Pythagorean versus Just

I have been looking at ways to tune a conventional keyboard other than equal tempered. I see that the Yamaha CP1 provides a menu of alternate tunings, including Pythagorean. What this must mean is that eleven of the twelve fifths is 2 cents sharper than an equal tempered fifth... but then the last fifth must be 22 cents flat. I didn't see a mention in the CP1 user manual of which fifth was flat.

I was then charting out what just intonation should look like. For example in a C major scale, the series F-C-G-D would all be just tuned fifths. A-E-B would also be just tuned fifths. But the D-A interval would not be a just tuned fifth. Instead, it's the F-A, C-E, G-B intervals that are just major thirds, i.e. 14 cents flatter than equal tempered.

Starting say from F as a reference point, D would be 6 cents sharper than equal tempered, while A would be 14 cents flat. Thus the D-A interval is 20 cents flatter than equal tempered.

So that is my big AHA! Pythagorean tuning and just tuning are almost identical! They each have this funky flat fifth. In Pythagorean tuning, you try to avoid that flat fifth. In just tuning, it's that flat fifth that makes the major thirds just! OK, the exact size of that flat fifth is slightly different - about two cents. Hey, in Pythagorean tuning, one could just spread that two cents out across the eleven fifths so the twelfth fifth came out what's need for just thirds that cross that gap.

Here's a picture that might help:

http://i140.photobucket.com/albums/r6/kukulaj/justpythagorean.jpg

I expect this has been observed a thousand times already... but it's a big surprise for me!

Jim

--- In MakeMicroMusic@yahoogroups.com, "Jim K" <kukulaj@...> wrote:
>
> http://i140.photobucket.com/albums/r6/kukulaj/justpythagorean.jpg
>

Another way to interpret this picture - I have (re)discovered the schisma - 5 * (3**8) = 32805 is might close to 2**15 = 32768.

Jim

Jim K>"In just tuning, it's that flat fifth that makes the major thirds just!
OK, the exact size of that flat fifth is slightly different - about two cents."

My two cents: the academic world seems to have made the whole 12TET vs.
mean-tone vs. pythagorean tuning system differentiation very confusing when it
is for the most part quite simple (unless you want to get deep into details on
why it is the way it is mathematically).
I view it as either you have fairly flat fifths in a perfect circle (12TET)
or all virtually perfect 5ths except one (Pythagorean or JI) or variations
in-between (Mean-tone).

What bugs me about it all is that to do this you end up with a fifth of
around 40/27. Which isn't just sour, but so sour that (at least to me), it's
virtually unusable. Same goes with the tri-tone and sour 6th that result from
it in the 7 note scale.

I've experimented with using several different size slight sharp or flat
5ths within a scale to maximize the average purity of all possible dyads.
Technically the result is an "irregularly tempered scale" as calculated by a
computer by scanning all possible dyad constructions of a 7-note scale and
looking for one that gives all nearly-pure dyads. Note musicians in the
pre-computer era would have had a tough time doing it this way...and I'm fairly
sure the result can't be easily reduced back to something near "Pythagorean
tuning".
I'm also quite convinced that, at least with the assumption of dyads being
the basic building block of consonance (including for triads) and errors of to
about 7 cents as being indistinguishable to the human ear...this is virtually
the most consonant 7-tone non-adaptive scale possible.

For seven tones (approximating JI diatonic) this gives me an irregularly
tempered scale of about:

1
1.12 (about 9/8)
1.2 (6/5)
1.338 (about 4/3)
1.5 (3/2)
1.674 (about 5/3)
1.792 (about 9/5)
2/1 (octave)

>>>>NOTE I WOULD HIGHLY RECOMMEND THE ABOVE SCALE TO ANYONE COMPOSING IN 7-TONE
>>>>JI-DIATONIC AS I CAN THINK OF VERY FEW CASES WHERE IT WOULD SOUND LESS PURE AND
>>>>TONS WHERE IT WOULD SOUND MORE PURE!<<<<<<<<<<

Best yet...the dyads have all tones within about 7 cents of perfect from
JI-diatonic intervals (this means VERY near perfect triads) and those nasty 6ths
and tri-tone intervals are completely gone (you are left with very good
approximates of 8/5 and 7/5 instead)!

Now (perhaps a challenge for you or anyone else) is to try and arrange
several different sized 5ths so that many such 7-tone scales can be strung
together into a 12-tone scale where no dyad is more that about 7 cents off
pure. I had enough trouble "cracking" 7-tones...but it would be awesome if
someone managed 12!

[Non-text portions of this message have been removed]

I very much enjoyed reading your post and find your challenge exciting (to the point that I was worried I wouldn't be able to sleep at 5am this morning.) However, I think that you have made some mistakes and have underestimated the power of the technology and techniques of previous ages.

--- In MakeMicroMusic@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Jim K>"In just tuning, it's that flat fifth that makes the major thirds just!
> OK, the exact size of that flat fifth is slightly different - about two cents."
>
> Michael: My two cents: the academic world seems to have made the whole 12TET vs.
> mean-tone vs. pythagorean tuning system differentiation very confusing when it
> is for the most part quite simple (unless you want to get deep into details on
> why it is the way it is mathematically).
> I view it as either you have fairly flat fifths in a perfect circle (12TET)
> or all virtually perfect 5ths except one (Pythagorean or JI) or variations
> in-between (Mean-tone).

Mark: fairly close, but 12 edo (12 tet) is in between the usual meantones (quarter comma and sixth comma) and pythagorean. 12 edo can be regarded as being 11th comma meantone (to an accuracy to closer than 1 cent. You can give an exact mathematical expression for the fraction of a comma that 12 edo subtracts from the pure 3/2 fifth).

Michael: What bugs me about it all is that to do this you end up with a fifth of
> around 40/27. Which isn't just sour, but so sour that (at least to me), it's
> virtually unusable. Same goes with the tri-tone and sour 6th that result from
> it in the 7 note scale.
>

Mark: for my taste the pythagorean wolf 5th (almost 2 cents flatter than 40/27) is still usable, but I have unusual tastes.

Michael: I've experimented with using several different size slight sharp or flat
> 5ths within a scale to maximize the average purity of all possible dyads.
> Technically the result is an "irregularly tempered scale" as calculated by a
> computer by scanning all possible dyad constructions of a 7-note scale and
> looking for one that gives all nearly-pure dyads. Note musicians in the
> pre-computer era would have had a tough time doing it this way...and I'm fairly
> sure the result can't be easily reduced back to something near "Pythagorean
> tuning".

Mark: When exactly was the "pre computer era"? People have been using mechanical computers for millennia. I personally made myself a mechanical computer that was 4 cardboard disks which was a great help in solving a related problem of how to have 8 pure 5/4 major thirds in all possible patterns in a system of 12 notes while reducing the deviations from 3/2 fifths as much as possible.

Link to picture of my mechanical computer:
http://www.facebook.com/photo.php?pid=6942582&l=48e6b494aa&id=536589255

In the picture, the computer is being used to solve the problem of how to arrange the notes to allow pure major thirds with roots on the notes E, G, G# A, B flat, B, C# and D, while reducing deviations from just 3/2 fifths. I found that this mechanical computer made a difficult problem easy to solve. People could use wood or other materials instead of cardboard. I am tempted to make another mechanical computer to solve your 7 note problem (with 7 disks) or to attempt to solve the 12 note problem (with 12 disks).

Michael: I'm also quite convinced that, at least with the assumption of dyads being
> the basic building block of consonance (including for triads) and errors of to
> about 7 cents as being indistinguishable to the human ear...this is virtually
> the most consonant 7-tone non-adaptive scale possible.

Mark: The intonation known as Quarter Comma Meantone, when limitted to 7 notes, meets the requirements you list (all dyad intervals within 7 cents of the pure intervals 16/15, 9/8, 6/5, 5/4, 4/3, 7/5). It is directly derived from Pythagorean intonation. You just flatten all of the "perfect" fifths from 3/2 to the fourth root of 5.

Michael:
>
> For seven tones (approximating JI diatonic) this gives me an irregularly
> tempered scale of about:
>
> 1
> 1.12 (about 9/8)
> 1.2 (6/5)
> 1.338 (about 4/3)
> 1.5 (3/2)
> 1.674 (about 5/3)
> 1.792 (about 9/5)
> 2/1 (octave)
>
>
> >>>>NOTE I WOULD HIGHLY RECOMMEND THE ABOVE SCALE TO ANYONE COMPOSING IN 7-TONE
> >>>>JI-DIATONIC AS I CAN THINK OF VERY FEW CASES WHERE IT WOULD SOUND LESS PURE AND
> >>>>TONS WHERE IT WOULD SOUND MORE PURE!<<<<<<<<<<
>
> Best yet...the dyads have all tones within about 7 cents of perfect from
> JI-diatonic intervals (this means VERY near perfect triads) and those nasty 6ths
> and tri-tone intervals are completely gone (you are left with very good
> approximates of 8/5 and 7/5 instead)!

Mark: Thank you for that. I will enjoy trying out your scale sonically and investigating it mathematically.

Michael: Now (perhaps a challenge for you or anyone else) is to try and arrange
> several different sized 5ths so that many such 7-tone scales can be strung
> together into a 12-tone scale where no dyad is more that about 7 cents off
> pure. I had enough trouble "cracking" 7-tones...but it would be awesome if
> someone managed 12!
>
> [Non-text portions of this message have been removed]

Mark: I find the challenge very exciting. My first concern is whether it is ever possible to have 3 nearly pure major thirds stacked on top of each other without distorting the octave. Are you allowed to change the size of the octave within a 7 cent margin within your challenge? What about intervals larger than an octave? To make the task more specific, Which intervals are we aiming for? Particularly, does a minor 2nd (1 semitone) have to approximate to 16/15? or can it be more like 25/24? or 17/16?

>"Mark: When exactly was the "pre computer era"? People have been using
>mechanical computers for millennia. I personally made myself a mechanical
>computer that was 4 cardboard disks which was a great help in solving a related
>problem of how to have 8 pure 5/4 major thirds in all possible patterns in a
>system of 12 notes while reducing the deviations from 3/2 fifths as much as
>possible."

I mean, around pre 1980's. Correct me if I'm wrong but I don't see a whole
lot being done using irregular temperaments and multiple generators before then
(before then, as I've seen, it's been by and large about taking circles formed
by a single generator). The option of creating scales by comparing errors of 1)
individual dyads or 2) triads or greater ALA adaptive JI just were not around to
the best of my knowledge. Another example: Sethares "tuning to timbre"
algorithm in the Basic and Matlab programming languages: implementing such a
proof of concept I strongly suspect would take ages without computers.

>"Mark: fairly close, but 12 edo (12 tet) is in between the usual meantones
>(quarter comma and sixth comma) and pythagorean. 12 edo can be regarded as being
>11th comma meantone (to an accuracy to closer than 1 cent. You can give an exact
>mathematical expression for the fraction of a comma that 12 edo subtracts from
>the pure 3/2 fifth)."
Bizarre...because 12TET doesn't ever seem to build up a comma that has to be
"narrowed down/up" at the end. I guess that's why I made the mistake of never
comparing it directly to mean-tone.

>"In the picture, the computer is being used to solve the problem of how to
>arrange the notes to allow pure major thirds with roots on the notes E, G, G# A,
>B flat, B, C# and D, while reducing deviations from just 3/2 fifths. I found
>that this mechanical computer made a difficult problem easy to solve. People
>could use wood or other materials instead of cardboard. I am tempted to make
>another mechanical computer to solve your 7 note problem (with 7 disks) or to
>attempt to solve the 12 note problem (with 12 disks)."

Fascinating...now how to these discs work exactly? I'd be fascinated to see
what such a system comes up as the answers for 7 and 12 discs. Your above
problem doesn't sound to hard as it's optimizing the accuracy of "just" interval
sizes. My computer program, meanwhile, considers all interval sizes: second
(major/minor), third (major/minor), fourth, etc.

>"Mark: The intonation known as Quarter Comma Meantone, when limitted to 7 notes,
>meets the requirements you list (all dyad intervals within 7 cents of the pure
>intervals 16/15, 9/8, 6/5, 5/4, 4/3, 7/5). It is directly derived from
>Pythagorean intonation. You just flatten all of the "perfect" fifths from 3/2 to
>the fourth root of 5."
Alas, that assumes a single generator. The funny thing is, looking at the
numbers, the two (1/4 comma and my scale) do seem to be within a few cents of
each other so, I stand corrected, they are virtually the same scale. I knew the
two had somewhat similar numbers...but had no idea they were that
close...especially since that same program also came up with the completely
different answer

------best dyad combination scale for not allowing half-steps (wider intervals
only for better critical band dissonance ratings)----------
1.12
1.2222
1.338
1.5
1.674
1.83
2

>"Mark: I find the challenge very exciting. My first concern is whether it is
>ever possible to have 3 nearly pure major thirds stacked on top of each other
>without distorting the octave."
>
Of course, so long as the modified-octave itself is not distorted by more
than 8 cents. That includes up to 8 cents larger than the octave.

>"Are you allowed to change the size of the octave within a 7 cent margin within
>your challenge?"
Make it within 8 cents...and it's close enough.

>"Particularly, does a minor 2nd (1 semitone) have to approximate to 16/15?"
No...it can be anything...actually the minor 2nd is the one dyad I will let
slide/change a lot and 17/16 and 16/15 are already pretty near the maximum
critical band dissonance point at about 20/19.

________________________________
From: Mark <mark.barnes3@...>
To: MakeMicroMusic@yahoogroups.com
Sent: Sat, August 28, 2010 11:49:42 AM
Subject: Re: [MMM] Pythagorean versus Just

I very much enjoyed reading your post and find your challenge exciting (to the
point that I was worried I wouldn't be able to sleep at 5am this morning.)
However, I think that you have made some mistakes and have underestimated the
power of the technology and techniques of previous ages.

--- In MakeMicroMusic@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Jim K>"In just tuning, it's that flat fifth that makes the major thirds just!
> OK, the exact size of that flat fifth is slightly different - about two
cents."
>
> Michael: My two cents: the academic world seems to have made the whole 12TET
>vs.
>
> mean-tone vs. pythagorean tuning system differentiation very confusing when it

> is for the most part quite simple (unless you want to get deep into details on

> why it is the way it is mathematically).
> I view it as either you have fairly flat fifths in a perfect circle
>(12TET)
>
> or all virtually perfect 5ths except one (Pythagorean or JI) or variations
> in-between (Mean-tone).

Mark: fairly close, but 12 edo (12 tet) is in between the usual meantones
(quarter comma and sixth comma) and pythagorean. 12 edo can be regarded as being
11th comma meantone (to an accuracy to closer than 1 cent. You can give an exact
mathematical expression for the fraction of a comma that 12 edo subtracts from
the pure 3/2 fifth).

Michael: What bugs me about it all is that to do this you end up with a fifth
of

> around 40/27. Which isn't just sour, but so sour that (at least to me), it's

> virtually unusable. Same goes with the tri-tone and sour 6th that result from

> it in the 7 note scale.
>

Mark: for my taste the pythagorean wolf 5th (almost 2 cents flatter than 40/27)
is still usable, but I have unusual tastes.

Michael: I've experimented with using several different size slight sharp or
flat

> 5ths within a scale to maximize the average purity of all possible dyads.
> Technically the result is an "irregularly tempered scale" as calculated by
>a
>
> computer by scanning all possible dyad constructions of a 7-note scale and
> looking for one that gives all nearly-pure dyads. Note musicians in the
> pre-computer era would have had a tough time doing it this way...and I'm fairly
>
> sure the result can't be easily reduced back to something near "Pythagorean
> tuning".

Mark: When exactly was the "pre computer era"? People have been using mechanical
computers for millennia. I personally made myself a mechanical computer that was
4 cardboard disks which was a great help in solving a related problem of how to
have 8 pure 5/4 major thirds in all possible patterns in a system of 12 notes
while reducing the deviations from 3/2 fifths as much as possible.

Link to picture of my mechanical computer:
http://www.facebook.com/photo.php?pid=6942582&l=48e6b494aa&id=536589255

In the picture, the computer is being used to solve the problem of how to
arrange the notes to allow pure major thirds with roots on the notes E, G, G# A,
B flat, B, C# and D, while reducing deviations from just 3/2 fifths. I found
that this mechanical computer made a difficult problem easy to solve. People
could use wood or other materials instead of cardboard. I am tempted to make
another mechanical computer to solve your 7 note problem (with 7 disks) or to
attempt to solve the 12 note problem (with 12 disks).

Michael: I'm also quite convinced that, at least with the assumption of dyads
being

> the basic building block of consonance (including for triads) and errors of to

> about 7 cents as being indistinguishable to the human ear...this is virtually
> the most consonant 7-tone non-adaptive scale possible.

Mark: The intonation known as Quarter Comma Meantone, when limitted to 7 notes,
meets the requirements you list (all dyad intervals within 7 cents of the pure
intervals 16/15, 9/8, 6/5, 5/4, 4/3, 7/5). It is directly derived from
Pythagorean intonation. You just flatten all of the "perfect" fifths from 3/2 to
the fourth root of 5.

Michael:
>
> For seven tones (approximating JI diatonic) this gives me an irregularly
> tempered scale of about:
>
> 1
> 1.12 (about 9/8)
> 1.2 (6/5)
> 1.338 (about 4/3)
> 1.5 (3/2)
> 1.674 (about 5/3)
> 1.792 (about 9/5)
> 2/1 (octave)
>
>
> >>>>NOTE I WOULD HIGHLY RECOMMEND THE ABOVE SCALE TO ANYONE COMPOSING IN 7-TONE
>
> >>>>JI-DIATONIC AS I CAN THINK OF VERY FEW CASES WHERE IT WOULD SOUND LESS PURE
>AND
>
> >>>>TONS WHERE IT WOULD SOUND MORE PURE!<<<<<<<<<<
>
> Best yet...the dyads have all tones within about 7 cents of perfect from
> JI-diatonic intervals (this means VERY near perfect triads) and those nasty
>6ths
>
> and tri-tone intervals are completely gone (you are left with very good
> approximates of 8/5 and 7/5 instead)!

Mark: Thank you for that. I will enjoy trying out your scale sonically and
investigating it mathematically.

Michael: Now (perhaps a challenge for you or anyone else) is to try and arrange

> several different sized 5ths so that many such 7-tone scales can be strung
> together into a 12-tone scale where no dyad is more that about 7 cents off
> pure. I had enough trouble "cracking" 7-tones...but it would be awesome if
> someone managed 12!
>
> [Non-text portions of this message have been removed]

Mark: I find the challenge very exciting. My first concern is whether it is ever
possible to have 3 nearly pure major thirds stacked on top of each other without
distorting the octave. Are you allowed to change the size of the octave within a
7 cent margin within your challenge? What about intervals larger than an octave?
To make the task more specific, Which intervals are we aiming for? Particularly,
does a minor 2nd (1 semitone) have to approximate to 16/15? or can it be more
like 25/24? or 17/16?

[Non-text portions of this message have been removed]

It has been my impression that Just Intonation, in the broadest sense, is
merely the use of specific ratios between pitches to form scales.
Generally, instead of defining a just intonation scale by stacking up an
interval (like the 3/2 fifth) and then filling it into every octave, one
would define a just intonation scale as a set of ratios relative to the
root. For example, a just major scale could look like this:
1/1
9/8
5/4
4/3
3/2
5/3
15/8

Also, the wolf (bad) fifth being described is only apparent in a twelve tone
system where one has to substitute (for example) a Gb for an F#. If you
stack up 5 fifths from D, you get F#. However, if you stack down 5 fifths
from D you get Gb, a completely different pitch if you are stacking up a
fifth other than 700 cents (like about 702 cents, i.e., the just fifth in
Pythagorean tuning). If you are only given 12 notes to work with, you have
to choose one versus the other. In one situation, the fifth above B will
sound terrible because it will be Gb instead of F#, and in the other, the
note a fourth above Db will sound terrible because you will be using F#
instead of Gb. Given the opportunity to continue stacking up fifths however
and use them all, you will not run into wolf fifths, because they are a
product of *musical keyboards*; they are not inherent to the tuning itself.

John

[Non-text portions of this message have been removed]

--- In MakeMicroMusic@yahoogroups.com, John Moriarty <JlMoriart@...> wrote:
> a just major scale could look like this:
> 1/1
> 9/8
> 5/4
> 4/3
> 3/2
> 5/3
> 15/8

I'm not sure exactly what John is responding to. What he has outlined is exactly the sort of thinking that I am working with. I can shuffle the note he has listed:

4/3
1/1
3/2
9/8

5/3
5/4
15/8

to make it clearer that the just major scale is really two stacks of perfect fifths. That funny hole in the middle 40/27 is 21.5 cents flatter than a perfect fifth - that's almost exactly the wolf tone of the Pythagorean tuning - just a schisma off! Of course the wolf tone is not a natural part of Pythagorean tuning but just an artifact of trying to shoehorn it into the 12 notes of a conventional keyboard.

Jim

Hi Jim,

First of all, both Pythagorean and Just Intonation are not for 12 fixed
notes per octave.
Both have potentially infinite notes per octave.

In Pythagorean one avoids the wolf fifth indeed, the wolf fifth shifts out
of the way depending on the music / chords to allow for all fifths to
allways be 3/2.

In Just Intonation it's a whole different story, it's nothing like
Pythagorean.
First of all, the "just" scale you're mentioning, 1/1 16/15 9/8 6/5 5/4 4/3
45/32 3/2 8/5 5/3 9/5 15/8 2/1, is not universally agreed upon to be "just".
I for instance think it is complete nonsense, and that it sounds horrible,
and way worse than 12tet.
Even for this (in my opinion wrong) scale, some people think it fits one key
(so it only works for music that is in one "key" and doesn't modulate), and
other people think that even for music in one key this scale doesn't contain
enough notes (which is slightly more logical thinking, though the scale is
still wrong).

As for the "just" major scale where C-E-G, F-A-C and G-B-D are 1/1 5/4 3/2
major triads.
Yes these are just triads (not the only just triads there are though), but
the resulting major scale of 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1 sounds
horrible melodically.
Try comparing 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1 with 1/1 9/8 5/4 4/3 3/2
27/16 15/8 2/1.
The first one doesn't sound "right", second one does.

As for the flat fifth in Just Intonation (also called the "wolf fifth"), I
think it can fuction perfectly well in music (though many wil disagree with
me on this).
Just not above the fundamental bass.

But seriously, to make Just Intonation work it's a serious study.
And even then you're more likely to make things terribly out of tune and
would be better of with 12tet.

Marcel
www.develde.net

I have been looking at ways to tune a conventional keyboard other than equal
> tempered. I see that the Yamaha CP1 provides a menu of alternate tunings,
> including Pythagorean. What this must mean is that eleven of the twelve
> fifths is 2 cents sharper than an equal tempered fifth... but then the last
> fifth must be 22 cents flat. I didn't see a mention in the CP1 user manual
> of which fifth was flat.
>
> I was then charting out what just intonation should look like. For example
> in a C major scale, the series F-C-G-D would all be just tuned fifths. A-E-B
> would also be just tuned fifths. But the D-A interval would not be a just
> tuned fifth. Instead, it's the F-A, C-E, G-B intervals that are just major
> thirds, i.e. 14 cents flatter than equal tempered.
>
> Starting say from F as a reference point, D would be 6 cents sharper than
> equal tempered, while A would be 14 cents flat. Thus the D-A interval is 20
> cents flatter than equal tempered.
>
> So that is my big AHA! Pythagorean tuning and just tuning are almost
> identical! They each have this funky flat fifth. In Pythagorean tuning, you
> try to avoid that flat fifth. In just tuning, it's that flat fifth that
> makes the major thirds just! OK, the exact size of that flat fifth is
> slightly different - about two cents. Hey, in Pythagorean tuning, one could
> just spread that two cents out across the eleven fifths so the twelfth fifth
> came out what's need for just thirds that cross that gap.
>
> Here's a picture that might help:
>
> http://i140.photobucket.com/albums/r6/kukulaj/justpythagorean.jpg
>
> I expect this has been observed a thousand times already... but it's a big
> surprise for me!
>
> Jim
>

[Non-text portions of this message have been removed]

--- In MakeMicroMusic@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
> Try comparing 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1
> with 1/1 9/8 5/4 4/3 3/2 27/16 15/8 2/1.
> The first one doesn't sound "right", second one does.

Thanks for your observations, Marcel.

I think what really draws me into all this is a fascination with the way tuning is intertwined with its musical use. No doubt some scales are so unbalanced they won't have much use at all.

My little project of the moment got kicked off by meeting an excellent jazz pianist, which got me thinking - how could all this microtonal stuff get out more into the mainstream. What could I offer my new pianist friend? Modern electronic pianos work pretty well but also allow easy alternate tunings.

Thirty years ago I was designing alternative keyboards to allow all sorts of wild tunings with many more than 12 notes per octave. Just to build one of those new keyboards would be a huge project and then somebody would have to take the time to learn to play it.

So why not use a conventional keyboard as a gateway to microtonality?

My basic idea is to introduce a switch, so the tuning of the piano can be altered on the fly during a performance, much like an orchestral harp can switch keys by hitting a pedal.

What I had never realized was: the syntonic comma and the Pythagorean comma are very close. I always think of the basic microtonality step as being a syntonic comma. My usual playground is 53-edo. Both a syntonic comma and a Pythagorean comma are represented by one (micro)step in 53-edo. So I should have known all along how close they are. Anyway, if a pedal gave a performer a systematic way to shift tunings up or down a syntonic comma, that might be a nice tool.

It might be too coarse, though. The subtler colorings of various meantone tunings are probably more usable for music that is just stepping incrementally out of the conventions of Bach, Mahler, etc.

For a less incremental step, here is one of my 53-edo experiments

http://soundclick.com/share?songid=7607461

Enjoy!
Jim

>"Yes these are just triads (not the only just triads there are though), but
the resulting major scale of 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1 sounds
horrible melodically."

This is the kinda of example where I favor tunings like 1/4 comma mean-tone
a whole lot. The fact the error is evenly distributed among all notes seems to
make it a lot easier to follow melodically than something like JI diatonic where
there are jumps in error as a result of that wolf fifth. Correct me if I'm wrong
but I think that's an issue with almost all JI...it quite often makes some tones
pure at the expense of others...rather than getting all notes "equally
pure/impure" with the highest average level of purity...for example.
And the side-effects, to me at least, are bad melodic feel and one or more
wolf-fifths. The only way I've found around the "wolves" is to accept a second
type of 5th like 22/15 as "ok" and also start accepting some seriously
extended-JI type chord.

This goes back to composition in that the gains in melodic and chord/harmony
ability appear to me greatly increased by temperament and by irregular
temperament even greater because there need be no sudden "comma elimination"
that way.

[Non-text portions of this message have been removed]

Hi Michael,

This is the kinda of example where I favor tunings like 1/4 comma mean-tone
> a whole lot. The fact the error is evenly distributed among all notes seems
> to
> make it a lot easier to follow melodically than something like JI diatonic
> where
> there are jumps in error as a result of that wolf fifth. Correct me if I'm
> wrong
> but I think that's an issue with almost all JI...
>

You're not wrong here, as it's indeed an issue with ALMOST all JI :)
But if you have a listen to the JI examples at my website
www.develde.netyou'll here that the M-JI versions do not have harmonic
or melodic problems
at all.
Truly none.

I've also uploaded a classic 5-limit JI version (no wolfs, all chords low
ratio 5-limit) for comparison.
Here you'll hear clearly why JI has gotten such a bad name. It sounds
horrible melodically, and while the chords are in themselves low ratio
5-limit, they sound all wrong too because they're completely out of context
most of the time.

Really worth a listen:
M-JI:
https://sites.google.com/site/develdenet/files/Beethoven_Drei_Equale_no1_%28M-JI_09-08-2010%29.mp3
Classic 5-limit JI:
https://sites.google.com/site/develdenet/files/Beethoven_Drei_Equale_No1_%28classic-5-limit-JI%29.mp3

> it quite often makes some tones
> pure at the expense of others...
>

Well yes, the wrong way of doing JI does that. But it does not need to be
so.

> rather than getting all notes "equally
> pure/impure" with the highest average level of purity...for example.
>

There is no system that will give all notes equally pure/impure unless you
first know what pure should be.
12tet doesn't get all notes equally pure/impure for instance.

> And the side-effects, to me at least, are bad melodic feel and one or more
> wolf-fifths. The only way I've found around the "wolves" is to accept a
> second
> type of 5th like 22/15 as "ok" and also start accepting some seriously
> extended-JI type chord.
>

I've found wolves sound perfectly natural in their right place.
Infact, they'll sound wrong when they're removed to me.
But for instance a wolf above the fundamental bass.. yes that'll sound baad.

>
> This goes back to composition in that the gains in melodic and
> chord/harmony
> ability appear to me greatly increased by temperament and by irregular
> temperament even greater because there need be no sudden "comma
> elimination"
> that way.
>

It is relevant to wrongly done JI.
But I don't think it's the case compared to correctly done JI.

Marcel

[Non-text portions of this message have been removed]

Hi Jim,

I think what really draws me into all this is a fascination with the way
> tuning is intertwined with its musical use.
>

Yes I think it is too.
What would music be without tuning.
What's the most defining character of a note musically? It's its pitch, in
relation to past and present other notes.
The pure octave is 2/1, the pure fifth is 3/2 etc. Tuning is at the very
heart of music.

> My little project of the moment got kicked off by meeting an excellent jazz
> pianist, which got me thinking - how could all this microtonal stuff get out
> more into the mainstream. What could I offer my new pianist friend?
>

You could offer him nothing at the moment I think :)
I think more reseatch is needed before tuning becomes practical and ready
for the mainstream.
I wouldn't recomend any microtonal scale to a practicing musician and the
results will surely be waay out of tune and much worse than 12tet.

>
> So why not use a conventional keyboard as a gateway to microtonality?
>
> My basic idea is to introduce a switch, so the tuning of the piano can be
> altered on the fly during a performance, much like an orchestral harp can
> switch keys by hitting a pedal.
>

Yes this is my thought too :)
But I think it would be hard to do so for a real piano with it's high string
tension etc. Would be a very big project to do it right.
And again, one would first need to know exactly what you'd want it to do
tuning wise, otherwise you'll end up with a very expensive piano that can
only make out of tune music. Not something the world is waiting for :)

But one can do this allready with the computer and sampled pianos or
pianoteq.
I prefer an outside software program to do the tuning by pitchbends
(pitchbent notes are then sent to instrument of choice) and use 2 keyboards,
one for playing notes, one for switching the scale (could use a foot pedal
board for this).

>
> What I had never realized was: the syntonic comma and the Pythagorean comma
> are very close. I always think of the basic microtonality step as being a
> syntonic comma. My usual playground is 53-edo. Both a syntonic comma and a
> Pythagorean comma are represented by one (micro)step in 53-edo. So I should
> have known all along how close they are.
>

53 edo is a temperament of 53 pure fifths.
So it's a temperament of pythagorean taken to 53 notes per octaves.

I don't think 53edo is good for JI (used to think this but have since
changed my mind on this).
JI has many many more commas than only the syntonic one, and possible
pitches fill the entire spectra.

As for switching. One would need to change the entire scale with a switch.
In the case of Pythagorean one wouldn't want to switch the scale up or down
a syntonic comma, but nicer would be to change the syntonic comma in a 12
tone subset of pythagorean to be a fifth up or down I think.
Same sort of think would apply to JI.
And one would need to indicate with the score of when the switches must
happen.

Marcel

Marcel

[Non-text portions of this message have been removed]

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

> I think more reseatch is needed
> before tuning becomes practical and ready for the mainstream.
> I wouldn't recomend any microtonal scale
> to a practicing musician and the
> results will surely be waay out of tune and much worse than 12tet.

Surely you are too pessimistic! There are classical temperaments other than 12tet with a solid history of use. There are non-European scales. Then there are folks like Harry Partch or La Monte Young or Michael Harrison. OK, none of these are mainstream, but I don't care much about mainstream! I just like building bridges between theory and practice... exploring new musical possibilities.

> > My basic idea is to introduce a switch,
> > so the tuning of the piano can be
> > altered on the fly during a performance,
> > much like an orchestral harp can
> > switch keys by hitting a pedal.

> But one can do this allready with the computer and sampled pianos or
> pianoteq.

Ah, thanks, I didn't know about pianoteq. That looks interesting. I see that I can load a scala file to tweak the tuning. Do you know, can I somehow flip from one tuning to another on the fly with pianoteq?

> I don't think 53edo is good for JI

My current exploration of 53-edo is very much not JI. I exploit the commas that 53-edo tempers, e.g. the kleisma. The tempering really wraps the network of intervals into a kind of torus. I try to have some fun with paths that run around the torus, that exploit the topological non-triviality. JI doesn't temper anything, so there is no toroidal structure for paths to run around!

Here is a diagram that illustrates a scale that makes a loop via the tempering of the kleisma:

http://i140.photobucket.com/albums/r6/kukulaj/twelve.jpg

> As for switching.
> One would need to change the entire scale with a switch.
> In the case of Pythagorean
> one wouldn't want to switch the scale up or down
> a syntonic comma,
> but nicer would be to change the syntonic comma in a 12
> tone subset of pythagorean to be a fifth up or down I think.

What I am trying to do is to leave as many of the 12 notes the same across any particular switch. Maybe just switch one of the 12, or anyway as few as possible. It's like a key change. Going from C Major to G Major, it's just F that gets sharpened, the other 6 notes of the scale are unaltered. Similarly, with a Pythagorean tuning, at one time one might have D-A-E-...-F-C-G all perfect fifths, with G-D the flat fifth. Hit a switch, and just D gets sharpened by a Pythagorean comma, so now G-D is also a perfect fifth, but D-A becomes the flat fifth.

The broader class of non-Pythagorean JI opens up an infinite variety of choices one might make to pick 12 notes per octave. I am just exploring the switching game. Mostly folks would want to switch a selection of notes up or down a fifth, i.e. whatever role was played by the C note in one choice might now be played by F. What's fun is to think through - in making that shift of harmonic relationships, one or more notes can keep the same pitch - it's like the fixed reference point of the tuning switch. It's just an added dimension to the tuning puzzle - how two tunings might best be related, if one is switching from one to another on the fly in performance.

> And one would need to indicate with the score of
> when the switches must happen.

Yes, exactly, just like a convention change of key signature.

Thanks,
Jim

--- In MakeMicroMusic@yahoogroups.com, "Jim K" <kukulaj@...> wrote:
>

> My current exploration of 53-edo is very much not JI. I exploit the commas that 53-edo tempers, e.g. the kleisma. The tempering really wraps the network of intervals into a kind of torus. I try to have some fun with paths that run around the torus, that exploit the topological non-triviality. JI doesn't temper anything, so there is no toroidal structure for paths to run around!
>
> Here is a diagram that illustrates a scale that makes a loop via the tempering of the kleisma:
>
> http://i140.photobucket.com/albums/r6/kukulaj/twelve.jpg
>

I should point out... this "Hanson-Kleismatic" scale has twelve notes per octave, but it is not a very good candidate for mapping to a conventional keyboard! What "fifths" there are, i.e. the 3/2 frequency ratio - are actually mostly 8 "half-steps" apart. I think this would drive any pianist crazy! This scale is just too outlandish for a conventional keyboard!

Jim

I would like to throw out the idea the intervals are possible of contain notions beyond good or terrible. i think each can convey something very specific that can only be expressed with that interval. I don't want to fall asleep every time i listen to a piece of music for instance nor do i want to climb the walls .
actually i want to run the full gamut of experience possible.

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

On 29/08/10 8:14 AM, John Moriarty wrote:
> It has been my impression that Just Intonation, in the broadest sense, is
> merely the use of specific ratios between pitches to form scales.
> Generally, instead of defining a just intonation scale by stacking up an
> interval (like the 3/2 fifth) and then filling it into every octave, one
> would define a just intonation scale as a set of ratios relative to the
> root. For example, a just major scale could look like this:
> 1/1
> 9/8
> 5/4
> 4/3
> 3/2
> 5/3
> 15/8
>
> Also, the wolf (bad) fifth being described is only apparent in a twelve tone
> system where one has to substitute (for example) a Gb for an F#. If you
> stack up 5 fifths from D, you get F#. However, if you stack down 5 fifths
> from D you get Gb, a completely different pitch if you are stacking up a
> fifth other than 700 cents (like about 702 cents, i.e., the just fifth in
> Pythagorean tuning). If you are only given 12 notes to work with, you have
> to choose one versus the other. In one situation, the fifth above B will
> sound terrible because it will be Gb instead of F#, and in the other, the
> note a fourth above Db will sound terrible because you will be using F#
> instead of Gb. Given the opportunity to continue stacking up fifths however
> and use them all, you will not run into wolf fifths, because they are a
> product of *musical keyboards*; they are not inherent to the tuning itself.
>
> John
>
>
> [Non-text portions of this message have been removed]
>
>
>
> ------------------------------------
>
> Yahoo! Groups Links
>
>
>
>
>

I thought Hanson's scale makes 11and 15 tone scales not 12

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

On 30/08/10 12:41 PM, Jim K wrote:
>
> --- In MakeMicroMusic@yahoogroups.com, "Jim K"<kukulaj@...> wrote:
>> My current exploration of 53-edo is very much not JI. I exploit the commas that 53-edo tempers, e.g. the kleisma. The tempering really wraps the network of intervals into a kind of torus. I try to have some fun with paths that run around the torus, that exploit the topological non-triviality. JI doesn't temper anything, so there is no toroidal structure for paths to run around!
>>
>> Here is a diagram that illustrates a scale that makes a loop via the tempering of the kleisma:
>>
>> http://i140.photobucket.com/albums/r6/kukulaj/twelve.jpg
>>
> I should point out... this "Hanson-Kleismatic" scale has twelve notes per octave, but it is not a very good candidate for mapping to a conventional keyboard! What "fifths" there are, i.e. the 3/2 frequency ratio - are actually mostly 8 "half-steps" apart. I think this would drive any pianist crazy! This scale is just too outlandish for a conventional keyboard!
>
> Jim
>
>
>
> ------------------------------------
>
> Yahoo! Groups Links
>
>
>
>
>

thats why serial technique can be useful if you want to express anxiety in a piece, but if all you write is serial music then all you express is the one emotion.

Sent from my iPad

On Aug 30, 2010, at 1:38 AM, Kraig Grady <kraiggrady@...> wrote:

> I would like to throw out the idea the intervals are possible
> of contain notions beyond good or terrible. i think each can
> convey something very specific that can only be expressed with
> that interval. I don't want to fall asleep every time i listen
> to a piece of music for instance nor do i want to climb the walls .
> actually i want to run the full gamut of experience possible.
>
> /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> Mesotonal Music from:
> _'''''''_ ^North/Western Hemisphere:
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
>
> _'''''''_ ^South/Eastern Hemisphere:
> Austronesian Outpost of Anaphoria
> <http://anaphoriasouth.blogspot.com/>
>
> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>
> a momentary antenna as i turn to water
> this evaporates - an island once again
>
> On 29/08/10 8:14 AM, John Moriarty wrote:
> > It has been my impression that Just Intonation, in the broadest sense, is
> > merely the use of specific ratios between pitches to form scales.
> > Generally, instead of defining a just intonation scale by stacking up an
> > interval (like the 3/2 fifth) and then filling it into every octave, one
> > would define a just intonation scale as a set of ratios relative to the
> > root. For example, a just major scale could look like this:
> > 1/1
> > 9/8
> > 5/4
> > 4/3
> > 3/2
> > 5/3
> > 15/8
> >
> > Also, the wolf (bad) fifth being described is only apparent in a twelve tone
> > system where one has to substitute (for example) a Gb for an F#. If you
> > stack up 5 fifths from D, you get F#. However, if you stack down 5 fifths
> > from D you get Gb, a completely different pitch if you are stacking up a
> > fifth other than 700 cents (like about 702 cents, i.e., the just fifth in
> > Pythagorean tuning). If you are only given 12 notes to work with, you have
> > to choose one versus the other. In one situation, the fifth above B will
> > sound terrible because it will be Gb instead of F#, and in the other, the
> > note a fourth above Db will sound terrible because you will be using F#
> > instead of Gb. Given the opportunity to continue stacking up fifths however
> > and use them all, you will not run into wolf fifths, because they are a
> > product of *musical keyboards*; they are not inherent to the tuning itself.
> >
> > John
> >
> >
> > [Non-text portions of this message have been removed]
> >
> >
> >
> > ------------------------------------
> >
> > Yahoo! Groups Links
> >
> >
> >
> >
> >
>

[Non-text portions of this message have been removed]

On Sun, Aug 29, 2010 at 8:22 PM, Marcel de Velde <m.develde@...> wrote:
>
> Hi Michael,
> Really worth a listen:
> M-JI:
> https://sites.google.com/site/develdenet/files/Beethoven_Drei_Equale_no1_%28M-JI_09-08-2010%29.mp3
> Classic 5-limit JI:
> https://sites.google.com/site/develdenet/files/Beethoven_Drei_Equale_No1_%28classic-5-limit-JI%29.mp3

Marcel, how are you tuning your minor triads here vs the "classic"
version? Are yours the 3-limit approximation to 16:19:24, and the
classic one 10:12:15?

-Mike

I have the same problem with much of this type of work and much derived from it. The lack of breath.
It is nice that Cage was able to produce much just like it via random methods but at the same time what good is the method if it fails to move us much beyond it. His earlier music might be the most successful at this

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

On 30/08/10 3:54 PM, Dante Rosati wrote:
> thats why serial technique can be useful if you want to express anxiety in a piece, but if all you write is serial music then all you express is the one emotion.
>
> Sent from my iPad
>
> On Aug 30, 2010, at 1:38 AM, Kraig Grady<kraiggrady@...> wrote:
>
>> I would like to throw out the idea the intervals are possible
>> of contain notions beyond good or terrible. i think each can
>> convey something very specific that can only be expressed with
>> that interval. I don't want to fall asleep every time i listen
>> to a piece of music for instance nor do i want to climb the walls .
>> actually i want to run the full gamut of experience possible.
>>
>> /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
>> Mesotonal Music from:
>> _'''''''_ ^North/Western Hemisphere:
>> North American Embassy of Anaphoria Island<http://anaphoria.com/>
>>
>> _'''''''_ ^South/Eastern Hemisphere:
>> Austronesian Outpost of Anaphoria
>> <http://anaphoriasouth.blogspot.com/>
>>
>> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>>
>> a momentary antenna as i turn to water
>> this evaporates - an island once again
>>
>> On 29/08/10 8:14 AM, John Moriarty wrote:
>>> It has been my impression that Just Intonation, in the broadest sense, is
>>> merely the use of specific ratios between pitches to form scales.
>>> Generally, instead of defining a just intonation scale by stacking up an
>>> interval (like the 3/2 fifth) and then filling it into every octave, one
>>> would define a just intonation scale as a set of ratios relative to the
>>> root. For example, a just major scale could look like this:
>>> 1/1
>>> 9/8
>>> 5/4
>>> 4/3
>>> 3/2
>>> 5/3
>>> 15/8
>>>
>>> Also, the wolf (bad) fifth being described is only apparent in a twelve tone
>>> system where one has to substitute (for example) a Gb for an F#. If you
>>> stack up 5 fifths from D, you get F#. However, if you stack down 5 fifths
>>> from D you get Gb, a completely different pitch if you are stacking up a
>>> fifth other than 700 cents (like about 702 cents, i.e., the just fifth in
>>> Pythagorean tuning). If you are only given 12 notes to work with, you have
>>> to choose one versus the other. In one situation, the fifth above B will
>>> sound terrible because it will be Gb instead of F#, and in the other, the
>>> note a fourth above Db will sound terrible because you will be using F#
>>> instead of Gb. Given the opportunity to continue stacking up fifths however
>>> and use them all, you will not run into wolf fifths, because they are a
>>> product of *musical keyboards*; they are not inherent to the tuning itself.
>>>
>>> John
>>>
>>>
>>> [Non-text portions of this message have been removed]
>>>
>>>
>>>
>>> ------------------------------------
>>>
>>> Yahoo! Groups Links
>>>
>>>
>>>
>>>
>>>
>
> [Non-text portions of this message have been removed]
>
>
>
> ------------------------------------
>
> Yahoo! Groups Links
>
>
>
>
>

Not quite true. Everything depends on the selection of intervals which create the tone row (series), and which are combined together to make chords (if there's some harmony used in the piece). By controlled use of intervals it's possible to create quite consonant serial music (do you know that Bach quotation in Berg's Violin concerto? - all that is part of his 12tone row), even a music in gray zone between tonality and atonality. Thus is possible to make a music which can be called "tonal 12tone serialism" or "12tone serial tonality", "atonal modality", "modal serialism", "serial modalism"...

This is exactly what I'm interested all my life since I have started to compose. This gray zone between tonality - atonality - modality - seriality... Fortunately my teacher of composition invented that intervallic rows method and I could learn it and use as one of my working methods. It's not limited to 12tone rows, it can be applied also in older music styles, or in modality...

Classical dodecaphony can be pretty boring, because composers wanted to make all tones equal (which is anyway impossible). But when we add into it careful work with intervals and rows created from certain selected intervals only, it starts to be interesting.

Daniel Forro

>
> On 30/08/10 3:54 PM, Dante Rosati wrote:
>> thats why serial technique can be useful if you want to express >> anxiety in a piece, but if all you write is serial music then all >> you express is the one emotion.
>>
>> Sent from my iPad
>>
>> On Aug 30, 2010, at 1:38 AM, Kraig >> Grady<kraiggrady@...> wrote:
>>
>>> I would like to throw out the idea the intervals are possible
>>> of contain notions beyond good or terrible. i think each can
>>> convey something very specific that can only be expressed with
>>> that interval. I don't want to fall asleep every time i listen
>>> to a piece of music for instance nor do i want to climb the walls .
>>> actually i want to run the full gamut of experience possible.
>>>
>>> /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
>>

Hi Daniel~
I found working with the Eikosany a good way to investigate those places you mentioned.
I t gave me also a relatively more consonant and less ambiguous environment to do so. Perhaps these questions are more important where you are than where i am.
I am really not sure why i strayed away from it. I can't really say what it is I am now doing or why i am doing it, so i am not proposing an alternative.

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

On 30/08/10 8:15 PM, Daniel Forr� wrote:
> Not quite true. Everything depends on the selection of intervals
> which create the tone row (series), and which are combined together
> to make chords (if there's some harmony used in the piece). By
> controlled use of intervals it's possible to create quite consonant
> serial music (do you know that Bach quotation in Berg's Violin
> concerto? - all that is part of his 12tone row), even a music in gray
> zone between tonality and atonality. Thus is possible to make a music
> which can be called "tonal 12tone serialism" or "12tone serial
> tonality", "atonal modality", "modal serialism", "serial modalism"...
>
> This is exactly what I'm interested all my life since I have started
> to compose. This gray zone between tonality - atonality - modality -
> seriality... Fortunately my teacher of composition invented that
> intervallic rows method and I could learn it and use as one of my
> working methods. It's not limited to 12tone rows, it can be applied
> also in older music styles, or in modality...
>
> Classical dodecaphony can be pretty boring, because composers wanted
> to make all tones equal (which is anyway impossible). But when we add
> into it careful work with intervals and rows created from certain
> selected intervals only, it starts to be interesting.
>
> Daniel Forro
>
>
>> On 30/08/10 3:54 PM, Dante Rosati wrote:
>>> thats why serial technique can be useful if you want to express
>>> anxiety in a piece, but if all you write is serial music then all
>>> you express is the one emotion.
>>>
>>> Sent from my iPad
>>>
>>> On Aug 30, 2010, at 1:38 AM, Kraig
>>> Grady<kraiggrady@...> wrote:
>>>
>>>> I would like to throw out the idea the intervals are possible
>>>> of contain notions beyond good or terrible. i think each can
>>>> convey something very specific that can only be expressed with
>>>> that interval. I don't want to fall asleep every time i listen
>>>> to a piece of music for instance nor do i want to climb the walls .
>>>> actually i want to run the full gamut of experience possible.
>>>>
>>>> /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
>
>
> ------------------------------------
>
> Yahoo! Groups Links
>
>
>
>
>

berg did not become the poster boy, webern did. can you suggest some music you find interesting based on "intervals and rows created from certain selected intervals only"?

Sent from my iPad

On Aug 30, 2010, at 6:15 AM, Daniel Forró <dan.for@...> wrote:

> Not quite true. Everything depends on the selection of intervals
> which create the tone row (series), and which are combined together
> to make chords (if there's some harmony used in the piece). By
> controlled use of intervals it's possible to create quite consonant
> serial music (do you know that Bach quotation in Berg's Violin
> concerto? - all that is part of his 12tone row), even a music in gray
> zone between tonality and atonality. Thus is possible to make a music
> which can be called "tonal 12tone serialism" or "12tone serial
> tonality", "atonal modality", "modal serialism", "serial modalism"...
>
> This is exactly what I'm interested all my life since I have started
> to compose. This gray zone between tonality - atonality - modality -
> seriality... Fortunately my teacher of composition invented that
> intervallic rows method and I could learn it and use as one of my
> working methods. It's not limited to 12tone rows, it can be applied
> also in older music styles, or in modality...
>
> Classical dodecaphony can be pretty boring, because composers wanted
> to make all tones equal (which is anyway impossible). But when we add
> into it careful work with intervals and rows created from certain
> selected intervals only, it starts to be interesting.
>
> Daniel Forro
>
> >
> > On 30/08/10 3:54 PM, Dante Rosati wrote:
> >> thats why serial technique can be useful if you want to express
> >> anxiety in a piece, but if all you write is serial music then all
> >> you express is the one emotion.
> >>
> >> Sent from my iPad
> >>
> >> On Aug 30, 2010, at 1:38 AM, Kraig
> >> Grady<kraiggrady@...> wrote:
> >>
> >>> I would like to throw out the idea the intervals are possible
> >>> of contain notions beyond good or terrible. i think each can
> >>> convey something very specific that can only be expressed with
> >>> that interval. I don't want to fall asleep every time i listen
> >>> to a piece of music for instance nor do i want to climb the walls .
> >>> actually i want to run the full gamut of experience possible.
> >>>
> >>> /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> >>
>
>

[Non-text portions of this message have been removed]

Hi Mike,

> Really worth a listen:
> > M-JI:
> >
> https://sites.google.com/site/develdenet/files/Beethoven_Drei_Equale_no1_%28M-JI_09-08-2010%29.mp3
> > Classic 5-limit JI:
> >
> https://sites.google.com/site/develdenet/files/Beethoven_Drei_Equale_No1_%28classic-5-limit-JI%29.mp3
>
> Marcel, how are you tuning your minor triads here vs the "classic"
> version? Are yours the 3-limit approximation to 16:19:24, and the
> classic one 10:12:15?
>
> -Mike

I'm using different minor chords depending on the musical structure.
1/1 19/16 3/2
1/1 32/27 3/2
1/1 6/5 3/2
1/1 32/27 40/27
I've used all of these in the piece. (though writing them here all with
thesame 1/1 point is confusing, they all have their specific place in my
system)
The opening chord is 1/1 3/2 2/1 19/8 btw.

I can't yet say I did it 100% correct (not that even if I said such a thing
it would hold any meaning anymore after saying it wrongly so many times
before haha)
But I do think I've "hit the jackpot" with my theory at long last. And that
if not all notes, then by far most notes are correct in my version.

I've bought the domain name www.justintonation.com (transfering it to my
server now) and am writing a "e-book" in which I'm describing my theory
(with many nice images and sound examples), I'll have the first version
online in about 1-2 months.
Really have too much to explain to type it here now.

Btw, Drei Equale no2 is not at all what it seems either.
It's very different from classic 5-limit JI aswell.

Marcel

[Non-text portions of this message have been removed]

Hi Jim,

Surely you are too pessimistic! There are classical temperaments other than
> 12tet with a solid history of use. There are non-European scales. Then there
> are folks like Harry Partch or La Monte Young or Michael Harrison. OK, none
> of these are mainstream, but I don't care much about mainstream! I just like
> building bridges between theory and practice... exploring new musical
> possibilities.
>

Well, I don't know about Partch etc. Better not get into that before I get
the list angry at me again hehe :)

But you're right about temperaments other than 12tet.
They can make music slightly more in tune overall (or slightly more out of
tune).
I was thinking more along the lines of JI or Pythagorean with what I said.

> Ah, thanks, I didn't know about pianoteq. That looks interesting. I see
> that I can load a scala file to tweak the tuning. Do you know, can I somehow
> flip from one tuning to another on the fly with pianoteq?
>

I don't think that's possible with pianoteq.
However, if I'm correct the new version of pianoteq will respond correctly
to pitch bends (thanks to h-pi talking to them), and then one could use for
instance Scala to do the tuning for pianoteq with MIDI relay, and use Scala
to do the switching.

>
> What I am trying to do is to leave as many of the 12 notes the same across
> any particular switch. Maybe just switch one of the 12, or anyway as few as
> possible. It's like a key change. Going from C Major to G Major, it's just F
> that gets sharpened, the other 6 notes of the scale are unaltered.
> Similarly, with a Pythagorean tuning, at one time one might have
> D-A-E-...-F-C-G all perfect fifths, with G-D the flat fifth. Hit a switch,
> and just D gets sharpened by a Pythagorean comma, so now G-D is also a
> perfect fifth, but D-A becomes the flat fifth.
>

Aah ok yes that would be the way to switch I think so too.
Btw I belief there's also a script available for the Kontakt sampler written
by a tuning list member that implements this way of switching.

Cheers,
Marcel

[Non-text portions of this message have been removed]

On 30 Aug 2010, at 11:21 PM, Dante Rosati wrote:

> berg did not become the poster boy, webern did.

???

> can you suggest some music you find interesting based on "intervals
> and rows created from certain selected intervals only"?

This quotation looks strange when it's put together this way. I
meant: "careful work with intervals and careful work with rows
created from certain selected intervals only".

I have no time now for deeper research in this field to send you
quite satisfying answer, but at least some composers found work with
intervals essential for creating basic atmosphere of composition. I
think Bach was aware of it, and many of his fugues themes are based
on certain intervals, arranged in such way that it can't be only by
chance. And because fugue is monothematic work, it gives certain
character to whole piece.

Probably the first composer using interval method quite intentionally
was Debussy. In his diatonic, chromatic or modal works he often
emphasized certain interval or more intervals.

In 12tone music Alban Berg started to create more hierarchical tone
rows with certain character thanks to selection of limited number of
intervals. So his rows are on the opposite side of 12tone series
possibilities - the most boring are all-intervallic rows, because
they sound neutral, cool and sterile - no interval is repeated.
Hierarchical tone rows have always distinct character.

Then of course works of my composition teacher Alois Pinos, and more
of my works use intervallic 12tone rows... I don't know if some other
composers use this method.

Daniel Forro

>
> On Aug 30, 2010, at 6:15 AM, Daniel Forró <dan.for@...> wrote:
>
>> Not quite true. Everything depends on the selection of intervals
>> which create the tone row (series), and which are combined together
>> to make chords (if there's some harmony used in the piece). By
>> controlled use of intervals it's possible to create quite consonant
>> serial music (do you know that Bach quotation in Berg's Violin
>> concerto? - all that is part of his 12tone row), even a music in gray
>> zone between tonality and atonality. Thus is possible to make a music
>> which can be called "tonal 12tone serialism" or "12tone serial
>> tonality", "atonal modality", "modal serialism", "serial modalism"...
>>
>> This is exactly what I'm interested all my life since I have started
>> to compose. This gray zone between tonality - atonality - modality -
>> seriality... Fortunately my teacher of composition invented that
>> intervallic rows method and I could learn it and use as one of my
>> working methods. It's not limited to 12tone rows, it can be applied
>> also in older music styles, or in modality...
>>
>> Classical dodecaphony can be pretty boring, because composers wanted
>> to make all tones equal (which is anyway impossible). But when we add
>> into it careful work with intervals and rows created from certain
>> selected intervals only, it starts to be interesting.
>>
>> Daniel Forro
>>

[Non-text portions of this message have been removed]

--- In MakeMicroMusic@yahoogroups.com, Daniel Forró <dan.for@...> wrote:
> Then of course works of my composition teacher Alois Pinos, and more
> of my works use intervallic 12tone rows... I don't know if some other
> composers use this method.

Luigi Dallapiccola composed many dodecaphonic music with generous use of 3rds and 6ths.

Hudson

i meant that the serial tradition took webern as its paradigm, not berg or
schoenberg, both of which were "hopelessly romantic". webern pointed towards
the "music of the future" to composers like Boulez and Stockhausen in the
50s at Darmstadt.

On Mon, Aug 30, 2010 at 11:37 AM, Daniel Forr� <dan.for@...> wrote:

>
>
>
>
> On 30 Aug 2010, at 11:21 PM, Dante Rosati wrote:
>
> > berg did not become the poster boy, webern did.
>
> ???
>
> > can you suggest some music you find interesting based on "intervals
> > and rows created from certain selected intervals only"?
>
> This quotation looks strange when it's put together this way. I
> meant: "careful work with intervals and careful work with rows
> created from certain selected intervals only".
>
> I have no time now for deeper research in this field to send you
> quite satisfying answer, but at least some composers found work with
> intervals essential for creating basic atmosphere of composition. I
> think Bach was aware of it, and many of his fugues themes are based
> on certain intervals, arranged in such way that it can't be only by
> chance. And because fugue is monothematic work, it gives certain
> character to whole piece.
>
> Probably the first composer using interval method quite intentionally
> was Debussy. In his diatonic, chromatic or modal works he often
> emphasized certain interval or more intervals.
>
> In 12tone music Alban Berg started to create more hierarchical tone
> rows with certain character thanks to selection of limited number of
> intervals. So his rows are on the opposite side of 12tone series
> possibilities - the most boring are all-intervallic rows, because
> they sound neutral, cool and sterile - no interval is repeated.
> Hierarchical tone rows have always distinct character.
>
> Then of course works of my composition teacher Alois Pinos, and more
> of my works use intervallic 12tone rows... I don't know if some other
> composers use this method.
>
> Daniel Forro
>
>
> >
> > On Aug 30, 2010, at 6:15 AM, Daniel Forr� <dan.for@...<dan.for%40tiscali.cz>>
> wrote:
> >
> >> Not quite true. Everything depends on the selection of intervals
> >> which create the tone row (series), and which are combined together
> >> to make chords (if there's some harmony used in the piece). By
> >> controlled use of intervals it's possible to create quite consonant
> >> serial music (do you know that Bach quotation in Berg's Violin
> >> concerto? - all that is part of his 12tone row), even a music in gray
> >> zone between tonality and atonality. Thus is possible to make a music
> >> which can be called "tonal 12tone serialism" or "12tone serial
> >> tonality", "atonal modality", "modal serialism", "serial modalism"...
> >>
> >> This is exactly what I'm interested all my life since I have started
> >> to compose. This gray zone between tonality - atonality - modality -
> >> seriality... Fortunately my teacher of composition invented that
> >> intervallic rows method and I could learn it and use as one of my
> >> working methods. It's not limited to 12tone rows, it can be applied
> >> also in older music styles, or in modality...
> >>
> >> Classical dodecaphony can be pretty boring, because composers wanted
> >> to make all tones equal (which is anyway impossible). But when we add
> >> into it careful work with intervals and rows created from certain
> >> selected intervals only, it starts to be interesting.
> >>
> >> Daniel Forro
> >>
>
> [Non-text portions of this message have been removed]
>
>
>

[Non-text portions of this message have been removed]

--- In MakeMicroMusic@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> I thought Hanson's scale makes 11and 15 tone scales not 12

I confess, I have barely scratched the surface of Hanson's ideas!

But compare figure 5 from:

http://anaphoria.com/hanson.PDF

with my diagram:

http://i140.photobucket.com/albums/r6/kukulaj/twelve.jpg

Hanson talks about a basic pattern of 19 tones, while I have only 12. But it should be easy to see how his 19 is just a natural extension of my 12, or my 12 is a natural restriction of his 19.

The basic formula for my subset of tones is stack of minor thirds, which I just picked to start at 25 which is totally arbitrary - nice though that my arbitrary choice is very close to Hanson's! A minor third, i.e. 6/5 ratio, is 14 53-edo microsteps, or Mercators in Hanson's terminology. So the stack of minor thirds goes

25, 39, 0, 14, 28, 42, 3, 17, 31, 45, 6, 20

The fun of this, the kleismic "coincidence", is that perfect fifths appear in this sequence too (3/2 is 31 Mercators, 4/3 is 22 Mercators):

25-3, 39-17, 0-31, 14-45, 28-6, 42-20.

The stack of thirds can be extended in either direction, i.e. one could extend to 11 at the front or 34 at the back. More fifths will get picked up as one does this.

I just stopped with 12 because it seems elegant - every note appears in just one perfect fifth, at the top or bottom. I confess: I don't understand exactly why Hanson's chain of minor thirds has 19 notes. It makes his keyboard work; I think that's the basic idea. I was not thinking about keyboards when I came up with my 12 - I was just thinking about composition: how might I explore the tempering of the kleisma in 53-edo?

I do think that one would need an unconventional keyboard to play such an unconventional scale!

Jim

Hi Jim~ If you look at page 5 of http://anaphoria.com/MOSedo.PDF
you can see under 5)19 how the 11 and 15 tone scale can be found in 19 which is different to how the 12 comes about as an MOS of the 8)19. not reason one can not use both. in fact any of those presented under 19.

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

On 31/08/10 11:40 AM, Jim K wrote:
> --- In MakeMicroMusic@yahoogroups.com, Kraig Grady<kraiggrady@...> wrote:
>> I thought Hanson's scale makes 11and 15 tone scales not 12
> I confess, I have barely scratched the surface of Hanson's ideas!
>
> But compare figure 5 from:
>
> http://anaphoria.com/hanson.PDF
>
> with my diagram:
>
> http://i140.photobucket.com/albums/r6/kukulaj/twelve.jpg
>
> Hanson talks about a basic pattern of 19 tones, while I have only 12. But it should be easy to see how his 19 is just a natural extension of my 12, or my 12 is a natural restriction of his 19.
>
> The basic formula for my subset of tones is stack of minor thirds, which I just picked to start at 25 which is totally arbitrary - nice though that my arbitrary choice is very close to Hanson's! A minor third, i.e. 6/5 ratio, is 14 53-edo microsteps, or Mercators in Hanson's terminology. So the stack of minor thirds goes
>
> 25, 39, 0, 14, 28, 42, 3, 17, 31, 45, 6, 20
>
> The fun of this, the kleismic "coincidence", is that perfect fifths appear in this sequence too (3/2 is 31 Mercators, 4/3 is 22 Mercators):
>
> 25-3, 39-17, 0-31, 14-45, 28-6, 42-20.
>
> The stack of thirds can be extended in either direction, i.e. one could extend to 11 at the front or 34 at the back. More fifths will get picked up as one does this.
>
> I just stopped with 12 because it seems elegant - every note appears in just one perfect fifth, at the top or bottom. I confess: I don't understand exactly why Hanson's chain of minor thirds has 19 notes. It makes his keyboard work; I think that's the basic idea. I was not thinking about keyboards when I came up with my 12 - I was just thinking about composition: how might I explore the tempering of the kleisma in 53-edo?
>
> I do think that one would need an unconventional keyboard to play such an unconventional scale!
>
> Jim
>
>
>
>
> ------------------------------------
>
> Yahoo! Groups Links
>
>
>
>
>

--- In MakeMicroMusic@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> >"Mark: When exactly was the "pre computer era"?
Michael: "I mean, around pre 1980's. Correct me if I'm wrong but I don't see a whole
> lot being done using irregular temperaments and multiple generators before then
> (before then, as I've seen, it's been by and large about taking circles formed
> by a single generator). The option of creating scales by comparing errors of 1)
> individual dyads or 2) triads or greater ALA adaptive JI just were not around to
> the best of my knowledge. Another example: Sethares "tuning to timbre"
> algorithm in the Basic and Matlab programming languages: implementing such a
> proof of concept I strongly suspect would take ages without computers."

Mark: I was thinking on a longer time scale and greater width of geography. Although I don't always trust what I read of music history, there used to be a tuning system called "Ordinair" in which the musician tunes 12 notes to a meantone such as quarter comma or six comma, then, leaving 7 notes unchanged as a major scale, retunes the other notes individually to give more please harmonies when modulating to other keys. This is a general technique rather than a specific scale, but it gives irrular temperaments with up to 7 different sizes of fifth and was popular before the time of Johan Sebastian Bach.

Bach and other musicians such as Kimberger and Werckmeister developed more irregular temperaments, such as the "Well temperaments" used for Bach's "The Well Tempered Clavier".

Mechanical computers such as the one I made can be used to make complicated arithmatic visual in a way that makes it much easier to understand. The one I made relies on the use of logarhythms. I have the impression that logarhythmic tables have been availlable since the 16th or 17th century.

Also, competent musicians can tune the strings of a harp or a zither, or the pipes of an organ, or the frets of a lute, to the intervals of an irregular temperament by ear without having to bother with the arithmatic at all.

...In the picture, the computer is being used to solve the problem of how to
> >arrange the notes to allow pure major thirds with roots on the notes E, G, G# A,
> >B flat, B, C# and D, while reducing deviations from just 3/2 fifths. I found
> >that this mechanical computer made a difficult problem easy to solve. People
> >could use wood or other materials instead of cardboard. I am tempted to make
> >another mechanical computer to solve your 7 note problem (with 7 disks) or to
> >attempt to solve the 12 note problem (with 12 disks)."
>
> Michael: Fascinating...now how to these discs work exactly? I'd be fascinated to see
> what such a system comes up as the answers for 7 and 12 discs. Your above
> problem doesn't sound to hard as it's optimizing the accuracy of "just" interval
> sizes. My computer program, meanwhile, considers all interval sizes: second
> (major/minor), third (major/minor), fourth, etc.
>
Mark: Basically, it works like a slide rule: You mark a reference point on each disk (In my picture, this is a filled in circle with another circle arroung it), then mark intervals relative to this point, as fractions of an octave, so a 12 tet minor third is 90 degrees, and a just 5/4 major third is about 115.89 degrees and so on. What intervals you mark out depends on what problem you are trying to solve. A more permanent system would be to have disks marked out in cents, or with almost all commonly used intervals and then have movable colour markers.

Moving one disk relative to another shows how the intervals fit together.

To solve your 7 note problem, each disk would be marked out with all the intervals you find acceptable (such as 15/14, 16/15, 10/9 and so on) aswell as the ones you desire (such as 6/5, 5/4, 7/5 and 3/2). The disks could also be marked out with an area fo 7 cents or so each side of every interval (but not the reference point).

Then, slide the disks relative to each other to some starting point (For example a natural major scale, so the reference points on the disks would be seperated by the intervals tone, tone, semitone, tone, tone, tone).

Next, you move the disks relative to each other while observing how the reference point on each disk matches up with the desired and acceptable intervals on the other disks.

If you can make all the reference points fit into desired or accpetable areas on all the other disks then you have solved the problem.

You can then measure the angles, or, if the disks are marked out in cents, read off the intervals directly.

It might help to explain things if I filmed the process, but I have returned my best camera to the shop to have it fixed.

> >"Mark: The intonation known as Quarter Comma Meantone, when limitted to 7 notes,
> >meets the requirements you list (all dyad intervals within 7 cents of the pure
> >intervals 16/15, 9/8, 6/5, 5/4, 4/3, 7/5). It is directly derived from
> >Pythagorean intonation. You just flatten all of the "perfect" fifths from 3/2 to
> >the fourth root of 5."
> Alas, that assumes a single generator. The funny thing is, looking at the
> numbers, the two (1/4 comma and my scale) do seem to be within a few cents of
> each other so, I stand corrected, they are virtually the same scale. I knew the
> two had somewhat similar numbers...but had no idea they were that
> close...especially since that same program also came up with the completely
> different answer

Mark: I appologise. I myself was wrong here. Whereas Quarter Comma does approximate to the intervals 16/15 (or better still, 17/14), 6/5, 5/4, 4/3 and 7/5 (and their respective 7ths, 3rds and 5ths), very well, it fails when it comes to the intervals 9/8 and 10/9. The 7 note quarter comma tone is halways between 10/9 and 9/8, making it wrong by 11 and a half cents.

Your scale approximates the tones better and has better minor 3rds, while having almost as good major thirds. I have very much enjoyed exploring the mathematics of your scale. I would write more but I am running out of time.

>
> ------best dyad combination scale for not allowing half-steps (wider intervals
> only for better critical band dissonance ratings)----------
> 1.12
> 1.2222
> 1.338
> 1.5
> 1.674
> 1.83
> 2
>
>
> >"Mark: I find the challenge very exciting. My first concern is whether it is
> >ever possible to have 3 nearly pure major thirds stacked on top of each other
> >without distorting the octave."
> >
> Of course, so long as the modified-octave itself is not distorted by more
> than 8 cents. That includes up to 8 cents larger than the octave.
>
> >"Are you allowed to change the size of the octave within a 7 cent margin within
> >your challenge?"
> Make it within 8 cents...and it's close enough.
>
> >"Particularly, does a minor 2nd (1 semitone) have to approximate to 16/15?"
> No...it can be anything...actually the minor 2nd is the one dyad I will let
> slide/change a lot and 17/16 and 16/15 are already pretty near the maximum
> critical band dissonance point at about 20/19.
>
>
>
>
>
>
>
>
>
>
> ________________________________
> From: Mark <mark.barnes3@...>
> To: MakeMicroMusic@yahoogroups.com
> Sent: Sat, August 28, 2010 11:49:42 AM
> Subject: Re: [MMM] Pythagorean versus Just
>
>
>
> I very much enjoyed reading your post and find your challenge exciting (to the
> point that I was worried I wouldn't be able to sleep at 5am this morning.)
> However, I think that you have made some mistakes and have underestimated the
> power of the technology and techniques of previous ages.
>
> --- In MakeMicroMusic@yahoogroups.com, Michael <djtrancendance@> wrote:
> >
> > Jim K>"In just tuning, it's that flat fifth that makes the major thirds just!
> > OK, the exact size of that flat fifth is slightly different - about two
> cents."
> >
> > Michael: My two cents: the academic world seems to have made the whole 12TET
> >vs.
> >
> > mean-tone vs. pythagorean tuning system differentiation very confusing when it
>
> > is for the most part quite simple (unless you want to get deep into details on
>
> > why it is the way it is mathematically).
> > I view it as either you have fairly flat fifths in a perfect circle
> >(12TET)
> >
> > or all virtually perfect 5ths except one (Pythagorean or JI) or variations
> > in-between (Mean-tone).
>
> Mark: fairly close, but 12 edo (12 tet) is in between the usual meantones
> (quarter comma and sixth comma) and pythagorean. 12 edo can be regarded as being
> 11th comma meantone (to an accuracy to closer than 1 cent. You can give an exact
> mathematical expression for the fraction of a comma that 12 edo subtracts from
> the pure 3/2 fifth).
>
> Michael: What bugs me about it all is that to do this you end up with a fifth
> of
>
> > around 40/27. Which isn't just sour, but so sour that (at least to me), it's
>
> > virtually unusable. Same goes with the tri-tone and sour 6th that result from
>
> > it in the 7 note scale.
> >
>
> Mark: for my taste the pythagorean wolf 5th (almost 2 cents flatter than 40/27)
> is still usable, but I have unusual tastes.
>
> Michael: I've experimented with using several different size slight sharp or
> flat
>
> > 5ths within a scale to maximize the average purity of all possible dyads.
> > Technically the result is an "irregularly tempered scale" as calculated by
> >a
> >
> > computer by scanning all possible dyad constructions of a 7-note scale and
> > looking for one that gives all nearly-pure dyads. Note musicians in the
> > pre-computer era would have had a tough time doing it this way...and I'm fairly
> >
> > sure the result can't be easily reduced back to something near "Pythagorean
> > tuning".
>
> Mark: When exactly was the "pre computer era"? People have been using mechanical
> computers for millennia. I personally made myself a mechanical computer that was
> 4 cardboard disks which was a great help in solving a related problem of how to
> have 8 pure 5/4 major thirds in all possible patterns in a system of 12 notes
> while reducing the deviations from 3/2 fifths as much as possible.
>
> Link to picture of my mechanical computer:
> http://www.facebook.com/photo.php?pid=6942582&l=48e6b494aa&id=536589255
>
> In the picture, the computer is being used to solve the problem of how to
> arrange the notes to allow pure major thirds with roots on the notes E, G, G# A,
> B flat, B, C# and D, while reducing deviations from just 3/2 fifths. I found
> that this mechanical computer made a difficult problem easy to solve. People
> could use wood or other materials instead of cardboard. I am tempted to make
> another mechanical computer to solve your 7 note problem (with 7 disks) or to
> attempt to solve the 12 note problem (with 12 disks).
>
> Michael: I'm also quite convinced that, at least with the assumption of dyads
> being
>
> > the basic building block of consonance (including for triads) and errors of to
>
> > about 7 cents as being indistinguishable to the human ear...this is virtually
> > the most consonant 7-tone non-adaptive scale possible.
>
> Mark: The intonation known as Quarter Comma Meantone, when limitted to 7 notes,
> meets the requirements you list (all dyad intervals within 7 cents of the pure
> intervals 16/15, 9/8, 6/5, 5/4, 4/3, 7/5). It is directly derived from
> Pythagorean intonation. You just flatten all of the "perfect" fifths from 3/2 to
> the fourth root of 5.
>
> Michael:
> >
> > For seven tones (approximating JI diatonic) this gives me an irregularly
> > tempered scale of about:
> >
> > 1
> > 1.12 (about 9/8)
> > 1.2 (6/5)
> > 1.338 (about 4/3)
> > 1.5 (3/2)
> > 1.674 (about 5/3)
> > 1.792 (about 9/5)
> > 2/1 (octave)
> >
> >
> > >>>>NOTE I WOULD HIGHLY RECOMMEND THE ABOVE SCALE TO ANYONE COMPOSING IN 7-TONE
> >
> > >>>>JI-DIATONIC AS I CAN THINK OF VERY FEW CASES WHERE IT WOULD SOUND LESS PURE
> >AND
> >
> > >>>>TONS WHERE IT WOULD SOUND MORE PURE!<<<<<<<<<<
> >
> > Best yet...the dyads have all tones within about 7 cents of perfect from
> > JI-diatonic intervals (this means VERY near perfect triads) and those nasty
> >6ths
> >
> > and tri-tone intervals are completely gone (you are left with very good
> > approximates of 8/5 and 7/5 instead)!
>
> Mark: Thank you for that. I will enjoy trying out your scale sonically and
> investigating it mathematically.
>
> Michael: Now (perhaps a challenge for you or anyone else) is to try and arrange
>
> > several different sized 5ths so that many such 7-tone scales can be strung
> > together into a 12-tone scale where no dyad is more that about 7 cents off
> > pure. I had enough trouble "cracking" 7-tones...but it would be awesome if
> > someone managed 12!
> >
> > [Non-text portions of this message have been removed]
>
> Mark: I find the challenge very exciting. My first concern is whether it is ever
> possible to have 3 nearly pure major thirds stacked on top of each other without
> distorting the octave. Are you allowed to change the size of the octave within a
> 7 cent margin within your challenge? What about intervals larger than an octave?
> To make the task more specific, Which intervals are we aiming for? Particularly,
> does a minor 2nd (1 semitone) have to approximate to 16/15? or can it be more
> like 25/24? or 17/16?
>
>
>
>
> [Non-text portions of this message have been removed]
>

Mark>"This is a general technique rather than a specific scale, but it gives
irregular temperaments with up to 7 different sizes of fifth and was popular
before the time of Johan Sebastian Bach."
Interesting...I guess my main question is how does it calculate the
intervals...or does it just "feel for them" by ear? It sounds a bit like
adaptive JI in some ways. But it does indeed follow the same up to "7 different
sizes of fifth" logic my program does.

>"Bach and other musicians such as Kimberger and Werckmeister developed more
>irregular temperaments, such as the "Well temperaments" used for Bach's "The
>Well Tempered Clavier"."
I looked at the Wiki for it and it seems to say
"The syntonic comma was distributed between four intervals, with most of
the comma accommodated in the sol♯ to mi♭ diminished sixth, which expands to
nearly a minor sixth. It is this interval that is usually called the "wolf",
because it is so far out of consonance."
-http://en.wikipedia.org/wiki/Well_temperament

...in other words (same problem I noted before): the consonance/dissonance
is not evenly distributed at all. 1/4 comma mean-tone indeed seems evenly
distributed (perhaps a lucky fluke...since it appears to seek to optimize the
lowest possible maximum dissonance between thirds and fifths AKA 'mini-max
dissonance' and ends up also optimizing several other intervals 'by chance').
So it seems 1/4 comma mean-tone has one part of the puzzle ('mini-max') and
well-tempered the other ('irregular tempering')...but neither has both. Plus
1/4 comma, if I understand it well, misses mini-max in a way because the octave
is well over 7 cents off if you keep just multiplying that 1/4 comma-off fifth.

Again my point becomes...it seems very hard if not virtually impossible to
make scales with optimum mini-max dyadic dissonance among all intervals without
using irregular temperament and doing so in a way that calculates all possible
dyads against all others without using a computer since there are so many
possibilities when you deal with alternating generators (IE 1.494 * 1.51 * 1.5 *
etc.).

>"Also, competent musicians can tune the strings of a harp or a zither, or the
>pipes of an organ, or the frets of a lute, to the intervals of an irregular
>temperament by ear without having to bother with the arithmatic at all. "

I can believe it. But that still doesn't seem to explain how one can run a
dyadic comparison/optimization among, say, about 50 possible dyads within a two
octave range for a 7-tone scale. As I understand it 1/4 comma meantone only
does the "mini-max" dyadic optimization between the 5th and 3rd...so that's two
variables instead of 50. So I pity the fool who has to calculate the "optimum
limit" on all 50 by hand (possible yes...but very very time consuming)...and
have my doubts anyone in history would have the patience to do so and/or
construct and equation set clever enough to greatly simplify the problem.

>"Moving one disk relative to another shows how the intervals fit
>together."..."If you can make all the reference points fit into desired or
>accpetable areas on all the other disks then you have solved the problem."
Ah so if two lines on two disks are greater than a certain rotation % apart
you know they are off tune....say....more than 8 cents off? Makes sense...but
it seems you'd still be drawing lines somewhat at random and re-calculating the
fractional/"just intonation" values for each until you found something that
visually matches. So it seems to quickly tell you how well a scale works but
not how to make a scale the works well...so to speak. That echoes the "If you
can make all the reference points fit into desired or acceptable areas"
statement you made.

Funny thing...my program does almost exactly the same process...only it
solves the "random reference points" issue by running a loop of all possible
reference points "for each disk" (say, it has 5th values n1*n2...*n7 with each n
ranging from 3/2 minus 7 cents to 3/2 plus 7 cents independently).
By hands I could imagine it could take hours or more. By having the computer
loop through all possible values, it takes less than a minute. I cheat. :-)

>"Mark: I appologise. I myself was wrong here. Whereas Quarter Comma does
>approximate to the intervals 16/15 (or better still, 17/14), 6/5, 5/4, 4/3 and
>7/5 (and their respective 7ths, 3rds and 5ths), very well, it fails when it
>comes to the intervals 9/8 and 10/9. The 7 note quarter comma tone is halways
>between 10/9 and 9/8, making it wrong by 11 and a half cents."
Actually, I somewhat ignored seconds in my program/scale as well as I've
found that I can barely hear a difference between 10/9 and 9/8 and everything in
between. But it's interesting they apparently came out better nonetheless. :-D

>"Your scale approximates the tones better and has better minor 3rds, while
>having almost as good major thirds. I have very much enjoyed exploring the
>mathematics of your scale. I would write more but I am running out of time."
Well it should be impossible to achieve a lower mini-max than that scale
assuming the alternative generators (hence irregular temperament) must be within
8 cents of 3/2...because the program reviews all possible errors of all possible
7-tone scales using that generator range and compares the results of all dyads
to find the minimax. Glad you enjoyed it!

-Michael

[Non-text portions of this message have been removed]

Sorry for the many mistakes in my last reply. I ran out of time in the Library and didn't edit properly.
While I very much like your scale, with it's mixture of just and tempered intervals ( I enjoy scales with a variety of different sounding versions of the same interval), I think you have made a mistake and need to check your intervals with a calculator.

I you are aiming for a 7 note scale inwhich the 3rds and 4ths deviate from 6/5, 5/4, 4/3 and 7/5 as little as possible (and 6ths and 5ths match the same way), and by this you mean that the largest deviation is as small as possible, it is easy to prove with simple algebra that a 7 note quarter comma meantone scale is the best approximation possible.

Your scale is only better if you are also trying to approximate to 9/8 and 10/9.

This may not be true if you apply a different importance to the deviations from different just intervals.

>"Your scale is only better if you are also trying to approximate to 9/8 and
>10/9."
I'm not doing either...I don't sense a need to as the maximum log distance
between those (IE "midpoint") is less than 8 cents.

>"I you are aiming for a 7 note scale inwhich the 3rds and 4ths deviate from
>6/5, 5/4, 4/3 and 7/5 as little as possible (and 6ths and 5ths match the same
>way), and by this you mean that the largest deviation is as small as possible,
>it is easy to prove with simple algebra that a 7 note quarter comma meantone
>scale is the best approximation possible."

I'm aiming for not just 3rds and 4ths/5ths...but also 6th, and 7ths and 8ths
(IE meeting the octave exactly, no comma popping out at you).
That's perhaps the largest difference between what I'm trying to do and how
1/4 comma meantone supposedly works. I'm not focussing "mostly" on optimizing
major/minor triads (ALA much of meantone and diatonic 7-tone JI)...I'm trying to
focus on optimizing all possible dyadic combinations. Personally I think just
optimizing major/minor 3rds and 5ths is lousy because it hurts accuracy of
other, more interesting chords in other to perfect those triads.

[Non-text portions of this message have been removed]

--- In MakeMicroMusic@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> >"Your scale is only better if you are also trying to approximate to 9/8 and
> >10/9."
> I'm not doing either...I don't sense a need to as the maximum log distance
> between those (IE "midpoint") is less than 8 cents.
>
>
> >"I you are aiming for a 7 note scale inwhich the 3rds and 4ths deviate from
> >6/5, 5/4, 4/3 and 7/5 as little as possible (and 6ths and 5ths match the same
> >way), and by this you mean that the largest deviation is as small as possible,
> >it is easy to prove with simple algebra that a 7 note quarter comma meantone
> >scale is the best approximation possible."
>
> I'm aiming for not just 3rds and 4ths/5ths...but also 6th, and 7ths and 8ths
> (IE meeting the octave exactly, no comma popping out at you).
> That's perhaps the largest difference between what I'm trying to do and how
> 1/4 comma meantone supposedly works. I'm not focussing "mostly" on optimizing
> major/minor triads (ALA much of meantone and diatonic 7-tone JI)...I'm trying to
> focus on optimizing all possible dyadic combinations. Personally I think just
> optimizing major/minor 3rds and 5ths is lousy because it hurts accuracy of
> other, more interesting chords in other to perfect those triads.
>
> [Non-text portions of this message have been removed]
>
Mark: The octave is perfect in quarter comma. The sixths (as dyads) are closer to 8/5 and 5/3 in 7 note quarter comma than in your scale.
As for the 7ths, this maybe just how I think (pitches in a circle with one revolution equalling an octave, instead of a straight line), but it seems to me that if you don't care about the accuracy of 9/8 and 10/9 then you also don't care about the accuracy of 16/9 and 9/5.
Quarter comma gives these to an accuracy of about 11 cents.
What sort of 7th dyads are you aiming for?
Also what 6ths?
I have assumed 5/3, but you might prefer 27/16.

Mark>"The octave is perfect in quarter comma."
I don't see how that could be unless you scale it down and end up with a comma
in order to force the octave into place.
1.4953(1/4 comma mean-tone generator)^12 = 124.95 while the octave is at 128
(2^12). So the "commatic" difference between those is about 128/125...a whole
lot more than 20 cents and certainly not "perfect". People (in practice) may
push quarter comma's 12 note up to match the octave...but taking the "spiral of
5ths" by which creates the meantone scale certainly doesn't get us there. And
any interval that crosses from below to over that (in practice) modified to fit
2/1 octave is going to be thrown off a bit in the same way they'd be thrown off
by a pythagorean 12-tone scale with the final tone scaled to fit the octave.

>"As for the 7ths, this maybe just how I think (pitches in a circle with one
>revolution equalling an octave, instead of a straight line), but it seems to me
>that if you don't care about the accuracy of 9/8 and 10/9 then you also don't
>care about the accuracy of 16/9 and 9/5."

That assumption seems to assume a constant interval size for the second, so
that any 6th plus any second in the scale can be added to produce a single size
of 7th, for example.
Mine has no such constant size, hence it is an >>irregular<< temperament. All
the fifths are different sizes as well. Close but not the same. Trying to
treat it like a scale generated by taking the power of a single interval won't
work because that's not how it's generated. It's about as much like meantone as
JI diatonic is in that way: JI diatonic can't be created by taking a power a
single generator either.
http://en.wikipedia.org/wiki/Regular_temperament

>"16/9 and 9/5....Quarter comma gives these to an accuracy of about 11 cents."
11 cents...ugh...not very close IMVHO...it's almost a 12TET-ish error
margin. Try actually calculating the difference in dyads for the desired
sevenths in my scale...I'm quite sure the difference for mine is far under 11
cents.

For 7th: 16/9, 9/5, and 15/8
For the 6th: 8/5 and 5/3

>"Also what 6ths? I have assumed 5/3, but you might prefer 27/16."
In general, I'm going for JI (from the root tone) diatonic values. That
means 27/16 is out. So is 13/7 for the seventh. I'm not trying to approximate
12TET-like ratios, but strict JI ones in this case.

[Non-text portions of this message have been removed]

On Wed, Sep 8, 2010 at 3:00 PM, Michael <djtrancendance@...> wrote:
>
> Mark>"The octave is perfect in quarter comma."
> I don't see how that could be unless you scale it down and end up with a comma
> in order to force the octave into place.
> 1.4953(1/4 comma mean-tone generator)^12 = 124.95 while the octave is at 128
> (2^12). So the "commatic" difference between those is about 128/125...a whole
> lot more than 20 cents and certainly not "perfect". People (in practice) may
> push quarter comma's 12 note up to match the octave...but taking the "spiral of
> 5ths" by which creates the meantone scale certainly doesn't get us there.

Because quarter comma meantone has two generators: the flat fifth, and
the octave. The octave is perfect because it's one of the generators,
by definition (unless you're tempering the octave). If the generator
ended up -exactly- intersecting the octave, it wouldn't be a rank-2
temperament. It would be a rank-1 temperament, or an equal
temperament.

You would find that if you took the 1/4 comma meantone generator up to
31 octaves instead of 12, it would almost exactly close... but it
still wouldn't close. If you end up changing the fifth slightly to
"aim it" so that it does exactly close, you would have a rank-1
temperament. It would be 31-equal in this case, and both the fifth and
the octave would be formed by the same generator. This is why 31-equal
and 1/4 comma meantone are almost "the same thing."

-Mike

MikeB>"You would find that if you took the 1/4 comma meantone generator up to
31 octaves instead of 12, it would almost exactly close... but it still wouldn't
close."
Tried it...to me 1.4953^31 is so close to perfect it becomes un-noticable.

>"This is why 31-equal and 1/4 comma meantone are almost "the same thing."
Got it. The information I was missing is that 1/4 comma had two generators
that nearly match at a point (^31, in this case)...I always thought it had just
one generator.

[Non-text portions of this message have been removed]

Does this thread belong on MMM? -Carl

--- In MakeMicroMusic@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Does this thread belong on MMM? -Carl

My sources say no.

we mustn't have any post post that interfere with the non comments on the music:)

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

On 9/09/10 6:00 PM, jonszanto wrote:
> --- In MakeMicroMusic@yahoogroups.com, "Carl Lumma"<carl@...> wrote:
>> Does this thread belong on MMM? -Carl
> My sources say no.
>
>
>
>
> ------------------------------------
>
> Yahoo! Groups Links
>
>
>
>
>

Michael, I'm sorry but you clearly have not understood how these things work. The generator of quarter-comma meantone is the pure fifth flatted by one quarter comma ( 696.57843 cents). 31 of these, modulo 2, results in a "wolf" fifth which happens to be, due to the quite heavy flatness of the tempered generator fifth... almost a perfect 3/2! less than .7 cents sharp of pure. The octave is pure.

It's not your fault, the fascist hegemony of 12-tET damages the learning process of us all and the alternative world, though blessed, suffers from the very individualism that gives it birth, so pedagogy tends to be on the level of egregious suck. If we could just sit down for a couple of hours with some beers and a joint, you'd make a quantum leap in technical understanding because you've obviously got the main ingredient.

--- In MakeMicroMusic@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Mark>"The octave is perfect in quarter comma."
> I don't see how that could be unless you scale it down and end up with a comma
> in order to force the octave into place.
> 1.4953(1/4 comma mean-tone generator)^12 = 124.95 while the octave is at 128
> (2^12). So the "commatic" difference between those is about 128/125...a whole
> lot more than 20 cents and certainly not "perfect". People (in practice) may
> push quarter comma's 12 note up to match the octave...but taking the "spiral of
> 5ths" by which creates the meantone scale certainly doesn't get us there. And
> any interval that crosses from below to over that (in practice) modified to fit
> 2/1 octave is going to be thrown off a bit in the same way they'd be thrown off
> by a pythagorean 12-tone scale with the final tone scaled to fit the octave.
>
>
> >"As for the 7ths, this maybe just how I think (pitches in a circle with one
> >revolution equalling an octave, instead of a straight line), but it seems to me
> >that if you don't care about the accuracy of 9/8 and 10/9 then you also don't
> >care about the accuracy of 16/9 and 9/5."
>
> That assumption seems to assume a constant interval size for the second, so
> that any 6th plus any second in the scale can be added to produce a single size
> of 7th, for example.
> Mine has no such constant size, hence it is an >>irregular<< temperament. All
> the fifths are different sizes as well. Close but not the same. Trying to
> treat it like a scale generated by taking the power of a single interval won't
> work because that's not how it's generated. It's about as much like meantone as
> JI diatonic is in that way: JI diatonic can't be created by taking a power a
> single generator either.
> http://en.wikipedia.org/wiki/Regular_temperament
>
>
> >"16/9 and 9/5....Quarter comma gives these to an accuracy of about 11 cents."
> 11 cents...ugh...not very close IMVHO...it's almost a 12TET-ish error
> margin. Try actually calculating the difference in dyads for the desired
> sevenths in my scale...I'm quite sure the difference for mine is far under 11
> cents.
>
> For 7th: 16/9, 9/5, and 15/8
> For the 6th: 8/5 and 5/3
>
> >"Also what 6ths? I have assumed 5/3, but you might prefer 27/16."
> In general, I'm going for JI (from the root tone) diatonic values. That
> means 27/16 is out. So is 13/7 for the seventh. I'm not trying to approximate
> 12TET-like ratios, but strict JI ones in this case.
>
> [Non-text portions of this message have been removed]
>

Cameron>"Michael, I'm sorry but you clearly have not understood how these things
work. The generator of quarter-comma meantone is the pure fifth flatted by one
quarter comma ( 696.57843 cents). 31 of these, modulo 2, results in a "wolf"
fifth which happens to be, due to the quite heavy flatness of the tempered
generator fifth... almost a perfect 3/2! less than .7 cents sharp of pure. The
octave is pure."

If I have it right...the real thing I missed was not the pure fifth flatted
by one quarter comma ( 696.57843 cents) as the generator, but apparently that it
takes 31 (and not 12!) of them to make the "virtually" pure octave. (Even) if
you look at http://en.wikipedia.org/wiki/1/4-comma_meantone, they give an
example with 12 notes. That's what confused me: I thought it was 12 intervals
of 696.57 that generated 1/4 comma mean-tone, now it seems obvious it is 31 (and
the reason it is very close to 31TET thus also seems obvious). Now does it
sound like I have it or...what's missing?

>"results in a "wolf" fifth which happens to be... almost a perfect 3/2! less
>than .7 cents sharp of pure"
You see, before I thought the "wolf" described the rather significant error
you get when rounding the (quarter-comma 5th)^12 to the octave. Using
(quarter-comma 5th)^31...I'm surprised people even call it a 'Wolf' because the
error is ridiculously small.

Mean-tone = 31-tone scale, not 12-tone scale. That seems to be the gist of
it.

[Non-text portions of this message have been removed]

On Fri, Sep 10, 2010 at 1:54 PM, Michael <djtrancendance@...> wrote:
>
> If I have it right...the real thing I missed was not the pure fifth flatted
> by one quarter comma ( 696.57843 cents) as the generator, but apparently that it
> takes 31 (and not 12!) of them to make the "virtually" pure octave. (Even) if
> you look at http://en.wikipedia.org/wiki/1/4-comma_meantone, they give an
> example with 12 notes. That's what confused me: I thought it was 12 intervals
> of 696.57 that generated 1/4 comma mean-tone, now it seems obvious it is 31 (and
> the reason it is very close to 31TET thus also seems obvious). Now does it
> sound like I have it or...what's missing?

What you're missing is that in 1/4-comma meantone, you don't have just
one generator. The octave isn't "generated" by 31 fifths. The octave
counts as a -separate- generator. Separate. There are two generators
and they aren't related: the fifth, and the octave. This is because
it's a linear temperament, not an equal temperament.

1/4-comma meantone is not 31-tet. 31-tet is just an equal temperament
that "is almost" 1/4-comma meantone is what you get if you use two
generators, a fifth and an octave, and temper the fifth by 1/4 of a
syntonic comma.

> Mean-tone = 31-tone scale, not 12-tone scale. That seems to be the gist of
> it.

No. A meantone is anything where 81/80 is eliminated. 12-tet is a
meantone, 19-tet is a meantone, 31-tet is a meantone, 26-tet is a
meantone, 43-tet is a meantone, 88-tet is a meantone. 53-tet is not a
meantone, 34-tet is not a meantone, 22-tet is not a meantone.

Anything where you can go Cmaj -> Em -> Am -> Dm -> Gmaj -> Cmaj, and
end up at the same Cmaj where you started, is a meantone.

-Mike

MikeB>"1/4-comma meantone is not 31-tet"
Yes, I got that. Note I said "and the reason it is very CLOSE to 31TET"...I
never said it WAS 31TET. :-S

> Mean-tone = 31-tone scale, not 12-tone scale. That seems to be the gist of
> it.
>"No. A meantone is anything where 81/80 is eliminated."
When I said meantone I meant short for (1/4 comma) Meantone. As in quarter
comma meantone is a 31-tone scale, not a 12-tone scale. I wonder why you
didn't ask "do you mean 1/4 comma Meantone or all Meantone scales"? if you
weren't clear on that.

>"The octave isn't "generated" by 31 fifths. The octave counts as a -separate-
>generator. Separate."
I get that. Essentially taking 31 1/4 comma mean-tone fifths almost 100%
meets the octave...but not quite (misses by a fraction of a cent, as Cameron
said). Using the octave as a separate generator makes it 100% perfect. I was
getting at that it appears the reason it works so well is the 31 fifths
"essentially" generate the octave....meaning they get darn close to it. So if
you use the octave as a period and "adjust" that last fifths, you don't end up
destroying the dyadic relationships that note has with other notes.

[Non-text portions of this message have been removed]

Mike wrote:
>What you're missing is that in 1/4-comma meantone, you don't have just
>one generator. The octave isn't "generated" by 31 fifths. The octave
>counts as a -separate- generator. Separate. There are two generators
>and they aren't related: the fifth, and the octave. This is because
>it's a linear temperament, not an equal temperament.

Thanks Mike. -Carl

--- In MakeMicroMusic@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> MikeB>"1/4-comma meantone is not 31-tet"
> Yes, I got that. Note I said "and the reason it is very CLOSE to 31TET"...I
> never said it WAS 31TET. :-S
>
> > Mean-tone = 31-tone scale, not 12-tone scale. That seems to be the gist of
> > it.
> >"No. A meantone is anything where 81/80 is eliminated."
> When I said meantone I meant short for (1/4 comma) Meantone. As in quarter
> comma meantone is a 31-tone scale, not a 12-tone scale.

But quarter-comma meantone is not a 31-tone scale- it's not a scale at all, it's a temperament. Historical meantone scales were generated but going up and down a half-dozen fifths, octave-reducing when desired of course. Or down a few fifths and up a few more from a central tone; there are different schemes. There's also another by-ear way of tuning a quarter-comma meantone scale using mostly Just thirds. You don't have to go to twelve notes, and you don't have to stop at twelve, either. Handel's organ supposedly had 4 extra keys to the octave, probably 16 tones of quarter-comma meantone (don't know if anyone knows for sure how he tuned). At any rate, having split keys is a typically meantone thing, increases your number of key signatures with Just intervals.

Notice that there is a direct comparison with the "regular temperament paradigm" and historical quarter-comma meantone practice. For example there are 21, 31, and so on, tone scales of "Miracle temperament". Taking Miracle out all the way to where it reasonably closes will give you a scale that's realistically indistinguishable from 72 equal divisions of the octave, just as taking quarter-comma meantone all the way out to where it realistically closes will give you a scale very close to 31 equal divisions of the octave. Also notice that 72 equal divisions of the octave is not Miracle, and 31-edo is not quarter-comma meantone. You could for whatever reason have a temperament which was based on 8/7 tempered a tiny bit sharp (just an example). Let's say you tempered it a cent sharp, because you dig the extra slow beating, whatever, and took it out all the way (modulo 2) to where it realistically speaking closes. You'd be even closer, much closer, to 31-edo than quarter-comma meantone is. So is 31-edo an "8/7 system"? No. 31 is not an 8/7 system, and it's not a meantone system. You can just use it those ways, and other ways.
>
>
> >"The octave isn't "generated" by 31 fifths. The octave counts as a -separate-
> >generator. Separate."
> I get that. Essentially taking 31 1/4 comma mean-tone fifths almost 100%
> meets the octave...but not quite (misses by a fraction of a cent, as Cameron
> said).

That's not what I said. It is just- as I said- your leftover "wolf" fifth happens to be just that amount wide of the generating (1/4-comma flat) fifth so as to handily be practically identical to a Just fifth.

Using the octave as a separate generator makes it 100% perfect. I >was
> getting at that it appears the reason it works so well is the 31 >fifths
> "essentially" generate the octave....meaning they get darn close to >it.
> So if
> you use the octave as a period and "adjust" that last fifths, you don't end up
> destroying the dyadic relationships that note has with other notes.

So, it's like hands going around the face of the clock in jumps of a certain amounts of hours. Depending on how many hours the hands jump, you may or may not wind up eventually hitting all the hours. Jumps of six hours will hit just 12 and 6 repeatedly, and nothing else, on a 12-hour clock. The 12-tone tritone will do the same- tritone, octave, forever. It's not a generator. Jumps of five hours will hit every hour and return to twelve. In equal divisions of the octave, it's the scale steps which are coprime numbers (numbers made of different primes, like 12 is made of 2 and 3 while 5 is a prime) which will do this, which means you're set when your equal division of octave is itself a prime, for every scale step will eventually generate the entire thing.
Only steps 1, 5, 7 and 11 will do this in 12-edo, for all other steps share primes with 12. 31 is a prime, and a pretty big number as far as number of tones, so you see it could be generated by a big pile of different intervals and that it is especially silly to say "it's a meantone". The only way you can say that is to take the Harry Partch stance that all equal temperaments are Pythagorean in nature, which is pure bullshit, as many equal temperaments are not generated by an approximate fifth, and many can be generated by a huge number of alternative, non-Pythagorean, intervals.

-Cameron Bobro

Mark:

> >"As for the 7ths, this maybe just how I think (pitches in a circle with one
> >revolution equalling an octave, instead of a straight line), but it seems to me
> >that if you don't care about the accuracy of 9/8 and 10/9 then you also don't
> >care about the accuracy of 16/9 and 9/5."
>

Michael: That assumption seems to assume a constant interval size for the second, so
> that any 6th plus any second in the scale can be added to produce a single size
> of 7th, for example.
> Mine has no such constant size, hence it is an >>irregular<< temperament.

Mark: No. It doesn't assume a constant interval size for the major 2nd. What it assumes is that the octave is exactly 2/1. I don't think you specified the octave size in your scale (atleast to start with. Later I think you said 2/1 was your ideal size for the octave and you may have said that 2/1 is what you use in your scale).

The point is that if you divide 2/1 by 16/9 you get 9/8 and if you divide 2/1 by 9/5 you get 10/9. Thus, fixing the size of the 7ths fixes the size of the 2nds. (Similarly 2/1 divided by 15/8 is 16/15)

This sort of thing also applies with stretched or compressed octaves, or Bohlen Pierce scales, though in those cases the mathematics is slightly different.

There are many things I would like to reply to in your posts, but unfortunately I am short of time. I am hoping to make mechanical instruments capable of playing irregular intonations and temperaments such as yours because I think I have been missing out on a lot of fun.

Mark>"No. It doesn't assume a constant interval size for the major 2nd. What it
assumes is that the octave is exactly 2/1. I don't think you specified the
octave size in your scale (atleast to start with. Later I think you said 2/1 was
your ideal size for the octave and you may have said that 2/1 is what you use in
your scale)."

Indeed, 2/1 is what I use in my scale. But it's a direct part of the scale,
not a separately generated value (as 2/1 in 1/4 comma meantone is seperately
generated).

>"The point is that if you divide 2/1 by 16/9 you get 9/8 and if you divide 2/1
>by 9/5 you get 10/9."
Right, but that fixes the size of two 2nds and 7ths...out of 7 notes in the
scale. As for the others, they vary. For example one of the other "9/5's" may
be more like 11/6...in which case 2/1 over 11/6 is 12/11 and not 10/9. I
understand the "octave inverse of 7th = second" idea, but am quite sure it
doesn't apply to my scale as the 7ths vary in size and, thus...so do the "octave
inverse" seconds.

Also (again going back to a mistake I made a while ago concerning how I
though 1/4 comma meantone was generated), the scale I created does not have two
consistent generators as mean-tone has a slightly flat 5th and a 2/1 as separate
constant generators.

The bizarre thing, again, is that 1/4 comma meantone acheives a very similar
overall result to my scale despite being generated through constant generators.
Correct me if I'm wrong but it appears to try and "balance" between a pure third
and fifth, thus making incredibly pure triads possible. However, this has a
side effect of making the 6ths and 4ths very pure (due to the "inverse octave"
effect you described). So, musically, I'd consider the two quite close in
possible usage.

>"Thus, fixing the size of the 7ths fixes the size of the 2nds. (Similarly 2/1
>divided by 15/8 is 16/15)"
Right, and I don't fix the size of either.

>"I am hoping to make mechanical instruments capable of playing irregular
>intonations and temperaments such as yours because I think I have been missing
>out on a lot of fun."
Thank you! I've become into irregular temperaments due to their ability to
get more mini-max accuracy in many cases than regular ones, thus keeping the
maximum impurity of possible chords down and hopefully allowing more options far
as chords for us musicians to use as resting places.

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Side note: if you are going to try my above "1/4 comma meantone like" scale,
I'd highly recommend opting for it's "superset" tuning which I call "Dimension"
----------------Dimension Tuning--------------------------------
Nearest Just:Actual(in cents)
#1 1 : 1
#2 9/8 : 196.1984
#3 6/5: 315.64128
#4 11/9 : 347.4048
#5 5/4: 386.3137 (extra unused tone/"mystery tone")
#6 4/3: 504.09373
#7 3/2: 701.955
#8 5/3: 891.959
#9 9/5: 1009.88476
#10 11/6: 1046.212378
2/1 (PERIOD)

Known modes (all modes are "mini-max" as individual scales! ....ah the
flexibility of irregular temperament)

1,2,3,6,7,8,9 (1/4 comma mean-tone diatonic approximation, has two
"semitones"...IE the mode we have been discussing)

1.2.4,6,7,8,10 (7TET-like spaced scale with mostly non-extended JI diatonic
intervals but also 7 or so very sharp 11-limit quasi-JI (IE within 8 cents of
JI) intervals giving it a somewhat Arabic feel and musical capability of making
some very clear-sounding clustered chords)

1,2,4,6,7,8,9 (Mix of properties from the above modes. Has one "semi-tone"
with almost completely non-extended JI diatonic intervals but a few 11-limit
ones due to the 11/9 included in it)

Maybe you can help me find some more modes utilizing the 5/4? :-)

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