Yahoo Tuning Groups Ultimate Backup tuning 29 in a fifth?

Hi tuners,

yesterday I made an experiment by taking two pure fifths and splitting each
of them into different intervals -- i.e. one became 4:5:6, the other was
6:7:9. When I took that as a single 4:5:6:7:9 chord and I started
subtracting one-step intervals again and again, after a while, I found two
small intervals which I could temper out. So the four intervals used were
1296/1225, 25/24, 3136/3125, and 420175/419904. This means that one mapping
for the 3/2 was "2 7 2 0" and another was "5 3 0 2". When I subtracted these
to get the 1/1, I got a mapping of "3 -4 -2 2". And when I then subtracted
multiples of the fifth's mappings to get the required amounts of tempering
for the large and small intervals, I got "29 0 -6 14" and "0 29 10 -4". So
it turns out that's actually 29 equal steps to a fifth, where the large
interval has a size of 4 steps and the small interval size is 3 steps. I
don't have a chance to try it out but it seems it must work pretty well
because the two commas I tempered out are really small ... Anyone who has
something to say about 29-ED3/2? :-D

Petr

🔗Andreas Sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, Petr Paï¿½ï¿½zek <p.parizek@...> wrote:

>When I took that as a single 4:5:6:7:9 chord ....

Hi Petr,
or expand that into George's HTT-29 approximation:
/tuning/topicId_77885.html#78000
that represents....

> ....an "extended JI" style:
> one with very thickly saturated sonorities
> like 4:5:6:7:9:11:13:15...

> I got "29 0 -6 14" and "0 29 10 -4".
> So it turns out that's actually 29 equal steps to a fifth,...

> Anyone who has something to say about 29-ED3/2? :-D
Here i can say about 29-EDO,
that it approximates well my

septenarian29.scl

scale that contains the desired exact JI interpolation:
"
That fits exactly to hepatonic C-major JI,
but meets also as well the harmonic-overtone series:

F :G :A- :A# :C :C# :D+ :E- :F' == 11*(8:9:10:11:12:13:14:15:16)

over the fundamental F and also an 5th below the partials over C:

C :D :E- :E+ :F+ :G :G# :A+ :B- :C' == 33*(8:9:10:11:12:13:14:15:16)

out of the harmonic overtone-series scale:
http://upload.wikimedia.org/wikipedia/de/b/b2/Barocktrompetenskala.png

bye
A.S.

🔗Tony Taylor <leopold_plumtree@...>

Isn't that basically 50-edo stretched out?

--- On Tue, 5/5/09, Petr Pařízek <p.parizek@...> wrote:

From: Petr Pařízek <p.parizek@...>
Subject: [tuning] 29 in a fifth?
To: "Tuning List" <tuning@yahoogroups.com>
Date: Tuesday, May 5, 2009, 10:02 AM

Hi tuners,

Petr

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🔗Petr Pařízek <p.parizek@...>

Tony Taylor wrote:

> Isn't that basically 50-edo stretched out?

Well, quite heavily stretched :-) ... Octaves don’t work this way.

Petr

--- On Tue, 5/5/09, Petr Pařízek <p.parizek@...> wrote:

From: Petr Pařízek <p.parizek@chello..cz>
Subject: [tuning] 29 in a fifth?
To: "Tuning List" <tuning@yahoogroups.com>
Date: Tuesday, May 5, 2009, 10:02 AM

Hi tuners,

Petr

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🔗Andreas Sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, Petr Paøízek <p.parizek@...> wrote/asked:
> ... Anyone who has
> something to say about 29-EDO ?
>
Hi Petr,

Considering the sequence of gaining better 3-limit approximations
http://www.research.att.com/~njas/sequences/A060528
yields:
" A list of equal temperaments (equal divisions of the octave) whose nearest scale steps are closer and closer approximations to the ratios of two tones of musical harmony: the perfect 4th, 4/3 and its complement the perfect 5th, 3/2:

1, 2, 3, 5, 7, 12, 29, 41, 53, ..... "

therefore 29-EDO is located in the gap just inbetween the EDOs

12-EDO http://en.wikipedia.org/wiki/Equal_temperament
and
41-EDO http://en.wikipedia.org/wiki/41_equal_temperament

Alternative:
The so called 'Schismic-temperaments'
deliver another view:
http://x31eq.com/schismic.htm
"The pattern of singly positive scales is 5+12n.
That gives the set 5, 17, 29, 41, 53, 65, 77, ..."
http://x31eq.com/schv12.htm
"It does still work for a 29 note schismic scale..."

hope that helps
bye
A.S.

🔗Petr Parízek <p.parizek@...>

🔗Andreas Sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, Petr Parï¿½zek <p.parizek@...> wrote:
>
> But anyway, I didn't say "29-EDO",
> I actually said "29-ED3/2",
> which should have meant
> dividing the pure fifth into 29 equal steps.
>
Sorry Petr,
that i confused that both different ones.

~702Cents/29=~24.2Cents, thats almost an PC.
take that 50 times as rule of the thumb rough estimate:

50*702/29 = ~1210.3Cents

an almost 1/2 comma overstechted octave.

Or more precisely by more accurate calculation:

(3/2)^(50/29) = ~2.01189643... yields ~1210.26724...Cents

Here some corrsponding links:
http://en.wikipedia.org/wiki/Pseudo-octave
http://www.mmk.ei.tum.de/persons/ter/top/octstretch.html
http://eamusic.dartmouth.edu/~book/MATCpages/applethtml/ch1_octave_stretch.html
http://en.wikipedia.org/wiki/Piano_acoustics
Hence that works only limited on pianos due to
http://upload.wikimedia.org/wikipedia/commons/a/ae/Railsback2.png
barely well for the octaves #2 and #6
but rather badly in the range inbetween,
producing for the middle octaves
somewhat ugly discordances,
that sounds out of tune
like cheap ringing-bell dissonaces,
almost an Helmholz "roughness"
http://dactyl.som.ohio-state.edu/Music829B/roughness.html
http://staff.science.uva.nl/~ahoningh/publicaties/measures_of_consonance.pdf
http://books.google.de/books?id=-3eEuKXS4HcC&pg=PA337&lpg=PA337&dq=helmholtz+roughness&source=bl&ots=UJQME2Uib2&sig=GSoCOvC2ZeBcsMr3cb7ZXf58TSo&hl=de&ei=zTkDSsvJKNLt_Abdv8SiBw&sa=X&oi=book_result&ct=result&resnum=5

kind regards
A.S.

🔗Tony <leopold_plumtree@...>

It's a stretch on any level. I wasn't referring to the octave stretch; simply that in both cases you have 29 equal divisions to differently-sized fifths.

--- In tuning@yahoogroups.com, Petr PaÅÃzek <p.parizek@...> wrote:
>
> Tony Taylor wrote:
>
> > Isn't that basically 50-edo stretched out?
>
> Well, quite heavily stretched :-) ... Octaves donât work this way.
>
> Petr
>
>
>
>
>
>
>
> --- On Tue, 5/5/09, Petr PaÅÃzek <p.parizek@...> wrote:
>
>
> From: Petr PaÅÃzek <p.parizek@...>
> Subject: [tuning] 29 in a fifth?
> To: "Tuning List" <tuning@yahoogroups.com>
> Date: Tuesday, May 5, 2009, 10:02 AM
>
> Hi tuners,
>
> yesterday I made an experiment by taking two pure fifths and splitting each
> of them into different intervals -- i.e. one became 4:5:6, the other was
> 6:7:9. When I took that as a single 4:5:6:7:9 chord and I started
> subtracting one-step intervals again and again, after a while, I found two
> small intervals which I could temper out. So the four intervals used were
> 1296/1225, 25/24, 3136/3125, and 420175/419904. This means that one mapping
> for the 3/2 was "2 7 2 0" and another was "5 3 0 2". When I subtracted these
> to get the 1/1, I got a mapping of "3 -4 -2 2". And when I then subtracted
> multiples of the fifth's mappings to get the required amounts of tempering
> for the large and small intervals, I got "29 0 -6 14" and "0 29 10 -4". So
> it turns out that's actually 29 equal steps to a fifth, where the large
> interval has a size of 4 steps and the small interval size is 3 steps. I
> don't have a chance to try it out but it seems it must work pretty well
> because the two commas I tempered out are really small ... Anyone who has
> something to say about 29-ED3/2? :-D
>
> Petr
>
>
>
>
>
> ------------------------------------
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