Yahoo Tuning Groups Ultimate Backup tuning interesting arithmetic

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
> 13/11 * 14/11 = 3/2 * 364/363
> 19/15 * 19/16 = 3/2 * 361/360
> 13/11 * 19/15 = 3/2 * 494/495
>
> ...what can be made of this pseudo-tempering? 13/11 certainly
> looks like it should be quite recognizable as a minor 3rd...
> ~~~T~~~

Gene used 14/11 in his "grail" WT. For me, it's simply way
the heck too sharp for common-practice music. 13/11 is a
beautiful minor 3rd, but even with a 696-cent 5th, it'll
produce get a 407-cent major 3rd, which is also sharper than
I'd like.

It happens I just found another 2.3.17.19 WT last night.
Though more unequal than my previous "vrwt", it avoids the
problem of that tuning, the 694-cent 5th. Here the flattest
5th is 696 cents, the sharpest 3rd remains 24/19, and there
is still no harmonic waste. It also seems to have a more
traditional key contrast pattern than my vrwt.

!
{2 3 17 19} well temperament.
12
!
1024/969
272/243
384/323
64/51
4/3
24/17
256/171
512/323
256/153
16/9
32/17
2/1
!
! Convex backbone:
!
! 32/19-----24/19-----36/19
! /|\ /|\ /|\
! / | \ / | \ / | \
! /17/12-----17/16-----51/32\
! / .' '. \ /.' '.\ /.' '.\
! 4/3-------1/1-------3/2-------9/8
!

I've been playing with it in pianoteq, and I think I might
throw it on my Yamaha U5.

-Carl

🔗Tom Dent <stringph@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@> wrote:
> >
> > 13/11 * 14/11 = 3/2 * 364/363
> > 19/15 * 19/16 = 3/2 * 361/360
> > 13/11 * 19/15 = 3/2 * 494/495
> >
> > ...what can be made of this pseudo-tempering? 13/11 certainly
> > looks like it should be quite recognizable as a minor 3rd...
> > ~~~T~~~
>
> Gene used 14/11 in his "grail" WT. For me, it's simply way
> the heck too sharp for common-practice music. 13/11 is a
> beautiful minor 3rd, but even with a 696-cent 5th, it'll
> produce a 407-cent major 3rd, which is also sharper than
> I'd like.
>
> It happens I just found another 2.3.17.19 WT last night.
> Though more unequal than my previous "vrwt", it avoids the
> problem of that tuning, the 694-cent 5th. Here the flattest
> 5th is 696 cents, the sharpest 3rd remains 24/19, and there
> is still no harmonic waste. It also seems to have a more
> traditional key contrast pattern than my vrwt.
>
> !
> {2 3 17 19} well temperament.
> 12
> !
> 1024/969
> 272/243
> 384/323
> 64/51
> 4/3
> 24/17
> 256/171
> 512/323
> 256/153
> 16/9
> 32/17
> 2/1
> !
> ! Convex backbone:
> !
> ! 32/19-----24/19-----36/19
> ! /|\ /|\ /|\
> ! / | \ / | \ / | \
> ! /17/12-----17/16-----51/32\
> ! / .' '. \ /.' '.\ /.' '.\
> ! 4/3-------1/1-------3/2-------9/8
> !

Hmm I'll see what I make of that. My current interest is in making
triads like B-D#-F# and F#-A#-C# sound somehow nicer in modified
meantone / temp. ordinaire. I agree that 14/11 is pretty alien, though
somehow 11/7 as a minor 6th isn't quite so bad. Now I can live with
19/15 for a major third in some remote keys, on the understanding that
they are treated *as* remote - ie briefly touched on - so putting
19/15 together with 13/11 might work for F#-A#-C#, and inversely for
F-Ab-C. But then C#-E# becomes somewhat worse...
~~~T~~~

🔗Tom Dent <stringph@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> It happens I just found another 2.3.17.19 WT last night.
> Though more unequal than my previous "vrwt", it avoids the
> problem of that tuning, the 694-cent 5th. Here the flattest
> 5th is 696 cents, the sharpest 3rd remains 24/19, and there
> is still no harmonic waste. It also seems to have a more
> traditional key contrast pattern than my vrwt.
>
> !
> {2 3 17 19} well temperament.
> 12
> !
> 1024/969
> 272/243
> 384/323
> 64/51
> 4/3
> 24/17
> 256/171
> 512/323
> 256/153
> 16/9
> 32/17
> 2/1
> !
> ! Convex backbone:
> !
> ! 32/19-----24/19-----36/19
> ! /|\ /|\ /|\
> ! / | \ / | \ / | \
> ! /17/12-----17/16-----51/32\
> ! / .' '. \ /.' '.\ /.' '.\
> ! 4/3-------1/1-------3/2-------9/8
> !
>
> I've been playing with it in pianoteq, and I think I might
> throw it on my Yamaha U5.
>
> -Carl

I've looked into this one - I would want to put A somewhat higher,
essentially D major is the worst triad here which I don't think is
such a good result.
How about setting A=57/43, then A-E has the same tempering as C-G and
D-A is a lot better. It takes one point of your 'convex backbone' out,
but doesn't spoil any 17/12's or other potential consonances.
~~~T~~~

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
> > It happens I just found another 2.3.17.19 WT last night.
> > Though more unequal than my previous "vrwt", it avoids the
> > problem of that tuning, the 694-cent 5th. Here the flattest
> > 5th is 696 cents, the sharpest 3rd remains 24/19, and there
> > is still no harmonic waste. It also seems to have a more
> > traditional key contrast pattern than my vrwt.
> >
> > !
> > {2 3 17 19} well temperament.
> > 12
> > !
> > 1024/969
> > 272/243
> > 384/323
> > 64/51
> > 4/3
> > 24/17
> > 256/171
> > 512/323
> > 256/153
> > 16/9
> > 32/17
> > 2/1
> > !
>
> I've looked into this one - I would want to put A somewhat
> higher, essentially D major is the worst triad here which
> I don't think is such a good result.
> How about setting A=57/43, then A-E has the same tempering
> as C-G and D-A is a lot better. It takes one point of your
> 'convex backbone' out, but doesn't spoil any 17/12's or other
> potential consonances.
> ~~~T~~~

Hi Tom- You want *A* to be 57/43??

-Carl

🔗Tom Dent <stringph@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@> wrote:
> >
> > > It happens I just found another 2.3.17.19 WT last night.
> > > Though more unequal than my previous "vrwt", it avoids the
> > > problem of that tuning, the 694-cent 5th. Here the flattest
> > > 5th is 696 cents, the sharpest 3rd remains 24/19, and there
> > > is still no harmonic waste. It also seems to have a more
> > > traditional key contrast pattern than my vrwt.
> > >
> > > !
> > > {2 3 17 19} well temperament.
> > > 12
> > > !
> > > 1024/969
> > > 272/243
> > > 384/323
> > > 64/51
> > > 4/3
> > > 24/17
> > > 256/171
> > > 512/323
> > > 256/153
> > > 16/9
> > > 32/17
> > > 2/1
> > > !
> >
> > I've looked into this one - I would want to put A somewhat
> > higher, essentially D major is the worst triad here which
> > I don't think is such a good result.
> > How about setting A=57/43, then A-E has the same tempering
> > as C-G and D-A is a lot better. It takes one point of your
> > 'convex backbone' out, but doesn't spoil any 17/12's or other
> > potential consonances.
> > ~~~T~~~
>
> Hi Tom- You want *A* to be 57/43??
>
> -Carl
>
Actually, no, spot the elementary typing error... 43 would kind of
ruin the whole [2 3 17 19] idea, wouldn't it. Anyway the basis of the
suggestion was to raise A so that F-A was equal to G-B rather than to C-E.
If I were going to incorporate three 17/12's in a WT I would look
rather at A-Eb or D-G# (or their inversions) than C-F#, which will be
significantly narrower than 600c if C / G / D major are to be better
than ET.
Anyway in this game (irregular WTs) there is a tradeoff between
satisfying 'musical' expectations and nice-looking mathematical
structure, Carl's 'backbone' does remarkably well considering the
mathematical restrictions.
~~~T~~~

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
> > > > !
> > > > {2 3 17 19} well temperament.
> > > > 12
> > > > !
> > > > 1024/969
> > > > 272/243
> > > > 384/323
> > > > 64/51
> > > > 4/3
> > > > 24/17
> > > > 256/171
> > > > 512/323
> > > > 256/153
> > > > 16/9
> > > > 32/17
> > > > 2/1
> > > > !
> > >
> > > I've looked into this one - I would want to put A somewhat
> > > higher, essentially D major is the worst triad here which
> > > I don't think is such a good result.
> > > How about setting A=57/43, then A-E has the same tempering
> > > as C-G and D-A is a lot better. It takes one point of your
> > > 'convex backbone' out, but doesn't spoil any 17/12's or other
> > > potential consonances.
> > > ~~~T~~~
> >
> > Hi Tom- You want *A* to be 57/43??
> >
> > -Carl
> >
> Actually, no, spot the elementary typing error... 43 would
> kind of ruin the whole [2 3 17 19] idea, wouldn't it.

Sorry, I'm not in the mood to guess. Give your suggestion
and I'm happy to tell you what I make of it. I don't agree
that D is the worst triad, so I thought we may be talking
about different modes or something.

-Carl

🔗Tom Dent <stringph@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@> wrote:
> >
> > > > > !
> > > > > {2 3 17 19} well temperament.
> > > > > 12
> > > > > !
> > > > > 1024/969
> > > > > 272/243
> > > > > 384/323
> > > > > 64/51
> > > > > 4/3
> > > > > 24/17
> > > > > 256/171
> > > > > 512/323
> > > > > 256/153
> > > > > 16/9
> > > > > 32/17
> > > > > 2/1
> > > > > !
> > > >
> > > > I've looked into this one - I would want to put A somewhat
> > > > higher, essentially D major is the worst triad here which
> > > > I don't think is such a good result.
> > > > How about setting A=57/43, then A-E has the same tempering
> > > > as C-G[,] and D-A is a lot better. It takes one point of
your
> > > > 'convex backbone' out, but doesn't spoil any 17/12's or
other
> > > > potential consonances.
> > > > ~~~T~~~
> > >
>
> Sorry, I'm not in the mood to guess. Give your suggestion
> and I'm happy to tell you what I make of it. I don't agree
> that D is the worst triad, so I thought we may be talking
> about different modes or something.
>
> -Carl

57/34 then. D major as you specified above has a 32/27 in it, which
means the largest possible deviation in a major triad given that no
fifths are wide.
~~~T~~~

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
> > > > > > !
> > > > > > {2 3 17 19} well temperament.
> > > > > > 12
> > > > > > !
> > > > > > 1024/969
> > > > > > 272/243
> > > > > > 384/323
> > > > > > 64/51
> > > > > > 4/3
> > > > > > 24/17
> > > > > > 256/171
> > > > > > 512/323
> > > > > > 256/153
> > > > > > 16/9
> > > > > > 32/17
> > > > > > 2/1
> > > > > > !
> >
> > Sorry, I'm not in the mood to guess. Give your suggestion
> > and I'm happy to tell you what I make of it. I don't agree
> > that D is the worst triad, so I thought we may be talking
> > about different modes or something.
> >
> > -Carl
>
> 57/34 then. D major as you specified above has a 32/27 in it,
> which means the largest possible deviation in a major triad
> given that no fifths are wide.
> ~~~T~~~

F#-A is 32/27, but the deviation of the minor third in a
major triad does not have the same impact on the quality
of that triad as the deviation of the major third.

Here is a tally of 19-limit consonances...

old new
19/18 2 1
18/17 3 3
17/16 3 2
19/17 1 0
9/8 4 3
19/16 4 5
24/19 3 2
4/3 7 6
17/12 4 4
----------------------
31 26

We shouldn't make too big of a deal out of this sum (though
the difference does double when we consider inversions).

In logland, the new scale has only one 696-cent 5th instead
of two, which is good. Both scales have six major 3rds worse
than 400 cents, but the new scale has only two that are
really bad, compared to three in the old scale.

The cycle-of-5ths pattern of major 3rds is slightly more
round in the new scale.

Really the way to do this is to put a large number of RWTs
on an x/y plot by number of 19-limit consonances and
total deviation from 12-ET, and then observe if there are
any sudden jumps.

-Carl

🔗Carl Lumma <carl@...>

🔗chrisvaisvil@...

Is 7 limit 7 tones per octave? - I am still learning

Sent via BlackBerry from T-Mobile

-----Original Message-----
From: "Carl Lumma" <carl@lumma.org>

Date: Tue, 23 Sep 2008 18:16:10
To: <tuning@yahoogroups.com>
Subject: [tuning] Re: interesting arithmetic

--- In tuning@yahoogroups.com, "Chris Vaisvil" <chrisvaisvil@...> wrote:
>
> out of curiousity - has anyone tried using the Pythagorean
> tuning without tempering the comma?
>
> it obviously means doing without "true" octaves.... but I
> wonder if it might work.

I think you're talking about the 7th root of 3/2 tuning
again. The octaves get stretched to 1203 cents or something.

-Carl

🔗Kalle Aho <kalleaho@...>

🔗Michael Sheiman <djtrancendance@...>

Anyone here please feel free to correct me if I am wrong...

But wouldn't a non-octave or non-tritave scale force all the 2x/1 and 3x/1 overtones to drift further and further away from root notes? For example the even harmonics "C5" and "C8" would not work well together in a non octave or tri-tave based scale...that is, unless you are looking for "bluer"/less-consonant sounding tones on purpose.

For the record, I have dabbled with tritave scales but usually lean heavily on the 5th IE 3/2 tritave...and that is also known as the octave. Often it seems there is only so much you can bend the rules...beside using different terminology to acheive very similar answers.
__________________________________________
Mostly, Just Intonation seems ideal if a bit
unadventurous...but (far as "hacked scales") I have found you CAN very easily put two completely different (say, 5 note) scales on each side of the stereo spectrum and use this binaurality to get the flexibility of 10 note scale tonal expression with 5 note scale stability.

Your idea poses an interesting side question to me...can you get a very natural sound by shifting every other overtone (octave or tritave) to the opposite side of the stereo spectrum and/or stretching both the scale AND overtones to match each other?
______________________________
As I recall Bill Sethares has done a lot with simultaneous scale and overtone stretching...then again it takes very complex software to stretch overtones.
Maybe I'll try to implement a "pythagorean comma harmonic stretcher" in the next version of my open-source SpectraL mixing software. :-)

--- On Tue, 9/23/08, Chris Vaisvil
<chrisvaisvil@...m> wrote:
From: Chris Vaisvil <chrisvaisvil@...>
Subject: Re: [tuning] Re: interesting arithmetic
To: tuning@yahoogroups.com
Date: Tuesday, September 23, 2008, 8:46 AM

out of curiousity - has anyone tried using the Pythagorean tuning without tempering the comma?

it obviously means doing without "true" octaves.... but I wonder if it might work.

🔗chrisvaisvil@...

Thanks for the reference. My thought is why try to force a replication of notes ever octave? I am saying do not justly tune. Make each octave unique.
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: "Kalle Aho" <kalleaho@mappi.helsinki.fi>

Date: Tue, 23 Sep 2008 19:45:37
To: <tuning@yahoogroups.com>
Subject: [tuning] Re: interesting arithmetic

--- In tuning@yahoogroups.com, chrisvaisvil@... wrote:
>
> Is 7 limit 7 tones per octave? - I am still learning

http://en.wikipedia.org/wiki/Limit_(music)

Kalle

🔗Carl Lumma <carl@...>

Hi Chris,

> Is 7 limit 7 tones per octave? - I am still learning

Short answer: No.

Medium answer: "7-limit" refers to the complexity of the
harmony in a scale. 4:5:6 triads (also spelled 1/1 5/4 3/2)
are considered "5-limit" because the biggest number in any
of the ratios is 5.

Long answer:
http://en.wikipedia.org/wiki/Limit_(music)

In the message you quoted though,

> I think you're talking about the 7th root of 3/2 tuning
> again. The octaves get stretched to 1203 cents or something.
>
> -Carl

the thing was "7th root". The square root of 9 is 3.
That is also the "2th root of 9". A 12-ET semitone has
size 12th root of 2. Since human hearing is logarithmic
with respect to pitch, to divide an interval equally we
take the nth root of that interval.

In this case, "7th root of 3/2" means dividing the
perfect fifth into 7 equal steps. You can do it without
the fancy roots by using cents. Cents are a logarithmic
measure already, so you just divide by 7 instead of
taking the 7th root. 702 cents / 7 = 100.3 cents.
If you take 12 consecutive ones of those, you get
1203-cent octaves.

Peace!

-Carl

🔗chrisvaisvil@...

Thanks Carl - that helpped on a lot of levels!
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: "Carl Lumma" <carl@lumma.org>

Date: Tue, 23 Sep 2008 21:02:16
To: <tuning@yahoogroups.com>
Subject: [tuning] Re: interesting arithmetic

Hi Chris,

> Is 7 limit 7 tones per octave? - I am still learning

Short answer: No.

Medium answer: "7-limit" refers to the complexity of the
harmony in a scale. 4:5:6 triads (also spelled 1/1 5/4 3/2)
are considered "5-limit" because the biggest number in any
of the ratios is 5.

Long answer:
http://en.wikipedia.org/wiki/Limit_(music)

In the message you quoted though,

> I think you're talking about the 7th root of 3/2 tuning
> again. The octaves get stretched to 1203 cents or something.
>
> -Carl

Peace!

-Carl

🔗Tom Dent <stringph@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@> wrote:
> >
> > > !
> > > {2 3 17 19} well temperament.
> > > 12
> > > !
> > > 1024/969
> > > 272/243
> > > 384/323
> > > 64/51
> > > 4/3
> > > 24/17
> > > 256/171
> > > 512/323
> > > 256/153
> > > 16/9
> > > 32/17
> > > 2/1
> > > !
> >
> > [A=]57/34 then. D major as you specified above has a 32/27 in it,
> > which means the largest possible deviation in a major triad
> > given that no fifths are wide.
> > ~~~T~~~
>
> F#-A is 32/27, but the deviation of the minor third in a
> major triad does not have the same impact on the quality
> of that triad as the deviation of the major third.

So the theory goes - I meant the effect of a 32/27 on the relative
qualities of the 5th and major 3rd. The 3rd can't be narrow enough for
comfort unless the 5th is already fairly flat, and I find the
combination of a quite sharp 3rd and fairly flat 5th (say 0 404 698)
more jarring than either one on its own. I'd happily forget about the
m3 deviation per se, but it equals the sum of M3+5 deviations.

> Here is a tally of 19-limit consonances...
>
> old new
> 19/18 2 1
> 18/17 3 3
> 17/16 3 2
> 19/17 1 0
> 9/8 4 3
> 19/16 4 5
> 24/19 3 2
> 4/3 7 6
> 17/12 4 4
> ----------------------
> 31 26
>
> We shouldn't make too big of a deal out of this sum

I'd only count the last 4 or 5 lines of the table as 'consonance'
myself, but I guess you can make whatever table you want, this one
would reflect the mathematical viewpoint.

> In logland, the new scale has only one 696-cent 5th instead
> of two, which is good. Both scales have six major 3rds worse
> than 400 cents

As you expect, really, unless you take extra special measures!

> but the new scale has only two that are
> really bad, compared to three in the old scale.

So 24/19 is bad in log-land but good as a consonance? Or am I
over-/misinterpreting?

> Really the way to do this is to put a large number of RWTs
> on an x/y plot by number of 19-limit consonances and
> total deviation from 12-ET, and then observe if there are
> any sudden jumps.
>
> -Carl

*Total* deviation is always the same number, if you stick to the 'pure
and narrow' (ie no wide fifths). Perhaps you mean maximum deviation
(of 5ths or M3s or triads)?
Or total deviation over all the worst 5ths/M3s/triads?
~~~T~~~

🔗Carl Lumma <carl@...>

Tom wrote:

> > F#-A is 32/27, but the deviation of the minor third in a
> > major triad does not have the same impact on the quality
> > of that triad as the deviation of the major third.
>
> So the theory goes - I meant the effect of a 32/27 on the
> relative qualities of the 5th and major 3rd. The 3rd can't
> be narrow enough for comfort unless the 5th is already fairly
> flat, and I find the combination of a quite sharp 3rd and
> fairly flat 5th (say 0 404 698) more jarring than either one
> on its own. I'd happily forget about the m3 deviation per se,
> but it equals the sum of M3+5 deviations.

Well in that case, D isn't the worst Maj triad in the scale
I posted.

> > Here is a tally of 19-limit consonances...
> >
> > old new
> > 19/18 2 1
> > 18/17 3 3
> > 17/16 3 2
> > 19/17 1 0
> > 9/8 4 3
> > 19/16 4 5
> > 24/19 3 2
> > 4/3 7 6
> > 17/12 4 4
> > ----------------------
> > 31 26
> >
> > We shouldn't make too big of a deal out of this sum
>
> I'd only count the last 4 or 5 lines of the table as
> 'consonance' myself, but I guess you can make whatever
> table you want, this one would reflect the mathematical
> viewpoint.

There are tiny dips near 17/16 and 17/8 when continuously
tuning a synthesizer. The 17/8 dip is greater when
it's a min 9th on top of a major 7th chord, and greater
still when you have 2 or more of: 10 12 15 to go with it.
There may be a dip around 17/9 (here represented as 18/17)
in some circumstances.
There's a tiny dip around 19/16, and 16:19:24 may even
be the harmonic entropy minimum of the minor triad.
9/8 and 9/4 are probably my favorite consonances of all
time, especially when paired with a 5/4 approximation.

> > but the new scale has only two that are
> > really bad, compared to three in the old scale.
>
> So 24/19 is bad in log-land but good as a consonance? Or am I
> over-/misinterpreting?

Above, 24/19 also stands in for 19/12. I'm dubious that
24/19 is better in any substantive way than other nearby
thirds, but if I have to have sharp thirds I might as
well have 24/19s.

> > Really the way to do this is to put a large number of RWTs
> > on an x/y plot by number of 19-limit consonances and
> > total deviation from 12-ET, and then observe if there are
> > any sudden jumps.
>
> *Total* deviation is always the same number, if you stick
> to the 'pure and narrow' (ie no wide fifths). Perhaps you
> mean maximum deviation (of 5ths or M3s or triads)?
> Or total deviation over all the worst 5ths/M3s/triads?

I mean total abs deviation, sorry. Max deviation probably
approximates that fairly well. Scala finds things like
the biggest semitone / smallest semitone, and something
called "interval standard deviation".

At any rate, what do you think of the idea? I'm willing
to be you're more handy with the appropriate tools (e.g.
Matlab) than I.

-Carl

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Tom wrote:
>
> > > F#-A is 32/27, but the deviation of the minor third in a
> > > major triad does not have the same impact on the quality
> > > of that triad as the deviation of the major third.
> >
> > So the theory goes - I meant the effect of a 32/27 on the
> > relative qualities of the 5th and major 3rd. The 3rd can't
> > be narrow enough for comfort unless the 5th is already fairly
> > flat, and I find the combination of a quite sharp 3rd and
> > fairly flat 5th (say 0 404 698) more jarring than either one
> > on its own. I'd happily forget about the m3 deviation per se,
> > but it equals the sum of M3+5 deviations.
>
> Well in that case, D isn't the worst Maj triad in the scale
> I posted.

Actually meant to chop the last phrase of the quote off the
quote there. It equals the M3+5 deviations only if you
preserve the signs, and doing so doesn't jive with the
psychoacoustics.

-Carl

🔗Tom Dent <stringph@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> > Tom wrote:
> >
> > > I'd happily forget about the m3 deviation per se,
> > > but it equals the sum of M3+5 deviations.
> >
> > Well in that case, D isn't the worst Maj triad in the scale
> > I posted.
>
> Actually meant to chop the last phrase of the quote off the
> quote there. It equals the M3+5 deviations only if you
> preserve the signs, and doing so doesn't jive with the
> psychoacoustics.
>
> -Carl

I guess I should be more explicit (George is the expert on this sort
of thing isn't he?) - I meant adding absolute deviations.
Of course we always have
m3 signed deviation = 5th sig.dev - M3 sig.dev.
but I agree that adding up signed deviations this way doesn't tell us
about the triad's sound, so I take the appropriate signs for
'well-temperament' (ie no fifths wide and no M3s narrow) getting
- m3 absolute deviation = - 5th abs.dev. - M3 abs.dev.

so: sum of M3 and 5th abs.dev.'s = m3 abs.dev. in 'WT's.

(Also psychoacoustically, and subjectively and personally, I find
major triads where the 3rd is pulling strongly upwards and the 5th
strongly downwards quite discomfiting, but that may be something yet
unquantified...)

For the same arithmetic reason, the sum of signed deviations, and the
sum of absolute deviations, are always the same in a wide-fifth-less
'WT', so your minimization has to operate over the maximum deviation,
or sum of squared deviations, or something equally nonlinear.

The problem in programming with n-limit RWT's is to get an algorithm
which finds them in a systematic way. Perhaps you could start with the
Pythagorean 12-note scale and implement some genetic algorithm where
the mutations involve moving one or more notes by some comma or
other... but I've never done genetic programming myself, so who knows.
~~~T~~~

🔗Carl Lumma <carl@...>

Hi Tom,

> so: sum of M3 and 5th abs.dev.'s = m3 abs.dev. in 'WT's.

Of course.

> (Also psychoacoustically, and subjectively and personally,
> I find major triads where the 3rd is pulling strongly upwards
> and the 5th strongly downwards quite discomfiting, but that
> may be something yet unquantified...)

0-405-702 sounds marginally worse to me than 0-402-696,
and 0-702-1605 sounds clearly worse than 0-696-1602. Do you
disagree?

I think the catch is, cents error starts to not work at
this level of subtly. You have to pay attention to harmonic
entropy. I'll do sum of dyadic entropy...

0-405-702 = 13.2697735
0-402-696 = 13.2975758

0-702-1605 = 13.1649794
0-696-1602 = 13.1871984

Nope, that still agrees with you. Well, I'm stumped.
Maybe I have an aberrant hatred of sharp M3s, or maybe
triadic harmonic entropy would vindicate me. Unfortunately
I don't know how to calculate it.

> The problem in programming with n-limit RWT's is to get an
> algorithm which finds them in a systematic way.

Yes. I was originally hoping to use Tonescape to find them
as periodicity blocks in the 3.17.19 lattice, but the unison
vectors one needs are things like 81/64, and if I understood
correctly, Tonescape doesn't allow UVs outside of the space
you're working in.

I came up with the present RWT by manually going through a
list of intervals near the origin of the 3.17.19 lattice and
near 12-ET in size, that I generated in Scheme. It wasn't
a great approach.

So now I'm thinking of searching possible chains of 5ths,
under the constraint that there be 6 or 7 just 5ths.
Then the remaining 5 or 6 intervals can be drawn from a
list that is composed of 5ths of the appropriate size,
that are created out of stacked 19-limit 3rds, or a perfect
4th + a 19-limit 2nd, or something like that. Then you
just deal with permutations on the chain of fifths. Still
not something I'd normally look forward to doing.

-Carl

🔗Tom Dent <stringph@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> 0-405-702 sounds marginally worse to me than 0-402-696,
> and 0-702-1605 sounds clearly worse than 0-696-1602. Do you
> disagree?

Definitely!

> I think the catch is, cents error starts to not work at
> this level of subtly. You have to pay attention to harmonic
> entropy. I'll do sum of dyadic entropy...
>
> 0-405-702 = 13.2697735
> 0-402-696 = 13.2975758
>
> 0-702-1605 = 13.1649794
> 0-696-1602 = 13.1871984
>
> Nope, that still agrees with you. Well, I'm stumped.
> Maybe I have an aberrant hatred of sharp M3s, or maybe
> triadic harmonic entropy would vindicate me. Unfortunately
> I don't know how to calculate it.

Maybe you just like flat fifths. For me it isn't even close because
you're offering 3 cents difference on an already bad 3rd versus 6
cents of mud in the 5th.

It might also be interesting to run dyadic entropy on the sequence
0-408-702
0-406-700
0-404-698
0-402-696
(etc)
to see if there is any optimum 'solution' to a major triad with a
32/27... perhaps 0-409-703 would win because of 19/15.

> searching possible chains of 5ths,
> under the constraint that there be 6 or 7 just 5ths.
> Then the remaining 5 or 6 intervals can be drawn from a
> list that is composed of 5ths of the appropriate size,
> that are created out of stacked 19-limit 3rds, or a perfect
> 4th + a 19-limit 2nd, or something like that. Then you
> just deal with permutations on the chain of fifths.

So equivalent to starting with a Pythagorean comma and chopping
19-limit bits off it. You have to be clever/persistent to find
suitable 'bits' and how they can be permuted usefully. So far we have
512/513, 323/324, and the ugly/useful leftover 4617/4624...
~~~T~~~

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> > 0-405-702 sounds marginally worse to me than 0-402-696,
> > and 0-702-1605 sounds clearly worse than 0-696-1602. Do you
> > disagree?
>
> Definitely!

What timbre are you using? Harpsichord?

> It might also be interesting to run dyadic entropy on the sequence
> 0-408-702
> 0-406-700
> 0-404-698
> 0-402-696
> (etc)
> to see if there is any optimum 'solution' to a major triad with a
> 32/27... perhaps 0-409-703 would win because of 19/15.

Here's an Excel spreadsheet I made:
http://lumma.org/stuff/DyadicHarmonicEntropyCalc.xls

> > searching possible chains of 5ths,
> > under the constraint that there be 6 or 7 just 5ths.
> > Then the remaining 5 or 6 intervals can be drawn from a
> > list that is composed of 5ths of the appropriate size,
> > that are created out of stacked 19-limit 3rds, or a perfect
> > 4th + a 19-limit 2nd, or something like that. Then you
> > just deal with permutations on the chain of fifths.
>
> So equivalent to starting with a Pythagorean comma and chopping
> 19-limit bits off it. You have to be clever/persistent to find
> suitable 'bits' and how they can be permuted usefully. So far we
> have 512/513, 323/324, and the ugly/useful leftover 4617/4624...

Yup.

-Carl