A new well-temperament

Hey,

I made what I think is an elegant discovery, and phoned Margo
Schulter, who was equally excited!!!

I always wanted a rationally-based 12-note well-temperament,
starting from three rational thirds filling an octave, and then
tempering them each to four fifths, so as to have a more-or-less
traditional well-temperament bearing.

My discovery: 5/4 * 24/19 * 19/15 = 2 !!!!!! (cool, huh?)

There it is, an elegant chain of three rational thirds, the largest
being only sightly larger than pythagorean, yet smaller than A-flat
to C in 1/6 comma meantone, thus allowing for a large amount of
Baroque music to be performed reasonably well in this scheme.
Scarlatti, I think, would really do well in this tuning.

I think this is cool, because it is an elegant set of rational
thirds, but it also is usable for an awful lot of common practive
music, *and*, it sort of sums up the development of Western Art
music tuning from the Renaissance to now: 1/4 comma meantone in the
C-G-D-A-E fifths, a neo-Neidhardt type sound in the E-B-F#-C#-G#
fifths, and finally, slightly larger than pure neo-pythagorean
(looking backwards *and* fowards, that is!) Ab-Eb-Bb-F-C fifths.

Tune it up, and let me know your subjective thoughts !

! akj_temperament.scl
!
temperament based on 5/4, 24/19, and 19/15 filling the octave
12
!
89.64400
193.15700
293.06500
386.31400
497.68700
588.53400
696.57800
790.75400
889.73500
995.37600
1087.42400
2/1

Cheers,
Aaron.

--- In tuning@yahoogroups.com, "akjmicro" <akjmicro@c...> wrote:
> Hey,
>
> I made what I think is an elegant discovery, and phoned Margo
> Schulter, who was equally excited!!!
>
> I always wanted a rationally-based 12-note well-temperament,
> starting from three rational thirds filling an octave, and then
> tempering them each to four fifths, so as to have a more-or-less
> traditional well-temperament bearing.
>
> My discovery: 5/4 * 24/19 * 19/15 = 2 !!!!!! (cool, huh?)

It's a 19-limit augmented triad in its own right, but note also that
(19/15)/(24/19) = 361/360 and (14/11)/(19/15) = 210/209. Tempering by
one or both of these would make sense, but of course the result would
no longer be rationally based by my thinking, though it seems I don't
know what you mean by "rationally based".

> Tune it up, and let me know your subjective thoughts !

One thought is that it isn't what I'd call rationally based; in fact
you've got 12 different sizes of major third, extending all the way
from 5/4 to 19/15. Scala tells us it is a constant structure and
strictly proper, but does not call it a well-temperament. That seems
ungracious, since the circle of fifths is pretty even. Certainly it
could be used as a well-temperament.

--- In tuning@yahoogroups.com, "akjmicro" <akjmicro@c...> wrote:
> Hey,
>
> I made what I think is an elegant discovery, and phoned Margo
> Schulter, who was equally excited!!!
>
> I always wanted a rationally-based 12-note well-temperament,
> starting from three rational thirds filling an octave, and then
> tempering them each to four fifths, so as to have a more-or-less
> traditional well-temperament bearing.
>
> My discovery: 5/4 * 24/19 * 19/15 = 2 !!!!!! (cool, huh?)

Yes, very interesting!

> There it is, an elegant chain of three rational thirds, the largest
> being only sightly larger than pythagorean, yet smaller than A-flat
> to C in 1/6 comma meantone, thus allowing for a large amount of
> Baroque music to be performed reasonably well in this scheme.
> Scarlatti, I think, would really do well in this tuning.
>
> I think this is cool, because it is an elegant set of rational
> thirds, but it also is usable for an awful lot of common practive
> music, *and*, it sort of sums up the development of Western Art
> music tuning from the Renaissance to now: 1/4 comma meantone in the
> C-G-D-A-E fifths, a neo-Neidhardt type sound in the E-B-F#-C#-G#
> fifths, and finally, slightly larger than pure neo-pythagorean
> (looking backwards *and* fowards, that is!) Ab-Eb-Bb-F-C fifths.
>
> Tune it up, and let me know your subjective thoughts !
>
> ! akj_temperament.scl
> !
> temperament based on 5/4, 24/19, and 19/15 filling the octave
> 12
> !
> 89.64400
> 193.15700
> 293.06500
> 386.31400
> 497.68700
> 588.53400
> 696.57800
> 790.75400
> 889.73500
> 995.37600
> 1087.42400
> 2/1

Just analyzing it on a spreadsheet I had already prepared, I can see
that it has a very nice contrast between the common and remote keys.

Strictly speaking, it's not really a well-temperament, since it has
some fifths wider than just, but they're only very slightly wider.
(The total error in the 24 major and minor triads of a well-
temperament should be no greater than the theoretical minimum, which
dictates that no fifth should be wider than just, nor any major 3rd
be narrower than just.)

One flaw I find is that consonance is skewed toward the flat keys,
i.e., the F major triad has less total error than G major, Bb less
than D, Eb less than A, etc., and your two most dissonant major
triads are Db and Ab (with F#/Gb not among them). Many well-
temperaments (including Werckmeiser III) also have this flaw.

But overall, it's very similar to my #2 (well-temperament), which
you'll find here (near the end of the message):

/tuning/topicId_38919.html#38970

I'm not sure whether making adjustments to your tuning per my
comments would require eliminating the rational major 3rds. It might
be worth a try.

--George

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@y...> wrote:

> But overall, it's very similar to my #2 (well-temperament), which
> you'll find here (near the end of the message):
>
> /tuning/topicId_38919.html#38970

Scala sorts it out as similar to the Prinz well-temperament of 1808.

> I'm not sure whether making adjustments to your tuning per my
> comments would require eliminating the rational major 3rds. It might
> be worth a try.

I tried going in the opposite direction; inspired by Aaron I found the
Fokker blocks for {81/80, 361/360, 513/512}. It turns out there are 72
of these, which is too many to list, so below I just give the one with
the lowest average Tenney height. Scala wants to compare this to
"Agricola's Monochord, Rudimenta musices (1539)." It's got one flat
40/27 fifth of dubious utility, but the rest are excellent. Major
thirds range from 5/4 to 19/15, as with Aaron's well-temperament.

! akj19_12.scl
Fokker block from 81/80, 361/360 and 513/512
12
!
19/18
9/8
19/16
19/15
4/3
38/27
3/2
19/12
27/16
16/9
19/10
2

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

> I tried going in the opposite direction; inspired by Aaron I found the
> Fokker blocks for {81/80, 361/360, 513/512}.

I might add that (19/15)/(81/64) = 1216/1215,
(81/84)/(24/19) = 513/512, (513/512)*(1216/1215) = 361/360,
(513/512)/(1216/1215) = 32768/32805. These are the most important of
the {2,3,5,19}-commas under 5 cents in size.

From: "Gene Ward Smith" <gwsmith@s>
> --- In tuning@yahoogroups.com, "akjmicro" <akjmicro@c...> wrote:
> > Hey,
> >
> > I made what I think is an elegant discovery, and phoned Margo
> > Schulter, who was equally excited!!!
> >
> > I always wanted a rationally-based 12-note well-temperament,
> > starting from three rational thirds filling an octave, and then
> > tempering them each to four fifths, so as to have a more-or-less
> > traditional well-temperament bearing.
> >
> > My discovery: 5/4 * 24/19 * 19/15 = 2 !!!!!! (cool, huh?)
>
> It's a 19-limit augmented triad in its own right, but note also that
> (19/15)/(24/19) = 361/360 and (14/11)/(19/15) = 210/209. Tempering by
> one or both of these would make sense, but of course the result would
> no longer be rationally based by my thinking, though it seems I don't
> know what you mean by "rationally based".
>
> > Tune it up, and let me know your subjective thoughts !
>
> One thought is that it isn't what I'd call rationally based; in fact
> you've got 12 different sizes of major third, extending all the way
> from 5/4 to 19/15. Scala tells us it is a constant structure and
> strictly proper, but does not call it a well-temperament. That seems
> ungracious, since the circle of fifths is pretty even. Certainly it
> could be used as a well-temperament.

Wanna know why it's not a well-temperament by definition? Well, this tuning
has fifths that are about 0.3 cents larger than the pure 3/2. As far as I
know, well-temperaments are made to distribute the pythag. comma flattening
in more than one fifth. Note that I say "flattening". So if some fifths in
his temperament are larger than 3/2, then there is some tempering in the
opposite direction. A well-temperament may have the fifths pure or slightly
flatter but a sharper fifth is normally not expected to occur in a
well-temperament as it's not so useful in terms of distributing the comma.
Yet there's another thing I'd like to say here. Of course, I don't want to
demote someone's invention but, in my personal view at least, it seems
rather unuseful to me to combine lots of primes with lots of tempering. I
think that it's enough to either start with a single prime and do the
tempering where needed (like the Baroque well-temperaments that use the
prime of 3 as the main element, if we exclude 2 for octave equivalence) or
make it all rational (like some of the recently posted tunings). I'm not
saying I don't like it, I'm just suggesting that I probably won't find such
a semi-rational well-temperament useful for my own musical purposes.
Petr

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@w...> wrote:

> Wanna know why it's not a well-temperament by definition? Well, this
tuning
> has fifths that are about 0.3 cents larger than the pure 3/2. As far
as I
> know, well-temperaments are made to distribute the pythag. comma
flattening
> in more than one fifth. Note that I say "flattening".

A temperament ordinare has some fifths which are sharp; it seems a
little pedantic not to call this a well-temperament, but maybe the
best thing would be for Scala to come up with criteria for classifying
things as temperaments ordinaire.

So if some fifths in
> his temperament are larger than 3/2, then there is some tempering in the
> opposite direction. A well-temperament may have the fifths pure or
slightly
> flatter but a sharper fifth is normally not expected to occur in a
> well-temperament as it's not so useful in terms of distributing the
comma.

I don't buy the theory that it's not useful, but then I've concocted
some extreme examples of the opposite method, with sharp fifths. I
think circulating temperaments with sharp fifths, even quite sharp as
with my grail and bifrost temperaments, are actually quite useful.

> Yet there's another thing I'd like to say here. Of course, I don't
want to
> demote someone's invention but, in my personal view at least, it seems
> rather unuseful to me to combine lots of primes with lots of
tempering. I
> think that it's enough to either start with a single prime and do the
> tempering where needed (like the Baroque well-temperaments that use the
> prime of 3 as the main element, if we exclude 2 for octave
equivalence) or
> make it all rational (like some of the recently posted tunings).

I think it can be quite useful. My bifrost temperament manages to
ciculate while having four pure fifths and three pure major thirds,
plus three nearly pure 14/11 thirds.

I'm not
> saying I don't like it, I'm just suggesting that I probably won't
find such
> a semi-rational well-temperament useful for my own musical purpose.

Have you tried any?

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

> I think circulating temperaments with sharp fifths,
> even quite sharp as with my grail and bifrost temperaments,
> are actually quite useful.

and my Mahler 7th MIDI sounds damn good in bifrost!

(i have CDs available, if anyone's interested ... it should
be purchased with the book i wrote to go along with it.)

http://tonalsoft.com/monzo/mahler/mahler7th.htm

-monz

My Outlook Express has gone somewhat mad so I'm gonna mark my newest words
with series of open and close parens.

From: "Gene Ward Smith" <gwsmith@s>
--- In tuning@yahoogroups.com, Petr Par�zek <p.parizek@w...> wrote:

> Wanna know why it's not a well-temperament by definition? Well, this
tuning
> has fifths that are about 0.3 cents larger than the pure 3/2. As far
as I
> know, well-temperaments are made to distribute the pythag. comma
flattening
> in more than one fifth. Note that I say "flattening".

A temperament ordinare has some fifths which are sharp; it seems a
little pedantic not to call this a well-temperament, but maybe the
best thing would be for Scala to come up with criteria for classifying
things as temperaments ordinaire.

((((((((((((((((
Well, if this is the case, then I agree. I didn't know about "temperament
ordinaire".
))))))))))))))))

So if some fifths in
> his temperament are larger than 3/2, then there is some tempering in the
> opposite direction. A well-temperament may have the fifths pure or
slightly
> flatter but a sharper fifth is normally not expected to occur in a
> well-temperament as it's not so useful in terms of distributing the
comma.

I don't buy the theory that it's not useful, but then I've concocted
some extreme examples of the opposite method, with sharp fifths. I
think circulating temperaments with sharp fifths, even quite sharp as
with my grail and bifrost temperaments, are actually quite useful.

((((((((((((((((
I was inexact. They are, of course, useful in the result as a whole. I was
referring to the original perspective of tempering(i.e. distributing the
comma).
))))))))))))))))

> Yet there's another thing I'd like to say here. Of course, I don't
want to
> demote someone's invention but, in my personal view at least, it seems
> rather unuseful to me to combine lots of primes with lots of
tempering. I
> think that it's enough to either start with a single prime and do the
> tempering where needed (like the Baroque well-temperaments that use the
> prime of 3 as the main element, if we exclude 2 for octave
equivalence) or
> make it all rational (like some of the recently posted tunings).

I think it can be quite useful. My bifrost temperament manages to
ciculate while having four pure fifths and three pure major thirds,
plus three nearly pure 14/11 thirds.

((((((((((((((((
Is this a rational scale, or are there some irrational intervals in it?
))))))))))))))))

I'm not
> saying I don't like it, I'm just suggesting that I probably won't
find such
> a semi-rational well-temperament useful for my own musical purpose.

Have you tried any?

((((((((((((((((
I have, of course. And, interestingly, they were my own scales. Again,
another of my inexact expressions. A scale like this is certainly useful as
a whole. But if someone finds a different scale which possesses more or less
equally good sonic qualities but is tuned in a less complex way, he probably
goes away from the previous one (at least as far as my experience can
serve).
))))))))))))))))

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@w...> wrote:

> I think it can be quite useful. My bifrost temperament manages to
> ciculate while having four pure fifths and three pure major thirds,
> plus three nearly pure 14/11 thirds.
>
> ((((((((((((((((
> Is this a rational scale, or are there some irrational intervals in it?
> ))))))))))))))))

It's got some rational intervals, but mostly it is irrational; the
scale is

45/64*5^(1/4), 1/2*5^(1/2), 16/45*5^(3/4), 5/4,
2/5*5^(3/4), 15/16*5^(1/4), 5^(1/4), 1/2*10^(1/2),
1/2*5^(3/4), 8/15*5^(3/4), 5/4*5^(1/4), 2

It has six 5^(1/4) 1/4-comma meantone fifths, followed by two pure 3/2
fifths on either side. The remaining two fifths are of the size needed
to make up the seven octaves, sqrt(2048/2025 * sqrt(5)). In 1578-equal
(a strong 7 and 11 limit temperament) it would be 6 fifths of 916
steps, with two on each side of 923 steps, and then two more of 929
steps. This means you could figure out how to notate it in Sagittal;
in case someone had the strange idea that they wanted to do this. In
1578 equal, 6125/4096 is 916 steps, an interesting 7-limit version of
the 1/4-comma meantone fifth, and 385/256 is 929 steps; you could make
an 11-limit version of bifrost using these but I don't see much point
in it unless you really did want to notate.

If anyone can think of a use for the fact that a 1/4-comma meantone
fifth is almost exactly (5/4)^3/(8/7)^2 = (35/32)^2 (5/4) it would be
interesting to hear it.