Yahoo Tuning Groups Ultimate Backup tuning Re: [tuning] Re: Why does a perfect minor triad sound worse than a major one even with pure s

Marcel>"With normal sounds the overtones simply do not overlapp as nicely with the minor chords as with the 4:5:6 major chord."

Very good point for "normal" instruments with overtone...but that still doesn't explain why the major chord also sounds more pure even when played with pure sine waves (IE no overtones).

--- On Fri, 5/15/09, Marcel de Velde <m.develde@gmail.com> wrote:

From: Marcel de Velde <m.develde@...>
Subject: Re: [tuning] Re: Why does a perfect minor triad sound worse than a major one even with pure s
To: tuning@yahoogroups.com
Date: Friday, May 15, 2009, 6:07 PM

I'm not so sure myself about the 16:19:24 , 1/1 19/16 3/2 minor triad.
I've tried to use it but in the end it didn't make sense to me.
It's also very close to 54:64:81 , 1/1 32/27 3/2 the pythagorean minor triad which makes much more sense.

I especially like the 32/27 in there like in the V7 of 1/1 5/4 3/2 16/9.
But it's also in 27:32:40 , 1/1 32/27 40/27.

The only minor triads I'm sure of are used in common practice music a lot are these 2:

10:12:15 , 1/1 6/5 3/2
and 27:32:40 , 1/1 32/27 40/27 (this is normally the ii chord like D F A where I - C E G and V - G B D are 4:5:6)

For these 2 minor triads it's easy to see why they are less consonant than 4:5:6 major chord.

With normal sounds the overtones simply do not overlapp as nicely with the minor chords as with the 4:5:6 major chord.

Marcel

🔗Marcel de Velde <m.develde@...>

>
> Very good point for "normal" instruments with overtone...but that still
> doesn't explain why the major chord also sounds more pure even when played
> with pure sine waves (IE no overtones).

I don't think the difference in consonance is thesame when you play 4:5:6
and 10:12:15 with pure sines vs instrument with overtones.

As for the difference in consonance that remains I can only speculate.
Where as in 4:5:6 the lowest tone is 4:5 and 2:3 in major.
The lowest tone is 5:6 and 2:3 in minor.

Where the 2:3 is the "dominant" in-sync relation. The 4:5 of the major fits
in more nicely with this than the 5:6 of the minor.
The major chord has a more in sync sound as a result of this, which is
percieved as consonance even with sines it seems to me.
More easily seen the 4:5:6 is offcourse more in sync as a whole than
10:12:15.
You could say 10:12:15 is simply 4:5:6 in reverse.
Yes this may be so but sound doesn't work in reverse it seems to me.
Sound is overtones from low to high, not from high to low.
So it seems wrong to me to expect the major chord to be equally consonant in
reverse.
How weird is the harmonic series in reverse. It sounds terrible.

Marcel

🔗djtrancendance@...

Marcel>"How weird is the harmonic series in reverse. It sounds terrible."

Then again, how weird are the tones F and A relative the the rest of Diatonic JI?

Note that (only) in Pentatonic JI (which exclude these 2 notes from diatonic) it is virtually impossible to create sour chords! :-(

Look at diatonic JI:
1/1 9/8 10/8 4/3 12/8 5/3 15/8
...then look at the interval gaps between each notes
0 9/8 10/9 X16/15X 9/8 10/9 X9/8X
...and note how the gaps indicated with X's are "reverse order".

Furthermore, see how the F and B are in reverse harmonic compared to all the other tones?
Now look at common music practice. Notice how most chords in classical theory avoid putting E and F or B and C together in chords (and in
jazz they only do it several octaves apart to help relieve dissonance)?

This is one problem I've >always< had with diatonic JI...the fact the E to F and B to C semitone gaps become fairly unusable for consonant chords.

Why not make certain notes from pentatonic decent but less-that-perfect in their purity and use the extra gaps created by doing so to shift the tones F and A to achieve something not so terrible impure relative to the tones around them so those "chord-less semi-tones" (E-F and B-C) can actually be used clearly in harmony within the same octave?

In short: I agree whole-heartedly with you; reverse harmonic series are bad news. The problem is...I see them all over the place in standard JI.

-Michael

🔗Marcel de Velde <m.develde@...>

>
> Marcel>"How weird is the harmonic series in reverse. It sounds terrible."
>
> Then again, how weird are the tones F and A relative the the rest of
> Diatonic JI?
>

Not weird at all to me.
Not yet fully explained, I agree but this goed for JI and music as a whole.
Your example is a random pick from this.

>
>
> Note that (only) in Pentatonic JI (which exclude these 2 notes from
> diatonic) it is virtually impossible to create sour chords! :-(
>
> Look at diatonic JI:
> 1/1 9/8 10/8 4/3 12/8 5/3 15/8
> ...then look at the interval gaps between each notes
> 0 9/8 10/9 X16/15X 9/8 10/9 X9/8X
> ...and note how the gaps indicated with X's are "reverse order".
>

Well if you take F as 1/1 it is no longer so.
There are a zillion patterns in JI, I don't see why your example / pattern
is relevant any more than all the others.
There are a zillion things not well understood in JI.

> Furthermore, see how the F and B are in reverse harmonic compared to all
> the other tones?
> Now look at common music practice. Notice how most chords in classical
> theory avoid putting E and F or B and C together in chords (and in
> jazz they only do it several octaves apart to help relieve dissonance)?
>
> This is one problem I've >always< had with diatonic JI...the fact the E to
> F and B to C semitone gaps become fairly unusable for consonant chords.
>
> Why not make certain notes from pentatonic decent but less-that-perfect in
> their purity and use the extra gaps created by doing so to shift the tones F
> and A to achieve something not so terrible impure relative to the tones
> around them so those "chord-less semi-tones" (E-F and B-C) can actually be
> used clearly in harmony within the same octave?
>

Sorry I can't follow you here.
The things you say are not specific to JI at all, you seem to disagree with
music as a whole.
You seem to want to make something where no matter what you play it's pretty
and consonant and makes musical sense somehow.
This is not how music works.
Sure you can kinda do it with 5 tempered tones, but really pretty music
doesn't behave like this at all.
Much better aproach would be to learn some music theory and use 12tet it
seems to me if your end goal is pretty music :)

>
> In short: I agree whole-heartedly with you; reverse harmonic series are bad
> news. The problem is...I see them all over the place in standard JI.
>

Hmm I don't know if for instance 1/1 6/5 3/2 should be seen as 1/1 5/4 3/2
in reverse in a musical sense.
But this wasn't what I ment anyway.
I ment to say that reverse harmonic series is bad news in a consonance way.
Wether less consonant reverse harmonic segments appear in actual music
nontheless is irrelevant.
The thread was about the difference in consonance between 1/1 5/4 3/2 and
for instance 1/1 6/5 3/2.

🔗Marcel de Velde <m.develde@...>

Btw Michael, I think the most consonant 8-notes per octave is 2 series of 4
notes connected by pure fifths, one a pure third apart from the other.1/1
3/2 9/8 27/16
5/4 15/8 45/32 135/128

You can actually play all notes together in very consonant ways, but not in
one octave.
play for instance 1/1 3/2 9/4 27/8 5/1 15/2 45/4 135/8
nice huh :)

Here the interval matrix:
1/1 : 135/128 9/8 5/4 45/32 3/2 27/16 15/8 2/1
135/128: 16/15 32/27 4/3 64/45 8/5 16/9 256/135 2/1
9/8 : 10/9 5/4 4/3 3/2 5/3 16/9 15/8 2/1
5/4 : 9/8 6/5 27/20 3/2 8/5 27/16 9/5 2/1
45/32 : 16/15 6/5 4/3 64/45 3/2 8/5 16/9 2/1
3/2 : 9/8 5/4 4/3 45/32 3/2 5/3 15/8 2/1
27/16 : 10/9 32/27 5/4 4/3 40/27 5/3 16/9 2/1
15/8 : 16/15 9/8 6/5 4/3 3/2 8/5 9/5 2/1
2/1

you can take it even further to 12 tones like this:
1/1 3/2 9/4 27/8 5/1 15/2 45/4 135/8 25/1 75/2 225/4 675/8

or 16 tones:
1/1 3/2 9/4 27/8 5/1 15/2 45/4 135/8 25/1 75/2 225/4 675/8 125/1 375/2
1125/4 3375/8

Nice pattern huh :)
You can continue this from 1 hertz to above hearing and nowhere do 2 tones
sound together with a beating ratio of greater than X/8.
All sound much better than doing these things pythagorean.

Please come back to JI land ;-)
it's much more interesting than phi and sounds better and has the potential
to fully explain common practice music and create completely new music.

🔗Marcel de Velde <m.develde@...>

🔗djtrancendance@...

Btw Michael, I think the most consonant 8-notes per octave is 2
series of 4 notes connected by pure fifths, one a pure third apart from
the other.1/1 3/2 9/8 27/165/4 15/8 45/32 135/128

Not bad, this turns out like diatonic JI, minus the 4/3 (F) note which I think is off tune and adds the 45/32 = 1.40625 note which is almost indistinguishably close to the 1.411764 (24/17) tone in my approximately 8-tone per octave "arithmetic series" scale which generates a JI scale using a PHI-like generation formula. The "arithmetic mean" scale I found is
1
17/16
9/8 (in common with yours)
5/4 (in common with yours)
24/17 (almost exactly in common with yours)
3/2 (period for my scale) (in common with yours)
51/32
27/16 (in common with yours)
15/8 (in common with yours)
etc.

>"Please come back to JI land ;-) it's much more interesting than phi"

If you look at the two formulas I use the generate my scale (which has much in common with yours):
A) (1/octave)^x + 1
B) (1 + 1/octave) - (1/octave)^x + 1

...you'll notice a huge similarity between the way it is generated and my latest PHI scale generation formula
A) (1/PHI)^x + 1

B) (1+PHI) - (1/PHI)^x + 1

So...ironically, the generation method I've found works best with PHI also seems to work best with JI. :-)
So, let's just say I'm back in JI land, but using some new knowledge I have gained from "messing around" with PHI. And, for the record, I don't think either system is superior just...different. My above JI scale (and yours as well to an extent) seem to point to the idea of at least consonant 7 note per octave chords that are impossible in traditional diatonic JI. And I think PHI scales using the same generation have the same capability, admittedly minus the fact they require special timbres to work so well as JI ones.

I think we can all learn a lot both from PHI-based scales and JI...

-Michael

🔗Michael Sheiman <djtrancendance@...>

Me>"Look at diatonic JI:
1/1 9/8 10/8 4/3 12/8 5/3 15/8
...then look at the interval gaps between each notes

0 9/8 10/9 X16/15X 9/8 10/9 X9/8X
...and note how the gaps indicated with X's are "reverse order"."

Marcel> "Well if you take F as 1/1 it is no longer so."
But, if I'm interpreting this well that means changing the root of the scale. Which would require you take use sort of adaptive JI to switch the root.

Me>"(why not) shift the
tones F and A to achieve something not so terrible impure relative to
the tones around them so those "chord-less semi-tones" (E-F and B-C)
can actually be used clearly in harmony within the same octave?"

Marcel>"Sorry I can't follow you here. The things you say are not specific to JI at all, you seem to disagree with."
Well, they are specific to diatonic music as a whole.

Marcel>"There are a zillion things not well understood in JI."
Unfortunately I swear much of the reason they continue not to be understood is that so many people insist on basing JI on diatonic intervals. Even Elrich's decatonic scales seem to filter down to such rules.

Marcel> "The things you say are not specific to JI at all, you seem to disagree with music as a whole."
Then that would mean all "considered as music" scales must have the exact notes F and B in them. Tons of scales lack equivalents of those two tones, the most obvious being C pentatonic which is C D E G A and lacks both F and B. I don't believe I disagree with music all simply by saying F and B are "off".

>"Much better approach would be to learn some music theory and use 12tet it seems to me if your end goal is pretty music :)"
But don't you see my point is straight from knowledge of chord theory? Music theory tells us to align chords to fit within diatonic scales formed from the root note. That's why EFA, for example, is not a chord...and neither is EFB or EFC. What are some valid chords? How about E G B or F A C or F G# C....these work but, as you probably noticed, never use E and F or B and C at the same time within the same octave.

Which is why you don't hear people use semi-tone gaps in their chords. In fact it is >because< (and not in spite of) theory that I'm coming to this conclusion. In addition the whole statement about B and C or E and F never being used on the same octave in combination in chords was a take off Carl's saying that to me. So it sounds like you'd have to accuse him of needing to learn some theory as well.

>"Whether less consonant reverse harmonic segments appear in actual music nonetheless is irrelevant."
But is it? Is it really "ok" to just settle for the fact we can never use B5 and C5 or E5 and G5 together to make a consonant chord? You made a very relevant statement about reverse harmonic series' sounding considerably worse than regular ones...and I was simply giving an example of how reverse harmonic series pose a significant obstacle to chord theory (IE they force you to avoid certain intervals religiously in making good-sounding chords).

>"The thread was about the difference in consonance between 1/1 5/4 3/2 and for instance 1/1 6/5 3/2."
Right...and you brought up a great point (the terrible sound of a reversed harmonic series) and I brought up a possible application of your point. Is that really such a crime?

-Michael

🔗Marcel de Velde <m.develde@...>

Hi Michael,

Marcel> "The things you say are not specific to JI at all, you seem to
> disagree with music as a whole."
> Then that would mean all "considered as music" scales must have the
> exact notes F and B in them. Tons of scales lack equivalents of those two
> tones, the most obvious being C pentatonic which is C D E G A and lacks both
> F and B. I don't believe I disagree with music all simply by saying F and B
> are "off".
>

No sorry I was referring to you trying to put as many notes together in an
as consonant possible way where any number of tones you play it's a
consonant chord.
Sorry if I wasn't clear, my whole message wasn't very clear, was very late
lastnight for me :)

>
> >"Much better approach would be to learn some music theory and use 12tet it
> seems to me if your end goal is pretty music :)"
> But don't you see my point is straight from knowledge of chord theory?
> Music theory tells us to align chords to fit within diatonic scales formed
> from the root note. That's why EFA, for example, is not a chord...and
> neither is EFB or EFC. What are some valid chords? How about E G B or F A
> C or F G# C....these work but, as you probably noticed, never use E and F or
> B and C at the same time within the same octave.
>
> Which is why you don't hear people use semi-tone gaps in their
> chords. In fact it is >because< (and not in spite of) theory that I'm
> coming to this conclusion. In addition the whole statement about B and C or
> E and F never being used on the same octave in combination in chords was a
> take off Carl's saying that to me. So it sounds like you'd have to accuse
> him of needing to learn some theory as well.
>

M i think you can hit them in thesame octave. But consonant it isn't no.
But so many chords aren't consonant.
I think the only consonant chord is 1/1 5/4 3/2 and it's way more consonant
offcourse as 1/1 3/1 5/1
I beleive Rameau (still have to finish that book) thinks 1/1 5/4 3/2 is the
only consonant chord too and dissonance starts with the V7 chord 1/1 5/4 3/2
16/9 (though not sure if he agrees on this beeing 16/9 perhaps he thought it
should be 225/128)

>
> >"Whether less consonant reverse harmonic segments appear in actual music
> nonetheless is irrelevant."
> But is it? Is it really "ok" to just settle for the fact we can never
> use B5 and C5 or E5 and G5 together to make a consonant chord? You made a
> very relevant statement about reverse harmonic series' sounding considerably
> worse than regular ones...and I was simply giving an example of how reverse
> harmonic series pose a significant obstacle to chord theory (IE they force
> you to avoid certain intervals religiously in making good-sounding chords).
>

I don't think consonance and good sounding are equal. I think extremely
dissonant chords can be very good sounding if in the right musical context.

Also don't think JI is for diatonic music only.
Though chromatic music is even harder to translate in JI than diatonic
music.

> >"The thread was about the difference in consonance between 1/1 5/4 3/2 and
> for instance 1/1 6/5 3/2."
> Right...and you brought up a great point (the terrible sound of a
> reversed harmonic series) and I brought up a possible application of your
> point. Is that really such a crime?
>

No crimes commited on this list except perhaps by me yesterday posting too
many messages without thinking :)

Would be great to have you back in JI land though! :)
I myself am thinking now that the best way to figure out JI is to start with
normal 12tet classical music and try to translate it to JI.
Some music works out perfectly easily, other music gives trouble and this is
where i think the most can be learned by solving the trouble.

🔗Cameron Bobro <misterbobro@...>

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
> Btw Michael, I think the most consonant 8-notes per octave is 2 >series of 4
> notes connected by pure fifths, one a pure third apart from the other.1/1
> 3/2 9/8 27/16
> 5/4 15/8 45/32 135/128
>
> You can actually play all notes together in very consonant ways, >but not in
> one octave.
> play for instance 1/1 3/2 9/4 27/8 5/1 15/2 45/4 135/8
> nice huh :)
>
> Here the interval matrix:
> 1/1 : 135/128 9/8 5/4 45/32 3/2 27/16 15/8 2/1
> 135/128: 16/15 32/27 4/3 64/45 8/5 16/9 256/135 2/1
> 9/8 : 10/9 5/4 4/3 3/2 5/3 16/9 15/8 2/1
> 5/4 : 9/8 6/5 27/20 3/2 8/5 27/16 9/5 2/1
> 45/32 : 16/15 6/5 4/3 64/45 3/2 8/5 16/9 2/1
> 3/2 : 9/8 5/4 4/3 45/32 3/2 5/3 15/8 2/1
> 27/16 : 10/9 32/27 5/4 4/3 40/27 5/3 16/9 2/1
> 15/8 : 16/15 9/8 6/5 4/3 3/2 8/5 9/5 2/1
> 2/1
>
> you can take it even further to 12 tones like this:
> 1/1 3/2 9/4 27/8 5/1 15/2 45/4 135/8 25/1 75/2 225/4 675/8
>
> or 16 tones:
> 1/1 3/2 9/4 27/8 5/1 15/2 45/4 135/8 25/1 75/2 225/4 675/8 125/1 375/2
> 1125/4 3375/8
>
> Nice pattern huh :)
> You can continue this from 1 hertz to above hearing and nowhere do >2 tones
> sound together with a beating ratio of greater than X/8.
> All sound much better than doing these things pythagorean.
>

Yes, it's a very nice pattern indeed, very smooth sounding. The earliest example of this I know of is from 1739. This is just a small subset of an entire approach, especially explored and detailed since the late 19th century, and by strange coincidence right here at the tuning list you have some people who may very well be some of the knowledgable in the world on the topic, they've been working on it and documenting it for decades (including precisely the scales you just posted).

> Please come back to JI land ;-)
> it's much more interesting than phi and sounds better and has the >potential
> to fully explain common practice music and create completely new >music.
>

55:60:63:68:72:76:89

is also known as "Michael's phi scale".

-Cameron Bobro

🔗Andreas Sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
> The 4:5 of the major fits in more nicely with this
> than the 5:6 of the minor.

Fully agreed Marcel,

that meets also Euler's
'Gradus-suavits' considerations:
http://home.datacomm.ch/straub/mamuth/mamufaq.html
Formula of consonance:
"
Gs: Ratio (inteval) degree in 'Gradus-suavitatis' ranking
2: 1/2 (octave) 2
3: 3/2 (fifth) 4
4: 4/3 (forth) 5
5: 5/4 (major third), 5/3 (major sixth)
8: 6/5 (minor third), 9/8 (major whole tone),
8: 8/5 (minor sixth) 8
10: 10/9 (minor whole tone), 9/5 (minor seventh),
10: 15/8 (major seventh) 10
11: 16/15 (diatonic semitone) 11
14: 81/64 (pythagorean major third), 45/32 (tritone) 14
"
http://vladimir_ladma.sweb.cz/english/music/articles/links/mfreson.htm
For more details see the references:
http://www.sonic-arts.org/monzo/euler/euler-en.htm
http://www.georghajdu.de/fileadmin/material/articles/LowEnergy.pdf
http://eamusic.dartmouth.edu/masters/courses/harmony.html

Online calculator for G(m,n)
http://www.mathematik.com/Piano/index.html

Or more formally :
http://www.soi.city.ac.uk/project/DOC_TechReport/TR_2008_DOC_02.pdf
http://arxiv.org/pdf/math/0402204

bye
A.S.

🔗Michael Sheiman <djtrancendance@...>

Marcel>"However, the reason for this might be because, as Carl has pointed out,
there is an uncertainty around each interval which is why we hear for
instance 6/5, 7/6, 19/16 etc...all as "minor thirds". "

Agreed. In fact the arithmetic series scale I've posted about several times contains 24/17 in the midst of many x/16 type fractions. However, the fact 24/17 is so close to x/16 makes it able to be heard as 23/16 (not exactly sure what the technical terms for the C to F# interval is...something like an augmented fourth?).

A rule of thumb I usually follow is if the root has a different common denominator in a triad than the rest of the chord then the other 2 tones must share a common denominator to provide the mind enough of a cue to "fit it in place" to the harmonic series. And if a note other than the root has a different denominator than the harmonic series all 3 notes approach on, the only requirement is the other two notes must be very close to the series itself (within about 20 cents or each other and the third tone).

And agree with you, there is no one minor third but several possible near-neighbors. Furthermore I have found personally, that some degree of temperament actually produces some desirable and relaxed slow-beating where the amplitude fades a bit...at least to my ears, having a large group of notes near the x/16 harmonic series segment sounds a lot more flowing than the mechanical beating of a perfect x/16 harmonic series: some variation/imperfection just seems to make things more natural, at least to me.

-Michael