Yahoo Tuning Groups Ultimate Backup tuning Re: periodic table/"lousy thirds"

[Dave Keenan:]
>Also note that some ETs in this table are still not "useful" because they
have such lousy thirds, e.g. 17-tET, 33-tET.

I personally find the 2dt, 364c 33-tET major third (or the '<5/4, & >2/7ths
of an octave group') here a lot less 'offending' than the 2dt, 424c major
third (or the '>1/3rd of an octave group'). With that in mind, and without
much altering the "periodic table," I like something along the lines of a
'6 by 7' out of '5 by 7' major third parameter:

where I've added a (perhaps unnecessary) "d" for diatonic, as in the
chromatic set, the chromatic semitone ("cs") would be: dt-ds=cs, and the
difference between the augmented fourth and the flatted fifth would be:
ds-cs.

Although I've stated my personal preference (if forced to choose, and in
the particular context of the WWhWWWh diatonic scale) for 'neutral' thirds
over 'augmented' thirds; this 1/21st of on octave major third parameter is
probably both too wide on the 'neutral' side, and not quite wide enough on
the 'augmented' side... I'm curious as to where others might draw the
extremities of this '_major_ third boarder': 17/14 - 9/7?, 27/22 - 14/11?,
79/64 - 81/64... 5/4 +/- 2 or 3c?

Dan

🔗Dave Keenan <d.keenan@uq.net.au>

[Dan Stearns TD 227.9]

>I personally find the 2dt, 364c 33-tET major third (or the '<5/4, & >2/7ths
>of an octave group') here a lot less 'offending' than the 2dt, 424c major
>third (or the '>1/3rd of an octave group'). With that in mind, and without
>much altering the "periodic table," I like something along the lines of a
>'6 by 7' out of '5 by 7' major third parameter:
>
>1=dt, (dt/2)...dt=ds| 6 7
>2=dt, (dt/2)...dt=ds| 12 13 14
>3=dt, (dt/2)...dt=ds| 18 19 20 21
>4=dt, (dt/2)...dt=ds| 24 25 26 27 28
>5=dt, (dt/2)...dt=ds| 30 31 32 33 34 35
>6=dt, (dt/2)...dt=ds| 36 37 38 39 40 41 42
>7=dt, (dt/2)...dt=ds| 42 ...

If you're going to include 7 and 14-tET as having usable major thirds (43c
narrow and 42c wide!), then surely you must include 3, 9, 10, 15, 16, 17,
22, 23, 29. Personally, I wouldn't include 7, 10, 14, 17 or 23 but would
include 3, 9, 15, 16, 22, 29. Explained further below.

>Although I've stated my personal preference (if forced to choose, and in
>the particular context of the WWhWWWh diatonic scale) for 'neutral' thirds
>over 'augmented' thirds; this 1/21st of on octave major third parameter is
>probably both too wide on the 'neutral' side, and not quite wide enough on
>the 'augmented' side... I'm curious as to where others might draw the
>extremities of this '_major_ third boarder': 17/14 - 9/7?, 27/22 - 14/11?,
>79/64 - 81/64... 5/4 +/- 2 or 3c?

Thanks for reminding me about the terms diatonic and chromatic semitone.

I guess I can only reject 33-tET because of the _combination_ of marginal
fifths and marginal major thirds, and the fact that they go in opposite
directions so that its consistent 5:6 minor third is completely useless.

-----------------------------------------------------------------------

As an approximation of a 4:5 major third, I would say that anything outside
of 9:11 (347.4c, 39c narrow) to 7:9 (435.1c, 49c wide) is definitely not a
candidate! I'd prefer to halve those margins. Say -20 to +25 cents. I note
that your 1/21st of an octave is 57 cents. Is that the total width off the
acceptable band, or do you mean +-57c!?

I find that ETs with usable major thirds can be expressed as 3y + 7x where
x and y are natural-numbers and x <= y. They are more reasonable when x <=
y/2. But I have no idea what x and y might represent with regard to
notating the major thirds in a given ET (as s and t related to notating the
fifths).

The degenerate case of 0-tET is included to better show the pattern (I
should have done this with the usable fifths ET table too). In zero-tone
equal temperament every interval is perfect. :-) At least you'll never find
out if they aren't, since there are no notes.

So here's my table of ETs whose major thirds have errors between -17.1 and
+13.7c:

0
3
6 13
9 16
12 19 26
15 22 29
18 25 32 39
21 28 35 42
24 31 38 45 52
27 34 41 48 55
30 37 44 51 58 65
33 40 47 54 61 68
36 43 50 57 64 71 78
39 46 53 60 67 74 ...
42 49 56 63 70 77 ...
45 52 59 66 73 ...
48 55 62 69 76 ...
51 58 65 72 ...
54 61 68 75 ...
57 64 71 ...
60 67 74 ...
63 70 ...
66 73...
69 ...
72 ...

(i.e. 3y + 7x where y >= 0 and 0 <= x <= y/2)
If an ET appears more than once it means it has more than one major third
size in the range. The one nearest a line drawn thru 0, 31 and 59-tET is
the best. Those above that line are narrow, those below are wide. Note that
all ETs from 24 onward appear at least once.

Here's a repeat of my table of ET's with reasonable fifths (errors between
-12.6 and +13.0c):

12
17 19
22 24 26
27 29 31 33
32 34 36 38 40
37 39 41 43 45 47
42 44 46 48 50 52
47 49 51 53 55 57 59
54 56 58 60 62 64 66
59 61 63 65 67 69 71 73
64 66 68 70 72 74 76 78 80
69 71 73 75 77 79 ...
74 76 78 ...

(i.e. 5t + 2s where t >= 0 and t/9 <= s <= 6t/7)
Those above a line drawn thru 0, 41 and 53 have narrow fifths, those below
are wide (same directions as the thirds table). All ET's from 36 onward
appear at least once.

The ETs that are common to both are:
0, 12, 19, 22, 24, 26, 27, 29, 31, 32, 33, 34, and 36 and above.

Then we have to look at the minor third created by the chosen fifth and
major third. If the ET appears on the same side of the imaginary centerline
line in both cases (thirds and fifths) or close to the centerline in both
cases, then the minor third will be ok. 32 and 33-tET violate this
requirement. So we're left with:

0, 12, 19, 22, 24, 26, 27, 29, 31, 34, and 36 and above,

as the magic numbers (we haven't looked at ratios of 7). Many people have
produced very similar lists by many different means.

Regards,
-- Dave Keenan
http://dkeenan.com

🔗Graham Breed <g.breed@xxx.xx.xxx>

Dave Keenan wrote:

>I find that ETs with usable major thirds can be expressed as 3y + 7x where
>x and y are natural-numbers and x <= y. They are more reasonable when x <=
>y/2. But I have no idea what x and y might represent with regard to
>notating the major thirds in a given ET (as s and t related to notating the
>fifths).

A major third is y+2x in general. I've checked this for 12,19,31,41 and
53-equal. With 31 and 53, there are two ways of getting the octave, and
it's the one with the smallest x that gives this relation.

>So here's my table of ETs whose major thirds have errors between -17.1 and
>+13.7c:

Everything except 1 2 4 5 7 8 10 11 14 17 20 and 23 is listed. As x=y for
10 and 20, should the condition be x<y rather than x<=y? It has two
effects:

It removes all temperaments with fewer than 13 notes that are not divisible
by 3. As a third of an octave is a good approximation to a major third,
such small ETs with good major thirds will, not surprisingly, be divisible
by 3.

7, 10, 14, 17 and 20 are all in the pattern 7x+3y where x>=y. This is also
the pattern of scales with good neutral thirds. It should not be surprising
that an ET with not many notes and a good neutral third will not also have a
good major third.

That only leaves 23, which is x=1, y=2. So, the more reasonable scales
would be 2x<y rather than 2x<=y (or x<=y/2).

For just octaves and major thirds, x should be tuned to an enharmonic diesis
of 41.1 cents (2^7/5^3) and y should be 304.2 cents (5^7/2^16). Hence the
ideal ratio y/x is 7.4. Setting y=7 and x=1 gives 28-equal, and y=15, x=2
gives 59-equal. Sure enough, both have very good major thirds.

Oh, the wonders of numbers!

Graham
http://www.cix.co.uk/~gbreed/tuning.htm

🔗D. Stearns <stearns@xxxxxxx.xxxx>

[Dave Keenan:]
>If you're going to include 7 and 14-tET as having usable major thirds (43c
narrow and 42c wide!),

Actually the '6 by 7' gives all the multiples of 7 (<6*7) a 2/7ths, and
only a 2/7ths third... as they would be all be dt=ds (an exterior set).

>then surely you must include 3, 9, 10, 15, 16, 17, 22, 23, 29.

No, it was a combination of ET's falling in the >3/5ths and <4/7ths fifth
on the "periodic table":

12
/ \
17 19
/ \ / \
22 24 26
/ \ / \ / \
27 29 31 33
/ \ / \ / \ / \
32 36...

that also had a major third >2/7ths (but not >1/3rd), where dt-ds>0 that I
was attempting to 'distinguish.'

>I guess I can only reject 33-tET because of the _combination_ of marginal
fifths and marginal major thirds, and the fact that they go in opposite
directions so that its consistent 5:6 minor third is completely useless.

I don't see why is this minor third would be "completely useless?" 33-tET
(@: 4+1+4+1+4+1+3+1+4+1+4+1+4) would have a 691c fifth, a 364c major third,
and a 327c minor third - no?

>I note that your 1/21st of an octave is 57 cents. Is that the total width
off the acceptable band, or do you mean +-57c!?

Yes to the former (_but_ this is in the specific context of trying to
simply readdress the "periodic table..."), and obviously no to the latter.

>As an approximation of a 4:5 major third, I would say that anything
outside of 9:11 (347.4c, 39c narrow) to 7:9 (435.1c, >49c wide) is
definitely not a candidate! I'd prefer to halve those margins. Say -20 to
+25 cents.

I pretty much agree with this. (I had looked at 4/13ths and 8/23rds as
'simple' ET's to express this...)

>I find that ETs with usable major thirds can be expressed as 3y + 7x where
x and y are natural-numbers and x <= y. They are more reasonable when x <=
y/2. But I have no idea what x and y might represent with regard to
notating the major thirds in a given ET (as s and t related to notating the
fifths).

A simple (and personally appealing) expression of the latter was really all
that I was interested in.

Dan

🔗Paul Hahn <Paul-Hahn@xxxxxxx.xxxxx.xxxx>

🔗D. Stearns <stearns@xxxxxxx.xxxx>

[Dave Keenan wrote:]
>I guess I can only reject 33-tET because of the _combination_ of marginal
fifths and marginal major thirds, and the fact that they go in opposite
directions so that its consistent 5:6 minor third is completely useless.

[I wrote:]
>I don't see why this minor third would be "completely useless..." 33-tET
(@: 4+1+4+1+4+1+3+1+4+1+4+1+4) would have a 691c fifth, a 364c major third,
and a 327c minor third - no?

[Paul Hahn wrote:]
>The problem is that the 400c major third in 33TET is closer to the just
5:4 than the 364c one. It is impossible to have the closest available
approximations to 3:2, 5:4, and 6:5 in a single 33TET triad. This is what
Paul Erlich and I mean when we say that 33TET is inconsistent at the
5-limit.

I understand this, and while it may be a "problem," I don't really see any
reason why that (aside from personal preference) should read: "completely
useless..." in the case of 33-tET (@: 4+1+4+1+4+1+3+1+4+1+4+1+4).

Dan

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

Dave Keenan wrote,
>>I guess I can only reject 33-tET because of the _combination_ of marginal
>>fifths and marginal major thirds, and the fact that they go in opposite
>>directions so that its consistent 5:6 minor third is completely useless.

Dan Stearns wrote,
>I don't see why this minor third would be "completely useless..." 33-tET
>(@: 4+1+4+1+4+1+3+1+4+1+4+1+4) would have a 691c fifth, a 364c major third,

>and a 327c minor third - no?

Paul Hahn wrote,
>>The problem is that the 400c major third in 33TET is closer to the just
>>5:4 than the 364c one. It is impossible to have the closest available
>>approximations to 3:2, 5:4, and 6:5 in a single 33TET triad. This is what

>>Paul Erlich and I mean when we say that 33TET is inconsistent at the
>>5-limit.

>I understand this, and while it may be a "problem," I don't really see any
>reason why that (aside from personal preference) should read: "completely
>useless..." in the case of 33-tET (@: 4+1+4+1+4+1+3+1+4+1+4+1+4).

For ETs >35, inconsistency is not that big a deal, since the second-worst
approximation is usually acceptable, but in 33-tET, I would say that the
364� interval is, for my purposes (namely, getting across the character of
5:4 in a consonant fashion), unacceptable as a major third. Your purposes
may of course be different.

🔗David C Keenan <d.keenan@xx.xxx.xxx>

[Graham Breed TD 229.1]

>A major third is y+2x in general. I've checked this for 12,19,31,41 and
>53-equal. With 31 and 53, there are two ways of getting the octave, and
>it's the one with the smallest x that gives this relation.

Excellent!

>Everything except 1 2 4 5 7 8 10 11 14 17 20 and 23 is listed. As x=y for
>10 and 20, should the condition be x<y rather than x<=y?

Not important, since my criteria for "usable" were vague. I intended to
include 10 and 20-tET with their -26c errors. If you wonder why I didn't
also allow greater than the +13.6c errors in the other direction, it's
because there _are_ none greater until you get to 14-tET with +42c. I
decided that was not "usable". The tighter criteria for "reasonable" (what
the table showed) are more important.

>...
>That only leaves 23, which is x=1, y=2. So, the more reasonable scales
>would be 2x<y rather than 2x<=y (or x<=y/2).

23 has x=2, y=3 in (3y + 7x). So x <= y/2 eliminates it.

>For just octaves and major thirds, x should be tuned to an enharmonic diesis
>of 41.1 cents (2^7/5^3) and y should be 304.2 cents (5^7/2^16). Hence the
>ideal ratio y/x is 7.4. Setting y=7 and x=1 gives 28-equal, and y=15, x=2
>gives 59-equal.
> Sure enough, both have very good major thirds.

That's great! One _minor_ nit, (pun intended), according to Scala 2^7/5^3
(41.1c) is called a minor diesis, and according to
http://home.earthlink.net/~kgann/Octave.html
an enharmonic diesis is 525/512 = 3*5^2*7/2^9 (43.4c).

Can't find a name for the 5^7/2^16 (304.2c) minor third. It needs one. The
"double-diesis-flat third"? "Breed's minor third"?

So the corresponding intervals relating to fifths are 3^2/2^3 (203.9c) the
major whole tone (or Pythagorean whole tone), and 2^8/3^5 (90.2c) the
Pythagorean semitone (or Pythagorean limma). The ideal t/s ratio is
therefore 2.26 and thus we have 53-tET with excellent fifths where t = 9
and s = 4.

I find it interesting that the diatonic scale falls out of this formula for
ET's (EDO's) with reasonable fifths, N = 5T + 2s where T >= 0 and T/9 <= s
<= 6T/7. i.e. one definition of the diatonic scale might be "that scale
having the maximum notes, with two different step sizes arranged maximally
evenly, which can be constructed in every equal division of the octave that
has reasonable fifths". Does any other scale with any number of notes fit
this description.

This suggests the question of whether the analogous thing for major thirds
might be of interest. i.e. "that scale having the maximum notes, with two
different step sizes arranged maximally evenly, which can be constructed in
every equal division of the octave that has reasonable major thirds".

So I change x to d and y to B, where B is the number of ET steps in the
approximate Breed minor thirds and d is the number of ET steps in the
approximate minor dieses. And we have N = 3B + 7d, where B >= 0 and 0 <= d
<= B/2.

So, analogous to the diatonic TTsTTTs we get this weird decatonic scale
that goes
dBddBdddBd.
Just as the diatonic can also be constructed as a chain of 6 fifths (3T+s
each), the above can also be constructed as a chain of 9 major thirds (B+2d
each). Notice that ddB, dBd and Bdd are all major thirds above.

In some such ET's we have the degenerate case where d = 0 and the scale
becomes merely BBB. The simplest non-degenerate case is in 13-tET where B =
2 and d = 1. Of the familiar ET's that also have reasonable fifths, the
ones that support a nondegenerate version are:
19-tET B=4 d=1 T=3 s=2 (B+2d = 2T)
22-tET B=5 d=1 T=4 s=1 (B+2d = 2T-1)
26-tET B=4 d=2 T=4 s=3 (B+2d = 2T)
29-tET B=5 d=2 T=5 s=2 (B+2d = 2T-1, B=T, d=s)
31-tET B=8 d=1 T=5 s=3 (B+2d = 2T)
34-tET B=9 d=1 T=6 s=2 (B+2d = 2T-1)
37 and above.

Consider the weird decatonic as cast in 19-tET

C C# Db D D# Eb E E# F F# Gb G G# Ab A A# Bb B Cb(C)
* * * * * * * * * * (*)

Here's a 5-limit lattice of it

E# Cb
C# G#
A E
C G
Ab Eb
E# Cb

10 triads with 10 notes.

Paul E., you might want to examine its available harmonies in 22-tET. Does
it have any hope of being accepted melodically in any of them?

> Oh, the wonders of numbers!

Indeed.

Regards,
-- Dave Keenan
http://dkeenan.com

🔗David J. Finnamore <dfin@xxxxxxxxx.xxxx>

David C Keenan wrote:
> I find it interesting that the diatonic scale falls out of this formula for
> ET's (EDO's) with reasonable fifths ...

I like the designation EDO as far as it goes. Is it in use
among tuners generally? One advantage of it is that it
applies equally well (_there's_ a pair of puns!) to octaves
that are not strictly 2:1, such as the stretched tunings
used on pianos sometimes, and the near octaves proposed for
use with 19t-ET and such.

We all run into the problems of vagueness and
inappropriateness in the term ET from time to time when:

1) the interval being divided is other than 2:1 frequency
2) the inherent scales do not represent a tempering of any
previously recognized set of intervals

The drawbacks I can see with "EDO" are:

1) the seemingly inevitable association of the term "octave"
with the ratio 2:1; "octave" comes from the root word for
"eight" and so carries irrelevant connotations for ETs that
don't support diatonic scales.
2) it only applies to divisions of the, um, you-know-what

What about a generalized term EDI (Equal Divisions of the
Interval)? In addition to being preceded by the number of
divisions, it could be followed by an integer, ratio, or
decimal (or symbol) to indicate divisions of n(:1), n:m, or
a non-rational interval, respectively.

Thus, dear old 12t-ET would be expressed 12 EDI 2
19 equal divisions of 3:1 would be 19 EDI 3
7 equal divisions of 3:2 would be 7 EDI 3:2
31 equal divisions of pi:1 would be 31 EDI pi

88 CET and the like are perfectly clear as is.

Would this conflict with anyone's previously established
conventions? It would sure make things clearer on paper (or
monitor :-) from my perspective.

David

🔗D.Stearns <stearns@xxxxxxx.xxxx>

[Dave Keenan TD 228. 17]
>As an approximation of a 4:5 major third, I would say that anything outside of 9:11 (347.4c, 39c narrow) to 7:9 (435.1c, >49c wide) is definitely not a candidate! I'd prefer to halve those margins. Say -20 to +25 cents.

While my computer was down I took some time to go over some old piles of scribbling... One of the things I came upon was a pretty simple way to work with these types of (N+1/D+1 to N-1/D-1) interval borders (or "margins"). Using Dave's 4:5, -20 or +25 cents as an example (and calling 4:5 by generic numerator over denominator terms, in this case D:N), let {([x2*y)*2]+1}:{[(x1*y)*2]+1}, where x2 and x1 are [D-(D/a)/b] and [N-(D/a)/b], a and b are 'weights*' (2 and 1 in this 4:5 example**), and y is [(logN-logD)*(12/log2)], be the "-20 margin," and {[(x1*y)*2]-1}:{[(x2*y)*2]-1} be the "+25 margin..." This would give a ~365� to ~410� 4:5 interval border.

Dan

_______________
*i.e., a simple way to alter D*_1,_2,...+1:N*_1,_2,...+1 and D*_1,_2,...-1:N*_1,_2,...-1 a bit...

**If D:N=2:3 and a=2.4, the 1/35th of an octave fifth border of 5 and 7e is approximated with a difference of -.01025�.

🔗A440A@aol.com

>[Dave Keenan TD 228. 17]
>
>>As an approximation of a 4:5 major third, I would say that anything outside
>of 9:11 (347.4c, 39c narrow) to 7:9 (435.1c, >49c wide) is definitely not
>a candidate! I'd prefer to halve those margins. Say -20 to +25 cents.

Greetings,
I suppose that the spectra will actually determine the "lousiness" of the
third( see Sethares) in any given application. The historical record that
shows us that since the Werckmiester rules were in place, (1681) the syntonic
comma, (21.5 c) seems to be the limit of acceptable tempering for this
interval. Not to say that it can't be used wider with today's ears,
expectation, and equipment, but I find with my acoustic customers, the 21
cent M3 pushes them about as far as they can harmonically go. (yes, the
dim4ths tuned in some of the latter meantones were used musically, but they
didn't stay in vogue very long.)
In my tuning work, I have learned that there is a dissonance acceptance
curve to the customer base. Whereas I would convert approx. 1/3 of new
listeners to "other than ET tuning" when using Thomas Young's temperament of
1799 for the first departure, I have now found that the latter "Victorian"
style of tuning, which usually limits the F#-A# third to 17 cents, is much
more successful, (like 95% of first time). Temperament initiates love the
different sound, there are voices in a Steinway tuned this way that simply
don't exist in normal 12 ET.
Bob Segar just finished a week long recording project here using a tuning
that was
documented to have been in use in the Broadwood factory in 1885. There were
two pianos in the studio, and I had kept one of them in this Victorian style,
just for experiment, and they chose it over the usual tracking piano, without
knowing that the tuning was different. It sounded better on everything,
until the last day when the decision was made to transpose one cut down to
Ab. This was too much, and I yanked it back into ET over the lunch break.
The comment was, "hmm, ok, that sounds alright, but it sure takes the life
out of this piano........"
I recently had a successful presentation to the Piano Technician's Guild's
national convention on this subject, and actually sold a fair number of our
temperament CDs, (Beethoven on a Kirnberger and Young tuned Steinway D).
Numerous techs told me later that they are finally ready to begin trying the
older temperaments, and I foresee a greatly heightened appreciation for
tonality as we get the sounds out in the public.
Hope this post is not too "organic" for the number fanatics here, and I
will miss H'stick's spiritual torch. I thought his were some of the most
valuable postings on the list. Could we have done more to keep his interest?

Regards to all,
Ed Foote
Precision Piano Works
Nashville, Tn.
http://www.uk-piano.org/edfoote/index.html

🔗David Beardsley <xouoxno@xxxx.xxxx>

🔗A440A@xxx.xxx

🔗azrael <azrael@xxxx.xxxx>

I am one of those people who has a copy of Ed's "Beethoven in the
Temperaments" CD. I was amazed at how much I liked it. It soothed me and
receives lots of play in our household. There's something "truthful" about
it. Thank you, Ed.
Azrael

> I recently had a successful presentation to the Piano Technician's
Guild's
> national convention on this subject, and actually sold a fair number of our
>temperament CDs, (Beethoven on a Kirnberger and Young tuned Steinway D).
>Numerous techs told me later that they are finally ready to begin trying the
>older temperaments, and I foresee a greatly heightened appreciation for
>tonality as we get the sounds out in the public.
> Hope this post is not too "organic" for the number fanatics here, and I
>will miss H'stick's spiritual torch. I thought his were some of the most
>valuable postings on the list. Could we have done more to keep his
interest?
>
>Regards to all,
>Ed Foote
>Precision Piano Works
>Nashville, Tn.
>http://www.uk-piano.org/edfoote/index.html
>
>--------------------------- ONElist Sponsor ----------------------------
>
>ONElist: your connection to like-minds and kindred spirits.
>
>------------------------------------------------------------------------
>You do not need web access to participate. You may subscribe through
>email. Send an empty email to one of these addresses:
> tuning-subscribe@onelist.com - subscribe to the tuning list.
> tuning-unsubscribe@onelist.com - unsubscribe from the tuning list.
> tuning-digest@onelist.com - switch your subscription to digest mode.
> tuning-normal@onelist.com - switch your subscription to normal mode.
>
>
>
>

🔗David Beardsley <xouoxno@xxxx.xxxx>

A440A@aol.com wrote:

> From: A440A@aol.com
>
> Greetings,
> Inre my kudos to Neil Haverstick, David Beardsley writes:
>
> >Other than organizing the tuning list cd, what did he do that was so
> valuable?
>
> The emotional aspects of tuning and performance were common themes with
> Neil, and were often lost in the tidal waves of numbers and diagrams,
> diamonds and charts. If a scientist confronts the human/emotional facets of
> music, there is little there that he can measure or quantify, so the
> "value" is going to be invisible.
> The emotions that are the goal of music are ineffable, they come from a
> spiritual world which the scientist, by definition, is unable enter. Perhaps
> this explains why, to some, Neil Haversticks postings had little value, while
> to others, his words were stimulative and reassuring at the same time.

Ah. OK. I thought his comments were kind of the lite side,I didn't feel he had
anything really say about the emotional
aspects of tuning intervals. I always thought it was kind
of strange that he would go on about this when he was so into
equal temperaments. I don't know what his recent views
on JI are, but he used to say it wasn't for him because he
*had to modulate*.

If one is going to discuss emotions in music, they really
shouldn't write off JI. A man full of contradictions...

--
* D a v i d B e a r d s l e y
* xouoxno@virtulink.com
*
* J u x t a p o s i t i o n N e t R a d i o
* M E L A v i r t u a l d r e a m house monitor
*
* http://www.virtulink.com/immp/lookhere.htm