life (as I like
to quip), and the high string on a normal steel 6-string often hurts
my fingers. Classical/nylon guitars don't bother me as much, nor
do baritone or bass guitars. Since it's piezo I guess nylon would
work as well (?). Thoughts appreciated.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:

> I'd like to consult with
> you a bit on tuning at some point, maybe off-list. I'm thinking
> of 22-ET, but the Roz tuning sounds interesting.

Sure. Whenever you like.

> >These prices include a strap and sock and hard case.

I also include the nut driver for tuning it.

> I'll pay in advance anyway. What's your e-mail address for
> purposes of PayPal?

Same as usual. d period keenan at bigpond period net period au.

> >When shipping outside Australia I figure I should ship it with the
> >ballast chamber empty and open. I'll assemble it to test it, and then
> >take it apart enough to give access. I wouldn't want customs to have
> >to take it apart just to prove there's nothing in there but a little
> >bit of Aussie beach. So I'm afraid you'll have to source your own wet
> >sand and complete the assembly. Instructions will of course be
provided.
> >
> >Assistance with designing a guitar or bass for your desired microtonal
> >tuning will be provided free of charge, as always, since that's pure
> >fun. :-) The design discussion can be carried out on one of the tuning
> >lists if you think others would be interested, or by email.
>
> Well this is just great news.
>
> I was thinking about buying one of these
> http://blackbirdguitars.com
> and maybe I still will, but I think a Choob has a lot of things
> going for it as a first step into guitarland.
>
> By the way, I asked my friend who's a guitarist 'seduced into being
> a guitar tech' about piezo vs. magnetic, and he says piezos:
> * tend to sound brittle
> * don't overdrive as well
> I'm not worried about either of these, and at any rate observations
> made on a normal guitar may not apply to choobs.

How right you are.

> Oh: string material. Are these steel? I have soft hands, owing
> to never having done an honest day's work in my life (as I like
> to quip), and the high string on a normal steel 6-string often hurts
> my fingers. Classical/nylon guitars don't bother me as much, nor
> do baritone or bass guitars. Since it's piezo I guess nylon would
> work as well (?). Thoughts appreciated.

There is no problem with using classical guitar strings. I only used
steel because I wanted an electric guitar sound. Nylon doesn't have
the same sustain, as you would appreciate. There would also be nothing
to stop you changing from nylon to steel at some later date, except
that the saddle-setback may need to change slightly to maintain
accurate intonation on the high frets and this is not adjustable
except by cutting the old saddle off the bridge and gluing on a new one.

Should we move this to MMM? If so, you might repost your previous
message there (that contains my previous one quoted). Or start over
with your original question if you want and we'll reconstruct the
whole conversation there. I don't want to be posting commercial
advertising -- but since it was in response to an unprompted question...

As you pointed out, its a bit off-topic for this list. I'd like to
talk on MMM about the rapid prototyping option that could be of
interest if you are finding it hard to decide on a tuning.

-- Dave Keenan

>> By the way, I asked my friend who's a guitarist 'seduced into being
>> a guitar tech' about piezo vs. magnetic, and he says piezos:
>> * tend to sound brittle
>> * don't overdrive as well
>> I'm not worried about either of these, and at any rate observations
>> made on a normal guitar may not apply to choobs.
>
>How right you are.

I'm curious as to why it might be, though, especially the
second point.

>> Oh: string material. Are these steel? I have soft hands, owing
>> to never having done an honest day's work in my life (as I like
>> to quip), and the high string on a normal steel 6-string often hurts
>> my fingers. Classical/nylon guitars don't bother me as much, nor
>> do baritone or bass guitars. Since it's piezo I guess nylon would
>> work as well (?). Thoughts appreciated.
>
>There is no problem with using classical guitar strings. I only used
>steel because I wanted an electric guitar sound. Nylon doesn't have
>the same sustain, as you would appreciate. There would also be nothing
>to stop you changing from nylon to steel at some later date, except
>that the saddle-setback may need to change slightly to maintain
>accurate intonation on the high frets and this is not adjustable
>except by cutting the old saddle off the bridge and gluing on a new one.
>
>Should we move this to MMM? If so, you might repost your previous
>message there (that contains my previous one quoted). Or start over
>with your original question if you want and we'll reconstruct the
>whole conversation there. I don't want to be posting commercial
>advertising -- but since it was in response to an unprompted question...

Sure, will do.

-Carl

--- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:
>
> Hi Dave,
>
>
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@> wrote:
> >
> > I just noticed that the following percentages are approximately
> > proportional to the square of the prime number, in case that's of
any
> > use. Approx p^2/500. They were based on considering the lowest
> > consistent power we could tolerate for each prime.
>
>
> That's similar to the method i used. I wanted one number which
> shows me the "score" of an EDO in the approximation-to-JI game.
> So i created a function which divides the percentage error
> for each prime by that prime, then adds the results:

Wouldn't a score based on the maximum error over
the q-diamond make more sense? Say, for each odd
q require that the max error is less than q^2/500.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:

> > Î" "Delfi unit: 1/665 part of an octave
> > Used in Byzantine music theory?
> Was it or wasn't it used in Byzantine theory? Where,
> if anywhere, are citations?

http://www.google.de/search?hl=de&q=delfi-unit+665+&btnG=Google-Suche&meta=
shows mostly references in modern greek language,

that can be even translated by
http://babelfish.altavista.com/

Warning:
the output yields horrible pidgin-english :-(

A.S.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:

> Wouldn't a score based on the maximum error over
> the q-diamond make more sense? Say, for each odd
> q require that the max error is less than q^2/500.

I searched for 27-limit consistent systems with
max percentage (patent) error less than q^2/5,
for q to 31, and got 6079, 11664, 36269, 47933.
Raising the cutoff to q^2/4 added 8539 and 30631.
The consistency limits of these are:

6079: 29
8539: 27
11664: 27
30631: 35
36269: 29
47933: 27

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <genewardsmith@> wrote:
>
> > Wouldn't a score based on the maximum error over
> > the q-diamond make more sense? Say, for each odd
> > q require that the max error is less than q^2/500.

Sounds good to me. But the relative error will always be less than 50%
(although not necc. consistent) so this doesn't affect q>21 does it?

> I searched for 27-limit consistent systems with
> max percentage (patent) error less than q^2/5,
> for q to 31, and got 6079, 11664, 36269, 47933.
> Raising the cutoff to q^2/4 added 8539 and 30631.
> The consistency limits of these are:
>
> 6079: 29
> 8539: 27
> 11664: 27
> 30631: 35
> 36269: 29
> 47933: 27

Awesome! Thanks! Can you post their percentage errors?

-- Dave Keenan

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
> Raising the cutoff to q^2/4 added 8539 and 30631.

How much do we have to raise that cutoff before we get a 35-limit
consistent one below 30631-EDO? And what is it and what are its
percent errors? Thanks.

-- Dave Keenan

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <genewardsmith@> wrote:
> > Raising the cutoff to q^2/4 added 8539 and 30631.
>
> How much do we have to raise that cutoff before we get a 35-limit
> consistent one below 30631-EDO? And what is it and what are its
> percent errors? Thanks.

20203 and 28342 are 35-limit consistent and might
serve, I suppose.

Here are some percentage errors for primes 3 to 19:

6079: [1.295840167, -.08915866667, 8.943222167, 11.51869183,
2.695023333, 31.33927133, -15.13519025]
8539: [.5208790000, 5.599449250, -.3678885833, -8.659257583, -
5.475633750, 15.48049542, -5.303430583]
11664: [-.2604960000, 3.069576000, 1.218888000, 18.95983200,
7.112124000, -16.65716400, 17.34825600]
30631: [1.363079500, 2.052277000, -8.860016750, 15.00663742, -
16.90065425, -7.428017500, -26.76638883]
36269: [-.4956763333, -1.006464750, 4.433885250, -12.54000675, -
24.81404083, -18.97775425, -8.299556167]
47933: [-.7549447500, 2.061119000, 5.652099583, 6.419027583, -
17.70325467, -35.63419108, 9.051348167]

Here are more percentages, for 35-consistent ets:

20203: [.2592718333, 8.670454167, .8518931667, 10.30016283,
1.636443000, -1.176824750, 12.04435517]
28342: [-.7203591667, -8.606520667, -5.318848667, -21.09589533,
13.75059367, 12.81766950, 23.84270750]
30631: [1.363079500, 2.052277000, -8.860016750, 15.00663742, -
16.90065425, -7.428017500, -26.76638883]
34368: [.8763840000, -2.477360000, -17.39593600, 25.41227200,
28.77460800, 7.709888000, 22.72297600]
37375: [2.653625000, -6.254083333, 10.97890625, -25.67662500,
6.553083333, 7.633843750, -29.08086458]
39170: [1.883424167, 7.651206667, -9.227799167, 6.348804167, -
22.37586250, 8.052699167, -32.07043750]
45473: [.01894708333, -3.626471750, 14.96440642, 26.59791558, -
9.530382917, -19.77696558, -.7806198333]

Note 34368 is divisible by 12.

Thanks Gene,

That's just the information we needed. Those >8.6% errors in 5's of
20203 1n 28342 are the killers and aanything as large as 30631 or
34368 is just too much of a jump up from 2460. So I guess we have to
give up on 31-limit consistency and fall back to 29 or 27 limit.

6079, 8539, 11664 are looking good. Of which 11664 is divisible by 12.
George tells me 6079 is 29-limit consistent.

What about some sort of criteria for being "almost 31 limit
consistent". Is it possible that the consistency limits could change
when these are cast as EDA's instead of EDO's?

-- Dave Keenan

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@>
> wrote:
> >
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> > <genewardsmith@> wrote:
> > > Raising the cutoff to q^2/4 added 8539 and 30631.
> >
> > How much do we have to raise that cutoff before we get a 35-limit
> > consistent one below 30631-EDO? And what is it and what are its
> > percent errors? Thanks.
>
> 20203 and 28342 are 35-limit consistent and might
> serve, I suppose.
>
> Here are some percentage errors for primes 3 to 19:
>
> 6079: [1.295840167, -.08915866667, 8.943222167, 11.51869183,
> 2.695023333, 31.33927133, -15.13519025]
> 8539: [.5208790000, 5.599449250, -.3678885833, -8.659257583, -
> 5.475633750, 15.48049542, -5.303430583]
> 11664: [-.2604960000, 3.069576000, 1.218888000, 18.95983200,
> 7.112124000, -16.65716400, 17.34825600]
> 30631: [1.363079500, 2.052277000, -8.860016750, 15.00663742, -
> 16.90065425, -7.428017500, -26.76638883]
> 36269: [-.4956763333, -1.006464750, 4.433885250, -12.54000675, -
> 24.81404083, -18.97775425, -8.299556167]
> 47933: [-.7549447500, 2.061119000, 5.652099583, 6.419027583, -
> 17.70325467, -35.63419108, 9.051348167]
>
> Here are more percentages, for 35-consistent ets:
>
> 20203: [.2592718333, 8.670454167, .8518931667, 10.30016283,
> 1.636443000, -1.176824750, 12.04435517]
> 28342: [-.7203591667, -8.606520667, -5.318848667, -21.09589533,
> 13.75059367, 12.81766950, 23.84270750]
> 30631: [1.363079500, 2.052277000, -8.860016750, 15.00663742, -
> 16.90065425, -7.428017500, -26.76638883]
> 34368: [.8763840000, -2.477360000, -17.39593600, 25.41227200,
> 28.77460800, 7.709888000, 22.72297600]
> 37375: [2.653625000, -6.254083333, 10.97890625, -25.67662500,
> 6.553083333, 7.633843750, -29.08086458]
> 39170: [1.883424167, 7.651206667, -9.227799167, 6.348804167, -
> 22.37586250, 8.052699167, -32.07043750]
> 45473: [.01894708333, -3.626471750, 14.96440642, 26.59791558, -
> 9.530382917, -19.77696558, -.7806198333]
>
> Note 34368 is divisible by 12.
>

Hi Dave and Gene,

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:

> Thanks Gene,
>
> That's just the information we needed. Those >8.6% errors in
> 5's of 20203 1n 28342 are the killers and aanything as large
> as 30631 or 34368 is just too much of a jump up from 2460.
> So I guess we have to give up on 31-limit consistency and
> fall back to 29 or 27 limit.
>
> 6079, 8539, 11664 are looking good. Of which 11664 is
> divisible by 12. George tells me 6079 is 29-limit consistent.

I'm only looking at prime-limits.

8539 looks mighty good to me thru the 19-limit, and
quite good all the way thru the 31-limit.

30631 does look good indeed all the way thru the 41-limit.

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

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

> 30631 does look good indeed all the way thru the 41-limit.

I've searched up to 100000-edo so far, and for 5-digit
cardinalities good thru the 41-limit, 58973 looks very
impressive. Any comments on that one, Gene?

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

--- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "monz" <monz@> wrote:
>
> > 30631 does look good indeed all the way thru the 41-limit.
>
>
> I've searched up to 100000-edo so far, and for 5-digit
> cardinalities good thru the 41-limit, 58973 looks very
> impressive. Any comments on that one, Gene?

Here are the percent errors for 58973:

prime .. % error
.. 3 .. - 00.6445
.. 5 .. + 06.5540
.. 7 .. + 14.1819
. 11 .. + 06.0846
. 13 .. + 03.1498
. 17 .. - 05.3863
. 19 .. + 02.9250
. 23 .. + 01.9235
. 29 .. - 28.6774
. 31 .. - 18.0988
. 37 .. + 09.3331
. 41 .. - 08.5632

For your purposes, Dave, i suppose that the error for 7
may be too large, but otherwise this one looks very good.

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

Just wanted to explain the ranking method i'm using
for this, as it's different from what i posted the
other day:

I simply take the absolute percent error for each prime,
and multiply each one by an arbitrary exponent-limit,
then add them all together. Lowest score wins.

I've chosen as my exponent-limits: 3^12, 5^6, 7^3, 11^2,
and 1 for all the rest up to 41. I personally feel that
5^5, 7^2 and 11^1 are all OK, but something in my gut
tells me that i need to rank them a little more
importantly than that, so i increased each of those
by one.

I know this cardinality is already higher than what
most folks want, but it (58973) and 8539 both look
like they could be very useful as interval measurements.

Thru the 41-limit, 8539 is only weak for 37 --
all the other primes look very good. It's 809-EDA.

58973 is excellent for all of them, its weakest
approximation (29) still coming in under 29%, and it
looks to me (i haven't checked) as if it's consistent
thru 41 too. It's 5587-EDA.

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

--- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "monz" <monz@> wrote:
> >
> > --- In tuning-math@yahoogroups.com, "monz" <monz@> wrote:
> >
> > > 30631 does look good indeed all the way thru the 41-limit.
> >
> >
> > I've searched up to 100000-edo so far, and for 5-digit
> > cardinalities good thru the 41-limit, 58973 looks very
> > impressive. Any comments on that one, Gene?
>
> Here are the percent errors for 58973:
>
> prime .. % error
> .. 3 .. - 00.6445
> .. 5 .. + 06.5540
> .. 7 .. + 14.1819
> . 11 .. + 06.0846
> . 13 .. + 03.1498
> . 17 .. - 05.3863
> . 19 .. + 02.9250
> . 23 .. + 01.9235
> . 29 .. - 28.6774
> . 31 .. - 18.0988
> . 37 .. + 09.3331
> . 41 .. - 08.5632
>
>
> For your purposes, Dave, i suppose that the error for 7
> may be too large, but otherwise this one looks very good.

--- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "monz" <monz@> wrote:
>
> > 30631 does look good indeed all the way thru the 41-limit.
>
>
> I've searched up to 100000-edo so far, and for 5-digit
> cardinalities good thru the 41-limit, 58973 looks very
> impressive. Any comments on that one, Gene?

It is very impressive. Sadly, its prime factorization,
17 * 3469, is less useful than one might hope for, and
the percentage errors in the 5 and especially the 7
limits may be larger than is wanted. Hpwever the very
low error of 3 is a big plus.

Below I list, for odd limits up to 69, the logflat
badness, the maximum percentage error over the entire
tonality diamond, and whether or not it is consistent.
Low logflat badness, under 1 for example, is good, and
we can see that it's terrific in the 27 limit, with
a percentage error under 20%, and remains consistent
up to the 47 limit.

3: 380.079290 .644497 true
5: 17.481037 7.198476 true
7: 5.771011 14.826347 true
9: 5.771011 15.470845 true
11: 2.410890 15.470845 true
13: 1.392017 15.470845 true
15: 1.392017 15.470845 true
17: 1.220839 19.568145 true
19: .939878 19.568145 true
21: .939878 19.568145 true
23: .772469 19.568145 true
25: .772469 19.568145 true
27: .772469 19.568145 true
29: 1.452505 42.859284 true
31: 1.285613 42.859284 true
33: 1.285613 42.859284 true
35: 1.285613 49.413264 true
37: 1.341341 49.413264 true
39: 1.341341 49.413264 true
41: 1.234235 49.413264 true
43: 1.150315 49.413264 true
45: 1.150315 49.413264 true
47: 1.082940 49.413264 true
49: 1.082940 57.041135 false
51: 1.082940 57.041135 false
53: 1.311734 63.067005 false
55: 1.311734 63.067005 false
57: 1.311734 63.067005 false
59: 1.253049 63.067005 false
61: 1.373991 72.004114 false
63: 1.373991 72.004114 false
65: 1.373991 72.004114 false
67: 1.518510 82.486191 false
69: 1.518510 82.486191 false

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

> I simply take the absolute percent error for each prime,
> and multiply each one by an arbitrary exponent-limit,
> then add them all together. Lowest score wins.

Might make more sense over the whole diamond instead.

> It's 5587-EDA.

Which means what?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "monz" <monz@> wrote:
>
> > I simply take the absolute percent error for each prime,
> > and multiply each one by an arbitrary exponent-limit,
> > then add them all together. Lowest score wins.
>
> Might make more sense over the whole diamond instead.

It would take me some work to configure my spreadsheet
to figure that out, and anyway, i think it's important
to have the exponents of 5 and especially 3 go pretty high,
higher than they go for tonality diamonds in a given
odd-limit.

> > It's 5587-EDA.
>
> Which means what?

It's logarithmically 1/5587 of the pythagorean apotome [-11 7>.
5587 "equal divisions of the apotome" ... at least, i *think*
that's what "EDA" means.

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

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "monz" <monz@> wrote:
> >
> > --- In tuning-math@yahoogroups.com, "monz" <monz@> wrote:
> >
> > > 30631 does look good indeed all the way thru the 41-limit.
> >
> >
> > I've searched up to 100000-edo so far, and for 5-digit
> > cardinalities good thru the 41-limit, 58973 looks very
> > impressive. Any comments on that one, Gene?
>
> It is very impressive. Sadly, its prime factorization,
> 17 * 3469, is less useful than one might hope for,

That's true. But i think that kind of divisibility is
important for a *tuning* which is to be used for composition,
but i don't think it's necessarily important for a division
which is to be used as a unit of interval measurement --
and certainly *no one* is going to use 58973 as a tuning.

To my way of thinking, what's important here is that the
division gives good integer values for a wide range of
JI intervals. Of course, if it gives good interval values
for some important tempered intervals too, then so much
the better.

> and the percentage errors in the 5 and especially
> the 7 limits may be larger than is wanted. Hpwever the
> very low error of 3 is a big plus.

This is exactly why i like using the method i'm using,
which multiplies the error by some exponent value.
Since i'm using 3^12, then only tunings which give a
very low error for 3 are going to have a decent score.

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

--- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <genewardsmith@> wrote:
> >
> > --- In tuning-math@yahoogroups.com, "monz" <monz@> wrote:
> >
> > > I simply take the absolute percent error for each prime,
> > > and multiply each one by an arbitrary exponent-limit,
> > > then add them all together. Lowest score wins.
> >
> > Might make more sense over the whole diamond instead.
>
>
> It would take me some work to configure my spreadsheet
> to figure that out, and anyway, i think it's important
> to have the exponents of 5 and especially 3 go pretty high,
> higher than they go for tonality diamonds in a given
> odd-limit.
>
>
> > > It's 5587-EDA.
> >
> > Which means what?
>
>
> It's logarithmically 1/5587 of the pythagorean apotome [-11 7>.
> 5587 "equal divisions of the apotome" ... at least, i *think*
> that's what "EDA" means.

Uh, well, one degree of 58973-edo is not *exactly*
1/5587 apotome, but pretty darn close. The error
is ~ -4.5% ... which is approximately the same amount
as this division's error for all the primes except 7
from 5 thru 23.

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

--- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:
>
> Just wanted to explain the ranking method i'm using
> for this, as it's different from what i posted the
> other day:
>
> I simply take the absolute percent error for each prime,
> and multiply each one by an arbitrary exponent-limit,
> then add them all together. Lowest score wins.
>
> I've chosen as my exponent-limits: 3^12, 5^6, 7^3, 11^2,
> and 1 for all the rest up to 41. I personally feel that
> 5^5, 7^2 and 11^1 are all OK, but something in my gut
> tells me that i need to rank them a little more
> importantly than that, so i increased each of those
> by one.

Here are the top 15 edos between cardinality 1 and 99,
ranked from lowest score to highest (for the 41-limit):

53, 87, 65, 41, 84, 94, 31, 77, 24, 46, 72, 22, 12, 19, 89

53 scores quite a bit better than any of the rest.

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

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

> This is exactly why i like using the method i'm using,
> which multiplies the error by some exponent value.
> Since i'm using 3^12, then only tunings which give a
> very low error for 3 are going to have a decent score.

I tried another approach. There are 526 27-limit
consistent n less than 100000. I took this list,
and then computed the maximum percentage error
over the q-limit diamond for each q to 27, divided
by q. I then sorted the list according to this,
and got the following as my top 20:

16808: 1.657071
8539: 1.700212
8269: 1.830615
14348: 1.867826
66288: 2.005204
11664: 2.095180
90893: 2.106371
58973: 2.118050
20203: 2.128624
87445: 2.159419
95242: 2.174214
49039: 2.197463
11934: 2.202312
77781: 2.217031
67512: 2.244990
36269: 2.249763
73321: 2.264368
67242: 2.280986
45861: 2.281726
74085: 2.305438

Here the floating point number is such that if
we call it r, then the maximum percentage error
in the q-limit is bounded by r*q. Of the above,
the ones divisible by 12 are 66288, 11664, 11934,
and 67512.

I then did the same thing, only this time with the
bound being r*q^2. Divisible by 12 among the top
20 on that list are 54624, 11664, 66288, 76716,
66900 and 42348.

47933: .145534
54624: .148794
11664: .161006
66288: .162697
36269: .173059
6079: .184322
90893: .202853
30631: .222741
8539: .223991
76716: .234831
51499: .248991
17743: .251893
66900: .261454
27730: .264285
15230: .264287
70637: .272379
60703: .276143
65847: .286500
24605: .286875
42348: .295353

I note I am sometimes getting different numbers
standing out than I am used to; 47933 is a good
11-limit system in particular, but its virtues
don't really show up in the way I was looking at
things before. On the other hand 54624, for instance,
is an exceptional 11-limit system.

--- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:
> Thru the 41-limit, 8539 is only weak for 37 --
> all the other primes look very good. It's 809-EDA.

Thanks Monz and Gene!

The almost 35-limit consistency of 8539-EDO/809-EDA looks like making
it a winner for our Sagittal problem, although for a MIDI standard it
would have to be 11664-EDO/1105-EDA. Approximating one by the other is
subliminal anyway, so I don't see that it matters.

-- Dave Keenan

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:

> 16808: 1.657071
> 8539: 1.700212
> 8269: 1.830615
> 14348: 1.867826

14348 doesn't look suitable for Dave and George, but
it's quite interesting. Here is a list of logflat
badness and maximum percent error over the diamond
up to 101. Note that while it is an excellent system
for the 17-limit, and consistent up to 29, it is *also*
hot stuff at higher limits up to 69. The patent val
gets just a gnat's eyebrow over 50% error for a range
from the 31 to the 69 limit, which is pretty much
as good as consistency.

3: 602.047061 4.196035 true
5: 5.026142 4.196035 true
7: 2.758931 11.353866 true
9: 2.758931 15.549901 true
11: 1.740801 15.905649 true
13: 1.185799 17.484481 true
15: 1.185799 17.484481 true
17: .861889 17.484481 true
19: 1.392892 35.488700 true
21: 1.392892 35.488700 true
23: 1.174057 35.488700 true
25: 1.174057 35.488700 true
27: 1.174057 35.488700 true
29: 1.165074 40.224220 true
31: 1.309033 50.265666 false
33: 1.309033 50.265666 false
35: 1.309033 50.265666 false
37: 1.199946 50.265666 false
39: 1.199946 50.265666 false
41: 1.116017 50.265666 false
43: 1.049602 50.265666 false
45: 1.049602 50.265666 false
47: .995830 50.265666 false
49: .995830 50.265666 false
51: .995830 50.265666 false
53: .951461 50.265666 false
55: .951461 50.265666 false
57: .951461 50.265666 false
59: .914262 50.265666 false
61: .882650 50.265666 false
63: .882650 50.265666 false
65: .882650 50.265666 false
67: .855469 50.265666 false
69: .855469 50.265666 false
71: 1.224062 73.964744 false
73: 1.193615 73.964744 false
75: 1.193615 73.964744 false
77: 1.193615 73.964744 false
79: 1.166722 73.964744 false
81: 1.166722 73.964744 false
83: 1.142799 73.964744 false
85: 1.142799 73.964744 false
87: 1.142799 78.160778 false
89: 1.185002 78.160778 false
91: 1.185002 78.160778 false
93: 1.185002 78.160778 false
95: 1.185002 78.160778 false
97: 1.197727 80.381874 false
99: 1.197727 80.381874 false
101: 1.178772 80.381874 false

Hi Dave,

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "monz" <monz@> wrote:
> > Thru the 41-limit, 8539 is only weak for 37 --
> > all the other primes look very good. It's 809-EDA.
>
> Thanks Monz and Gene!
>
> The almost 35-limit consistency of 8539-EDO/809-EDA looks
> like making it a winner for our Sagittal problem, although
> for a MIDI standard it would have to be 11664-EDO/1105-EDA.
> Approximating one by the other is subliminal anyway, so I
> don't see that it matters.

But hmm ... i found something interesting. Even tho
8539-edo does not divide well by 12, it *does* give a
pretty good approximation to the 1/11-comma meantone 5th,
which is nearly the same as the 12-edo 5th: 4981 degrees
of 5839 with only +8.25% error.

One 12-edo semitone maps to 712 degrees of 8539, with
a whopping -41.67% error.

But in the meantone mapping, the chromatic-semitone/apotome
(3^7) is 711 degrees and the diatonic-semitone/limma (3^-5)
is 712 degrees.

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

Any ideas on a name for one degree of 8539-edo?

I want to make a webpage for it in the Encyclopedia,
but since the plan is to use it as a unit of interval
measurement, it should have a name.

I thought of "hepticent", since it's very close
to 1/7 cent ... but i'm reluctant to use numerical
prefixes in cases that are only approximate.
And the denominator of 1200/8539 cent is prime,
so there's no hope of an accurate division prefix.

Anyway, i'd much prefer a short one-syllable word
to one with three syllables.

Suggestions?

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

While we're at it, how about a name for one degree
of 58973-edo?

This one is ~1/49 cent.
And again, a short word is preferred.

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

--- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:
>
> Any ideas on a name for one degree of 8539-edo?

No, but you could call one degree of 6079-edo
a threnode.

Thanks to Marc Jones for this ...

Wow -- if you want an EDO for use as an integer interval
measurement unit for the 7-limit, and don't mind the
high cardinality, i doubt if it gets any better than this:

103169-edo

percent error for the prime-factors:

3 .. - 0.3763 %
5 .. - 0.0379 %
7 .. - 0.0046 %

See my webpage on Marc's EDOs:
http://tonalsoft.com/enc/j/marc-jones-edolist.aspx

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

George and I have been referring to it as a "tina", pronounced "teena".

--- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:
>
> Any ideas on a name for one degree of 8539-edo?
>
> I want to make a webpage for it in the Encyclopedia,
> but since the plan is to use it as a unit of interval
> measurement, it should have a name.
>
> I thought of "hepticent", since it's very close
> to 1/7 cent ... but i'm reluctant to use numerical
> prefixes in cases that are only approximate.
> And the denominator of 1200/8539 cent is prime,
> so there's no hope of an accurate division prefix.
>
> Anyway, i'd much prefer a short one-syllable word
> to one with three syllables.
>
> Suggestions?
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:
>
> George and I have been referring to it as a "tina",
> pronounced "teena".

OK, then "tina" it is!

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

(You responded in the nick of time, because i had
already made the page with the name "hepticent" and
was about to announce it here when i saw your post.)

Note that in my bid to help ensure its use, i've spent
hours creating a list of many common 31-limit JI intervals
with their tina values. And i quote from the webpage:

>> "tina values are given both in integer and floating-point
>> form, expressly to point out how unnecessary it is
>> to use the decimal places for most of the cases"

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

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@> wrote:
>and we would likewise need the tina to be an integral division
> of the apotome, not the octave.
sorry but from:
http://www.research.att.com/~njas/sequences/A028507
can be concluded that none EDA meets ever an integral EDO exactly.
> Any suggestions?

190 537
correspondents to the convergent at the notably gain on the first '53' in:
http://www.research.att.com/~njas/sequences/table?a=28507&fmt=5
alike the less eminent '23' case earlier
before that in the 665 case respectively.

3^7 / 2^11 ~ 18 051 deg or ~ EDApotome := (2187/2048)^(~deg/18051)

3^12 / 2^19 ~ 3 724.999 998 76... deg ~3725 EDPyth. : (TUs = 720 EDP)
==>> 1 TU = ~ 745/144 = ~ 5.173 611 111 111 ... deg

3^53 / 2^84 ~ 573.999 986... deg ~574 EDMercator's 53 subdivision

3^665 / 2^1054 ~ 11.999 993 991 ... deg ~ "Satanic-comma"^(~1/12)

Definition:
n-EDS := (3^665/2^1054)^(1/n)

How about other n-EDSes different from 12 for the higher limits than3?

A.S.

--- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:
>
> Thanks to Marc Jones for this ...
>
> Wow -- if you want an EDO for use as an integer interval
> measurement unit for the 7-limit, and don't mind the
> high cardinality, i doubt if it gets any better than this:
>
>
> 103169-edo

When we were discussing Paul's 2401/2400 periodicity
business, it turned out this dominated, in some sense,
the ets under it. It's very striking in that it
is the first et to have a 7-limit logflat badness
figure under 171. Paul told me Mark Jones called it
Halloween 69, from 10/31/69.

Here's an article on decreasing logflat badness:

/tuning-math/message/14339

Here's a previous article on it, including
the preposterous ennbuster temperament:

/tuning-math/message/15630

and earlier:

/tuning-math/message/6413

Here are integer sequence with it and other strong
7-limit systems in it:

http://www.research.att.com/~njas/sequences/A117555
http://www.research.att.com/~njas/sequences/A117556

2401/2400 periodicity:

/tuning-math/message/1869

--- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@> wrote:
> >
> > George and I have been referring to it as a "tina",
> > pronounced "teena".
>
>
> OK, then "tina" it is!
>
> http://tonalsoft.com/enc/t/tina.aspx

That's good Monz. Except that "first suggested by" thing could be left
out. It was first suggested to _me_ by George Secor in email in
September 2004, but we wanted to see what others might come up with
independently (George's maths might have been wrong).

A search of tuning-math revealed that, in a sense, Paul Hahn (the
consistency master) suggested it long ago.
/tuning-math/message/13711

But even if we limit it to _public_ suggestions, specifically for use
as a _measurement_unit_, then Gene suggested it here before you did
/tuning-math/message/16490

-- Dave Keenan

--- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:
> 103169-edo

Monz,

If we're talking about the pitch discriminating abilities of Homo
Sapiens here, then any random numbered EDO in the vicinity would be
just as good, since in absolute terms you're talking 0.01 cents.

-- Dave K

--- In tuning-math@yahoogroups.com, "Andreas Sparschuh"
<a_sparschuh@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@> wrote:
> >and we would likewise need the tina to be an integral division
> > of the apotome, not the octave.
> sorry but from:
> http://www.research.att.com/~njas/sequences/A028507
> can be concluded that none EDA meets ever an integral EDO exactly.

Sure. I was well aware of that. But for Sagittal design purposes we
needed an EDA, not an EDO or an ED-anything else. We'll be using
809-EDA. But thanks for your efforts.

-- Dave Keenan

--- In tuning-math@yahoogroups.com, I wrote:
> But even if we limit it to _public_ suggestions, specifically for use
> as a _measurement_unit_, then Gene suggested it here before you did
> /tuning-math/message/16490

And apparently Erv Wilson before that, although somewhat cryptically.
http://anaphoria.com/sieve.PDF
Thanks Kraig (on MMM).

As Gene said (on tuning) it's no more special than a number of others,
except for some very specific requirements in one aspect of Saqgittal
design.

-- Dave K

Hi Dave,

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "monz" <monz@> wrote:
> > 103169-edo
>
> Monz,
>
> If we're talking about the pitch discriminating abilities
> of Homo Sapiens here, then any random numbered EDO in the
> vicinity would be just as good, since in absolute terms
> you're talking 0.01 cents.

Yes, i realize that. The point i'm making is that its
*relative* error (i.e., percent of a degree) is extremely
low for prime-factors 3, 5, and 7 -- thus using units of
103169-edo is essentially the same as 7-limit JI, but
with the advantage of being logarithmically addable
integer values.

Anyway, that was just something i noticed while checking
out the error of various EDOs. It's not a tuning i'm
particularly interested in. I think 8539-edo is the
holy grail i've been searching for, and 311-edo or even
53-edo as backups if i don't want such big numbers.

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

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

> Yes, i realize that. The point i'm making is that its
> *relative* error (i.e., percent of a degree) is extremely
> low for prime-factors 3, 5, and 7 -- thus using units of
> 103169-edo is essentially the same as 7-limit JI, but
> with the advantage of being logarithmically addable
> integer values.

I've actually used it for programming purposes
when floating point arithmetic was causing
problems.

>Anyway, that was just something i noticed while checking
>out the error of various EDOs. It's not a tuning i'm
>particularly interested in. I think 8539-edo is the
>holy grail i've been searching for, and 311-edo or even
>53-edo as backups if i don't want such big numbers.

311 is hard to beat in the consistency dept. It's the
smallest ET to be 41-limit consistent, and no ET up to
12000 is consistent at a higher limit.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:

> 311 is hard to beat in the consistency dept. It's the
> smallest ET to be 41-limit consistent, and no ET up to
> 12000 is consistent at a higher limit.

It's pretty dramatic--311 is holding the fort as
the smallest consistent et, and suddenly when you
reach 43 there is a quantum leap, and 17461 is
the smallest. Sadly, it's not that interesting--
it's barely consistent from 23 to 45. The sequence
of 43-consistent edos goes 17461, 20203, 20567...,
and 20203 is of course very interesting. With a 3
error of about a quarter of a percent, and a very,
very strong 21-limit, it seems it might interest
the same people who are fond of 8539. It's also
consistent through 45. 20567, which is consistent
through the 57 limit, is slated for the smallest
spot when we get above 45.

I think I may do an integer sequence for this.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:

> The sequence
> of 43-consistent edos goes 17461, 20203, 20567...,
> and 20203 is of course very interesting. With a 3
> error of about a quarter of a percent, and a very,
> very strong 21-limit, it seems it might interest
> the same people who are fond of 8539.

Here's a side-by-side of the percentage error of
8539 and 20203 over the q-diamond for each q to 59.
You can see, I think, the strong family likeness.

q 8539 20203

3: .520634 .259793
5: 5.599776 8.669899
7: 5.967721 8.669899
9: 5.967721 8.669899
11: 14.258930 10.300867
13: 14.258930 10.300867
15: 14.779564 10.300867
17: 24.139011 11.479045
19: 24.139011 13.222768
21: 24.139011 13.222768
23: 39.104877 48.958356
25: 39.104877 48.958356
27: 39.104877 48.958356
29: 60.417462 48.958356
31: 60.417462 48.958356
33: 60.417462 48.958356
35: 60.417462 48.958356
37: 77.742829 48.958356
39: 77.742829 48.958356
41: 77.742829 48.958356
43: 77.742829 48.958356
45: 77.742829 48.958356
47: 77.742829 83.637222
49: 77.742829 83.637222
51: 77.742829 83.637222
53: 77.742829 83.637222
55: 77.742829 83.637222
57: 77.742829 83.637222
59: 77.742829 83.637222

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <genewardsmith@> wrote:
>
> > The sequence
> > of 43-consistent edos goes 17461, 20203, 20567...,
> > and 20203 is of course very interesting. With a 3
> > error of about a quarter of a percent, and a very,
> > very strong 21-limit, it seems it might interest
> > the same people who are fond of 8539.

It looks good. But for Sagittal we want 5^6 (maybe even 5^7)
consistency more than we want 29 or greater prime consistency (and
it's a little on the large side too).

-- Dave Keenan

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:
>
> ... for Sagittal design purposes we
> needed an EDA, ... We'll be using
> 809-EDA.
>
How about 18051-EDA for gaining better 3-limit approximation of the
PC, or at least 63-EDA ( approx. = ~ 665 EDO ) ?

PC-approximation quality in the EDAs: 63, 809, 819=63*13 and 18051:

63 * ln(3^12/2^19)/ln(3^7/2^11) = 13.0006648... ~13deg of 63-EDA

809 * ln(3^12/2^19)/ln(3^7/2^11) = 166.945045... ~167deg of 809-EDA

819 * ln(3^12/2^19)/ln(3^7/2^11) = 169.008642... ~13^2deg of 63*13EDA
yields no better rating in quality than already 63-EDA

but finally:

18 051*ln(3^12/2^19)/ln(3^7/2^11) = ~ 3 724.999 9989...deg 18051-EDA

works even well, when high precision demanded,
with the factorization of 3725 = 149 * 5^2

c.f:
/tuning-math/message/16529

A.S.

--- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:
>
> Hi Dave,
>
>
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@> wrote:
> >
> > --- In tuning-math@yahoogroups.com, "monz" <monz@> wrote:
> > > Thru the 41-limit, 8539 is only weak for 37 --
> > > all the other primes look very good. It's 809-EDA.
> >
> > Thanks Monz and Gene!
> >
> > The almost 35-limit consistency of 8539-EDO/809-EDA looks
> > like making it a winner for our Sagittal problem, although
> > for a MIDI standard it would have to be 11664-EDO/1105-EDA.
> > Approximating one by the other is subliminal anyway, so I
> > don't see that it matters.
>
>
> But hmm ... i found something interesting. Even tho
> 8539-edo does not divide well by 12, it *does* give a
> pretty good approximation to the 1/11-comma meantone 5th,
> which is nearly the same as the 12-edo 5th: 4981 degrees
> of 5839 with only +8.25% error.
>
> One 12-edo semitone maps to 712 degrees of 8539, with
> a whopping -41.67% error.
>
> But in the meantone mapping, the chromatic-semitone/apotome
> (3^7) is 711 degrees and the diatonic-semitone/limma (3^-5)
> is 712 degrees.

Actually, it turns out that 8539-edo gives a pretty good
approximation of the meantone 5ths of 12-edo, 31-edo, and
55-edo, which are probably the three most important EDO
meantones.

EDO ....... % error

12-edo ... + 08.3333
31-edo ... + 12.9032
55-edo ... + 14.5455

It's not as good for any of the other important EDO meantones.

And possibly its second-biggest disadvantage (other than the
fact that 8539 is prime) is that its error for the 53-edo 5th
is 49%.

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