Yahoo Tuning Groups Ultimate Backup tuning "In tune" equal temperaments

"How would you compare it to 22 edo?"

with 22 edo - as in my last release - the distance from 12 edo fret
lines change with the note you play which makes it more complicated to
implement than 24.

This part I didn't follow completely

I think you are saying there are nice MOS scales in 24 et?

"and if if we add 5 to the mix, then 81/80 and 121/120 are commas, and
adding either to 243/242 gives a rank three temperament which if we
toss 7 becomes a 2,3,5,11 version of mohajira for which 24edo is a
good tuning. In this (now rank two) temperament, five 11/9 neutral
thirds come to 11/4, and then eight to a 5. You've got MOS of size 7,
10 and 17 notes, which as you can see is enough to encompass a lot of
{3,11} harmony and a little less {3,5,11} harmony."

I'm thinking perhaps a hybrid of 3 limit, "true" 5 limit thirds, a few
from 7 limit, 9 limit and 11 limit might be playable.

Essentially if I can mentally "map" to close to a former 12 fret or
between frets with adjustment by ear I seem to be ok.

Biggest draw back is the difficulty in doing barres that are exactly
perpendicular to the strings.

On Fri, Jul 2, 2010 at 1:33 AM, genewardsmith
<genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
> >
> > Gene,
> >
> > by eyeball it seems 24 edo is close to a number, not all, but a number
> > of 11 limit intervals.
>
> Indeed it is. In fact, if you drop 5 and 7 and just look at 3 and 11, it's a standout. It's also got a controversial role in Middle Eastern music. Some {2,3,11} commas it tempers out are the Pythagorean comma, 243/242, and 264627/262144. 243/242 says that two 11/9s give 3/2, and points to neutral third systems, and if if we add 5 to the mix, then 81/80 and 121/120 are commas, and adding either to 243/242 gives a rank three temperament which if we toss 7 becomes a 2,3,5,11 version of mohajira for which 24edo is a good tuning. In this (now rank two) temperament, five 11/9 neutral thirds come to 11/4, and then eight to a 5. You've got MOS of size 7, 10 and 17 notes, which as you can see is enough to encompass a lot of {3,11} harmony and a little less {3,5,11} harmony.
>
> > 24 edo is very playable on my fretless guitar and I've been messing
> > with it a bit so any information / opinions / thoughts would be useful
> > to me.
>
> How would you compare it to 22 edo?
>
>

🔗genewardsmith <genewardsmith@...>

🔗cityoftheasleep <igliashon@...>

🔗Marcel de Velde <m.develde@...>

Hi Gene,

It's been suggested that 256/255, or 6.776 cents, makes a good standard
> cutoff limit for deciding if an approximation to a JI interval is in tune or
> not. While it may be objected that this figure is arbitrary and takes no
> account of the differences between kinds of intervals, it does have the
> merit of being precisely defined, easily evaluated, and representative of a
> definite opinion. So with no further ado, here are ets "in tune" to various
> odd limits:
>
> 3 limit: 12, 17, 29, 31 ...
> 5 limit: 31, 34, 41, 46 ...
> 7 limit: 31, 41, 53, 68 ...
> 9 limit: 53, 72, 77, 80 ...
> 11 limit: 72, 80, 87, 99 ...
> 13 limit: 80, 87, 94, 103 ...
> 15 limit: 87, 94, 103, 111 ...
> 17 limit: 94, 103, 111, 113 ...
> 19 limit: 94, 111, 113, 121 ...
> 21 limit: 94, 111, 113, 121 ...
> 23 limit: 94, 111, 113, 128 ...
>

These are equal divisions of the octave.
If 6.776 cents is a good standard cutoff limit for deciding if an
approximation to ANY JI interval is in tune ot not, then one doesn't need to
use equal divisions of the octave only, as the 6.776 cents would apply to
the 2/1 JI octave aswell.
So one could use any Equal Temperament, EDO or non EDO.

But, I think what may work even better in practice, is not use 6.776 cents
for any JI interval, but use harmonic entropy for determining the maximum
deviation from JI.
So different intervals or different prime or odd limits would get different
maximum deviations.
For instance maximum deviation from 2/1 could be 2 cents, maximum deviation
from 3/2 could be 3 cents, maximum deviation 5/4 could be 5 cents, etc.

Though, I personally think what would ultimately decide what's "in tune"
would be a Just Intonation music theory.
In other words, what's "in tune" depends on the music. Only music theory has
the potential to say wether a "minor third" should be a 75/64 or 32/27 or
6/5 or something like that in a specific case. To use a temperament with a
higher resolution than 12tet, it will soon become important to know which
one of those it should be to sound "in tune". Otherwise one may play the
tempered version of the "wrong" JI interval.

Marcel

🔗genewardsmith <genewardsmith@...>

🔗Carl Lumma <carl@...>

🔗Marcel de Velde <m.develde@...>

🔗genewardsmith <genewardsmith@...>

🔗Marcel de Velde <m.develde@...>

> Actually, we've pretty well investigated adjusting the octave to death on
> the tuning-math list. It's almost like a hobby over there, though the
> reasons are sounder than that--in many ways, it makes the theory simpler.
>

Ah ok I didn't know that.
Did any interesting scales come out of it?

Btw what I find an effective temperament myself for 5-limit JI is 2 12edo
scales of which one is transposed by -13.686 cents.
Which gives 12edo for all 3-limit intervals (with offcourse pure octave),
and the 5/4 pure aswell.
My most recent experimentation in putting common practice music in 5-limit
JI have made me belief that there is little 2 stacked 5-limit intervals in
common practice music, so no 5/4 25/16, or 5/4 and 6/5 in thesame song
(there's very specific harmony needed to do such a thing it seems to me
now).
So an infinite scale of stacked 3/2 fifths and a parallel by 5/4 second
chain of stacked fifths would cover most music it seems to me.

The tempered 24 note scale that results from this is:

86.314
100.000
186.314
200.000
286.314
300.000
386.314
400.000
486.314
500.000
586.314
600.000
686.314
700.000
786.314
800.000
886.314
900.000
986.314
1000.000
1086.314
1100.000
1186.314
1200.000

The most out of tune practical interval in this scale is the 81/64, which is
7.82 cents too low in this scale.
But for the most part, it should be a great improvement of 12tet for common
practice music.
The only catch is, that one needs to know which notes to hit, as there's no
easy translation for pieces written in 12edo :)

Marcel

🔗genewardsmith <genewardsmith@...>

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

> Btw what I find an effective temperament myself for 5-limit JI is 2 12edo
> scales of which one is transposed by -13.686 cents.
> Which gives 12edo for all 3-limit intervals (with offcourse pure octave),
> and the 5/4 pure aswell.

That's called compton temperament. Your tuning favors major thirds over minor thirds/major sixths, and splitting the difference more nearly should be considered. Also, moving to the 7-limit, it tempers out 225/224 wth good results if you are willing to move to three parallel tracks, which to do the 5-limit correctly you would need anyway.

> My most recent experimentation in putting common practice music in 5-limit
> JI have made me belief that there is little 2 stacked 5-limit intervals in
> common practice music, so no 5/4 25/16, or 5/4 and 6/5 in thesame song
> (there's very specific harmony needed to do such a thing it seems to me
> now).

This is, alas, quite false and is why I said two tracks is a little skimpy even for the 5-limit. Shifting from major to minor or the reverse is completely routine in common practice music. There are also
diminished and augmented chords to consider.

> So an infinite scale of stacked 3/2 fifths and a parallel by 5/4 second
> chain of stacked fifths would cover most music it seems to me.

If you actually tried this you would find it doesn't. But compton is an interesting alternative to meantone as a basis for retuning common practice music, and one which is quite a bit more in tune. In terms of 7-limit JI it is closely allied to the marvel planar temperament, with 240edo being an excellent tuning for both.

🔗Marcel de Velde <m.develde@...>

> That's called compton temperament.
>

Aah ok, thanks for the name.

> Your tuning favors major thirds over minor thirds/major sixths, and
> splitting the difference more nearly should be considered.
>

No I've found that it doesn't do this.
In major, to play a minor chord on the I, IV or V it's a 1/1 32/27 3/2 minor
I think.
For instance 3/2 2/1 64/27 -> 3/2 15/8 9/4 in major.
To go to real minor would be to go to 5/3 2/1 5/2 or 5/4 3/2 15/8.
It works out in practice extremely well (and in my theory aswell)

Also, moving to the 7-limit, it tempers out 225/224 wth good results if you
> are willing to move to three parallel tracks, which to do the 5-limit
> correctly you would need anyway.
>

Yes with 3 parallel tracks indeed.
But I'm thinking 3 parallel tracks are not needed to do 5-limit JI, 2 are
enough.

>
>
> > My most recent experimentation in putting common practice music in
> 5-limit
> > JI have made me belief that there is little 2 stacked 5-limit intervals
> in
> > common practice music, so no 5/4 25/16, or 5/4 and 6/5 in thesame song
> > (there's very specific harmony needed to do such a thing it seems to me
> > now).
>
> This is, alas, quite false and is why I said two tracks is a little skimpy
> even for the 5-limit. Shifting from major to minor or the reverse is
> completely routine in common practice music. There are also
> diminished and augmented chords to consider.
>

I know shifting from major to minor is routine. But it can work as I
described above with 2 parallel tracks.

>
>
> > So an infinite scale of stacked 3/2 fifths and a parallel by 5/4 second
> > chain of stacked fifths would cover most music it seems to me.
>
> If you actually tried this you would find it doesn't. But compton is an
> interesting alternative to meantone as a basis for retuning common practice
> music, and one which is quite a bit more in tune. In terms of 7-limit JI it
> is closely allied to the marvel planar temperament, with 240edo being an
> excellent tuning for both.
>

I still like 53edo best of temperaments.
But the advantage of compton is that is uses only 24 tones. And that one can
use it with any 12edo instrument if one's willing to use 2 of those
instruments detuned by 14cents.

Marcel

🔗Marcel de Velde <m.develde@...>

Would like to add some to what I said.

I only recently made this view on 5-limit (2-plane).
I do think that some chord movements go into 3-plane, but as far as I found
now, the basic triadic major minor movements are in 2-plane.
Which surprised me aswell.
I discovered this through work on my JI algorythms for automatic computer
music.
I'm now looking again at common practice music and how this all works out
(better than anything else I've tried so far, by far)

I'm not sure how this works with strong dissonant chords like diminished
chords. Agreed they don't look pretty on a 2-plane.
And my algorhythmic work also gives the possiblity of a 7-limit interval in
the diminished chord, I'm not sure about any of this now.

But the normal major/minor traidic music, it seems to be able to work really
really well in a 2-plane 5-limit model.
I'll make a new thread on this in a while, with some musical examples.
Also keeping my fingers crossed it will keep working out as well as it seems
to do now, as it would make things a lot easyer, and I see great potential
for a solid music theory based on this.

Marcel

On 4 July 2010 23:18, Marcel de Velde <m.develde@...> wrote:

>
> That's called compton temperament.
>>
>
> Aah ok, thanks for the name.
>
>
>> Your tuning favors major thirds over minor thirds/major sixths, and
>> splitting the difference more nearly should be considered.
>>
>
> No I've found that it doesn't do this.
> In major, to play a minor chord on the I, IV or V it's a 1/1 32/27 3/2
> minor I think.
> For instance 3/2 2/1 64/27 -> 3/2 15/8 9/4 in major.
> To go to real minor would be to go to 5/3 2/1 5/2 or 5/4 3/2 15/8.
> It works out in practice extremely well (and in my theory aswell)
>
> Also, moving to the 7-limit, it tempers out 225/224 wth good results if you
>> are willing to move to three parallel tracks, which to do the 5-limit
>> correctly you would need anyway.
>>
>
> Yes with 3 parallel tracks indeed.
> But I'm thinking 3 parallel tracks are not needed to do 5-limit JI, 2 are
> enough.
>
>
>>
>>
>> > My most recent experimentation in putting common practice music in
>> 5-limit
>> > JI have made me belief that there is little 2 stacked 5-limit intervals
>> in
>> > common practice music, so no 5/4 25/16, or 5/4 and 6/5 in thesame song
>> > (there's very specific harmony needed to do such a thing it seems to me
>> > now).
>>
>> This is, alas, quite false and is why I said two tracks is a little skimpy
>> even for the 5-limit. Shifting from major to minor or the reverse is
>> completely routine in common practice music. There are also
>> diminished and augmented chords to consider.
>>
>
> I know shifting from major to minor is routine. But it can work as I
> described above with 2 parallel tracks.
>
>
>>
>>
>> > So an infinite scale of stacked 3/2 fifths and a parallel by 5/4 second
>> > chain of stacked fifths would cover most music it seems to me.
>>
>> If you actually tried this you would find it doesn't. But compton is an
>> interesting alternative to meantone as a basis for retuning common practice
>> music, and one which is quite a bit more in tune. In terms of 7-limit JI it
>> is closely allied to the marvel planar temperament, with 240edo being an
>> excellent tuning for both.
>>
>
> I still like 53edo best of temperaments.
> But the advantage of compton is that is uses only 24 tones. And that one
> can use it with any 12edo instrument if one's willing to use 2 of those
> instruments detuned by 14cents.
>
> Marcel
>

🔗genewardsmith <genewardsmith@...>

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

> > Your tuning favors major thirds over minor thirds/major sixths, and
> > splitting the difference more nearly should be considered.
> >
>
> No I've found that it doesn't do this.

That it does is simply my way of describing a mathematical fact, based on the standard definition of what a just "minor third" and "major sixth" are. Redefining things means you should make note of the fact that this is what you are doing, and not imply your definitions are the standard ones. You are tuning major thirds exactly, which means just minor thirds, and this means 6/5 and nothing else, must be tuned flat by the same 1.955 cent schisma as fifths are, and major sixths therefore must be sharp by the same amount. If you split the difference and make the major third and minor third both a cent flat, you will separate the two tracks by 14 2/3 cent and get better tuning results, if you define "better" to include both major and minor thirds.

> In major, to play a minor chord on the I, IV or V it's a 1/1 32/27 3/2 minor
> I think.

That's not a just minor third, and it's not even that close to being one. Closer to 13/11, between 7/6 and 6/5. If you want to use Pythagorean intervals for minor thirds, you are stuck with them as major thirds at least part of the time, as the interval from 32/27 to 3/2 is 81/64. Of course if you are using 12 equal your descriptions are not correct, as the minor third in question isn't 32/27 at all, it's 300 cents, and the fifth is 700 cents. In any case, even if you massacre the minor third like this, what do you plan to do about the major sixth?

🔗Marcel de Velde <m.develde@...>

Hi Gene,

> That it does is simply my way of describing a mathematical fact, based on
> the standard definition of what a just "minor third" and "major sixth" are.
> Redefining things means you should make note of the fact that this is what
> you are doing, and not imply your definitions are the standard ones. You are
> tuning major thirds exactly, which means just minor thirds, and this means
> 6/5 and nothing else, must be tuned flat by the same 1.955 cent schisma as
> fifths are, and major sixths therefore must be sharp by the same amount. If
> you split the difference and make the major third and minor third both a
> cent flat, you will separate the two tracks by 14 2/3 cent and get better
> tuning results, if you define "better" to include both major and minor
> thirds.

A major third is 2 whole steps in 12edo.
A minor third is 1 whole step + 1 half step in 12edo.
A "Just" interval is a relatively low limit (prime or odd) rational
interval.

A minor third can be described as being 6/5 in Just Intonation, but also as
32/27.
Both 6/5 and 32/27 are relatively low limit rational intervals.
Just Intonation describes "tuning just" or tuning "in tune" in rational
intervals, more than any historical ratios attached to these notions.
To tune "in tune" has so far eluded everybody. It is not known with any
certainty how to tune common practice music "in tune".
If I chose to say 32/27 is a Just Intonation interval for the minor third in
certain cases I am perfectly justified to do so.
Btw, it has been used to describe the upper minor third in the dominant 7th
chord 1/1 5/4 3/2 16/9 (between 3/2 and 16/9) for a long time allready.
I'm saying that in true minor mode, most the minor thirds in minor triads
will be 6/5 indeed (one can chose to write it as 5/3 2/1 5/2 for instance,
or as 1/1 6/5 3/2 with the 1/1 beeing on the 5/3 for instance of the
relative major). And the major third in minor in major triads most often
beeing 81/64.
And the major third of the major triad in major mode beeing 5/4 most often
and the minor third of the minor triad beeing 32/27 often in major mode, all
depending on function and location etc.
On top of this, there are also the wolf minor triad and like also wolf major
triad.
So I will call the Just minor third both 6/5 and 32/27, and Just major third
both 5/4 and 81/64.

> > In major, to play a minor chord on the I, IV or V it's a 1/1 32/27 3/2
> minor
> > I think.
>
> That's not a just minor third, and it's not even that close to being one.
> Closer to 13/11, between 7/6 and 6/5. If you want to use Pythagorean
> intervals for minor thirds, you are stuck with them as major thirds at least
> part of the time, as the interval from 32/27 to 3/2 is 81/64. Of course if
> you are using 12 equal your descriptions are not correct, as the minor third
> in question isn't 32/27 at all, it's 300 cents, and the fifth is 700 cents.
> In any case, even if you massacre the minor third like this, what do you
> plan to do about the major sixth?

Major sixth can be 5/4 and 27/16 in major mode.

And the well known I-vi-ii-V comma pump can go like this:

1/1 5/4 3/2
1/1 5/4 5/3
9/8 4/3 5/3
9/8 4/3 3/2 15/8
1/1 5/4 3/2

or
1/1 5/4 3/2
1/1 5/4 27/16
9/8 4/3 27/16
9/8 4/3 3/2 15/8
1/1 5/4 3/2

There are other alternatives though the above seem most likely to me.

Marcel

🔗Marcel de Velde <m.develde@...>

🔗genewardsmith <genewardsmith@...>

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

> A "Just" interval is a relatively low limit (prime or odd) rational
> interval.
>
> A minor third can be described as being 6/5 in Just Intonation, but also as
> 32/27.

32/27 may be described as a minor third in just intonation, though since it's 27 limit that's stretching a point. It cannot be described as a "just minor third", and doing so is simply wrong. The phrase means a 6/5 frequency ratio, and nothing else. Correct would be to call it a Pythagorean minor third.

> Both 6/5 and 32/27 are relatively low limit rational intervals.

No, they are not. 6/5 is 5-limit and is therefore a low limit rational interval. 32/27 is 27 limit, and that is not what I would call low limit. Bear in mind here that "low limit" means low odd limit, not low prime limit.

> Just Intonation describes "tuning just" or tuning "in tune" in rational
> intervals, more than any historical ratios attached to these notions.

And your 1-32/27-3/2 Pythagorean minor triad is not what people mean when they say "just minor triad", sorry.

> To tune "in tune" has so far eluded everybody.

Sez you, but with what evidential support? While 1-6/5-3/2 may not have quite the blatantly in-tune quality of 1-5/4-3/2, its quality of "in tuneness" comes through clearly enough. Anyway, you would at least want to have 4/3-5/3-2 and 3/2-15/8-9/4 as well as 1-5/4-3/2, and there goes your 24 note scale.

It is not known with any
> certainty how to tune common practice music "in tune".

Depends on the time period, and whether you get to use adaptive tuning.

> If I chose to say 32/27 is a Just Intonation interval for the minor third in
> certain cases I am perfectly justified to do so.

You are allowed to call it just intonation, but if you call it a "just minor third" I will give you an "F".

> Btw, it has been used to describe the upper minor third in the dominant 7th
> chord 1/1 5/4 3/2 16/9 (between 3/2 and 16/9) for a long time allready.

That's the dominant 7th of 5-limit JI from the point of view of functional harmony, but sadly, musical terminology is not based on just intonation, it's based on meantone. In meantone there isn't any difference between 32/27 and 6/5 since they are 81/80 apart, so the whole issue becomes moot.

🔗Marcel de Velde <m.develde@...>

On 5 July 2010 04:28, genewardsmith <genewardsmith@...> wrote:

> It is not known with any
> > certainty how to tune common practice music "in tune".
>
> Depends on the time period, and whether you get to use adaptive tuning.

Oef.. if I'd call anything out of tune it would be adaptive JI.. haha
I hate adaptive JI.

> If I chose to say 32/27 is a Just Intonation interval for the minor third
> in
> > certain cases I am perfectly justified to do so.
>
> You are allowed to call it just intonation, but if you call it a "just
> minor third" I will give you an "F".

As far as I'm concerned we're all still students. And one student can't give
grades to other students :)

> Btw, it has been used to describe the upper minor third in the dominant
> 7th
> > chord 1/1 5/4 3/2 16/9 (between 3/2 and 16/9) for a long time allready.
>
> That's the dominant 7th of 5-limit JI from the point of view of functional
> harmony, but sadly, musical terminology is not based on just intonation,
> it's based on meantone. In meantone there isn't any difference between 32/27
> and 6/5 since they are 81/80 apart, so the whole issue becomes moot.

Well that was my point.
Since normal music terminology doesn't differentiate between 32/27 and 6/5,
and calls them both minor thirds. Then by that logic I'm calling both 32/27
and 6/5 Just minor thirds.
And perhaps 1215/1024 aswell.
But indeed it's all a bit moot.
I'd be ok with naming every single intervals a different name. Perhaps even
thesame JI interval with different names depending on function etc.
But I'd rather work out how JI works first :)
So I agree it's not that clear what is a Just minor third, eg if it includes
32/27 in the name.
But I'm glad we can agree that in 5-limit Just Intonation a minor third can
be 32/27.

Marcel

🔗genewardsmith <genewardsmith@...>

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

> > Btw, it has been used to describe the upper minor third in the dominant
> > 7th
> > > chord 1/1 5/4 3/2 16/9 (between 3/2 and 16/9) for a long time allready.
> >
> > That's the dominant 7th of 5-limit JI from the point of view of functional
> > harmony, but sadly, musical terminology is not based on just intonation,
> > it's based on meantone. In meantone there isn't any difference between 32/27
> > and 6/5 since they are 81/80 apart, so the whole issue becomes moot.
>
>
> Well that was my point.
> Since normal music terminology doesn't differentiate between 32/27 and 6/5,
> and calls them both minor thirds.

The point is not whether you get to call 32/27 a minor third. The point is not whether you can call it a consonant interval of just intonation. The point is whether you get to call it a "just minor third". You don't.

Then by that logic I'm calling both 32/27
> and 6/5 Just minor thirds.

Since this is the American holiday, the 4th of July, I'll quote this:

(From The Weekly Standard [Raleigh, North Carolina], 29 October 1862.)

OLD ABE GETS OFF ANOTHER JOKE. -- A couple of Abolitionists having called upon Old Abe to persuade him to issue his Emancipation Proclamation -- that is, before he issued it -- he got off the following good thing and knock down argument against his own act:

"You remember the slave who asked his master -- if I should call a sheep's tail a leg, how many legs would it have? 'Five.' 'No, only four, for my calling the tail a leg would not make it so.' Now, gentlemen, if I say to the slaves, 'you are free,' they will be no more free than at present."

> So I agree it's not that clear what is a Just minor third

Sorry, but it IS clear. A just minor third is 6/5, and nothing else. Calling a sheep's leg a just minor third will not make it one.

🔗Marcel de Velde <m.develde@...>

Hi Gene,

The point is not whether you get to call 32/27 a minor third. The point is
> not whether you can call it a consonant interval of just intonation. The
> point is whether you get to call it a "just minor third". You don't.
>

Your description of the "just minor third" seems to be to attach a single
ratio to this name, and make it the simplest ratio that comes close, 6/5 in
this case.
That's just your description, it is not written in gold anywhere.
Btw the 32/27 just minor third is indeed less consonant than 6/5, but not
very dissonant either. I'd call it a mild dissonance, it has some tension
and wishes to resolve yes. No reason to not call it a just minor third,
since in normal music theory it's often refered to as a minor third also in
minor triads.

> Sorry, but it IS clear. A just minor third is 6/5, and nothing else.
> Calling a sheep's leg a just minor third will not make it one.
>

Liked your story :)
But it's not a good example I think as things as not as clear cut as with
the sheeps tail.
If 32/27 were the sheeps tail, then the sheep would also use it's tailleg
for walking running etc :)

Ah but we can go on forever about names.
Even though we can't agree that 32/27 can possibly be called a just minor
third.
We seem to agree that 32/27 can occur in just intonation in places that are
called a minor third in normal music theory.
I'm happy to keep it at that for now and focus on tuning things :)

Cheers,
Marcel

🔗Afmmjr@...

Hello Gene,

I am sympathetic to your assertion:
"Sorry, but it IS clear. A just minor third is 6/5, and nothing else.
Calling a
sheep's leg a just minor third will not make it one."

For the nonce, please indulge a focus on the major third:
If the "just" (honest) major third of 5/4 (incidentally, an interval
detested by "just intonation" composer La Monte Young) is the only just major
third, then the 81/64 is the ditone. I try to be consistent with this in
all my writings. Each interval (5/4 and 81/64) is caused by a different
brain function: 5/4 is right hemisphere (proportional), while 81/64 is left
hemisphere (additive).

Do we have another name for a 32/27? For many musicians, any ratio of
positive integers may be considered just, or simply be the exclusivity of
"non-tempered" intervals (no irrationals).

Perhaps a comparison with a different word: ethnicity.
The word "ethnicity" would refer to a Korean child adopted by a non-Korean
Morman family in Utah. Her ethnicity would be Mormon culture. In
distinction, her ethnic stock would still be Korean.

Mayor Bloomberg of NYC said on television confused the word ethnic with
ethnicity, getting it wrong. And yet, he is leads the Department of
Education of NYC. Therefore, he must be right, no?

all the very best

Johnny

🔗Margo Schulter <mschulter@...>

Hello, all.

Please forgive me for participating in a thread on
equal temperaments when that is really not my main
concern: but the meaning of "just" (and therefore
also "near-just") is a very important concern to me,
and it can make a critical difference as to how
tunings, styles, and world musical traditions are
valued or otherwise.

The term "just" has many meanings, and a usage which
would be strange or even humorously incongruous in
one situation can be quite natural and convenient
in another. Language should fit the context, just
as intonation should.

One good starting point is Safi al-Din al-Urmawi,
the great 13th-century music theorist who discussed
a wide range of just tunings for maqam music, many
of them still in very active use, in just or tempered
forms, by many of us in various parts of the world.

Safi al-Din reasonably teaches that multiplex and
superparticular ratios with a 3-odd limit are the
most proper "consonances," but that more complex
superparticular ratios may have a degree of concord,
and may be classified into different groups. Further,
he uses a range of ratios in his tunings. He notes
that as the integers get larger, the sense of consonance
generally gets "weaker."

More broadly, "just" can simply mean "rational"; but
often implies that an interval has a rational size
deemed harmonious or ideal for a given style.

Please let me be one of the first people to praise
the beauties of 16th-century European counterpoint
and meantone temperament while asserting that other
world musical styles and tuning systems can be
equally beautiful. What is or isn't perceived as
"just" in a meantone context shouldn't govern our
usage in other contexts.

For example, if asked to give an example of a "just"
third, I might immediately mention 7/6, which tends
to be present in a pure or near-pure form in lots of
my favorite tunings.

However, in a 16th-century European context, speaking
of "a just 7/6" would likely be incongruous, because
while Praetorius (actually writing early in the next
century) actually used the term "wolf" specifically
for an augmented second in 1/4-comma meantone almost
identical to this ratio, even Vicentino in 1555 found
this "minimal third" leaning to the dissonant side,
and thus generally best put aside, as likewise the
near-7/4 of his 31-note meantone cycle -- although
he liked and recommended a near-11:9 as leaning
toward consonance!

There is indeed a certain incongruity in speaking
of "a just 32/27" _in such a context_, since meantone
was intended precisely to move away from the
Pythagorean intonation which still plays a very
respectable role in world music, and toward more
specifically the ratios of 5/4 and 6/5. But must
we all, in various parts of the world and pursuing
many different musical styles and ideals, be bound
by 16th-century European conventions and judgment.
If so, we maybe shouldn't speak of a "just 7/4" either.

In music where major and minor thirds are expected
to be at or close to 5/4 and 6/5, it's natural to
take these ratios as the reference point for "just,"
not only because they are rational, but because
they are the sonorous ideal.

In a different context, 32/27 could actually be
considered a just approximation of another ideal
tuning for the minor third in some styles: 19/16,
for example in 16:19:24, which may have a special
stability because it is based in an octave
multiple of the fundamental.

And in a setting where thirds -- major, minor,
or Zalzalian (i.e. middle or neutral) -- are expected
generally to have rather complex ratios, I can
and do speak of a "just" or "near-just" 13/11, 14/11,
33/26, 33/28, 17/14, 21/17, 16/13, 11/9, etc.

Here usage follows style and intention. In a meantone
around 1/5-comma in the 18th century, an interval
around 14/11 would arise as a diminished fourth,
viewed as anything but in tune outside of some
special idioms where a diminished fourth is permitted
and indeed desired (for example, between two upper
voices both forming conventional concords with the
bass -- or, at least, that is a 16th-century pattern).

In a 21st-century temperament around 704 cents used
with a main focus on the intervals formed by a
17-note chain of fifths, a "near-just" 13/11 or 14/11
is a regular major or minor third -- quite unlike
the situation in meantone! Language follows style.

I begin with the premise that participants in this
group may come from any world musical tradition or
stylistic aspiration.

When discussing European meantone conventions, or
corresponding conventions in other world musics,
it is appropriate to follow meantone conventions.
In the many other contexts that can and do arise,
a "just 32/27" is fine, and I would be happy to
use it myself.

Of course, this doesn't mean that 32/27 is as
simple as 7/6 or 13/11, but the ratio itself
informs us of this greater complexity.

As to 32/27, there is also the "isoharmonic"
phenomenon, where a sonority such as
17:22:27:32:37 if precisely tuned can have
a special kind of coherence and euphony.
Here tuning that 32/27 "just" might be
especially important! And let's not forget
the related differential coherence of Jacques
Dudon, brilliantly illustrated in the Ethno2
tuning collection.

This semantic territory has been explored
many times on this list, and agreeement to
disagree is probably the best outcome.
The term "rational intonation" or RI is
available as a neutral ground: Joe Monzo
and I were part of a process on the list
about a decade ago that resulting in the
coining (or reinvention?) of this term.

Most appreciatively,

Margo Schulter
mschulter@...

🔗Marcel de Velde <m.develde@...>

Hello Margo,

Thanks for your thoughts on this.
And I agree with them for the most part.

This semantic territory has been explored
> many times on this list, and agreeement to
> disagree is probably the best outcome.
> The term "rational intonation" or RI is
> available as a neutral ground: Joe Monzo
> and I were part of a process on the list
> about a decade ago that resulting in the
> coining (or reinvention?) of this term.
>
> Most appreciatively,
>
> Margo Schulter
>

I agree that the agreement to disagree on this subject is the best outcome,
atleast for now :)

But I'd also like to explain which meaning the words Just Intonation have
for me personally.
One can for instance tune a piano to exactly 12edo and say it is "in tune".
But one would mean it is "in tune" to the 12edo, a certain concept.
But one cannot say this piano is tuned "Just".

Just Intonation has taken on 2 meanings.
One is a system of tuning (with little though some agreement) where for
instance the major scale is 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1 (though there's
some disagreement here too), and in this sense it's a system similar to
12edo in that it's a made up and stated system of how to tune, and the name
Just Intonation was attached to it, just like 12tet / 12edo was attached to
the modern western tuning.

The other meaning of "Just Intonation" is quite different. It is an
objective description of tuning "Just" or "in tune", and currently there is
no consensus on how to tune for instance common practice music "in tune" in
an objective sense.
It is infact hard to describe even, what is "in tune" in an objective sense.
Many belief that there is no such thing possible even. As an example take
comma pumps.
I belief such a thing IS possible, and that there is only one way to tune
something "in tune" in an objective sense.
I belief that what our ear hears as perfectly "in tune" has a mathematical
logic behind it. (and that this cannot be reduced to fixed small ratios liek
6/5 for minor third, but instead is a complex music theoretical system)
And this mathematical logic behind the perfect "in tune" tuning is what I
understand behind the words "Just Intonation".
Quite literally the words Just Intonation have this meaning, to tune Just.

Now I personally don't care what people have used / mis-used / abused this
name for, or which ratios people have attached to the word "Just".
If I belief I'm finding out about this mathematical system behind objective
perfect "in tune" tuning, I will use the words "Just" and "Just Intonation"
to describe it, as it's the true meaning of those words. Rational intonation
doesn't cover it in my opinion, and why should I even search for other words
when Just Intonation is the word that truly describes my intentions.

When people search the internet for "Just Intonation", they hope to find THE
pure tuning system that is "in tune" and "correct".
Wether such a thing exists or not, as long as I belief that what I'm doing
is just that, I feel that I'm more than entitled to use the term Just
Intonation.
And I will continue to do so :)

What I can do is differentiate from other systems that are called "Just
Intonation" by an additive to the word.
For instance Marcel's Just Intonation, or something like that. (that should
make some people happy).

Kind regards,
Marcel

🔗genewardsmith <genewardsmith@...>

🔗Carl Lumma <carl@...>

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:

> The term "just" has many meanings

But the phrase "just minor third" does not. Do you ever use it without being conscious of the fact that people will treat it as synonymous with "6/5 third"?

> For example, if asked to give an example of a "just"
> third, I might immediately mention 7/6

Same here, but I never call that a "just minor third", I call it a "just subminor third" or "just septimal minor third" but NEVER a "just minor third", and I notice you don't either.

> This semantic territory has been explored
> many times on this list

I've never seen this particular issue arise before. It's usually about what just intonation" means.

and agreeement to
> disagree is probably the best outcome.

A terrible idea, frankly, if it lets people think they can call any rationally tuned interval in the vicinity of 300 cents a "just minor third".

> The term "rational intonation" or RI is
> available as a neutral ground:

Not really; calling 32/27 an "RI minor third" is fine but doesn't address the point at issue.

🔗Marcel de Velde <m.develde@...>

🔗Margo Schulter <mschulter@...>

Hello, Gene, and thank you for the opportunity to join you in
exploring some points of usage.

>> The term "just" has many meanings

> But the phrase "just minor third" does not. Do you ever use it
> without being conscious of the fact that people will treat it
> as synonymous with "6/5 third"?

Reflecting on my own usage, I realize that one of my tendencies
may be to avoid this issue by speaking of "a minor third at a
just 6/5, 7/6, 13/11, or whatever," with the justness as a
property of the ratio.

>> For example, if asked to give an example of a "just"
>> third, I might immediately mention 7/6

> Same here, but I never call that a "just minor third", I call
> it a "just subminor third" or "just septimal minor third" but
> NEVER a "just minor third", and I notice you don't either.

Yes, I'd say "a just 7/6 minor third" or "a just septimal minor"
would be my style, with "subminor" especially expressive in a
meantone environment, although I tend to say "septimal" in lots
of contexts.

And I'd likewise call 6/5 a "pental minor third," or for those
not familiar with this useful adjective, a "5-limit minor third."

For newcomers to this issue, I'd explain that a minor third can
range from around 7:6 to 6:5, and a major third from around 5:4
to 9:7. A small major third is most harmoniously placed below a
large minor third (4:5:6), and a small minor third below a large
major third (6:7:9). These two divisions of the fifth, the pental
or 5-limit and septimal or 7-limit, are most audibly pure or
"just," although the term "just" can also be apply to many, or
indeed any, of the other rational ratios a major or minor third
may take on. There's a vast middle range between pental and
septimal, with some of us often tuning regular thirds in this
region.

Examples in this middle range are a just 32/27 or 13/11 -- and
also, in the "upper middle" range, 19/16, with some people
finding 16:19:24 notably stable. A related point, of course, is
that in some styles thirds are considered basic stable concords,
and in others as relatively concordant or "assonant" adjuncts to
simpler concords like 2:3:4.

>> This semantic territory has been explored
>> many times on this list

> I've never seen this particular issue arise before. It's
> usually about what just intonation" means.

I agree that this specific point might be novel and distinct from
the usual "What is just intonation?" dialogues, and might, it
occurs to me, connect that discussion with the related one about
interval naming, also a theme raised here and elsewhere over the
years.

>> and agreeement to
>> disagree is probably the best outcome.

> A terrible idea, frankly, if it lets people think they can call
> any rationally tuned interval in the vicinity of 300 cents a
> "just minor third".

Here it is curious, and possibly revealing from your standpoint,
that I tend to say, "a minor third at such-and-such a just
ratio." That phraseology may avoid the conflict, while specifying
the interval promotes good communications, however tastes may
differ.

>> The term "rational intonation" or RI is
>> available as a neutral ground:

> Not really; calling 32/27 an "RI minor third" is fine but
> doesn't address the point at issue.

Yes, I agree. However usage develops on the "just minor third"
question, specifying the ratio should promote mutual
comprehension.

Most appreciatively,

Margo
mschulter@...

🔗genewardsmith <genewardsmith@...>

🔗Afmmjr@...

🔗Marcel de Velde <m.develde@...>

🔗Margo Schulter <mschulter@...>

🔗Mike Battaglia <battaglia01@...>

🔗genewardsmith <genewardsmith@...>

🔗Marcel de Velde <m.develde@...>

Hi Gene,

> Call me crazy..
> > But after more research.. I think 1215/1024 may be the main just minor
> > third.
> > 1/1 1215/1024 3/2 being the main just minor triad.
> > I know it's very close to 19/16, but if think it should be 1215/1024
> > nontheless.
>
> I think it should be 39858075/33554432. That has even more factors of 3 and
> 5 in the numerator and 2 in the denominator, and is even closer to 19/16.
>
>
> > If true this would be a great show of the power of the octave and fifth.
>
> Mine even more so!

Haha, ok that's like calling me crazy :)
I expected that.
But you'll come back from that statement :D

1/1 1215/1024 3/2 is consonant in that it harmonizes with the 1/1 root.
1/1 32/27 3/2 is dissonant to 1/1 root, because of the /27. But it also
occurs.
1/1 6/5 3/2 is also dissonant to 1/1 root because of /5.
There is also 1/1 32/27 40/27, offcourse also dissonant to 1/1 root (but it
does harmonize as 1/1 5/4 27/16 to 1/1 root)

There is also 1/1 5/4 3/2, which is consonant to 1/1 root.
1/1 81/32 3/2, which is less consonant to 1/1 root.
1/1 2048/1215 3/2, which is dissonant to 1/1 root (occurs for instance as
405/256 2/1 1215/1024 to the 1/1 root, add 45/32 and you have the German
Sixth chord)

Here a chord which contains 5/4, 81/64 and 2048/1215:
1/1 5/4 405/256 2/1
Here the diminished chord:
9/8 4/3 405/256 15/8

etc etc.
It completely works out to put common practice music in JI in this way.
And not only does it work out.. you can "feel" and "read" the emotional
content and tensions / dissonance consonance etc in the ratios.
Trust me, I've tried a LOT of things to put common practice music in JI.
And the above way works out like magic, completely on another level than
anything I've tried or heard before.

Yes I know, at first it seems crazy and counter-intuitive.
But a crazy and counter-intuitive things was apparently needed. For millenia
people have tried to figure out how musical tuning works and failed.
All the simple and intuitive JI solutions have been tried and failed, and
temperaments offer nothing.
I urge you take investigate my above system seriously for your own pleasure
:)

Marcel

🔗Marcel de Velde <m.develde@...>

🔗genewardsmith <genewardsmith@...>

🔗Marcel de Velde <m.develde@...>

Just to update..

The piece I was hoping to conclusively say 1215/1024 is the main minor third
isn't conclusive.
I could still use 19/16 in it's place..
or 32/27, but 32/27 may sound too low to my ears.. though it's so close I
can't say for sure.

In similar ways 19/16 and 1215/1024 make sense, and in different ways each
makes sense on it's own.
Need more time.. though 6/5 is for sure out the window forgood now. If not
for theory that doesn't work out well enough, then because of my ears
because that 6/5 sure sounds bad in actual common practice music compared to
1215/1024 and 19/16.

I can think up harmonic progressions that seem to need the minor third as
1215/1024, but I don't understand them well enough to rely on yet.

I'm actually keeping my fingers crossed for 19/16 as it would open up a big
world of strong microtonal intervals.
Starting to get the feel for the power of 3/2, in melody and harmony.
It could well be that because of 12edo, the only harmonic subset that comes
close enough to be used is 1/1 17/16 9/8 19/16 5/4 and or something like
that, and the power of the fifth is used to transpose it all over. And
western music theory is based on this.
I kind of like this way of thinking.
The most noticable thing I hear in for instance arabian music is that the
"rest points" in melodies are most often in different places than in western
music.
It could be that it is still resting on fifths etc, just in a different
harmonic subset or something like that.

Marcel

Hi Gene,
>
> Thanks for listening.
>
>
>> I think you can tweak the tuning of an occasional minor third in various
>> ways without a drastic overall effect on the piece as a whole.
>
>
> Well I've tried it with 6/5 minor thirds in the past and they stick out as
> too high to my ears.
> Especially now in comparison.
>
> But this piece isn't the best example for my 1215/1024 minor thirds as it
> allows other minor thirds without messing up other chords in held notes etc.
> Working on a piece now that does not allow other minor thirds without
> messing up the piece as a whole. Will post when ready.
>
> Marcel
>

🔗rick <rick_ballan@...>

🔗Marcel de Velde <m.develde@...>

🔗rick <rick_ballan@...>

🔗genewardsmith <genewardsmith@...>

🔗Marcel de Velde <m.develde@...>

Ah about the tempered diminished opinions differ, lets agree to disagree :)
We've been there before :) And I'm against tempering.

Btw by about the same logic one can construct the 7-tone diatonic scale from
the 15/8 2/1 (17/8) 9/4 (19/8) 5/2 harmonic segment duplicated by a fourth
(or fifth depending on view) to get 15/8 2/1 9/4 5/2 4/3 3/2 5/3.
One could see this as a 3+5 scale.

One could construct a 3+7 scale (not using the prime 5 at all) in a similar
way:
21/16 11/8 (23/16) 3/2 (25/16) 13/8 (27/16) 7/4, which gives the following
scale when duplicated by a fourth and with 3/2 as the 1/1:

1/1 13/12 7/6 11/9 4/3 13/9 14/9 7/4 11/6 2/1
|
0: 1/1 C Dbb unison, perfect prime
1: 13/12 tridecimal 2/3-tone
2: 7/6 D# Eb septimal minor third
3: 11/9 undecimal neutral third
4: 4/3 F Gbb perfect fourth
5: 13/9 tridecimal diminished fifth
6: 14/9 septimal minor sixth
7: 7/4 A# Bb harmonic seventh
8: 11/6 21/4-tone, undecimal neutral seventh
9: 2/1 C Dbb octave

Looks like potentially a GREAT arabic like scale.
One would not harmonize this as western music (though one could move the
original segment in fifths /fourths just like western music). I have some
ideas about harmonies in this scale and melodic movements / restpoints etc
but need to investigate further.

Marcel

🔗Marcel de Velde <m.develde@...>

Well actually.. this is the first time any scale that differs a lot from
12edo (or any subset of it) sounds good to me and is kind of functional.
Really worth a try I think.
In melodies put all the accents / important rhythmic steps on 3/2 and 7/4
related intervals. The 11 and 13 intervals lead up or down to 3 and 7
intervals, but are fairly dissonant themselves (less so than chromatic steps
in 12edo, but still a lot in that direction).
I've also found one could omit certain steps by choice like either 7/4 or
11/6, and 11/9.

Perhaps it makes more sense to put the scale with the 4/3 on 1/1 btw.

I'm almost tempted to post a few musical examples.
It seems quite possible to do very microtonal sounding counterpoint like
things (which don't sound all out of tune in a wrong way like it usually
does when doing counterpoint like things with very non 12edo scales)

Marcel

Btw by about the same logic one can construct the 7-tone diatonic scale from
> the 15/8 2/1 (17/8) 9/4 (19/8) 5/2 harmonic segment duplicated by a fourth
> (or fifth depending on view) to get 15/8 2/1 9/4 5/2 4/3 3/2 5/3.
> One could see this as a 3+5 scale.
>
> One could construct a 3+7 scale (not using the prime 5 at all) in a similar
> way:
> 21/16 11/8 (23/16) 3/2 (25/16) 13/8 (27/16) 7/4, which gives the following
> scale when duplicated by a fourth and with 3/2 as the 1/1:
>
> 1/1 13/12 7/6 11/9 4/3 13/9 14/9 7/4 11/6 2/1
> |
> 0: 1/1 C Dbb unison, perfect prime
> 1: 13/12 tridecimal 2/3-tone
> 2: 7/6 D# Eb septimal minor third
> 3: 11/9 undecimal neutral third
> 4: 4/3 F Gbb perfect fourth
> 5: 13/9 tridecimal diminished fifth
> 6: 14/9 septimal minor sixth
> 7: 7/4 A# Bb harmonic seventh
> 8: 11/6 21/4-tone, undecimal neutral seventh
> 9: 2/1 C Dbb octave
>
> Looks like potentially a GREAT arabic like scale.
> One would not harmonize this as western music (though one could move the
> original segment in fifths /fourths just like western music). I have some
> ideas about harmonies in this scale and melodic movements / restpoints etc
> but need to investigate further.
>
> Marcel
>

🔗genewardsmith <genewardsmith@...>

🔗Marcel de Velde <m.develde@...>

🔗Margo Schulter <mschulter@...>

[Please forgive me for the curious subject line,
which it seems easier not to change in order to
let people follow the thread.]

Dear Marcel,

Thank you for your JI tuning, which indeed has many of the
intervals which I use, in just or tempered form, for Near Eastern
music.

> 1/1 13/12 7/6 11/9 4/3 13/9 14/9 7/4 11/6 2/1

As Gene has said, this is a nice selection, and I realize from
your comments that you might have various ideas about making it
part of a larger tuning set. People might also want to take a
look at a classic presentation by George Secor on how to use
ratios like these in some less traditional stylistic contexts:

<http://xenharmony.wikispaces.com/space/showimage/17puzzle.pdf>

To illustrate the great aptness of some of these intervals for an
Arab, or often I would guess especially for an Iranian style, I
thought that I might expand your set to one possible 12-note
tuning, and point out a few available Near Eastern and related
modes. By the way, this 12-note set also includes a number of the
scales or modes discussed in Secor's article above.

! variant-on-marcel_12.scl
!
Expansion of Marcel de Velde's JI tuning, TL #90805 (9 July 2010)
12
!
28/27
13/12
7/6
11/9
4/3
13/9
3/2
14/9
13/8
7/4
11/6
2/1

Something that immediately occurred to me on seeing your set was
Persian shur, which I'll write with some Persian note notation:

Ap Bp
13/9 13/8
1/1 13/12 7/6 4/3 3/2 14/9 7/4 2/1
D Ep F G A Bb C D

Here a koron (ASCII p) shows a step lowered by often about a
third of a tone, say 70 cents, although this may vary. In modern
practice, the 13/9 or A-koron (Ap) is a kind of lowered fifth
often favored in passages descending toward the resting note or
final D, the 1/1. While a minor sixth is standard, a small
neutral sixth is favored as a variation in certain modern styles,
so it's good to have both steps available.

Interestingly, a tuning of the great philosopher and music
theorist Ibn Sina in the early 11th century has a very similar
pattern, with the second and sixth steps in an alternative
interpretation being 14/13 and 21/13:

1/1 13/12 7/6 4/3 3/2 13/8 7/4 2/1

A fair caution: some leading writers on Persian music say that a
step as small as 63 cents (3/2-14/9, or 28:27) would occur only
as a special ornament, nor as a regular interval in a mode; but
some frettings do seem to show steps not too much larger than
this, around 70-75 cents. I use it often on keyboard, but Iranian
musicians must judge how wisely.

Now let's move to the 13/12 step, where we find another mode with
likely Persian affinities, Buzurg, of which Jacques Dudon and I
are alike enamoured:

14/13 16/13 4/3 56/39 3/2 21/13 24/13 2/1

Note that two theorists of the 13th century, Safi al-Din
al-Urmawi and Qutb al-Din al-Shirazi, give the lower fifth as
"Buzurg," but various upper tetrachords might be used. Here I've
given a symmetrical "Buzurg" tetrachord 3/2-21/13-24/13-2/1 with
the steps 14:13-8:7:13:12.

Another mode you'll hear a great deal about here, and which
Jacques Dudon has explored over the last almost two decades, is
what he has termed Mohajira, from an Arabic root for "migration."
On the 1/1 step of the tuning, we have:

1/1 13/12 11/9 4/3 3/2 13/8 11/6 2/1
Z T Z T Z T Z

A special property of Dudon's Mohajira is that it alternates
whole tones (T) with Zalzalian or neutral steps (Z), the latter
justly named after the lutenist Mansur Zalzal, who in the 8th
century is said to have introduced a neutral third fret.

A beautiful variation on this, but with 7/4 here in place of
Dudon's 16/9, is this:

1/1 13/12 11/9 4/3 3/2 13/8 7/4 2/1

With the 16/9, this would be Jacques' Ibina scale.

For something with a characteristically Arab flavor, we go back
to the 13/12 step, and find this pleasant version of the
"Arab fundamental scale" used in Maqam Rast, with two tetrachords
each consisting of a whole tone plus a larger and smaller
Zalzalian or neutral second. Note that the simple ratio of 9/8 is
here "virtually tempered" at 44/39, not quite five cents wider:

|-----------------| |--------------------|
1/1 44/39 16/13 4/3 3/2 22/13 24/13 2/1
C D Ed F G A Bd C

Here ASCII "d" shows an Arab half-flat, which like the Persian
koron actually can vary quite a bit in size but marks a Zalzalian
or neutral interval. In Rast, we generally expect the neutral
third and seventh to be rather on the large size, or somewhat
wider than around 351 and 1053 cents (with ~351 cents equal to
half of a pure 3/2 fifth), here 16/13 and 24/13 at 359 and 1061
cents. This tuning is likely to please lots of Arab tastes, and
to be if anything a bit "low," but acceptable, to Turkish
tastes.

If we start this same mode on its 3/2 step -- or the 13/8 step of
the 12-note tuning, then we get another beautiful version of
Rast, here with the fifth "virtually tempered" by not quite five
cents wide at 176/117 (as George Secor once noted, JI tunings I
devise -- or here, as with yours, expand -- often seem to feature
a lot of these ratios):

|-----------------|-------------------|
1/1 44/39 16/13 4/3 176/117 64/39 16/9 2/1
G A Bd C D Ed F G

Here the two Rast tetrachords are conjunct rather than disjunct,
resulting in a minor rather than Zalzalian seventh. This form is
in fact rather similar to al-Farabi's account of Zalzal's tuning
for the 'oud or lute as reported around the 10th century, but
with the third and sixth a bit higher than that theorist's 27/22
and 18/11 (355 and 853 cents).

Thank you again for your scale, and please forgive me for a
rather commonplace 12-note expansion which may be overly
symmetrical by comparison to your original concept. But I did
want to share my enthusiasm.

With many thanks,

Margo Schulter
mschulter@...

🔗Marcel de Velde <m.develde@...>

Hello Margo,

Thanks for your reply!

It's amazing to see how much the 11 and 13 derived ratios are common in
arabian music theory.
I had no idea.
And yes, many of those scales do resemble my idea a lot it seems.
Good to be in somewhat in this company, as the microtonal music I enjoy most
up till now is arabian music :)

Though actually getting this kind of music out of my own scale is still
rather difficult.
At times I feel I'm getting the hang of it. At other times I'm at a complete
loss and everything sounds wrong to me.
I guess it's also due to the fact that I want to understand how it works,
how to use the scale theoretically, and I don't know this well enough yet
(by far).
Unlike western common practice music, which I do understand how it functions
for a large part.

What I have discovered so far in playing, is that the scale I gave seems too
limited to me now.
I guess it needs at least 3 segments to work (similar to the I - IV - V idea
in western music)

The 3 segments would be:
- 7/6 11/9 4/3 13/9 14/9
- 7/4 11/6 2/1 13/12 7/6
- 21/16 11/8 3/2 13/8 7/4

Making the full scale:
1/1 13/12 7/6 11/9 21/16 4/3 11/8 13/9 3/2 14/9 13/8 7/4 11/6 2/1, with the
real fundamental basses beeing 1/1, 4/3 and 16/9, though they're probably
not played as the harmonic 1/1 point is not part of the harmonic segments.
It seems the harmonic 3/2 point of the segments has a good rest/resolution
function aswell (as it does in western music too), and the 7/4 perhaps also
somewhat? As I said, still trying to make sense of the function of the
tones.
Btw the scale is sadly 13 tones /oct, not 12 which would have been handy..
One could also add 16/9 if one uses the 1/1 of the harmonic segments (16/9
making the 1/1 of 7/6 11/9 4/3 13/9 14/9) which makes it 14 tones.

I'll try to give a more worthy indepth reply to your email soon, bedtime now
:)

Kind regards,
Marcel

🔗Marcel de Velde <m.develde@...>

Couldn't sleep and have been thinking and thinking..
This 19th harmonic as the minor third just doesn't make enough sense to me..
I've been there before many many times. Every time I can make it work a
little better, but not because of the 19th harmonic, but because of
completely different lines of thinking..
19th harmonic, harmonic segments.. I'm leaving those for what they are at
the moment.
Maybe one day I'll revisit but I hope not as it's a disaster to me for
actually making music haha :)

But for the minor third and common practice music.
I'm putting my money on my best working, sounding and logical discovery yet.
Which is the minor third as 1215/1024 and 32/27 in major mode (on the i, ii,
iv, v etc) with major third as 5/4
And the minor third as 6/5 in minor mode (which is the relevant minor mode
on vi) with major third as 81/64 and 405/256.
I can make music with this, and possibly write good algorithms for analysis
and creation.

It just doesn't make enough sense to me either when music that's in one key
switches tones a lot (like 10/9 then 9/8 then 10/9 again etc or 5/3 then
27/16)
I'm not sure wether I trust common practice music theory to tell me wether
something is in one key or not.
Beethoven's Drei Equale no1 seems to me to be in one key (except maybe for
the ending, though I think it still is)
Drei Equale no2 on the other seems to have one part in it that modulates to
a fifth below (it actually fools me into thinking one part is repeating when
it's actually a fifth below!)

So, after all this work.
I'm into a fixed 12 tone scale per key.

It is:
1/1 135/128 9/8 1215/1024 5/4 4/3 45/32 3/2 405/256 5/3 16/9 15/8 2/1
It has 7 pure fifths on 16/9 and on 5/3
Major key is on 1/1, minor key is on 5/3.
Scale moves with modulations offcourse.

As for arabic music.
I really think that before I can really make that work, I need to really
deeply understand common practice music tuning.
Then I can hopefully build on that and get to the really hard stuff.

Marcel

🔗rick <rick_ballan@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "rick" <rick_ballan@> wrote:
> And for the dim, the tempered would be the 'ideal' since it reaches the 8ve exactly.
> >
>
> Except it sounds kind of boring.
>
Tell that to J.S. Bach, or Beethoven. The diminished is a modulation or passing chord and *never* appears on its own or out of context. You don't program it in Scala and compare it to other 'tunings' without understanding what it is. If we lower any note of a dim 7 chord by a semitone we get a dom 7 chord. Combining them gives the 7(b9). Since the diminished has the same interval content in all inversions then it is ideal for chord voicing and modulating to different keys with tonics minor thirds apart. Particularly relative major to minor or vice-versa. Eg G7 -> Cma becomes Abdim -> Cmaj becomes G#dim = E7 -> Amin.

One of its most useful properties is that the tritone between its maj 3rd and b7 degrees descends chromatically if we go through the cycle of fifths, inverting each time. Eg C7 with E-Bb (3rd-7th)-> F7 with Eb-A (7th-3rd) etc...These can then be modified to fit the particular chord progression Eg C7 to Fma7 becomes E-Bb to E-A. We jazz musicians call these *guide tones*. Contrary to old fashioned views of harmony, the fifth is quite dispensable.

Finally, only the tritone tuned as Sqrt2 gives the 8ve when squared. It is in a manner of speaking a '2 tET'. Only the dim7 tuned to the 4th rt 2 gives the 8ve when multiplied by itself 4 times. It is a '4 tET'. Its harmonic function is totally distinct from the minor chord and only these can be retuned to JI without running into trouble all over the place.

-Rick

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, "rick" <rick_ballan@...> wrote:

> > --- In tuning@yahoogroups.com, "rick" <rick_ballan@> wrote:
> > And for the dim, the tempered would be the 'ideal' since it reaches the 8ve exactly.
> > >
> >
> > Except it sounds kind of boring.

> Tell that to J.S. Bach, or Beethoven.

I would except

(1) They are dead, and
(2) Neither one ever used it.

The diminished is a modulation or passing chord and *never* appears on its own or out of context.

(Nods head). Which diminished seventh chord are we talking about here, by the way?

As for appearing alone or out of context, it will do that if and when you tell it to. That's the joy of being a composer. If you are in, for example, myna temperament, a version of it, related to the meantone chord consisting of three minor thirds and an augmented second, is quite likely to appear all over the place.

>You don't program it in Scala and compare it to other 'tunings' without understanding what it is.

Which as far as I can tell is something you need to work on. This is the tuning list. We are a tough audience, and will not necessarily buy the idea that the diminished seventh chord of common practice harmony has always been, and must and shall forever be, composed of 300 cent minor thirds. Much less that it also must and shall be so in xenharmonic contexts.

> If we lower any note of a dim 7 chord by a semitone we get a dom 7 chord.

What's a "semitone"? 100 cents?

> It is a '4 tET'. Its harmonic function is totally distinct from the minor chord and only these can be retuned to JI without running into trouble all over the place.

Um, yeah. Of course there are various ways of doing what you claim cannot be done.

🔗rick <rick_ballan@...>

You know me too well by now to know I'm not going to give up that easily Marcel (lol). I think it's extremely idealistic to believe that there is a one-one correspondence between interval and ratio or that there cannot be a 'give' to each interval. Even the slightest temperature change can make an instrument go out of tune in the middle of a performance and the sky doesn't come crumbling down. The idea that there is something special about the whole-numbers, that they are somehow the 'ideal' against which all others are compared, is an old myth going right back to Pythagoras and one which I now suspect has stood in the way of music theory for centuries. (This is made even worse when mathematicians and scientists, who have only a rudimentary understanding of musical harmony, believe that their own explanations are sufficient). As I've shown recently, the underlying maths of GCD's is modelled quite beautifully by the approximate GCD's of the actual waves themselves. These have all the desirable properties of the integers but with many added advantages. Beats, for eg, are automatically explained by these ~ GCD's since the operation of subtraction is contained inside that of division (for pure consecutive harmonics the GCD = difference, which translates to ~GCD's). Since all larger intervals are mapped back onto simpler JI's then it also provides a practical method for grouping intervals in principle.

But I do find it strange that you seem to love Beethoven who wrote all his music in 12 tET. Don't you think this is hypocritical?

-Rick

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
> Ah about the tempered diminished opinions differ, lets agree to disagree :)
> We've been there before :) And I'm against tempering.
>
> Btw by about the same logic one can construct the 7-tone diatonic scale from
> the 15/8 2/1 (17/8) 9/4 (19/8) 5/2 harmonic segment duplicated by a fourth
> (or fifth depending on view) to get 15/8 2/1 9/4 5/2 4/3 3/2 5/3.
> One could see this as a 3+5 scale.
>
> One could construct a 3+7 scale (not using the prime 5 at all) in a similar
> way:
> 21/16 11/8 (23/16) 3/2 (25/16) 13/8 (27/16) 7/4, which gives the following
> scale when duplicated by a fourth and with 3/2 as the 1/1:
>
> 1/1 13/12 7/6 11/9 4/3 13/9 14/9 7/4 11/6 2/1
> |
> 0: 1/1 C Dbb unison, perfect prime
> 1: 13/12 tridecimal 2/3-tone
> 2: 7/6 D# Eb septimal minor third
> 3: 11/9 undecimal neutral third
> 4: 4/3 F Gbb perfect fourth
> 5: 13/9 tridecimal diminished fifth
> 6: 14/9 septimal minor sixth
> 7: 7/4 A# Bb harmonic seventh
> 8: 11/6 21/4-tone, undecimal neutral seventh
> 9: 2/1 C Dbb octave
>
> Looks like potentially a GREAT arabic like scale.
> One would not harmonize this as western music (though one could move the
> original segment in fifths /fourths just like western music). I have some
> ideas about harmonies in this scale and melodic movements / restpoints etc
> but need to investigate further.
>
> Marcel
>

🔗genewardsmith <genewardsmith@...>

🔗jlmoriart <JlMoriart@...>

🔗Daniel Forró <dan.for@...>

On 10 Jul 2010, at 4:15 PM, rick wrote:

>>
> Tell that to J.S. Bach, or Beethoven. The diminished is a > modulation or passing chord and *never* appears on its own or out > of context.

There's million examples in classical music where these chords are used on its own, or out of context, or chained.

> You don't program it in Scala and compare it to other 'tunings' > without understanding what it is. If we lower any note of a dim 7 > chord by a semitone we get a dom 7 chord.

That's case only for the lowest note. Lowering the other notes gives dom7 inversions.
And it's almost always connected with lot of enharmonic changes to get properly spelled chord.

> Combining them gives the 7(b9).

Combining what?

> Since the diminished has the same interval content in all inversions

... not true in the score and theory, only on the keyboard or fretboard ...

> then it is ideal for chord voicing

????

> and modulating to different keys with tonics minor thirds apart.

This was true maybe for early Baroque when they use it for modulations to 8 keys. But any new tonic chord (major or minor} can follow dim7 chord. So we can modulate to all keys with the help of it. Such progressions can be found in Vivaldi, Bach, Handel... and of course later until Late Romantism. Especially enharmonic-chromatic type of modulation is surprising - great example is in Lacrimosa from Mozart's Requiem.

> Particularly relative major to minor or vice-versa. Eg G7 -> Cma > becomes Abdim -> Cmaj becomes G#dim = E7 -> Amin.

Diminished seventh chord used this way is always on the VIIth grade of the scale in functional harmony (and of course it can precede any grade as borrowed VII7 chord), so before Cmi or C you must use Bdim7 to keep the proper spelling. Abdim must be spelled Ab-Cb-Ebb-Gbb which is not possible in C tonality. If you want to use exactly same notes in C, it must be spelled Ab-B-D-F, which is the last inversion of Bdim7 (B-D-F-Ab).
Which you wrote well in the case of G#dim before Ami.

>
> One of its most useful properties is that the tritone between its > maj 3rd and b7 degrees descends chromatically if we go through the > cycle of fifths, inverting each time. Eg C7 with E-Bb (3rd-7th)-> > F7 with Eb-A (7th-3rd) etc...These can then be modified to fit the > particular chord progression Eg C7 to Fma7 becomes E-Bb to E-A. We > jazz musicians call these *guide tones*.

Here you talk about dom7 chord.

These progressions have been overused since Baroque times, and disused in jazz, so I have experimented many years ago with chains of mi6 chords, which contain also tritone. Guide tones go up in this case, and you get progressions like Fmi6 - Cmi6 - Gmi6 - Dmi6 - Ami6 - Emi6 ...

And in last years when I want to use chromatically shifted tritone, I prefer to harmonize it with tritone progression and alternate major and minor chords - like Cmi6 - F#7 - Dmi6 - Ab7.... or C7 - F#mi6 - D7 - G#mi6... Much more interesting.

> Contrary to old fashioned views of harmony, the fifth is quite > dispensable.

Which old fashioned? Which period or theory? Classical functional harmony allows omitting of fifth in seventh and 9th chords, and 5th and 3rd in higher triadic chords 11th and 13th.

Daniel Forro

🔗rick <rick_ballan@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "rick" <rick_ballan@> wrote:
>
> > > --- In tuning@yahoogroups.com, "rick" <rick_ballan@> wrote:
> > > And for the dim, the tempered would be the 'ideal' since it reaches the 8ve exactly.
> > > >
> > >
> > > Except it sounds kind of boring.
>
> > Tell that to J.S. Bach, or Beethoven.
>
> I would except
>
> (1) They are dead, and
> (2) Neither one ever used it.

Well let me see. Page 1 Bach's LautenMusik, praludium I. Bar 3, E dim7/D bass, bar 8, G#dim7, then later a whole passage Edim, Fdim/E bass, Edim, Edim, D#dim/E bass...Now you get to say that Bach played in another temperament or that we can only tune irrationals to a few decimal places. And Beethoven uses it even in the opening variation of the fifth symph! Are you being funny?

> The diminished is a modulation or passing chord and *never* appears on its own or out of context.
>
> (Nods head). Which diminished seventh chord are we talking about here, by the way?

Well 12 ET of course.
>
> As for appearing alone or out of context, it will do that if and when you tell it to. That's the joy of being a composer. If you are in, for example, myna temperament, a version of it, related to the meantone chord consisting of three minor thirds and an augmented second, is quite likely to appear all over the place.

Sure, but strictly speaking it won't be the same under inversion; two min 3rds, aug 2, min 3rd etc...Of course because of harmonic entropy the ear will probably still hear it as being perfectly symmetrical. But my original point to Marcel was that despite his desire to retune Beethoven to pure JI, his first tuning sounded out because the Opus Drei is full of puns on the ambiguous diminished7 chord which doesn't suit JI.
>
> >You don't program it in Scala and compare it to other 'tunings' without understanding what it is.
>
> Which as far as I can tell is something you need to work on. This is the tuning list. We are a tough audience, and will not necessarily buy the idea that the diminished seventh chord of common practice harmony has always been, and must and shall forever be, composed of 300 cent minor thirds. Much less that it also must and shall be so in xenharmonic contexts.

Nonsense. There is only one number that is the fourth root of 2 and all diminished's will be sufficient approximations to this. The cent value is just an artificial construct. Saying that 2 + 2 = 5 "just because we can" doesn't make it so. It's not creative but destructive. OTOH by finding the limits that intervals can be detuned we can at least conceive of a system which might allow 2 + 2 ~ 5. As for being a tough audience, well you improvise a four hour solo gig. I'm not coming from nowhere.
>
> > If we lower any note of a dim 7 chord by a semitone we get a dom 7 chord.
>
> What's a "semitone"? 100 cents?

In this context what else would it be? I was explaining how dim's are used to substitute for the V7 chord in common western harmony. Understanding traditional harmony is not a 'prejudice' to be overcome. As far as I'm concerned it should be a necessary prerequisite.
>
> > It is a '4 tET'. Its harmonic function is totally distinct from the minor chord and only these can be retuned to JI without running into trouble all over the place.
>
> Um, yeah. Of course there are various ways of doing what you claim cannot be done.

Did I say it cannot be done? No. I might have implied that there is no point in doing it when its done perfectly already. The 12 tET system evolved out of years of accumulated experience, weighing certain practical considerations against aesthetic ones etc... And the use of the same note/key to serve every function equally is actually quite a beautiful and efficient solution to the problem.
>

🔗Afmmjr@...

🔗Marcel de Velde <m.develde@...>

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, "rick" <rick_ballan@...> wrote:

> Well let me see. Page 1 Bach's LautenMusik, praludium I. Bar 3, E dim7/D bass, bar 8, G#dim7, then later a whole passage Edim, Fdim/E bass, Edim, Edim, D#dim/E bass...Now you get to say that Bach played in another temperament

Which is exactly what he did. He was very fussy about tuning things his way, and his way was not 12 equal. If you think it should have been, tell it to Bach.

> ... or that we can only tune irrationals to a few decimal places. And > Beethoven uses it even in the opening variation of the fifth symph! > Are you being funny?

No, I am being accurate. You should try it. An orchestra in Beethoven's day wasn't even using 12et as an ideal, much less actually playing in it.

> Nonsense. There is only one number that is the fourth root of 2 and all diminished's will be sufficient approximations to this.

Sufficient for whom? How close is "sufficient"? 300 cents is 33 cents sharp of 7/6 and 16 cents flat of 6/5.

> Did I say it cannot be done? No. I might have implied that there is no point in doing it when its done perfectly already. The 12 tET system evolved out of years of accumulated experience, weighing certain practical considerations against aesthetic ones etc... And the use of the same note/key to serve every function equally is actually quite a beautiful and efficient solution to the problem.
> >

And one, which as I said when I inadvertently started this thread, has a dim7 chord which is BORING. Being boring is sort of characteristic of 12et; the augmented triad is another boring version of an interesting chord, and the triads are nothing to write home about. It's the first equal temperament you can actually get away with using for 5-limit and to some extent 7-limit music, but that doesn't make it a magical solution to all tuning problems.

🔗Marcel de Velde <m.develde@...>

🔗genewardsmith <genewardsmith@...>

🔗Marcel de Velde <m.develde@...>

On 11 July 2010 05:10, Marcel de Velde <m.develde@...> wrote:

> Ah it's not Kirnberger I.
> Not sure what tuning it is.
> But Kirnberger 1 put the fifth+Schisma between 256/143 and 45/32, and the
> wolf fifth between 9/8 and 5/3.
> I'm actually not quite sure where to put the Schisma myself, though the 9/8
> 5/3 are certain.
> I'm going to investigate for a while and make a new thread when I'm done
> with that and have lots of music retuned.
>
> Marcel
>

Ah I may aswell admit I made a stupid mistake :)
I somehow was so crazy to think I had 7 notes connected by 3/2 fifths on
16/9, and 7 notes connected by 3/2 fifths on 5/3, and this in a 12 note
scale ;)
lol
Too little sleep or something..
The scale I think I was meaning to do is:
1/1 135/128 9/8 32/27 5/4 4/3 45/32 3/2 405/256 5/3 16/9 15/8 2/1

6 notes connected by 3/2 fifths on 32/27, and 6 notes connected by 3/2
fifths on 5/3.
The wolf fifth between 9/8 and 5/4.
The fifth+Schisma between 405/256 and 32/27.
(Which is kinda funny, because this makes the German sixth 405/256 - 2/1 -
64/27 - 45/16, a very special chord, and perhaps it can do something
special, like go straight to I where that would normally be against the
rules? (look at the 135/128 steps from that chord to both 3/2 and 5/2,
exception from normal counterpoint rules because of this?))

Btw, before I said that I couldn't accept 1/1 6/5 3/2 as a minor third
because the 6/5 sounded too high.
And that it should be 1/1 1215/1024 3/2 or 1/1 32/27 3/2 which sound "right"
to me.
I based that on the beginning of the Drei Equale no1.
If playing it as 1/1 3/2 2/1 12/5 -> 3/2 15/8 9/4, then the 12/5 sounds way
too high to me and sounds right as a 1215/512 or 64/27.
But.. I've found that if playing it as 1/1 3/2 2/1 12/5 -> 3/2 243/128 9/4
then it sounds "right" too!
So it's the 12/5 in combination with 15/8 that was bothering my ears!
Non of that in my new scale.
In my new scale there's major mode on 1/1, and minor mode (relevant minor
key) on 5/3.
So one scale for both major and minor.
When you put the 5/3 on 1/1 you'll see the minor scale from it's root:
1/1 16/15 9/8 6/5 81/64 27/20 64/45 3/2 8/5 27/16 9/5 243/128 2/1

Ok I'm done now :)
This thread has gone in a different direction allready and I'm replying to
myself :)

Marcel

🔗Marcel de Velde <m.develde@...>

🔗rick <rick_ballan@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "rick" <rick_ballan@> wrote:
> > (This is made even worse when mathematicians and scientists, who have only a rudimentary understanding of musical harmony, believe that their own explanations are sufficient).
>
> Darn those mathematicians and scientists anyway. If people like Euler, Huygens, Fokker or Borodin hadn't meddled, think of how much farther on music would be! And let's not even get into the philosophical meddlers like Boethius or Avicenna, who must share their portion of blame.

I'm not a conspiracy theorist Gene. My point here was to take to task the underlying assumption that the scientific method, still prevalent today, is independent from politics or personal agendas. Nietzsche makes a similar point about philosophers when he says that every philosophy, while they all claim to be *the* most objective theory to date, are merely autobiographies of their author, his own included. Now, now one would doubt that Eulers ID or Totient function are invaluable tools for wave theory in general and musical harmony in particular, but his paper on musical harmony (I forget its Latin title) which attempts to identify chords with LCM's, is back to front. If we invert his results to time periods instead of freq's then we get GCD's of periodic functions. Similarly, while it is standard to derive the harmonic series by applying boundary conditions to the wave equation, a very basic fact that is generally NOT considered is that the same harmonies apply A) between boundary conditions and B) without them. This apparently simple distinction changes the entire view of waves and their place in the scheme of things. This is because boundary conditions are no longer given a priori but can be defined in terms of the class of given wavelengths. IOW it is waves that are a priori and natural, not man-made measuring sticks. Yet most physicists still believe that musical harmony was covered centuries ago by Newtonian mechanics. It is not their equations I doubt but their *interpretation* of them which has been (unconsciously) biased to paint a mechanical 'point-particle' view of the universe, contrary to all the evidence. Why is it called a "particle accelerator" for eg and not a "wave-particle accelerator", a "particle zoo" and not a "wave-particle" etc?
>
> > As I've shown recently, the underlying maths of GCD's is modelled quite beautifully by the approximate GCD's of the actual waves themselves.
>
> Write it up and send it to the Journal of Number Theory. That way it will be peer reviewed. Include proofs.

Yes I'm intending to.
>
> > But I do find it strange that you seem to love Beethoven who wrote all his music in 12 tET. Don't you think this is hypocritical?
>
> It might be, but Beethoven never wrote any music for 12et.

I don't know what you mean here. He was a pianist. Sure orchestra string sections are fretless but the rest of the instruments aren't.
>

🔗rick <rick_ballan@...>

Well its true that 4 is a subset of 12. There are 3 x 4 = 12 diminished chords in 12 tET, 4 x 3 augmented's etc...It's also true what you say about 12 music. But my point was also that this 4 is itself a kind of ET and is independent. For eg, in a (albeit very boring musical universe) of 4 tET we would have only 4 keys and 4 intervals + inversions. Therefore trying to retune it to JI is going to destroy that internal logic. Besides, it would be a subset of any ET which is divisible by 4. At any rate, the dim7 chord is very valuable in standard harmony even when its not played. In a basic blues for eg we have the tritone between the maj3 and b7 moving chromatically.

--- In tuning@yahoogroups.com, "jlmoriart" <JlMoriart@...> wrote:
>
> > the diminished has the same interval content in all
> inversions
>
> From C to Eb is a minor third, from Eb to Gb is a minor third, from Gb to Bbb is a minor third, but from Bbb to C is an augmented second. Isn't this nature of diminished chords (where they can resolve in so many directions and represent so many different chords at once) a product of twelve equal divisions of the octave where the enharmonic equivalents of an augmented second and a minor third *cause* things to act this way?
> And doesn't this mean that the coolness of these versatilities should be attributed to the tuning (12-edo) and not the chord (diminished)?
>
> Not that it's not a valid compositional technique or anything. It's awesome that each tuning, and its own enharmonic equivalents, can lead to different patterns and possibilities. Giant Steps is (I think) reliant on the fact that three major thirds are equal to four minor thirds which is unique to 12-edo.
>
> John Moriarty
>

🔗jonszanto <jszanto@...>

🔗rick <rick_ballan@...>

Oh I see what you're saying Gene. I thought you'd gone temporarily mad. I once got into trouble from Claudio for saying that the well tempered clavier was 12 tET (which at the time I thought was true). I'll just say that there are many performances of say Bach on modern tempered instruments (Segovia and John Williams for eg) and the music still 'comes through'. Besides, as a guitarist myself I know that strings go out of tune all the time without ruining a performance. I imagine that there is a 'zone' where all major thirds, fifths etc...fit, which is no doubt a motive behind my own theory of approx GCD's. It's also probably what attracted me to harmonic entropy.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "rick" <rick_ballan@> wrote:
>
> > Well let me see. Page 1 Bach's LautenMusik, praludium I. Bar 3, E dim7/D bass, bar 8, G#dim7, then later a whole passage Edim, Fdim/E bass, Edim, Edim, D#dim/E bass...Now you get to say that Bach played in another temperament
>
> Which is exactly what he did. He was very fussy about tuning things his way, and his way was not 12 equal. If you think it should have been, tell it to Bach.
>
> > ... or that we can only tune irrationals to a few decimal places. And > Beethoven uses it even in the opening variation of the fifth symph! > Are you being funny?
>
> No, I am being accurate. You should try it. An orchestra in Beethoven's day wasn't even using 12et as an ideal, much less actually playing in it.
>
> > Nonsense. There is only one number that is the fourth root of 2 and all diminished's will be sufficient approximations to this.
>
> Sufficient for whom? How close is "sufficient"? 300 cents is 33 cents sharp of 7/6 and 16 cents flat of 6/5.
>
> > Did I say it cannot be done? No. I might have implied that there is no point in doing it when its done perfectly already. The 12 tET system evolved out of years of accumulated experience, weighing certain practical considerations against aesthetic ones etc... And the use of the same note/key to serve every function equally is actually quite a beautiful and efficient solution to the problem.
> > >
>
> And one, which as I said when I inadvertently started this thread, has a dim7 chord which is BORING. Being boring is sort of characteristic of 12et; the augmented triad is another boring version of an interesting chord, and the triads are nothing to write home about. It's the first equal temperament you can actually get away with using for 5-limit and to some extent 7-limit music, but that doesn't make it a magical solution to all tuning problems.
>

🔗rick <rick_ballan@...>

We're just talking about different language systems Daniel. Sure, the diminished can appear by itself or in succession in a composition to build tension effect (used in the Dastardly Dan cartoon for eg where he ties the victim to a railway track). My point to Gene was that in tonality it almost invariably appears as a substitute for the V7 in a cycle.

As for the spellings, one must keep in mind that jazz is fundamentally a practice based art form. Charts are often just guidelines, even excuses, to jam. I once heard a story of an Australian musician joining the Duke Ellington orchestra and asking "where's the charts?". The band members all laughed and one of them showed him to a room which had boxes and boxes of them and said "good luck!". IOW they all learned to play from memory.

There was even a push in jazz during the 60's against using any written language at all. It's claim was that the music should speak for itself. Miles Davis never allowed writing on his album covers.

To allot of jazz musicians, a G# and Ab need to be the same because the harmonies can get so remote from any tonic that it doesn't matter anyway. Or it might be that an Ab dim to Amin is used because it is not intended as an E7(b9) type chord etc...And over analysing is even frowned upon because, in certain jazz circles, it is seen as unmusical, a "brain thing".

Rick

--- In tuning@yahoogroups.com, Daniel Forró <dan.for@...> wrote:
>
>
> On 10 Jul 2010, at 4:15 PM, rick wrote:
>
> >>
> > Tell that to J.S. Bach, or Beethoven. The diminished is a
> > modulation or passing chord and *never* appears on its own or out
> > of context.
>
>
> There's million examples in classical music where these chords are
> used on its own, or out of context, or chained.
>
>
> > You don't program it in Scala and compare it to other 'tunings'
> > without understanding what it is. If we lower any note of a dim 7
> > chord by a semitone we get a dom 7 chord.
>
> That's case only for the lowest note. Lowering the other notes gives
> dom7 inversions.
> And it's almost always connected with lot of enharmonic changes to
> get properly spelled chord.
>
> > Combining them gives the 7(b9).
>
> Combining what?
>
>
> > Since the diminished has the same interval content in all inversions
>
> ... not true in the score and theory, only on the keyboard or
> fretboard ...
>
>
> > then it is ideal for chord voicing
>
> ????
>
> > and modulating to different keys with tonics minor thirds apart.
>
>
> This was true maybe for early Baroque when they use it for
> modulations to 8 keys. But any new tonic chord (major or minor} can
> follow dim7 chord. So we can modulate to all keys with the help of
> it. Such progressions can be found in Vivaldi, Bach, Handel... and of
> course later until Late Romantism. Especially enharmonic-chromatic
> type of modulation is surprising - great example is in Lacrimosa from
> Mozart's Requiem.
>
> > Particularly relative major to minor or vice-versa. Eg G7 -> Cma
> > becomes Abdim -> Cmaj becomes G#dim = E7 -> Amin.
>
>
> Diminished seventh chord used this way is always on the VIIth grade
> of the scale in functional harmony (and of course it can precede any
> grade as borrowed VII7 chord), so before Cmi or C you must use Bdim7
> to keep the proper spelling. Abdim must be spelled Ab-Cb-Ebb-Gbb
> which is not possible in C tonality. If you want to use exactly same
> notes in C, it must be spelled Ab-B-D-F, which is the last inversion
> of Bdim7 (B-D-F-Ab).
> Which you wrote well in the case of G#dim before Ami.
>
> >
> > One of its most useful properties is that the tritone between its
> > maj 3rd and b7 degrees descends chromatically if we go through the
> > cycle of fifths, inverting each time. Eg C7 with E-Bb (3rd-7th)->
> > F7 with Eb-A (7th-3rd) etc...These can then be modified to fit the
> > particular chord progression Eg C7 to Fma7 becomes E-Bb to E-A. We
> > jazz musicians call these *guide tones*.
>
> Here you talk about dom7 chord.
>
> These progressions have been overused since Baroque times, and
> disused in jazz, so I have experimented many years ago with chains of
> mi6 chords, which contain also tritone. Guide tones go up in this
> case, and you get progressions like Fmi6 - Cmi6 - Gmi6 - Dmi6 - Ami6
> - Emi6 ...
>
> And in last years when I want to use chromatically shifted tritone,
> I prefer to harmonize it with tritone progression and alternate major
> and minor chords - like Cmi6 - F#7 - Dmi6 - Ab7.... or C7 - F#mi6 -
> D7 - G#mi6... Much more interesting.
>
>
> > Contrary to old fashioned views of harmony, the fifth is quite
> > dispensable.
>
> Which old fashioned? Which period or theory? Classical functional
> harmony allows omitting of fifth in seventh and 9th chords, and 5th
> and 3rd in higher triadic chords 11th and 13th.
>
> Daniel Forro
>

🔗genewardsmith <genewardsmith@...>

🔗rick <rick_ballan@...>

🔗Marcel de Velde <m.develde@...>

Hi Gene,

Ah, the Ramis monochord! That keeps turning up--it turned up not once but
> twice in my survey of 12-note, 5-limit Fokker blocks, with two different
> pairs of commas.
>

Ah cool :)

> Again, it's a sort of modified schismatic MOS--modified so that all the
> triads are just instead of so close to just you would find it difficult if
> not impossible to tell the difference.
>

Yes I agree, I can't reliably tell a 32/27 from a 1215/1024.
53 tet I consider just in practicle sense aswell.

>
> The Graham complexity of a triad in schismatic is 9, so there are 12-9=3
> major and 3 minor triads, which this scale converts to JI ones. You do a lot
> better with 17 notes, and I wonder if you could tweak by schismas so that
> all 16 of its triads are just.
>

What's a Graham complexity?

Btw, you know I consider things like 9/8 4/3 5/3 just ;)
As well as 135/128 4/3 405/256, 5/4 405/256 15/8, etc.
Only thing I'm not sure about yet is the third+Schisma, perhaps the Schisma
can move in one key (unlike the syntonic comma), but if I had to make a
guess I'd say it doesn't right now and 405/256 64/27 is fine.
I think the difference between all these triads is in how they "feel" and
function harmonically, and especially in how they work in counterpoint.

As far as what's a "key" or in one "tonic".
I'm allready finding this doesn't correspond to normal music theory exactly
(though almost it seems)
I'm finding Beethoven's Drei Equale no1 is in D minor untill the ending
which is in G minor.
There's a change of key a fifth down aswell in Drei Equale no2 about
halfway, and then it goes back again to original key.
And in Bach's BWV924 which sais it's in C major, I find to be completely in
G major even though it starts and ends with a C major chord.

Marcel

🔗genewardsmith <genewardsmith@...>

🔗Herman Miller <hmiller@...>

genewardsmith wrote:
> > --- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
> > >> Scala now sais: ramis.scl : equal in key 5 tamil_vi.scl : equal in
>> key 10
> > Ah, the Ramis monochord! That keeps turning up--it turned up not once
> but twice in my survey of 12-note, 5-limit Fokker blocks, with two
> different pairs of commas. Again, it's a sort of modified schismatic
> MOS--modified so that all the triads are just instead of so close to
> just you would find it difficult if not impossible to tell the
> difference.

Interesting. I see that one of the preset tunings for Pianoteq is "Monochord of Ramos de Pareja (Ramis de Pareia), Musica Practica (1482)". I tried a couple of Bach fugues on the F.E. Blanchet harpsichord in Pianoteq and they sound nice in this tuning. Some piano music with a piano timbre is a little sour, but it could just be that they're in a bad key for this tuning. (That 40/27 between G and D has got to make it hard to find a good key.)

> The Graham complexity of a triad in schismatic is 9, so there are
> 12-9=3 major and 3 minor triads, which this scale converts to JI
> ones. You do a lot better with 17 notes, and I wonder if you could
> tweak by schismas so that all 16 of its triads are just.

You can get 15 of them just, at least (e.g.)

C G D A E B F# C#

Ab Eb Bb F C G D A E

I don't see any way around that C#-Ab, but at least it's only off by a schisma.

🔗Marcel de Velde <m.develde@...>

🔗Andy <a_sparschuh@...>

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

>...diatonic scale from the 15/8 2/1 (17/8) 9/4 (19/8) 5/2 harmonic...

Hi Marcel,
that good old 15:16:17:18:19:20 idea originates back to to

/tuning/topicId_82928.html#90056
"
Sylvestro Ganassi's temperament (1543)
1/1 20/19 10/9 20/17 5/4 4/3 24/17 3/2 30/19 5/3 30/17 15/8 2/1
"
and also was used later by Alexander Malcoms's in his ~[1721] tuning.

Here comes an further refinement of that both earlier attempts:

by the Ansatz: C#0 : D_0 : Eb0 : E_0 == 16 : 17 : 18 : 19

interpolated by a sequential chain, that consists in a dozen of 5ths:

C#0: 16
G#0: 24 48 := 3*C#1
Eb0: 18 36 72 := 3*G#0
Bb0: 27 54 := 3*D#0
F_2: 81 := 3*Bb0
C_4: 243 := 3*F_2 middle C_4
G_2: 91 182 364 728 (< 729 := 3*C_4 )
D_0: 17 34 68 136 272 (< 273 := 3*G_2 )
A_0: 25+7/17 50+14/17 (< 51 := 3*D0 )
E_0: 19 38 76 (<76+4/17 := 3*A_0)
B_1: 57 := 3*E_0
F#3: 171 := 3*B_1
C#0: 16 32 64 128 256 512 (< 513 := 3*F#3 )

that divides the PC=3^12/2^19 into 5 epimoric subfators

C 728/729 G 272/273 D 288/289 A 323/324 E-B-F# 512/513 C#-G#-Eb-Bb-F-C

or in chromatic ascending order:

C 243 Hz middle_C4
# 256
D 272
# 288
E 304
F 324
# 342
G 364
# 384
A 406+10/17 Hz ~J.S. Bach's coeval Baroque "Cammer-thon"
# 432
B 456
c 486

in order to get rid of the fractional valure at the note 'A'
arise each pitch by an major-semitone: 17/16 ~+104.955...Cents

C 258.1875 = 258+3/16 middle_C4
# 272
D 289
# 306
E 323
F 344.25
# 363.375 = 363+3/8
G 386.75 = 386+3/4
# 408
A 432 early 19th-century normal-pitch (G. Verdi)
# 459
B 484.5
c 516.375

Now all the 12 frequencies of that pitches do possess commonly the
http://en.wikipedia.org/wiki/Dyadic_rational
property.

For gaining todays standard normal-pitch A4=440Hz
arise them all by another increment of factor 55/54 ~+31.766...Cents

C 262 + 31/32 middle_C4
# 277 + 1/27
D 294 + 19/54
# 311 + 2/3
E 328 + 53/54
F 350 + 5/8
# 370 + 5/48
G 393 + 197/216
# 415 + 5/9
A 440 Hz
# 467 + 1/2
B 493 + 17/36
c 525 + 15/16 tenor_C5

Or simply let 'scala' do the job for any other desired pitch-level:
Hence here comes the whole thing 'scala'-file format:

! Sp19limWell.scl
!
Sparschuh's 19-limit well-temperament [2010] with epimoric 5ths & 3rds
12
!
256/243 ! C# |8,-5>
272/243 ! D |4,-5,0,0,0,0,1>
32/27 ! Eb |4,-3>
304/243 ! E |4,-5,0,0,0,0,0,1>
4/3 ! F |2,-1>
38/27 ! F# |1,-3,0,0,0,0,0,1>
364/243 ! G |2,-5,0,1,0,1>
128/81 ! G# |7,-4>
256/153 ! A |8,-2,0,0,0,0,-1>
16/9 ! Bb |4,-2>
2/1 ! C' |1>
!
![eof]

here once again the tempering of the 5ths in ratios, monzo's and cents

C [begin]
728/729 |3,-6,0,-1,0,-1> ~-2.376...Cents
G
272/273 |4,-1,0,-1,0,-1,1> ~-6.353...Cents
D
288/289 |5,2,0,0,0,0,-2> ~-6.0008...Cents
A
323/324 |-2,-4,0,0,0,0,1,1> ~-5.351...Cents
E
B
F#
512/513 |9,-3,0,0,0,0,0,-1> ~-3.378...Cents
C#
G#
Eb
Bb
F
C [cycle of 5ths returns back to the initial C.]

Control of the Pythagorean-Comma 2^19/3^12 = |19,12> ~--23.460...C
(728/729)(272/273)(288/289)(323/324)(512/513)=2^19/3^12 = |19,-12>
is simply done by summing up the corresponding 5 epimoric monzos all together.

Now attend the analysis of the corresponding 3rds:
Here the deviations of the 3rds turn also out to be all epimoric sharpend above 5/4 = |-2,0,1> ~+386.313...Cents,
with some specific addional increments, that deliver
four different epimoric subpartitions of the diesis:

1.) Stack of 3rds: Bb-D-F#-Bb
-----------------------------
Bb
136/135 |3,-3,-1,0,0,1> ~+12.776...Cents
D
171/170 |-1,2,-1,0,0,-1,1> ~+10.153...Cents
F#
96/95 |5,1,-1,0,0,0,0,-1> ~+18.128...Cents
Bb
Control for Bb-D-F#-Bb by adding up the 3 monzos into the 'diesis'
(136/135)(171/171)(96/95) = 128/125 = |7,0,-3> ~41.0588...C ok.

2.) Stack of 3rds: F-A-C#-F
---------------------------
F
256/255 |8,-1,-1,0,0,0,-1> ~+6.775...Cents
A
136/135 |3,-3,-1,0,0,0,1> ~+12.776...Cents
C#
81/80 |-4,4,-1> ~+21.506...Cents the SC
F
Control for F-A-C#-F by adding up the 3 monzos into the 'diesis'
(256/255)(136/135)(81/80) = 128/125 = |7,0,-3> ~41.0588...C ok.

3.) Stack of 3rds: C-E-G#-C
----------------------------
C
1216/1215 |6,-5,-1,0,0,0,0,1> ~+1.424...Cents Erasthostens's Comma
E
96/95 |5,1,-1,0,0,0,0,-1> ~+18.128...Cents
G#
81/80 |-4,4,-1> ~+21.506...Cents the SC
Control for C-E-G#-C by adding up the 3 monzos into the 'diesis'
(1216/1215)(96/95)(81/80) = 128/125 = |7,0,-3> ~41.0588...C ok.

4.) Stack of 3rds: G-B-Eb-G
---------------------------
G
456/455 |3,1,-1,-1,0,-1,0,1> ~+3.8007...Cents
B
96/95 |5,1,-1,0,0,0,0,-1> ~+18.128...Cents
Eb
91/90 |-1,0,-1,1,0,1> ~+19.129...Cents
G
Control for G-B-Eb-G by adding up the 3 monzos into the 'diesis'
(456/455)(96/95)(91/90) = 128/125 = |7,0,-3> ~41.0588...C ok.

Consider all that four stacks more concise in 'Sorge'-matrix terms:

Summary of the 3rds sharpnesses:
1.) Bb (136/135) D (171/171) F# (96/95) Bb
2.) F (256/255) A( 136/135) C# (81/80) F
3.) C (1216/1215) E (96/95) G# (81/80) C
4.) G (456/455) B (96/95) Eb (91/90) G

Conclusion:
That sounds slightly different against the ordinary 12-ET,
which has all 3rds the same sharp of about one-third diesis:
(128/125)^(1/3) = ~(126.99473.../125.99473...) = ~+13.686..Cents
without any 'key-characteristics'
Especially the sonority of the almost pure JI C-major chord triad

C:E:G = 4 : 5*(1215/1216 ~+1.424...Cent) : 6*(728/729 ~-2.376...Cent)

Quest.
Are there any thoghts about that particular 19-limit well-tuning?

bye
Andy

🔗Andy <a_sparschuh@...>

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

> Am I right in finding this is Kirnberger I tuning???
> 1/1 135/128 9/8 1215/1024 5/4 4/3 45/32 3/2 405/256 5/3 16/9 15/8 2/1
> I'm reading it was used by "some of the best" piano tuners in Italy a > few centuries back, and recomended by Schiassi.

Hi Marcel,
there exists still a letter of Kirnberger to Forkel:

http://harpsichords.pbworks.com/f/Kirn_1871.html
"
The following reproduces the content of the letter as it appeared in the AMZ, omitting Bellermann's footnotes. The temperament in question is defined near the end and is identified by Kirnberger in the final table as "II".
"
Quote from the original text:
"
Oder wenn man von C nach e 80 : 81 in vier Quinten vertheilen will, kann es folgender Art geschehen:

Or if one prefers to split 80:81 into four fifths from C to e, it can happen by the following method:

C-G 216 : 323 temperirte Quinte = 2/3 - 1/324
216 : 324 reine Quinte
-------------------------------------
G-d 215 1/3 : 322 temperirte Quinte = 2/3 - 1/323
215 1/3 : 323 reine Quinte
-------------------------------------
A-e 214 2/3 : 321 temperirte Quinte = 2/3 - 1/322
214 2/3 : 322 reine Quinte
-------------------------------------
D-A 214 : 320 temperirte Quinte = 2/3 - 1/321
214 : 321 reine Quinte
-------------------------------------

An partial translation by Thomas Dent can be found in:
http://harpsichords.pbworks.com/f/K_III.html

Summary of Ki3:

C 323/324 G 322/323 D 321/322 A 320/321 E B
F# 32768/32805 Db Ab Eb Bb F C

Here comes my epimoric refinement of that rational cycle of 5hts:
by subdivding the schimsma into two epimoric subfactors:

32805/32768 = (6561/6560)*(1025/1025)

/tuning-math/message/17405
"
1; 41-limit
(1025/1024)*(6561/6560) := (41*5^2/2^11)*(3^8/41/5/2^5)
~1.68983327...Cents + ~0.263887517...Cents
"

inbetween the 5ths

Eb: 1/1
Bb: 3
F : 9
C : 27
G :(E/27=5 10 20 .. 320 < A/9=321 < D/3=322 <) G=323 (<C*3 =324 .. 81)
D :(E/9=15 30 60 .. 960 < A/3=963 <) D=966
A :(E/3=45 90 180 .. 2880 <) A=2889
E : 135
B : 405
F# (or enharmonic = Gb): 1025/27 ... 65600/27 ( {6560/6561} * 1215)
Db: 1/9 ... 1024/9 (<1025/1025)
Ab: 1/3
Eb: 1/1

or as more concise abstract

C 323/324 G 322/323 D 321/322 A 320/321 E B
B 6560/6561 F#=Gb 1024/1025 Db Ab Eb Bb F C

attend here the more smooth subdivision of the s:='schisma'
inbetween over the 5ths: .... B - (F#=Gb) - Db ....

that vields in Leonhard Euler's lowest absolute-pitch
the frequencies in Hertzians [Hz]:

c' 216 middle_C4
#' 227 + 5/9
d' 241.5
#' 256
e' 270
f' 288
#' 303 + 19/27
g' 323
#' 341 + 1/3
a' 361 + 1/8 Hz
#' 384
b' 405
c" 432 tenor_C5

and finally in the corresponding 'scala'-file

!Sp_refi_Ki_3.scl
Sparschuh's [2010] refined epimoric Kirnberger III variant
12
!
256/243 ! 2^8/3^5 the limma
161/144 ! (10/9)*(161/160) = (9/8)*(161/162) by halfing the SC
32/27 ! 2^5/3^3 Pyth. minor 3rd
5/4
4/3
1025/729 ! (1024/729)*(1025/1024) = (45/32)*(6560/6561) halfing the s
323/216 ! (3/2)*(323/324)
128/81 ! 2^7/3^4
107/64 ! (5/3)*(321/320)
16/9
15/8
2/1
!
![eof]

Quest:
Who can discern the difference against K's original scheme?

bye
Andy

"In tune" equal temperaments

🔗genewardsmith <genewardsmith@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗genewardsmith <genewardsmith@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗genewardsmith <genewardsmith@...>

🔗cityoftheasleep <igliashon@...>

🔗Marcel de Velde <m.develde@...>

🔗genewardsmith <genewardsmith@...>

🔗Carl Lumma <carl@...>

🔗Marcel de Velde <m.develde@...>

🔗genewardsmith <genewardsmith@...>

🔗Marcel de Velde <m.develde@...>

🔗genewardsmith <genewardsmith@...>

🔗Marcel de Velde <m.develde@...>

🔗Marcel de Velde <m.develde@...>

🔗genewardsmith <genewardsmith@...>

🔗Marcel de Velde <m.develde@...>

🔗Marcel de Velde <m.develde@...>

🔗genewardsmith <genewardsmith@...>

🔗Marcel de Velde <m.develde@...>

🔗genewardsmith <genewardsmith@...>

🔗Marcel de Velde <m.develde@...>

🔗Afmmjr@...

🔗Margo Schulter <mschulter@...>

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🔗genewardsmith <genewardsmith@...>

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🔗genewardsmith <genewardsmith@...>

🔗Afmmjr@...

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🔗Marcel de Velde <m.develde@...>

🔗Marcel de Velde <m.develde@...>

🔗genewardsmith <genewardsmith@...>

🔗Marcel de Velde <m.develde@...>

🔗Marcel de Velde <m.develde@...>

🔗rick <rick_ballan@...>

🔗Marcel de Velde <m.develde@...>

🔗rick <rick_ballan@...>

🔗genewardsmith <genewardsmith@...>

🔗Marcel de Velde <m.develde@...>

🔗Marcel de Velde <m.develde@...>

🔗genewardsmith <genewardsmith@...>

🔗genewardsmith <genewardsmith@...>

🔗Marcel de Velde <m.develde@...>

🔗Marcel de Velde <m.develde@...>

🔗Margo Schulter <mschulter@...>

🔗Marcel de Velde <m.develde@...>

🔗Marcel de Velde <m.develde@...>

🔗rick <rick_ballan@...>

🔗genewardsmith <genewardsmith@...>

🔗rick <rick_ballan@...>

🔗genewardsmith <genewardsmith@...>

🔗jlmoriart <JlMoriart@...>

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🔗rick <rick_ballan@...>

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🔗Marcel de Velde <m.develde@...>

🔗genewardsmith <genewardsmith@...>

🔗genewardsmith <genewardsmith@...>

🔗Marcel de Velde <m.develde@...>

🔗genewardsmith <genewardsmith@...>

🔗Marcel de Velde <m.develde@...>

🔗Marcel de Velde <m.develde@...>

🔗Marcel de Velde <m.develde@...>

🔗rick <rick_ballan@...>

🔗rick <rick_ballan@...>

🔗jonszanto <jszanto@...>

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🔗rick <rick_ballan@...>

🔗genewardsmith <genewardsmith@...>

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🔗Marcel de Velde <m.develde@...>

🔗genewardsmith <genewardsmith@...>