Natural Pan Temperament with pure sine wave tones

Michael and all,

Michael said: "The other solution is to do the reverse and match your timbre to fit the scale, something I'd highly recommend as it brings you many more options."

Good idea, very good idea. I should rename my book "The Mathematics of Music that uses only pure sine wave tones". I am 87.7146% sure that my system is perfect when only pure sine wave tones are used. All bets are covered and there is nothing you could add to it.

The notes in my system are: 1/1, 15/14, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 9/5, 15/8, 2/1.

With my system (using pure sine wave tones) all intervals wider than an octave are good. The only (37) intervals that are bad (if the tonic is 'E' or 1/1 and within a one octave range) are:

E/F
E/F#
F/F#
F/G
F/G#
F/A
F/A#
F/C
F/D
F#/G
F#/G#
F#/A
F#/A#
F#/C
F#/C#
G/G#
G/A
G/A#
G/C#
G/D#
G#/A
G#/A#
G#/C
G#/D
A/A#
A/B
A/D
A/D#
A/F
A#/B
A#/C
A#/C#
A#/D
A#/D#
A#/F
A#/F#
B/C
B/C#
C/C#
C/D
C/D#
C/F
C/F#
C/G#
C#/D
C#/D#
C#/F
C#/F#
C#/G
C#/A#
D/D#
D/E
D/F
D/G#
D/A
D#/E
D#/F
D#/G
D#/A
D#/A#
D#/C

If you avoid these 37 intervals when constructing a chord you can't go wrong. Michael pointed out that the 33/16 interval is clearly dissonant using a piano "voice". Using sine wave tones however the beating is obvious but to my ear not too unpleasant.

If I'm right then this is big, bigger than Pythagoras and Harry Partch rolled into one.

If you play keyboards then check out my system using the sine voice, it should be consistent. Also it seems to me that using pure sine wave tones this system is more versatile than any other.

The next step is using synthetic tones that have exactly 16 harmonics. The frequencies of the harmonics are x, 2x, 3x, 4x...up to 16x. The amplitude of each harmonic is y, y/2, y/3, y/4... up to y/16.

I hope to have some maths on this fairly soon. Hopefully these "16 harmonic" synthetic tones should be close enough to most natural tones to have an approximate system for quantifying melody and harmony for most instruments.

How all this works is explained in my book: "The Mathematics of Music" which you can download from the "Files" section on the left of your screen. Alternatively go to www.johnsmusic7.com and click the link near the bottom of the page. A tune I wrote and recorded is also available on this web site.

John.

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
> Good idea, very good idea. I should rename my book "The Mathematics of Music that uses only pure sine wave tones".

... and then be sure to add an exhaustive listing of all the pieces of music, down the ages, that have been made only using sine waves.

As to the following:

> If I'm right then this is big, bigger than Pythagoras and Harry Partch rolled into one.

Yeah, if you say so.

Jon,

I said "If I'm right".

John.

--- In tuning@yahoogroups.com, "jonszanto" <jszanto@...> wrote:
>
> --- In tuning@yahoogroups.com, "john777music" <jfos777@> wrote:
> > Good idea, very good idea. I should rename my book "The Mathematics of Music that uses only pure sine wave tones".
>
> ... and then be sure to add an exhaustive listing of all the pieces of music, down the ages, that have been made only using sine waves.
>
> As to the following:
>
> > If I'm right then this is big, bigger than Pythagoras and Harry Partch rolled into one.
>
> Yeah, if you say so.
>

> The notes in my system are: 1/1, 15/14, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 9/5, 15/8, 2/1.

What about 10/9? What about 45/32? What about 16/9? What about 16/15?

> With my system (using pure sine wave tones) all intervals wider than an octave are good. The only (37) intervals that are bad (if the tonic is 'E' or 1/1 and within a one octave range) are:
> E/F#
> F#/G#

Uh, E F# G# B? Emaj add2?

> F#/G

How about E F# G B? Em add2?

> A/B
E A B, Esus. Now you're even saying that fundamental common practice
chords are invalid.

> A/D#
No B7 allowed today?

> B/C#
Emaj6?

Anyway, the point is: your system should be describing how music
works, not how you believe it should work. For you to say that Esus is
an "invalid chord" in your system serves as a detriment to your
system, not to the Esus chord.

> If I'm right then this is big, bigger than Pythagoras and Harry Partch rolled into one.

I suggest familiarizing yourself with both Sethares' theory of
roughness-based consonance as well as Erlich's ideas on harmony
entropy before throwing out ridiculous statements like that.

-Mike

John,
If you need something to create timbres matched to your scale and not just be stuck with sine waves try this program (which Bill Sethares obviously had a hand in creating)
http://homepages.cae.wisc.edu/~sethares/software/TFSdocs/index.html

>"With my system (using pure sine wave tones) all intervals wider than an
octave are good. The only (37) intervals that are bad (if the tonic is
'E' or 1/1 and within a one octave range) are:
E/F"
The odd thing is....I don't see it as a realistic example to have a full 12 tone system (like the one you sent) and judge the consonance of intervals in the whole system. Something more realistic would be to pick out the bad intervals in, say, 7-8 tone subsets.

>"If you avoid these 37 intervals when constructing a chord you can't go
wrong. Michael pointed out that the 33/16 interval is clearly dissonant
using a piano "voice". Using sine wave tones however the beating is
obvious but to my ear not too unpleasant."
True...it's not too bad...here low critical band dissonance helps level out the high periodic dissonance.

>"If I'm right then this is big, bigger than Pythagoras and Harry Partch rolled into one."
Pythagorus comes across to me as very narrow minded; reducing everything to circles of 5ths or near-5ths. Partch is more interesting, but still seems to boil down to straight o-tonal and u-tonal relationships so far as I see it.
If you want to pick a tuning/"scale-smith" icon to compete against, I'd recommend people such as Erv Wilson, William Sethares, and Ptolemy...all of who break far away from mean-tone, 12TET, and other temperament systems based on "circle of xths" logic. Wilson and Ptolemy also turn out to both consider critical band roughness and periodicity in their models: Ptolemy's Homolon or "Smooth" system merges on 7TET intervals and thus avoid critical band "rough"-ness while using very periodic tetra-chords while Wilson's Moment of Symmetry system balances between equal-spaced critical band smoothness and the sort of slightly-jagged major/minor harmonic series stacked intervals that encourage high periodicity. To be honest though, MOS seems to fall apart so far as consonance at around 6 notes to my ears and Ptolemy's system doesn't seem to do as well as it should with making consonant chords that go between different octaves IE "G5 B5 E6". For sure,
there is lots of progress left to be made/found.

>"If you play keyboards then check out my system using the sine voice, it
should be consistent. Also it seems to me that using pure sine wave
tones this system is more versatile than any other."
In a way it is....if a tuning system fails with sine waves, it's guaranteed to fail with all other waveforms.

>"The next step is using synthetic tones that have exactly 16 harmonics.
The frequencies of the harmonics are x, 2x, 3x, 4x...up to 16x. The
amplitude of each harmonic is y, y/2, y/3, y/4... up to y/16."
Right, but I'm still worried you are going to wind up simply merging your scale ever closer to plain old diatonic-JI if you use many overtones over 5 or so in your "overtone roughness" analysis...unless you take the amplitude of overtones into account and make overtones over around 5 much lower on volume than the other ones.

>"Anyway, the point is: your system should be describing how music
works, not how you believe it should work. For you to say that Esus is
an "invalid chord" in your system serves as a detriment to your
system, not to the Esus chord."

Oddly enough, I think perhaps both of you are being too narrow-minded on this issue.
Let me sum it up this way...what makes a good system (IMVHO) includes
A) An excellent "consonant chord"-to-note ratio IE as many chords as possible per notes available to make sure it has both great harmonic flexibility and ease of use. I say ease of use since more good chords means less chances to go wrong and less chances to make mistakes in playing chords under the scale...a good scale, to me, needs less theory because the scale itself resolves many conflicts even without the need for musicians to "calculate" how to avoid sour chords and thus hit sweet ones.
B) At least 6 notes so as to be as or nearly as melodically flexible as traditional 7-tone scales...at least if you want to compete with popular music's typical 7-tone scales.

You know what...many of my scales can't play certain Western chords because they simply use different interval sets...yet they put different chords in their places so it's "changing" rather than "dropping" chords.
-------------------------------------------------------------------------------------------------------
John> If I'm right then this is big, bigger than Pythagoras and Harry Partch rolled into one.

Mike B>"I suggest familiarizing yourself with both Sethares' theory of
roughness-based consonance as well as Erlich's ideas on harmony
entropy before throwing out ridiculous statements like that."
Ah, a follow-up on my statements about Sethares' relevance to any use of instruments beside sine waves and critical band dissonance...yes Sethares work is incredibly important.
Erlich (at least to me) comes across as kind of a grandfather of Just Intonation. Again, I think any theory that doesn't take both of those into mind and balance them (plus perhaps add more levels of interpretation on top of that) is incomplete.
Also note I've referenced both of those two in many of my tips...heck, I referenced both at once when I explained why I thought that dissonance levels out around 12/11 and 11/10 and that both intervals should be considered fine for chords (about 12/11 has the lowest point between about 13/12-10/9 in Erlich's curve while about 10/9 has the lowest point in Sethares' curve so the two curves "balance each other out" within that range).

__,_._,__

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
> I said "If I'm right".

Yes, I read that correctly.

Even if you say "bigger than Partch's contribution to tuning theory" that's a whole lot of hubris. If one factors in all of Partch's other contributions, you don't stand a chance at this point.

I'm not saying your investigations aren't worthy, I just think you need some perspective.

Cheers,
Jon

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

> Erlich (at least to me) comes across as kind of a grandfather of Just Intonation.

That is just amazingly weird.

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
>
> Michael and all,
>
> Michael said: "The other solution is to do the reverse and match
> your timbre to fit the scale, something I'd highly recommend as it
> brings you many more options."
>
> Good idea, very good idea. I should rename my book "The Mathematics
> of Music that uses only pure sine wave tones".

Well, as I think I wrote, most existing acoustical instruments will be fine, since the overtone stgructure of their timbres contains many of the intervals. Renaming your book to the title above I would not call necessary - rename I would, though, since (as I also have written before) "THE mathematics of musc" sounds quite, so to say, boastful.

> I am 87.7146% sure that my system is perfect when only pure sine
> wave tones are used. All bets are covered and there is nothing you
> could add to it.
>
> The notes in my system are: 1/1, 15/14, 9/8, 6/5, 5/4, 4/3, 7/5,
> 3/2, 8/5, 5/3, 9/5, 15/8, 2/1.
>
> With my system (using pure sine wave tones) all intervals wider
> than an octave are good. The only (37) intervals that are bad (if
> the tonic is 'E' or 1/1 and within a one octave range) are:
>
<snip>
>
> If you avoid these 37 intervals when constructing a chord you can't
> go wrong. Michael pointed out that the 33/16 interval is clearly
> dissonant using a piano "voice". Using sine wave tones however the
> beating is obvious but to my ear not too unpleasant.
>

A system where you have to avoid certain intervals I would not call perfect. Anyway, this tuning, as I see it, is basically a kind of just intonation scale. It sure sounds fine - as long as you do not try to modulate to another key. For example, a "wolf" fifth between d and a, one of the known flaws of the common just intonation scale, is also present in your system. Calling this "bigger than Pythagoras and Harry Partch" is sure absolutely not appropriate.
--
Hans Straub

>    Oddly enough, I think perhaps both of you are being too narrow-minded on this issue.
>    Let me sum it up this way...what makes a good system (IMVHO) includes
> A) An excellent "consonant chord"-to-note ratio IE as many chords as possible per notes available to make sure it has both great harmonic flexibility and ease of use.  I say ease of use since more good chords means less chances to go wrong and less chances to make mistakes in playing chords under the scale...a good scale, to me, needs less theory because the scale itself resolves many conflicts even without the need for musicians to "calculate" how to avoid sour chords and thus hit sweet ones.

You and John have a unique definition of "consonance" that as far as I
know nobody else in the world subscribes to. Chords like C E F A are
used all of the time, but you guys act like they're "dissonant"
because of the beating.

Furthermore - what's wrong with beating? Beating can sound amazing in
a musical context.

What it really boils down to is that in reality there is no single
quantitative dimension of "consonance," but rather several interacting
features. But this is really a topic for another day. The point is
that if your "system" dismisses as invalid things that musicians are
already doing that sound great, then it just isn't a good system.

> B) At least 6 notes so as to be as or nearly as melodically flexible as traditional 7-tone scales...at least if you want to compete with popular music's typical 7-tone scales.

OK, so the pentatonic scale isn't a "valid system" then?

>     Ah, a follow-up on my statements about Sethares' relevance to any use of instruments beside sine waves and critical band dissonance...yes Sethares work is incredibly important.
>     Erlich (at least to me) comes across as kind of a grandfather of Just Intonation.

I don't know why you say that. If you were going to assign Paul a
"label" the first thing that comes to mind would be regular
temperament rather than JI.

>  Again, I think any theory that doesn't take both of those into mind and balance them (plus perhaps add more levels of interpretation on top of that) is incomplete.
> Also note I've referenced both of those two in many of my tips...heck, I referenced both at once when I explained why I thought that dissonance levels out around 12/11 and 11/10 and that both intervals should be considered fine for chords (about 12/11 has the lowest point between about 13/12-10/9 in Erlich's curve while about 10/9 has the lowest point in Sethares' curve so the two curves "balance each other out" within that range).

A theory doesn't necessarily needs to have an opinion on what is
"good" and "bad." Any theory that does that is going to have Godel
statement somewhere that screws it up.

I think that very "rough" intervals can be heard as harmonic, and
their "roughness" doesn't detract from the pleasantness of the chord.
And even more importantly, any theory that basically comes out of the
following train of thought:

1) The entire matter of whether a listener will like or dislike a
sound/chord/whatever can be simplistically reduced to how "consonant"
it is
2) This entire dimension of "consonance" results from a single
physical or psychoacoustic or mathematical principle
3) Musical elements that don't fit into this and are thus "dissonant"
have a very limited or no musical use

is making so many unfounded assumptions that it's going to be useless.

-Mike

> > Erlich (at least to me) comes across as kind of a grandfather of Just Intonation.
>
> That is just amazingly weird.

Hahahaha!

-Mike

On 14 April 2010 11:51, Mike Battaglia <battaglia01@...> wrote:

> You and John have a unique definition of "consonance" that as far as I
> know nobody else in the world subscribes to. Chords like C E F A are
> used all of the time, but you guys act like they're "dissonant"
> because of the beating.

C E F A is dissonant by very common definitions. If you start
classifying minor seconds as consonances, it gets hard do find any
dissonance in 12-equal.

> Furthermore - what's wrong with beating? Beating can sound amazing in
> a musical context.

What's wrong with dissonance? Dissonance can sound amazing in a
musical context.

Graham

> C E F A is dissonant by very common definitions. If you start
> classifying minor seconds as consonances, it gets hard do find any
> dissonance in 12-equal.

I have not heard too many people classify a major7 chord as a
dissonant chord. I also think that a blanket sweep pronunciation of
the minor second as a "dissonant" interval is a bit rash as well.

If you'd like an even more trivial example, how about C D E G? Would
you say that's dissonant?

> > Furthermore - what's wrong with beating? Beating can sound amazing in
> > a musical context.
>
> What's wrong with dissonance? Dissonance can sound amazing in a
> musical context.

Hold on now, I'm the one who's saying that!

-Mike

Dissonance and consonance aren't "value judgements" like good or bad, they are simply descriptions, more like "blonde or brunette".

A major 7 is called a dissonance, that's how it is classified, as is a minor second. Mostly it's a matter of traditional words, in 12-tET. Because the M3 is audibly dissonant for crying out loud.

Originally these terms were obviously mostly about whether the sonority called for further motion, resolution, within stylistic boundries. Major 7 is Ti, and it goes to Do- a major (ka-ching) element of Western music.

You can easily demonstrate that the terms dissonant and consonant were and remain mostly about resolution by playing a major tetrad, with either aug.6 or min.7! in quarter-comma meantone. It is far more harmonious and near Just, with a great deal of concordance (blending) than any 12-tET multiple sonority, but it was called a dissonance then and now.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > C E F A is dissonant by very common definitions. If you start
> > classifying minor seconds as consonances, it gets hard do find any
> > dissonance in 12-equal.
>
> I have not heard too many people classify a major7 chord as a
> dissonant chord. I also think that a blanket sweep pronunciation of
> the minor second as a "dissonant" interval is a bit rash as well.
>
> If you'd like an even more trivial example, how about C D E G? Would
> you say that's dissonant?
>
> > > Furthermore - what's wrong with beating? Beating can sound amazing in
> > > a musical context.
> >
> > What's wrong with dissonance? Dissonance can sound amazing in a
> > musical context.
>
> Hold on now, I'm the one who's saying that!
>
> -Mike
>

Also,

> C E F A is dissonant by very common definitions. If you start
> classifying minor seconds as consonances, it gets hard do find any
> dissonance in 12-equal.

I didn't say that E-F by itself wasn't dissonant, but that C-E-F-A
wasn't dissonant. There is a huge difference between the isolated
interval and its context within a chord. 7:9 is pretty dissonant by
itself, but it sounds great when you play 7:8:9:10:11:12. 81:40 sounds
awfully dissonant by itself, but I would be surprised if most people
could even tell that it wasn't 2:1 in the chord C Eb+ G Bb+ D F+ A C+.

You get the point here. To say that any chord that contains a
"dissonant" interval is itself dissonant is a bit oversimplistic.

-Mike

> Originally these terms were obviously mostly about whether the sonority called for further motion, resolution, within stylistic boundries. Major 7 is Ti, and it goes to Do- a major (ka-ching) element of Western music.
>
> You can easily demonstrate that the terms dissonant and consonant were and remain mostly about resolution by playing a major tetrad, with either aug.6 or min.7! in quarter-comma meantone. It is far more harmonious and near Just, with a great deal of concordance (blending) than any 12-tET multiple sonority, but it was called a dissonance then and now.

What I'm saying is - I just studied jazz for four years. A maj7 chord
is hardly viewed as dissonant in that circle. In fact, it's one of the
most consonant chords there is.

So when people say things like "a major 7 chord is considered
dissonant," my question is - by who? Where's the committee that gets
to decide these things? Neither I nor the musical group I've
surrounded myself with for the past four years of my life thinks of
that chord as dissonant. Nobody in the musical circle I am a part of
would really consider the major 7 chord to be in need of resolution
either. It's usually used as the I chord (and far more often than a
straight major triad).

It is possible to create music in whic a maj7 basically functions as a
consonant I chord with no need to resolve anywhere, and I've been
studying it intensely for a while now. So to just say it "is"
dissonant because it "needs" to resolve is a bit simplistic. To put it
in the form of a zen koan, what does a chord need?

However, a lot of people here come from a background in which chords
like that may be considered dissonant. OK, fine, I get it. Which goes
back to my original point: you can't just mix together all of these
different concepts of "dissonant" and "consonant" together and expect
a coherent theory to result from it. Different definitions of
consonance usually overlap somewhat, and so for casual conversation
it's usually fine to just disregard their differences. But if we're
talking about a fundamental theory of music that's going to make
Partch and Pythagoras give thumbs up signs from their graves, then
it's pointless to do that.

-Mike

> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > > C E F A is dissonant by very common definitions. If you start
> > > classifying minor seconds as consonances, it gets hard do find any
> > > dissonance in 12-equal.
> >
> > I have not heard too many people classify a major7 chord as a
> > dissonant chord. I also think that a blanket sweep pronunciation of
> > the minor second as a "dissonant" interval is a bit rash as well.
> >
> > If you'd like an even more trivial example, how about C D E G? Would
> > you say that's dissonant?
> >
> > > > Furthermore - what's wrong with beating? Beating can sound amazing in
> > > > a musical context.
> > >
> > > What's wrong with dissonance? Dissonance can sound amazing in a
> > > musical context.
> >
> > Hold on now, I'm the one who's saying that!
> >
> > -Mike
> >
>
>

"If you avoid these 37 intervals when constructing a chord you can't go
wrong. Michael pointed out that the 33/16 interval is clearly dissonant
using a piano "voice". Using sine wave tones however the beating is obvious
but to my ear not too unpleasant."

What I don't understand is the desire to create musical "training wheels".

It is as if composers are incapable of selecting notes for a chord properly.

If you want to create music mindlessly without effort there is already a
solution for you:

http://research.microsoft.com/en-us/um/redmond/projects/songsmith/

Now I shall delete this silly thread.

Chris

On Tue, Apr 13, 2010 at 8:45 PM, john777music <jfos777@...> wrote:

>
>
> Michael and all,
>
>

Yes, I understand what you mean. And there's no jazz that even comes near to treating "dissonances" as consonant to the extent I do, it's simply not possible without heavy-duty microtonality. My point was about traditional terminology. "That's just the way it is".

To give you an idea as to how much I think there is any INHERENT correctness in this terminology, let me quote a Yiddish saying:
a language is a dialect with an army and a navy.

:-)

Nevertheless is is good to be aware of different connotations and denotations, historical and current.

The idea that a m2 within a vertical sonority makes it "dissonant" on a, let's say, "physical", level, is demonstratably wrong of course.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Originally these terms were obviously mostly about whether the sonority called for further motion, resolution, within stylistic boundries. Major 7 is Ti, and it goes to Do- a major (ka-ching) element of Western music.
> >
> > You can easily demonstrate that the terms dissonant and consonant were and remain mostly about resolution by playing a major tetrad, with either aug.6 or min.7! in quarter-comma meantone. It is far more harmonious and near Just, with a great deal of concordance (blending) than any 12-tET multiple sonority, but it was called a dissonance then and now.
>
> What I'm saying is - I just studied jazz for four years. A maj7 chord
> is hardly viewed as dissonant in that circle. In fact, it's one of the
> most consonant chords there is.
>
> So when people say things like "a major 7 chord is considered
> dissonant," my question is - by who? Where's the committee that gets
> to decide these things? Neither I nor the musical group I've
> surrounded myself with for the past four years of my life thinks of
> that chord as dissonant. Nobody in the musical circle I am a part of
> would really consider the major 7 chord to be in need of resolution
> either. It's usually used as the I chord (and far more often than a
> straight major triad).
>
> It is possible to create music in whic a maj7 basically functions as a
> consonant I chord with no need to resolve anywhere, and I've been
> studying it intensely for a while now. So to just say it "is"
> dissonant because it "needs" to resolve is a bit simplistic. To put it
> in the form of a zen koan, what does a chord need?
>
> However, a lot of people here come from a background in which chords
> like that may be considered dissonant. OK, fine, I get it. Which goes
> back to my original point: you can't just mix together all of these
> different concepts of "dissonant" and "consonant" together and expect
> a coherent theory to result from it. Different definitions of
> consonance usually overlap somewhat, and so for casual conversation
> it's usually fine to just disregard their differences. But if we're
> talking about a fundamental theory of music that's going to make
> Partch and Pythagoras give thumbs up signs from their graves, then
> it's pointless to do that.
>
> -Mike
>
> > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> > >
> > > > C E F A is dissonant by very common definitions. If you start
> > > > classifying minor seconds as consonances, it gets hard do find any
> > > > dissonance in 12-equal.
> > >
> > > I have not heard too many people classify a major7 chord as a
> > > dissonant chord. I also think that a blanket sweep pronunciation of
> > > the minor second as a "dissonant" interval is a bit rash as well.
> > >
> > > If you'd like an even more trivial example, how about C D E G? Would
> > > you say that's dissonant?
> > >
> > > > > Furthermore - what's wrong with beating? Beating can sound amazing in
> > > > > a musical context.
> > > >
> > > > What's wrong with dissonance? Dissonance can sound amazing in a
> > > > musical context.
> > >
> > > Hold on now, I'm the one who's saying that!
> > >
> > > -Mike
> > >
> >
> >
>

Hi Mike,

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> I have not heard too many people classify a major7 chord as a
> dissonant chord. I also think that a blanket sweep pronunciation of
> the minor second as a "dissonant" interval is a bit rash as well.
>
> If you'd like an even more trivial example, how about C D E G? Would
> you say that's dissonant?

if a piece of music establishes major and minor triads as the basic
harmonies then I at least hear both of these chords as unstable and
in need of a resolution.

I don't believe in a simple reduction of the consonance/dissonance of
a chord to the consonance/dissonance of its' intervals. For example,
there is a dramatic difference between major and minor 7-limit
tetrads even though they have the same (consonant) interval content
but in different order.

But I wonder if at least in older jazz the intervals traditionally
classified as dissonances tend to move (when they move) to intervals
traditionally classified as consonances even when using seventh
and ninth chords and beyond. This is just a hunch, I don't know
enough about jazz to claim anything.

Kalle Aho

Dear Mike,

On 14.04.2010, at 13:17, Mike Battaglia wrote:
> What I'm saying is - I just studied jazz for four years. A maj7 chord
> is hardly viewed as dissonant in that circle. In fact, it's one of the
> most consonant chords there is.

Consonance and dissonance are no clear-cut categories like black and white. Instead, there can be different degrees of dissonance.

There exists much literature on this subject (including microtonal music), see below for a few references. Most of this literature discusses different models of consonance/dissonance degrees of dyads, but some authors also cover chords. For example, Sethares extends the notion of dissonance curves (for dyads) to dissonance surfaces (for triads, Sec. 6.8) and proposes some strait-forward formula to compute these surfaces (and higher-dimensional plots): the total dissonance is the sum of the dissonances between all simultaneously sounding partials. Concerning the major 7th chord and also the 9th chord you where mentioning, I understand these are relatively consonant because the major 7th and the 9th are actually harmonic partials, and so the sum of the dissonances is relatively low compared with more complex chords.

References

Sethares, William A. 2005. Tuning, timbre, spectrum, scale. 2nd ed. London: Springer.

Tenney, James. 1988. A History of 'Consonance' and 'Dissonance'. New York: Excelsior Music Publishing Company.

Partch, Harry. 1974. Genesis of a Music: An Account of a Creative Work, Its Roots and Its Fulfillments. 2nd ed. New York, NY: DaCapo Press.

Erlich, Paul. 2004. Harmonic Entropy. http://www.soundofindia.com/showarticle.asp?in_article_id=1905806937.

Helmholtz, Hermann von. 1980. On the Sensation of Tone. Dover Publications, orig. 1863.

Best wishes,
Torsten

--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-586219
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

John>"If you avoid these 37 intervals when constructing a chord you can't go
wrong. Michael pointed out that the 33/16 interval is clearly dissonant
using a piano "voice". Using sine wave tones however the beating is
obvious but to my ear not too unpleasant."

Chris>"What I don't
understand is the desire to create musical "training wheels".
I actually side with John here. The most often criticism I hear of micro-tonality is that "12TET is hard enough", meaning the learning curve for micro-tonal is high and many people view it as lots of extra effort for very little reward. Making things like less sour chords possible in new scales makes those scales much easier to digest for the general public. You see it as training wheels, I see it as putting full gear on when I ride my motorcycle.

>"It is as if composers are incapable of selecting notes for a chord
properly.
If you want to create music mindlessly without effort there is
already a solution for you:"
Counter-example...my brother is a professional jazz guitarist and definitely not a lazy mindless musician. He won't touch micro-tonality (and Lord knows I've tried, showing him the works of Neil Haverstick and several others on this list)...and even he gave the "12TET is hard enough" excuse. He also told me he was afraid of not being able to find good chords and having to "start from scratch". And you'd better believe he is NOT that type of person to use Microsoft Songsmith. :-)

So I think there exist two options to lessen the harshness of the micro-tonal learning curve
1) Make scales with very few ways to make sour chords and thus lessen the amount of new theory needed to learn them...what type of chords (including those not in common practice) does not matter in this so long as they are not sour.
2) Avoid making musicians learn any new theory at all and make all "common practice" 12TET chords accessible in all keys (thus indirectly forcing the scale to sound a lot like 12TET)

....and, as you might have guessed...out of these two I strongly side with 1)

Mike B>"So when people say things like "a major 7 chord is considered
dissonant," my question is - by who? Where's the committee that gets
to decide
these things?"

I think it's more a question of compositional technique than anything else.
It's the same sort of thing the makes a chord sound strong in a major key and weak in its relative minor...even with the same notes.

To perhaps over-simplify it..."predictable" chords sound consonant...if you hear a bunch of not-so-just chords and the a Just one pops out it surprises you and that surprise comes across to some people as dissonance. If you play a bunch of diminished chords and then a major one...it's the major chord that sounds "weird".

I tend to side toward voicing my "opinions" tuning and rhythm theory and not composition because composition is so subjective and often depends on "surprises" like those explain above that can make even the same type of chords change to act as consonant and dissonant depending on the context. If someone asks me "is blue or red better" I'd answer "it depends what you are trying to match it with". :-)

> C E F A is dissonant by very common definitions. If you start
> classifying minor seconds as consonances, it gets hard do find any
> dissonance in 12-equal.

CEFA is one of my favorite example chords because it uses the half-step which is supposedly un-usable in chords and does it with pretty good consonance. Indeed having a dissonant interval in a chord does not make that chord dissonant...only makes it more likely to be dissonant (IE there are very few consonant chords possible using the minor second and the fact common practice scales space the seconds at maximal distance apart to try to (not so successfully IMVHO) combat this issue is no accident).

MikeB> "I didn't say that E-F by itself wasn't dissonant, but that C-E-F-A wasn't dissonant. "
Here (as I've said before) the relatively far-spaced C E and F A intervals seem to compensate for the overly close E-F interval and draw attention away from the E-F dyad. Then again, I agree that doing something like chaining minor seconds within a chord IE C EFF# is too dense to be consonant. I think there's a sweet spot somewhere around the not-used-in-common-practice 12/11 interval where you can get things like 8:10:11:12 (with the consecutive "smaller than major second" 11/10 and 12/11 intervals) and still be consonant enough for it to count as a true chord and not just a bunch of densely arranged "neighboring tones".

Yeah here I go again :-)...I think a huge problem is that scale-smiths keep trying to get excellent chord capability out of 15/14 and 16/15 when they should break free of common-practice and make scales using 12/11 and 11/10 as "(new) minor seconds" instead. Maybe the "classical" JI minor second isn't always the best solution...

Cameron>"Dissonance and consonance aren't "value judgements" like good or bad,
they are simply descriptions, more like "blonde or brunette"."
And (muahahaha), I would gladly take the "bad" brunette anyhow. ;-)

Cameron>"A major 7 is called a dissonance, that's how it is classified, as is a
minor second. Mostly it's a matter of traditional words, in 12-tET.
Because the M3 is audibly dissonant for crying out loud."
Yes the 12TET major third is notoriously off pure. But I think/agree that says more nasty things about 12TET inaccuracy than about the major 7th chord...and also bad things about how many people throughout history have mis-interpreted consonance of chords due to problems with 12TET intervals.

>"You and John have a unique definition of "consonance" that as far as I
know nobody else in the world subscribes to. Chords like C E F A are
used all of the time, but you guys act like they're "dissonant"
because of the beating."

I NEVER said C E F A was dissonant, look at my wording'
> Let me sum it up this way...what makes a good system (IMVHO)
includes
> A) An excellent "consonant chord"-to-note ratio IE as many chords
as possible per notes available to make sure it has both great harmonic
flexibility and ease of use.
And I've used C E F A as an example of a consonant chord many times before...even used it as an example of how non-extended JI can be a flawed theory in assuming a chord so high up the series "has" to be dissonant. The re-state (for, what, the 5th time)...I think EF is dissonant but CEFA is not since the C E and F A intervals are spaced far enough that their consonance counters the EF's dissonance. Why do I avoid the 16/15 or 15/14 EF interval then and use 12/11 and 11/10 instead? Because, so far as I've found, you can't make anything with too many dissonant dyads IE a chord with EFF# work while making a chord with 10:11:12 in it work seems a lot easier to do.

>"Furthermore - what's wrong with beating? Beating can sound amazing in a musical context."
Right, but the problem is learning curve of memorizing, say, tons of new chords in order to make sure you get the level of beating you want when you want it. Hence my mention of musicians I know hating micro-tonality because of the often-taken-as-hideous learning curve.

>"What it really boils down to is that in reality there is no single quantitative dimension of "consonance, " but rather several interacting features."
Well I would say the two most established are periodicity and critical band dissonance...and if you look at my theories, I use both of those together extensively.

> B) At least 6 notes so as to be as or nearly as melodically
flexible as traditional 7-tone scales...at least if you want to compete
with popular music's typical 7-tone scales.
>"OK, so the pentatonic scale isn't a "valid system" then?"

I had said :at least if you want to compete
with popular music's typical 7-tone scales." And it's not valid as I see it in the sense of not being able to compete with typical 7-tone scales and get the public's attention on that level. It's an issue of the musical flexibility of having both a few more tones and exponentially more harmonies available from those extra tones.

>"If you were going to assign Paul a "label" the first thing that comes to mind would be regular temperament rather than JI."
I say JI because his theory of harmonic entropy consistantly hits dead-on JI ratios and the levels of consonance marked for each interval is fairly consistent with what the odd-limit consonance for those ratios is according to JI theory.

>"I think that very "rough" intervals can be heard as harmonic, and their "roughness" doesn't detract from the pleasantness of the chord."
I would counter to say they can, but only when balanced by a certain degree of consonant intervals.

>"And even more importantly, any theory that basically comes out of the following train of thought:
1) The entire matter of whether a listener will like or dislike a
sound/chord/ whatever can be simplistically reduced to how "consonant"
it is"
Not at all true, I'm simply saying they should meet a minimum level of consonance to avoid coming across as "un-calculated noise" to many listeners. Considering my standard definition for meeting "ok" consonance levels for held/sustained dyads is about 12/11 I don't see why that would make a hard standard to meet at all (many musicians say 6/5 is where consonance "stops" for dyads; I consider my definition of where dissonance really gets to people pretty liberal).

>"2) This entire dimension of "consonance" results from a single physical or psychoacoustic or mathematical principle"
I am starting to question how much people are listening to from me. I consistently name periodicity theory and critical band dissonance as basis for any theories of consonance I have...so that's two, not one. If you consider things I've discussed like mirroring around intervals and chaining/"squaring" certain super-particular intervals plus other things I've mentioned...there are several theories that translate into what makes consonance work. Ultimately yes...I believe there is an ultimate answer, but it sure as heck is NOT composed of just one principle.

>"3) Musical elements that don't fit into this and are thus "dissonant" have a very limited or no musical use"
Well, many of them DO have very limited musical use so far as sustained harmony. Within a single octave, beside C EF A, C EF B and transposed versions of those chords...how many chords can you find with half-steps that can be used as sustained chords without getting an annoyed "hey, this guy doesn't know music!" reaction from the audience?
Agreed, far as melody and neighboring tones you can get a bit more adventurous, using more dissonant chords and even "accidentals" for short "stabs" while still making the music sound confident and professional.

But I'm mainly talking about consonance so far as being used in sustained chords in harmony and, sadly, attitudes like "odd dissonances are ok" I swear are sadly those that keep much micro-tonal music from reaching the public ear and making a difference in what kinds of music theory are taught in schools.

________________________________
From: Mike Battaglia <battaglia01@...>
To: tuning@yahoogroups.com
Sent: Wed, April 14, 2010 2:51:05 AM
Subject: Re: [tuning] Natural Pan Temperament with pure sine wave tones

> Oddly enough, I think perhaps both of you are being too narrow-minded on this issue.
> Let me sum it up this way...what makes a good system (IMVHO) includes
> A) An excellent "consonant chord"-to-note ratio IE as many chords as possible per notes available to make sure it has both great harmonic flexibility and ease of use. I say ease of use since more good chords means less chances to go wrong and less chances to make mistakes in playing chords under the scale...a good scale, to me, needs less theory because the scale itself resolves many conflicts even without the need for musicians to "calculate" how to avoid sour chords and thus hit sweet ones.

You and John have a unique definition of "consonance" that as far as I
know nobody else in the world subscribes to. Chords like C E F A are
used all of the time, but you guys act like they're "dissonant"
because of the beating.

Furthermore - what's wrong with beating? Beating can sound amazing in
a musical context.

What it really boils down to is that in reality there is no single
quantitative dimension of "consonance, " but rather several interacting
features. But this is really a topic for another day. The point is
that if your "system" dismisses as invalid things that musicians are
already doing that sound great, then it just isn't a good system.

> B) At least 6 notes so as to be as or nearly as melodically flexible as traditional 7-tone scales...at least if you want to compete with popular music's typical 7-tone scales.

OK, so the pentatonic scale isn't a "valid system" then?

> Ah, a follow-up on my statements about Sethares' relevance to any use of instruments beside sine waves and critical band dissonance.. .yes Sethares work is incredibly important.
> Erlich (at least to me) comes across as kind of a grandfather of Just Intonation.

I don't know why you say that. If you were going to assign Paul a
"label" the first thing that comes to mind would be regular
temperament rather than JI.

> Again, I think any theory that doesn't take both of those into mind and balance them (plus perhaps add more levels of interpretation on top of that) is incomplete.
> Also note I've referenced both of those two in many of my tips...heck, I referenced both at once when I explained why I thought that dissonance levels out around 12/11 and 11/10 and that both intervals should be considered fine for chords (about 12/11 has the lowest point between about 13/12-10/9 in Erlich's curve while about 10/9 has the lowest point in Sethares' curve so the two curves "balance each other out" within that range).

A theory doesn't necessarily needs to have an opinion on what is
"good" and "bad." Any theory that does that is going to have Godel
statement somewhere that screws it up.

I think that very "rough" intervals can be heard as harmonic, and
their "roughness" doesn't detract from the pleasantness of the chord.
And even more importantly, any theory that basically comes out of the
following train of thought:

1) The entire matter of whether a listener will like or dislike a
sound/chord/ whatever can be simplistically reduced to how "consonant"
it is
2) This entire dimension of "consonance" results from a single
physical or psychoacoustic or mathematical principle
3) Musical elements that don't fit into this and are thus "dissonant"
have a very limited or no musical use

is making so many unfounded assumptions that it's going to be useless.

-Mike

Well, you know - if you combine all the oil colors from a paint set you get
brown. If you don't know what you are doing in music (and don't let a
computer do it for you) you will get crap.

I don't see the problem with this. This is why everyone is not a Salvador
Dali or The Beatles. Talent is involved.

If you want to convert your brother I think your best route is to come up
with a compelling riff on say a 22 edo guitar that can't be done in 12 tet.

On Wed, Apr 14, 2010 at 10:13 AM, Michael <djtrancendance@...> wrote:

>
>
> John>"If you avoid these 37 intervals when constructing a chord you can't
> go wrong. Michael pointed out that the 33/16 interval is clearly dissonant
> using a piano "voice". Using sine wave tones however the beating is obvious
> but to my ear not too unpleasant."
>
>
> Chris>"What I don't understand is the desire to create musical "training
> wheels".
> I actually side with John here. The most often criticism I hear of
> micro-tonality is that "12TET is hard enough", meaning the learning curve
> for micro-tonal is high and many people view it as lots of extra effort for
> very little reward. Making things like less sour chords possible in new
> scales makes those scales much easier to digest for the general public. You
> see it as training wheels, I see it as putting full gear on when I ride my
> motorcycle.
>
>

>"Well, you know - if you combine all the oil colors from a paint set you
get brown. If you don't know what you are doing in music (and don't let a computer do it for you) you will get crap."
I agree, but I consider tuning like a tool. If you give a professional driver an old Toyota Tercel and give an average guy a very new Ferrari with advanced traction control, automatically adjustable spoilers to increase grip on turns, anti-lock brakes...the average guy actually has a good chance of winning because there are not many ways he can go wrong and much more power available for when he does keep things under control. Someone watching would thing the average guy was a much better driver than he really is.
Meanwhile, of course, the professional driver would still have an obvious advantage if he also had the Ferrari...because he has his driving style so he can add more of what can go right "even" if the Ferrari takes care of most of what could go wrong for him.

Point is I'd be more interested to see the typical playing field changed from bad to good musicians plus a few great ones to mostly good to great musicians and a few fantastic ones. I'd like to give all "drivers" the Ferrari. :-)

>"I don't see the problem with this. This is why everyone is not a
Salvador Dali or The Beatles. Talent is involved. "
I agree talent is involved in composition IE how you use the tools. If you gave a composer tones 6-12 of the harmonic series as a scale he would be virtually guaranteed to make very consonant music, but fairly likely not to get the same level of emotion as said-above musicians. Now you might say "oh, you see, he's bad"...but then try putting him in something like 22TET (without knowing where the octa-tonic scale are) and see how much worse he sounds. Tools matter...a lot.

>"If you want to convert your brother I think your best route is to come
up with a compelling riff on say a 22 edo guitar that can't be done in
12 tet."
Cool idea...do you have a link?

Actually I'm working on just that. And I might be able to borrow a 22edo
guitar for a while to compose with. In the mean time I have successfully
used fractal tune smithy to on the fly re-tune the (really good) guitar
sounds in my Roland GR-20 whilst playing my Fender Mustang. Right now I
cant get an easily reproducible set up with FTS but I am working on it.

If I come up with something good I will be sure to spam everyone - even if
it isn't good I'll probably spam everyone anyway :-)

Chris

> >"If you want to convert your brother I think your best route is to come up
> with a compelling riff on say a 22 edo guitar that can't be done in 12 tet."
> Cool idea...do you have a link?
>
>
>

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:

> You can easily demonstrate that the terms dissonant and consonant were and remain mostly about resolution by playing a major tetrad, with either aug.6 or min.7! in quarter-comma meantone. It is far more harmonious and near Just, with a great deal of concordance (blending) than any 12-tET multiple sonority, but it was called a dissonance then and now.

And I still think its possible part of the reason people spelled an augmented sixth chord as they did was because they wanted it performed that way. The chord does date back to the Renaissance, after all.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Michael <djtrancendance@> wrote:
>
> > Erlich (at least to me) comes across as kind of a grandfather of Just Intonation.
>
> That is just amazingly weird.

Your ability to pun has been sorely missed!

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:

> If you want to convert your brother I think your best route is to come up
> with a compelling riff on say a 22 edo guitar that can't be done in 12 tet.

Try a porcupine temperament comma pump, or check with Paul Erlich, the Grandfather of Just Intonation.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

> >"If you want to convert your brother I think your best route is to come
> up with a compelling riff on say a 22 edo guitar that can't be done in
> 12 tet."
> Cool idea...do you have a link?

http://www.io.com/~hmiller/midi/porcupine-22.mid

http://www.archive.org/details/NightOnPorcupineMountain

http://www.lumma.org/tuning/erlich/decatonic-swing.mp3

"Try a porcupine temperament comma pump"

Sorry, but in my family my wife wears those. ;-)

I will chk out the links though.

CHris

On Wed, Apr 14, 2010 at 12:44 PM, genewardsmith <genewardsmith@...
> wrote:

>
>
>
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Chris Vaisvil
> <chrisvaisvil@...> wrote:
>
> > If you want to convert your brother I think your best route is to come up
> > with a compelling riff on say a 22 edo guitar that can't be done in 12
> tet.
>
> Try a porcupine temperament comma pump, or check with Paul Erlich, the
> Grandfather of Just Intonation.
>
>
>
>

Gene>"or check with Paul Erlich, the Grandfather of Just Intonation."
Lol...shoot you are never going to let me live that one down.
I can't help it the first time I asked about just-intonation vs. Sethares work in critical band theory Carl told me Erlich was the closest thing he could thing of to an authority on the "periodicity theory" behind JI. But yeah, it should have really been "Godfather" not "Grandfather"...I honestly have no clue how old Paul is, nor do I care. :-D

John here again,

Natural Pan "Temperament" is a misnomer. For years I thought that "temperament" meant a tuning system's nature or character and only discovered relatively recently that it implies some notes are 'tempered' so as to fit a scale. It implies imperfection. Interestingly the latin word 'temperamentum' means 'correct mixture'. So if you like call it 'Natural Pan Tuning'. 'NPT' for short whichever word you prefer.

There are 61 'bad' intervals (listed earlier) in NPT and not 37 (using sine waves). I said 37 because, with my system, *all* intervals that are one or two notes apart are dissonant. There are 24 of these. So you only have to learn the other 37.

With regard to chords that contain very narrow (dissonant) intervals it seems to me that no matter how much you 'dress up' the chord (adding wider and stronger intervals), the dissonance of the 'bad' interval can still be heard if you listen carefully.

Michael, it seems to me that we've been working along parallel lines: the 1/x + 1/y part of my formula refers to periodicity and the x/y part (note x<y) deals with dissonance (I think the term you use is "critical band dissonance" or something like that). If you haven't done so already take a look at my chapter on chords. This needs some minor revision but the jist of it you might find interesting.

Jon and Hans, my apologies, I couldn't resist the temptation to create a bit of a buzz.

John.

John's NPT scale>"1/1
15/14 9/8
6/5 5/4 4/3 7/5 3/2
8/5 5/3
9/5 15/8 2/1

You might find this interesting. All this time I've been mostly avoiding the 6/5 interval because it does not fit into the x/12 and x/18 harmonic series my latest scales use extensively (because they allow x/2, x/3,x/4,x/6, and x/9 type chords. But then I figured out that 6/5 forms an interval with its nearest note in my scale,10/9, of 27/25...which is incredibly close (well under 13 cents) to the relatively low limit (for a minor-second type interval) 13/12.

>"Michael, it seems to me that we've been working along parallel lines:
the 1/x + 1/y part of my formula refers to periodicity and the x/y part
(note x<y) deals with dissonance (I think the term you use is
"critical band dissonance" or something like that)."
I'm going to have to recheck this one....how exactly does the x/y part related to critical band dissonance?
I think a common loophole to watch out for is the idea smaller fractions = less critical band dissonance: a really easy counter example is the 27/25 fraction above, which has its root tones almost as far apart as 13/12. Sure things like 3/2 and 4/3 have root tones far apart and very little critical band dissonance, but so does 187/87.

John,

Now this might really make you wonder.

I found a subset of your 12-tone scale that's fairly near my latest scale.
That subset is
1/1 9/8 6/5 4/3 3/2 5/3 9/5 2/1
which is very near my latest scale version of
1/1 10/9 6/5 4/3 3/2 5/3 11/6 2/1

So I tried testing them against each other to see which chords turned out better...and some turned out better than others in each scale.
The one thing that got me about your scale was that the 9/8 and 6/5 clashed due to too much critical band dissonance (they are a merely 16/15 interval apart).

But what I came up with as a scale with the best of both scale qualities was

1/1 (10/9) 6/5 4/3 3/2 5/3 (9/5) 2/1

This scale has its smallest interval set at 27/25 which is about 13/12 and contains two 27/25 interval maximally separated just like the half steps in standard 7-tone diatonic JI.

The only thing a bit sour about it I found was the interval between 9/5 and 10/9 is about (1.234567) and the near-by relatively low-limit interval 11/9 is about 1.22222 and the other one (16/13) is at about 1.230769. So there's a bit of compromise there...I wish I could figure out a trick through which to get that interval down closer to an x/12 or lower format for better periodicity.
----------------------------
But I'm pretty sure said "combined" version of parts of our scales works better than the individual scales themselves. Within an 8 cent "error" boundary, SCALA found 101 possible chords in that scale...more than it found in any of my scales. And those likely doesn't include all the avant-garde chords with consecutive close intervals such as 10:11:12.

Amusingly, in some ways, we may indeed be along the same path....any ideas to improve this "combined" scale?

Michael,

if my formulas are correct for pure sine wave tones then the next step is to use these formulas for working out the consonance values of complex tone intervals. The first thing to do is to work out how 'strong' each harmonic is. Then when pairing each of the harmonics of two notes 'weight' each calculation according to the strength value of the weaker harmonic (the strength of the pairing is only as strong as the weaker element). I'm working on this at the moment and should have something in a day or two. This is where my theory on chords comes in (the harmonic series of one note is a 'chord').

If I'm on the right track then I will have an approximate system that covers all possible intervals for "ideal notes" (i.e. the harmonics are exactly x, 2x, 3x, 4x, 5x etc, and the 'initial' amplitudes of the harmonics (before they are paired with all the others to give 'overall' amplitudes) are exactly y, y/2, y/3, y/4 etc.). At the moment I'm just working with the first 16 harmonics which should be enough to get a rough idea. So with instruments whose timbres are close to "ideal notes" we'll have a 'rough guide' to consonance. This could be the holy grail of music theory.

John.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> John,
>
> Now this might really make you wonder.
>
> I found a subset of your 12-tone scale that's fairly near my latest scale.
> That subset is
> 1/1 9/8 6/5 4/3 3/2 5/3 9/5 2/1
> which is very near my latest scale version of
> 1/1 10/9 6/5 4/3 3/2 5/3 11/6 2/1
>
> So I tried testing them against each other to see which chords turned out better...and some turned out better than others in each scale.
> The one thing that got me about your scale was that the 9/8 and 6/5 clashed due to too much critical band dissonance (they are a merely 16/15 interval apart).
>
> But what I came up with as a scale with the best of both scale qualities was
>
> 1/1 (10/9) 6/5 4/3 3/2 5/3 (9/5) 2/1
>
> This scale has its smallest interval set at 27/25 which is about 13/12 and contains two 27/25 interval maximally separated just like the half steps in standard 7-tone diatonic JI.
>
> The only thing a bit sour about it I found was the interval between 9/5 and 10/9 is about (1.234567) and the near-by relatively low-limit interval 11/9 is about 1.22222 and the other one (16/13) is at about 1.230769. So there's a bit of compromise there...I wish I could figure out a trick through which to get that interval down closer to an x/12 or lower format for better periodicity.
> ----------------------------
> But I'm pretty sure said "combined" version of parts of our scales works better than the individual scales themselves. Within an 8 cent "error" boundary, SCALA found 101 possible chords in that scale...more than it found in any of my scales. And those likely doesn't include all the avant-garde chords with consecutive close intervals such as 10:11:12.
>
> Amusingly, in some ways, we may indeed be along the same path....any ideas to improve this "combined" scale?
>

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

> You might find this interesting. All this time I've been mostly avoiding the 6/5 interval because it does not fit into the x/12 and x/18 harmonic series my latest scales use extensively (because they allow x/2, x/3,x/4,x/6, and x/9 type chords. But then I figured out that 6/5 forms an interval with its nearest note in my scale,10/9, of 27/25...which is incredibly close (well under 13 cents) to the relatively low limit (for a minor-second type interval) 13/12.

You are in effect talking about the comma 325/324. This is tempered out in a great variety of 13-limit temperaments of various ranks; I'm fond of 46 equal so I'll suggest to start out with tuning the scale to the nearest notes of 46.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

> I found a subset of your 12-tone scale that's fairly near my latest scale.
> That subset is
> 1/1 9/8 6/5 4/3 3/2 5/3 9/5 2/1
> which is very near my latest scale version of
> 1/1 10/9 6/5 4/3 3/2 5/3 11/6 2/1

A wimpier version of "very near" than 325/324, since these are 81/80 and 55/54.

> The only thing a bit sour about it I found was the interval between 9/5 and 10/9 is about (1.234567)

It's 81/50, which is 1.62.

>"So with instruments whose timbres are close to "ideal notes" we'll have
a 'rough guide' to consonance. This could be the holy grail of music
theory."
Even though many people would call me overly optimistic about how much can be done with scales, I don't think there is a "Holy Grail". It is like tuning the engine of a gas-powered car...you are always trading "torque for horsepower" or vice-versa...and the question becomes which is more valuable for you.

As for me, I'll settle for things like edging-on-sour 5ths and bittersweet-sweet 4th and semi-sweet 3rds for the sake of sweet 2nds and minor seconds. You apparently, try to make anything a minor 3rd or over very sweet and the price you pay is often "ok" seconds and fairly rotten minor seconds. There's no Holy Grail, only smarter ways to optimize. And, sad thing is, if you think your scale is the solution to everything, very few will take you seriously.
I pose my scale as a very strong (thought not perfect) solution to purifying small intervals and causing super particular or very near super-particular relationships among them...and I strongly suggest you pose your scale as very strong at purifying intervals of minor 3rd and up.

>"This is where my theory on chords comes in (the harmonic series of one note is a 'chord')."
>"The first thing to do is to work out how 'strong' each harmonic is.
Then when pairing each of the harmonics of two notes 'weight' each
calculation according to the strength value of the weaker harmonic"

Hate to say it, but I seriously think that system is far too generic to work. How would it handle the chord of C E F A? My guess is it would call it dissonant when it's not.
Hate to sound like a Puritan...but anything which pretty much sums a chord by its "weakest link" is bound not only to fail, but to contradict critical band theory. It would almost be like trying to prove 45/43 is a just interval.
Plug all the dyads possible in a pure C E F A into your formula and then try it with a pure C D# G B...then repeat the analysis by adding values from Plompt and Llevelt's dissonance curve (which is used directly by Sethares). I strongly suspect you'll see your formula's results will contradict his...and the ears of most people (to whom C E F A is a fairly consonant chord). Please try this experiment with your formula...and then we'll see where we can go from there far as a "harmonic-series-timbre-matching" theory for your scale.
-----------------
Also you completely ignored my original question
Me> "I found a subset of your 12-tone scale that's fairly near my latest scale.
> That subset is
> 1/1 9/8 6/5 4/3 3/2 5/3 9/5 2/1
> which is very near my latest scale version of
> 1/1 10/9 6/5 4/3 3/2 5/3 11/6 2/1

I'm looking for a scale which compromises between your scale's good ability to purify wider intervals and my scale's to generate super-particular or near-super-particular small ones. it won't be a perfect scale either, of course, but it will hopefully approach being near-perfect for most situations (including chords using things like consecutive near-minor seconds, which are far as I know impossible to make consonant in 12TET)..

-Michael

Me>> I found a subset of your 12-tone scale that's fairly near my latest scale.
>> That subset is
>> 1/1 9/8 6/5 4/3 3/2 5/3 9/5 2/1
>> which is very near my latest scale version of
>> 1/1 10/9 6/5 4/3 3/2 5/3 11/6 2/1
Gene>A wimpier version of "very near" than 325/324, since these are 81/80 and 55/54.
What's 81/80 and 55/54...the difference between 10/9 and 9/8 vs. 9/5 and 11/6? By "very near", I meant they share exact notes except for those two, and those two are more like an 1/8th note away rather than something like a 1/4 note which, even standing alone and not in harmony, is tolerably different in pitch.

Gene> The only thing a bit sour about it I found was the interval between 9/5 and 10/9 is about (1.234567)

>It's 81/50, which is 1.62.
I actually meant going from 9/5 to the octave plus 10/9. When I analyze chords I also take into account what happens to chords which reach both before and past 2/1.

Me> "I found a subset of your 12-tone scale that's fairly near my latest scale.
> That subset is
> 1/1 9/8 6/5 4/3 3/2 5/3 9/5 2/1
> which is very near my latest scale version of
> 1/1 10/9 6/5 4/3 3/2 5/3 11/6 2/1"

I started noticing something bizarre about the new version of the scale.
There were TONS of symmetries. Mirroring in multiple directions. In fact, I built that scale into a 9-tone scale called the "infinity scale" (because it forms continuous loops around itself in a sense) where every tones reflects around 2/1 and 3/2. No I didn't force this...it just jumped out at me after I made the scale and fine tuned it by ear. Look at the results from SCALA

9-tone "Infinity" scale

PITCH INTERVAL
1/1 16/15
16/15 25/24
10/9 27/25
6/5 10/9
4/3 9/8
3/2 10/9
5/3 27/25
9/5 25/24
15/8 16/15
2/1 16/15

Also note, even in the very basic 3:4:5:6 chord 4/3 and 5/3 mirror around both 3/2 and 2/1.

Does anyone else agree this make the idea of mirroring tones around simple intervals very suspect to being related to making scales easier for the mind to find patterns in and hear as pleasant (even if you hate my scale, lol)?

-Michael

> > If you'd like an even more trivial example, how about C D E G? Would
> > you say that's dissonant?
>
> if a piece of music establishes major and minor triads as the basic
> harmonies then I at least hear both of these chords as unstable and
> in need of a resolution.

Sure, because the composer is using little musical tricks to mess with
your mind and get you to hear that chord as "unfamiliar" or "out of
place" or something. I get it. What I'm saying is, how exactly that
plays out is going to be very dependent on

1) The specific tricks that the composer has used
2) The person's cultural and musical background
3) The person's state of mind and tolerance for new things
4) All kinds of subconscious stuff that Freud would have a field day with

That being said, to just write C-D-E-G off as an inherently dissonant
chord is a bit rash.

> I don't believe in a simple reduction of the consonance/dissonance of
> a chord to the consonance/dissonance of its' intervals. For example,
> there is a dramatic difference between major and minor 7-limit
> tetrads even though they have the same (consonant) interval content
> but in different order.

How about the consonance/dissonance of all of its subsets? That is,
all of its dyads, but then all of its triads as well? Besides, the
brain doesn't know that we conceive of a sound coming in as a unified
"chord," it just hears a bunch of different harmonic relationships
going on.

> But I wonder if at least in older jazz the intervals traditionally
> classified as dissonances tend to move (when they move) to intervals
> traditionally classified as consonances even when using seventh
> and ninth chords and beyond. This is just a hunch, I don't know
> enough about jazz to claim anything.

That is partially it, but also partially that it manages to manipulate
all of the above elements to get you to hear intervals previously
considered "dissonant" as consonant. It also really reveals how much
stuff labeled as inherently "dissonant" can have a very consonant use
that was previously unknown.

Again I don't really just mean just "jazz" per se, but all of that
early 20th century American pop music (and some late Romantic music
too). Think Leroy Anderson and Gershwin or something. Hell, for some
of that stuff using maj7 chords over the I chord is necessary (or a
related variant, like a maj 6/9 chord or something). Playing a
straight major chord where a maj7 chord can sound just as terrible as
the reverse scenario, playing a maj7 chord in common practice music.

I view all of that stuff as a beautiful, intuitive, and logical
expansion of the usual traditional diatonic common practice music. I
wish it were studied more often as well, and outside of the
trivialized "jazz" moniker.

-Mike

> Kalle Aho
>
>

> I think it's more a question of compositional technique than anything else.
> It's the same sort of thing the makes a chord sound strong in a major key and weak in its relative minor...even with the same notes.
>
> To perhaps over-simplify it..."predictable" chords sound consonant...if you hear a bunch of not-so-just chords and the a Just one pops out it surprises you and that surprise comes across to some people as dissonance. If you play a bunch of diminished chords and then a major one...it's the major chord that sounds "weird".

Sure, I understand that. And you also understand that this
"prediction" mechanism evolves through life and is extremely
culture-dependent.

So my point is: if you come up with a system of music saying "do this"
and "avoid this," or even "these intervals are 'consonant'" and "these
are 'dissonant'" then you are disregarding everything that you just
said. And systems like that are fine as a way of writing music or as
compositional algorithms, but they are hardly suitable as a
groundbreaking theory that explains the foundations of all music.
Certainly nothing bigger than Partch and Pythagoras rolled into one.

-Mike

> MikeB> "I didn't say that E-F by itself wasn't dissonant, but that C-E-F-A wasn't dissonant. "
>      Here (as I've said before) the relatively far-spaced C E and F A intervals seem to compensate for the overly close E-F interval and draw attention away from the E-F dyad.  Then again, I agree that doing something like chaining minor seconds within a chord IE C EFF# is too dense to be consonant.  I think there's a sweet spot somewhere around the not-used-in-common-practice 12/11 interval where you can get things like 8:10:11:12 (with the consecutive "smaller than major second" 11/10 and 12/11 intervals) and still be consonant enough for it to count as a true chord and not just a bunch of densely arranged "neighboring tones".

E-F-F# can be made to be consonant. One of my favorite tricks in
12-equal these days is to use a cluster of 5 half steps to represent
16:17:18:19:20. Works very well in the higher register. Works very
well if you know how to finesse it.

Let's say you're in F#. The cluster F# G G# A A# can work very well if
you play it over like an F# lydian aug #2 type thing, but just keep
expanding it mentally along the 5-limit lattice. That means throwing
in a Dx as well as an E#.

You can seriously get a cluster of like 5-6 half 12-tet steps all in a
row that sounds extremely consonant and I swear it represents a chunk
of the harmonic series. It's kind of like tempering out the difference
between 17/16 and 135/128, as well as 19/16 and 75/64. It just takes
all of these different concepts and smashes them all together. A very
"mystical" sound.

But the point is - while I won't be so bold as to say that ANYTHING
can be made consonant - there is a lot more leeway than simple rules
saying "Never put two half steps in a row." When the theory has
evolved enough that we can see how every single harmonic structure
affects the average guy's psychology, and how chord progressions and
such are processed, and the rules are like "never provide the listener
more than xxx pieces of schematic information at once" - maybe things
will be different. But for now they aren't.

-Mike

MikeB>"So my point is: if you come up with a system of music saying "do this" and "avoid this," or even "these intervals are 'consonant'" and "these are 'dissonant'" then you are disregarding everything that you just
said."
First of all...in the systems I'm working on virtually all intervals are at least semi-consonant. My example concerned "better" and "worse" far as consonance...not "good" or "bad". In the diminished chord progression with a surprise major chord example, the major chord sticks out as "worse" because it's a surprise...meanwhile the piece as a whole would most likely have an, on the average, a more tense feel than one with primarily major chords. The goal is to provide a system where you can get more dissonance if you want it...but almost never so much you make the audience think you are out of key or just don't know music theory.

>"And systems like that are fine as a way of writing music or as compositional algorithms, but they are hardly suitable as a
groundbreaking theory that explains the foundations of all music.
Certainly nothing bigger than Partch and Pythagoras rolled into one."

Hmm...are you talking to myself or John? Because I never made that quote, John did. What I'm doing is not aiming to be "perfect" or "better" than anything else, just different. The goal of my system is to make as many chords as possible accessible in a relatively small 7-tone scale and with very few sour points so it is easy to learn escapes the usual excuse of "12TET is hard enough". It also focuses on purifying smaller intervals foremost and larger ones second-most (hence the "super particular" in the name).

Personal note...I'm not even a big fan of Pythagorean scale theories...and much more a fan of Ptolemic theories. +100 for tetra-chord-based scale systems. :-)

_,_._,___

MikeB>"One of my favorite tricks in 12-equal these days is to use a cluster of 5 half steps to represent 16:17:18:19: 20.
Cool...though I do hear my brain trying to tune to 8:9:10 and treating the other tones like "noise" IE I don't see much gain in actual tonal variety.

>"Let's say you're in F#. The cluster F# G G# A A# can work very well if
you play it over like an F# lydian aug #2 type thing, but just keep
expanding it mentally along the 5-limit lattice. That means throwing
in a Dx as well as an E#."

These are all cool tricks for things like my own self-indulgent music composition...but how do you expect an average musician to find uses for them and/or even find out they exist?
The other thing is, ok, those are two ways to use consecutive half-steps...but I doubt there are that many more so, unless you can show they are, it kind of throws a rod in the concept of versatility/variety of half steps in harmony. In the above example it sounds like you're stressing certain notes in the cluster by putting a chord under it to boost the more harmonic roots in the cluster with the overtones of the Lydian Aug 2...thus getting a more tense Lydian Aug 2 type of sound...is that correct?

>"You can seriously get a cluster of like 5-6 half 12-tet steps all in a row that sounds extremely consonant and I swear it represents a chunk of the harmonic series"
How so? Virtually every time I've tried that sort of thing I ended up with something that sounded to me (and friends I showed it to) just like someone smacking their arm across the keyboard (not smooth at all).

>"But the point is - while I won't be so bold as to say that ANYTHING can be made consonant - there is a lot more leeway than simple rules saying "Never put two half steps in a row.""
I'll agree with that and even use the knowledge. :-) But at the same time it still seems 1/2 step clusters used in the right way come across as harmonic chords with extra tones simply adding noise or "texture" to those chords, rather than distinct new chords with added tonal variety.

>"and the rules are like "never provide the listener more than xxx pieces of schematic information at once" - maybe things will be different. But for now they aren't."
Right. The marketability for such ideas, no matter how carefully thought out is low. Heck, I think much a micro-tonality faces that issue...people see it as facing a high learning curve and often ending up with music that send "tons of schematic information" and yet gives very little added benefit far as emotional flexibility IE you hear a lot more and a lot as different, but very few of it makes enough sense to really nod your head to.

It's funny because things like 22TET octa-tonic scales seem to almost make it: sounding very clear despite employing a large deal of new information...but the learning curve is pretty nasty and the numbers of chords that work / the number possible seems smaller even than under 12TET, plus there are more keys to keep track of. I think systems like 22TET have great potential for listeners...but only so-so potential for recruiting new composers to micro-tonality.
Also, honestly, it amazes me just how much music theory is needed to teach classical music in 12TET or mean-tone...you'd think if these musical systems were so elegant musicians would just have to memorize a handful of chords/intervals to avoid instead on memorizing hundreds of chords knowing hundreds more will simply sound out of tune. I can indeed see where musicians get that annoying "12TET is hard enough" line from.

Let me interrupt with this link

I released Revisionist Phiter under the pseudonym Charlie (et al) and it
has almost 1,800 DLs from this site

http://charlie-buddha-ghandi-revisionist-phiter-mp3-download.kohit.net/_/893130

Now - look towards the bottom and look for
Charlie - Over Silver Ho 1854
Charlie - Over Silver Ho 1743

Apparently this piece with your silver tuning was uploaded twice and has
close to 4,000 DLs

On Thu, Apr 15, 2010 at 11:25 AM, Michael <djtrancendance@...> wrote:

>
>
> MikeB>"One of my favorite tricks in 12-equal these days is to use a cluster
> of 5 half steps to represent 16:17:18:19: 20.
> Cool...though I do hear my brain trying to tune to 8:9:10 and treating
> the other tones like "noise" IE I don't see much gain in actual tonal
> variety.
>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > > If you'd like an even more trivial example, how about C D E G?
> > > Would you say that's dissonant?
> >
> > if a piece of music establishes major and minor triads as the
> > basic harmonies then I at least hear both of these chords as
> > unstable and in need of a resolution.
>
> Sure, because the composer is using little musical tricks to mess
> with your mind and get you to hear that chord as "unfamiliar"
> or "out of place" or something. I get it.

But these chords (or simultaneities) are not unfamiliar or out of
place in common practice music. They just sound unstable in that
context. About messing with your mind: using seventh chords (or
beyond) as basic harmonies is just as much a trick as triadic harmony
is. All music messes with your mind in my opinion.

> What I'm saying is, how exactly that plays out is going to be very
> dependent on
>
> 1) The specific tricks that the composer has used
> 2) The person's cultural and musical background
> 3) The person's state of mind and tolerance for new things
> 4) All kinds of subconscious stuff that Freud would have a field
> day with
>
> That being said, to just write C-D-E-G off as an inherently
> dissonant chord is a bit rash.

I agree but I'd say the same about its' "inherent consonance" too.

> > I don't believe in a simple reduction of the
> > consonance/dissonance of a chord to the consonance/dissonance of
> > its' intervals. For example, there is a dramatic difference
> > between major and minor 7-limit tetrads even though they have the
> > same (consonant) interval content but in different order.
>
> How about the consonance/dissonance of all of its subsets? That is,
> all of its dyads, but then all of its triads as well?

(I assume we are talking about a sensory aspect of
consonance/dissonance that is independent of the musical context.)

Yes, and not only the proper subsets but also the complete chord's
fit to a harmonic series should be considered. As a harmonic series
subset the 7-limit minor tetrad is 70:84:105:120 which results in a
complex waveform. Interpreting the minor tetrad as 10:12:15:17 makes
it sound smoother even if it then goes beyond 7-limit.

It might even be interpreted as 24:29:36:41! (Incidentally, the
ratios 24:29, 41:24 and 41:29 are extremely well approximated in 22-
equal.) Now, I'm not suggesting these numbers as such are
particularly meaningful for auditory system but with steady tones the
resulting waveform is audibly more periodic than 70:84:105:120. The
dyads might still be heard as approximate 7-limit intervals which
would make this a case of harmonic tempering! Also, 10:12:15:17 is
not readily extendable into "9-limit" minor pentad while that could
be interpreted as 24:29:32:36:41. But enough of this. :)

> Besides, the brain doesn't know that we conceive of a sound coming
> in as a unified "chord," it just hears a bunch of different
> harmonic relationships going on.

I disagree a little: if partials have close onsets the auditory system
tends to group them together. That's why there are formulas for the
preparation of dissonances: because the dissonant interval is
introduced so that the voices don't enter into it simultaneously its'
dissonance is much reduced.

> > But I wonder if at least in older jazz the intervals traditionally
> > classified as dissonances tend to move (when they move) to
> > intervals traditionally classified as consonances even when using
> > seventh and ninth chords and beyond. This is just a hunch, I
> > don't know enough about jazz to claim anything.
>
> That is partially it, but also partially that it manages to
> manipulate all of the above elements to get you to hear intervals
> previously considered "dissonant" as consonant. It also really
> reveals how much stuff labeled as inherently "dissonant" can have a
> very consonant use that was previously unknown.

Maybe in such music there is a less clear cut difference between
consonance and dissonance but surely some chords are more unstable
than others (like dim7).

> Again I don't really just mean just "jazz" per se, but all of that
> early 20th century American pop music (and some late Romantic music
> too). Think Leroy Anderson and Gershwin or something. Hell, for some
> of that stuff using maj7 chords over the I chord is necessary (or a
> related variant, like a maj 6/9 chord or something). Playing a
> straight major chord where a maj7 chord can sound just as terrible
> as the reverse scenario, playing a maj7 chord in common practice
> music.
>
> I view all of that stuff as a beautiful, intuitive, and logical
> expansion of the usual traditional diatonic common practice music.
> I wish it were studied more often as well, and outside of the
> trivialized "jazz" moniker.

I believe that this harmonic language was very much in use also in
much of the music I was exposed in childhood: certain kinds of 70's
and 80's popular music and the music of TV shows from that time. I
don't think it is an accident that I instantly loved Debussy's music
when I heard it (other later french stuff like Duruflé is also close
to my heart) while common practice classical music sounded foreign
and kind of bland at first. I also never understood why modulation
must be "carefully prepared". :D

Kalle Aho

Ah...I always suspected the Silver Scale had better symmetries than PHI and (boom)...there you go. ;-)

I guess people like the gains in tonal color and symmetries more than enough to compensate for the huge lack of periodicity in some parts...well, that and the downright brilliant arrangement of "skewed" melodies you used. Listening to it now...it sounds grossly un-periodic yet also un-canilly ordered and oddly quite confident...reminds me a lot of the feel I get from Sethares "Ten Fingers" where he timbre-matches a guitar to work with 10TET.

I feel a bit torn b/c I wonder if there's actually a larger appreciation than I thought for scales built almost solely to optimize section-based symmetry like the skill more-so than compromisingbetween optimizing section-based symmetry and JI periodicity (as my new scales do).

***********************************************************************
Another odd flashback I remember recognizing way back when....
Using the Silver Section splitting algorithm on the 2/1 octave instead of the silver ratio you get

(1/(2/1))^x + 1 AKA
(1/2)^x + 1
...which generates the tones.....
3/2 (for x = 1)
5/4 (for x = 2)
9/8 (for x = 3)
2/1

Now taking the mirror of this (the same way the Silver Scale does) by doing
2/1 / ((1/(2/1))^x + 1) AKA
2 / (0.5^x + 1) OR 2 / the result from the last generation IE 2 /
(9/8)...2 / (5/4)...etc.
...gives...
8/5 (for x = 2)
16/9 (for x = 3)

----------------------
Giving an extended "Octave Sections" scale of
1/1
9/8
5/4
3/2
8/5
7/4
16/9
2/1
See any parallels between these ratios and those used in many common JI scales?
I think symmetry and "sections" may actually play a huge role in what makes music "digestible".

-Michael

________________________________
From: Chris Vaisvil <chrisvaisvil@...>
To: tuning@yahoogroups.com
Sent: Thu, April 15, 2010 10:55:40 AM
Subject: Re: [tuning] Natural Pan Temperament with pure sine wave tones

Let me interrupt with this link

I released Revisionist Phiter under the pseudonym Charlie (et al) and it has almost 1,800 DLs from this site

http://charlie- buddha-ghandi- revisionist- phiter-mp3- download. kohit.net/ _/893130

Now - look towards the bottom and look for
Charlie - Over Silver Ho 1854
Charlie - Over Silver Ho 1743

Apparently this piece with your silver tuning was uploaded twice and has close to 4,000 DLs

On Thu, Apr 15, 2010 at 11:25 AM, Michael <djtrancendance@ yahoo.com> wrote:

>
>
>
>
>
>
>
>
>
>
>
>
>
>
> >
>
>>
>
>>
>
>MikeB>"One of my favorite tricks in 12-equal these days is to use a cluster of 5 half steps to represent 16:17:18:19: 20.
>> Cool...though I do hear my brain trying to tune to 8:9:10 and treating the other tones like "noise" IE I don't see much gain in actual tonal variety.
>

>     These are all cool tricks for things like my own self-indulgent music composition...but how do you expect an average musician to find uses for them and/or even find out they exist?

I don't know, how do you expect the average musician to find uses for
anything? The average musician wouldn't be able to make sense out of
any of this if it weren't for the theory already in place. You keep
exploring, messing with new sounds, going by subtle intuitive
inklings, questioning areas of arbitrariness in the existing theory.

> The other thing is, ok, those are two ways to use consecutive half-steps...but I doubt there are that many more so, unless you can show they are, it kind of throws a rod in the concept of versatility/variety of half steps in harmony.  In the above example it sounds like you're stressing certain notes in the cluster by putting a chord under it to boost the more harmonic roots in the cluster with the overtones of the Lydian Aug 2...thus getting a more tense Lydian Aug 2 type of sound...is that correct?

Sort of. I'm not really sure how it works. The long story:

I started messing around with the concept of a "#15" a while ago,
which in JI would most likely be tuned 135/128. I figured that since
there's a clear progression from major -> major7 -> major9 -> maj9#11
-> maj9#11/13, that maj9#11/13/#15 would be next. And it sort of works
that way, once you figure out how to "massage" the situation so that
you don't just hear that note as incredibly weird and foreign. This is
more of an art than a science (so far).

So I figured out how to make that interval sound harmonic and then I
started expanding it into a more dissonant sound, like Ebmaj/Gbmaj or
something. So that G sticking out over the Gb is incredibly weird
without a more direct "bridge" linking it to the Eb, but I got used to
it. And then I started experimenting more, trying to figure out what
the "lydian aug version", would be then lydian aug #2. And THEN, if
you view the #2 as just a fifth above the #5, I tried for one more
fifth above that and got a #6. I also tried "smushing" all of the
above modes together to get like an "omni-lydian" mode.

As you know the #6 is very close to 7/4 and suddenly it all snapped in
for me that way. So at that point I had like 11 half steps in a row
(imagine we're in Gb): C, Db, D, Eb, E, F, Gb, G, Ab, A, Bb. Sweet.
The only note that doesn't really fit is the Cb. But that's like the
entire 12-tet chromatic scale being used in a consonant way. And you
can very clearly hear hints of 14:15:16:17:18:19:20 in there, but it's
sort of tempered and also has a different 5-limit perception as well.

The point? The point is that this is something new that hasn't been
explored all that much in 12-tet (as you know, 17/16 and 19/16 have
great approximations in 12-tet). And it also sounds good. I'll come up
with some sound clips of me noodling around on it for you to listen
to.

But the point is, I don't have any "theory" saying that stuff like
this isn't possible. But some people do. So I always smirk when I hear
these "no more then xxx half steps" myths being floated around.

-Mike

MikeB>> These are all cool tricks for things like my own self-indulgent music composition. ..but how do you expect an average musician to find
uses for them and/or even find out they exist?
>"I don't know, how do you expect the average musician to find uses for
anything?"
Anything that can be shown quickly as an extension to the type of mood a musician always is going for is fair game in my book. The problem is the game of adding texture using closely spaced tones I'm guessing is rarely attractive to musicians: most of them would much rather just play with the timbre of sounds and add effects to get the same type of effect only with easier listen-ability.

>"So at that point I had like 11 half steps in a row (imagine we're in Gb): C, Db, D, Eb, E, F, Gb, G, Ab, A, Bb. Sweet.
The only note that doesn't really fit is the Cb. But that's like the entire 12-tet chromatic scale being used in a consonant way. And you can very clearly hear hints of 14:15:16:17: 18:19:20 in there, but it's
sort of tempered and also has a different 5-limit perception as well."

Ah...very interesting, so you are using enharmonic tones and not just the standard 12TET ones and/or using chords from different modes typically used in typical 12TET progressions stacked on top of each other to find the above "scale"?

>"But the point is, I don't have any "theory" saying that stuff like this isn't possible. But some people do. So I always smirk when I hear
these "no more then xxx half steps" myths being floated around."

You definitely have my attention here....now it's time for some sound clips.... ;-)

Sorry, Kalle, I missed this one. My response:

> > > if a piece of music establishes major and minor triads as the
> > > basic harmonies then I at least hear both of these chords as
> > > unstable and in need of a resolution.
> >
> > Sure, because the composer is using little musical tricks to mess
> > with your mind and get you to hear that chord as "unfamiliar"
> > or "out of place" or something. I get it.
>
> But these chords (or simultaneities) are not unfamiliar or out of
> place in common practice music. They just sound unstable in that
> context. About messing with your mind: using seventh chords (or
> beyond) as basic harmonies is just as much a trick as triadic harmony
> is. All music messes with your mind in my opinion.

Agreed, but what is wrong is to take the historical "tricks" that
people used and posit them as the foundation for the definition of
consonance itself. They're just historical tricks.

> > That being said, to just write C-D-E-G off as an inherently
> > dissonant chord is a bit rash.
>
> I agree but I'd say the same about its' "inherent consonance" too.

Fair enough.

> (I assume we are talking about a sensory aspect of
> consonance/dissonance that is independent of the musical context.)
>
> Yes, and not only the proper subsets but also the complete chord's
> fit to a harmonic series should be considered.

Certainly, as one of those subsets will be the chord itself. HOWEVER,
chords can appear consonant even if the entire chord doesn't resolve
too well to produce a unified fundamental. Then you get this sort of
"fractured fundamental" or "multiple fundamental" approach.

A classic example is the 5-limit minor chord. Another example is a
maj6 chord: 12:15:18:20. If this is spelled C-E-G-A, chances are
you're not going to hear that F pop out on the bottom at all. Why?
Because the strength of the C-E-G triad is overwhelming.

> As a harmonic series
> subset the 7-limit minor tetrad is 70:84:105:120 which results in a
> complex waveform. Interpreting the minor tetrad as 10:12:15:17 makes
> it sound smoother even if it then goes beyond 7-limit.

Yes, but that doesn't eliminate chords that have complex overall JI
representations from being able to be harmonically processed
successfully by the brain. A Cmaj7#9 chord would most likely be tuned
something like C E F# B D# (the F# added for harmonic stability,
technically would make it a #11 chord as well). That in JI would be
32:40:45:60:75. Not so consonant. However, you can still make sense of
the chord as a series of interlinking harmonic relationships: you hear
the Bmaj triad on top, the C-E dyad on the bottom, the E-B dyad
linking the two together, etc. Perhaps if you're harmonically advanced
enough you can hear the C-F# as 45:32 directly too and C-B as 15:8.
Then you hear the less important relationships: the E-F# as 9:8, and
the E-D# as 15:8, and you get the full "map."

It's my theory that once you come up with this "map" of the chord, you
can start to get a "glimpse" of the full 32:40:45:60:75 structure -
even if the direct waveform would otherwise be too complicated to
understand. For example, 32:40:45:60:75 looks like it would be more
dissonant than 31:39:45:59:71, because the numbers in the first are
higher, but -- the first one will sound MUCH more "consonant" than the
second. The first one is made up of a system of interlocking 5-limit
dyads and thus the "substructure" can be attained and lead to the
overall "superstructure." The second one is made of higher-limit
primes with no apparent relation to one another.

This getting a "glimpse" corresponds to a perceptual shift whereby the
listener hears the whole chord as one unified structure instead of
several unrelated polytonal structures. I think that this may take a
few repeated listens to happen and that it is somewhat
listener-dependent. It would explain why a piece of music like the
Rite of Spring can sound so chaotic and random upon the first listen,
but then after listening a few times the huge polychords contained
therein start to become "coherent."

I think this theory could use some updating though, because of what
you wrote here:

> It might even be interpreted as 24:29:36:41! (Incidentally, the
> ratios 24:29, 41:24 and 41:29 are extremely well approximated in 22-
> equal.) Now, I'm not suggesting these numbers as such are
> particularly meaningful for auditory system but with steady tones the
> resulting waveform is audibly more periodic than 70:84:105:120. The
> dyads might still be heard as approximate 7-limit intervals which
> would make this a case of harmonic tempering! Also, 10:12:15:17 is
> not readily extendable into "9-limit" minor pentad while that could
> be interpreted as 24:29:32:36:41. But enough of this. :)

And also agreed that the interpretation of m6 as 10:12:15:17 is very
underused (and extremely resonant as well). I came across that version
a while ago and it's my new favorite tuning for that chord. I suppose
it's a case of using JI as temperament, basically, like you said.

But it does make me question my "map" theory a bit - while my somewhat
overused C-E-F#-B-D# analogy supports it, stuff like this doesn't. You
can have conflicting "maps" for the same chord. How does that work?
Does the brain just come up with a skewed non-JI representation for
how things are working?

> > Besides, the brain doesn't know that we conceive of a sound coming
> > in as a unified "chord," it just hears a bunch of different
> > harmonic relationships going on.
>
> I disagree a little: if partials have close onsets the auditory system
> tends to group them together. That's why there are formulas for the
> preparation of dissonances: because the dissonant interval is
> introduced so that the voices don't enter into it simultaneously its'
> dissonance is much reduced.

That is true and I concede your point. But I don't think it's really
destructive to my point either, which is that - even though we might
call chord like C-Eb-G 10:12:15, the brain doesn't really just hear it
as that singular chunk, but multiple interlocking chunks as well.

> Maybe in such music there is a less clear cut difference between
> consonance and dissonance but surely some chords are more unstable
> than others (like dim7).

It's just successful at getting you to raise your own tolerance for
dissonance. Once that tolerance is raised, chords that previously
sounded weird and out of place suddenly "settle in" and sound
perfectly natural.

> I believe that this harmonic language was very much in use also in
> much of the music I was exposed in childhood: certain kinds of 70's
> and 80's popular music and the music of TV shows from that time. I
> don't think it is an accident that I instantly loved Debussy's music
> when I heard it (other later french stuff like Duruflé is also close
> to my heart) while common practice classical music sounded foreign
> and kind of bland at first. I also never understood why modulation
> must be "carefully prepared". :D

Debussy is probably my favorite classical composer and I still am not
the biggest fan of regular common practice music (although I
appreciate it more as I get older). It's also funny that that harmonic
language is to me second nature, but I still hear people on this list
talking about certain chords from it as being "dissonant" or
"demanding resolution" or whatever, as if all of the musical
developments of the 20th century had never happened.

-Mike

Mike,

sorry to not have gotten back to this until now, but I've been ill
and haven't had the energy to respond:

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> > But these chords (or simultaneities) are not unfamiliar or out of
> > place in common practice music. They just sound unstable in that
> > context. About messing with your mind: using seventh chords (or
> > beyond) as basic harmonies is just as much a trick as triadic
> > harmony is. All music messes with your mind in my opinion.
>
> Agreed, but what is wrong is to take the historical "tricks" that
> people used and posit them as the foundation for the definition of
> consonance itself. They're just historical tricks.

Maybe, but there are reasons why I understand their attitude:

in the diatonic scale major and minor triads are the largest harmonic
structures where dyadic reductionism of consonance actually works.
That is, the dyads in the triads sound consonant also outside the
chords in all inversions and octave compounds. On the other hand,
while a chord like 8:10:12:15 sounds consonant, a bare 8:15 dyad and
particularly its' inversion 15:16 sound dissonant played alone. To
me, maj7 chord has a somewhat melancholic sound. Perhaps the 8:15
doesn't really lose all of its' dissonant quality in maj7 chord. In
fact, I don't think it is necessary to think of seventh chords as
consonant to use them as the basic sonorities. If there is always at
least one dissonant interval to resolve by voice leading the music
attains drive, emotion and "virtual causality". I'm pretty sure
Vincent D'Indy taught Cole Porter the common practice notion of
dissonance.

It is debatable whether 8:9 or 9:10 sound consonant also outside
chords but since the diatonic scale is normally considered a no-7s
system there is a jump in relative dissonance with the 9-limit
ratios. So it is at least understandable why there is such a clear-cut
classification into consonances and dissonances in common practice
theory.

One thing about 8:10:12:15: I don't think it is particularly important
to have very well-tuned 8:15s in tempered maj7 chords for them to
sound right. That would support understanding it as a chord with one
dissonance in it.

> > (I assume we are talking about a sensory aspect of
> > consonance/dissonance that is independent of the musical context.)
> >
> > Yes, and not only the proper subsets but also the complete chord's
> > fit to a harmonic series should be considered.
>
> Certainly, as one of those subsets will be the chord itself.
> HOWEVER, chords can appear consonant even if the entire chord
> doesn't resolve too well to produce a unified fundamental. Then you
> get this sort of "fractured fundamental" or "multiple fundamental"
> approach.
>
> A classic example is the 5-limit minor chord. Another example is a
> maj6 chord: 12:15:18:20. If this is spelled C-E-G-A, chances are
> you're not going to hear that F pop out on the bottom at all. Why?
> Because the strength of the C-E-G triad is overwhelming.

Yes and this multiple fundamental thing happens already as low in the
harmonic series as 10:12:15.

> > As a harmonic series subset the 7-limit minor tetrad is
> > 70:84:105:120 which results in a complex waveform. Interpreting
> > the minor tetrad as 10:12:15:17 makes it sound smoother even if
> > it then goes beyond 7-limit.
>
> Yes, but that doesn't eliminate chords that have complex overall JI
> representations from being able to be harmonically processed
> successfully by the brain.

Agreed. Chords higher up in the harmonic series must be played with
very steady tones if the complete period is to have any effect on
consonance. But likewise, I think dead steady electronic tones make
the 7-limit minor tetrad sound less consonant than acoustic instument
tones.

> A Cmaj7#9 chord would most likely be
> tuned something like C E F# B D# (the F# added for harmonic
> stability, technically would make it a #11 chord as well). That in
> JI would be 32:40:45:60:75. Not so consonant. However, you can
> still make sense of the chord as a series of interlinking harmonic
> relationships: you hear the Bmaj triad on top, the C-E dyad on the
> bottom, the E-B dyad linking the two together, etc. Perhaps if
> you're harmonically advanced enough you can hear the C-F# as 45:32
> directly too and C-B as 15:8. Then you hear the less important
> relationships: the E-F# as 9:8, and the E-D# as 15:8, and you get
> the full "map."

I'm skeptical about our abilities to literally hear these numerical
relationships. I suppose that young children and very untutored
listeners hear just an undifferentiated wash of sound when C E F# B D#
is played. The rest of us hear out tones, intervals, dyads and triads
with varying skill. Musical context, voice leading and such also
play a role in what is heard.

> It's my theory that once you come up with this "map" of the chord,
> you can start to get a "glimpse" of the full 32:40:45:60:75
> structure - even if the direct waveform would otherwise be too
> complicated to understand. For example, 32:40:45:60:75 looks like
> it would be more dissonant than 31:39:45:59:71, because the numbers
> in the first are higher, but -- the first one will sound MUCH
> more "consonant" than the second. The first one is made up of a
> system of interlocking 5-limit dyads and thus the "substructure"
> can be attained and lead to the overall "superstructure." The
> second one is made of higher-limit primes with no apparent relation
> to one another.

I think that subset consonance explains why the overall impression is
more consonant in the first chord. I'm not so sure if hearing the
structure is important for consonance.

> This getting a "glimpse" corresponds to a perceptual shift whereby
> the listener hears the whole chord as one unified structure instead
> of several unrelated polytonal structures. I think that this may
> take a few repeated listens to happen and that it is somewhat
> listener-dependent. It would explain why a piece of music like the
> Rite of Spring can sound so chaotic and random upon the first
> listen, but then after listening a few times the huge polychords
> contained therein start to become "coherent."
>
> I think this theory could use some updating though, because of what
> you wrote here:
>
> > It might even be interpreted as 24:29:36:41! (Incidentally, the
> > ratios 24:29, 41:24 and 41:29 are extremely well approximated in
> > 22-equal.) Now, I'm not suggesting these numbers as such are
> > particularly meaningful for auditory system but with steady tones
> > the resulting waveform is audibly more periodic than
> > 70:84:105:120. The dyads might still be heard as approximate 7-
> > limit intervals which would make this a case of harmonic
> > tempering! Also, 10:12:15:17 is not readily extendable into "9-
> > limit" minor pentad while that could be interpreted as
> > 24:29:32:36:41. But enough of this. :)
>
> And also agreed that the interpretation of m6 as 10:12:15:17 is very
> underused (and extremely resonant as well). I came across that
> version a while ago and it's my new favorite tuning for that chord.
> I suppose it's a case of using JI as temperament, basically, like
> you said.
>
> But it does make me question my "map" theory a bit - while my
> somewhat overused C-E-F#-B-D# analogy supports it, stuff like this
> doesn't. You can have conflicting "maps" for the same chord. How
> does that work? Does the brain just come up with a skewed non-JI
> representation for how things are working?

I still believe it doesn't need to make sense in terms of JI math to
sound coherent.

> > > Besides, the brain doesn't know that we conceive of a sound
> > > coming in as a unified "chord," it just hears a bunch of
> > > different harmonic relationships going on.
> >
> > I disagree a little: if partials have close onsets the auditory
> > system tends to group them together. That's why there are
> > formulas for the preparation of dissonances: because the
> > dissonant interval is introduced so that the voices don't enter
> > into it simultaneously its' dissonance is much reduced.
>
> That is true and I concede your point. But I don't think it's really
> destructive to my point either, which is that - even though we might
> call chord like C-Eb-G 10:12:15, the brain doesn't really just hear
> it as that singular chunk, but multiple interlocking chunks as well.

Yes, I don't believe that "exclusive allocation" applies when
listening to music. Multiple interpretations of the same data are
often processed at the same time.

> > Maybe in such music there is a less clear cut difference between
> > consonance and dissonance but surely some chords are more unstable
> > than others (like dim7).
>
> It's just successful at getting you to raise your own tolerance for
> dissonance. Once that tolerance is raised, chords that previously
> sounded weird and out of place suddenly "settle in" and sound
> perfectly natural.

What sort of motor for musical movement do you propose other than
the "virtual causality" of dissonances resolving to consonances?

> > I believe that this harmonic language was very much in use also in
> > much of the music I was exposed in childhood: certain kinds of
> > 70's and 80's popular music and the music of TV shows from that
> > time. I don't think it is an accident that I instantly loved
> > Debussy's music when I heard it (other later french stuff like
> > Duruflé is also close to my heart) while common practice
> > classical music sounded foreign and kind of bland at first. I
> > also never understood why modulation must be "carefully
> > prepared". :D
>
> Debussy is probably my favorite classical composer

Mine too!

> and I still am not the biggest fan of regular common practice music
> (although I appreciate it more as I get older). It's also funny
> that that harmonic language is to me second nature, but I still
> hear people on this list talking about certain chords from it as
> being "dissonant" or "demanding resolution" or whatever, as if all
> of the musical developments of the 20th century had never happened.

Yes, one thing I don't understand about music history: there is this
meta-narrative of enculturation where more and more complex chords
and intervals are accepted as consonant and more and more daring
treatments of dissonances are slowly accepted and all of this has
been painstakingly earned over centuries. But a child born in our
culture doesn't have to go through all these phases to enjoy the
music of our times (at least it happens really fast!).

Kalle Aho

> > Agreed, but what is wrong is to take the historical "tricks" that
> > people used and posit them as the foundation for the definition of
> > consonance itself. They're just historical tricks.
>
> Maybe, but there are reasons why I understand their attitude:
>
> in the diatonic scale major and minor triads are the largest harmonic
> structures where dyadic reductionism of consonance actually works.
> That is, the dyads in the triads sound consonant also outside the
> chords in all inversions and octave compounds. On the other hand,
> while a chord like 8:10:12:15 sounds consonant, a bare 8:15 dyad and
> particularly its' inversion 15:16 sound dissonant played alone.

A bare 8:15 dyad to me sounds "consonant." But I suppose that this is
rather a value judgment than an immutable fact of nature. When I hear
8:15 by itself, it has a few properties that I suppose one could call
"dissonant." On the other hand, I hear it as sort of the "distilled
essence" of the maj7 chord. This is a technique commonly used in jazz,
and to play a bare maj7 in the left hand while improvising in the
right is referred to as a "shell" voicing.

But I do see your point. I just think, however, that if people raised
their tolerance for "dissonance" a bit, suddenly things wouldn't seem
all that dissonant after all. When you're used to a pristine
environment where every chord is a major triad and every dyad
consonant, your tolerance for things like maj7 chords and other crazy
stacked extended harmonies will be... understandably low.

> One thing about 8:10:12:15: I don't think it is particularly important
> to have very well-tuned 8:15s in tempered maj7 chords for them to
> sound right. That would support understanding it as a chord with one
> dissonance in it.

Well that fits in with my idea of the brain creating a JI "map" of the
chord. You don't really hear 8:15 directly, but as a major third on
top of a fifth (or the other way around). Another example is the F# in
C-E-G-B-D-F#. Normally that F# would be 45/32, but you don't really
hear it that way "directly" over the C. You instead hear it as a major
third on top of a major second (or some octave inversion thereof). But
I was messing around with some very flat-fifth meantone tuning where
that F# become so flat it was more like F half-#. Suddenly my brain
snapped it in as 11/8 over the root.

So there are three intervals here: C-D (9/8), D-F# (5/4), C-F#
(11/8?!). So the map doesn't even have to be consistent to work.

I think part of the reason why 15/8 doesn't sound very dissonant to me
is that I automatically "imagine" it as part of a major7 chord and
thus create my own little "map" to the note. Perhaps if I stopped
doing that it would be more dissonant.

> Agreed. Chords higher up in the harmonic series must be played with
> very steady tones if the complete period is to have any effect on
> consonance. But likewise, I think dead steady electronic tones make
> the 7-limit minor tetrad sound less consonant than acoustic instument
> tones.

I think electronic tones make EVERYTHING sound less consonant than
acoustic instrument tones, and I'm not sure why :)

> I'm skeptical about our abilities to literally hear these numerical
> relationships. I suppose that young children and very untutored
> listeners hear just an undifferentiated wash of sound when C E F# B D#
> is played. The rest of us hear out tones, intervals, dyads and triads
> with varying skill. Musical context, voice leading and such also
> play a role in what is heard.

What are your thoughts now that I have clarified my meaning above?
Again I don't think we hear the 75/32 from C-D# "directly," if you
will, but I think that we do get a sense of how the D# is "related" to
the C via a series of more consonant intervals (thirds and fifths).
For the 12-tet chord C Eb G C Eb, we get a different way that the Eb
is "related" to C, even though they're tuned the same interval.

> I think that subset consonance explains why the overall impression is
> more consonant in the first chord. I'm not so sure if hearing the
> structure is important for consonance.

Psychoacoustically speaking, it must be. Each of these intervals
produces its own fundamental, and then I believe the brain figures out
how the fundamentals themselves relate to each other. Why? Because
each "note" in the chord is itself a harmonic series placed to a
fundamental, and the whole concept of chords involves seeing how these
fundamentals then relate to each other. Especially if the timbre has
no fundamental of its own - let's say you play 4:5:6 with a timbre
that has been sent through a high-pass filter so that no fundamental
remains. So it's really a "phantom" 4, a "phantom" 5, and a "phantom"
6. Chances are you'll still hear the "double phantom" 2 and 1 popping
out anyway.

> Yes, one thing I don't understand about music history: there is this
> meta-narrative of enculturation where more and more complex chords
> and intervals are accepted as consonant and more and more daring
> treatments of dissonances are slowly accepted and all of this has
> been painstakingly earned over centuries. But a child born in our
> culture doesn't have to go through all these phases to enjoy the
> music of our times (at least it happens really fast!).

Indeed, and hopefully xenharmonic stuff will be next!

-Mike

MikeB>"A bare 8:15 dyad to me sounds "consonant." But I suppose that this is
rather a value judgment than an immutable fact of nature. When I hear
8:15 by itself, it has a few properties that I suppose one could call
"dissonant.""

Funny...turns out 15/8 to sounds fairly consonant/non-conflicting, yet also a bit stressed/un-resolved. Kind of like a singer stretching her voice a bit to hit a high note. So it sounds mathematically consonant to me, but the non-mathematical mood over it sounds a bit tense. I could see how a composer could use it in either way.

>"Well that fits in with my idea of the brain creating a JI "map" of the chord. You don't really hear 8:15 directly, but as a major third on
top of a fifth (or the other way around)."
You know Igs fooled the heck out of me with this. One of his songs had a ton of triads with a "horrific" sounding dyad, a 16/11 representing the gap between the low and high ends of the chord, but very normal sounding dyads representing the gaps between the first and second note and the second and third. Indeed it seems if the brain gets "2 out of 3" intervals as fairly pure, it draws a line itself to a pure third interval even the actual interval is impure.

>"I think electronic tones make EVERYTHING sound less consonant than acoustic instrument tones, and I'm not sure why :)"
Unless you're a master pad/string programmer...most electronic sounds have less intelligent amplitude and frequency modulation on their partials and tend to sound "organ-like" in comparison...while acoustic sounds naturally have such movements build in to the resonance of the instruments.

>"I'm skeptical about our abilities to literally hear these numerical relationships"
I don't think we can hear them directly...but I think we can tell when the relationships cause, say, so much critical band or periodicity-based dissonance that the brain can't focus. It's like trying to read letters in the distance...there's a certain point where you "jump" from not being able to read the letters to reading them and not much of a slack distance where you can "kind of read the letters".

Kalle> "Yes, one thing I don't understand about music history: there is this
> meta-narrative of enculturation where more and more complex chords
> and intervals are accepted as consonant and more and more daring
> treatments of dissonances are slowly accepted and all of this has
> been painstakingly earned over centuries. But a child born in our
> culture doesn't have to go through all these phases to enjoy the
> music of our times (at least it happens really fast!).

Let me put it this way...when I have my first child I will gladly by a MIDI keyboard (probably with a plastic bag-type cover for obvious reasons) hooked up to a computer and place it in the crib...tuning it to things like Ptolemy's scales along with things like Pentatonic scales (in both playing "out of key" is fairly hard to do) and letting him/her mess around with it. I'm betting you once they get the idea that things like a 9:11:12 chord aren't so alien the child will avoid getting into the pattern of "only accepting one definition of consonance".
I am pretty convinced chords like 10:11:12 or 11:15:18 and certainly not only ones like 5:6:7 represent the near limits of the human mind to easily map chords. One major thing I think keeps a lot of people from accepting things like extended JI or sometimes even something like 5-limit JI over 12TET is that the "range" of dissonance in 12TET is gradual while in things like JI often it's either "x chord is perfectly pure" or "x chord is way off pure" with very little in between to help the mind transition from one to the other. I generally think making systems (or composing in ways) that have the most dissonant and least dissonant chords within a not-so-broad range of dissonance difference will make people less "surprised" by the extra dissonance. In fact I think it's a huge part of modern music theory...the idea of drifting so and from dissonance within a not-so-broad range of dissonance...not only in things like chords, but also transitions, moving
to and from less-consonant/tenser-sounding bridges only after getting the mind used to the theme, etc.

Mike,

sorry again for the late response. This time my excuse is that
I've been renovating our new home. :)

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> > Maybe, but there are reasons why I understand their attitude:
> > in the diatonic scale major and minor triads are the largest
> > harmonic structures where dyadic reductionism of consonance
> > actually works. That is, the dyads in the triads sound consonant
> > also outside the chords in all inversions and octave compounds.
> > On the other hand, while a chord like 8:10:12:15 sounds
> > consonant, a bare 8:15 dyad and particularly its' inversion 15:16
> > sound dissonant played alone.
>
> A bare 8:15 dyad to me sounds "consonant." But I suppose that this
> is rather a value judgment than an immutable fact of nature. When I
> hear 8:15 by itself, it has a few properties that I suppose one
> could call "dissonant." On the other hand, I hear it as sort of
> the "distilled essence" of the maj7 chord. This is a technique
> commonly used in jazz, and to play a bare maj7 in the left hand
> while improvising in the right is referred to as a "shell" voicing.

I don't think it should be a value judgment. There is an unfortunate
tendency to equate 'dissonant' with 'bad' and 'consonant' with 'good'
but there is no logical connection here.

Perhaps the shell voicings take advantage of the internalized diatonic
template so that playing a bare maj7 interval implies a full maj7
chord.

> But I do see your point. I just think, however, that if people
> raised their tolerance for "dissonance" a bit, suddenly things
> wouldn't seem all that dissonant after all. When you're used to a
> pristine environment where every chord is a major triad and every
> dyad consonant, your tolerance for things like maj7 chords and
> other crazy stacked extended harmonies will be... understandably
> low.

Actually very little common practice music is like that. Something
like Mozart might sound non-offensive to an average listener but his
music has dissonances everywhere. I think people are actually
extremely tolerant when it comes to dissonance, it's just that when
there is no audible logic in the treatment of dissonances (as in some
contemporary music) they get frustrated.

> I think part of the reason why 15/8 doesn't sound very dissonant to
> me is that I automatically "imagine" it as part of a major7 chord
> and thus create my own little "map" to the note. Perhaps if I
> stopped doing that it would be more dissonant.

Yes, see above what I said about the diatonic template.

> > I'm skeptical about our abilities to literally hear these
> > numerical relationships. I suppose that young children and very
> > untutored listeners hear just an undifferentiated wash of sound
> > when C E F# B D# is played. The rest of us hear out tones,
> > intervals, dyads and triads with varying skill. Musical context,
> > voice leading and such also play a role in what is heard.
>
> What are your thoughts now that I have clarified my meaning above?
> Again I don't think we hear the 75/32 from C-D# "directly," if you
> will, but I think that we do get a sense of how the D# is "related"
> to the C via a series of more consonant intervals (thirds and
> fifths). For the 12-tet chord C Eb G C Eb, we get a different way
> that the Eb is "related" to C, even though they're tuned the same
> interval.

I'd say we hear all intervals directly but I don't really know if
we hear them also indirectly through relationships. I think intervals
heard as dissonant have much wider tolerance for mistuning than
consonances. Their tuning can vary wildly, the only requirement being
that they are recognized categorically as an instance of "that
interval". Example: 12-equal maj7 is much closer to 9:17 than 8:15
but sounds fine. Perhaps the auditory system doesn't really recognize
the 12-equal major seventh chord as a 8:10:12:15 harmonic series
pattern. But in a 8:9:10:11:12:13:14:15 sonority, 8:15 tolerates less
mistuning because the auditory system recognizes the harmonic series
pattern. Here the 8:15 is unambiguous and thus according to the
harmonic entropy idea, consonant.

> > I think that subset consonance explains why the overall
> > impression is more consonant in the first chord. I'm not so sure
> > if hearing the structure is important for consonance.
>
> Psychoacoustically speaking, it must be. Each of these intervals
> produces its own fundamental, and then I believe the brain figures
> out how the fundamentals themselves relate to each other. Why?
> Because each "note" in the chord is itself a harmonic series placed
> to a fundamental, and the whole concept of chords involves seeing
> how these fundamentals then relate to each other. Especially if
> the timbre has no fundamental of its own - let's say you play 4:5:6
> with a timbre that has been sent through a high-pass filter so that
> no fundamental remains. So it's really a "phantom" 4, a "phantom"
> 5, and a "phantom" 6. Chances are you'll still hear the "double
> phantom" 2 and 1 popping out anyway.

First, you are confusing two "fundamentals" here. The "fundamental"
of your high-pass filtered timbre is just the first harmonic. But
that is not what we hear when we hear the fundamental of an interval,
a chord or a high-pass filtered tone. There is nothing particularly
special about this "fundamental", it is just one of the harmonics.
Pitch is not the first harmonic. Instead pitch at least correlates
with the *fundamental frequency* of a periodic vibration.
Fundamental frequency is a quantity i.e. property-like entity while
harmonics are thing-like entities. There is nothing phantom (or
virtual) about the fundamental frequency even in the "missing
fundamental" situations but I do approve the use of "phantom"
in relation to chords as chords are at least partly some kind
of "fake sources" from the point of view of the auditory system.

Second, the auditory system probably finds the pitch of the 4:5:6
complex directly, not by first finding 4, 5 and 6.

Kalle Aho

Kalle>"I think people are actually extremely tolerant when it comes to dissonance, it's just that when
there is no audible logic in the treatment of dissonances (as in some contemporary music) they get frustrated."

I agree with a lot of this. The problem seems to start when you go, say, from high consonance to much higher dissonance or vice-versa where the brain has trouble predicting where the dissonance range is. In many (if not most) cases I've found producing a scale with lots of reasonably pure intervals sounds much more relaxed/balanced than one with some perfectly pure intervals and some dead-sour intervals...because the composer can suddenly switch consonance/dissonance level by a huge amount. If you play chords that are consistently fairly dissonant the mind will begin to expect dissonance and relax a bit...but it you take a bunch of consistently consonant chords and suddenly switch to a consonant one the brain or vice-versa becomes confused. Note, especially in Igs aka "City of the Asleep"'s music he uncovers consonances in dissonant tunings and manages to meld both high dissonance and consonance...but he is consistently very careful to fade
gently between the two (something a whole lot of micro-tonalists seem to fail at).

>"There is an unfortunate tendency to equate 'dissonant' with 'bad' and 'consonant' with 'good' but there is no logical connection here. "
I think it's fair to say there is a point where, say, even with chords of about equal dissonance that flow well into each other there is such a thing as too much dissonance. However I think such a "limit" on minimum dissonance is at a considerably higher level of dissonance than most standard chords in 12TET. The problem, I believe, is when people think "dissonance" they think of a fairly (though not terribly) sour chord being played uncontrollably in the middle of a bunch of sweet-sounding chords IE by an inexperienced musician. That and the fact many people like larger chords (IE 4+ notes) and adding more notes to chords often adds to dissonance...hence to enable larger chords you often need to move the minimum consonance level to a higher level of consonance. Another note is that musicians like Bach obviously use a lot of string sections...and chords played with strings (IE cello, violin, viola...) often sound more consonant than the same
chords with other instruments due to how the overtones of those stringed instruments slide up/downwards (frequency modulation) and often at one point in this slide align was other root tones and overtones they wouldn't align with if they were not modulated.

-Michael