Yahoo Tuning Groups Ultimate Backup tuning well temperament

I have discovered a well temperament that could, possibly, replace 12 ET. As far as I can tell it is just as flexible.

1/1 135/128 9/8 1215/1024 512/405 4/3 64/45 3/2 405/256 2048/1215 16/9 256/135 2/1

The above ratios can be derived in the following manner, thus making the scale suitable for tuning by ear

Starting from C

3/2 3/2 5/4 3/2 3/2 3/2

Starting from C again, and retuning F# when it is arrived at

4/3 4/3 8/5 4/3 4/3 4/3

Experimenting with this tuning has convinced me that the interplay of the overtones is much more important than the proximity to a given interval, so that a sharp third could sound better than one that is closer to 5/4, if it's overtones are in harmony with the other notes in the scale.

For example, I believe that the overtone fifth of 16384/10935 is more resonant than the 12 ET fifth, even though they are almost the same size.

F#-C# might be considered to be a wolf interval, yet I would encourage everyone to determine that by ear, and not by theory.

I have retuned numerous mid files to the scale, including both books of the WTC, and everything sounds good in all keys.

Thanks,

Raintree

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🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

This scale is not too extraordinary to replace 12-tET in my opinion. I think
Secor 5/23 TX is way better.

Oz.

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@coolgoose.com>
To: <tuning@yahoogroups.com>
Sent: 14 Mart 2007 ï¿½arï¿½amba 22:45
Subject: [tuning] Re: well temperament

> --- In tuning@yahoogroups.com, Raintree Goldbach
> <thefractalgourmet@...> wrote:
> >
> > I have discovered a well temperament that could, possibly, replace
> 12 ET. As far as I can tell it is just as flexible.
>
> I wouldn't call it a well-temperament, as one of the fifths is too
> flat for the scale to count as circulating. Ordinarily, I might tell
> you what other scales are similar, but there's a defective scl file
> in my directory which prevents this. The latest Scala is supposed to
> be able to locate such files, but it doesn't work for me. If someone
> wants to help test Scala, try putting a defective file (say, with the
> wrong count of number of notes) in a directory, and then attempting
> to run "compare scale". Does it work for you? Does Scala tell you
> where the defect is?
>
> Here's the scale in scl format:
>
> ! raintree.scl
> Raintree scale
> 12
> !
> 135/128
> 9/8
> 1215/1024
> 512/405
> 4/3
> 64/45
> 3/2
> 405/256
> 2048/1215
> 16/9
> 256/135
> 2/1
>
>

🔗Herman Miller <hmiller@IO.COM>

Gene Ward Smith wrote:
> --- In tuning@yahoogroups.com, Raintree Goldbach > <thefractalgourmet@...> wrote:
>> I have discovered a well temperament that could, possibly, replace > 12 ET. As far as I can tell it is just as flexible.
> > I wouldn't call it a well-temperament, as one of the fifths is too > flat for the scale to count as circulating. Ordinarily, I might tell > you what other scales are similar, but there's a defective scl file > in my directory which prevents this.

It's closest to duoden12.scl.

subset of coul_27.scl size: 27
subset of duodenarium.scl size: 117
Scales with the same size and same key:
closest average absolute: duoden12.scl diff. 0.1628 cents
closest root mean square: duoden12.scl diff. 0.5640 cents
closest highest absolute: duoden12.scl diff. 1.9537 cents
Scales with the same size and other key:
closest highest absolute: meanschis7.scl diff. 1.1167 cents in key 8
Scales with any size: idem
Number of files compared: 2933

! duoden12.scl
!
Almost equal 12-tone subset of Duodenarium 12
!
135/128
9/8
1215/1024
512/405
4/3
64/45
3/2
405/256
2048/1215
3645/2048
256/135
2/1

As you can see, only one note is different (3645/2048 instead of 16/9).

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Raintree Goldbach <thefractalgourmet@yahoo.com>

Gene et al,

thanks for the input.

while researching the duodenarium i found this interesting post from the tuning list

Message: 9159 - Contents - Hide Contents

Date: Thu, 15 Jan 2004 07:53:37

Subject: The Four Mistyschism Scales

From: Gene Ward Smith

A hoary eccelesiastical joke has it that mysticism should really be
spelled "misty schism", so this is how I am spelling the four Fokker
blocks which come from misty and the schisma.

The two most notable things about these scales is that Scala classifies all four of them
as well-temperments, and that this concludes the classification, though I'll check the latter claim.

Mistyschism2 is the same scale as
Scala's duoden12.scl, about which we read "Almost equal 12-tone
subset of Duodenarium".

! mistyschism1.scl
Mistyschism scale 2048/2025 67108864/66430125
12
!
524288/492075
9/8
1215/1024
512/405
4/3
64/45
3/2
262144/164025
2048/1215
3645/2048
256/135
2

! mistyschism2.scl
Mistyschism scale 2048/2025 67108864/66430125 = duoden12.scl
12
!
135/128
9/8
1215/1024
512/405
4/3
64/45
3/2
405/256
2048/1215
3645/2048
256/135
2

! mistyschism3.scl
Mistyschism scale 2048/2025 67108864/66430125
12
!
135/128
9/8
1215/1024
512/405
4/3
1476225/1048576
3/2
405/256
2048/1215
3645/2048
256/135
2

! mistyschism4.scl
Mistyschism scale 2048/2025 67108864/66430125
12
!
524288/492075
9/8
1215/1024
512/405
4/3
64/45
3/2
405/256
2048/1215
3645/2048
256/135
2

i take it that you determined at that time that they are not well temperaments?

all i can say for say for certain is that every midi file that i retune to the scale sounds better, and i have yet to find any music that sounds good in 12 ET that doesn't sound good in the modified mystical tuning.

is not the ear the final arbiter?

also, i believe that too much emphasis is placed on proximity, when the emphasis should rather be the "harmony of the overtones"

the only sacrosanct interval in most music written to date is the octave, of which there is, and always will be, only one.

every other interval has, perhaps, an infinite number of valid ratios, and at the very least several.

once you accept the "flexibility" of the fifth and the third, etc, then everything becomes easier.

if anyone can find a midi file that doesn't sound good in the tuning, yet sounds good in 12 ET, i would very much be interested in hearing it, as i have yet to find one, and i have been searching for several weeks for such, especially those in the keys of F# major and F# minor.

thanks,

Raintree

Herman Miller <hmiller@IO.COM> wrote:
Gene Ward Smith wrote:
> --- In tuning@yahoogroups.com, Raintree Goldbach
> <thefractalgourmet@...> wrote:
>> I have discovered a well temperament that could, possibly, replace
> 12 ET. As far as I can tell it is just as flexible.
>
> I wouldn't call it a well-temperament, as one of the fifths is too
> flat for the scale to count as circulating. Ordinarily, I might tell
> you what other scales are similar, but there's a defective scl file
> in my directory which prevents this.

It's closest to duoden12.scl.

! duoden12.scl
!
Almost equal 12-tone subset of Duodenarium

12
!
135/128
9/8
1215/1024
512/405
4/3
64/45
3/2
405/256
2048/1215
3645/2048
256/135
2/1

As you can see, only one note is different (3645/2048 instead of 16/9).

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🔗Gene Ward Smith <genewardsmith@coolgoose.com>

--- In tuning@yahoogroups.com, Raintree Goldbach
<thefractalgourmet@...> wrote:

> Mistyschism2 is the same scale as
> Scala's duoden12.scl, about which we read "Almost equal 12-tone
> subset of Duodenarium".

So, it's a Fokker block.

> i take it that you determined at that time that they are not well
temperaments?

I think one of the fifths is too flat to count, and Scala agrees with
me, no longer counting these as well-temperaments. The schisdia
series is similar, by the way.

> all i can say for say for certain is that every midi file that i
retune to the scale sounds better, and i have yet to find any music
that sounds good in 12 ET that doesn't sound good in the modified
mystical tuning.

Have you tried something in B major or F# major?

> is not the ear the final arbiter?

You get to decide what you like, but not what everyone likes. We
could try posting some midi files and see what people in general
think.

> also, i believe that too much emphasis is placed on proximity,
when the emphasis should rather be the "harmony of the overtones"

Why? And what does that mean?

> the only sacrosanct interval in most music written to date is the
octave, of which there is, and always will be, only one.

> every other interval has, perhaps, an infinite number of valid
ratios, and at the very least several.

If the octave has only one valid ratio, then the same is true of 3/2.
But what makes you think it is true? If you sharpen an octave by a
schisma, it somehow isn't valid?

> once you accept the "flexibility" of the fifth and the third,
etc, then everything becomes easier.

Once you accept the flexibility of everything, you have adopted the
point of view taken on tuning by many and tuning-math by even more.

🔗Tom Dent <stringph@gmail.com>

--- In tuning@yahoogroups.com, Raintree Goldbach
<thefractalgourmet@...> wrote:
>
>
> Starting from C
>
> 3/2 3/2 5/4 3/2 3/2 3/2
>
> Starting from C again, and retuning F# when it is arrived at
>
> 4/3 4/3 8/5 4/3 4/3 4/3
>

You seem to have reinvented the Urbino clavichord or van Zwolle tuning
where there is a comma between B and F# and everything else pure. To
be more precise a slight transposition. Or the Erlangen clavichord
tuning, which manages to put a couple of schismas into the game but
basically has the whole comma between A and E. Or so-called
'Kirnberger I'.

> Experimenting with this tuning has convinced me that the interplay
of the overtones is much more important than the proximity to a given
interval (...)
> For example, I believe that the overtone fifth of 16384/10935 is
more resonant than the 12 ET fifth, even though they are almost the
same size.
>

Good luck in telling the difference (0.001 cent!!!) Can this be
determined by ear - experimenting? - and not by theory?

> F#-C# might be considered to be a wolf interval, yet I would
encourage everyone to determine that by ear, and not by theory.

But I already know what a fifth flat by a comma sounds like on a
harpsichord, I would run a mile from it in Baroque music.

> I have retuned numerous mid files to the scale, including both
books of the WTC, and everything sounds good in all keys.
>

As well as your personal taste, that might well depend very strongly
on what timbre you used - as experiments here recently demonstrated!!

~~~T~~~

🔗Herman Miller <hmiller@IO.COM>

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Raintree Goldbach <thefractalgourmet@yahoo.com>

even if it were only just as good as 12-tET, the ease of tuning, and that by ear alone, should suggest that the scale would be of great practical value.

is there another scale that is easier to tune that even begins to approach well temperament?

what is Secor 5/23 TX by the way?

Ozan Yarman <ozanyarman@ozanyarman.com> wrote:
This scale is not too extraordinary to replace 12-tET in my opinion. I think
Secor 5/23 TX is way better.

Oz.

----- Original Message -----
From: "Gene Ward Smith"
To:
Sent: 14 Mart 2007 ï¿½arï¿½amba 22:45
Subject: [tuning] Re: well temperament

> --- In tuning@yahoogroups.com, Raintree Goldbach
> wrote:
> >
> > I have discovered a well temperament that could, possibly, replace
> 12 ET. As far as I can tell it is just as flexible.
>
> I wouldn't call it a well-temperament, as one of the fifths is too
> flat for the scale to count as circulating. Ordinarily, I might tell
> you what other scales are similar, but there's a defective scl file
> in my directory which prevents this. The latest Scala is supposed to
> be able to locate such files, but it doesn't work for me. If someone
> wants to help test Scala, try putting a defective file (say, with the
> wrong count of number of notes) in a directory, and then attempting
> to run "compare scale". Does it work for you? Does Scala tell you
> where the defect is?
>
> Here's the scale in scl format:
>
> ! raintree.scl
> Raintree scale
> 12
> !
> 135/128
> 9/8
> 1215/1024
> 512/405
> 4/3
> 64/45
> 3/2
> 405/256
> 2048/1215
> 16/9
> 256/135
> 2/1
>
>

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Gene Ward Smith <genewardsmith@coolgoose.com> wrote:

>If the octave has only one valid ratio, then the same is true of 3/2

for most, if not all, of the classical music repertoire the octave is inviolate, yet why should that apply to the fifth? if it did then tempering would not exist. certainly, in olden times, no one ever tempered the octave. it is valid to temper the octave, yet not for Mozart or Bach,etc.

>I think one of the fifths is too flat to count, and Scala agrees with
me, no longer counting these as well-temperaments. The schisdia
series is similar, by the way.

perhaps, yet flat implies that there is only one true standard to measure the fifth by, whereas i believe that the size of the fifth and the third, of all intervals other than the octave, should be relative. the ideal fifth and/or third depends on the size of the interval(s) it is relating to, so that the overtones are in tune as well, to the greatest degree possible.

in the scale in question the primary chord is 1/1 512/405 3/2, which sounds better, to my ear, than the standard 1/1 5/4 3/2. the comma between 5/4 and 512/405 is 2048/2025, which, i believe, is 1/12 of a pythagorean comma smaller than a syntonic comma.

therefore, when you say that the fifth is flat, it is not the syntonic comma, which everyone knows produces a wolf, rather it is the 2048/2025 comma.

if 1/1 512/405 3/2 is tolerably consanant, at the very least, then there is a certain logic to the 1/1 5/4 6075/4096 chord that occurs in F# major, because the fifth and the third are both flat by 2048/205, which is why they sound consonant together, in my opinion.

"Have you tried something in B major or F# major?"

extensively, i have tested every key numerous times, every style of music, different timbres, etc, and have yet to hear a single objectionable tone.

> also, i believe that too much emphasis is placed on proximity,
when the emphasis should rather be the "harmony of the overtones"

>Why? And what does that mean?

when the overtones reinforce one another. this effect cannot occur with an infinite gradation. that is why the proximity to 5/4 does not make a note more in tune, necessarily.
it is more important to make sure that the overtones are in concert. perhaps someone could give a more technically precise definition of what i am saying. in 12-tET, for instance, the overtones do not reinforce one another, even though the harmonies are tolerable, and the resultant lack of cohesiveness is palpable.

Gene Ward Smith <genewardsmith@coolgoose.com> wrote:
--- In tuning@yahoogroups.com, Raintree Goldbach
<thefractalgourmet@...> wrote:

> Mistyschism2 is the same scale as
> Scala's duoden12.scl, about which we read "Almost equal 12-tone
> subset of Duodenarium".

So, it's a Fokker block.

> i take it that you determined at that time that they are not well
temperaments?

I think one of the fifths is too flat to count, and Scala agrees with
me, no longer counting these as well-temperaments. The schisdia
series is similar, by the way.

Have you tried something in B major or F# major?

> is not the ear the final arbiter?

You get to decide what you like, but not what everyone likes. We
could try posting some midi files and see what people in general
think.

> also, i believe that too much emphasis is placed on proximity,
when the emphasis should rather be the "harmony of the overtones"

Why? And what does that mean?

> the only sacrosanct interval in most music written to date is the
octave, of which there is, and always will be, only one.

> every other interval has, perhaps, an infinite number of valid
ratios, and at the very least several.

If the octave has only one valid ratio, then the same is true of 3/2.
But what makes you think it is true? If you sharpen an octave by a
schisma, it somehow isn't valid?

> once you accept the "flexibility" of the fifth and the third,
etc, then everything becomes easier.

Once you accept the flexibility of everything, you have adopted the
point of view taken on tuning by many and tuning-math by even more.

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🔗Ozan Yarman <ozanyarman@ozanyarman.com>

----- Original Message -----
From: Raintree Goldbach
To: tuning@yahoogroups.com
Sent: 16 Mart 2007 Cuma 16:09
Subject: Re: [tuning] Re: well temperament

even if it were only just as good as 12-tET, the ease of tuning, and that by ear alone, should suggest that the scale would be of great practical value.

There is no question of the majority of scales proposed here of having practical value. They are, after all, promulgated to invite readers to consider utilizing them.

is there another scale that is easier to tune that even begins to approach well temperament?

Yours is, as Gene has suggested, not a well temperament as far as the eye can tell, for it does not circulate in the classical sense. I would surmise that Lehman's Bach tuning is easier to set and is an excellent well-temperament.

what is Secor 5/23 TX by the way?

You mean you have never heard of the fantastic proportional beating 12-tone temperament extraordinaire by the very George Secor of the tuning list? I have my grand piano (or at least the lower half) adjusted to it:

George Secor's rational 5/23-comma temperament extraordinaire
|
0: 1/1 0.000 unison, perfect prime
1: 390/371 86.466
2: 3321/2968 194.552
3: 4397/3710 294.120
4: 929/742 389.111
5: 2476/1855 499.911
6: 8325/5936 585.551
7: 555/371 697.282
8: 9365/5936 789.345
9: 621/371 891.809
10: 9904/5565 997.956
11: 5559/2968 1086.401
12: 2/1 1200.000 octave

Oz.

🔗Raintree Goldbach <thefractalgourmet@yahoo.com>

Oz,

Lehman's temperament is wonderful, yet much more complicated to set.

http://www-personal.umich.edu/~bpl/larips/practical.html

Recipe [Chart]

Get A from the tuning fork or other appropriate source.
Tenor F-A (below middle C) beating wide 3 beats per second. It has 3 beats per second (or double the metronome speed of 89) if we are starting from A=440; or 2.8 beats (double the metronome 84) if starting from A=415. This establishes the boundary for the four intervening 5ths, F-C-G-D-A: our regular size of 5ths, 1/6 Pythagorean comma. All four of those 5ths will now become equally spaced within that region, tempered 1/6 comma each.
Tenor F fourth down to C temporarily pure. (See why we're doing this temporary step...)
Tenor A fifth down to D temporarily pure.
Tenor G from that C and D so the fourth D-G beats 1.5 times as fast as the fifth C-G. (Triplets against duplets.)
Middle C from the tenor F and that G, likewise: the fourth G-C beating as triplets against the duplets of the fifth F-C.
Middle D from tenor G and A likewise.
Correct the lower D and C to match these two.
Middle E from tenor A slightly flat, same quality as these other fifths. It should beat as a major third from middle C at 4.5 times per second. (4.2 if you started from an A=415.) Also check it as a 10th from tenor C, the same 4.5 times per second. An excellent test for this particular F-A and C-E is found at bar 21 of the WTC book 1 C major prelude. These two major thirds should have exactly the same character as one another, although the beat rate is different (since the pitch is different). Coincidentally (?) this is at the golden mean of this piece.... The A-E fifth should beat exactly half as fast as our original F-A major 3rd, sharing the same A.
Copy middle E down to tenor E. For especially careful accuracy, fuss with both these E's until all four of the following points are true:
(1) Whatever speed the tenor D-A 5th is, the fifth A-E across middle C should beat in 3/2 ratio from that speed (i.e. triplets vs duplets).
(2) Likewise, whatever speed the tenor F-A major 3rd is, the middle C-E major 3rd is triplets against its duplets. [We are up a fifth on the keyboard, and all frequencies are 3/2 as much.]
(3) Playing the tenor E as a 4th against tenor A, it should beat exactly half as fast as our original F-A major 3rd. Likewise, the A-E 5th we are testing across middle C should be exactly half as fast as our original F-A major 3rd.
(4) The octave E must of course be pure! Tune the intervening B pure with both of them.

The above procedure has put all the naturals (except B) into regular 1/6 comma temperament, the same as the first half of "Vallotti": F-C-G-D-A-E all tempered identically. Bach's temperament finishes differently, as follows:
Pure fifths E-B-F#-C#.
Pure fifths F-Bb-Eb. This is a step to set the Eb exactly; we will move this temporary Bb soon.
Tune G# pure to C# and then slightly lower it so the C#-G# fifth and the Ab-Eb fifth have the same quality: very slow beat from each.
Lower Bb slightly so it has a slow beat from Eb like that of C#-G# and Ab-Eb.
Make all octaves and unisons pure to finish the instrument: all the bass, then all the treble.
Summary
1/6 comma fifths F-C-G-D-A-E
pure fifths E-B-F#-C#
1/12 comma fifths C#-G#-D#-A#
1/12 comma wide diminished sixth resulting at A#-F
as compared to, starting from C

3/2
3/2
5/4
3/2
3/2
3/2

starting from C again, and retuning F# when it is arrived at

4/3
4/3
8/5
4/3
4/3
4/3

if you retune any composition that sounds good in 12-tET it will sound good in the mystical tuning.

the Secor is exceptional, to say the least, very impressive!

thanks,

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🔗Gene Ward Smith <genewardsmith@coolgoose.com>

--- In tuning@yahoogroups.com, Raintree Goldbach
<thefractalgourmet@...> wrote:

> in the scale in question the primary chord is 1/1 512/405 3/2,
which sounds better, to my ear, than the standard 1/1 5/4 3/2. the
comma between 5/4 and 512/405 is 2048/2025, which, i believe, is 1/12
of a pythagorean comma smaller than a syntonic comma.

It's also 81/64 lowered by a schisma of 32805/32768, which means
instead of comma of difference, you get a comma lowered by a schisma,
which is the diaschisma, 2048/2025.

It's simply a fact of life that people differ in how they react to
intervals, and that you prefer 1-512/405-3/2 does not mean most
people would. To me, it lacks the sweet sound of a more accurately
tuned major third.

> therefore, when you say that the fifth is flat, it is not the
syntonic comma, which everyone knows produces a wolf, rather it is
the 2048/2025 comma.

A fifth flat by a diaschisma of 19.55 cents is very flat; it's even
flatter than the fifth of 7-et. I guess Ozan could try 51-et, for
which it is the second-best fifth, if he thought it would work, which
he wouldn't. But one guy who might go for it is Herman Miller, as
your 6075/4096 is right smack in the optimal range for 7-limit
mavilla temperament, and even on the sharp side for 5-limit mavilla.

> > also, i believe that too much emphasis is placed on proximity,
> when the emphasis should rather be the "harmony of the overtones"
>
> >Why? And what does that mean?
>
> when the overtones reinforce one another. this effect cannot
occur with an infinite gradation. that is why the proximity to 5/4
does not make a note more in tune, necessarily.
> it is more important to make sure that the overtones are in
concert.

This isn't making any sense to me at all. Did you listen to my midi
examples? I thought your scale came out less than stellar there,
frankly.

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

Here's a Fokker block, derived from the Kirnberger atom and the
schisma, which showed up in the course of my survey of 12-note 5-
limit Fokker blocls. It's Kirnerger's quasi 12edo scale, consisting
of 11 fifths flat by a schisma, and one flat by a schisma and an
atom. Perhaps it has whatever voodoo exact 5-limit ratios are
supposed to have. It's certainly a well-temperament, at any rate.

! kirneq.scl
Kirnberger atom and schisma Fokker block quasi-equal
12
!
156348578434374084375/147573952589676412928
134217728/119574225
1307544150375/1099511627776
18014398509481984/14297995284350625
10935/8192
1709671705179880612640625/1208925819614629174706176
16384/10935
14297995284350625/9007199254740992
2199023255552/1307544150375
119574225/67108864
295147905179352825856/156348578434374084375
2

Here's a scale which can lay claim to no such voodoo, but which I
think counts as circulating. It's the Rainbird scale, tempered by
tuning it to 123-et. This is an old trick of mine; 123-et shrinks the
diaschisma, 2048/2025, to half of its size. Hence, what used to be
pretty out there now becomes reasonable. Note that in the course of
shrinking 2048/2025, it enlarges the schisma until the two are equal;
consequently it now has three flat fifths of 692.7 cents. It woudl
now be reasonable now to go back and replace the near-just fifths
with exact 3/2s, and the flat fifths with (65536/19683)^(1/3), at 694
cents, and get another circulating temperament.

! rain123.scl
Raintree scale tuned to 123-et
12
!
97.560976
204.878049
302.439024
400.000000
497.560976
604.878049
702.439024
800.000000
897.560976
995.121951
1102.439024
1200.000000

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

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

> This is an old trick of mine; 123-et shrinks the
> diaschisma, 2048/2025, to half of its size. Hence, what used to be
> pretty out there now becomes reasonable. Note that in the course of
> shrinking 2048/2025, it enlarges the schisma until the two are
>equal...

This means that 123 tempers out (2048/2025)/(32805/32768) =
67108864/66430125, the misty comma. Notable ets tempering this out
are 87, 99, and 111; and the 7-limit temperament you get from these
three is 7-limit misty. I actually started composing something in
misty, and never finised it, sad to relate.

Anyway, these all have a sharp fifth, and misty has a generally sharp
tendency. 123 is less sharp, but for this game of retuning scales to
make them circulate, a flat fifth would seem to be better. 12 does
that, but that's kind of pointless; 123 is the best we can find in a
patent val. But! Our old friend 159, using its second-best major
third, works just fine. As a well-temperament, the main complaint to
be made about it, I think, is that the major thirds are mostly 400
cents. Here's Raintree in 159-edo, something Ozan might find
interesting:

! rain159.scl
Raintree scale tuned to 159-edo
12
!
98.113208
203.773585
301.886792
400.000000
498.113208
603.773585
701.886792
800.000000
898.113208
996.226415
1101.886792
1200.000000

🔗Herman Miller <hmiller@IO.COM>

Gene Ward Smith wrote:

> A fifth flat by a diaschisma of 19.55 cents is very flat; it's even > flatter than the fifth of 7-et. I guess Ozan could try 51-et, for > which it is the second-best fifth, if he thought it would work, which > he wouldn't. But one guy who might go for it is Herman Miller, as > your 6075/4096 is right smack in the optimal range for 7-limit > mavilla temperament, and even on the sharp side for 5-limit mavilla.

That depends on what you're calling 7-limit mavila. 9&16 <<1, -3, 5, -7, 5, 20]] is a nice temperament, but <<1, -3, -4, -7, -9, -1]] is a little problematic. But if you think 682.4 cents (6075/4096) is a flat fifth, I've written music in 16-ET with 675-cent fifths. Not the best tuning in the world, but it's not the worst either.

>> > also, i believe that too much emphasis is placed on proximity, >> when the emphasis should rather be the "harmony of the overtones"
>>
>>> Why? And what does that mean?
>> >> when the overtones reinforce one another. this effect cannot > occur with an infinite gradation. that is why the proximity to 5/4 > does not make a note more in tune, necessarily.
>> it is more important to make sure that the overtones are in > concert. > > This isn't making any sense to me at all. Did you listen to my midi > examples? I thought your scale came out less than stellar there, > frankly.

Really? I thought they sounded perfectly fine (at least as good as 12-ET at any rate, if not a little better in places). Better than a lot of the rational 12-note scales out there, but I still prefer a more traditional WT like Sorge 1758.

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

I see Brad's directions are a bit tedious. Yes, yours is easier to set, but is still not a well temperament.

Oz.

----- Original Message -----
From: Raintree Goldbach
To: tuning@yahoogroups.com
Sent: 16 Mart 2007 Cuma 20:40
Subject: [tuning] well temperament

Oz,

Lehman's temperament is wonderful, yet much more complicated to set.

http://www-personal.umich.edu/~bpl/larips/practical.html

Recipe [Chart]
a.. Get A from the tuning fork or other appropriate source.
b.. Tenor F-A (below middle C) beating wide 3 beats per second. It has 3 beats per second (or double the metronome speed of 89) if we are starting from A=440; or 2.8 beats (double the metronome 84) if starting from A=415. This establishes the boundary for the four intervening 5ths, F-C-G-D-A: our regular size of 5ths, 1/6 Pythagorean comma. All four of those 5ths will now become equally spaced within that region, tempered 1/6 comma each.
c.. Tenor F fourth down to C temporarily pure. (See why we're doing this temporary step...)
d.. Tenor A fifth down to D temporarily pure.
e.. Tenor G from that C and D so the fourth D-G beats 1.5 times as fast as the fifth C-G. (Triplets against duplets.)
f.. Middle C from the tenor F and that G, likewise: the fourth G-C beating as triplets against the duplets of the fifth F-C.
g.. Middle D from tenor G and A likewise.
h.. Correct the lower D and C to match these two.
i.. Middle E from tenor A slightly flat, same quality as these other fifths. It should beat as a major third from middle C at 4.5 times per second. (4.2 if you started from an A=415.) Also check it as a 10th from tenor C, the same 4.5 times per second. An excellent test for this particular F-A and C-E is found at bar 21 of the WTC book 1 C major prelude. These two major thirds should have exactly the same character as one another, although the beat rate is different (since the pitch is different). Coincidentally (?) this is at the golden mean of this piece.... The A-E fifth should beat exactly half as fast as our original F-A major 3rd, sharing the same A.
j.. Copy middle E down to tenor E. For especially careful accuracy, fuss with both these E's until all four of the following points are true:
a.. (1) Whatever speed the tenor D-A 5th is, the fifth A-E across middle C should beat in 3/2 ratio from that speed (i.e. triplets vs duplets).
b.. (2) Likewise, whatever speed the tenor F-A major 3rd is, the middle C-E major 3rd is triplets against its duplets. [We are up a fifth on the keyboard, and all frequencies are 3/2 as much.]
c.. (3) Playing the tenor E as a 4th against tenor A, it should beat exactly half as fast as our original F-A major 3rd. Likewise, the A-E 5th we are testing across middle C should be exactly half as fast as our original F-A major 3rd.
d.. (4) The octave E must of course be pure! Tune the intervening B pure with both of them.
a.. The above procedure has put all the naturals (except B) into regular 1/6 comma temperament, the same as the first half of "Vallotti": F-C-G-D-A-E all tempered identically. Bach's temperament finishes differently, as follows:
b.. Pure fifths E-B-F#-C#.
c.. Pure fifths F-Bb-Eb. This is a step to set the Eb exactly; we will move this temporary Bb soon.
d.. Tune G# pure to C# and then slightly lower it so the C#-G# fifth and the Ab-Eb fifth have the same quality: very slow beat from each.
e.. Lower Bb slightly so it has a slow beat from Eb like that of C#-G# and Ab-Eb.
f.. Make all octaves and unisons pure to finish the instrument: all the bass, then all the treble.
Summary
a.. 1/6 comma fifths F-C-G-D-A-E
b.. pure fifths E-B-F#-C#
c.. 1/12 comma fifths C#-G#-D#-A#
d.. 1/12 comma wide diminished sixth resulting at A#-F
as compared to, starting from C

3/2
3/2
5/4
3/2
3/2
3/2

starting from C again, and retuning F# when it is arrived at

4/3
4/3
8/5
4/3
4/3
4/3

if you retune any composition that sounds good in 12-tET it will sound good in the mystical tuning.

the Secor is exceptional, to say the least, very impressive!

thanks,

🔗Ozan Yarman <ozanyarman@ozanyarman.com>

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@coolgoose.com>
To: <tuning@yahoogroups.com>
Sent: 16 Mart 2007 Cuma 21:57
Subject: [tuning] Re: well temperament

> --- In tuning@yahoogroups.com, Raintree Goldbach
> <thefractalgourmet@...> wrote:
>
> > in the scale in question the primary chord is 1/1 512/405 3/2,
> which sounds better, to my ear, than the standard 1/1 5/4 3/2. the
> comma between 5/4 and 512/405 is 2048/2025, which, i believe, is 1/12
> of a pythagorean comma smaller than a syntonic comma.
>
> It's also 81/64 lowered by a schisma of 32805/32768, which means
> instead of comma of difference, you get a comma lowered by a schisma,
> which is the diaschisma, 2048/2025.
>
> It's simply a fact of life that people differ in how they react to
> intervals, and that you prefer 1-512/405-3/2 does not mean most
> people would. To me, it lacks the sweet sound of a more accurately
> tuned major third.
>
> > therefore, when you say that the fifth is flat, it is not the
> syntonic comma, which everyone knows produces a wolf, rather it is
> the 2048/2025 comma.
>
> A fifth flat by a diaschisma of 19.55 cents is very flat; it's even
> flatter than the fifth of 7-et. I guess Ozan could try 51-et, for
> which it is the second-best fifth, if he thought it would work, which
> he wouldn't.

Of course I would not.

SNIP

Oz.

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Tom Dent <stringph@gmail.com>

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Herman Miller <hmiller@IO.COM>

🔗Joe <tamahome02000@yahoo.com>

Commented a note out? Did you change the size of the scale at the top?

Joe

----- Original Message ----
From: Herman Miller <hmiller@IO.COM>
To: tuning@yahoogroups.com
Sent: Sunday, March 18, 2007 9:57:54 PM
Subject: Re: [tuning] Scala problem

Gene Ward Smith wrote:

> --- In tuning@yahoogroups. com, Herman Miller <hmiller@... > wrote:

>> I tried the latest version of Scala (2.23p); since I had duoden12.scl

> in

>> my recent files list, I saved a copy of it as "duodenum.scl" ,

> commented

>> a note out, and tried a "compare scale". I did get the error message:

>> ** Read error for duodenum.scl

> I'm still wanting to know the details of your OS. Also, reports from

> anyone else who could try this.

Did you not get the private reply I sent? I'm running Windows XP with

the latest version of the Gtk+ (gtk+-2.10.6- 1-setup.zip) . I still use

FAT32 if it makes a difference; this old system I'm running it on was

originally a Windows ME system (from shortly before XP came out) and I

never bothered upgrading to NTFS.

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____________________________________________________________________________________
Now that's room service! Choose from over 150,000 hotels
in 45,000 destinations on Yahoo! Travel to find your fit.
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🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Raintree Goldbach <thefractalgourmet@yahoo.com>

🔗Carl Lumma <clumma@yahoo.com>

🔗Raintree Goldbach <thefractalgourmet@yahoo.com>

🔗Cameron Bobro <misterbobro@yahoo.com>

🔗Andy <a_sparschuh@...>

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
> Or the Erlangen clavichord
> tuning, which manages to put a couple of schismas into the game but
> basically has the whole comma between A and E. Or so-called
> 'Kirnberger I'.

Hi Tom,
http://groenewald-berlin.de/text/text_T080.html
has the ratios of that medivial one:
"
Apotome -1287/2048 - 113.685 cent
diat Halbton - 16/15 - 111.731 cent 32805/32768
(Schisma)

gr. Chroma - 135/128 - 92.179cent

Leimma - 256/243 - 90.225 cent 2048/2025
(Diaschisma)

15/8 x 16/15 = 2/1 (c) 4/3 x 3/2 : 2
= 1/1 (c)

16/9 x 135/128 = 15/8 (h) 16/9 x 3/2 : 2
= 4/3 (f)

2048/1215x 135/128 = 16/9 (b) 32/27 x 3/2 :
= 16/9 (b)

128/81 x 16/15 = 2048/1215 (a) 128/81 x 3/2 : 2
= 32/27 (es)

3/2 x 256/243 = 128/81 gis) 256/243 x 3/2
= 128/81 (gis)

1024/729x 2187/2048 = 3/2 (g) 1024/729 x 3/2 : 2
= 256/243 (cis)

4/3 x 256/243 = 1024/729 (fis) 15/8 x 3/2 : 32805/32768
: 2 = 1024/729 (fis)

5/4 x 16/15 = 4/3 (f) 5/4 x 3/2
= 15/8 (h)

32/27 x 135/128 = 5/4 (e) 2048/1215x 3/2 : 2048/2025
: 2 = 5/4 (e)

4096/3645x 135/128 = 32/27 (es) 4096/3645x 3/2 :
= 2048/1215 (a)

256/243 x 16/15 =4096/3645 (d) 3/2 x 3/2 : 32805/32768
: 2 =4096/3645 (d)

1/1 x 256/243 = 256/243 (cis) 1/1 x 3/2
= 3/2 (g)

That's in the 'Scala' file-format
!ErlangenMonochord.scl
The Erlangen Monochord: Â´Pro clavichordiis faciendisÂ´[1454~1480]
by A.Sparschuh
12
!
256/243 ! C# |8,-5>
4096/3645 ! D |12,-6,-1>
32/27 ! Eb |5,-3>
5/4 ! E |-2,0,1>
4/3 ! F |2,-1>
1024/729 ! F# |10,-6>
3/2 ! G |-1,1>
128/81 ! G# |7,-4>
2048/1215 ! A |11,-5,-1>
16/9 ! Bb |4,-2>
15/8 ! B |-3,1,1>
2/1
!
![eof]

bye
Andy