The problem with 12TET and scales that approximate it

One thing proves painfully obvious in the below messages by Andreas
and Claudio:

Certain intervals are painfully out of tune in 12TET...there always
seems to be 2-4 "very sour intervals" in every key. 13 cents off is
so far off it might as well be a different note altogether.

********************************************************
Which begs the question is there a scale designed to be between
about 0-5 cents from a "perfect interval" for all possible intervals?
************************************************************

Indeed 12-TET seems to do a lousy job at this...either the intervals
seem more perfect than they have to be to be "close enough" or
blatantly sour.

Just diatonic, if I have it right, preserves some intervals 100%
perfect at the expense of having others sound a bit more sour.

Isn't there a way to compromise and make it so all intervals are
"fairly in-tune"...rather than the usual either "near-perfectly
in-tune" or "obviously out of tune (IE over 7 cents difference from
perfect intervals)"?

-Michael
-------------------------------------------------------------

From: Andreas Sparschuh <a_sparschuh@...>
Subject: [tuning] Re: What is the Most "Out-Of-Tune" or
"Dissonant" Interval in Equal Temperament
To: tuning@yahoogroups.com
Date: Thursday, January 22, 2009, 10:15 AM

--- In tuning@yahoogroups. com, "Claudio Di Veroli" <dvc@...> wrote:
> My answer...:
>
> "The MAJOR THIRD:
1200Cents * ln(5/4)/ln(2) = ~386.313714. ...Cents

>and then the
>MINOR SIXTH!":
1200Cents * ln(8/5)/ln(2) = ~813.686286. ..Cents

Agreed, due to ~14Cents off, out of tune.

Dear Claudio,

Quest:
How about to add here also some other 5-limit intervals
that are even worser out of tune:
Minor-3rd:

1200Cents * ln(6/5)/ln(2) = ~315.641287. ..Cents

and the MAJOR-6th:

1200Cents * ln(5/3)/ln(2) = ~884.358713. ..Cents

both with @ even ~16Cents out of tune versus 12-ET, so too?

bye
A.S.

--- In tuning@yahoogroups.com, "djtrancendance" <djtrancendance@...>
wrote:
>
> One thing proves painfully obvious in the below messages by
> Andreas and Claudio:
>
> Certain intervals are painfully out of tune in 12TET...there always
> seems to be 2-4 "very sour intervals" in every key. 13 cents off is
> so far off it might as well be a different note altogether.
>
> ********************************************************
> Which begs the question is there a scale designed to be between
> about 0-5 cents from a "perfect interval" for all possible intervals?
> ************************************************************
> ....
http://mto.societymusictheory.org/issues/mto.93.0.3/mto.93.0.3.lindley7.gif
http://en.wikipedia.org/wiki/53_equal_temperament

"Mercator's Comma is of such small value to begin with (~3.615 cents),
but 53 equal temperament flattens each fifth by only 1 / 53 of that
comma. "

~3,615...Cents / 53 = ~0,0682...Cents = insiginificant ~(1/15)Cents

"However, unlike most Pythagorean forms of tuning, 53-TET contains
additional intervals that are very close to just intonation. For
instance, the interval of 17 steps is also a major third, but only 1.4
cents narrower than the very pure just interval 5/4. 53-TET is very
good as an approximation to any interval in 5-limit just intonation."

bye
A.S.

Andreas,

Does it make sense to map a 53 TET to a "normal" 12-TET keyboard?

Thanks,

chris

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...>
wrote/aked:
>
> Does it make sense to map a 53 TET to a "normal" 12-TET keyboard?
>
http://www.math.tu-dresden.de/~mutabor/
does so, but only in German documentation.

sorry
A.S.

Thanks - I looked at the site - it looks like a programming language that
modifies a stream of midi data?

On Fri, Jan 23, 2009 at 2:32 PM, Andreas Sparschuh <a_sparschuh@...>wrote:

> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Chris Vaisvil
> <chrisvaisvil@...>
> wrote/aked:
>
> >
> > Does it make sense to map a 53 TET to a "normal" 12-TET keyboard?
> >
> http://www.math.tu-dresden.de/~mutabor/<http://www.math.tu-dresden.de/%7Emutabor/>
> does so, but only in German documentation.
>
> sorry
> A.S.
>
>
>

On Jan 23, 2009, at 10:20 PM, Chris Vaisvil wrote:
> Thanks - I looked at the site - it looks like a programming > language that modifies a stream of midi data?

Exactly so. It is a programming language for creating tunings which change on the fly. Programs analyse the MIDI input and create MIDI output with added pitchbend messages (spread over multiple channels, the usual trick).

Best
Torsten

>
>
> On Fri, Jan 23, 2009 at 2:32 PM, Andreas Sparschuh > <a_sparschuh@...> wrote:
> --- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...>
> wrote/aked:
>
>
> >
> > Does it make sense to map a 53 TET to a "normal" 12-TET keyboard?
> >
> http://www.math.tu-dresden.de/~mutabor/
> does so, but only in German documentation.
>
> sorry
> A.S.
>
>
>
>