Pythagoras scale Vs. 12-note scale

Hi all! =)

Well, this is my first letter to this group and it's
mainly why I joined the mailing list in the first place. It's
the issue of the best tuning system - or what pitches
the ear /really/ wants to hear. I'm sorry this post
is ~10k big, but I had to make sure I made every point.

What I'm trying to do is take a completely objective
approach and give evidence for and against the two
most likely candidates which are:
The even tempered scale (I'll call it 12-note from now on)
(2^(x/12)), where x=0 to 12. Which makes the notes
powers of the number: 1.059463094359...

and the:
Pythagoras system or perfect fifth scale.
(((3^n)/(2^(n+m))) - notes C, G, D, A etc.
or ((4^n)/(3^(n+m))) - notes F, Bb, Eb, Ab etc.
(where m is a fudge factor to keep the ratio between
1 and 2)

By an amazing mathematical coincidence, these two
methods of obtaining the notes of the scale are extremely
close - for most intents and purposes, pitches are no more
than 1% out between the two systems. In fact, so close,
that it's almost fruitless to try and listen to hear which one
sounds better.

What is assumed (and for now which I'm arguing
against), is that the 12 note system has always been
an admittedly good but imperfect 'compromise' when
put up against the original Pythagoras pitch ratios (see
far end of doc for a list of these pitches numerically).
I'm wondering if there's any truth in the possibility that
the 12-note system contains the perfect pitches after all,
and therefore /not/ a compromise.
In other words I'm wondering if the two notes say...
(for example) D# and Eb - are, for all intents and
purposes the same (or should be the same) note - giving
only 12 possible 'golden' notes.

I have researched all over the net to find more information,
but have found few clear explantions on the two systems
and their quirks and 'failures'.

Anyway, here are all the issues I've come across in my
search for logic:

*********************
*********************Pythagoras System - is it flawed?

*******Issue number 1: 'Unsavoury pitches'
[case for attack (...ing Pythagoras)]
I said earlier that the most any pitch was different in the
two systems was by about 1% and was therefore very hard to
tell apart... OK, well this applies to the
immediate 5ths like F, G and D etc, but if you progress in
5ths along the scale....
C,G,D,A,E,B,F#,C#,G#, D#... (let's stop at) A#.
This pitch ratio (1.80203) for example sounds out of
place in any tune and has a discrepancy of 1.1% when
compared to the 12-note's version (1.7818).
It gets (generally) worse if you go further out aswell.....
which leads us to the dreaded 'wolf' note.....................

[case for defense] <see defense for: 'and he cried "Wolf" '>

*******Issue number 2: 'and he cried "Wolf" '

...[case for attack]
The infamous 'wolf' note (or B# - 1.01364 (and massive
1.3% discrepancy compared to root C)) clashes if it's played
in any song. The point here is that if /this/ note sounds
wrong, then all the notes that precede it must also be
wrong (to an increasingly less, but still prelavent degree).

...[case for defense]
I imagine the possible defense Pythagoras fans would use
is that the more 'exotic' notes later on (including the wolf
note and beyond) clash, but only if played with the 'simpler'
notes like G, D and A etc. (and they'd argue that good tunes
and melodies don't contain these two extremes in any decent
counterpoint/chord combinations/progressions.)

*******Issue number 3: 'But why stop at the wolf note....'
...[case for attack (...ing pythagoras)]
Notes obtained from pythag's scale are related to the last
note in the sequence.
Because of this, you could theoretically go on forever with
supposedly infinitely 'useful' notes. The 12 note system
is more definitive about just using the 'golden' 12 notes for
all tune possibilites.

...[case for defense]
Ah, but the simpler (pythagoras) notes in the beginning are
the most useful and thus are more likely to be used in any
decent melody. Each note afterwards gets more and more
'exotic', and therefore should be used less often.
This argument is related to the 'wolf' note business.

*******Issue number 4: 'oh, so you're saying ~1.498 =G ? '
...[case for attack (...ing 12-note)]
A nice method of attack against the 12 note system is the
pure note G (perfect fifth) which seems to require a pure number
like.... hmmm..... 1.5?? It isn't of course. 12-note G = ~1.4983.
1.498 seems to be unsound because it means that extra low
frequency beat patterns (when played with the note C) are
created and it is thus imperfect.

...[case for defense]
These 'beats' are negligible and/or possibly completely irrelevant.
For example:
Pythag C = 1.000000
Pythag E = 1.265625 (81/64)
The waveform for the note E (in the pythagoras scale!!) for
example repeats 81 times before the sound reloops exactly.
This gives way to many extra frequencies when played
with C etc. The note B has to repeat 243 times and the note C#
has to repeat a whopping 2187 (!) times before that reloops too...

Even if extra unheard 'beats' are a by-product (which is silly
anyway when you're looking at 2187 repeats before the loop),
this does not have to imply that the pitch is slightly 'off'.
Maybe 1.498 is the way the real 'perfect 5th' or G is meant to
sound.

Besides, if ratios were so good, then the number 1.66666 (5/3)
would sound good (when obviously it sounds too low for the
note A (1.68189)). This lends more weight to the argument
that the pythagoras/'just tuning' systems are actually spurious.

The final thing to say is that irrational numbers like e, pi and
the square root of 0.5 (~0.7071....) also seem too complex
to be useful, but are in fact very important in mathematics.

**************************************************
**************************************************

Inherent positive traits (non-arguable):

12 note system:
A scale composed up of any value /other/ than 12 in
the formula 2^(x/12), sounds dreadful.
I am very much for the theory that the number 12 is a very
special number. Every single number replacing x in the formula
gives a possibly perfect sounding note. All important pitches
covered and no extra added flawed notes. Too much of a
coincidence surely?
Try other possibilites like 2^(x/16) or 2^(x/8) or 2^(x/10) or
whatever, and you'll get some (possibly) perfect notes, some
'OK-ish' notes and also some bad (wasted) ones.

Pythagoras system:
Despite the questionable notes later on in the series, the pythag
pitches start off incredibly well with seemingly perfect pitches
just like the 12 note system. Using the various powers of 3/4,
2/3 and their various factors/powers also seems an appealing
idea, mathematically.

=================================

I know this text is a comparison on the 12-note system vs. Pythagoras,
but here's a quick mention of the 'just intonation' system (which is
related to the pythagoras system in the sense that they are ratios too).

5/3=1.6666 : Sounds very strange for the note A. Too high though of course for G#.
7/6=1.1666 : too low for D# (1.1892) and too high for D (1.1224)
7/4=1.75 : too low for A# (1.7818) and too high for A (1.6818)
8/7=1.1428 : too high for D
etc. etc...

All in all, I think the 12-note (equal tempered) scale has being
given a raw deal and would appreciate any feedback on where
I've made a good point or where I've gone wrong.
(See end of this text for pitch ratios btw)

Lastly, I hear that some of the more exotic scales contain 50 or
more notes. hmmmm, I bet that if ever any 2 given notes (taken from
the pool of 50 in this case) sound good together, that's because
they're probably close to the notes from the 12-note/Pythag scale
anyway ;-) Either this or...
......the (surplus) 'off' pitches are played in between the quarter
beats (or for sound effect) where they are less prominent and
therefore don't intrude. "I know, let's make an instrument with
70,000 note pitches! - better than even 70!" ;)
In other words, I believe scales with more than 12 notes are
redundant - at least in terms of producing decent sounding chords.
Only the 12 notes are essential (apart from for sound slides,
sound fx and possibly 8th/16th 'in-between' beats - all of which
aren't related to chords and counterpoint, but are more used as
effects or devices in music). It's hard to prove this, as I know
so many of you probably think that scales with more than 12
notes are valid (and obviously I'm not 100% sure myself even),
but maybe here's some evidence: Rarely do you hear tunes in many
channels of rich melodic counterpoint done in scales with more
notes than the Pythagoras/12-note scales.

Many thanks,

Daniel (dspwhite@email.com)

--------------------------
12-note pitches:

C = 1.0 (1.00000)
C#/Db = 1.05946 (0.94387)
D = 1.12246 (0.89089)
D#/Eb = 1.18921 (0.84089)
E = 1.25992 (0.79370)
F = 1.33484 (0.74915)
F#/Gb = 1.41421 (0.70710)
G = 1.49831 (0.66741)
G#/Ab = 1.58740 (0.62996)
A = 1.68189 (0.59460)
A#/Bb = 1.78179 (0.56123)
B = 1.88775 (0.52973)

Pythagoras' pitches:

B# 531441/524288 (1.01364)
E# 177147/131072 (1.35152)
A# 59049/32768 (1.80203)
D# 19683/16384 (1.20135)
G# 6561/4096 (1.60180)
C# 2187/2048 (1.06787)
F# 729/512 (1.42382)
B 243/128 (1.89843)
E 81/64 (1.26562)
A 27/16 (1.6875 )
D 9/8 (1.125 )
G 3/2 (1.5 )
C * 1 * (1 )
F 4/3 (1.33333)
Bb 16/9 (1.77777)
Eb 32/27 (1.18518)
Ab 128/81 (1.58024)
Db 256/243 (1.05349)
Gb 1024/729 (1.40466)
Cb 2048/2187 (0.93644)
Fb 8192/6561 (1.24859)
Bbb 32768/19683 (1.66478)
Ebb 65536/59049 (1.10985)
Abb 262144/177147 (1.47981)
Dbb 524288/531441 (0.98654)

"Out there somewhere is a mathematical
formula for creating the perfect music"

Visit www.mp3.com/soundburst for catchy
melodic music :)

Make a difference, help support the relief efforts in the U.S.
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In-Reply-To: <GONGAPMDBBBEKBAA@mailcity.com>
Daniel White wrote:

> Well, this is my first letter to this group and it's
> mainly why I joined the mailing list in the first place. It's
> the issue of the best tuning system - or what pitches
> the ear /really/ wants to hear. I'm sorry this post
> is ~10k big, but I had to make sure I made every point.

Hello Daniel! Great to have you on board. I have some points to pick
out.

> I have researched all over the net to find more information,
> but have found few clear explantions on the two systems
> and their quirks and 'failures'.

Does that include Margo's pages?
<http://www.medieval.org/emfaq/harmony/pyth.html>

> Besides, if ratios were so good, then the number 1.66666 (5/3)
> would sound good (when obviously it sounds too low for the
> note A (1.68189)). This lends more weight to the argument
> that the pythagoras/'just tuning' systems are actually spurious.

Say what?

> Inherent positive traits (non-arguable):
>
> 12 note system:
> A scale composed up of any value /other/ than 12 in
> the formula 2^(x/12), sounds dreadful.

That's certainly arguable.

> I am very much for the theory that the number 12 is a very
> special number. Every single number replacing x in the formula
> gives a possibly perfect sounding note. All important pitches
> covered and no extra added flawed notes. Too much of a
> coincidence surely?

It's not a coincidence, but it isn't quite true. What about 1, 2 or 6?

> Try other possibilites like 2^(x/16) or 2^(x/8) or 2^(x/10) or
> whatever, and you'll get some (possibly) perfect notes, some
> 'OK-ish' notes and also some bad (wasted) ones.

Have you tried 19, 22, 29 or 31?

> 5/3=1.6666 : Sounds very strange for the note A. Too high though of
> course for G#.

It may sound very strange to you, but it's roughly where A was for a few
centuries.

> 7/6=1.1666 : too low for D# (1.1892) and too high for D (1.1224)
> 7/4=1.75 : too low for A# (1.7818) and too high for A (1.6818)
> 8/7=1.1428 : too high for D
> etc. etc...

These are what we call "7-limit ratios". As you see, they aren't anything
you expect, but can be good to play with.

> Lastly, I hear that some of the more exotic scales contain 50 or
> more notes. hmmmm, I bet that if ever any 2 given notes (taken from
> the pool of 50 in this case) sound good together, that's because
> they're probably close to the notes from the 12-note/Pythag scale
> anyway ;-)

Well, maybe. See <http://x31eq.com/schismic.htm>.

If you're trying to prove that no tuning can beat 12-equal, because no
other tuning is as close to 12-equal, well, that'll be difficult to argue
with.

> It's hard to prove this, as I know
> so many of you probably think that scales with more than 12
> notes are valid (and obviously I'm not 100% sure myself even),

I suppose the best example I have to hand is
<http://x31eq.com/magicpump.mp3>.

> but maybe here's some evidence: Rarely do you hear tunes in many
> channels of rich melodic counterpoint done in scales with more
> notes than the Pythagoras/12-note scales.

Some of Gesualdo's music requires more than 12 notes, and certainly isn't
Pythagorean.

Graham

[Daniel White wrote:]
>Hi all! =)

Hi, Daniel. Welcome to the list!

>Well, this is my first letter to this group and it's
>mainly why I joined the mailing list in the first place. It's
>the issue of the best tuning system - or what pitches
>the ear /really/ wants to hear.

Hmmm... are you focusing more on individual pitches, or on the intervals
they form? In my listening experience (highly subjective, of course),
the latter is more important than the former.

Are you familiar with converting ratios to cents and back? If not,
I would encourage you to consider this. See, for example,

/tuning/topicId_4284.html#4284

>Inherent positive traits (non-arguable):
>12 note system:
>A scale composed up of any value /other/ than 12 in
>the formula 2^(x/12), sounds dreadful.
>I am very much for the theory that the number 12 is a very
>special number. Every single number replacing x in the formula
>gives a possibly perfect sounding note. All important pitches
>covered and no extra added flawed notes. Too much of a
>coincidence surely?
>Try other possibilites like 2^(x/16) or 2^(x/8) or 2^(x/10) or
>whatever, and you'll get some (possibly) perfect notes, some
>'OK-ish' notes and also some bad (wasted) ones.

Er... there are many other equal tempered scales that are very good.
See Joe Monzo's huge list, for example,

http://www.ixpres.com/interval/dict/eqtemp.htm

(and if you poke around in Joe's dictionary, I'm sure you'll find a
wealth of tuning info of all sorts)

>I believe scales with more than 12 notes are
>redundant - at least in terms of producing decent sounding chords.

What is your criterion for what makes a chord "decent" or "not decent"?
Have you, may I ask, experienced the sound of a major third, in a
harmonic timbre, tuned to exactly 1.25 frequency ratio, as compared to
the 12-tET major third and/or the Pythagorean major third? To my ear,
the latter two options are pretty much "not decent", now that I'm used
to a justly tuned 4:5.

The 12-tET scale has many well-known strengths, which have been
discussed on this list at great length. It also has many well-known
weaknesses, starting, perhaps, with its very wide major third.

If you are hoping to make new converts to 12-tET, this list may not be
a good place to do it ;-> as it is pretty much a haven _from_ the 12-tET
world all around us.

I think you may find much of interest on this list, however. Again,
welcome.

JdL

--- In tuning@y..., "Daniel White" <soundburst@l...> wrote:

/tuning/topicId_29489.html#29489

> >
> All in all, I think the 12-note (equal tempered) scale has being
> given a raw deal

Huh?? I can think of at least a couple of people who've used it...

_________ _______ ______
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:
> --- In tuning@y..., "Daniel White" <soundburst@l...> wrote:
>
> /tuning/topicId_29489.html#29489
>
> > >
> > All in all, I think the 12-note (equal tempered) scale has being
> > given a raw deal
>
> Huh?? I can think of at least a couple of people who've used it...
>
> _________ _______ ______
> Joseph Pehrson

Hardy-haw-haw! That was a good one, Joseph. Welcome to the list,
Daniel. If you're open-minded, there's an awful lot you can learn
here. I sure have had my head opened pretty wide at times.

Just a quick note. If you think 1% accuracy is an adequate criterion
for good intonation, this is not a healthy environment to hang around
in very long if you wish to hold onto that idea. After all, a whole
cotton-pickin' semitone in 12-tET is roughly only a 6% difference
(1.059463094359...as you posted).

1% is therefore (extrapolating linearly for a quick approximation)
about 1/6 semitone or somewhere around 16.67 cents. That's considered
pretty far out around here and I for one have absolutely NO trouble
discriminating that even melodically, not to mention harmonically!

If you're an intonationally accurate musician (the only kind I
consider worth listening to), then 1% is a lot. If you're not, why
bother to discuss intonation unless you want to improve it?

--- In tuning@y..., "Daniel White" <soundburst@l...> wrote:

> Well, this is my first letter to this group and it's
> mainly why I joined the mailing list in the first place. It's
> the issue of the best tuning system - or what pitches
> the ear /really/ wants to hear.

Your question presupposes that there is a best tuning system. The
first thing that needs to be discussed is whether this is true, and I
for one think it is false.

> What I'm trying to do is take a completely objective
> approach and give evidence for and against the two
> most likely candidates which are:
> The even tempered scale (I'll call it 12-note from now on)
> (2^(x/12)), where x=0 to 12. Which makes the notes
> powers of the number: 1.059463094359...

> and the:
> Pythagoras system or perfect fifth scale.

Why do you think these are the most likely? They both sound out of
tune to me when playing diatonic music, where my ears prefer a
flatter fifth.

Hi all!

...Wow...! Well, you've all certainly given me
food for thought. There are so many things
I want to say and reply to each of you, but
for now I'll resist until I've looked
at those links and thought things through
a bit. I also have a program which can
create a pitch at any frequency I like, So I
can test out a few things etc.

Many thanks for each response! :)
Look forward to writing again soon!

Chyrio,

Daniel (dspwhite@email.com)

(/me tries to keep as open a mind as possible ;)

Make a difference, help support the relief efforts in the U.S.
http://clubs.lycos.com/live/events/september11.asp

> From: "Daniel White" <soundburst@lycos.com>
> Subject: Re: Pythagoras scale Vs. 12-note scale
>
> Hi all!
>
> ...Wow...! Well, you've all certainly given me
> food for thought. There are so many things
> I want to say and reply to each of you, but
> for now I'll resist until I've looked
> at those links and thought things through
> a bit. I also have a program which can
> create a pitch at any frequency I like, So I
> can test out a few things etc.

Hi Daniel and welcome,

I started a reply and canned it since I realized
so much of what I was going to say would be said
by others and probably better.

Tune-em-up-and-listen-to-em is absolutely what
you want to do.

I too saw your post as somehow reaching a
conclusion based on some very constrained
assumptions. You are correct that if the only
thing you wanted to have "in tune" was pure
fifths and fourths, then 12 is an extremely
compact system and has no glitches. If you can
tolerate its approximations to other intervals,
as billions do daily, then you can choose the
easy path and stay with it.

But if you aren't satisfied with its
approximations for the music you want to do
(whether new compositions or old) then you may
want to look elsewhere. Probably the main thing
is, as JdL pointed out, that the major third
is generally treated as a consonance over the
past 500 years of Western music and the 12et
version of it is very jangly. Flexible pitched
instruments will adjust the pitches such that
the harmony is in tune (fairly close to beat
free).

Similarly, if you are interested in resources
that are approximated very badly or not at all
in 12eq, as are many on this list, then you
must look elsewhere. Lots of world and folk
musics use notes that just aren't on
the piano.

As an example you mentioned, 7/6 is basically
not represented in 12tet, as its nearest
approximation is nearly 40 cents sharp and
is also used for the more traditional "minor
third". You said "7/6 is too flat for D#"
without saying what D# was supposed to be
used for, is D# a fifth up from G#, a major
third up from B, or the top of an F#
diminshed-seventh? In what key?

The fact that there is one D# which
is equal to Eb and its good for everything
is a feature of 12tet, but it comes at the
cost that, if you would like D# to be close
to 7/6 (as it is in 31tet), you don't have it.

This is only touching the base of approximating
JI with ets, which is one vigorously discussed
approach. But there are people who say, "why not
just have a palette of in-tune notes and extend
the pallette as far as you need for a
composition". Search for Harry Partch, Kraig
Grady (or Anaphoria), or the Just Intonation
Network for more background information on
this approach in Just Intonation (and
David Dotys Just Intonation Primer is a good
introduction to this material).

You could also pick an et for interesting
structural properties similar to those of 12.
As you point out, 12 is a small and readilly
divisable number (1, 2, 3, 4, 6). It does not
have 5, 7, 8, 9 etc... Are these interesting
capabilities? We have a member who has done
a lot of work in 20tet. 24tet adds 8et as a
subset to the familiar divisions of 12. 72tet
gives you gobs of near Just goodies, plays
in-tune with standard instruments and includes
8 and 9 et as new subsets.

And there are an infinite number of other
ways to get involved with a 'found' or
'determined' tuning which you can't get
in 12tet.

Have fun!

Bob Valentine

Again, a warm welcome, Daniel, and congrats on a very informative and
hospitable introduction to tuning, Mr. Valentine!

Sincerely,

Bob Wendell

--- In tuning@y..., Robert C Valentine <BVAL@I...> wrote:
> > From: "Daniel White" <soundburst@l...>
> > Subject: Re: Pythagoras scale Vs. 12-note scale
> >
> > Hi all!
> >
> > ...Wow...! Well, you've all certainly given me
> > food for thought. There are so many things
> > I want to say and reply to each of you, but
> > for now I'll resist until I've looked
> > at those links and thought things through
> > a bit. I also have a program which can
> > create a pitch at any frequency I like, So I
> > can test out a few things etc.
>
> Hi Daniel and welcome,
>
> I started a reply and canned it since I realized
> so much of what I was going to say would be said
> by others and probably better.
>
> Tune-em-up-and-listen-to-em is absolutely what
> you want to do.
>
> I too saw your post as somehow reaching a
> conclusion based on some very constrained
> assumptions. You are correct that if the only
> thing you wanted to have "in tune" was pure
> fifths and fourths, then 12 is an extremely
> compact system and has no glitches. If you can
> tolerate its approximations to other intervals,
> as billions do daily, then you can choose the
> easy path and stay with it.
>
> But if you aren't satisfied with its
> approximations for the music you want to do
> (whether new compositions or old) then you may
> want to look elsewhere. Probably the main thing
> is, as JdL pointed out, that the major third
> is generally treated as a consonance over the
> past 500 years of Western music and the 12et
> version of it is very jangly. Flexible pitched
> instruments will adjust the pitches such that
> the harmony is in tune (fairly close to beat
> free).
>
> Similarly, if you are interested in resources
> that are approximated very badly or not at all
> in 12eq, as are many on this list, then you
> must look elsewhere. Lots of world and folk
> musics use notes that just aren't on
> the piano.
>
> As an example you mentioned, 7/6 is basically
> not represented in 12tet, as its nearest
> approximation is nearly 40 cents sharp and
> is also used for the more traditional "minor
> third". You said "7/6 is too flat for D#"
> without saying what D# was supposed to be
> used for, is D# a fifth up from G#, a major
> third up from B, or the top of an F#
> diminshed-seventh? In what key?
>
> The fact that there is one D# which
> is equal to Eb and its good for everything
> is a feature of 12tet, but it comes at the
> cost that, if you would like D# to be close
> to 7/6 (as it is in 31tet), you don't have it.
>
> This is only touching the base of approximating
> JI with ets, which is one vigorously discussed
> approach. But there are people who say, "why not
> just have a palette of in-tune notes and extend
> the pallette as far as you need for a
> composition". Search for Harry Partch, Kraig
> Grady (or Anaphoria), or the Just Intonation
> Network for more background information on
> this approach in Just Intonation (and
> David Dotys Just Intonation Primer is a good
> introduction to this material).
>
> You could also pick an et for interesting
> structural properties similar to those of 12.
> As you point out, 12 is a small and readilly
> divisable number (1, 2, 3, 4, 6). It does not
> have 5, 7, 8, 9 etc... Are these interesting
> capabilities? We have a member who has done
> a lot of work in 20tet. 24tet adds 8et as a
> subset to the familiar divisions of 12. 72tet
> gives you gobs of near Just goodies, plays
> in-tune with standard instruments and includes
> 8 and 9 et as new subsets.
>
> And there are an infinite number of other
> ways to get involved with a 'found' or
> 'determined' tuning which you can't get
> in 12tet.
>
> Have fun!
>
> Bob Valentine

--- In tuning@y..., Robert C Valentine <BVAL@I...> wrote:

> This is only touching the base of approximating
> JI with ets, which is one vigorously discussed
> approach. But there are people who say, "why not
> just have a palette of in-tune notes and extend
> the pallette as far as you need for a
> composition". Search for Harry Partch, Kraig
> Grady (or Anaphoria), or the Just Intonation
> Network for more background information on
> this approach in Just Intonation (and
> David Dotys Just Intonation Primer is a good
> introduction to this material).

These are all practicioners of strict JI. Of course, extended
meantone is another example of this approach, with many advocates in
the 16th century and today, and has the advantage that the standard
nomenclature for note-names is perfectly applicable and useful.

Meantone being only one example of a "Middle Path" between strict JI
and ETs.

Hi all,

Graham said,

>Does that include Margo's pages?
><http://www.medieval.org/emfaq/harmony/pyth.html>

Thanks for this.
I noticed at the url:
http://www.medieval.org/emfaq/harmony/pyth2.html
...that it made a thing of the sequence:
Eb Bb F C G D A E B F# C# G#

What's interesting is how it goes 3 fifths below C
and 8 fifths above to make the scale.
Why not 6 - 6 equally both ways?
Also it made a point of how Eb and G# clash and make
a 'wolf' note. If this is the case, then I can only
assume that all the other notes (in relation to each
other) must /also/ be off (to lesser, but still
prelavent degrees).
Plus the major third of the pythag system is even
sharper than the 12-note version, which is a bad
thing for many of us (see later).

Of course, the reason why they devised the 12-note
system was to stop certain note clashes... but maybe
these 'clashes' would have sounded OK if they used
the proper harmonic notes (even if they do need to
be 'double' flat/sharp notes). (I'm assuming a
very versatile (like computer) instrument that
easily has all the pythagorean notes accessible.)

>> Inherent positive traits (non-arguable):
>>
>> 12 note system:
>> A scale composed up of any value /other/ than 12 in
>> the formula 2^(x/12), sounds dreadful.

>That's certainly arguable.

ummm... ok :) Maybe it is. See comments later for my
thoughts on this.

>> I am very much for the theory that the number 12 is a very
>> special number. Every single number replacing x in the formula
>> gives a possibly perfect sounding note. All important pitches
>> covered and no extra added flawed notes. Too much of a
>> coincidence surely?

>It's not a coincidence, but it isn't quite true. What about 1, 2 or 6?

1, 2, 6 (and 4) don't cover all the notes though. 6
covers the whole tone scale though if you like to
make all your music in whole tones ;) Note the part
saying 'All important pitches covered' :)

>> Try other possibilites like 2^(x/16) or 2^(x/8) or 2^(x/10) or
>> whatever, and you'll get some (possibly) perfect notes, some
>> 'OK-ish' notes and also some bad (wasted) ones.

> Have you tried 19, 22, 29 or 31?

Hmmm... well, I've heard a few (see later for my
comments), but I have a question. In any given tune
(with 31 seperate notes), how many of those are
used overall? (only count those that are used
in structural chords - not any 'decoration' of
any sort)

Like I said before, maybe the best notes in these
unorthodox scales are those that are the ones
closest to 12-note pitches anyway. Others would
clearly be off, especially those that are right
in between semitones (i.e. the dreadful 'in-between
semi-tone' or quarter-tone notes ;)

I imagine that these 19, 22, 29 and 31 based
scales are mostly used in atonal music, which
don't necessarily rely on melody, sensible
chords or chord progression, but rather rely on
'textural colour' and what-not to create its 'tune' :)

>> 5/3=1.6666 : Sounds very strange for the note A. Too high though of
>> course for G#.

> It may sound very strange to you, but it's
> roughly where A was for a few centuries.

Wow - they never even knew what they were missing
out on ;)

Also, John said,

>>or what pitches the ear /really/ wants to hear.

>Hmmm... are you focusing more on individual pitches, or on the intervals
>they form? In my listening experience (highly subjective, of course),
>the latter is more important than the former.

Yup, the intervals.

>> It's hard to prove this, as I know
>> so many of you probably think that scales with more than 12
>> notes are valid (and obviously I'm not 100% sure myself even),

> I suppose the best example I have to hand is
> <http://x31eq.com/magicpump.mp3>.

erm... 'interesting' piece =) What do you think of
it aesthetically? Can't each of those notes be
replaced with similar off pitch notes taken from
completely random scales like say... 2^(x/177.3)
and with the tune still being able to give its
'effect' across?

This certainly seems to be in the realm of atonal
music where the emphasis is on the 'textural'
sound rather than any harmony.

I looked at a few more tunes in these unorthodox
scales at:
ftp://ftp.io.com/pub/usr/hmiller/music/

..."jack" for example:
This known tune has been changed to fit the new
scale, but (perhaps) any additional notes that
sound good (which aren't present in the original
tune) only sound good because they are practically
the same pitches as the 12-note ones (which are very
few of them btw) and are part of a chord that at
that very moment is inherently 12-note. Sorry - that
was a bit of a mouthful :)

The "Porcupine adventure" sounds as though it could
be very good if it was played in the 12-note scale
for instance :)
And the Kaltan tune (which I quite liked) was
atmospheric, but alas, was mostly 12-note sounding,
and for notes which weren't, I can't help feeling
the tune would benefit from a total change to the
12-note (or similar) scale.

>>I believe scales with more than 12 notes are
>>redundant - at least in terms of producing decent sounding chords.

>What is your criterion for what makes a chord "decent" or "not decent"?

Ah - this is the fun bit :)
There are obviously clashing chords which /do/ have
their place in music like C, F, F#, and B which could
be (say...) appropriate for a scary movie, but... try
these out on your music keyboard a sec.. ;)

C#, F, C, Eb F# Bb
...or this beauty ;)
D, F#, B, Eb, F, C.

Too many notes? ok then, here's 5.
E, Ab, C#, F and Bb
It's very hard to make a bad chord with just 4 notes because most
of them could fit in /somehow/ to a tune (i.e. resolve somehow),
but I've tried:
F, A, C#, Bb. ......or maybe..... C, Eb, A, C#
hmm, that was harder than coming up with a /good/
chord ;)

All these above chords don't just simply clash,
they also manage to be extremely dull, ambiguous
and generally crud :)

OK, now take this stunner: C#, F#, C, Eb, G, Bb...
awful huh? But just lower the C and G so that C
becomes B and G becomes F#. A bit better
dontcha think? :)

To ease the pain of those ear-aching earlier
chords, here are two of my fave chords using
six notes.

C and deep C in the bass followed by Bb C D F G and A
or:
C and deep C in the bass followed by Bb C# E G and A

Once again I could be mistaken, but in terms of
the scales with more than 12 notes, almost any
chord seems plausible since they're 'random'
sounding anyway, so any chord ends up like a
'mish-mash' of sound - with one chord effectively
no worse than another.

But for the (IMO) meaningful 12 note scale,
there definitely seems to be a concept of what
chords sound good and what chords sound bad,
at least in terms of how the chord should
progress, but probably also as a single chord in
itself.

I'm not sure if there's a mathematical formula for
finding the best chords or the best chord
/continuations/ (which is dependent on what else
is happening in the tune, so is a very complex
issue), but I do know that certain
chords/combinations are better than others.

>The 12-tET scale has many well-known strengths, which have been
>discussed on this list at great length. It also has many well-known
>weaknesses, starting, perhaps, with its very wide major third.

hmmm... it's (too?) wide at 1.2599 but not
quite as wide as the pythagoras one at ~1.265.
See later for my comments on the major third.
What are its other main (alleged) weaknesess?

>Have you, may I ask, experienced the sound of a major third, in a
>harmonic timbre, tuned to exactly 1.25 frequency ratio, as compared to
>the 12-tET major third and/or the Pythagorean major third? To my ear,
>the latter two options are pretty much "not decent", now that I'm used
>to a justly tuned 4:5.

It's interesting that you have gotten used
to the new pitch. I'm curious, do you think
you could ever get used to the 'old' 12-note E
again?... and if so, would you be missing
anything that you appreciated in the 1.25
(5/4) E ?

Has anyone here gone the other way from 1.25
to 1.26?

I tried out the pitches myself as a chord (C,
E, G), using a modulating sample so I wouldn't
be biased by the 'beat' effect. And the
results are inconclusive.
At one point, the 1.26 E sounded better, but
then the 1.25 E seemed to sound better later.

I also tried the note E in a sequence:
C..D..E..F..E..D..C
And the 12-note E seemed to win. 1.25 E
seemed a bit flat when compared to F, but I'm
not 100% sure. I didn't have too many worries
about how the /F/ or /d/ should be, because
12-note F is very close to 4/3 (1.33333), and
12-note D is close to 9/8 (1.125) so those
wouldn't ruin the test in any way.

>If you are hoping to make new converts to 12-tET, this list may not be
>a good place to do it ;-> as it is pretty much a haven _from_ the 12-tET
>world all around us.

Argh - foiled ;) - (well I can try ;7
Justly tuned D, E, F and G are 9/8, 5/4, 4/3
and 3/2 respectively.
What's A and B? Or are those pitches only
available in the pythagoras/12-note tunings?

>I think you may find much of interest on this list, however. Again,
>welcome.

Ta :D

Also, Joseph and Bob said:

>> > All in all, I think the 12-note (equal tempered) scale has being
>> > given a raw deal
>>
>> Huh?? I can think of at least a couple of people who've used it...
>>
>> _________ _______ ______
>> Joseph Pehrson

>Hardy-haw-haw! That was a good one, Joseph. Welcome to the list,
>Daniel. If you're open-minded, there's an awful lot you can learn

:D

>here. I sure have had my head opened pretty wide at times. Just a
>quick note. If you think 1% accuracy is an adequate criterion
>for good intonation, this is not a healthy environment to hang around

I was referring in terms of the biggest difference
between the Pythagoras pitches and the 12-note
pitches. At that amount, the difference is very
discernable - agreed.

>in very long if you wish to hold onto that idea. After all, a whole
>cotton-pickin' semitone in 12-tET is roughly only a 6% difference
>(1.059463094359...as you posted). 1% is therefore (extrapolating
>linearly for a quick approximation)
>about 1/6 semitone or somewhere around 16.67 cents. That's considered
>pretty far out around here and I for one have absolutely NO trouble
>discriminating that even melodically, not to mention harmonically!

Yes, I can distinguish a difference of about 5
cents I guess, maybe less. And obviously, measurements
should be made in much finer increments than that
(which they are).
Btw, for the record: JustlyTuned E = 1.25 and Pythag
E = 1.265. That's a difference of 1.2%. Do any
people think the 1.265 pitch for the major third
is how it should sound? (it's even higher than
12-note E!)

Also Gene said:

>> Well, this is my first letter to this group and it's
>> mainly why I joined the mailing list in the first place. It's
>> the issue of the best tuning system - or what pitches
>> the ear /really/ wants to hear.

> Your question presupposes that there is a best tuning system.
> The first thing that needs to be discussed is whether this is
> true, and I for one think it is false.

Even if the 12-note system isn't perfect (which
I'm still not sure about), then maybe there's a
way to dynamically alter the pitches of a tune,
as the tune actually progresses. For example, all
the pitches change as soon as the tune has changed
key. The 12 pitches change back again when you
return to the original key etc.

> and the:
> Pythagoras system or perfect fifth scale. Why do you think these
> are the most likely?

Well, I used both of them mainly for the sake
of argument, but perhaps they are the most
mathematically elegant systems. Mind you, I heard
something about using the prime numbers or PI
which seemed interesting...

>They both sound out of tune to me when playing
>diatonic music, where my ears prefer a flatter fifth.

Do you mean the 'flatter fifth' as in 3/2 ?
Remember that the 12-note 5th (1.498...) is
actually flatter than the just/pythag 5th...

Also, Bob said:

>I too saw your post as somehow reaching a
>conclusion based on some very constrained
>assumptions. You are correct that if the only
>thing you wanted to have "in tune" was pure
>fifths and fourths, then 12 is an extremely
>compact system and has no glitches. If you can

Well, 12-note 4ths and fifths aren't (allegedly)
perfectly in tune (unlike Just tuning which
(allegedly) is perfect), unless you mean that
they're close enough...

>tolerate its approximations to other intervals,
>as billions do daily, then you can choose the
>easy path and stay with it.

This is my point - maybe they're not approximations
after all. I see no absolute proof that they are,
though I'm willing to be proved wrong :)

>You said "7/6 is too flat for D#"
>without saying what D# was supposed to be
>used for, is D# a fifth up from G#, a major
>third up from B, or the top of an F#
>diminshed-seventh? In what key?

Well, umm.. I meant it as an only pitch - one
of the 12 'golden' notes as it were. So I
could have just as easily said Eb instead
of D# (or 2^(3/12)).

>This is only touching the base of approximating
>JI with ets, which is one vigorously discussed
>approach. But there are people who say, "why not
>just have a palette of in-tune notes and extend
>the pallette as far as you need for a
>composition"

Ah - this is what I meant when I said
dynamically altering the pitches as you go
along, or having as many pitches as you need to.
I think that /if/ the 12-note system isn't
perfect, then this could be the perfect 'scale'
after all. It would need probably need a computer
though to help with the dynamic pitches.

Phew, that's about all I have to say for now :)

Cya all,

Daniel (dspwhite@email.com)

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

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--- In tuning@y..., "Daniel White" <soundburst@l...> wrote:
> Hi all,
>
> Graham said,
>
> >Does that include Margo's pages?
> ><http://www.medieval.org/emfaq/harmony/pyth.html>
>
> Thanks for this.
> I noticed at the url:
> http://www.medieval.org/emfaq/harmony/pyth2.html
> ...that it made a thing of the sequence:
> Eb Bb F C G D A E B F# C# G#
>
> What's interesting is how it goes 3 fifths below C
> and 8 fifths above to make the scale.
> Why not 6 - 6 equally both ways?

Thinking of "C" as a center is a very recent phenomenon -- much more
recent than Medieval music.

> Also it made a point of how Eb and G# clash and make
> a 'wolf' note. If this is the case, then I can only
> assume that all the other notes (in relation to each
> other) must /also/ be off (to lesser, but still
> prelavent degrees).

What do you mean? The only stable consonances in medieval music were
fifths and fourths. All the other fifths and fourths in this 12-tone
Pythagorean scale would have been perfect -- only Eb-G# would clash.

> Plus the major third of the pythag system is even
> sharper than the 12-note version, which is a bad
> thing for many of us (see later).

But a good think for medievalists like Margo Schulter . . . major
thirds were considered unstable and had to resolve in a particular
way (outwards to fifths) in Medieval music. Their inherent dissonance
and wideness only help make this effect more powerful.
>
> Of course, the reason why they devised the 12-note
> system was to stop certain note clashes... but maybe
> these 'clashes' would have sounded OK if they used
> the proper harmonic notes

I'm not sure how using the proper harmonic notes amounts to making
clashes sound OK rather than just eliminating the clashes
altogether . . . hmm, perhaps you're focusing closely on the 12-tone
Pythagorean system and ignoring the historical reality of extended
Pythaogean systems, especially in singing.
>
> Like I said before, maybe the best notes in these
> unorthodox scales are those that are the ones
> closest to 12-note pitches anyway. Others would
> clearly be off, especially those that are right
> in between semitones (i.e. the dreadful 'in-between
> semi-tone' or quarter-tone notes ;)

Depends on the context and the cultural conditioning of the listener.
Consider that Arabic music uses many of these quarter-tone notes, and
then consider that they can sound harmonically pleasing in chords
like 8:9:10:11:12 (the 11 is a "quarter-tone".

> I imagine that these 19, 22, 29 and 31 based
> scales are mostly used in atonal music, which
> don't necessarily rely on melody, sensible
> chords or chord progression, but rather rely on
> 'textural colour' and what-not to create its 'tune' :)

On the contrary, 19 and 31 were used as early as the 16th and 17th
centuries, and are favorites of _tonal_ microtonalists today. On the
other hand, atonal microtonalists are attracted to systems such as 24-
tET and 10.7669-tET.
>
> The "Porcupine adventure" sounds as though it could
> be very good if it was played in the 12-note scale
> for instance :)

Not sure if I'm to take you as joking because of the "smiley", but if
this is anything like Herman Miller's Mizarian Porcupine Overture, it
relies very strongly on the properties of 15-tET, and wouldn't work
at all in 12-tET -- certain chord progressions would end up a
semitone away from where they started! For another example of this
type of progression, go to

http://www.ixpres.com/interval/monzo/blackjack/blackjack.htm

scroll down to the bottom, and listen to Graham Breed's Blackjack
Pump progression.

> And the Kaltan tune (which I quite liked) was
> atmospheric, but alas, was mostly 12-note sounding,
> and for notes which weren't, I can't help feeling
> the tune would benefit from a total change to the
> 12-note (or similar) scale.

All of us have this reaction when we're first exposed to non-12
music -- but trust me, this feeling goes away -- it's just your
acculturation speaking.
>
> Once again I could be mistaken, but in terms of
> the scales with more than 12 notes, almost any
> chord seems plausible since they're 'random'
> sounding anyway,

You obviously haven't spent much time trying to make nice music in
alternative tunings! Well, welcome to the land where dozens of us (at
least) have done so, and we'll all tell you the same thing: you're
dead wrong!
>
> But for the (IMO) meaningful 12 note scale,
> there definitely seems to be a concept of what
> chords sound good and what chords sound bad,
> at least in terms of how the chord should
> progress, but probably also as a single chord in
> itself.

This is just as much true of 19- and 31-tET, as many musicians in the
Renaissance could have told you. And it's certainly true of most ET
scales. The ET scale where you might have the most trouble finding
any chords that sound better than others might be 11-tET, and maybe
18-tET.
>
> I'm not sure if there's a mathematical formula for
> finding the best chords

Well, we've thrown some around in the past, and generally speaking,
it has a lot to do with how well you're approximating _simple
ratios_. While 12-tET is awfully good at this, other ETs are better,
the most commonly mentioned being 19-, 22-, 31-, 41-, 53-, and 72-tET.
>
> >The 12-tET scale has many well-known strengths, which have been
> >discussed on this list at great length. It also has many well-
known
> >weaknesses, starting, perhaps, with its very wide major third.
>
> hmmm... it's (too?) wide at 1.2599 but not
> quite as wide as the pythagoras one at ~1.265.
> See later for my comments on the major third.
> What are its other main (alleged) weaknesess?

It's not capable of approximating the ratios of 7 (7:4, 7:5, 7:6),
with the possible exception of 7:5. Many composers experimenting with
Just and other-non-12-tET systems have found these ratios of 7 very
musically useful, in a new and initially "alien", but ultimately very
beautiful, way.
>
> I tried out the pitches myself as a chord (C,
> E, G), using a modulating sample so I wouldn't
> be biased by the 'beat' effect. And the
> results are inconclusive.
> At one point, the 1.26 E sounded better, but
> then the 1.25 E seemed to sound better later.

Give yourself a lot of time to listen, sleep on it, and listen again.
Get yourself to the point where you can hear the difference between
the two without looking -- and then take note of the _qualitative_
difference between the two.
>
> I also tried the note E in a sequence:
> C..D..E..F..E..D..C
> And the 12-note E seemed to win.

Well of course it did -- what did you do (or not do) to the other
pitches?

> 1.25 E
> seemed a bit flat when compared to F, but I'm
> not 100% sure.

It would sound flat because you're very used to 100-cent half steps.
In the couple of centuries when major thirds were tuned at or near
4:5, scales were tuned to _meantone temperament_ (please read up on
this), and half steps a bit larger where what everyone was accustomed
to hearing in their scales.

> I didn't have too many worries
> about how the /F/ or /d/ should be, because
> 12-note F is very close to 4/3 (1.33333), and
> 12-note D is close to 9/8 (1.125) so those
> wouldn't ruin the test in any way.

They do ruin the test. The great inequality between the whole step
from C to D, and the whole step from D to E, ruins the test. In
meantone temperament, these two whole steps would be the same size.
>
> Argh - foiled ;) - (well I can try ;7
> Justly tuned D, E, F and G are 9/8, 5/4, 4/3
> and 3/2 respectively.
> What's A and B? Or are those pitches only
> available in the pythagoras/12-note tunings?

Just intonation does not provide a fixed pitches for each traditional
note name. If you wish to think in terms of traditional note names,
you might want to focus on meantone temperament rather than just
intonation for the time being.

Side note: you simply must stop focusing on pitches and start
focusing on intervals, or else this whole subject will be "out-of-
focus" to you forever!
>
> Yes, I can distinguish a difference of about 5
> cents I guess, maybe less. And obviously, measurements
> should be made in much finer increments than that
> (which they are).
> Btw, for the record: JustlyTuned E = 1.25 and Pythag
> E = 1.265.

My friend, please stop saying E and start saying "major third". It
going to make clear communication a lot easier.

> That's a difference of 1.2%. Do any
> people think the 1.265 pitch for the major third
> is how it should sound?

In Medieval music before 1420 AD, it's how it should sound. In
Renaissance music, definitely go for the 1.25.

> Even if the 12-note system isn't perfect (which
> I'm still not sure about), then maybe there's a
> way to dynamically alter the pitches of a tune,
> as the tune actually progresses.

This procedure has been perfected to a very fine degree by our own
John deLaubenfels. See his pages at adaptune.com.

But one still might want to make music using more than 12 distinct
pitches per octave. Listen, for example, to my piece TIBIA, the third
piece at

http://stations.mp3s.com/stations/140/tuning_punks_a-i.html

This uses all (or at least most) of the pitches in 22-tone equal
temperament. Listen to it a few times until you get used to the
language -- and please forgive my sloppy performance.
>
> Ah - this is what I meant when I said
> dynamically altering the pitches as you go
> along, or having as many pitches as you need to.
> I think that /if/ the 12-note system isn't
> perfect, then this could be the perfect 'scale'
> after all. It would need probably need a computer
> though to help with the dynamic pitches.

Not really -- the Fokker keyboard with 31 tones per octave is easily
playable by a human and has been used to perform early as well as
modern music.

From: Paul Erlich

| --- In tuning@y..., "Daniel White" <soundburst@l...> wrote:
|
| > Eb Bb F C G D A E B F# C# G#
| >
| > What's interesting is how it goes 3 fifths below C
| > and 8 fifths above to make the scale.
| > Why not 6 - 6 equally both ways?
|
| Thinking of "C" as a center is a very recent phenomenon -- much more
| recent than Medieval music.

I've gone by D for over a decade now, while considering Dorian the "central"
mode. That's probably what they did back in yonder days.

~DaW~

_________________________________________________________
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[I wrote:]
>>The 12-tET scale has many well-known strengths, which have been
>>discussed on this list at great length. It also has many well-known
>>weaknesses, starting, perhaps, with its very wide major third.

[Daniel White:]
>hmmm... it's (too?) wide at 1.2599 but not
>quite as wide as the pythagoras one at ~1.265.
>See later for my comments on the major third.
>What are its other main (alleged) weaknesess?

Several, depending upon one's perspective. It lacks different "color"
from one key to another, unlike meantone tunings or well temperaments.
It is poor for 7-limit intervals.

May I bug you again about learning cents? ;-> The 12-tET major third
is 400 cents; a just 1.25 is 386.31 cents, and that error, 13.69 cents,
is easier (IMO) to visualize than frequency ratios are.

[JdL:]
>>Have you, may I ask, experienced the sound of a major third, in a
>>harmonic timbre, tuned to exactly 1.25 frequency ratio, as compared to
>>the 12-tET major third and/or the Pythagorean major third? To my ear,
>>the latter two options are pretty much "not decent", now that I'm used
>>to a justly tuned 4:5.

[Daniel:]
>It's interesting that you have gotten used
>to the new pitch. I'm curious, do you think
>you could ever get used to the 'old' 12-note E
>again?... and if so, would you be missing
>anything that you appreciated in the 1.25
>(5/4) E ?

Some list members prefer scales that are not heavily consonant (i.e.,
don't stress low-number JI ratios). I'm never going back, I don't
think.

>Has anyone here gone the other way from 1.25
>to 1.26?

I'm sure some have, since when it comes to taste everyone is different.

>This is my point - maybe they're not approximations
>after all. I see no absolute proof that they are,
>though I'm willing to be proved wrong :)

One can certainly say that 12-tET is "perfect"; all of its alleged
weaknesses can be called strengths, if they suit one's taste. I think
it's natural for all of us to come into this group with a bit of a chip
on our shoulder (_I_ did; I was shocked to find that 4:5:6:7 dominant
7ths weren't universally accepted), and certainly none of us has to give
up our favorite tuning techniques, but over time by learning about what
others do, and why they do it, we become more understanding of other
tuning approaches.

>Ah - this is what I meant when I said
>dynamically altering the pitches as you go
>along, or having as many pitches as you need to.
>I think that /if/ the 12-note system isn't
>perfect, then this could be the perfect 'scale'
>after all. It would need probably need a computer
>though to help with the dynamic pitches.

That's my game; see www.adaptune.com. If you have a MIDI sequence or
two you'd like to hear with adaptive tuning, I'd be glad to do it.

JdL

In-Reply-To: <9ra0rc+v1oq@eGroups.com>
Paul wrote:

> > I'm not sure if there's a mathematical formula for
> > finding the best chords
>
> Well, we've thrown some around in the past, and generally speaking,
> it has a lot to do with how well you're approximating _simple
> ratios_. While 12-tET is awfully good at this, other ETs are better,
> the most commonly mentioned being 19-, 22-, 31-, 41-, 53-, and 72-tET.

I don't think we're close to explaining the chords Daniel gave in
12-equal. All we can do is select chords as "consonances" and point to
singularities in tuning. Choosing between dissonances is beyond us.
Fortunately, I've never found optimizing consonances means losing good
dissonances, so I can erect an SEP field around the whole subject.

Graham

Daniel White:
> >> Try other possibilites like 2^(x/16) or 2^(x/8) or 2^(x/10) or
> >> whatever, and you'll get some (possibly) perfect notes, some
> >> 'OK-ish' notes and also some bad (wasted) ones.

Me:
> > Have you tried 19, 22, 29 or 31?

Daniel:
> Hmmm... well, I've heard a few (see later for my
> comments), but I have a question. In any given tune
> (with 31 seperate notes), how many of those are
> used overall? (only count those that are used
> in structural chords - not any 'decoration' of
> any sort)

The only example I have is Vicentino, and that only on paper. He uses 24
notes, which could work as two keyboards tuned with a usual 12 note
meantone (which isn't quite how he writes it). But the chords are
consistently 5-limit, to the extent that any exceptions are taken to be
typos.

I have played with neutral third scales. You might find something at
<http://x31eq.com/pelog.mid>, I can't remember.

> Like I said before, maybe the best notes in these
> unorthodox scales are those that are the ones
> closest to 12-note pitches anyway. Others would
> clearly be off, especially those that are right
> in between semitones (i.e. the dreadful 'in-between
> semi-tone' or quarter-tone notes ;)

It depends on what you mean by "best". And anyway, it's putting the cart
before the horse. 12-equal came into use by offering approximations to
the 12 notes already in use, which were *very* close to a subset of
31-equal.

> I imagine that these 19, 22, 29 and 31 based
> scales are mostly used in atonal music, which
> don't necessarily rely on melody, sensible
> chords or chord progression, but rather rely on
> 'textural colour' and what-not to create its 'tune' :)

This has already been answered.

Daniel:
> >> It's hard to prove this, as I know
> >> so many of you probably think that scales with more than 12
> >> notes are valid (and obviously I'm not 100% sure myself even),

Me:
> > I suppose the best example I have to hand is
> > <http://x31eq.com/magicpump.mp3>.

Daniel:
> erm... 'interesting' piece =) What do you think of
> it aesthetically? Can't each of those notes be
> replaced with similar off pitch notes taken from
> completely random scales like say... 2^(x/177.3)
> and with the tune still being able to give its
> 'effect' across?

I listened to it again to see if I still like it. And I do! I hope it
works best in the tuning(s) I chose for it. And I don't hear it as off
pitch, although theoretically most of it is.

> This certainly seems to be in the realm of atonal
> music where the emphasis is on the 'textural'
> sound rather than any harmony.

You think so? I don't hear it like that at all. I mean, it's based on a
repeating chord sequence that ends on a major triad. How tonal to you
want?

> Ah - this is the fun bit :)
> There are obviously clashing chords which /do/ have
> their place in music like C, F, F#, and B which could
> be (say...) appropriate for a scary movie, but... try
> these out on your music keyboard a sec.. ;)
>
> C#, F, C, Eb F# Bb
> ...or this beauty ;)
> D, F#, B, Eb, F, C.
>
> Too many notes? ok then, here's 5.
> E, Ab, C#, F and Bb
> It's very hard to make a bad chord with just 4 notes because most
> of them could fit in /somehow/ to a tune (i.e. resolve somehow),
> but I've tried:
> F, A, C#, Bb. ......or maybe..... C, Eb, A, C#
> hmm, that was harder than coming up with a /good/
> chord ;)
>
> All these above chords don't just simply clash,
> they also manage to be extremely dull, ambiguous
> and generally crud :)

Boy, it's a long time since I tuned up 12-equal. It's strange how out of
tune the chromatic stuff sounds. Anyway, yes, these chords are truly
awful.

> OK, now take this stunner: C#, F#, C, Eb, G, Bb...
> awful huh? But just lower the C and G so that C
> becomes B and G becomes F#. A bit better
> dontcha think? :)

Yes, that's a lot better.

> To ease the pain of those ear-aching earlier
> chords, here are two of my fave chords using
> six notes.
>
> C and deep C in the bass followed by Bb C D F G and A
> or:
> C and deep C in the bass followed by Bb C# E G and A

Excellent piano chords! Yes, this is what 12-equal is good at.

> Once again I could be mistaken, but in terms of
> the scales with more than 12 notes, almost any
> chord seems plausible since they're 'random'
> sounding anyway, so any chord ends up like a
> 'mish-mash' of sound - with one chord effectively
> no worse than another.

This goes completely against my experience. There's a big range of chords
out there, with 12-equal's sitting somewhere in the middle. You don't
get anything in 12-equal as radiant as a 4:5:6 tuned major triad. Or,
for that matter, a 5:6:7 diminished triad. (Yes, there are open fifths,
but they sound bare in comparison to 2:3:4:5 in 72-equal.) And nothing
remotely as sour as a car-horn triad (1/1:9/7:3/2 or 14:18:21) or
1/1:9/8:11/9 (72:81:88??) or a quartertone cluster.

And turn the vibrato off when you try these! The beatless sound is
exactly what you're supposed to be listening for!

Graham

--- In tuning@y..., graham@m... wrote:

> The only example I have is Vicentino, and that only on paper.

Harold Fortuin performed this nicely, using a trombone sound, last
week.
>
> I listened to it again to see if I still like it. And I do! I
hope it
> works best in the tuning(s) I chose for it. And I don't hear it as
off
> pitch, although theoretically most of it is.

What do you mean, theoretically most of it is? All I know is that,
last time I heard it, the chord progression sounded nice, but it was
completely buried under distortion, poor timing, and a melody that
didn't seem to fit the chords at all . . . sorry Graham, just being
honest.
>
> > This certainly seems to be in the realm of atonal
> > music where the emphasis is on the 'textural'
> > sound rather than any harmony.
>
> You think so? I don't hear it like that at all. I mean, it's
based on a
> repeating chord sequence that ends on a major triad. How tonal to
you
> want?

I'm afraid I can completely see where this guy is coming from. The
chord sequence is really buried in there.

On Thu, 25 Oct 2001 20:50:06 0000, "Daniel White" <soundburst@lycos.com>
wrote:

>And the Kaltan tune (which I quite liked) was
>atmospheric, but alas, was mostly 12-note sounding,
>and for notes which weren't, I can't help feeling
>the tune would benefit from a total change to the
>12-note (or similar) scale.

Aside from the tuning issues with the old Moog synthesizer, it _was_ in
12-TET. Specifically an 8-note subset of 12-TET: C Db Eb F G Ab A B.