Microtonality repression?

nobody EVER even bothered to mention the
origin of enharmonics to me at all! I had no idea why there were two
note names for "black keys..." Such a basic concept was never

Same here. I got 12 toned up the wazoo at a supposedly contemporary school
and for a time (until I found this forum) I thought I was the only one
thinking of structures outside of 12et. Now I'm at the point (as a studio
composer) where I can't even write in 12et. It's just old-school.

Looking back now it seems as if there was a concerted effort to repress
anything that wasn't 12et. Intructors lead me to believe that non-12et was a
waste of time. Why would they do that? How can they call themselves
contemporary? It baffles me.

Thanks everybody. This list is great!
I feel indebted,
andy

p.s. What is the basic concept of enharmonics. AFAIK its just to spell the
chords correctly. Is there more to it?

--- In tuning@egroups.com, AMiltonF@a... wrote:

http://www.egroups.com/message/tuning/17230

> Same here. I got 12 toned up the wazoo at a supposedly
contemporary school and for a time (until I found this forum) I
thought I was the only one thinking of structures outside of 12et.
Now I'm at thepoint (as a studio composer) where I can't even write
in
12et. It's just old-school.
>
> Looking back now it seems as if there was a concerted effort to
repress anything that wasn't 12et. Intructors lead me to believe
that non-12et was a waste of time. Why would they do that? How can
they call themselves contemporary? It baffles me.
>
> Thanks everybody. This list is great!
> I feel indebted,
> andy

Well, Andy this kind of attitude continues. Over the Holidays, I
visited with one of my old composition teachers (who will remain
nameless at the moment) who insisted that microtonal music was
basically "out of tune." In addition, he assured all of us that the
notion of building new instruments was currently a "dead issue." I
kept my mouth shut, but didn't mention Grady, others...

Additionally, I just received a message from a conductor friend, also
nameless for a moment (Reinhard, you know who this is, but MUM
please) that reads as follows:

"I don't believe that Haba deserves the space though as time has
proven microtonal music to be not a music but an unpopular accident
that occurs when one is not quite well enough prepared on his/her
instrument of choice."

So, that's where it stands at the moment... It takes great courage
to proceed...

_______ ___ ____ ___
Joseph Pehrson

Joseph Pehrson wrote:

> Additionally, I just received a message from a conductor friend, also
> nameless for a moment (Reinhard, you know who this is, but MUM
> please) that reads as follows:
>
> "I don't believe that Haba deserves the space though as time has
> proven microtonal music to be not a music but an unpopular accident
> that occurs when one is not quite well enough prepared on his/her
> instrument of choice."
>
> So, that's where it stands at the moment... It takes great courage
> to proceed...

I try to avoid people like that.

--
* D a v i d B e a r d s l e y
* 49/32 R a d i o "all microtonal, all the time"
* http://www.virtulink.com/immp/lookhere.htm

--- In tuning@egroups.com, AMiltonF@a... wrote:

http://www.egroups.com/message/tuning/17230

> p.s. What is the basic concept of enharmonics. AFAIK its just to
spell the chords correctly. Is there more to it?

I'm basically not the person to explain this to you... somebody like
Paul Erlich could do it much better...

BUT, as I understand it, our current "traditional" notational system
derives from Pythagorean tuning --- a string of fifths that went from
double flats to double sharps, basically (see Ellis at the back of
Helmholz, ON THE SENSATIONS OF TONE page 433):

Abb, Ebb, Bbb, Fb, Cb, Gb, Db, Ab, Eb, Bb, F, C, G, D, A, E, B, F#,
C#, G#, D#, A#, E#, B#, F##, C##, G##

And, since we're talking about a system that could NEVER return to
the same octave-equivalent pitch because of the Pythagorean comma
(i.e. the 3:2 ratio ofthe perfect fifth can never multiply around to
arrive at a 2:1 octave ratio) all these pitches were ACTUALLY
DIFFERENT!

This multiplicity lasted for some time, held over into meantone,
where there STILL was a difference, even on some keyboards, between
G# and Ab (although by that time, the G# became HIGHER than Ab,
different from the Pythagorean... due to compression of the fifths in
quarter-comma meantone, for example).

So, all these notes were really DIFFERENT at one time, and only
later, with 12-equal intervals per octave, and tempering by the
fifths to make that result, did everything become HOMOGENIZED so that
there was one "black note" with two different spellings, sharp and
flat....

This is pretty basic stuff... and some on this list can elaborate
more eloquently for you... but the point is, I never learned even THIS
much in school...

Hope this helps!

______ _____ ____ _
Joseph Pehrson

--- In tuning@egroups.com, "Joseph Pehrson" <josephpehrson@c...>
wrote:

http://www.egroups.com/message/tuning/17243

> This multiplicity lasted for some time, held over into meantone,
> where there STILL was a difference, even on some keyboards, between
> G# and Ab (although by that time, the G# became HIGHER than Ab,
> different from the Pythagorean... due to compression of the fifths
in quarter-comma meantone, for example).
>

Sorry, Andy, I meant in meantone the G# became LOWER from the Ab. I
couldn't tell "uptown" from "downtown..."
_______ _____ __ _
Joseph Pehrson

Wow, pretty stunning quotes here in 2001. But I guess it's kind of
nice in one way to know that the great collective Rollo is still alive
and well! Oh well, silly silly stuff...

--Dan Stearns

--- In tuning@egroups.com, "D.Stearns" <STEARNS@C...> wrote:

http://www.egroups.com/message/tuning/17246

> Wow, pretty stunning quotes here in 2001. But I guess it's kind of
> nice in one way to know that the great collective Rollo is still
alive and well! Oh well, silly silly stuff...
>
> --Dan Stearns

True, "intellectually" silly, but rather disturbing, since many of
the
people that hold such attitudes also hold the power to distribute and
"popularize" new music. It is, in many ways, the "musical
establishment" so to speak, and one can try ostrich activities, but
they still come back to haunt us, if we want to have our music
performed and distributed...

________ ____ ___ _
Joseph Pehrson

Joseph Pehrson wrote,

<< True, "intellectually" silly, but rather disturbing, since many of
the people that hold such attitudes also hold the power to distribute
and "popularize" new music. It is, in many ways, the "musical
establishment" so to speak, and one can try ostrich activities, but
they still come back to haunt us, if we want to have our music
performed and distributed... >>

Not an issue for me, but I can sympathize... but I think for the most
part there actually is a good amount of curiosity when it comes to
microtonality, and views like the ones that you quoted are probably
going to become less and less prevalent... well at least I would think
so anyway! Personally I like the ostrich defense myself.

--Dan Stearns

--- In tuning@egroups.com, "D.Stearns" <STEARNS@C...> wrote:

http://www.egroups.com/message/tuning/17251

>
> Not an issue for me, but I can sympathize... but I think for the
most part there actually is a good amount of curiosity when it comes
to microtonality, and views like the ones that you quoted are probably
> going to become less and less prevalent... well at least I would
think so anyway! Personally I like the ostrich defense myself.
>
> --Dan Stearns

OK... but what if we decided we would rather "fight than switch..."
or rather fight than hide??

Frankly, I think the vitriol that I have experienced in certain
quarters upon my Conversion to Micromusiasm, is a positive sign.

If I am becoming that disturbing, I must be doing SOMETHING right!

________ _____ ____ ____
Joseph Pehrson

Joseph Pehrson wrote,

<< OK... but what if we decided we would rather "fight than switch..."
or rather fight than hide?? >>

Oh I say fight for sure, but you know -- let your music do the
talking, lead by example... that type of thing.

Maybe this seems like an easy thing to say for someone who doesn't
have to depend on these people in any way to accomplish what he's
doing, I dunno... but I don't think you can argue with that type on a
'it's all out of tune and a waste of time' mindset very well with just
words. Music, music, music. And if that's not enough, then who the
hell cares what these people think?

--Dan Stearns

Andy wrote,

>Well, Andy this kind of attitude continues. Over the Holidays, I
>visited with one of my old composition teachers (who will remain
>nameless at the moment) who insisted that microtonal music was
>basically "out of tune." In addition, he assured all of us that the
>notion of building new instruments was currently a "dead issue." I
>kept my mouth shut, but didn't mention Grady, others...

Guess what? Classical music with the traditional instruementarium is
basically a "dead issue" culturally compared with the often-microtonal
electronica hitting the dance clubs.

Joseph Pehrson wrote,

>This multiplicity lasted for some time, held over into meantone,
>where there STILL was a difference, even on some keyboards, between
>G# and Ab (although by that time, the G# became HIGHER than Ab,
>different from the Pythagorean... due to compression of the fifths in
>quarter-comma meantone, for example).

It's the reverse -- G# was higher than Ab in Pythagorean; G# is lower than
Ab in meantone. Write that on your bulletin board. For your next quiz, name
the interval that approximates 7:4 in meantone and the one that approximates
7:4 in Pythagorean.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/17277

> Andy wrote,
>
> >Well, Andy this kind of attitude continues. Over the Holidays, I
> >visited with one of my old composition teachers (who will remain
> >nameless at the moment) who insisted that microtonal music was
> >basically "out of tune." In addition, he assured all of us that
the notion of building new instruments was currently a "dead issue."
I kept my mouth shut, but didn't mention Grady, others...
>

> Guess what? Classical music with the traditional instruementarium is
> basically a "dead issue" culturally compared with the
often-microtonal electronica hitting the dance clubs.

Hi Paul!

Now you're doing what *I* generally do. No, *I* was the one who said
that, not Andy!

Anyway, point well taken. Recently the Absolute Ensemble has been
performing in "Joe's Pub," part of the Public Theatre in Manhattan,
and these performances... of so-called "accessible" music (but
including, parts from L'Histoire du Soldat of all things) are quite
popular and, perhaps, more "vital" than some poorly attended,
serially-oriented (still!)new music concerts.

So, yes, the entire culture is changing, so perhaps depending on the
"old fogey" establishment is really just a waste of time after all...
______ ____ ____ _
Joseph Pehrson

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/17278

>
> It's the reverse -- G# was higher than Ab in Pythagorean; G# is
lower than Ab in meantone.

Caught it, Paul... right after I sent it, unfortunately...

For your next quiz, name the interval that approximates 7:4 in
meantone and the one that approximates 7:4 in Pythagorean.

Well, this is a fun quiz... but I'm not sure I'm getting anyplace
with it... According to Scala, the Pythagorean minor seventh is 996
cents and the 1/4 comma meantone minor seventh is 1006 cents...

So, both of these minor sevenths would approximate a 7/4 at 968 cents
better than any other intervals, no??

________ _____ ____ _
Joseph Pehrson

Joseph wrote,

>Well, this is a fun quiz... but I'm not sure I'm getting anyplace
>with it... According to Scala, the Pythagorean minor seventh is 996
>cents and the 1/4 comma meantone minor seventh is 1006 cents...

Right . . .

>So, both of these minor sevenths would approximate a 7/4 at 968 cents
>better than any other intervals, no??

No. You can do _much_ better within the notes you named in the chain of
fifths.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/17284

>>
> No. You can do _much_ better within the notes you named in the
chain of fifths.

I believe I used 35 pitches... so SCALA generates a Pythagorean scale
with 35 pitches like this (I hope):

0: 1/1 0.000 unison, perfect prime
1: 34.982 cents 34.982
2: 76.046 cents 76.046
3: 111.028 cents 111.028
4: 152.092 cents 152.092
5: 187.074 cents 187.074
6: 193.156 cents 193.156
7: 228.138 cents 228.138
8: 269.202 cents 269.202
9: 304.184 cents 304.184
10: 345.248 cents 345.248
11: 386.312 cents 386.312
12: 421.294 cents 421.294
13: 462.358 cents 462.358
14: 497.340 cents 497.340
15: 538.404 cents 538.404
16: 579.468 cents 579.468
17: 614.450 cents 614.450
18: 655.514 cents 655.514
19: 690.496 cents 690.496
20: 696.578 cents 696.578
21: 731.560 cents 731.560
22: 772.624 cents 772.624
23: 807.606 cents 807.606
24: 848.670 cents 848.670
25: 883.652 cents 883.652
26: 889.734 cents 889.734
27: 924.716 cents 924.716
28: 965.780 cents 965.780
29: 1000.762 cents 1000.762
30: 1041.826 cents 1041.826
31: 1082.890 cents 1082.890
32: 1117.872 cents 1117.872
33: 1158.936 cents 1158.936
34: 1193.918 cents 1193.918
35: 2/1 1200.000 octave

I see 965 cents there... which is pretty close to the just 7/4 968...
only 3 cents off (!)

Now... I don't know whether this comes from the sharp side of the
chain or the flat side...

I would just guess the interval is C-Bbb in Pythagorean, since the
flat fifth names are lower than the sharp ones and we need something
lower than 1000 cents, obviously...

Is that right?? and how would I figure that out (??)

Thanks for the puzzle!
_______ ____ ____ _
Joseph Pehrson

Joseph wrote,

>I believe I used 35 pitches... so SCALA generates a Pythagorean scale
>with 35 pitches like this (I hope):

That's not Pythagorean, it's meantone!

>I see 965 cents there... which is pretty close to the just 7/4 968...
>only 3 cents off (!)

That's the one (for meantone). But it also looks like you've put 1/1 at _one
end_ of your chain of meantone fifths rather than _in the middle_ . . .
which is generally not a good idea . . . you had a 50-50 chance of not
finding this interval at all the way you did it -- you got lucky!

>I would just guess the interval is C-Bbb in Pythagorean, since the
>flat fifth names are lower than the sharp ones and we need something
>lower than 1000 cents, obviously...

Again, we're in meantone . . . A flat will always lower a pitch by the same
amount -- can you figure out what amount that is?

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/17293

Hmmm.

Well, first off, it might be nice if I could get SCALA to do this
correctly...

Here are my "instructions..."

PYTHAGOREAN [scalenr.]

Create a Pythagorean scale in the current or given scale. The program
asks first the scale size and then the formal octave. Next the
position of the formal fifth in the scale must be given. If instead 0
is entered, the order of pitches is determined by their size
(monotonically ascending). This is also default when just the return
key is pressed. Otherwise the range is indicated where the formal
fifth must be in, for the resulting scale to be monotonic. Then the
formal fifth is entered. The default value for this is pitch memory
0. Subsequently the count downwards, i.e. the number of fifths that
are stacked in the downward direction is entered which is by default
0. This gives a scale in another key. Defining it here can prevent
pitches from becoming a floating point value if ratios overflow.

OK... so, first off I want a chain of 35 fifths, so that would be my
scale size... Now it says "the position of the formal fifth in the
scale..." I guess I just want them stacked according to width so I
think I should leave that at "0" for now...Now the "formal fifth."
This time I've decided to use the correct one 701.955 cents. Now the
"count downward..." I GUESS this is where I made my mistake before.
I suppose it should just be half of the total... 17. I'll try that.

0: 1/1 0.000 unison, perfect prime
1: 23.460 cents 23.460
2: 66.765 cents 66.765
3: 90.225 cents 90.225
4: 113.685 cents 113.685
5: 180.450 cents 180.450
6: 203.910 cents 203.910
7: 227.370 cents 227.370
8: 270.675 cents 270.675
9: 294.135 cents 294.135
10: 317.595 cents 317.595
11: 384.360 cents 384.360
12: 407.820 cents 407.820
13: 431.280 cents 431.280
14: 474.585 cents 474.585
15: 498.045 cents 498.045
16: 521.505 cents 521.505
17: 588.270 cents 588.270
18: 611.730 cents 611.730
19: 678.495 cents 678.495
20: 701.955 cents 701.955
21: 725.415 cents 725.415
22: 768.720 cents 768.720
23: 792.180 cents 792.180
24: 815.640 cents 815.640
25: 882.405 cents 882.405
26: 905.865 cents 905.865
27: 929.325 cents 929.325
28: 972.630 cents 972.630
29: 996.090 cents 996.090
30: 1019.550 cents 1019.550
31: 1086.315 cents 1086.315
32: 1109.775 cents 1109.775
33: 1133.235 cents 1133.235
34: 1176.540 cents 1176.540
35: 2/1 1200.000 octave

Ok... but now I don't get 965 cents at all?? What happened??
__________ _______ ______
Joseph Pehrson

AMiltonF@aol.com wrote:

Looking back now it seems as if there was a concerted effort to repress
anything that wasn't 12et. Intructors lead me to believe that non-12et
was a
waste of time. Why would they do that?

Fear.

Joseph wrote,

>Ok... but now I don't get 965 cents at all?? What happened??

The 965 cents was in meantone. In your latest correct table in Pythagorean,
you get something just as close to 7:4 -- see it? Now determine what
interval it is.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/17293

>
> Again, we're in meantone . . . A flat will always lower a pitch by
the same amount -- can you figure out what amount that is?

Well in this:

1/4-comma mean-tone scale. Pietro Aaron's temp. (1523). 6/5 beats
twice 3/2

0: 1/1 0.000 unison, perfect prime
1: 76.049 cents 76.049
2: 193.157 cents 193.157
3: 310.265 cents 310.265
4: 5/4 386.314 major third
5: 503.422 cents 503.422
6: 579.471 cents 579.471
7: 696.578 cents 696.578
8: 25/16 772.627 classic augmented fifth
9: 889.735 cents 889.735
10: 1006.843 cents 1006.843
11: 1082.892 cents 1082.892
12: 2/1 1200.000 octave

It looks like 76 cents, correct??

_________ ______ ______
Joseph Pehrson

I wrote,

>> Again, we're in meantone . . . A flat will always lower a pitch by
>> the same amount -- can you figure out what amount that is?

Joseph wrote,

>It looks like 76 cents, correct??

Correct. Now how much does a flat lower a pitch in Pythagorean?

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/17300

> Joseph wrote,
>
> >Ok... but now I don't get 965 cents at all?? What happened??
>
> The 965 cents was in meantone. In your latest correct table in
Pythagorean, you get something just as close to 7:4 -- see it? Now
determine what interval it is.

Whoa! ... I see a 972 as well....

Now how do I figure out if this is on the sharp or flat side of the
Pythagorean chain?? In 12-tET, it's closer to 1000 than 900, so I
would think it would be a flat... HOWEVER, in Pythagorean the sharps
are HIGHER than the flats... So I will say A##... (??)

buzzzz....
_________ ______ ______ _
Joseph Pehrson

Joseph wrote,

>Now how do I figure out if this is on the sharp or flat side of the
>Pythagorean chain?? In 12-tET, it's closer to 1000 than 900, so I
>would think it would be a flat... HOWEVER, in Pythagorean the sharps
>are HIGHER than the flats... So I will say A##... (??)

>buzzzz....

OK, first please work on the question of how much a flat lowers a note in
Pythagorean -- that'll be the same amount that a sharp raises a note in
Pythagorean -- then come back to this question.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

> I wrote,
>
> >> Again, we're in meantone . . . A flat will always lower a pitch
by
> >> the same amount -- can you figure out what amount that is?
>
> Joseph wrote,
>
> >It looks like 76 cents, correct??
>
> Correct. Now how much does a flat lower a pitch in Pythagorean?

12-tone Pythagorean scale
0: 1/1 0.000 unison, perfect prime
1: 2187/2048 113.685 apotome
2: 9/8 203.910 major whole tone
3: 32/27 294.135 Pythagorean minor third
4: 81/64 407.820 Pythagorean major third
5: 4/3 498.045 perfect fourth
6: 729/512 611.730 Pythagorean tritone
7: 3/2 701.955 perfect fifth
8: 6561/4096 815.640 Pythagorean augmented fifth
9: 27/16 905.865 Pythagorean major sixth
10: 16/9 996.090 Pythagorean minor seventh
11: 243/128 1109.775 Pythagorean major seventh
12: 2/1 1200.000 octave

Well, here it looks like 90 cents (??)

_______ __ __ __
JP

I wrote,

>> Correct. Now how much does a flat lower a pitch in Pythagorean?

Joseph wrote,

>Well, here it looks like 90 cents (??)

Sorry, Joseph, that's not correct. Guess you got lucky with the meantone
one. I suggest you step back and try to figure this out step-by-step rather
than trying to read it off 12-tone SCALA files.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/17306

> I wrote,
>
> >> Correct. Now how much does a flat lower a pitch in Pythagorean?
>
> Joseph wrote,
>
> >Well, here it looks like 90 cents (??)
>
> Sorry, Joseph, that's not correct. Guess you got lucky with the
meantone one. I suggest you step back and try to figure this out
step-by-step rather than trying to read it off 12-tone SCALA files.

I'm not quite certain how I would do that... It's not 113 is it??

_______ ___ __ _
Joseph Pehrson

Joseph wrote,

>It's not 113 is it??

Yup, it's 113. So -- can you answer the question -- what interval (using
conventional nomenclature such as "doubly diminished octave") approximates a
7:4 in Pythagorean, and which one does so in meantone?

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/17308

> Joseph wrote,
>
> >It's not 113 is it??
>
> Yup, it's 113. So -- can you answer the question -- what interval
(using conventional nomenclature such as "doubly diminished octave")
approximates a 7:4 in Pythagorean, and which one does so in meantone?

Hmmm. Well, this is quite some clue!

Getting to 972 cents Pythagorean...

1200-113=
1087-113=
974

So, I would say for the Pythagorean it's "doubly diminished octave!"

For meantone:

Getting to 965 cents meantone:

1200-76=
1125-76=
1049-76=
973

So, it kind of looks like "Augmented 7th" to me... (??)

_______ _____ ___ _
JP

Joseph wrote,

>1200-113=
>1087-113=
>974

>So, I would say for the Pythagorean it's "doubly diminished octave!"

Correct -- and sorry about the clue -- it was only an accident that that was
the right answer (or more like a Freudian slip).

>For meantone:

>Getting to 965 cents meantone:

>1200-76=
>1125-76=
>1049-76=
>973

>So, it kind of looks like "Augmented 7th" to me... (??)

Huh? By the logic above, didn't you just construct a "triply diminished
octave" in meantone? The triply diminished octave is indeed close to a 7:4,
but there's a simpler meantone interval which is the 965-cent one you saw
earlier. Can you figure out what it is?

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/17311

>
> Huh? By the logic above, didn't you just construct a "triply
diminished octave" in meantone? The triply diminished octave is
indeed close to a 7:4, but there's a simpler meantone interval which
is the 965-cent one you saw earlier. Can you figure out what it is?

Well, the minor seventh in meantone is 1006 cents... so, if I put
another flat on that I get 1006-76 or 930...

Would it be a diminished 7th??

_______ ______ _____
JP

Joseph Pehrson wrote,

>Well, the minor seventh in meantone is 1006 cents... so, if I put
>another flat on that I get 1006-76 or 930...

>Would it be a diminished 7th??

930 cents would indeed be a meantone diminshed 7th . . . but that's not very
close to the approximate 7:4 we wanted, which you found to be 965 cents . .
.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/17313

> Joseph Pehrson wrote,
>
> >Well, the minor seventh in meantone is 1006 cents... so, if I put
> >another flat on that I get 1006-76 or 930...
>
> >Would it be a diminished 7th??
>
> 930 cents would indeed be a meantone diminshed 7th . . . but that's
not very
> close to the approximate 7:4 we wanted, which you found to be 965
cents . .
> .

Bingo. 889, the meantone major sixth, with a sharp is --

889+76=
965!

So, it's an augmented sixth!

_________ ______ _____
Joseph Pehrson

Joseph Pehrson wrote,

>Bingo. 889, the meantone major sixth, with a sharp is --

>889+76=
>965!

>So, it's an augmented sixth!

Congratulations, Joseph! Now, see if you can digest this:

http://www.uq.net.au/~zzdkeena/Music/1ChainOfFifthsTunings.htm

chew on it for at least 24 hours (including sleep), and if you're completely
solid on it, move on to:

http://www.uq.net.au/~zzdkeena/Music/2ChainOfFifthsTunings.htm

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/17315

>
> Congratulations, Joseph! Now, see if you can digest this:
>
> http://www.uq.net.au/~zzdkeena/Music/1ChainOfFifthsTunings.htm
>
> chew on it for at least 24 hours (including sleep), and if you're
completely solid on it, move on to:
>
> http://www.uq.net.au/~zzdkeena/Music/2ChainOfFifthsTunings.htm

Thanks, Paul! This has been a great "interactive" tutorial!!!!!
______ _____ _____
Joseph Pehrson