Yahoo Tuning Groups Ultimate Backup tuning Most common Major scale in JI

I've made a discovery that is very important to me.
I hope some of you will find it interesting too that's why I'm sharing it.

For a long time I've had the belief that the common major scale in JI is 1/1
9/8 5/4 4/3 3/2 5/3 15/8 2/1
Somehow I got it in there and it stuck.
I had allready been deviating from this in my tuning of old compositions,
but I thought the compositions were doing something really weird.

The thing is, the Major scale I kept running into seemed to be C(4/3) D(3/2)
E(5/3) F(9/5) G(2/1) A(9/4) B(5/2) C(8/3)
4/3 3/2 5/3 9/5 2/1 9/4 5/2 8/3
(Or with C as 1/1: C(1/1) D(9/8) E(5/4) F(27/20) G(3/2) A(27/16) B(15/8)
C(2/1)
1/1 9/8 5/4 27/20 3/2 27/16 15/8 2/1)

After investigating it makes perfect sense.
I'm not saying this is the only major scale, but it's the most common one it
seems (perhaps by far).

Btw this makes the I-vi-ii-V7-I comma pump go the following way:
C(4/3) E(5/3) G(2/1)
C(4/3) E(5/3) A(9/4) wolf
D(3/2) F(9/5) A(9/4)
D(3/2) G(2/1) B(5/2)
G(1/1) D(3/2) F(9/5) B(5/2) V7
C(4/3) E(5/3) G(2/1)
It sounds great!

The other notes that can be used in the same harmonic root are:
C(4/3) D(9/8) Eb(8/5) E(5/3) F(9/5) F#(15/8) G(2/1) A(9/4) Bb(12/5) B(5/2)
C(8/3)

This scale is so natural, I can improvise very nice classical music in it
with complete ease (much easyer than in 12tet infact without the info JI
provides)

I'll write more on it later.

Marcel

🔗Mike Battaglia <battaglia01@...>

Have you ever considered the possibility that the JI major scale just might
have more than 7 notes?

-Mike

On Tue, Apr 27, 2010 at 6:52 PM, Marcel de Velde <m.develde@...>wrote:

>
>
> I've made a discovery that is very important to me.
> I hope some of you will find it interesting too that's why I'm sharing it.
>
> For a long time I've had the belief that the common major scale in JI is
> 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1
> Somehow I got it in there and it stuck.
> I had allready been deviating from this in my tuning of old compositions,
> but I thought the compositions were doing something really weird.
>
> The thing is, the Major scale I kept running into seemed to be C(4/3)
> D(3/2) E(5/3) F(9/5) G(2/1) A(9/4) B(5/2) C(8/3)
> 4/3 3/2 5/3 9/5 2/1 9/4 5/2 8/3
> (Or with C as 1/1: C(1/1) D(9/8) E(5/4) F(27/20) G(3/2) A(27/16) B(15/8)
> C(2/1)
> 1/1 9/8 5/4 27/20 3/2 27/16 15/8 2/1)
>
> After investigating it makes perfect sense.
> I'm not saying this is the only major scale, but it's the most common one
> it seems (perhaps by far).
>
> Btw this makes the I-vi-ii-V7-I comma pump go the following way:
> C(4/3) E(5/3) G(2/1)
> C(4/3) E(5/3) A(9/4) wolf
> D(3/2) F(9/5) A(9/4)
> D(3/2) G(2/1) B(5/2)
> G(1/1) D(3/2) F(9/5) B(5/2) V7
> C(4/3) E(5/3) G(2/1)
> It sounds great!
>
> The other notes that can be used in the same harmonic root are:
> C(4/3) D(9/8) Eb(8/5) E(5/3) F(9/5) F#(15/8) G(2/1) A(9/4) Bb(12/5) B(5/2)
> C(8/3)
>
> This scale is so natural, I can improvise very nice classical music in it
> with complete ease (much easyer than in 12tet infact without the info JI
> provides)
>
> I'll write more on it later.
>
> Marcel
>
>

🔗Chris Vaisvil <chrisvaisvil@...>

Marcel,

This statement comes from the wikipedia article on microtonal music so I
guess a grain of salt is required.

Composers such as Beethoven <http://en.wikipedia.org/wiki/Beethoven> and
Schubert <http://en.wikipedia.org/wiki/Schubert> made extensive use of the
enharmonic <http://en.wikipedia.org/wiki/Enharmonic> modulation cycles
possible only in a closed tuning of 12 pitches per octave, and not
open-ended tunings like meantone. This led to the demise of meantone
thinking in most of Europe by the outset of the Romantic period.

This would imply that Beethoven didn't compose with just intervals in mind.

What do you think?

thanks,

Chris

On Tue, Apr 27, 2010 at 6:52 PM, Marcel de Velde <m.develde@...>wrote:

🔗Mike Battaglia <battaglia01@...>

🔗Marcel de Velde <m.develde@...>

Hi Mike,

Have you ever considered the possibility that the JI major scale just might
> have more than 7 notes?
>
> -Mike
>

Yes I have. (if i understand what you mean, you mean different ratios for
the same note right?)
In the normal sense where the major scale has for instance 10/9 and 9/8, but
this was long ago and left it as it makes no sense in the end and doesn't
work / sounds horrible.

But in a different way my currect system does have such a thing, when
changing the "harmonic root" when playing the scale.
For instance the 4/3 - 9/5 wolf can become 4/3 - 16/9, or 27/20 - 9/5, but
then we'll have left the harmonic root of 1/1 (something which is very
common).
I'm still investigating the exact circumstances where the harmonic root
changes (though in actual music it is pretty clear usually, I wish to have a
deeper insight, hopefully soon).

But anyhow, it makes life a whole lot easyer to know that the most likely
major scale to start with is 1/1 9/8 5/4 27/20 3/2 27/16 15/8 2/1.
As untill now I was starting with 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1, and than
later after much pain work out the chords and end up with a different scale
which I thought wasn't the normal major scale, but some strange mode due to
the chord progressions.
It just makes life a lot easyer for me.
And for others who are not working with my system will probably find the 1/1
9/8 5/4 27/20 3/2 27/16 15/8 2/1 easy to work with.

As for some other things that bothered me with the classic ji major scale up
till now:
1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1
The 5/3 allways sounded low to me (I've said so before, over a year ago I
think on this list).
Too low.

When for instance playing the C major scale from C till C I hear possible
rhythms in it.
The default rhythms I hear in it are these:

C . d . e . f . G . a . b . c
1 - 2 - 3 - 4 - 1 - 2 - 3 - 4
With the emphasis on 1 (C and G)

Or:
C . . . D . e . F . g . A . b . C
1 - 2 - 1 - 2 - 1 - 2 - 1 - 2 - 1

And I find my ear expects repetitions at these rhythms.

With classic JI major scale these repetitions are not there.

But with my new major scale they are!
And it sounds beautifully rhythmically:

1/1 . 9/8 . 5/4 . 27/20 . 3/2 . 27/16 . 15/8 . 2/1
1 . 2 . 3 . 4 . 1 . 2 . 3 . 4
The difference in 4 I hear as correct rhythmically somehow as they have a
different function.
The main thing I expect to hear rhythmically repeating is 1 . 2 . 3, and it
does.

1/1 . . . 9/8 . 5/4 . 27/20 . 3/2 . 27/16 . 15/8 . 2/1
1 . 2 . 1 . 2 . 1 . 2 . 1 . 2 . 1
Perfect repetition here.

I can't recomend enough playing the scale with rhythm like this.
My ear atleast is telling me that this is 100% correct.
It's beautifull.

Marcel

🔗Mike Battaglia <battaglia01@...>

I see. Well, as you no doubt already know, this version of the major scale
completely destroys the IV chord. And you no doubt also know that
historically speaking, common practice music has strived to keep that chord
pure almost as a matter of paramount importance.

But let's disregard all of that and just go with your scale, which has come
out of an arbitrary mathematical operation that you appear to have invented
for no reason. We'll get used to the horrid wolf over the IV chord
eventually, I guess.

-Mike

On Tue, Apr 27, 2010 at 9:53 PM, Marcel de Velde <m.develde@...>wrote:

>
>
> Hi Mike,
>
>
> Have you ever considered the possibility that the JI major scale just might
>> have more than 7 notes?
>>
>> -Mike
>>
>
> Yes I have. (if i understand what you mean, you mean different ratios for
> the same note right?)
> In the normal sense where the major scale has for instance 10/9 and 9/8,
> but this was long ago and left it as it makes no sense in the end and
> doesn't work / sounds horrible.
>
> But in a different way my currect system does have such a thing, when
> changing the "harmonic root" when playing the scale.
> For instance the 4/3 - 9/5 wolf can become 4/3 - 16/9, or 27/20 - 9/5, but
> then we'll have left the harmonic root of 1/1 (something which is very
> common).
> I'm still investigating the exact circumstances where the harmonic root
> changes (though in actual music it is pretty clear usually, I wish to have a
> deeper insight, hopefully soon).
>
> But anyhow, it makes life a whole lot easyer to know that the most likely
> major scale to start with is 1/1 9/8 5/4 27/20 3/2 27/16 15/8 2/1.
> As untill now I was starting with 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1, and
> than later after much pain work out the chords and end up with a different
> scale which I thought wasn't the normal major scale, but some strange mode
> due to the chord progressions.
> It just makes life a lot easyer for me.
> And for others who are not working with my system will probably find the
> 1/1 9/8 5/4 27/20 3/2 27/16 15/8 2/1 easy to work with.
>
> As for some other things that bothered me with the classic ji major scale
> up till now:
>
> 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1
> The 5/3 allways sounded low to me (I've said so before, over a year ago I
> think on this list).
> Too low.
>
> When for instance playing the C major scale from C till C I hear possible
> rhythms in it.
> The default rhythms I hear in it are these:
>
> C . d . e . f . G . a . b . c
> 1 - 2 - 3 - 4 - 1 - 2 - 3 - 4
> With the emphasis on 1 (C and G)
>
> Or:
> C . . . D . e . F . g . A . b . C
> 1 - 2 - 1 - 2 - 1 - 2 - 1 - 2 - 1
>
> And I find my ear expects repetitions at these rhythms.
>
> With classic JI major scale these repetitions are not there.
>
> But with my new major scale they are!
> And it sounds beautifully rhythmically:
>
>
> 1/1 . 9/8 . 5/4 . 27/20 . 3/2 . 27/16 . 15/8 . 2/1
> 1 . 2 . 3 . 4 . 1 . 2 . 3 . 4
> The difference in 4 I hear as correct rhythmically somehow as they have a
> different function.
> The main thing I expect to hear rhythmically repeating is 1 . 2 . 3, and it
> does.
>
>
> 1/1 . . . 9/8 . 5/4 . 27/20 . 3/2 . 27/16 . 15/8 . 2/1
> 1 . 2 . 1 . 2 . 1 . 2 . 1 . 2 . 1
> Perfect repetition here.
>
> I can't recomend enough playing the scale with rhythm like this.
> My ear atleast is telling me that this is 100% correct.
> It's beautifull.
>
> Marcel
>
>
>
>
>
>
>

🔗Marcel de Velde <m.develde@...>

On 28 April 2010 04:02, Mike Battaglia <battaglia01@...> wrote:

> I see. Well, as you no doubt already know, this version of the major scale
> completely destroys the IV chord. And you no doubt also know that
> historically speaking, common practice music has strived to keep that chord
> pure almost as a matter of paramount importance.
>
> But let's disregard all of that and just go with your scale, which has come
> out of an arbitrary mathematical operation that you appear to have invented
> for no reason. We'll get used to the horrid wolf over the IV chord
> eventually, I guess.
>

Very friendly and positive again Mike ;)
If you'd give my theory a serious try you'de see that it works perfectly
well in practice.

The wolf IV depends on the situation.
It does not have to be dissonant. And it will not give an unacceptable wolf
IV, as when the musical context calls for it this will not be a wolf
anymore.
The 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1 is also a major scale.
As is 1/1 10/9 5/4 4/3 3/2 5/3 15/8 2/1.
I merely said the most used seems to be 1/1 9/8 5/4 27/20 3/2 27/16 15/8 2/1
(with extra notes 6/5 45/32 9/5)
Basically try any simple childrens or old holliday song in this scale and
good chance it'll work out (without any change of harmonic root needed)

They're all subsets / modes of 1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1
btw

Marcel

🔗Marcel de Velde <m.develde@...>

🔗Mike Battaglia <battaglia01@...>

🔗Marcel de Velde <m.develde@...>

🔗Cox Franklin <franklincox@...>

Chris,

the date of the demise of meantone approaches is still open to debate. See Barbieri's Enharmonic (and his article on temperament on Piano: An Encyclopedia), which is the most authoritative recent book on the subject. The majority of musicians, especially central Europe, were moving quickly toward equal temperament (I believe Barbieri mentions that in Germany ET had won out by about 1840), but there appear to have been significant "holdouts"; Italy and England appear to have moved toward ET a generation or so later.

For string players, I believe that there were at least three different approaches: equal temperament (favored by Spohr, although it is still difficult to tell how accurately string players actually played in ET; they would have needed in-tune pianos to tune ET accurately, it is still uncertain what the standards of piano tuning were at that time), holdovers from the meantone/syntonic tradition (such as Joachim, one of the leading violinists in Germany), and Pythagorean-ish tuning, with high leading tones (this is the approach that one finds in influential performers such as Sarasate and Casals in the early years of the era of recordings).

So although composers were writing in "functional ET" at the beginning of the Romantic era, it is unlikely that performers were actually performing consistently in ET. It is highly unlikely, though, that all that many performers would have been using JI (syntonic) tuning. According to Barbieri, even in the Baroque period, it was only the finest players who had mastered this. However, string players were definitely distinguishing between enharmonic notes such as C# and Db at the very least a decade or two into the 19th century.

best

Franklin

Dr. Franklin Cox

1107 Xenia Ave.

Yellow Springs, OH 45387

(937) 767-1165

franklincox@yahoo.com

--- On Wed, 4/28/10, Chris Vaisvil <chrisvaisvil@...> wrote:

From: Chris Vaisvil <chrisvaisvil@...>
Subject: Re: [tuning] Most common Major scale in JI
To: tuning@yahoogroups.com
Date: Wednesday, April 28, 2010, 12:40 AM

Marcel,

This statement comes from the wikipedia article on microtonal music so I guess a grain of salt is required.

Composers such as Beethoven and Schubert made extensive use of the enharmonic
modulation cycles possible only in a closed tuning of 12 pitches per
octave, and not open-ended tunings like meantone. This led to the
demise of meantone thinking in most of Europe by the outset of the
Romantic period.

This would imply that Beethoven didn't compose with just intervals in mind.

What do you think?

thanks,

Chris

On Tue, Apr 27, 2010 at 6:52 PM, Marcel de Velde <m.develde@gmail. com> wrote:

I've made a discovery that is very important to me.
I hope some of you will find it interesting too that's why I'm sharing it.

For a long time I've had the belief that the common major scale in JI is 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1

Somehow I got it in there and it stuck.
I had allready been deviating from this in my tuning of old compositions, but I thought the compositions were doing something really weird.

The thing is, the Major scale I kept running into seemed to be C(4/3) D(3/2) E(5/3) F(9/5) G(2/1) A(9/4) B(5/2) C(8/3)

4/3 3/2 5/3 9/5 2/1 9/4 5/2 8/3
(Or with C as 1/1: C(1/1) D(9/8) E(5/4) F(27/20) G(3/2) A(27/16) B(15/8) C(2/1)
1/1 9/8 5/4 27/20 3/2 27/16 15/8 2/1)

After investigating it makes perfect sense.
I'm not saying this is the only major scale, but it's the most common one it seems (perhaps by far).

It sounds great!

The other notes that can be used in the same harmonic root are:
C(4/3) D(9/8) Eb(8/5) E(5/3) F(9/5) F#(15/8) G(2/1) A(9/4) Bb(12/5) B(5/2) C(8/3)

This scale is so natural, I can improvise very nice classical music in it with complete ease (much easyer than in 12tet infact without the info JI provides)

I'll write more on it later.

Marcel

🔗Chris Vaisvil <chrisvaisvil@...>

I don't meant to be terribly contradictory - but what if I told you I
wrote a piece of music in a specific non-JI tuning?

Would you insist on disagreeing with me?

The piece I submitted to you was certainly written with 12 tet in
mind... though I am curious as to what you will interpret it to be.
And having pure stacked fifths will probably sound really nice - but
when I start collapsing and expanding intervals chromatically I'm at a
lost how that would be interpreted in JI - thus why I submitted it.

I also am curious how you'd interpret something like this piano sonata.

http://alonetone.com/vaisvil/tracks/piano-sonata-01-pianoteq.mp3

Though first things first.

I'm trying to learn here. Right now I see EDOs having a massive
advantage over other systems although JI sounds like a nice place to
go I have yet seen it work for the style of music I like to write. The
only other really attractive tuning systems were the horograms by Erv
Wilson (I think I have that right) but I've not yet run into a scala
file that I recognize as such.

Chris

On Tue, Apr 27, 2010 at 10:17 PM, Marcel de Velde <m.develde@gmail.com> wrote:
>
>
>
> Hi Chris,
>
>> Marcel,
>>
>> This statement comes from the wikipedia article on microtonal music so I guess a grain of salt is required.
>>
>> Composers such as Beethoven and Schubert made extensive use of the enharmonic modulation cycles possible only in a closed tuning of 12 pitches per octave, and not open-ended tunings like meantone. This led to the demise of meantone thinking in most of Europe by the outset of the Romantic period.
>>
>>
>> This would imply that Beethoven didn't compose with just intervals in mind.
>>
>> What do you think?
>>
>> thanks,
>>
>> Chris
>
> I have no idea :)
> But it doesn't matter to me really.
> In my opinion music is JI, no matter wether a composer thought so or not, if the composer wrote music, he wrote JI.
>
> Marcel
>

🔗Marcel de Velde <m.develde@...>

Hi Chris,

I don't meant to be terribly contradictory - but what if I told you I
> wrote a piece of music in a specific non-JI tuning?
>
> Would you insist on disagreeing with me?
>

Well, yes and no :)
I think the way we interpret harmony and melody is JI.
It's a language I think to our brain.
And 12tet for instance is something that makes absolutely no sense to me,
non rational intervals. Just isn't how music works I think on a fundamental
level.
But, on the other hand, if as an artist you choose to have a mistuning
deliberately, then that's something too.
However, the real harmonic and melodic meaning I think will still be heard
as JI by our brain, only slightly mistuned to our ear and less clear
perhaps.

>
> The piece I submitted to you was certainly written with 12 tet in
> mind... though I am curious as to what you will interpret it to be.
> And having pure stacked fifths will probably sound really nice - but
> when I start collapsing and expanding intervals chromatically I'm at a
> lost how that would be interpreted in JI - thus why I submitted it.
>
> I also am curious how you'd interpret something like this piano sonata.
>
> http://alonetone.com/vaisvil/tracks/piano-sonata-01-pianoteq.mp3
>

All chromatic music will work in JI.
Moderate chromatic music seems to me to work in harmonic-6-limit. Chromatics
comming from major to minor steps 25/24 mostly.
Extreme chromatics would be harmonic-7-limit.
12tet simply doesn't have enough keys to express anything that can't be done
in harmonic-7-limit I think.
Though interpreting chromatic music in JI is often extremely hard /
difficult.
I really like your piece. Please keep composing in 12tet :) As JI will give
you hell.

Marcel

🔗Marcel de Velde <m.develde@...>

🔗genewardsmith <genewardsmith@...>

🔗Cox Franklin <franklincox@...>

🔗Michael <djtrancendance@...>

>"You should care about this. How would you treat the music of Karlheinz Stockhausen, who wrote specifically for ET?"

Note I have few to no claims about my music history knowledge. However, simply from my knowledge of 12TET, 7TET, 10TET, 19TET, 34TET, 43TET, 1/4 comma meantone, JI Diatonic, 22TET, Ptolemy/Homalon and others plus loads of psychoacoustics from my DSP programming background I will say this much.
Musicians don't hear in one form or another...they hear a combination of things including
A) Root note periodicity...and/or temperament off it, too much of which produces beating
B) Overtone intersection / the formal definition of "consonance"...or lack of it
C) Lack of overtone intersection, which helps avoid dissonance...or near intersection which produces strong beating
D)
Character of intervals, which is highly subjective IE a mathematically consonant interval may have a tense mood despite the "consonance".
E) A sense of which chords are standard/tonic/dominant/etc. that give them a clue of where the "resolve" points in a piece are.
F) Periodic buzz...which some people like and others hate...often caused by "perfect" JI chords such as 4:5:6:7.

Can JI help to solve these things and help a musician gain control the balance of tensity? Of course. Can things like altering temperament, timbre, volume, and other psychoacoustic factors do the same thing? Of course. JI is just an "easy" way to quantify a whole bunch of the above phenomena and can be used to "estimate" what's going on in things like tempered scales...but by no means ensures say, a scale written in JI will best a scale written in some sort of temperament. For every few perfect intervals you make, you from things like commas from those stacked "ultra pure" intervals and imperfections in other intervals. And, as "Igs" has shown...make too many impure intervals in a scale and you're likely to indirectly create some very near pure ones.
A side question for your "common major scale" is what chords become more pure and which ones less pure vs. JI (and not just things like triads and 7ths, but suspended and augmented chords, 9ths, 11ths, 13ths. Diatonic JI, for example, does almost entirely perfect triads...but what if the composer wants something different? Give an honest assessment of what is gained and lost in your JI major scale...and I'm pretty sure you'll get people to take this idea much more seriously. :-)

--- On Tue, 4/27/10, Cox Franklin <franklincox@...> wrote:

From: Cox Franklin <franklincox@yahoo.com>
Subject: Re: [tuning] Most common Major scale in JI
To:
tuning@yahoogroups.com
Date: Tuesday, April 27, 2010, 8:27 PM

Marcel,

You should care about this. How would you treat the music of Karlheinz Stockhausen, who wrote specifically for ET? He even used a scale of tempi based on the 12th root of 2 and had special metronomes built for these tempi.

I think that eventually you'll have to come to the realization that composers consciously composed for other tuning systems than JI. I cannot comprehend what this would be a matter of indifference for you.

Franklin
I have no idea :)
But it doesn't matter to me
really.

In my opinion music is JI, no matter wether a composer thought so or not, if the composer wrote music, he wrote JI.

Marcel

🔗Cox Franklin <franklincox@...>

Michael,

that's all very nice, but Stockhausen had very fine hearing, was one of the great electronic composers of the century, and was a maniac for detail. He also personally supervised every detail of the production of the recordings. He knew what he wanted.

And, by the way, he was also composing serial, "atonal" music.

Or how about Milton Babbitt, whose entire serial compositional system is premised upon every interval of a given class (say M3rds) being the same size? He personally fed in precise frequencies for his complex electronic pieces to a computer, way back in the 1960's. He knew what he wanted, and it was 12 edo.

There are many people out there with finely-developed hearing for 12 edo, and even a number of people with perfect-pitch edo. Whatever individual theories of hearing might be, ideally they should correspond to facts.

Now 12 edo is not my favorite for traditional music; in fact, I think it's one of the things sucking the life out of Classical music performances. But when knowledgable, gifted composers such as Schoenberg, Boulez (one of the finest conductors of 20th century music), Stockhausen, and Babbitt insist on 12 edo for their music, it seems reasonable to take these admonitions seriously.

Franklin

--- On Wed, 4/28/10, Michael <djtrancendance@...> wrote:

From: Michael <djtrancendance@...>
Subject: Re: [tuning] Most common Major scale in JI
To: tuning@yahoogroups.com
Date: Wednesday, April 28, 2010, 4:13 AM

>"You should care about this. How would you treat the music of Karlheinz Stockhausen, who wrote specifically for ET?"

Musicians don't hear in one form or another...they hear a combination of things including

A) Root note periodicity. ..and/or temperament off it, too much of which produces beating

B) Overtone intersection / the formal definition of "consonance" ...or lack of it

C) Lack of overtone intersection, which helps avoid dissonance.. .or near intersection which produces strong beating

Character of intervals, which is highly subjective IE a mathematically consonant interval may have a tense mood despite the "consonance" .

E) A sense of which chords are standard/tonic/ dominant/ etc. that give them a clue of where the "resolve" points in a piece are.

F) Periodic buzz...which some people like and others hate...often caused by "perfect" JI chords such as 4:5:6:7.

🔗cameron <misterbobro@...>

Your JI version sounds better than the 12-tET to me. More like how an acoustic rendition might be performed.

But you guys really should pay more attention to what's going on around you. In recent posts, there have been some "regular tuning paradigm" tunings posted here as proposals for tunings which are small/usable but unusual (ie not 12-tET), and "pleasant" for want of a better word. At least three of them had a "fa" about a comma off, two down and one up.

-Cameron Bobro

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
> >> "We wish you a merry *CLANG*"
> >> "Happy birthday to you, happy birthday to you, happy birthday dear *BLAM*"
> >> "Jingle bells, jingle bells, jingle all the way, *HEAD BANG AGAINST WALL*"
> >>
> >> That would be my reaction.
> >>
> >> -Mike
> >
> > Why don't you actually try it out.
> > And then even post the results here specifying how you tuned it so I can
> > check you didn't mess up.
> > And perhaps then speak out like you did ;)
> >
> > Marcel
>
>
> Ok here "We wish you a merry christmas":
>
> /tuning/files/Marcel%20de%20Velde/major%20fun/WeWishYouAMerryXmas-JI.mid
>
> And 12tet for comparison:
> /tuning/files/Marcel%20de%20Velde/major%20fun/WeWishYouAMerryXmas-12tet.mid
>
>
> I'll do the happy birthday and jingle bells too later.
> Btw if the midi sound isn't clear use program 58 trombone or something
> like that.
> I'll do them all later with a clearer sound and check them a bit better.
>
> Anyhow... where's the *blam* head against wall blabla etc??
>
> Marcel
>

🔗cameron <misterbobro@...>

And, Igliashon recently expressed a downright preference for 27/20 over 4/3. I use the shadowy region between 13/10 and 17/13 myself.

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
> Your JI version sounds better than the 12-tET to me. More like how an acoustic rendition might be performed.
>
> But you guys really should pay more attention to what's going on around you. In recent posts, there have been some "regular tuning paradigm" tunings posted here as proposals for tunings which are small/usable but unusual (ie not 12-tET), and "pleasant" for want of a better word. At least three of them had a "fa" about a comma off, two down and one up.
>
> -Cameron Bobro
>
> --- In tuning@yahoogroups.com, Marcel de Velde <m.develde@> wrote:
> >
> > >> "We wish you a merry *CLANG*"
> > >> "Happy birthday to you, happy birthday to you, happy birthday dear *BLAM*"
> > >> "Jingle bells, jingle bells, jingle all the way, *HEAD BANG AGAINST WALL*"
> > >>
> > >> That would be my reaction.
> > >>
> > >> -Mike
> > >
> > > Why don't you actually try it out.
> > > And then even post the results here specifying how you tuned it so I can
> > > check you didn't mess up.
> > > And perhaps then speak out like you did ;)
> > >
> > > Marcel
> >
> >
> > Ok here "We wish you a merry christmas":
> >
> > /tuning/files/Marcel%20de%20Velde/major%20fun/WeWishYouAMerryXmas-JI.mid
> >
> > And 12tet for comparison:
> > /tuning/files/Marcel%20de%20Velde/major%20fun/WeWishYouAMerryXmas-12tet.mid
> >
> >
> > I'll do the happy birthday and jingle bells too later.
> > Btw if the midi sound isn't clear use program 58 trombone or something
> > like that.
> > I'll do them all later with a clearer sound and check them a bit better.
> >
> > Anyhow... where's the *blam* head against wall blabla etc??
> >
> > Marcel
> >
>

🔗Mike Battaglia <battaglia01@...>

Whoa, am I missing something here? Are you tuning it as you said? The IV
chord should have a fifth that's almost 20 cents flat. The JI version you
sent sounded fine, I could hear a few comma shifts but nothing noticeable

Can the timbre of an instrument really hide intervals that are THAT out of
tune?

-Mike

On Tue, Apr 27, 2010 at 11:19 PM, Marcel de Velde <m.develde@gmail.com>wrote:

>
>
> >> "We wish you a merry *CLANG*"
> >> "Happy birthday to you, happy birthday to you, happy birthday dear
> *BLAM*"
> >> "Jingle bells, jingle bells, jingle all the way, *HEAD BANG AGAINST
> WALL*"
> >>
> >> That would be my reaction.
> >>
> >> -Mike
> >
> > Why don't you actually try it out.
> > And then even post the results here specifying how you tuned it so I can
> > check you didn't mess up.
> > And perhaps then speak out like you did ;)
> >
> > Marcel
>
> Ok here "We wish you a merry christmas":
>
>
> /tuning/files/Marcel%20de%20Velde/major%20fun/WeWishYouAMerryXmas-JI.mid
>
> And 12tet for comparison:
>
> /tuning/files/Marcel%20de%20Velde/major%20fun/WeWishYouAMerryXmas-12tet.mid
>
> I'll do the happy birthday and jingle bells too later.
> Btw if the midi sound isn't clear use program 58 trombone or something
> like that.
> I'll do them all later with a clearer sound and check them a bit better.
>
> Anyhow... where's the *blam* head against wall blabla etc??
>
> Marcel
>
>

🔗cameron <misterbobro@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Your JI version sounds better than the 12-tET to me. More like how an acoustic rendition might be performed.
> >
> > But you guys really should pay more attention to what's going on around you. In recent posts, there have been some "regular tuning paradigm" tunings posted here as proposals for tunings which are small/usable but unusual (ie not 12-tET), and "pleasant" for want of a better word. At least three of them had a "fa" about a comma off, two down and one up.
>
> I suppose "you guys" is me. I'm generally a fan of a lot of tunings
> that come from "alternative" theories (I thought Mike's phi-based
> scales sounded amazing, and I'm not sure why). I just didn't think
> that a tuning with 27/20 used as the fourth, and allowing no
> possibility for comma shifts, was suitable as "THE JI TUNING FOR
> COMMON PRACTICE MUSIC." That's all.
>
> I am clearly losing my mind, though, because I didn't hear any wolf
> fourths in the MIDI that Marcel sent.
>
> -Mike
>

"you guys" is "the both y'all". You because you should know by now that a "mistuned" fourth can work sweetly in context, and Marcel because he should know by now that he's not actually proposing some unheard of or new thing. See Tartini, early 18th century for crying out loud, for historical references to allowing the syntonic comma to shift in certain places. IIRC, his fundamental principle was maintaining pure intervals in the upper voices and the bass would do the syntonic adapting (syntonic shifting being what Marcel is actually doing whether he wants to admit it or not). Marcel should take a look at Tartini's 7th-partial proposals, too. Like going to 7-limit for descending Te, but not for an ascending Si, IIRC.

-Cameron Bobro

🔗cameron <misterbobro@...>

🔗Mike Battaglia <battaglia01@...>

🔗cameron <misterbobro@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > "you guys" is "the both y'all". You because you should know by now that a "mistuned" fourth can work sweetly in context
>
> Yeah, but... I dunno about this context :P
>
> I love 27/20, wasn't I the guy pimping it a year ago with the C Eb+ G
> Bb+ D F+ chord I'm so fond of?

Say that's nice- do you do that in 72? I'd like to here some audio examples in, er, context.

> But do you really think it has a use in
> common practice music as the root of IV chord? And you're saying
> Tartini proposed this?

No, and (I'd have to check), I don't think so. Though it shouldn't be impossible that allowing the bass to shift syntonically might result in just such a thing... have to check it out when I get the chance.

>
> Keep in mind that Marcel isn't saying that 27/20 is allowed sometimes
> as the fourth degree of the scale, and 4/3 other times... he's saying
> that 27/20 is the fourth degree of the scale, period. And that all of
> the ridiculous wolfs that would occur are just something that we have
> to get used to, and that it sounds better when we do.

Well I've been arguing against his (insert Godwin's Law comment here) concept of tuning "law" from the beginning. I'm hoping that if he will understand CONTEXT that he'll get down to writing music in his tuning and cut out the historical revisionisms.

🔗Mike Battaglia <battaglia01@...>

> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > > "you guys" is "the both y'all". You because you should know by now that a "mistuned" fourth can work sweetly in context
> >
> > Yeah, but... I dunno about this context :P
> >
> > I love 27/20, wasn't I the guy pimping it a year ago with the C Eb+ G
> > Bb+ D F+ chord I'm so fond of?
>
> Say that's nice- do you do that in 72? I'd like to here some audio examples in, er, context.

Yeah, because I have no idea how to work in any system besides 72. 72
is easy - I load up a synth in sonar and come up with 6 midi channels
and detune each one 1/6 of a half step. Working in 31-tet or something
seems impossible.

If you want to load up scala and test it out, the chord works out to
40:48:60:72:90:108:135:162.

Someday I'll get my fancy online microtonal ajax/php-based
MIDI-er-izer out, and then I'll be able to send you URLs like
http://www.mikebattagliamusic.com/microtonalchordgenerator.php?chord=40:48:60:72:90:108:135:162.

but not today.

Another cool use for 27/20 is in the chord C Eb+ F+. So a 6:5 with a
9:8 on top of it. This works out to 20:24:27, isn't too bad at all.

You think that fourth is sharp, right? BUT WAIT! Just sit there and
listen to it for a while. I swear it sounds like oranges. The "citric"
fourth, I call it. It's awesome.

> > But do you really think it has a use in
> > common practice music as the root of IV chord? And you're saying
> > Tartini proposed this?
>
> No, and (I'd have to check), I don't think so. Though it shouldn't be impossible that allowing the bass to shift syntonically might result in just such a thing... have to check it out when I get the chance.

Right, well obviously the root shift by 81/80 isn't going to be the
end of the world. But again, Marcel isn't allowing 1/1 to shift to
81/80. So the IV chord would be (relative to IV) 1/1 5/4 40/27, with a
fifth of 680 cents.

Which can be a cool sound in certain contexts. I don't think "most of
common practice music" is the right context for it.

-Mike

🔗Marcel de Velde <m.develde@...>

Hi Mike,

Whoa, am I missing something here? Are you tuning it as you said? The IV
> chord should have a fifth that's almost 20 cents flat. The JI version you
> sent sounded fine, I could hear a few comma shifts but nothing noticeable
>
> Can the timbre of an instrument really hide intervals that are THAT out of
> tune?
>
> -Mike
>

There are no comma shifts at all as it's all in one "harmonic root" of C,
the Major scale is on F, the piece is completely in F major.
The reason you don't hear the wolfs is because wolfs sound perfectly good
where they're supposed to go.
The wolfs aren't a compromise in any way. To play a non wolf chord there in
this case would be too low and out of tune.
This would actually be noticable. It's just that when one pays a wolf where
it isn't supposed to go then it's way more noticable than when playing a non
wolf where a wolf is supposed to go because of the dissonance of the wolf.
But please do change the sound to trombone for instance, it'll still sound
right.

Also try the happy birthday and jingle bells. They'll work perfectly too.
Happy birthday goes like 1/1 1/1 9/8 1/1 4/3 5/4, 1/1 1/1 9/8 1/1 3/2 4/3,
1/1 1/1 2/1 5/3 4/3 5/4 9/8, 9/5 9/5 5/3 4/3 3/2 4/3.
Where 1/1 is the harmonic root of 1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/4 9/5 15/8
2/1.
And 4/3 is the "key root" for major.

I have no time most of the day away from the computer, but if you won't
render them I'll do so hopefully later today (and I'll change the sound to
more clear trombone or something like that).

It's really like I said.
A lot / perhaps most old simple childrens and holliday songs in major are
according to the scale I gave.

Marcel

🔗Marcel de Velde <m.develde@...>

I love 27/20, wasn't I the guy pimping it a year ago with the C Eb+ G
> Bb+ D F+ chord I'm so fond of? But do you really think it has a use in
> common practice music as the root of IV chord? And you're saying
> Tartini proposed this?
>

Ah ok I didn't know :)
I thought you had a problem with 27/20 in general.

Whoho, nooo I'm not at all saying that.
I'm saying that 27/20 is the fourth degree of the major mode that is on the
4/3 of the harmonic root of 1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1
And as such it is fixed.
But there are 56 unique 7-note modes (which don't have a 25/24 step / don't
switch major minor at from thesame degree) in my system.
I'm simply saying that what we call major mode is a lot of the time (perhaps
most of the time) a specific one of these modes.
I'm not excluding the other 55 modes, some of whom have the exact same major
notes in 12tet, but are different in JI (and who have the VI without wolf)
I'm allways saying musical context with everything.

Marcel

🔗Chris Vaisvil <chrisvaisvil@...>

Hi Marcel,

Again I'm not a tuning expert so I have only one example:

Gamelon music isn't near JI as far as I know - so then.... where does
that put the observation that we (universally) interpret melody and
harmony as JI?

On the other hand - I know that Indian music uses just intervals with
microtonal inflections so that is in your favour.

Perhaps someone else can bring up more examples if they see my line of thought.

The point is important because cultural bias would contradict a
sweeping statement such as
" I think the way we interpret harmony and melody is JI.
It's a language I think to our brain."

For the statement to be true it has to be true for all of humanity.

Chris

On Tue, Apr 27, 2010 at 11:09 PM, Marcel de Velde <m.develde@...> wrote:

> I think the way we interpret harmony and melody is JI.
> It's a language I think to our brain.
> And 12tet for instance is something that makes absolutely no sense to me, non rational intervals. Just isn't how music works I think on a fundamental level.
> But, on the other hand, if as an artist you choose to have a mistuning deliberately, then that's something too.
> However, the real harmonic and melodic meaning I think will still be heard as JI by our brain, only slightly mistuned to our ear and less clear perhaps.
>>

🔗Cox Franklin <franklincox@...>

Mike,
The problem is that that there is so much noise from the drum track in Marcel's version, and the voices he chose are so soupy, that one really can't here much of anything. There is a subtle difference in a few chords that would allow one to distinguish the different tuning approaches, but beyond that it's all mush.
The same problem exists in the real world as well. If string players use a wide vibrato, as long as they're in the ballpark, it really doesn't really matter what tuning system they are using. This was discovered by the Guarnari Quartet back in the 60's and has been imitated by chamber groups since then. In general, only the historically-informed groups such as the Quartet Mosäiques and Eroica Quartet attempt to play without the soupy vibrato. And the danger you run here is that you can actually hear the tunings, and they sound unusual, so some idiot will always come along and claim that the players are out of tune. Harnoncourt had a brutal putdown of this sort of idiocy in one of his books.
If Marcel were to use relatively pure sounds, I'm sure the wolfs will be more apparent--if, that is, his tunings are actually accurate. There is a certain tolerance for wolfs in pre-dominant-type chords, so not every chord actually has to be pure. But some of this was taken care of by the inversion--a ii chord, which is hopelessly out of tune in the syntonic scale, sounds tolerable as a 6 (1st inversion chord). And lo and behold, ii chords in root position were relatively rare before the latter part of the Baroque period, whereas IV chords in root position were common. One still finds this distinction in Sechter's harmony method, one of the most influential manuals of the 19th century (Sechter was the most powerful teacher in Austria-Hungary and was the teacher of Bruckner).
Franklin

Dr. Franklin Cox

1107 Xenia Ave.

Yellow Springs, OH 45387

(937) 767-1165

franklincox@yahoo.com

--- On Wed, 4/28/10, Mike Battaglia <battaglia01@...> wrote:

From: Mike Battaglia <battaglia01@...>
Subject: Re: [tuning] Re: Most common Major scale in JI
To: tuning@yahoogroups.com
Date: Wednesday, April 28, 2010, 7:55 AM

> Your JI version sounds better than the 12-tET to me. More like how an acoustic rendition might be performed.

> But you guys really should pay more attention to what's going on around you. In recent posts, there have been some "regular tuning paradigm" tunings posted here as proposals for tunings which are small/usable but unusual (ie not 12-tET), and "pleasant" for want of a better word. At least three of them had a "fa" about a comma off, two down and one up.

I suppose "you guys" is me. I'm generally a fan of a lot of tunings

that come from "alternative" theories (I thought Mike's phi-based

scales sounded amazing, and I'm not sure why). I just didn't think

that a tuning with 27/20 used as the fourth, and allowing no

possibility for comma shifts, was suitable as "THE JI TUNING FOR

COMMON PRACTICE MUSIC." That's all.

I am clearly losing my mind, though, because I didn't hear any wolf

fourths in the MIDI that Marcel sent.

-Mike

🔗Chris Vaisvil <chrisvaisvil@...>

🔗martinsj013 <martinsj@...>

🔗Cox Franklin <franklincox@...>

Very likely. Vibrato in earlier times was used much more sparingly than now. There have been ebbs and flows of it; it was used as an ornament in Baroque music, and a few Italian violinists apparently used it regularly. It's interesting that in the early Romantic period there was a strong reaction against "warbling". There's been a lot of ink spilt over when and how much vibrato has been used, but it's safe to say that only with the age of recordings did it become required of all players (except, of course, HIP performers).
Franklin

Dr. Franklin Cox

1107 Xenia Ave.

Yellow Springs, OH 45387

(937) 767-1165

franklincox@...

--- On Wed, 4/28/10, Chris Vaisvil <chrisvaisvil@...> wrote:

From: Chris Vaisvil <chrisvaisvil@...>
Subject: Re: [tuning] Most common Major scale in JI
To: tuning@yahoogroups.com
Date: Wednesday, April 28, 2010, 3:56 PM

Hi Franklin,

If I combine your two posts

"If string players use a wide vibrato, as long as they're in the

ballpark, it really doesn't really matter what tuning system they are

using. This was discovered by the Guarnari Quartet back in the 60's

and has been imitated by chamber groups since then. "

"For string players, I believe that there were at least three

different approaches: " [to tuning]

If I remember right vibrato was not always used? (I *think* this was

mentioned in theory class).

If that is correct (and I am thinking Renaissance and earlier did not

use vibrato) was vibrato introduced purposefully to blur the tuning as

a solution to the multiple competing tuning systems?

Thanks,

Chris

On Tue, Apr 27, 2010 at 10:47 PM, Cox Franklin <franklincox@ yahoo.com> wrote:

🔗cityoftheasleep <igliashon@...>

🔗Michael <djtrancendance@...>

🔗genewardsmith <genewardsmith@...>

🔗Carl Lumma <carl@...>

🔗Cox Franklin <franklincox@...>

As I said before, vibrato was used in the Baroque period as an ornament, i.e. on important notes, at cadences, etc. Certain Italian violinists played with a more continuous vibrato. This is indubitable. But continuous vibrato was almost certain not common. Recordings of Baroque music are attempts to recapture original performance styles, but they are not documentary evidence.

Clive Brown's Classical and Romantic Performing Practice 1750-1900 is one of the most important recent reference works. He believes that vibrato was used even less in the early Romantic period than earlier, at least in the parts of Europe covered in his book--there appears to have been a reaction against vibrato. When most people refer to "Romantic style" now they actually mean the big-vibrato modern style. This was almost certainly not commonly used in the Romantic period.

Franklin

Dr. Franklin Cox

1107 Xenia Ave.

Yellow Springs, OH 45387

(937) 767-1165

franklincox@...

--- On Wed, 4/28/10, Carl Lumma <carl@...> wrote:

From: Carl Lumma <carl@...>
Subject: [tuning] Re: Most common Major scale in JI
To: tuning@yahoogroups.com
Date: Wednesday, April 28, 2010, 7:27 PM

I wrote:

> Certainly into the Baroque, we believe [vibrato] not used.

At least, not in the chamber music I listen to... maybe

moreso in Baroque opera (which I don't know much about).

And of course I don't mean it was *never* used, just that

it wasn't the always-on kind of thing it later became.

-Carl

🔗Marcel de Velde <m.develde@...>

🔗Michael <djtrancendance@...>

🔗genewardsmith <genewardsmith@...>

🔗Michael <djtrancendance@...>

🔗cityoftheasleep <igliashon@...>

Marcel, have you given much thought as to how your theories might be applied practically in the live performance of music? Given that "common practice" music is, well, "commonly practiced"?

-Igs

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
> I love 27/20, wasn't I the guy pimping it a year ago with the C Eb+ G
> > Bb+ D F+ chord I'm so fond of? But do you really think it has a use in
> > common practice music as the root of IV chord? And you're saying
> > Tartini proposed this?
> >
>
> Ah ok I didn't know :)
> I thought you had a problem with 27/20 in general.
>
>
> >
> > Keep in mind that Marcel isn't saying that 27/20 is allowed sometimes
> > as the fourth degree of the scale, and 4/3 other times... he's saying
> > that 27/20 is the fourth degree of the scale, period. And that all of
> > the ridiculous wolfs that would occur are just something that we have
> > to get used to, and that it sounds better when we do.
> >
>
> Whoho, nooo I'm not at all saying that.
> I'm saying that 27/20 is the fourth degree of the major mode that is on the
> 4/3 of the harmonic root of 1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1
> And as such it is fixed.
> But there are 56 unique 7-note modes (which don't have a 25/24 step / don't
> switch major minor at from thesame degree) in my system.
> I'm simply saying that what we call major mode is a lot of the time (perhaps
> most of the time) a specific one of these modes.
> I'm not excluding the other 55 modes, some of whom have the exact same major
> notes in 12tet, but are different in JI (and who have the VI without wolf)
> I'm allways saying musical context with everything.
>
> Marcel
>

🔗Marcel de Velde <m.develde@...>

> I'm just wondering if you know of a scale with both 27/20 in it and a host
> of fairly pure thirds, seconds, thirds, and fifths (pure meaning, any dyads
> that sound relaxed...not at all restricted to "common practice theory" like
> intervals such as 6/5...or at least are within some 11 cents or so of
> relaxed intervals).

"Classic" JI has the following scale:
1/1 16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5 5/3 9/5 15/8 2/1
Which has several 27/20 in it, and many pure triads.

For un-common practice tones you can extend the scale to 7-limit in several
ways (get big scales).

My tonal-ji has the following scales:

6-limit harmonic model:
1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1

6-limit tonality:
1/1 25/24 16/15 10/9 9/8 75/64 6/5 5/4 4/3 27/20 25/18 45/32 3/2 25/16 8/5
5/3 27/16 16/9 9/5 15/8 2/1

For un-common practice:

7-limit harmonic model:
1/1 21/20 35/32 9/8 7/6 6/5 5/4 21/16 4/3 7/5 35/24 3/2 14/9 8/5 5/3 7/4 9/5
28/15 15/8 35/18 2/1

7-limit tonality:
1/1 28/27 25/24 21/20 16/15 35/32 10/9 9/8 7/6 75/64 6/5 175/144 56/45 5/4
35/27 21/16 4/3 27/20 175/128 25/18 7/5 45/32 35/24 3/2 14/9 25/16 63/40 8/5
175/108 105/64 5/3 27/16 7/4 16/9 9/5
175/96 28/15 15/8 35/18 63/32 2/1

Marcel

🔗genewardsmith <genewardsmith@...>

🔗Mike Battaglia <battaglia01@...>

On Wed, Apr 28, 2010 at 2:47 PM, Michael <djtrancendance@...> wrote:
> Like I said before, when I did a raw/blind test on finding intervals I sounded best just by sliding a sine wave by ear against a the first stationary sine wave....27/20 came out to my ear as better than it's 11/8 neighbor!
> Now what are some good scales which use 27/20?

I would wager that the reason for this is that when you play the two
intervals with sine waves, you hear that 27/20 as a sharp 4/3. But
since there are no partials to slam against each other and drive you
nuts when you play it, you just hear it as a chunk of a "stretched'
harmonic series. Similar to how the partials on a piano are stretched,
but a bit more stretched than that so to speak.

So I would imagine that part of the reason is that - you're just
enjoying the novel sensation of hearing a stretched harmonic series
without beating. You could also play that interval with a timbre whose
partials are stretched enough that 4/3 becomes 27/20, and it would
sound as good. I imagine that with sine waves, 11/8 either sounds

- Inharmonic, or
- Just not stretched, and more dissonant vs ~4/3

Just a half assed theory of mine.

-Mike

🔗Marcel de Velde <m.develde@...>

🔗Michael <djtrancendance@...>

🔗genewardsmith <genewardsmith@...>

🔗Marcel de Velde <m.develde@...>

🔗Billy <billygard@...>

The 24:27:30:32:36:40:45:48 tuning, which is another way to notate it, seems to be justified by the perfect tuning of the I, IV and V chords. In a simple diatonic melody that uses only those chords to harmonize, they would be the only chords you need. As you go beyond those three chords, you run into harmonies that involve modulations, and that is where the use of commas is justified. If you are doing the harmony for "Five Foot Two", you would be justified in using Pythagorean roots for the series of dominant 7th chords that follow a chain of just 5ths to the tonic. Here we aren't looking at conformance to a just diatonic scale so much as conformance to a series of just chords separated by 5ths. That would be a superset of the use of just the three chords that form a just major scale.

Billy

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
> I've made a discovery that is very important to me.
> I hope some of you will find it interesting too that's why I'm sharing it.
>
> For a long time I've had the belief that the common major scale in JI is 1/1
> 9/8 5/4 4/3 3/2 5/3 15/8 2/1
> Somehow I got it in there and it stuck.
> I had allready been deviating from this in my tuning of old compositions,
> but I thought the compositions were doing something really weird.
>

🔗Michael <djtrancendance@...>

Marcel>"If you want 27/20 plus 5 limit triads your suggestion of the Ellis
duodene was a good one, but I can't figure what, exactly he's after."

To be honest, what I'm after is not triads but ability to do many types different of intervals within about 11 cents of their pure forms IE close enough to maintain the basic sense of character...and do so with a few 12+ cents sour dyads as possible.
27/20 and 11/9 are two of my favorite not-so-standard-ish intervals which work this way...they aren't 4/3 or 6/5 but they maintain enough composure in the near-same frequency area IMVHO to be decent substitutes regardless of whether I hit, say, a standard major or minor triad or not. Same goes for the area around 9/5 for me...sometimes I temper a bit around 9/5 and end up rounding up within "tempering" to 11/6...neither interval sounds bad to me and it beats, say, forcing a near-exact 9/5 every time only to build up commas of sorts.

I'm not worried if you need to use "deviant from common practice theory" intervals every now and then so long as there aren't any bizarrely dissonant parts as an indirect result of trying to "over purify" certain chords and causing commas and such.

I have no incredible drive to, say, reduce chords to pretty little series like 4:5:6 or 3:5:4 or keep any sort over limit (well, unless you step over a common denominator of 11 or so) or stay within any certain limit...just so long as it's close enough to make the brain catch on. That seems to be to a huge extent how 12TET works and "cheats"...and I think it's a good idea for any sort of scale which doesn't royally distort many possible chords for the sake of getting a few chords "perfect".

________________________________
From: genewardsmith <genewardsmith@...>
To: tuning@yahoogroups.com
Sent: Wed, April 28, 2010 5:23:04 PM
Subject: [tuning] Re: 27/20 = explaining its odd beauty

--- In tuning@yahoogroups. com, Marcel de Velde <m.develde@. ..> wrote:

> What's the point at all in a temperament if one accepts a 27/20?
> It's the main thing that has been holding people back from prime-5-limit JI.
> If you want 27/20, prime-5-limit JI is THE tuning system.

If you want 27/20 plus 5 limit triads your suggestion of the Ellis duodene was a good one, but I can't figure what, exactly he's after.

🔗Marcel de Velde <m.develde@...>

Hi Billy,

The 24:27:30:32:36:40:45:48 tuning, which is another way to notate it, seems
> to be justified by the perfect tuning of the I, IV and V chords. In a simple
> diatonic melody that uses only those chords to harmonize, they would be the
> only chords you need. As you go beyond those three chords, you run into
> harmonies that involve modulations, and that is where the use of commas is
> justified.
>

I must strongly disagree here.
A modulation is a strong thing and I think your use of the word is wrong
here.
One can play whole pieces easily in one key without modulation, and use all
12 12tet notes in the octave (more in JI) and all sorts of chords.

But I think you mean with modulation here what I call a "change of harmonic
root".
This is done a lot, in many pieces, and "transposes" the scale. (but this
does not make it a modulation yet)
However this is not done in these simple childrens songs and holliday songs
mentioned in this thread.

One can in one scale use not only the "consonant" chords / triads, but also
the dissonant chords / traids.
To jump to a scale transposition every time a wolf is hit, as for instance a
passing chord between 2 consonant triads, or in elaborations etc, makes not
sense, and it sounds terrible.
Music doesn't comma shift, Ive experimented a lot with this.
And "harmonic-6-limit" music doesn't have to ever comma shift.
In my system, one can infact only change from one harmonic root to another
(transpose the scale) when this does not give a comma shift (it's a bit more
complicated but there's never a comma shift)
If you look at my serious case studies of the first 2 drei equale of
Beethoven at www.develde.net you'll see that there are wolves.
You welcome to try to get rid of the wolves, but I can allready tell you
that by doing so you'll make it out of tune and sound bad.

If you are doing the harmony for "Five Foot Two", you would be justified in
> using Pythagorean roots for the series of dominant 7th chords that follow a
> chain of just 5ths to the tonic. Here we aren't looking at conformance to a
> just diatonic scale so much as conformance to a series of just chords
> separated by 5ths. That would be a superset of the use of just the three
> chords that form a just major scale.
>
> Billy
>

No, this isn't at all how JI works I think.
I think your method makes no sense and will produce bad sounding results.

I'll post a lot of simple children and holliday song examples later today
(when the sun is off my computer screen), all in one "harmonic root" / fixed
scale.
And give some more logic on scales and modes.

Marcel

🔗Marcel de Velde <m.develde@...>

🔗Klaus Schmirler <KSchmir@...>

🔗martinsj013 <martinsj@...>

🔗Cox Franklin <franklincox@...>

Steve,
According to Barbieri, very fine Italian players could play in syntonic tuning and shift notes up or down a comma as needed; he shows examples of players marking in commas in the music, if I remember correctly. I've been playing the first few Bach suites this way for years, using Johnston's notation to keep track of the shifts, but I had considered it more of an experiment than a historically authentic approach. But evidently this was a real possibility in Bach's day, although I doubt that one of these virtuosi ever played his works. These players would even do things such as flattening the E and/or A strings by a comma, and Barbieri gives very precise diagrams for placing the fingers for the 10/9 or 9/8 seconds.
When playing with harpsichords, these players would use the same approach, hanging the fairly pure M or m 3rds on to whatever roots were given by the keyboard; players would tune their strings to the 5ths of the keyboard.
Barbieri believes a flip back to Pythagorean tuning (with the two 9/8's stacked above the dominant tone, resulting in a higher leading tone) was in the works in the latter part of the 18th century, which would explain why some manuals from ca. the 1790s show a C# above a Db. However, many manuals from the early years of the 19th century still distinguish the two tones, often with different fingerings given for "enharmonically equivalent" tones. By mid-century, most of the manuals I've seen no longer distinguish the tones. However, both the syntonic/meantone and "Pythagorean" (which I believe was less a precisely measured tuning approach than a matter of pushing sharps up and flats down) approaches were still alive, as evidenced by Joachim's testimony and recordings (for the former) and early recording stars such as Casals (for the latter).
There have been studies of violin intonation; an influential one claiming that violinists played with Pythagorean tuning was published in the 1940's, I believe. Most of the evidence I've seen shows a wide variety of tuning approaches early in the century, becoming more and more standardized (rough ET) toward the 1960's. Luckily, around that time historically informed performance practice started to really take off and saved us from uncritical acceptance of a single standard.
I have nothing against ET for modern music; I compose primarily post-tonal music involving ET microtones (24 edo, etc.), which requires learning ET intervals precisely. The tuning often sounds lovely, and has allowed orchestral players from different families to match up to an astonishing degree. My primary arguments with ET are primarily with it's being used for traditional (i.e., pre-20th century) music. First, no matter how wonderful it is to hear all members of an orchestra playing "in tune," the tuning system used to enable and enforce this has probably ended up sapping much (not all, of course) of the expressive substance out of the music. Every chord sounds fine now, whereas I don't think that all chords and keys in a tonal piece should be treated as equal. For example, one of the prime imperatives for a string player now is to play every A# and Bb with identical intonation; if you can't achieve this, you probably won't win any auditions.
For a fine Italian violinist of the Baroque period, this indifference to these distinctions would have marked the player as one of those "lousy tavern fiddlers playing in equal temperament" (i.e., unable to make the distinctions the elegant players knew how to produce; this is a rough quote from a discussion of tuning at the Royal Society in London in the late 17th century). Now the intonation of the "lousy tavern fiddlers" is being enforced as the norm.
Second, one of the primary means of achieving this lovely tonal blend has been the spread of vibrato to most instruments of the orchestra. This creates a chorus effect that blurs the pitch and makes everything sound lovely and sort of in tune. Again, I think this saps a lot of expressive weight out of the music, imposing a patina of cheesy beauty over every type of expression in every type of music. For some music, it is lovely, but I object to this vibrato being applied all the time. This is why for traditional music I often prefer older recordings by groups such as the Busch Quartet or artists such as Casals to more modern interpretations.
However, one has to survive as a professional, so I usually use the standard vibrato now when performing with other musicians; other musicians and the audience prefer it. I haven't met many players willing to do without it an tune very precisely. The Kepler Quartet is very unusual this way, and I admire their recording of Johnston's quartets tremendously. Also the Quatuor Mosäiques and Eroica Quartet.
best
Franklin

Dr. Franklin Cox

1107 Xenia Ave.

Yellow Springs, OH 45387

(937) 767-1165

franklincox@...

--- On Thu, 4/29/10, martinsj013 <martinsj@...> wrote:

From: martinsj013 <martinsj@...>
Subject: [tuning] Re: Most common Major scale in JI
To: tuning@yahoogroups.com
Date: Thursday, April 29, 2010, 2:37 PM

--- In tuning@yahoogroups. com, Cox Franklin <franklincox@ ...> wrote:

> the date of the demise of meantone approaches is still open to debate. ... For string players, I believe that there were at least three different approaches ... It is highly unlikely, though, that all that many performers would have been using JI (syntonic) tuning. According to Barbieri, even in the Baroque period, it was only the finest players who had mastered this. However, string players were definitely distinguishing between enharmonic notes such as C# and Db at the very least a decade or two into the 19th century.

Dr Cox,

With the last sentence (distinguishing C# and Db) am I right to assume you mean this indicates a meantone approach, not a JI one?

Despite all the interesting and esoteric tunings discussed in this group, I am most interested in knowing how performers actually perform "12tET" music today (as well as in the past) - my ears are not acute enough to answer this for myself. Are there any studies of this?

Steve M.

🔗Marcel de Velde <m.develde@...>

Hello Klaus!

> Also try the happy birthday and jingle bells. They'll work perfectly too.
> > Happy birthday goes like 1/1 1/1 9/8 1/1 4/3 5/4, 1/1 1/1 9/8 1/1 3/2
> > 4/3, 1/1 1/1 2/1 5/3 4/3 5/4 9/8, 9/5 9/5 5/3 4/3 3/2 4/3.
>
> Here's one reason for the resistance against your claims: Happy Birthday
> starts on the 5th degree; you are talking about a "mixolydian" scale.
>
>
> > Where 1/1 is the harmonic root of 1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/4 9/5
> > 15/8 2/1.
> > And 4/3 is the "key root" for major.
>
> Whereas this sounds remotely like George Russell's "lydian tonic" (where
> the bottom note of the tritone is the tonal root and teh dominant 7th
> chord needs a "modulation" to the true 4/3). Not sure how the two fit
> together.
>
> klaus

Thanks for paying attention :-)

I was indeed wrong about happy birthday.
I had allready discovered this.
Posting it and other later today.
It's in the classic way of doing major in JI.
Going like 3/2 3/2 5/3 3/2 2/1 15/8, 3/2 3/2 5/3 3/2 9/4 2/1, 3/2 3/2 3/1
5/2 2/1 15/8 5/3, 8/3 8/3 5/2 2/1 9/4 2/1.
I hadn't really checked it properly before writing the tuning last time.
(not how it's harmonized and not how the accent is on the happy BIRTHday to
.. etc)

Marcel

🔗Michael <djtrancendance@...>

Dr. Cox>"My primary arguments with ET are primarily with it's being used for traditional (i.e., pre-20th century) music. First, no matter how
wonderful it is to hear all members of an orchestra playing "in tune,"
the tuning system used to enable and enforce this has probably ended up
sapping much (not all, of course) of the expressive substance out of the music. Every chord sounds fine now, whereas I don't think that all
chords and keys in a tonal piece should be treated as equal."
I would consider that a very strong argument against playing original works in "new" scales. In other words, the fact when the intentional "disproportionately sour and sweet" parts in the originals are eliminated the piece of music in question loses a lot of it's genuine nature in using such "undesirable artifacts" the build and wind down tension. I figure in new songs in TET tunings and other "averaged feel" tunings, the composer can find other ways to build tension than, say, using wolf notes or other ways to build consonance than things like virtually perfect intervals...but in old songs it's doing a dis-service to how the original composer used "weird artifacts" of the scale to his/her advantage.

>"This creates a chorus effect that blurs the pitch and makes everything
sound lovely and sort of in tune. "
Putting vibrato over sounds indeed has this advantage. Come to think of it..."even" if you make a straight harmonic series scale I've heard slightly de-tuning some partials actually makes things sound less tense and more "rainbow-ish". Such partial "wavering" is, of course, a huge part of programming realistic string sounds and does give more slack far as giving a feeling of "perfect pitch" even if the pitches are not dead-on.
It can get a bit muddy and out-of-tune if the strings have to much de-tuning to the point you can't tell where the root tone is easily and it seems to, say, bounce between a 6/5 and a 13/8...but a bit of controlled partial de-tuning like that in many stringed instruments appears to unlock all sorts of flexibility.

🔗Marcel de Velde <m.develde@...>

A bit late (sore throat and fever means not so much behind computer)
but here are the examples of childrens and holliday songs in major in
JI.

In JI in classic major scale. (harmonic-6-limit diatonic scale-1
mode-1 (see modes below for explenation))
/tuning/files/Marcel%20de%20Velde/major%20fun/abc-JI-S1m1.mid

12tet for comparison:
/tuning/files/Marcel%20de%20Velde/major%20fun/abc-12tet.mid

Again classic major scale in JI:
/tuning/files/Marcel%20de%20Velde/major%20fun/HappyBirthday_03-JI-S1m1.mid

12tet for comparison:
/tuning/files/Marcel%20de%20Velde/major%20fun/HappyBirthday_03-12tet.mid

Scala couldn't retune all notes as there were too many midi notes, but
here it is anyhow (classic major scale in JI):
/tuning/files/Marcel%20de%20Velde/major%20fun/alle_eendjes_zwemmen_in_het_water-JI-S1m1.mid

12tet for comparison:
/tuning/files/Marcel%20de%20Velde/major%20fun/alle_eendjes_zwemmen_in_het_water-12tet.mid

Again classic major scale in JI:
/tuning/files/Marcel%20de%20Velde/major%20fun/het_regent_de_pannen_worden_nat-JI-S1m1.mid

12tet for comparison:
/tuning/files/Marcel%20de%20Velde/major%20fun/het_regent_de_pannen_worden_nat-12tet.mid

Again classic major scale in JI:
In this one the beginning and a part in the middle is out of tune.
Those parts are in a different harmonic root (almost certainly a fifth
up or down haven't checked). Haven't bothered to edit the song in
scala sequence format to shift the harmonic root in the right places,
but it's a nice example of in tune and out of tune now.
/tuning/files/Marcel%20de%20Velde/major%20fun/jingle-JI-S1m1.mid

12tet for comparison:
/tuning/files/Marcel%20de%20Velde/major%20fun/jingle-12tet.mid

Again classic major scale in JI:
/tuning/files/Marcel%20de%20Velde/major%20fun/

12tet for comparison:
/tuning/files/Marcel%20de%20Velde/major%20fun/

Classic major scale in JI:
/tuning/files/Marcel%20de%20Velde/major%20fun/vader_jacob-JI-S1m1.mid

An alternative major scale that started this thread. Harmonic-6-limit
diatonic scale-2 mode-4: (not sure which one I prefer)
/tuning/files/Marcel%20de%20Velde/major%20fun/vader_jacob-JI-S2m4.mid

12tet for comparison:
/tuning/files/Marcel%20de%20Velde/major%20fun/vader_jacob-12tet.mid

Again classic major scale in JI:
I posted this one before in the S2m4 major scale wrongly! Sorry. Had a
better look now and it's in the classic major scale S1m1.
/tuning/files/Marcel%20de%20Velde/major%20fun/WeWishYouAMerryXmas-JI-S1m1.mid

12tet for comparison:
/tuning/files/Marcel%20de%20Velde/major%20fun/WeWishYouAMerryXmas-12tet.mid

So.. yes. After a better look, all of these simple major songs are in
the classic major scale of 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1
And not the alternative major scale I started this thread with.
So I guess the alternative major scale is not the most commonly used one.
Though I do still see it's used a lot, my random pick of simple songs
has shown that the classic major scale deserves the title of most
common major scale I think.
The basic structre of I IV V is most used.
I will post some pieces by Bach which clearly do use the alternative
major scale soon though.

As for the S1m1 and S2m4 names.
I've decided to type out and name all possible harmonic-6-limit scales
and modes.
Here they are (not yet fully typed out)

Harmonic-6-limit scale:
1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1

Harmonic-6-limit 'diatonic' scales & modes:

Scale 1:
Mode 1: 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1
Mode 2: 9/8 5/4 4/3 3/2 5/3 15/8 2/1 9/4 <1/1 10/9 32/27 4/3 40/27
5/3 16/9 2/1>
Mode 3: 5/4 4/3 3/2 5/3 15/8 2/1 9/4 5/2 <1/1 16/15 6/5 4/3 3/2 8/5 9/5 2/1>
Mode 4: 4/3 3/2 5/3 15/8 2/1 9/4 5/2 8/3 <1/1 9/8 5/4 45/32 3/2 27/16 15/8 2/1>
Mode 5: 3/2 5/3 15/8 2/1 9/4 5/2 8/3 3/1 <1/1 10/9 5/4 4/3 3/2 5/3 16/9 2/1>
Mode 6: 5/3 15/8 2/1 9/4 5/2 8/3 3/1 10/3 <1/1 9/8 6/5 27/20 3/2 8/5 9/5 2/1>
Mode 7: 15/8 2/1 9/4 5/2 8/3 3/1 10/3 15/4 <1/1 16/15 6/5 4/3 64/45
8/5 16/9 2/1>

Scale 2:
Mode 1: 1/1 9/8 5/4 4/3 3/2 5/3 9/5 2/1
Mode 2: 9/8 5/4 4/3 3/2 5/3 9/5 2/1 9/4 <1/1 10/9 32/27 4/3 40/27 8/5 16/9 2/1>
Mode 3:
Mode 4: 4/3 3/2 5/3 9/5 2/1 9/4 5/2 8/3 <1/1 9/8 5/4 27/20 3/2 27/16 15/8 2/1>
Mode 5:
Mode 6:
Mode 7:

Scale 3:
Mode 1: 1/1 9/8 5/4 4/3 3/2 8/5 15/8 2/1
Mode 2:
Mode 3:
Mode 4:
Mode 5:
Mode 6:
Mode 7:

Scale 4:
Mode 1: 1/1 9/8 6/5 4/3 3/2 5/3 15/8 2/1
Mode 2:
Mode 3:
Mode 4:
Mode 5:
Mode 6:
Mode 7:

Scale 5:
Mode 1: 1/1 9/8 5/4 4/3 3/2 8/5 9/5 2/1
Mode 2:
Mode 3:
Mode 4:
Mode 5:
Mode 6:
Mode 7:

Scale 6:
Mode 1: 1/1 9/8 6/5 4/3 3/2 5/3 9/5 2/1
Mode 2:
Mode 3:
Mode 4:
Mode 5:
Mode 6:
Mode 7:

Scale 7:
Mode 1: 1/1 9/8 6/5 4/3 3/2 8/5 15/8 2/1
Mode 2:
Mode 3:
Mode 4:
Mode 5:
Mode 6:
Mode 7:

Scale 8:
Mode 1: 1/1 9/8 6/5 4/3 3/2 8/5 9/5 2/1
Mode 2:
Mode 3: 6/5 4/3 3/2 8/5 9/5 2/1 9/4 12/5 <1/1 10/9 5/4 4/3 3/2 5/3 15/8 2/1>
Mode 4:
Mode 5:
Mode 6:
Mode 7:

I'm also writing an algorithm now that'll make random music in each of
these 8 scales so you can hear their different moods.
Ones I've managed to program it I'll post the results.

Marcel

🔗Marcel de Velde <m.develde@...>

🔗a_sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, "Billy" <billygard@...> wrote:
> The 24:27:30:32:36:40:45:48 tuning, which is another way to notate..

> --- In tuning@yahoogroups.com, Marcel de Velde <m.develde@> wrote:
> > For a long time I've had the belief that the common major scale > > in JI is 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1 ....

Hi Billy & Marcel,
just interpolate that Heptatonics
24:27:30:32:36:40:45:48 = GGG:AAA:BBB\:CC:DD:EE\:GGb

into Isaac Newton's drawing:

http://www.sciencedirect.com/science?_ob=MiamiCaptionURL&_method=retrieve&_udi=B6WG9-4NS2GPC-2&_image=fig004&_ba=4&_user=2717328&_rdoc=1&_fmt=full&_orig=search&_cdi=6817&view=c&_acct=C000056831&_version=1&_urlVersion=0&_userid=2717328&md5=ecae25c10d0d7cfb603c20e7a93bf911

! Spa53tone256Hz.scl
Sparschuh's 24:27:30:32:36:40:45:48=GGG:AAA:BBB\:CC:DD:EE\:GGb in 53
!
53
!
! Generated by an generalizied 'Collatz-sequence' in 5hts
!
! 0. C.: unison=1...32................. c.'256Hz middle_C4 abs.pitch
! 1. G.: 3..........24................. g.'384
! 2. D.: 9..........36................. d.'288
! 3. A.: 27............................ a.'432
! 4. E.: 81............................ e.'324
! 5. B.: 243........................... b.'486
! 6. F#:(C#/3=11.39...182.24<)182.25... f#'364.5 729:=3^6
! 7. C#: (11.39*3) = 34.17............. c#'273.36
! 8. G#: (C#*3) = 102.51....205.02..... g#'410.04
! 9. D#: (6/5=1.2...9.6 <) 9.61........ d#'307.52 (< 307.53 := g#*3)
!10. A#: (D#*3)= 28.23 57.66 115.32.... a#'461.28
!11. F/: (A#*3)= 86.49................. f/'345.96
!12. C/: .............................. c/'259.47 := (F/*3)
!13. G/: 48.65...93.3...194.6.......... g/'389.20 788.4(<788.41:=C/*3)
!14. D/: (G/*3)=145.95................. d/'291.90
!15. A/: 54.73......................... a/'437.84 (< 437.85 := D/*3)
!16. E/: (A/*3)=164.19................. e/'328.38
!17. B/: (E/*3)=61.57.................. b/'492.56 (< 492.57 := E/*3)
!18. F& = F#/: (B/*3)=184.71........... f&'369.42
!19. C&: 138.53........................ c&'277.06 554.12(<554.13=F&*3)
!20. G&: 207.79........................ g&'415.58 (< 415.59 := C&*3)
!21. D&: 4.87.......................... d&'311.68 (< 623.37 := C&*3)
!22. A&: 14.61......................... a&'467.52
!23. F+ = F//: 43.83................... f+'350.64
!24. C+: 131.41........................ c+'262.98
!25. G+: 197.23........................ g+'394.46 (<394.47 := C+*3)
!26. D+: 18.49......................... d+'295.84 (<591.69 := G+*3)
!27. A+: 55.47......................... a+'443.76 Hz or 'cps'
!28. E+ = F- = E// = F\\: 1.30......... e-'332.80 enharmonics in 53
!29. C- = B+ = C\\ = B//: 3.9..c-'249.6 c-"499.20
!30. G-: 11.7.......................... g-'374.4
!31. D-: 17.55...35.1.................. d-'280.8
!32. A-: 52.65...105.3................. a-'421.2
!33. E-: 78.97 157.94 (<157.95:=A-*3).. e-'315.88
!34. B-: 118.45 236.9(<236.91:=E-*3)... b-'473.8
!35. GB =Gb\: 177.67................... gB'355.34 (<355.35 := b- *3)
!36. DB: 133.25 ....................... dB'266.50 533 (<533.01=dB*3)
!37. AB: .............................. aB'399.75 := dB*3
!38. EB: .............................. eB'299.81 ..1199.24(<1199.25)
!39. BB: .............................. bB'449.71 899.42 (< 899.43)
!40. F\: 5.27.......................... f\'337.28 1349.12 (<1349.13)
!41. C\: 15.81................c\'252.96 c\"505.92
!42. G\: 47.43......................... g\'379.44
!43. D\: 35.57 71.14 142.28 (<142.29).. d\'284.56
!44. A\: 106.7 (<106.71 := d\*3)....... a\'426.8
!45. E\: 5 10 20 40 80 160............. e\'320.00 (< 320.01 := A\*3)
!46. B\: 15 30......................... b\'480
!47. Gb: 45............................ gb'360
!48. Db: 135........................... db'270
!49. Ab: (25.31 50.62 101.24<) 101.25.. ab'405 := Db*3
!50. Eb: 9.49 ...75.92(<75.93:=25.31*3) eb'303.68
!51. Bb: 14.23 28.46 ( <28.47:= 9.49*3) bb'455.36
!52. F.: 10.67 21.34 42.68(<42.69=Bb*3) f.'341.44
!53. C.: 1...32 (< 32.01 := F.*3)...... c.'256
!
!
25947/25600 ! 1: c/
26298/25600 ! 2: c+
26650/25600 ! 3: dB
27000/25600 ! 4: db
27336/25600 ! 5: c#
27706/25600 ! 6: c&
28080/25600 ! 7: d-
28456/25600 ! 8: d\
28800/25600 ! 9: d. (9/8)
29190/25600 !10: d/
29584/25600 !11: d+
29981/25600 !12: eB
30368/25600 !13: eb
30752/25600 !14: d#
31168/25600 !15: d&
31588/25600 !16: e-
32000/25600 !17: e\ [5/4]
32400/25600 !18: e. (81/64)
32838/25600 !19: e/
33280/25600 !20: e+ = f-
33728/25600 !21: f\
34144/25600 !22: f. (4/3)*(3201/3200)
34596/25600 !23: f/
35064/25600 !24: f+
35534/25600 !25: gB
36000/25600 !26: gb [45/32]
36450/25600 !27: f# (729/512)
36942/25600 !28: f&
37400/25600 !29: g-
37944/25600 !30: g\
38400/25600 !31: g. (3/2)
38920/25600 !32: g/
39446/25600 !33: g+
39975/25600 !34: aB
40500/25600 !35: ab [405/256]
41004/25600 !36: g#
41558/25600 !37: g&
42120/25600 !38: a-
42680/25600 !39: a\ [5/3]*(3201/3200)
43200/25600 !40: a. (27/16)
43784/25600 !41: a/
44376/25600 !42: a+ = 443.76Hz
44971/25600 !43: bB
45536/25600 !44: bb
46128/25600 !45: a#
46752/25600 !46: a&
47380/25600 !47: b-
48000/25600 !48: b\ [15/8]
48600/25600 !49: b. (243/128)
49256/25600 !50: b/
49920/25600 !51: b+ = c-
50592/25600 !52: c\
2/1
!
!
![eof]

bye
A.S.