small ratio fundamentalism

If I may chime in here on the topic of ratio-fundamentalism. I am a bit of a ratio fundamentalist myself, in certain ways, but you can't simply say that music that is not based on (small number) ratios isn't music. Because even if it isn't based on small number ratios, it still (in terms of physics) results in ratios of some sort. Every ratio, small-fractional or not produces a difference tone, or some sort of relationship. That is music. I might add that even if it seems that "harmonicity" is the ideal goal of music, there are plenty of other avenues that have never been explored fully. Like the fact that in traditional Gamelan, the traditional tuners of the instruments never tuned the brass instruments exactly with themselves, believing that the "soul" was in the difference tone (or "beating") Those low low difference tones have been shown to have a psychoactive effect. So it kind of matters not only how well you tune something to itself, but to
what degree you intentionally mistune it, because the difference tone is itself a note.

Sincerely,
Mike Nolley

Mike Nolley wrote:
... in traditional Gamelan, the traditional tuners of the instruments never
tuned the brass instruments exactly with themselves, believing that the
"soul" was in the difference tone (or "beating") Those low low difference
tones have been shown to have a psychoactive effect.

Not only in Indonesia but in Italy as well, for centuries, sometimes
mistuning on purpose away from pure ratios was considered an interesting
effect.
As some members of the list surely know, ever since Renaissance times and
earlier, organs in Italy have had a stop called "Fiffaro", mistuned on
purpose, beating with the 8' Principal slowly, with speed proportional to
frequency.
The stop was by default off.
When set on, it would beat with the main Principal stop: the vibrato caused
by its mistuning was considered particularly beautiful for melodies.

Claudio

http://temper.braybaroque.ie

--Every ratio, small-fractional or not produces a difference tone, or
some sort of relationship. --That is music. I might add that even if it
seems that "harmonicity" is the ideal goal of --music, there are plenty
of other avenues that have never been explored fully.
   Exactly.  Which is a huge reason I am somewhat frustrated that so many people here who are so good with tuning math refuse to try/apply their skills toward such unexplored avenues.  Furthermore, thinking of harmonicity as simply being "related to the harmonic series" only really is a gross over-simplification of the phenomena of making overtones of several root tones match in a way that's easily interpreted by the mind/"relaxing".

--So it kind of matters not only how well you tune something to itself, but to
--what degree you intentionally mistune it, because the difference tone is itself a note.
    Exactly...I think it is quite evident this happens in the PHI tuning (PHI^y/2^x)...and some of you tuning wizards could probably find several more tunings that take advantage of this (if only you'd stop over-simplifying everything to small numbered ratios/ rational fractions.  I just hope people will give the general idea of breaking past small ratio fundamentalism a chance. :-)

-Michael

--- On Sat, 3/28/09, Mike Nolley <miken277@...> wrote:

From: Mike Nolley <miken277@...>
Subject: [tuning] small ratio fundamentalism
To: tuning@yahoogroups.com
Date: Saturday, March 28, 2009, 4:05 PM

If I may chime in here on the topic of ratio-fundamentalis m. I am a bit of a ratio fundamentalist myself, in certain ways, but you can't simply say that music that is not based on (small number) ratios isn't music. Because even if it isn't based on small number ratios, it still (in terms of physics) results in ratios of some sort. Every ratio, small-fractional or not produces a difference tone, or some sort of relationship. That is music. I might add that even if it seems that "harmonicity" is the ideal goal of music, there are plenty of other avenues that have never been explored fully. Like the fact that in traditional Gamelan, the traditional tuners of the instruments never tuned the brass instruments exactly with themselves, believing that the "soul" was in the difference tone (or "beating") Those low low difference tones have been shown to have a psychoactive effect. So it kind of matters not only how well you tune something to itself, but
to

what degree you intentionally mistune it, because the difference tone is itself a note.

Sincerely,

Mike Nolley

--- In tuning@yahoogroups.com, Mike Nolley <miken277@...> wrote:
>
>
> If I may chime in here on the topic of ratio-fundamentalism. I am a bit of a ratio fundamentalist myself, in certain ways, but you can't simply say that music that is not based on (small number) ratios isn't music. Because even if it isn't based on small number ratios, it still (in terms of physics) results in ratios of some sort. Every ratio, small-fractional or not produces a difference tone, or some sort of relationship. That is music. I might add that even if it seems that "harmonicity" is the ideal goal of music, there are plenty of other avenues that have never been explored fully. Like the fact that in traditional Gamelan, the traditional tuners of the instruments never tuned the brass instruments exactly with themselves, believing that the "soul" was in the difference tone (or "beating") Those low low difference tones have been shown to have a psychoactive effect. So it kind of matters not only how well you tune something to itself, but to
> what degree you intentionally mistune it, because the difference tone is itself a note.
>
> Sincerely,
> Mike Nolley
>
Hi Mike,

I recall that in certain parts of Africa (I forget where exactly) they tend to traditionally tune their melodic instruments sharp to our ears. I had an album by the jazz trumpeter Don Cherry somewhere which was played in this tuning, and with the theory behind it on the sleeve. However, wouldn't you agree that in order to deliberately 'mistune' we must first assume that we know what 'tuning' is to begin with? Or perhaps the Gamelan etc is not out of tune to their ears at all?
At any rate, some on this list tend to think of the small-numbered ratios as being somewhat restrictive (I'm guessing that they'll take the opportunity to jump on your choice of term 'fundamentalism'). But what I think they don't consider is that these are just the building blocks, that the beauty in standard modern tuning systems like 12 edo, and meantone etc...is how they solve the problem (in different ways) of equalising to all keys by 'mistuning' each one by the same degree. So 'mistuning' is not unique to world music. In fact the 12 edo intervals are 2^p/12, p = 0, 1, 2,...,11, which are not small-numbered ratios at all. Yet it would be absurd for anyone to claim that this isn't music. And as I have pointed out before on this list, there are certain ratios b/w large harmonics which far better approximate the 12 edo intervals than the small ones to which they were originally meant to approximate: eg, tempered maj 3 = 2^1/3 = 1.259921..., 5/4 = 1.25, but 645/512 = 1.25976562. So my point is that in the real world where there is a limit to what we can distinguish (the shizma), the 12 edo system might still come from the harmonic series after all.

Regards

-Rick

--that the beauty in standard modern tuning systems like 12 edo, and
meantone etc...is ---how they solve the problem (in different ways) of
equalising to all keys by 'mistuning' each --one by the same degree
  I agree...the beauty of 12TET as a tuning is that it makes of keys "out of tune by a predictable factor" IE the third in the key of C and D#, for example, are the same degree out of tune vs. 5-limit JI.

  However, there's a consistent underlying problem to which letting notes "beat beautifully" seems to be the solution.

  Certain notes, like the 3rd in 12TET, are consistently more out of tune vs. notes like the 5th.  For example, the 12TET 5th is virtually dead-on the perfect 5th, while the 3rd is off by about 13 cents (and this, of course, holds for any key in 12TET)!

  What would be really nice...is tuning where the 2nd, 3rd, 4th, 5th...in almost any key are ALL about the same degree mis-tuned vs. pure. 

    Diatonic JI has this problem too   For example (taken from wikipedia):
*********************************************************************

Note
A
B
C
D
E
F
G
A

Ratio
1/1
9/8
6/5
4/3
3/2
8/5
9/5
2/1

Cents
0.00
203.91
315.64
498.04
701.96
813.69
1017.60
1200.00

Step
 
T
s
t
T
s
T
t
 

Cent step
 
203.91
111.73
182.40
203.91
111.73
203.91
182.40
 

The major thirds are correct, and two minor thirds are right, but B-D is not.
If we compare with the scale above, we see that six notes can be
lined up, but one note, D, has changed its value. It is evidently not
possible to get all six chords mentioned correct.
There are other possibilities; instead of lowering D, we can raise A. But this breaks something else.**************************************************************

   The ultimate solution, in my opinion, is to have all possible intervals " 'broken' to about the same degree from perfect"...rather than have some interval perfect and others far from perfect.

-Michael

--- On Sun, 3/29/09, rick_ballan <rick_ballan@...> wrote:

From: rick_ballan <rick_ballan@...>
Subject: [tuning] Re: small ratio fundamentalism
To: tuning@yahoogroups.com
Date: Sunday, March 29, 2009, 1:51 AM

--- In tuning@yahoogroups. com, Mike Nolley <miken277@.. .> wrote:

>

>

> If I may chime in here on the topic of ratio-fundamentalis m. I am a bit of a ratio fundamentalist myself, in certain ways, but you can't simply say that music that is not based on (small number) ratios isn't music. Because even if it isn't based on small number ratios, it still (in terms of physics) results in ratios of some sort. Every ratio, small-fractional or not produces a difference tone, or some sort of relationship. That is music. I might add that even if it seems that "harmonicity" is the ideal goal of music, there are plenty of other avenues that have never been explored fully. Like the fact that in traditional Gamelan, the traditional tuners of the instruments never tuned the brass instruments exactly with themselves, believing that the "soul" was in the difference tone (or "beating") Those low low difference tones have been shown to have a psychoactive effect. So it kind of matters not only how well you tune something to itself, but
to

> what degree you intentionally mistune it, because the difference tone is itself a note.

>

> Sincerely,

> Mike Nolley

>

Hi Mike,

I recall that in certain parts of Africa (I forget where exactly) they tend to traditionally tune their melodic instruments sharp to our ears. I had an album by the jazz trumpeter Don Cherry somewhere which was played in this tuning, and with the theory behind it on the sleeve. However, wouldn't you agree that in order to deliberately 'mistune' we must first assume that we know what 'tuning' is to begin with? Or perhaps the Gamelan etc is not out of tune to their ears at all?

At any rate, some on this list tend to think of the small-numbered ratios as being somewhat restrictive (I'm guessing that they'll take the opportunity to jump on your choice of term 'fundamentalism' ). But what I think they don't consider is that these are just the building blocks, that the beauty in standard modern tuning systems like 12 edo, and meantone etc...is how they solve the problem (in different ways) of equalising to all keys by 'mistuning' each one by the same degree. So 'mistuning' is not unique to world music. In fact the 12 edo intervals are 2^p/12, p = 0, 1, 2,...,11, which are not small-numbered ratios at all. Yet it would be absurd for anyone to claim that this isn't music. And as I have pointed out before on this list, there are certain ratios b/w large harmonics which far better approximate the 12 edo intervals than the small ones to which they were originally meant to approximate: eg, tempered maj 3 = 2^1/3 = 1.259921..., 5/4 =
1.25, but 645/512 = 1.25976562. So my point is that in the real world where there is a limit to what we can distinguish (the shizma), the 12 edo system might still come from the harmonic series after all.

Regards

-Rick

Another example of the problem with 12TET having some intervals that are disproportionately out of tune with others...

Try the 12TET chord
cdga-(next octave)-deab

Sound fairly pure, doesn't it?
   That's because I made a point...of avoiding use of the (disproportionately out of tune) 3rd interval. 
   Taking the other disproportionately out of tune interval (the first A in the scale, the sour 6th interval) out of the chord also makes another huge jump in degree of consonance...and reveals the problem in 12TET of having some notes far more out of tune than others.

--- On Sun, 3/29/09, Michael Sheiman <djtrancendance@...> wrote:

From: Michael Sheiman <djtrancendance@...>
Subject: Re: [tuning] Re: small ratio fundamentalism
To: tuning@yahoogroups.com
Date: Sunday, March 29, 2009, 8:15 AM

--that the beauty in standard modern tuning systems like 12 edo, and
meantone etc...is ---how they solve the problem (in different ways) of
equalising to all keys by 'mistuning' each --one by the same degree
  I agree...the beauty of 12TET as a tuning is that it makes of keys "out of tune by a predictable factor" IE the third in the key of C and D#, for example, are the same degree out of tune vs. 5-limit JI.

  However, there's a consistent underlying problem to which letting notes "beat beautifully" seems to be the solution.

  Certain notes, like the 3rd in 12TET, are consistently more out of tune vs. notes like the 5th.  For example, the 12TET 5th is virtually dead-on the perfect 5th, while the 3rd is off by about 13 cents (and this, of course, holds for any key in 12TET)!

  What would be really nice...is tuning where the 2nd, 3rd, 4th, 5th...in almost any key are ALL about the same degree mis-tuned vs. pure. 

    Diatonic JI has this problem too   For example (taken from
wikipedia):
************ ********* ********* ********* ********* ********* ********* ***

Note
A
B
C
D
E
F
G
A

Ratio
1/1
9/8
6/5
4/3
3/2
8/5
9/5
2/1

Cents
0.00
203.91
315.64
498.04
701.96
813.69
1017.60
1200.00

Step
 
T
s
t
T
s
T
t
 

Cent step
 
203.91
111.73
182.40
203.91
111.73
203.91
182.40
 

The major thirds are correct, and two minor thirds are right, but B-D is not.
If we compare with the scale above, we see that six notes can be
lined up, but one note, D, has changed its value. It is evidently not
possible to get all six chords mentioned correct.
There are other possibilities; instead of lowering D, we can raise A. But this breaks something else.************ ********* ********* ********* ********* ********* *****

   The ultimate solution, in my opinion, is to have all possible intervals " 'broken' to about the same degree from perfect"...rather than have some interval perfect and others far from perfect.

-Michael

--- On Sun, 3/29/09, rick_ballan <rick_ballan@ yahoo.com. au> wrote:

From: rick_ballan <rick_ballan@ yahoo.com. au>
Subject: [tuning] Re: small ratio fundamentalism
To: tuning@yahoogroups. com
Date: Sunday, March 29, 2009, 1:51 AM

--- In tuning@yahoogroups. com, Mike Nolley <miken277@.. .> wrote:

>

>

> If I may chime in here on the topic of ratio-fundamentalis m. I am a bit of a ratio fundamentalist myself, in certain ways, but you can't simply say that music that is not based on (small number) ratios isn't music. Because even if it isn't based on small number ratios, it still (in terms of physics) results in ratios of some sort. Every ratio, small-fractional or not produces a difference tone, or some sort of relationship. That is music. I might add that even if it seems that "harmonicity" is the ideal goal of music, there are plenty of other avenues that have never been explored fully. Like the fact that in traditional Gamelan, the traditional tuners of the instruments never tuned the brass instruments exactly with themselves, believing that the "soul" was in the difference tone (or "beating") Those low low difference tones have been shown to have a psychoactive effect. So it kind of matters not only how well you tune something to itself,
but to

> what degree you intentionally mistune it, because the difference tone is itself a note.

>

> Sincerely,

> Mike Nolley

>

Hi Mike,

I recall that in certain parts of Africa (I forget where exactly) they tend to traditionally tune their melodic instruments sharp to our ears. I had an album by the jazz trumpeter Don Cherry somewhere which was played in this tuning, and with the theory behind it on the sleeve. However, wouldn't you agree that in order to deliberately 'mistune' we must first assume that we know what 'tuning' is to begin with? Or perhaps the Gamelan etc is not out of tune to their ears at all?

At any rate, some on this list tend to think of the small-numbered ratios as being somewhat restrictive (I'm guessing that they'll take the opportunity to jump on your choice of term 'fundamentalism' ). But what I think they don't consider is that these are just the building blocks, that the beauty in standard modern tuning systems like 12 edo, and meantone etc...is how they solve the problem (in different ways) of equalising to all keys by 'mistuning' each one by the same degree. So 'mistuning' is not unique to world music. In fact the 12 edo intervals are 2^p/12, p = 0, 1, 2,...,11, which are not small-numbered ratios at all. Yet it would be absurd for anyone to claim that this isn't music. And as I have pointed out before on this list, there are certain ratios b/w large harmonics which far better approximate the 12 edo intervals than the small ones to which they were originally meant to approximate: eg, tempered maj 3 = 2^1/3 = 1.259921..., 5/4 =
1.25, but 645/512 = 1.25976562. So my point is that in the real world where there is a limit to what we can distinguish (the shizma), the 12 edo system might still come from the harmonic series after all.

Regards

-Rick

The reason that I have mainly published very diatonic LucyTunings is because our target audience is the general listener punters, and anything too exotic is likely to make them switch off.
It is a matter of following that fine line between the familiar and the novel, so that the music is acceptably consonant and yet also contains "subtly different" intervals.

Maybe I should put up some of the more outrageous tunings to feed the "hardcore tunaniks";-)

On 29 Mar 2009, at 16:15, Michael Sheiman wrote:

> --that the beauty in standard modern tuning systems like 12 edo, and > meantone etc...is ---how they solve the problem (in different ways) > of equalising to all keys by 'mistuning' each --one by the same degree
> I agree...the beauty of 12TET as a tuning is that it makes of keys > "out of tune by a predictable factor" IE the third in the key of C > and D#, for example, are the same degree out of tune vs. 5-limit JI.
>
> However, there's a consistent underlying problem to which letting > notes "beat beautifully" seems to be the solution.
>
> Certain notes, like the 3rd in 12TET, are consistently more out of > tune vs. notes like the 5th. For example, the 12TET 5th is > virtually dead-on the perfect 5th, while the 3rd is off by about 13 > cents (and this, of course, holds for any key in 12TET)!
>
> What would be really nice...is tuning where the 2nd, 3rd, 4th, > 5th...in almost any key are ALL about the same degree mis-tuned vs. > pure.
>
> Diatonic JI has this problem too For example (taken from > wikipedia):
> *********************************************************************
>
> Note A B C D E F G A
> Ratio 1/1 9/8 6/5 4/3 3/2 8/5 9/5 2/1
> Cents 0.00 203.91 315.64 498.04 701.96 813.69 1017.60 1200.00
> Step T s t T s T t
> Cent step 203.91 111.73 182.40 203.91 111.73 203.91 182.40
> The major thirds are correct, and two minor thirds are right, but B-> D is not.
>
> If we compare with the scale above, we see that six notes can be > lined up, but one note, D, has changed its value. It is evidently > not possible to get all six chords mentioned correct.
>
> There are other possibilities; instead of lowering D, we can raise > A. But this breaks something else.
>
> **************************************************************
>
> The ultimate solution, in my opinion, is to have all possible > intervals " 'broken' to about the same degree from perfect"...rather > than have some interval perfect and others far from perfect.
>
> -Michael
>
>
>
Charles Lucy
lucy@...

- Promoting global harmony through LucyTuning -

for information on LucyTuning go to:
http://www.lucytune.com

For LucyTuned Lullabies go to:
http://www.lullabies.co.uk

Hi Michael,

What would be really nice...is tuning where the 2nd, 3rd, 4th, 5th...in
almost any key are ALL about the same degree mis-tuned vs. pure.

The problem of finding an "average temperament" was stated in early Baroque
times and the subject of published controversy in the 18th century,
especially in France, and centred in regular meantone variants.

You can only balance one interval vs another.
Thus if you include THREE types of intervals: fifths, major thirds and minor
thirds, the problem has NO solution.

With TWO types of intervals only, there are solutions.
Since the most consonant intervals within the octave are fifths and major
thirds, that was the subject of the historical controversies.

Some theoretician advocated an "homogeneous" meantone, with fifths tempered
by 1/5th Syntonic comma: fifths and major thirds were identically tempered
by 4.3 Cent each.

Others argued that mistuning in fifths was almost twice more noticeable
(because of beat frequencies due to the different ratios) and advocated an
"attenuated" meantone, with fifths tempered by 1/6th Syntonic comma: now the
major thirds were tempered by 7.2 Cent each, twice as much as the fifths by
3.6 Cent each.

Other theoreticians discussed intermediate solutions.

This hair-splitting ceased soon as Equal Temperament was adopted(except in
English-speaking countries) in the 2nd half of the 18th century (NOT in the
19th century) for its advantage in ensemble tuning, mainly.

Kind regards

Claudio

http://temper.braybaroque.ie/

(PS: All the above-and much more-is explained in an accessible way, yet with
full technical detail, in my U.T. book, pp.73-78 and elsewhere.)

--Thus if you include
THREE types of intervals: fifths, major thirds and minor thirds, the --problem
has NO solution.
  This is very informative.  The one thing that bugs me about this method is it seems to shout "some intervals matter much more than others, particularly the 5th" while my idea is "all intervals should be considered of equal importance."

   How about making the average of the major thirds and minor 2nds (the two approximate interval sizes used in the major scale) in tune starting from any note in the scale the same degree "out of tune"...wouldn't that indirectly take care of most combinations of tones in the major scale or is that also impossible in a 7-tone major scale?

  Note, when I say "error"...it can even be something so high as around 10 cents...it just has to be a consistent error, that's my main point...no need to reduce the error to something very tiny (like 6 cents or less) as you've shown people have tried to do in historic mean-tone.

-Michael

--- On Sun, 3/29/09, Claudio Di Veroli <dvc@...> wrote:

From: Claudio Di Veroli <dvc@...>
Subject: RE: [tuning] Re: small ratio fundamentalism
To: tuning@yahoogroups.com
Date: Sunday, March 29, 2009, 8:51 AM

Hi Michael,

 
  What would be really nice...is tuning where
the 2nd, 3rd, 4th, 5th...in almost any key are ALL about the same degree
mis-tuned vs. pure. 

The problem of finding
an "average temperament" was stated in early Baroque times and the subject of
published controversy in the 18th century, especially in France, and centred
in regular meantone variants.
 
You can only balance
one interval vs another.
Thus if you include
THREE types of intervals: fifths, major thirds and minor thirds, the problem
has NO solution.
 
With TWO types of
intervals only, there are solutions.
Since the most
consonant intervals within the octave are fifths and major thirds, that was
the subject of the historical controversies.
 
Some theoretician
advocated an "homogeneous" meantone, with fifths tempered by 1/5th
Syntonic comma: fifths and major thirds were identically tempered by 4.3 Cent
each.
 
Others argued that
mistuning in fifths was almost twice more noticeable (because of beat
frequencies due to the different ratios) and advocated an "attenuated"
meantone, with fifths tempered by 1/6th Syntonic comma: now the major thirds
were tempered by 7.2 Cent each, twice as much as the fifths by 3.6 Cent
each.
 
Other theoreticians
discussed intermediate solutions.
 
This hair-splitting
ceased soon as Equal Temperament was adopted(except in English-speaking
countries) in the 2nd half of the 18th century (NOT in the 19th
century) for its advantage in ensemble tuning,
mainly.
 
Kind
regards
 
Claudio
 
http://temper. braybaroque. ie/
 
(PS: All the above-and much
more-is explained in an accessible way, yet with full technical
detail, in my U.T. book, pp.73-78 and
elsewhere.)
 
 
 
 
 
 
 
 

Dear Claudio,
Although I find this list generally completely dominated with 'noise' these days, I still look occasionally - here I have a few remarks/questions, see below.

--- In tuning@yahoogroups.com, "Claudio Di Veroli" <dvc@...> wrote:

> What would be really nice...is tuning where the 2nd, 3rd, 4th, 5th...in
> almost any key are ALL about the same degree mis-tuned vs. pure.
>
> The problem of finding an "average temperament" was stated in early Baroque
> times and the subject of published controversy in the 18th century,
> especially in France, and centred in regular meantone variants.

I only know of Sauveur early in the 18th century as a theorist looking at his 'systems' - what sources are there later? (I remember vaguely some theorist advocating or describing 1/7 comma...)

> (...)
>
> With TWO types of intervals only, there are solutions.
> Since the most consonant intervals within the octave are fifths and major
> thirds, that was the subject of the historical controversies.

One can have optimum solutions for three types of interval if the problem is to minimise the largest absolute deviation - then immediately two intervals have the same deviation.
You actually pointed out that the major 6th 5:3 (not 5:4) is the most consonant interval within the octave after the fifth and its inversion. But what happens if you require that the deviations of 5:3 and 3:2 be equal? Just quarter-comma again! The pure 5:4 comes almost for free.
Although when one has three just intervals and one is the sum of the other two, it is obvious that when one is pure the deviations of the other two are equally large.
Thus if 7:4 is pure in any temperament, 3:2 and 7:6 must be equally mistuned.

> Others argued that mistuning in fifths was almost twice more noticeable
> (because of beat frequencies due to the different ratios) and advocated an
> "attenuated" meantone, with fifths tempered by 1/6th Syntonic comma: now the
> major thirds were tempered by 7.2 Cent each, twice as much as the fifths by
> 3.6 Cent each.

So I would be particularly interested which theorists you refer to here on 1/6 comma as keyboard tuning... I also seem to remember there was some question of what the sources were for the discussion of 'meantone circularized' on your p.76 (first ed.).

Unfortunately as I am moving house soon I don't have time to look at the recent revised version.
Best, Thomas

--Thus if you include THREE types of intervals: fifths, major thirds and
minor thirds, the --problem has NO solution.
This is very informative. The one thing that bugs me about this method is
it seems to shout "some intervals matter much more than others, particularly
the 5th" while my idea is "all intervals should be considered of equal
importance." How about making the average of the major thirds and minor
2nds (the two approximate interval sizes used in the major scale) in tune
starting from any note in the scale the same degree "out of tune"...wouldn't
that indirectly take care of most combinations of tones in the major scale
or is that also impossible in a 7-tone major scale?

If I understand you OK, it is possible but likely to mistune fifths a lot,
which is not much useful.
Please bear in mind that all the purpose of modern discussions such as the
ones in this list, is to enlarge the traditional range with new uses of JI
and meantone in instruments with either variable pitches or more than 12
pitches per octave, i.e. to address the needs of modern music with modern
sound production system.

If you are going to talk about traditional music on traditional 12-note
tunings, it is pointless to keep investigating as all the continuum (and
uni-dimensional) sets of JI and meantone solutions has been investigated.The
preferred results for meantone are those shown in my post.

Kind regards,

Claudio

_____

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf Of
Michael Sheiman
Sent: 29 March 2009 17:54
To: tuning@yahoogroups.com
Subject: RE: [tuning] Re: small ratio fundamentalism

--Thus if you include THREE types of intervals: fifths, major thirds and
minor thirds, the --problem has NO solution.
This is very informative. The one thing that bugs me about this method is
it seems to shout "some intervals matter much more than others, particularly
the 5th" while my idea is "all intervals should be considered of equal
importance."

How about making the average of the major thirds and minor 2nds (the two
approximate interval sizes used in the major scale) in tune starting from
any note in the scale the same degree "out of tune"...wouldn't that
indirectly take care of most combinations of tones in the major scale or is
that also impossible in a 7-tone major scale?

Note, when I say "error"...it can even be something so high as around 10
cents...it just has to be a consistent error, that's my main point...no need
to reduce the error to something very tiny (like 6 cents or less) as you've
shown people have tried to do in historic mean-tone.

-Michael

--- On Sun, 3/29/09, Claudio Di Veroli <dvc@...> wrote:

From: Claudio Di Veroli <dvc@...>
Subject: RE: [tuning] Re: small ratio fundamentalism
To: tuning@yahoogroups.com
Date: Sunday, March 29, 2009, 8:51 AM

Hi Michael,

What would be really nice...is tuning where the 2nd, 3rd, 4th, 5th...in
almost any key are ALL about the same degree mis-tuned vs. pure.

The problem of finding an "average temperament" was stated in early Baroque
times and the subject of published controversy in the 18th century,
especially in France, and centred in regular meantone variants.

You can only balance one interval vs another.
Thus if you include THREE types of intervals: fifths, major thirds and minor
thirds, the problem has NO solution.

With TWO types of intervals only, there are solutions.
Since the most consonant intervals within the octave are fifths and major
thirds, that was the subject of the historical controversies.

Some theoretician advocated an "homogeneous" meantone, with fifths tempered
by 1/5th Syntonic comma: fifths and major thirds were identically tempered
by 4.3 Cent each.

Others argued that mistuning in fifths was almost twice more noticeable
(because of beat frequencies due to the different ratios) and advocated an
"attenuated" meantone, with fifths tempered by 1/6th Syntonic comma: now the
major thirds were tempered by 7.2 Cent each, twice as much as the fifths by
3.6 Cent each.

Other theoreticians discussed intermediate solutions.

This hair-splitting ceased soon as Equal Temperament was adopted(except in
English-speaking countries) in the 2nd half of the 18th century (NOT in the
19th century) for its advantage in ensemble tuning, mainly.

Kind regards

Claudio

http://temper. braybaroque. ie/ <http://temper.braybaroque.ie/>

(PS: All the above-and much more-is explained in an accessible way, yet with
full technical detail, in my U.T. book, pp.73-78 and elsewhere.)

Hi Tom!

(I am taking the liberty to slightly alter the order of your sentences for
easier answers)
> Dear Claudio,
> > The problem of finding an "average temperament" was stated in early
Baroque
> > times and the subject of published controversy in the 18th century,
> > especially in France, and centred in regular meantone variants.
>I only know of Sauveur early in the 18th century as a theorist looking at
his 'systems' - what sources are there later? (I remember vaguely some
theorist advocating or describing 1/7 comma...)

>> Others argued that mistuning in fifths was almost twice more noticeable
>> (because of beat frequencies due to the different ratios) and advocated
an
>> "attenuated" meantone, with fifths tempered by 1/6th Syntonic comma: now
the
>> major thirds were tempered by 7.2 Cent each, twice as much as the fifths
by
>> 3.6 Cent each.
>So I would be particularly interested which theorists you refer to here on
1/6 comma as keyboard tuning...

Precisely one of the additional writings I quote in my 2nd edition (pages 82
and 149) is:
Barbieri, Patrizio. “Il ‘migliore’ sistema musicale temperato: ‘querelles’
fra Estève, Romieu e altri accademici francesi (c.1740-60)” in L’Organo
XXVII pp. 31-81. Bologna 1992.
The controversies scrutinised by Barbieri were historically of no
consequence. They obviously happened at a time when, outside
English-speaking countries and old church organs, meantone was already dead
in practical use.

> You actually pointed out that the major 6th 5:3 (not 5:4) is the most
consonant interval within the octave after the fifth and its inversion.
I wrote "perhaps": actually I find 5:3 and 5:4 equally consonant.
Mathematically it is difficult to argue one way or the other.
Historically however, as you know, most theoreticians disregarded 5:3 (see
my p.25) for not being superparticular.
Only with Helmholtz beat/consonance setup it became obvious that
superparticular was not such an important factor in consonance but merely a
consequence of ratios among very small integers. I am not surprised
therefore that Helmholtz was the first to notice that 5:3 was as consonant
as 5:4.
But tuners knew this in practice surely much before: I am sure there must be
some pre-Helmholtz source for this.
(Perhaps Montal in one of his many tuning checks? Should check ...)

> But what happens if you require that the deviations of 5:3 and 3:2 be
equal?
> Just quarter-comma again! The pure 5:4 comes almost for free.
This is obvious from my chart in p.81, but I did not put it in words there.
It is indeed very good argument to keep to standard pure-major-thirds 1/4
S.c. meantone!
Was this ever mentioned in Baroque sources? Not aware of any such
occurrence.
Their preference for the pure major thirds seem to arise from the
traditional two reasons: a) as a way to render JI practical and b) as a very
easy to tune system whereby the pure thirds guaranteed the precision.

>I also seem to remember there was some question of what the sources were
for the discussion of 'meantone circularized' on your p.76 (first ed.).
The matters were indeed muddy in mid-18th century; 1750 is a time of great
tuning confusion in the Western musical world:
1- We have the above querelles, where they believed-wrongly-that a regular
meantone with suitably larger fifths was the best way to get rid of wolves
(modifying meantone was the solution, which as we know is a completely
different tuning system).
2- We have a large amount of mutually-incompatible circular systems (from
very unequal such as the ordinaire such as quite equal as some Neidhardt
proposals).
3- Some performers of non-keyboard instruments would follow the keyboards
circular systems, others would follow meantone variants, others were
returning to Pythagorean!
4- Still others were suggesting Equal Temperament: d'Alembert for one put it
in writing that a good reason for E.T. was simply to put an end to the
different methods, that were causing mayhem in ensemble intonation.

>Unfortunately as I am moving house soon I don't have time to look at the
recent revised version.
You will like the new version!

Kind regards and happy removal!

Claudio

http://temper.braybaroque.ie/

--Please bear in mind that all the purpose of
modern discussions such as the ones in this --list, is to enlarge the
traditional range with new uses of JI and meantone in instruments
--with either variable pitches or more than 12 pitches per octave, i.e. to
address the needs --of modern music with modern sound production
system.
   It sure as heck doesn't say that in the group description...  When you say "variable pitches" do you basically mean things like "adaptive JI"? 
  ---more than 12 pitches per octave
   That would probably rule out a lot of the "macro-tonal" discussion that goes on here concerning things like 5 note scales (or anything less than 7-note scales).  Considering I believe we can learn a lot from such "macro-tonal" experiments, I see the idea of omitting them entirely as a bit tempting toward ignorance (and, at times, counter-productive to learning).

--If you are going to talk about traditional music
on traditional 12-note tunings, it is pointless --to keep investigating as
all the continuum (and uni-dimensional) sets of JI and meantone --solutions
has been investigated.

   Who said the solution to making approximately the same error between all intervals of the 7-tone scale and "perfect" dyads within them would involve mean-tone? 

   Mean-tone (if I have it right) involves basically stacking either 5ths (or something rather near 5ths) together to form a circle...and I'm guessing the best solution for keeping a more-or-less constant dyadic error from pure intervals (for all possible dyads in a 7-tone scale) would not be mean-tone but rather a scale with many different interval sizes. 

-Michael

--- On Sun, 3/29/09, Claudio Di Veroli <dvc@braybaroque.ie> wrote:

From: Claudio Di Veroli <dvc@...>
Subject: RE: [tuning] Re: small ratio fundamentalism
To: tuning@yahoogroups.com
Date: Sunday, March 29, 2009, 11:12 AM

--Thus if
you include THREE types of intervals: fifths, major
thirds and minor
thirds, the --problem has NO solution.
  This is
very informative.  The one thing that bugs me about this method is it
seems to shout "some intervals matter much more than others, particularly
the 5th" while my idea is "all intervals should be considered of equal
importance."  
How about making the average of the major thirds and minor 2nds (the two
approximate interval sizes used in the major scale) in tune starting from
any note in the scale the same degree "out of tune"...wouldn' t that
indirectly take care of most combinations of tones in the major scale or
is that also impossible in a 7-tone major
scale?
 
If I understand you OK, it is possible but likely
to mistune fifths a lot, which is not much
useful.
Please bear in mind that all the purpose of
modern discussions such as the ones in this list, is to enlarge the
traditional range with new uses of JI and meantone in instruments
with either variable pitches or more than 12 pitches per octave, i.e. to
address the needs of modern music with modern sound production
system.
 
If you are going to talk about traditional music
on traditional 12-note tunings, it is pointless to keep investigating as
all the continuum (and uni-dimensional) sets of JI and meantone solutions
has been investigated. The preferred results for meantone are those shown
in my post.
 
Kind regards,
 
Claudio
 
 
 
 
 

From: tuning@yahoogroups. com
[mailto:tuning@ yahoogroups. com] On Behalf Of Michael
Sheiman
Sent: 29 March 2009 17:54
To:
tuning@yahoogroups. com
Subject: RE: [tuning] Re: small ratio
fundamentalism

--Thus if you
include THREE types of intervals: fifths, major
thirds and minor
thirds, the --problem has NO solution.
  This
is very informative.  The one thing that bugs me about this method
is it seems to shout "some intervals matter much more than others,
particularly the 5th" while my idea is "all intervals should be
considered of equal importance."

  
How about making the average of the major thirds and minor 2nds (the two
approximate interval sizes used in the major scale) in tune starting
from any note in the scale the same degree "out of tune"...wouldn' t
that indirectly take care of most combinations of tones in the major
scale or is that also impossible in a 7-tone major scale?

 
Note, when I say "error"...it can even be something so high as around 10
cents...it just has to be a consistent error, that's my main point...no
need to reduce the error to something very tiny (like 6 cents or less)
as you've shown people have tried to do in historic
mean-tone.

-Michael

---
On Sun, 3/29/09, Claudio Di Veroli
<dvc@braybaroque. ie> wrote:

From:
Claudio Di Veroli <dvc@braybaroque. ie>
Subject: RE:
[tuning] Re: small ratio fundamentalism
To:
tuning@yahoogroups. com
Date: Sunday, March 29, 2009, 8:51
AM

Hi Michael,

 
  What would be really nice...is
tuning where the 2nd, 3rd, 4th, 5th...in almost any key are ALL
about the same degree mis-tuned vs. pure. 

The problem
of finding an "average temperament" was stated in early Baroque
times and the subject of published controversy in the 18th century,
especially in France, and centred in regular meantone
variants.
 
You can only
balance one interval vs another.
Thus if you
include THREE types of intervals: fifths, major thirds and minor
thirds, the problem has NO solution.
 
With TWO
types of intervals only, there are solutions.
Since the
most consonant intervals within the octave are fifths and major
thirds, that was the subject of the historical
controversies.
 
Some
theoretician advocated an "homogeneous" meantone, with fifths
tempered by 1/5th Syntonic comma: fifths and major thirds were
identically tempered by 4.3 Cent each.
 
Others
argued that mistuning in fifths was almost twice more noticeable
(because of beat frequencies due to the different ratios) and
advocated an "attenuated" meantone, with fifths tempered by 1/6th
Syntonic comma: now the major thirds were tempered by 7.2 Cent each,
twice as much as the fifths by 3.6 Cent each.
 
Other
theoreticians discussed intermediate solutions.
 
This
hair-splitting ceased soon as Equal Temperament was adopted(except
in English-speaking countries) in the 2nd half of the 18th
century (NOT in the 19th century) for its advantage in ensemble
tuning, mainly.
 
Kind
regards
 
Claudio
 
http://temper. braybaroque. ie/
 
(PS: All the
above-and much more-is explained in an accessible way, yet with full
technical detail, in my U.T. book, pp.73-78 and
elsewhere.)
 
 
 
 
 
 
 
 

Dear Michael,

There are two ways to produce new ideas about factual matters:
1. by building on existing knowledge, producing new knowledge (Helmholtz).
2. sometimes geniuses even show existing knowledge wrong and fix it
(Einstein).

If you disregard existing knowledge, you are out of alternative 1.
If you do not prove it wrong, you are out of alternative 2.

Let me try and understand your last post ...

> Who said the solution to making approximately the same error between all
intervals of the 7-tone scale and "perfect" dyads within them would involve
mean-tone?
Nobody said that. Who would advocate a solution to a problem that, in the
terms you state it, has been known for centuries (and verified by modern
science) to have no solution?

> Mean-tone (if I have it right) involves basically stacking either 5ths (or
something rather near 5ths) together to form a circle
- The Circle of Fifths has been used to define scales and temperaments since
medieval times until the present times. It is used for all temperaments, not
just meantone.
- Your definition above implies that meantone forms a circle. It is not so.
Circular temperaments form circles. Meantone is regular, not circular, it
forms a spiral, not a circle.

> ...and I'm guessing the best solution for keeping a more-or-less constant
dyadic error from pure intervals (for all possible dyads in a 7-tone scale)
would not be mean-tone but rather a scale with many different interval
sizes.
You are "guessing" best solutions for a problem that, in the term you state
it, has none.

Why don't you begin by giving us an example of what you are trying to
achieve?

There are relatively few intervals in the 7-tone scale:
Within an octave they make only 21 intervals, adding another octave you will
have a few more.
Quite a few involve ratios (or their approximation) between high integer
numbers, are thus dissonant and there is no point in trying to approach
them.
Once you discard the above, you have a manageable number.
Try and find out an example of solution and show it to us.
Mind that, in order to measure "error", you have
a) to define your goal with respect to which the error is measured
b) to work with interval sizes in cents, because the ratios used in JI
studies cannot be used to gauge deviations from purity

Regards,

Claudio

There
are two ways to produce new ideas about factual matters:
1. by building on existing knowledge, producing new knowledge
(Helmholtz).
2.
sometimes geniuses even show existing knowledge wrong and fix it
(Einstein).
 
--If you
disregard existing knowledge, you are
out of alternative 1.
--If you
do not prove it wrong, you are out of alternative 2.
A) I'm not disregarding existing knowledge beyond the limit of using rational ratios to produce or "generate" scales.  My scales obey the critical band and Helmholtz and P&L's theories respectively...and so would proposed ones with a fixed "error from perfect interval" limit. 
B) I don't see a need to show existing knowledge wrong.  Already there has been research that proves diatonic JI by William Sethares that shows where the frequencies in JI are using dissonance curve with 100% no use of mean-tone generation (there's more than one way to skin a car).  The only problem I have with his algorithm is it focuses on creating the "lowest average dissonance" rather than solving the situation of "where no one note has more than a certain level of dissonance vs. the others with no regard to average
dissonance".

************************************************************************************************
> Who said the solution to making approximately the same error
between all intervals of the
> 7-tone scale and "perfect" dyads within them would
involve mean-tone? 
Nobody said that. Who would advocate a solution to a problem that, in the terms you
state it, has been known for centuries (and verified by modern science) to have
no solution?

   Well, when you said "studied for centuries", you appeared to be alluding to the art of mean tone and only using mean-tone tunings as examples to prove/disprove the idea of "controlled errors from perfect intervals". 

   I will admit I don't know the solution to a scale with the same error from perfect intervals for all intervals (that's why I'm asking you).
    But I can think of at least one other way to attempt to solve the problem that have nothing to do with mean-tone.  One is modifying Sethares' dissonance algorithm (which builds on P&L's consonance theory and has nothing to do with past mean-tone research) in such a way that it does not count the dissonance values of frequencies that are either too near or too far from pure intervals away from each other).
  The idea would be to continuously
skew all the intervals to "maximum tolerable roughness" away from perfect intervals (you'd need to throw a list of all possible perfect intervals in for this to work: all 5-limit ones would probably be a nice start).

  I don't know enough about the math behind consonance equation to implement this, but maybe William Sethares or someone more experienced with his equation might be able to program all of this into a "revised dissonance curve" where the troughs in the graph are the ideal points for an "all ratios equally far from perfect" scale.

- The Circle
of Fifths has been used to define scales and temperaments since medieval times
until the present times. It is used for all temperaments, not just
meantone.

  I know, but it sure as heck isn't the only way to make a scale.  Firstly, there are things like the circle of thirds in 10TET.  And, yes, that works in kind of the same way...but then again many (albeit lesser known) scales are not tempered whatsoever.  Ptolemy's scales, for example, were not created by anything like the "circle of fifths".

- Your
definition above implies that meantone forms a circle. It is not so.
Circular -temperaments form circles. Meantone is regular, not circular,
it forms a spiral, not a circle.

    However, the "spiral" closes on itself very very near a circle near the octave with the comma being the error...thus it is essentially, though not completely, a circle.

  BTW, the reason I asked the question to you was that I was hoping you'd know and would be open-minded enough to take a shot of traveling outside of mean-tone logic to find the answer.  Maybe not?? :-(

    The closest thing I have to an answer is the above-mentioned part about modifying Sethares' dissonance algorithm...and then looking at the graph that it generates and finding the trough points, which should point to the "correct-error-from-perfect" intervals I keep mentioning.

-Michael

djtrancendance@... wrote:

> A) I'm not disregarding existing knowledge beyond the limit of using rational > ratios to produce or "generate" scales. My scales obey the critical band and > Helmholtz and P&L's theories respectively...and so would proposed ones with a > fixed "error from perfect interval" limit. Maybe the Dunning-Kruger effect's at work here and you don't know how much existing knowledge you're disregarding.

> B) I don't see a need to show existing knowledge wrong. Already there has been > research that proves diatonic JI by William Sethares that shows where the > frequencies in JI are using dissonance curve with 100% no use of mean-tone > generation (there's more than one way to skin a car). The only problem I have > with his algorithm is it focuses on creating the "lowest average dissonance" > rather than solving the situation of "where no one note has more than a certain > level of dissonance vs. the others with no regard to average dissonance".

Try searching for "minimax temperament".

>> Who said the solution to making approximately the same error between all > intervals of the
>> 7-tone scale and "perfect" dyads within them would involve mean-tone? > Nobody said that. Who would advocate a solution to a problem that, in the terms > you state it, has been known for centuries (and verified by modern science) to > have no solution?
> > Well, when you said "studied for centuries", you appeared to be alluding to > the art of mean tone and only using mean-tone tunings as examples to > prove/disprove the idea of "controlled errors from perfect intervals". Take a major triad relative to 4:5:6. The error in the fifth is the sum of the errors in the two thirds. If the thirds have the same error, the fifth has twice as much error. If the thirds have opposite errors, the fifth has no error. That's how we know there's no solution.

> - The Circle of Fifths has been used to define scales and temperaments since > medieval times until the present times. It is used for all temperaments, not > just meantone.
> > I know, but it sure as heck isn't the only way to make a scale. Firstly, > there are things like the circle of thirds in 10TET. And, yes, that works in > kind of the same way...but then again many (albeit lesser known) scales are not > tempered whatsoever. Ptolemy's scales, for example, were not created by > anything like the "circle of fifths".

Ptolemy's scales aren't temperaments. But otherwise you're right here. You could use the circle of fifths with miracle, for example, but it wouldn't make much sense and I don't think anybody has.

Graham

--Maybe the Dunning-Kruger effect's at work here and you don't
--know how much existing knowledge you're disregarding.
    Or maybe, you're simply not interested in working past mean-tone restrictions to develop a scale and more interested in insulting my lack of knowledge in mean-tone to help you avoid doing so.  Honestly, I never EVEN said I knew the answer...just that I am looking for a solution using a method other than mean-tone, so that was rather unnecessary of you.

--Try searching for "minimax temperament" .
>>Minimax simply means "minimizing the maximum error". The lowest value
of largest error
>> is acheived there. For example, 1/4-comma meantone is
the "minimax" meantone with >>respect to the 5-limit, as all 5-limit
intervals are approximated with a maximum error of >>1/4-comma, and no
other meantone can improve on that (see the "historical meantones"
>>chart at the bottom of Joe's meantone definition page to see why).
-Tonalsoft.com

   Good info this time around.....this is actually pretty spot on for what I am trying to achieve.  I am going to look into how they try and achieve this...any ideas for links?
   But, actually, I'm surprised just how close that is to perfect...perhaps using the modified theory I mentioned from Sethares (which would arbitrarily move notes to areas a bit outside any "spiral of 5ths")...but I'm going to have to try this it as, again, it sounds pretty near perfect mathematically and I wonder why I haven't heard of minimax tuning before.

--Take a major triad relative to 4:5:6. The error in the
---fifth is the sum of the errors in the two thirds. If the
---thirds have the same error, the fifth has twice as much
---error.
  Makes sense then that the above is the lowest possible error mean-tone
tuning.

   So what, for example if the thirds have errors of DIFFERENT amounts pointing in the opposite directions (IE the third is 1/15th comma too low and the second one 1/10th comma too high, thus making the fifth 1/30th comma too high).  Again...imagine you can move notes to wherever you want arbitrarily IE no need to match any "spiral of 5ths". 

    General note:   I'm not at all saying I know a lot about mean-tone (and I realize you do).

   However, I am saying, perhaps if we knock the mean-tone restriction off we can do things such as the "canceling errors" idea I mentioned above and get a bit closer to having a lower "minimax" error.  And, then again, maybe not...but I'd at least like you to give a counter-example without using a mean-tone construct to explain why not.

-Michael

--- On Mon, 3/30/09, Graham Breed <gbreed@...> wrote:

From: Graham Breed <gbreed@...>
Subject: Re: [tuning] Re: small ratio fundamentalism
To: tuning@yahoogroups.com
Date: Monday, March 30, 2009, 3:29 PM

djtrancendance@ yahoo.com wrote:

> A) I'm not disregarding existing knowledge beyond the limit of using rational

> ratios to produce or "generate" scales. My scales obey the critical band and

> Helmholtz and P&L's theories respectively. ..and so would proposed ones with a

> fixed "error from perfect interval" limit.

Maybe the Dunning-Kruger effect's at work here and you don't

know how much existing knowledge you're disregarding.

> B) I don't see a need to show existing knowledge wrong. Already there has been

> research that proves diatonic JI by William Sethares that shows where the

> frequencies in JI are using dissonance curve with 100% no use of mean-tone

> generation (there's more than one way to skin a car). The only problem I have

> with his algorithm is it focuses on creating the "lowest average dissonance"

> rather than solving the situation of "where no one note has more than a certain

> level of dissonance vs. the others with no regard to average dissonance".

Try searching for "minimax temperament" .

>> Who said the solution to making approximately the same error between all

> intervals of the

>> 7-tone scale and "perfect" dyads within them would involve mean-tone?

> Nobody said that. Who would advocate a solution to a problem that, in the terms

> you state it, has been known for centuries (and verified by modern science) to

> have no solution?

>

> Well, when you said "studied for centuries", you appeared to be alluding to

> the art of mean tone and only using mean-tone tunings as examples to

> prove/disprove the idea of "controlled errors from perfect intervals".

Take a major triad relative to 4:5:6. The error in the

fifth is the sum of the errors in the two thirds. If the

thirds have the same error, the fifth has twice as much

error. If the thirds have opposite errors, the fifth has no

error. That's how we know there's no solution.

> - The Circle of Fifths has been used to define scales and temperaments since

> medieval times until the present times. It is used for all temperaments, not

> just meantone.

>

> I know, but it sure as heck isn't the only way to make a scale. Firstly,

> there are things like the circle of thirds in 10TET. And, yes, that works in

> kind of the same way...but then again many (albeit lesser known) scales are not

> tempered whatsoever. Ptolemy's scales, for example, were not created by

> anything like the "circle of fifths".

Ptolemy's scales aren't temperaments. But otherwise you're

right here. You could use the circle of fifths with

miracle, for example, but it wouldn't make much sense and I

don't think anybody has.

Graham

djtrancendance@... wrote:

> But, actually, I'm surprised just how close that is to perfect...perhaps > using the modified theory I mentioned from Sethares (which would arbitrarily > move notes to areas a bit outside any "spiral of 5ths")...but I'm going to have > to try this it as, again, it sounds pretty near perfect mathematically and I > wonder why I haven't heard of minimax tuning before.

Because you weren't listening.

> --Take a major triad relative to 4:5:6. The error in the
> ---fifth is the sum of the errors in the two thirds. If the
> ---thirds have the same error, the fifth has twice as much
> ---error.
> Makes sense then that the above is the lowest possible error mean-tone tuning.

No, the minimax meantone is the one where major thirds have zero error. This has been known since at least the 17th century. It has the "mean tone" the class is named after.

> So what, for example if the thirds have errors of DIFFERENT amounts pointing > in the opposite directions (IE the third is 1/15th comma too low and the second > one 1/10th comma too high, thus making the fifth 1/30th comma too high). Yes. So the errors still aren't equal.

> Again...imagine you can move notes to wherever you want arbitrarily IE no need > to match any "spiral of 5ths". > > General note: I'm not at all saying I know a lot about mean-tone (and I > realize you do).

Why bring meantone into it?

> However, I am saying, perhaps if we knock the mean-tone restriction off we > can do things such as the "canceling errors" idea I mentioned above and get a > bit closer to having a lower "minimax" error. And, then again, maybe not...but > I'd at least like you to give a counter-example without using a mean-tone > construct to explain why not.

There are plenty of temperament classes (as I call them) with a lower minimax error than meantone. A good 5-limit example is kleismatic/hanson. It was in another thread recently. The generator is 317.0 cents and represents a 6:5 minor third. 5 generators represent a 3:2 perfect fifth. 6 generators represent a 5:4 major third. A 19 note scale in cents is

68.0 135.9 203.9 271.9 317.0 385.0 452.9 520.9 588.9 634.0 702.0 769.9 837.9 905.9 951.0 1018.9 1086.9 1154.9 1200.0

This is taking us closer to just intonation. Before you said you didn't want just intonation.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

>
> There are plenty of temperament classes (as I call them)
> with a lower minimax error than meantone. A good 5-limit
> example is kleismatic/hanson. It was in another thread
> recently. The generator is 317.0 cents and represents a 6:5
> minor third. 5 generators represent a 3:2 perfect fifth. 6
> generators represent a 5:4 major third. A 19 note scale in
> cents is
>
> 68.0 135.9 203.9 271.9 317.0 385.0 452.9 520.9 588.9 634.0
> 702.0 769.9 837.9 905.9 951.0 1018.9 1086.9 1154.9 1200.0
>
> This is taking us closer to just intonation. Before you
> said you didn't want just intonation.
>
>
> Graham

That's also a subset of 53-equal. An interesting one, as it doesn't use 53-equal's quite accurate approximation of a 7/4, but most of the tetrads are still going to work very well (for reasons, IMO, you probably wouldn't agree with, but perhaps you'll agree they sound much better than they "should" in terms of approximation).

> There are plenty of temperament classes (as I call them)

> with a lower minimax error than meantone.
    Right on!...I figured this had to be the case...just didn't know enough to find what temperaments would work as solutions.

> A good 5-limit

> example is kleismatic/hanson.
> A 19 note scale in

> cents is

>

> 68.0 135.9 203.9 271.9 317.0 385.0 452.9 520.9 588.9 634.0

> 702.0 769.9 837.9 905.9 951.0 1018.9 1086.9 1154.9 1200.0
    Now how would you go about deriving a consistent 7-note scale from within those 19 notes...or is the whole idea to keep switching tones used based on the intervals (kind of like adaptive JI, but with a limited, 19-note set)?

-Michael

--- On Tue, 3/31/09, Cameron Bobro <misterbobro@yahoo.com> wrote:

From: Cameron Bobro <misterbobro@...>
Subject: [tuning] Re: small ratio fundamentalism
To: tuning@yahoogroups.com
Date: Tuesday, March 31, 2009, 4:03 AM

--- In tuning@yahoogroups. com, Graham Breed <gbreed@...> wrote:

>

> There are plenty of temperament classes (as I call them)

> with a lower minimax error than meantone. A good 5-limit

> example is kleismatic/hanson. It was in another thread

> recently. The generator is 317.0 cents and represents a 6:5

> minor third. 5 generators represent a 3:2 perfect fifth. 6

> generators represent a 5:4 major third. A 19 note scale in

> cents is

>

> 68.0 135.9 203.9 271.9 317.0 385.0 452.9 520.9 588.9 634.0

> 702.0 769.9 837.9 905.9 951.0 1018.9 1086.9 1154.9 1200.0

>

> This is taking us closer to just intonation. Before you

> said you didn't want just intonation.

>

>

> Graham

That's also a subset of 53-equal. An interesting one, as it doesn't use 53-equal's quite accurate approximation of a 7/4, but most of the tetrads are still going to work very well (for reasons, IMO, you probably wouldn't agree with, but perhaps you'll agree they sound much better than they "should" in terms of approximation) .

> But, actually, I'm surprised just how close that is to perfect...perhaps

> using the modified theory I mentioned from Sethares (which would arbitrarily

> move notes to areas a bit outside any "spiral of 5ths")...but I'm going to have

> to try this it as, again, it sounds pretty near perfect mathematically and I

> wonder why I haven't heard of minimax tuning before.
--Because you weren't listening.
    I don't recall you mentioning minimax...at least not in this thread, and I don't read every single thread on this list.  Anyhow, I know it now.

> So what, for example if the thirds have errors of DIFFERENT amounts pointing

> in the opposite directions (IE the third is 1/15th comma too low and the second

> one 1/10th comma too high, thus making the fifth 1/30th comma too high).

---Yes. So the errors still aren't equal.
    They aren't...but they are significantly closer to equal.  Hence my point of making a "minimum error" scale rather than a "no error" scale.

--Why bring meantone into it?
Simple, because that's the only solution you seem to be mentioning.  Again, I want to think of any possible solution, not just mean-tone...and, for the record, I'm not convinced the mean-tone is the ultimate answer.

>There are plenty of temperament classes (as I call them)
>with a lower minimax error than meantone. A good 5-limit
>example is kleismatic/hanson.
   Thank you!  Finally, we are starting to answer the question I was asking in the first place: how to get the smallest error away from perfect intervals period (and realizing that the error will never be zero, but hopefully approach it to a degree indistinguishable by human hearing).

--This is
taking us closer to just intonation. Before you
---said you didn't want just intonation.
My mistake. :-)
I don't want JI to become the limiting requirement either (IE if an irrational ratio gives a lower minimax in some cases, so be it :-) )...but if JI gives lower minimax than mean-tone at least we're opening another option and getting closer.

    BTW, as a side note, the ultimate test is to take a 7-note scale that has the lowest minimax value mathematically possible (without any limitations such as JI, meantone, etc. being pre-requisite) and than pit my PHI scale against it.
    Hopefully it will allow me to more easily find out where the sour spots in my scale are vs. "a realistic 'tempered' version of adaptive JI".
   BTW, I sense one issue with adaptive JI I've found with all Adaptive-JI software I have tried...if you play ALL 7 notes within the octave at once, it sounds sour. Which is
why, even in adaptive JI, you still need music theory and to pick chord selectively to "knock out the kinks" in extreme combinations of closely spaced JI intervals.

  In the end of the day the goal I have in mind for both scales is the same...to have a scale where it's virtually impossible to play combinations of notes that create chords which are dissonant...and make a scale that "resolves music theory itself" so no music theory is needed and it becomes purely an issue of emotion as to the art of playing with such scales.  :-)

-Michael

--- On Tue, 3/31/09, Graham Breed <gbreed@...> wrote:

From: Graham Breed <gbreed@...>
Subject: Re: [tuning] Re: small ratio fundamentalism
To: tuning@...m
Date: Tuesday, March 31, 2009, 12:46 AM

djtrancendance@ yahoo.com wrote:

> But, actually, I'm surprised just how close that is to perfect...perhaps

> using the modified theory I mentioned from Sethares (which would arbitrarily

> move notes to areas a bit outside any "spiral of 5ths")...but I'm going to have

> to try this it as, again, it sounds pretty near perfect mathematically and I

> wonder why I haven't heard of minimax tuning before.

Because you weren't listening.

> --Take a major triad relative to 4:5:6. The error in the

> ---fifth is the sum of the errors in the two thirds. If the

> ---thirds have the same error, the fifth has twice as much

> ---error.

> Makes sense then that the above is the lowest possible error mean-tone tuning.

No, the minimax meantone is the one where major thirds have

zero error. This has been known since at least the 17th

century. It has the "mean tone" the class is named after.

> So what, for example if the thirds have errors of DIFFERENT amounts pointing

> in the opposite directions (IE the third is 1/15th comma too low and the second

> one 1/10th comma too high, thus making the fifth 1/30th comma too high).

Yes. So the errors still aren't equal.

> Again...imagine you can move notes to wherever you want arbitrarily IE no need

> to match any "spiral of 5ths".

>

> General note: I'm not at all saying I know a lot about mean-tone (and I

> realize you do).

Why bring meantone into it?

> However, I am saying, perhaps if we knock the mean-tone restriction off we

> can do things such as the "canceling errors" idea I mentioned above and get a

> bit closer to having a lower "minimax" error. And, then again, maybe not...but

> I'd at least like you to give a counter-example without using a mean-tone

> construct to explain why not.

There are plenty of temperament classes (as I call them)

with a lower minimax error than meantone. A good 5-limit

example is kleismatic/hanson. It was in another thread

recently. The generator is 317.0 cents and represents a 6:5

minor third. 5 generators represent a 3:2 perfect fifth. 6

generators represent a 5:4 major third. A 19 note scale in

cents is

68.0 135.9 203.9 271.9 317.0 385.0 452.9 520.9 588.9 634.0

702.0 769.9 837.9 905.9 951.0 1018.9 1086.9 1154.9 1200.0

This is taking us closer to just intonation. Before you

said you didn't want just intonation.

Graham

>I'm checking with the Scala files, and neither of the 9 note

>scales named after him are a consecutive set of octaves. So

>here's what I make the chain of pure octaves as frequency

>ratios:

>1.0000 X1.0407X X1.1021X 1.1672 1.2361 X1.3090X 1.3623 X1.4427X X1.5279X
    I plugged this into my sequencer.
    Ugh...sounds significantly different and worse than my version (and I'm talking about my original irrational-numbered PHI version, not the rationally-rounded version).  And, again (note the X's)...about half of the ratio values he uses are significantly different from mine.
    Virtually all the tones in his version sound "shadowed" IE "neither consonant nor dissonant" to me...while in my version only a few of (maybe 1/3rd or so) the tones sound at all shadowed.

    My PHI scale is approximately (IE the closest numbers in the PHI tuning to the following rational values).
18/17
 9/8
 19/16
 5/4
 4/3
 7/5
 28/19
 14/9
 21/13

    Compare these two with your ears and I'm
pretty sure you'll see why I find it obvious that my version may well have the ability to work as widely used scale even though O'Connell's version did not.  Yes, although I realize my scale still needs some serious work in a few places (namely the weird gap interval between 14/9 and 21/13) there really is a major difference between the two scales...

-Michael

--- On Tue, 3/31/09, Graham Breed <gbreed@...> wrote:

From: Graham Breed <gbreed@gmail.com>
Subject: Re: [tuning] Re: small ratio fundamentalism
To: tuning@yahoogroups.com
Date: Tuesday, March 31, 2009, 12:46 AM

djtrancendance@ yahoo.com wrote:

> But, actually, I'm surprised just how close that is to perfect...perhaps

> using the modified theory I mentioned from Sethares (which would arbitrarily

> move notes to areas a bit outside any "spiral of 5ths")...but I'm going to have

> to try this it as, again, it sounds pretty near perfect mathematically and I

> wonder why I haven't heard of minimax tuning before.

Because you weren't listening.

> --Take a major triad relative to 4:5:6. The error in the

> ---fifth is the sum of the errors in the two thirds. If the

> ---thirds have the same error, the fifth has twice as much

> ---error.

> Makes sense then that the above is the lowest possible error mean-tone tuning.

No, the minimax meantone is the one where major thirds have

zero error. This has been known since at least the 17th

century. It has the "mean tone" the class is named after.

> So what, for example if the thirds have errors of DIFFERENT amounts pointing

> in the opposite directions (IE the third is 1/15th comma too low and the second

> one 1/10th comma too high, thus making the fifth 1/30th comma too high).

Yes. So the errors still aren't equal.

> Again...imagine you can move notes to wherever you want arbitrarily IE no need

> to match any "spiral of 5ths".

>

> General note: I'm not at all saying I know a lot about mean-tone (and I

> realize you do).

Why bring meantone into it?

> However, I am saying, perhaps if we knock the mean-tone restriction off we

> can do things such as the "canceling errors" idea I mentioned above and get a

> bit closer to having a lower "minimax" error. And, then again, maybe not...but

> I'd at least like you to give a counter-example without using a mean-tone

> construct to explain why not.

There are plenty of temperament classes (as I call them)

with a lower minimax error than meantone. A good 5-limit

example is kleismatic/hanson. It was in another thread

recently. The generator is 317.0 cents and represents a 6:5

minor third. 5 generators represent a 3:2 perfect fifth. 6

generators represent a 5:4 major third. A 19 note scale in

cents is

68.0 135.9 203.9 271.9 317.0 385.0 452.9 520.9 588.9 634.0

702.0 769.9 837.9 905.9 951.0 1018.9 1086.9 1154.9 1200.0

This is taking us closer to just intonation. Before you

said you didn't want just intonation.

Graham

djtrancendance@... wrote:
> >I'm checking with the Scala files, and neither of the 9 note
> >scales named after him are a consecutive set of octaves. So
> >here's what I make the chain of pure octaves as frequency
> >ratios:
> >1.0000 X1.0407X X1.1021X 1.1672 1.2361 X1.3090X 1.3623 X1.4427X X1.5279X
> I plugged this into my sequencer.
> Ugh...sounds significantly different and worse than my version (and I'm > talking about my original irrational-numbered PHI version, not the > rationally-rounded version). And, again (note the X's)...about half of the > ratio values he uses are significantly different from mine.

So what is your version?

> Virtually all the tones in his version sound "shadowed" IE "neither > consonant nor dissonant" to me...while in my version only a few of (maybe 1/3rd > or so) the tones sound at all shadowed.
> > My PHI scale is approximately (IE the closest numbers in the PHI tuning to > the following rational values).
> 18/17
> 9/8
> 19/16
> 5/4
> 4/3
> 7/5
> 28/19
> 14/9
> 21/13

Once again, how do you get 7/5 from phi?

> Compare these two with your ears and I'm pretty sure you'll see why I find > it obvious that my version may well have the ability to work as widely used > scale even though O'Connell's version did not. Yes, although I realize my scale > still needs some serious work in a few places (namely the weird gap interval > between 14/9 and 21/13) there really is a major difference between the two scales...

Have you tried O'Connell's scale with O'Connell's timbre yet? I did, several years ago.

Graham

djtrancendance@... wrote:

> >There are plenty of temperament classes (as I call them)
> >with a lower minimax error than meantone. A good 5-limit
> >example is kleismatic/hanson.
> Thank you! Finally, we are starting to answer the question I was asking in > the first place: how to get the smallest error away from perfect intervals > period (and realizing that the error will never be zero, but hopefully approach > it to a degree indistinguishable by human hearing).

There was a sub-thread about it last year, in a wider thread you contributed to:

/tuning/topicId_79615.html#79623

> --This is taking us closer to just intonation. Before you
> ---said you didn't want just intonation.
> My mistake. :-)
> I don't want JI to become the limiting requirement either (IE if an irrational > ratio gives a lower minimax in some cases, so be it :-) )...but if JI gives > lower minimax than mean-tone at least we're opening another option and getting > closer.

JI always has an error of zero. Optimizing for the minimax error will only get you closer to JI.

> BTW, as a side note, the ultimate test is to take a 7-note scale that has > the lowest minimax value mathematically possible (without any limitations such > as JI, meantone, etc. being pre-requisite) and than pit my PHI scale against it.
> Hopefully it will allow me to more easily find out where the sour spots in > my scale are vs. "a realistic 'tempered' version of adaptive JI".
> BTW, I sense one issue with adaptive JI I've found with all Adaptive-JI > software I have tried...if you play ALL 7 notes within the octave at once, it > sounds sour. Which is why, even in adaptive JI, you still need music theory and > to pick chord selectively to "knock out the kinks" in extreme combinations of > closely spaced JI intervals.

Why would adaptive JI help when you play all the notes at the same time? It can make arbitrary chords with reasonable spacing sound in tune. Getting 7 notes to the octave to sound good together is difficult some intervals have to be within a critical band, and we've talked about this before.

> In the end of the day the goal I have in mind for both scales is the same...to > have a scale where it's virtually impossible to play combinations of notes that > create chords which are dissonant...and make a scale that "resolves music theory > itself" so no music theory is needed and it becomes purely an issue of emotion > as to the art of playing with such scales. :-)

How do you plan to express emotion without dissonance?

Graham

--Once again, how do you get 7/5 from phi?

    I rounded 1.38627 (the value from PHI) up to 7/5 (that a ratio of less than 1.01 difference!).  The other rational-numbered options included 32/23 (1.3913), 29/21 = 1.38905, 25/18 = 1.388888 and 18/13 = 1.38462.  7/5 seemed to be the interval that hit close enough around that area to be relatively indistinguishable from the original tone of 1.38627 and still satisfied my ear as being in tune relative to all of the other tones of the scale.  32/23 came in as my second best option and 25/18 as my third best...according to feeling it out between these ratios >>>BY EAR<<<.  Looking back I realize 7/5 is HUYGENS' TRITONE...but I surely didn't just stick it in there because it is Huygens' Tritone, if you get what I am
saying.

   One fact behind my methods of testing: once I get within a small margin of the goal frequencies I always use my ears as the primary guide rather than math.

  NOTE: the whole area in which I was testing above was >not near< either of the 1.3623 or 1.4427 tones in O'Connell's scale: in fact it falls right smack in-between them.  Yet more evidence his scales and mine are quite different, despite our using the same tuning.
***********************************************************************************************
--Have you tried O'Connell's scale with O'Connell's timbre
--yet? I did, several years ago.
  His timbre meaning...using phi^x to define the harmonics?  No I haven't, but I will try :-): perhaps by using the same timbre I used with my scale (the plain old harmonic series) on his scale I am getting worse results: especially when thinking of Sethares' work on matching
timbre and scale I could easily believe that could be a very valid issue.

   Note: I'm not saying O'Connell's work is bad or wrong...just that I strongly believe there are enough fundamental differences between his work and my own to warrant other people's taking a second look at ways to use PHI to make scales.  His work seems to have much more advanced math involved than my own...but, as we all know, superior math doesn't automatically guarantee superior sound (though it can help). :-)

-Michael

--JI always has an error of zero. Optimizing for the minimax
--error will only get you closer to JI.
   In case this is what you were thinking: I'm not talking about adaptive JI, and I know for a fact that diatonic 7-tone JI falls short.  Here are some examples from wikipedia of the sour spots in diatonic JI (which is by no means "zero error")

"A just diatonic scale may be derived as follows. Suppose we insist that the chords F-A-C, C-E-G, and G-B-D be just major triads (then A-C-E and E-G-B are just minor triads, but D-F-A is not)."
"Another way to do it is as
follows. We can insist that the chords D-F-A, A-C-E, and E-G-B be just minor triads (then F-A-C and C-E-G are just major triads, but G-B-D is not)."

   Either way, we are making some triads perfect and others fairly obviously sour...instead of making them all "fairly near perfect".  That's what I'm going for...I realize I may have confused minimax optimization with my goal.
-------------------------------------------------------------------------------------------------------------
--Getting 7 notes to the octave to
--sound good together is difficult some intervals have to be
--within a critical band, and we've talked about this before.
    I know, some of them will beat there's no avoiding
that.  The goal then would be to keep the beating under control...I agree we obviously can't eliminate it.  Heck, a minor second is somewhat within the critical band. 

   The "scary" thing is I've had much better luck
making a 7-note per 2/1 octave chord in, say, my PHI scale or, obviously, using around harmonics 7-14 in the harmonic series...than I ever have with making that kind of dense chord via adaptive JI.  And I'm looking to see if I can find a missing link through asking around about other research. 

********************************************************************************************
--How do you plan to express emotion without dissonance?

    As a random counter example...my PHI scale was actually created via a process of starting with a couple notes from the PHI tuning, composing with them, and then trying to match the notes I heard in my head to the nearest notes in the PHI tuning until I had about 9 notes in an octave.

   You can add dissonance to consonance contrast in ways other than using sour intervals.  Adding notes alone adds amplitude, which increases dissonance.  Changing the volume of each note played also effects dissonance.  In fact, Sethares' own dissonance formula uses two factors to determine the dissonance between notes: distance between notes/intervals AND amplitude.  So if you want dissonance...crank of the volume of notes...and, if you want more, crank up the
volumes of notes nearby each other. :-)

-Michael
--- On Tue, 3/31/09, Graham Breed <gbreed@gmail.com> wrote:

From: Graham Breed <gbreed@...>
Subject: Re: [tuning] Re: small ratio fundamentalism
To: tuning@yahoogroups.com
Date: Tuesday, March 31, 2009, 6:49 PM

djtrancendance@ yahoo.com wrote:

> >There are plenty of temperament classes (as I call them)

> >with a lower minimax error than meantone. A good 5-limit

> >example is kleismatic/hanson.

> Thank you! Finally, we are starting to answer the question I was asking in

> the first place: how to get the smallest error away from perfect intervals

> period (and realizing that the error will never be zero, but hopefully approach

> it to a degree indistinguishable by human hearing).

There was a sub-thread about it last year, in a wider thread

you contributed to:

http://launch. groups.yahoo. com/group/ tuning/message/ 79623

> --This is taking us closer to just intonation. Before you

> ---said you didn't want just intonation.

> My mistake. :-)

> I don't want JI to become the limiting requirement either (IE if an irrational

> ratio gives a lower minimax in some cases, so be it :-) )...but if JI gives

> lower minimax than mean-tone at least we're opening another option and getting

> closer.

JI always has an error of zero. Optimizing for the minimax

error will only get you closer to JI.

> BTW, as a side note, the ultimate test is to take a 7-note scale that has

> the lowest minimax value mathematically possible (without any limitations such

> as JI, meantone, etc. being pre-requisite) and than pit my PHI scale against it.

> Hopefully it will allow me to more easily find out where the sour spots in

> my scale are vs. "a realistic 'tempered' version of adaptive JI".

> BTW, I sense one issue with adaptive JI I've found with all Adaptive-JI

> software I have tried...if you play ALL 7 notes within the octave at once, it

> sounds sour. Which is why, even in adaptive JI, you still need music theory and

> to pick chord selectively to "knock out the kinks" in extreme combinations of

> closely spaced JI intervals.

Why would adaptive JI help when you play all the notes at

the same time? It can make arbitrary chords with reasonable

spacing sound in tune. Getting 7 notes to the octave to

sound good together is difficult some intervals have to be

within a critical band, and we've talked about this before.

> In the end of the day the goal I have in mind for both scales is the same...to

> have a scale where it's virtually impossible to play combinations of notes that

> create chords which are dissonant... and make a scale that "resolves music theory

> itself" so no music theory is needed and it becomes purely an issue of emotion

> as to the art of playing with such scales. :-)

How do you plan to express emotion without dissonance?

Graham