Yahoo Tuning Groups Ultimate Backup tuning new (old) music

The 5-limit typical scale is something like this, usually:

1/1 16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5 5/3 9/5 16/15 2/1

you can see it as a lattice:

5/3 5/4 15/16 45/32
4/3 1/1 3/2 9/8
16/15 8/5 6/5 9/5

thirds above and below are vertical, fifths are horizontal

this tuning is called the 5-limit duodene. I'm always amazed at it's beauty
and utility. in the quest for the 7- 11- 13- and beyond limits, people often
forget the simple power of pure thirds and fifths. many i guess find them
old and boring.

I can modulate this with SysEx messages to any key in TiMidity, so if I need
a pure 3:2 fifth from D, instead of 5/3 I can have 27/16 for instance....

-AKJ

On Fri, Sep 11, 2009 at 6:37 PM, Chris Vaisvil <chrisvaisvil@...>wrote:

>
>
> This is truly beautiful!!
>
> What extra intervals do you got with 5-limit?
>
> On Fri, Sep 11, 2009 at 4:12 PM, Aaron Johnson <aaron@...> wrote:
>
>>
>>
>> Hey all,
>>
>> I wrote this around Oct 2008, but only just sequenced and recorded it
>> last night.
>>
>> 5-limit JI, using SYSEX messages to TiMidity to change tuning tables
>> to stay pure (where it truly counts) throughout.
>>
>> 1'17"
>>
>> http://www.akjmusic.com/audio/AndanteDoloroso.ogg
>> http://www.akjmusic.com/audio/AndanteDoloroso.mp3
>>
>> --
>>
>> Aaron Krister Johnson
>> http://www.akjmusic.com
>> http://www.untwelve.org
>>
>
>
>
>

Aaron Krister Johnson
http://www.akjmusic.com
http://www.untwelve.org

🔗Petr Parízek <p.parizek@...>

AKJ wrote:

> 1/1 16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5 5/3 9/5 16/15 2/1

You have used 16/15 twice, the latter of which I think should have been 15/8 if you're mentioning duodene.

> 5/3 5/4 15/16 45/32

I don't see the reason why you're using 15/16 instead of 15/8 when all the rest are rising intervals and not falling.

> this tuning is called the 5-limit duodene. I'm always amazed at it's beauty and utility.

Yes, it contains 12 pure triads, 6 of which are major and 6 are minor. It can also be classified as an "Euler Genus" of "3^3 5^2" or "3 3 3 5 5". According to Manuel's scale archive, Euler called it the "Genus diatonico-chromaticum hodiernum correctum". The file "Euler.scl" containing the scale for the "Euler's monochord" is essentially the same scale, only in a different transposition (i.e. the fifths go from -1 to +3 and the thirds go from 0 to +2). This means thatt the degree from which you can invert everything compared to the 1/1 (the one which I was calling the "degree of symmetry") is #7 (i.e. the fifth) for the regular duodene and #10 (the augmented sixth) for the transposed version.

Another possibility is to use an Euler genus of "3^7 5^2", which may be viewed as an untempered version of the schismatic temperament and gives you a 24-tone scale.

Petr

🔗Petr Parízek <p.parizek@...>

🔗Aaron Johnson <aaron@...>

Thanks Petr, you caught my 2 typos...I wrote this fast last night.

Of course, I meant 15/8 instead of 15/16, both times.

Although, to be fair, when we cancel octaves, they are the same pitch! :)

AKJ

On Sat, Sep 12, 2009 at 5:21 AM, Petr Parízek <p.parizek@...> wrote:

>
>
> AKJ wrote:
>
> > 1/1 16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5 5/3 9/5 16/15 2/1
>
> You have used 16/15 twice, the latter of which I think should have been
> 15/8 if you’re mentioning duodene.
>
> > 5/3 5/4 15/16 45/32
>
> I don’t see the reason why you’re using 15/16 instead of 15/8 when all the
> rest are rising intervals and not falling.
>
> > this tuning is called the 5-limit duodene. I'm always amazed at it's
> beauty and utility.
>
> Yes, it contains 12 pure triads, 6 of which are major and 6 are minor. It
> can also be classified as an „Euler Genus“ of „3^3 5^2“ or „3 3 3 5 5“.
> According to Manuel’s scale archive, Euler called it the „Genus
> diatonico-chromaticum hodiernum correctum“. The file „Euler.scl“ containing
> the scale for the „Euler’s monochord“ is essentially the same scale, only in
> a different transposition (i.e. the fifths go from -1 to +3 and the thirds
> go from 0 to +2). This means thatt the degree from which you can invert
> everything compared to the 1/1 (the one which I was calling the „degree of
> symmetry“) is #7 (i.e. the fifth) for the regular duodene and #10 (the
> augmented sixth) for the transposed version.
>
> Another possibility is to use an Euler genus of „3^7 5^2“, which may be
> viewed as an untempered version of the schismatic temperament and gives you
> a 24-tone scale.
>
> Petr
>
>
>
>
>
>
>
>

Aaron Krister Johnson
http://www.akjmusic.com
http://www.untwelve.org

🔗Ozan Yarman <ozanyarman@...>

🔗Petr Pařízek <p.parizek@...>

🔗Aaron Johnson <aaron@...>

Hi Petr,

My ears are obviously not a sensitive as yours. I'm not hearing what you're
hearing at all. As far as I can tell, both 16/15 half steps, and I checked
the MIDI sysex messages, too.

As for 16/15 vs. 27/25, it's just the nature of the lattice of the
Duodene.....most folks prefer the 16/15 above the tonic anyway, I would
think (narrower leading tones from above, etc.) I suppose all of this
depends on if your 3-limit "core chain of fifths" is F-C-G-D or C-G-D-A at
the start.

AKJ

2009/9/13 Petr Pařízek <p.parizek@...>

>
>
> Hi Aaron,
>
> I've listened to the piece more than 5 times already. May I ask what was
> the reason for the particular JI rendering of the passage around 0:24,7? As
> far as I could hear it, the bottom voice goes like "Bb-A-F" as 25/27 and 8/5
> (meaning the intervals of consecutive tones, not from any refference pitch),
> while the top voice also goes like "Bb-A-F" but rather as 25/27 and 81/50.
> If both voices represent the same melodic phrase only in different
> transpositions, why should each of them be rendered differently in JI? And
> why not use 15/16 instead of 25/27 in the first place?
>
> Thanks.
>
> Petr
>
>
>
>
>

Aaron Krister Johnson
http://www.akjmusic.com
http://www.untwelve.org

🔗Petr Pařízek <p.parizek@...>

AKJ wrote:

> As far as I can tell, both 16/15 half steps, and I checked
> the MIDI sysex messages, too.

Either each of us is talking about a different portion of the piece or your instrument is doing something strange. I tried to mix your music with an 88Hz periodic sawtooth wave (to get a drone to which I could compare the Bb and A, which would have to be approx. 4/3 and 5/4 then) and this proved that it was definitely something wider than 15/16.

The other question I was raising was not so much about the usage of the duodene but rather about two different ways of rendering the same melodic progression, and I was interested why it was like that when the only difference in the „contents“ of each voice was actually a transposition into another octave.

Petr

🔗Aaron Johnson <aaron@...>

Petr,

I know exactly where you are talking about in the piece, but I will have to
do a test against an F (in both registers of the canonic passafe) tomorrow
when I get a chance. Not sure how you are arriving at 88hz test tone, but I
can confirm or disconfirm what you say by simply making a test version of
the piece with an F against both voices and find out what happens....

I just tested, out of curiosity, that it was the duodene by running slow
chromatic major thirds up a scale. I was thinking---"maybe TiMidity's
definition of 'pure intonation' is not the 5-limit duodene?" But it turned
out that it was indeed the standard Duodene, so that couldn't explain
it--i.e. it wasn't a case of me thinking I was doing something I wasn't
actually doing, and thus 'hearing' that way....I must confess, this has
happened to me! I once reveled in the pure harmonies of meantone, thinking
it was set on my tuning table with a piano sample, then realized moments
after the fact that it wasn't---strange shift of perception when that
happens. Something similar to the 'unreliability of witnesses'
phenomenon....I actually blushed with embarrassment, even though there was
no one else there to be ashamed in front of. It also attests to how subtle
some of these tuning difference can be in passive listening---I would never
mistake 5-equal for 12-equal, for instance.

Anyway, in this moment in my piece, I'm not hearing what you are talking
about. Does anyone else hear this? I'm certainly aware of different sized
intervals at different points in the piece, but nothing strikes me as out of
line in the passage you point out....and the different registers sound the
same (to me).

If indeed something is amiss, congrats on your sensitivity---it blew past me
for sure! Maybe I would notice easily only after you point it out (if it's
real)...But then again, I can't really tell between 81/64 and 19/15 under
anything but the most lab like settings, where Cameron Bobro simply nailed
it every time in a series of tests.
Lord, I hope I'm not going deaf.....

Best,
AKJ

2009/9/13 Petr Pařízek <p.parizek@...>

>
>
> AKJ wrote:
>
> > As far as I can tell, both 16/15 half steps, and I checked
> > the MIDI sysex messages, too.
>
> Either each of us is talking about a different portion of the piece or your
> instrument is doing something strange. I tried to mix your music with an
> 88Hz periodic sawtooth wave (to get a drone to which I could compare the Bb
> and A, which would have to be approx. 4/3 and 5/4 then) and this proved that
> it was definitely something wider than 15/16.
>
> The other question I was raising was not so much about the usage of the
> duodene but rather about two different ways of rendering the same melodic
> progression, and I was interested why it was like that when the only
> difference in the „contents“ of each voice was actually a transposition into
> another octave.
>
> Petr
>
>
>
>
>
>
>
>

Aaron Krister Johnson
http://www.akjmusic.com
http://www.untwelve.org

🔗Petr Pařízek <p.parizek@...>

🔗Aaron Johnson <aaron@...>

Hey Petr, everyone,

try these files out...

http://www.akjmusic.com/audio/AndanteDoloroso2.ogg
http://www.akjmusic.com/audio/AndanteDoloroso2.mp3

I fixed the issue you (Petr) noticed. There was definitely some beating with
the 'A' when I tested it against an F drone at the moment (around
0'24"-0'25") you pointed out, however, for whatever reason it was consistent
in both octaves (unlike how you reported it)...although this may have to do
with how I was playing it back, the SysEx message not always 'resetting' if
you don't go all the way back to the beginning, etc.

So anyway, these versions improve things, and are surely 16/15 both times.
However, there seems to be a melodic comma anomoly at 0'27"-0'28", the
moment with a C and G sounding...I have to dig and deeper with this to find
out exactly how this works with TiMidity's tuning tables, ie, when one
changes 'key', which pitches change and which stay the same, and function as
'pivots'. Furthermore, there is the undocumented issue of having four
choices: major, minor, passing major, and passing minor. What the difference
between 'passing major' and 'major' is unclear, but I would assume it has to
do with the functioning of pivots. The old artifact you heard may have had
something to do with my choosing 'passing' key signatures in the earlier
version. In this one, I just declared key signatures, but now we hear
(perhaps unavoidable in this JI rendering?) comma shift artifacts.

I have 2 future projects for realizing this piece: adaptive meantone (adjust
only 4th and 5ths to pure, but keep melodic basis), and custom JI tables so
I have precise control over pivots, without guessing how the preset tables
in TiMidity really work...

Best,
AKJ

2009/9/14 Petr Pařízek <p.parizek@...>

>
>
> AKJ wrote:
>
> > Not sure how you are arriving at 88hz test tone, but I can confirm or
> disconfirm
> > what you say by simply making a test version of the piece with an F
> against both voices
> > and find out what happens....
>
> I‘ve just realized that 87Hz is even closer to the F that I was looking
> for.
>
> Maybe it has something to do with what you said earlier, that you may
> change the starting „key“ of the scale during the piece, which might clearly
> explain why the upper voice plays an F in one moment and a comma higher F in
> another. If it’s like that, then it may well have something to do with this
> as well.
>
> Petr
>
>
>
>
>
>
>
>

Aaron Krister Johnson
http://www.akjmusic.com
http://www.untwelve.org

🔗Aaron Johnson <aaron@...>

Hey all,

I think I figured out how the 'passing major' or 'passing minor' works...My
working theory is this: I think this is TiMidity's way of allowing for JI
passages of music to be built on a Pythagorean basis if they modulate around
a circle of fifths. So for instance, specifying Eb major as a key gives you
a duodene transposed to 6/5, whereas Eb 'passing major' gives you a duodene
transposed to 32/37. If I have a chain of fifths modulation from C-F-Bb-Eb,
this comes in handy, allowing there to be a majority of pivot tones from the
old key, while keeping the pure 5-limit relationships of the duodene,
without immediately apparent audible commatic drift.

I redid my SysEx messages to reflect this idea, and here is the result:

http://www.akjmusic.com/audio/AndanteDoloroso3.ogg
http://www.akjmusic.com/audio/AndanteDoloroso3.mp3

Let me know if any of you still hear anything strange that I might have
missed....

Best,
AKJ

P.S. here is the online message where I first learned how TiMidity can use
SysEx
messages for MTS (and extensions):

http://www.mail-archive.com/timidity-talk@.../msg00081.html

On Tue, Sep 15, 2009 at 10:27 AM, Aaron Johnson <aaron@...m> wrote:

> Hey Petr, everyone,
>
> try these files out...
>
> http://www.akjmusic.com/audio/AndanteDoloroso2.ogg
> http://www.akjmusic.com/audio/AndanteDoloroso2.mp3
>
> I fixed the issue you (Petr) noticed. There was definitely some beating
> with the 'A' when I tested it against an F drone at the moment (around
> 0'24"-0'25") you pointed out, however, for whatever reason it was consistent
> in both octaves (unlike how you reported it)...although this may have to do
> with how I was playing it back, the SysEx message not always 'resetting' if
> you don't go all the way back to the beginning, etc.
>
> So anyway, these versions improve things, and are surely 16/15 both times.
> However, there seems to be a melodic comma anomoly at 0'27"-0'28", the
> moment with a C and G sounding...I have to dig and deeper with this to find
> out exactly how this works with TiMidity's tuning tables, ie, when one
> changes 'key', which pitches change and which stay the same, and function as
> 'pivots'. Furthermore, there is the undocumented issue of having four
> choices: major, minor, passing major, and passing minor. What the difference
> between 'passing major' and 'major' is unclear, but I would assume it has to
> do with the functioning of pivots. The old artifact you heard may have had
> something to do with my choosing 'passing' key signatures in the earlier
> version. In this one, I just declared key signatures, but now we hear
> (perhaps unavoidable in this JI rendering?) comma shift artifacts.
>
> I have 2 future projects for realizing this piece: adaptive meantone
> (adjust only 4th and 5ths to pure, but keep melodic basis), and custom JI
> tables so I have precise control over pivots, without guessing how the
> preset tables in TiMidity really work...
>
> Best,
> AKJ
>
> 2009/9/14 Petr Pařízek <p.parizek@...>
>
>
>>
>> AKJ wrote:
>>
>> > Not sure how you are arriving at 88hz test tone, but I can confirm or
>> disconfirm
>> > what you say by simply making a test version of the piece with an F
>> against both voices
>> > and find out what happens....
>>
>> I‘ve just realized that 87Hz is even closer to the F that I was looking
>> for.
>>
>> Maybe it has something to do with what you said earlier, that you may
>> change the starting „key“ of the scale during the piece, which might clearly
>> explain why the upper voice plays an F in one moment and a comma higher F in
>> another. If it’s like that, then it may well have something to do with this
>> as well.
>>
>> Petr
>>
>>
>>
>>
>>
>>
>>
>>
>
>
>
> --
>
> Aaron Krister Johnson
> http://www.akjmusic.com
> http://www.untwelve.org
>
>

Aaron Krister Johnson
http://www.akjmusic.com
http://www.untwelve.org

🔗Petr Pařízek <p.parizek@...>

🔗Aaron Johnson <aaron@...>

Yup, you got it...I understand completely, and agree...that's what I want to
do, but achieving it with the tuning SYSEX MIDI messages is another issue
altogether. I'm not sure the more primitive messages I've been sending can
do the trick. What I'm afraid I'll have to do is define a few table where
there are pivot frequencies.

But as I think throught this, and what you're saying, I realize there is
always going to be some kind of comma problem. If, as you say, I make the
'F' of g-minor a 9/5, then the melodic progression which temporarily leads
to a c-minor harmonic region would be descending like so: 9/10 8/9 15/16
9/8, and I'm not sure it would feel right to have c-minor's 'F' be a 27/20
above 'C'...does this make sense? This is the problem with JI---there are
always compromises, and there's no way to 'hide' the comma in certain chord
progressions....I'd like to try this though, and see how it works.

The real truth is that my piece is written within a 'meantone'
framework--the circle of 5ths type progression really screams that fact....I
supposed what I should do when I have time is try and adaptive meantone
version if I want to have those pure vertical sonorities.

best,
AKJ

2009/9/15 Petr Pařízek <p.parizek@...>

>
>
> Hi Aaron.
>
> Let's see if I've understood correctly. So, the initial key that I hear as
> G minor is "9/8, 6/5, 4/3, 3/2, 8/5, 16/9, 2/1" -- i.e. the passage around
> 0,19, which goes like "G F Eb D C", is played as "8/9, 9/10, 15/16, 8/9"
> (again, meaning intervals of consecutive tones). I just think it might be a
> bit more convenient to replace the F of 16/9 by an F of 9/5 because later
> you modulate to Bb major which requires the higher F anyway, so the two
> versions of F are audible as different pitches if you use the lower one at
> the beginning. But the rest seems okay. Hope my words are understandable.
>
> Petr
>
>
>
>
>
>

Aaron Krister Johnson
http://www.akjmusic.com
http://www.untwelve.org

🔗Petr Pařízek <p.parizek@...>

🔗Marcel de Velde <m.develde@...>

> But as I think throught this, and what you're saying, I realize there is
> always going to be some kind of comma problem. If, as you say, I make the
> 'F' of g-minor a 9/5, then the melodic progression which temporarily leads
> to a c-minor harmonic region would be descending like so: 9/10 8/9 15/16
> 9/8, and I'm not sure it would feel right to have c-minor's 'F' be a 27/20
> above 'C'...does this make sense? This is the problem with JI---there are
> always compromises, and there's no way to 'hide' the comma in certain chord
> progressions....I'd like to try this though, and see how it works.

Sorry but I have to object here.
The comma problems of JI are not problems of Just Intonation itself, but
lack of knowledge of the people on how to write something in JI.
JI has no compromises as it's the structure of music itself and music is not
a compromise.
There's simply a lack of knowledge on JI in humans, but nothing woring with
JI itself.

Here as an example the most simple comma problem that has often been
described as "proof" that JI doesn't work with everything.
C E G -> C E A -> D F A -> D G B - > C E G

Here 2 possible solutions with different fundamental bass:
C(1/1) C(2/1) E(5/2) G(3/1) -> C(1/1) C(2/1) E(5/2) A(10/3) -> F(4/3) D(9/4)
F(8/3) A(10/3) -> G(3/2) D(9/4) G(3/1) B(15/4) -> C(1/1) C(2/1) E(5/2)
G(3/1) C(4/1)

or:
C(1/1) C(2/1) E(5/2) G(3/1) -> C(1/1) C(2/1) E(5/2) A(27/8) -> D(9/8) D(9/4)
F(10935/4096) A(27/8) -> G(3/2) D(9/4) G(3/1) B(15/4) -> C(1/1) C(2/1)
E(5/2) G(3/1) C(4/1)

(not too sure where to put the fundamental basses though)

In any case, there is no "proof" that JI doesn't work or is a compromise.
I wish more people would do research to solve percieved JI problems.

-Marcel

🔗Aaron Johnson <aaron@...>

Marcel,

32/27 is not a simple 5-limit JI sonority. It's got a nice yet active third,
but it's not as smooth as 6/5 in the 5-limit. People use the 5-limit in
general for smoothness of vertical sonority. So your 'solution' isn't really
a solution if we take as a premise the desire to use consistently 5-limit
triads, not escape to 3-limit Pythagoreanism. Ditto the 1215/1024 that you
propose by way of 10935/8192....

If we accept the premise of allowing minor triads to be 3-limit, your
solution works. But the classical comma problem still stands when we ask
that every triad be 5-limit.

Best,
AKJ

On Tue, Sep 15, 2009 at 7:43 PM, Marcel de Velde <m.develde@...>wrote:

>
>
>
> But as I think throught this, and what you're saying, I realize there is
>> always going to be some kind of comma problem. If, as you say, I make the
>> 'F' of g-minor a 9/5, then the melodic progression which temporarily leads
>> to a c-minor harmonic region would be descending like so: 9/10 8/9 15/16
>> 9/8, and I'm not sure it would feel right to have c-minor's 'F' be a 27/20
>> above 'C'...does this make sense? This is the problem with JI---there are
>> always compromises, and there's no way to 'hide' the comma in certain chord
>> progressions....I'd like to try this though, and see how it works.
>
>
> Sorry but I have to object here.
> The comma problems of JI are not problems of Just Intonation itself, but
> lack of knowledge of the people on how to write something in JI.
> JI has no compromises as it's the structure of music itself and music is
> not a compromise.
> There's simply a lack of knowledge on JI in humans, but nothing woring with
> JI itself.
>
> Here as an example the most simple comma problem that has often been
> described as "proof" that JI doesn't work with everything.
> C E G -> C E A -> D F A -> D G B - > C E G
>
> Here 2 possible solutions with different fundamental bass:
> C(1/1) C(2/1) E(5/2) G(3/1) -> C(1/1) C(2/1) E(5/2) A(10/3) -> F(4/3)
> D(9/4) F(8/3) A(10/3) -> G(3/2) D(9/4) G(3/1) B(15/4) -> C(1/1) C(2/1)
> E(5/2) G(3/1) C(4/1)
>
> or:
> C(1/1) C(2/1) E(5/2) G(3/1) -> C(1/1) C(2/1) E(5/2) A(27/8) -> D(9/8)
> D(9/4) F(10935/4096) A(27/8) -> G(3/2) D(9/4) G(3/1) B(15/4) -> C(1/1)
> C(2/1) E(5/2) G(3/1) C(4/1)
>
> (not too sure where to put the fundamental basses though)
>
>
> In any case, there is no "proof" that JI doesn't work or is a compromise.
> I wish more people would do research to solve percieved JI problems.
>
> -Marcel
>
>
>
>

Aaron Krister Johnson
http://www.akjmusic.com
http://www.untwelve.org

🔗Aaron Johnson <aaron@...>

Marcel,

Also wanted to note that you have a problematic fifth in the thirds chord of
your first solution, and a problematic fourth in the 2nd chord of the second
solution.....

Best,
AKJ

On Tue, Sep 15, 2009 at 7:43 PM, Marcel de Velde <m.develde@...>wrote:

Aaron Krister Johnson
http://www.akjmusic.com
http://www.untwelve.org

🔗Aaron Johnson <aaron@...>

below should read "problematic 5th in the _third_ chord of your first
solution. NOt 'thirds'...sorry for the typo.

On Tue, Sep 15, 2009 at 11:10 PM, Aaron Johnson <aaron@...> wrote:

> Marcel,
>
> Also wanted to note that you have a problematic fifth in the thirds chord
> of your first solution, and a problematic fourth in the 2nd chord of the
> second solution.....
>
> Best,
> AKJ
>
> On Tue, Sep 15, 2009 at 7:43 PM, Marcel de Velde <m.develde@...>wrote:
>
>>
>>
>>
>> But as I think throught this, and what you're saying, I realize there is
>>> always going to be some kind of comma problem. If, as you say, I make the
>>> 'F' of g-minor a 9/5, then the melodic progression which temporarily leads
>>> to a c-minor harmonic region would be descending like so: 9/10 8/9 15/16
>>> 9/8, and I'm not sure it would feel right to have c-minor's 'F' be a 27/20
>>> above 'C'...does this make sense? This is the problem with JI---there are
>>> always compromises, and there's no way to 'hide' the comma in certain chord
>>> progressions....I'd like to try this though, and see how it works.
>>
>>
>> Sorry but I have to object here.
>> The comma problems of JI are not problems of Just Intonation itself, but
>> lack of knowledge of the people on how to write something in JI.
>> JI has no compromises as it's the structure of music itself and music is
>> not a compromise.
>> There's simply a lack of knowledge on JI in humans, but nothing woring
>> with JI itself.
>>
>> Here as an example the most simple comma problem that has often been
>> described as "proof" that JI doesn't work with everything.
>> C E G -> C E A -> D F A -> D G B - > C E G
>>
>> Here 2 possible solutions with different fundamental bass:
>> C(1/1) C(2/1) E(5/2) G(3/1) -> C(1/1) C(2/1) E(5/2) A(10/3) -> F(4/3)
>> D(9/4) F(8/3) A(10/3) -> G(3/2) D(9/4) G(3/1) B(15/4) -> C(1/1) C(2/1)
>> E(5/2) G(3/1) C(4/1)
>>
>> or:
>> C(1/1) C(2/1) E(5/2) G(3/1) -> C(1/1) C(2/1) E(5/2) A(27/8) -> D(9/8)
>> D(9/4) F(10935/4096) A(27/8) -> G(3/2) D(9/4) G(3/1) B(15/4) -> C(1/1)
>> C(2/1) E(5/2) G(3/1) C(4/1)
>>
>> (not too sure where to put the fundamental basses though)
>>
>>
>> In any case, there is no "proof" that JI doesn't work or is a compromise.
>> I wish more people would do research to solve percieved JI problems.
>>
>> -Marcel
>>
>>
>>
>>
>
>
>
> --
>
> Aaron Krister Johnson
> http://www.akjmusic.com
> http://www.untwelve.org
>
>

Aaron Krister Johnson
http://www.akjmusic.com
http://www.untwelve.org

🔗Petr Parízek <p.parizek@...>

Marcel wrote:

> C(1/1) C(2/1) E(5/2) G(3/1) -> C(1/1) C(2/1) E(5/2) A(10/3)
> -> F(4/3) D(9/4) F(8/3) A(10/3) -> G(3/2) D(9/4) G(3/1) B(15/4)
> -> C(1/1) C(2/1) E(5/2) G(3/1) C(4/1)

I can't accept this version because the third chord has a dissonant D-A fifth of 40/27, which is exactly the reason why people found this unusable.

> C(1/1) C(2/1) E(5/2) G(3/1) -> C(1/1) C(2/1) E(5/2) A(27/8)
> -> D(9/8) D(9/4) F(10935/4096) A(27/8) -> G(3/2) D(9/4) G(3/1) B(15/4)
> -> C(1/1) C(2/1) E(5/2) G(3/1) C(4/1)

Again, how can you say that "it works" if you use a dissonant E-A fourth of 27/20 in the third chord? AFAIK, you yourself have done lots of experiments with 5-limit JI, but this seems like if you had never "tasted" the terrible dissonance of these intervals. They're so much out of tune that they're just unusable for common 5-limit harmonies.

Further more, can you explain the reason for the inconsistent usage of 10935/4096 instead of 27/10 which would be the regular consistent choice?

Petr

🔗Petr Parízek <p.parizek@...>

🔗Marcel de Velde <m.develde@...>

Hi Petr,

> C(1/1) C(2/1) E(5/2) G(3/1) -> C(1/1) C(2/1) E(5/2) A(10/3)
> > -> F(4/3) D(9/4) F(8/3) A(10/3) -> G(3/2) D(9/4) G(3/1) B(15/4)
> > -> C(1/1) C(2/1) E(5/2) G(3/1) C(4/1)
>
> I can’t accept this version because the third chord has a dissonant D-A
> fifth of 40/27, which is exactly the reason why people found this unusable.
>

Yes but this fifth is not the fifth from the fundamental bass, there's a big
difference here.
A fifth of 40/27 from the fundemental bass is indeed unusable.
I gave 2 examples with different fundamental basses for this reason as one
solution doesn't cover all the uses of this progression.

> > C(1/1) C(2/1) E(5/2) G(3/1) -> C(1/1) C(2/1) E(5/2) A(27/8)
> > -> D(9/8) D(9/4) F(10935/4096) A(27/8) -> G(3/2) D(9/4) G(3/1) B(15/4)
> > -> C(1/1) C(2/1) E(5/2) G(3/1) C(4/1)
>
> Again, how can you say that „it works“ if you use a dissonant E-A fourth of
> 27/20 in the third chord? AFAIK, you yourself have done lots of experiments
> with 5-limit JI, but this seems like if you had never „tasted“ the terrible
> dissonance of these intervals. They’re so much out of tune that they’re just
> unusable for common 5-limit harmonies.
>

These intervals have tension yes but in the right places in the right way.
Have you tried playing my 2 examples?

> Further more, can you explain the reason for the inconsistent usage of
> 10935/4096 instead of 27/10 which would be the regular consistent choice?
>

To have 1/1 5/4 3/2 and 1/1 6/5 3/2 every time a triad can be found
somewhere in the notes, no matter of the true fundamental bass is impossible
and wrong.
It also sounds terrible, and it leads to many problems.

I've researches the minor third above the fundamental bass a lot and I'm
quite sure it is never 6/5.
I beleive it is 1215/1024. For instance found between 16/9 and 135/64.
I have a theory in developement to back it up, I will post it later.
Also please try playing my second example and play 27/10 instead of
10935/4096.
I hope your ears will object as much to this 27/10 as mine do.

-Marcel

🔗Marcel de Velde <m.develde@...>

>
> To have 1/1 5/4 3/2 and 1/1 6/5 3/2 every time a triad can be found
> somewhere in the notes, no matter of the true fundamental bass is impossible
> and wrong.
>

Just a simple example to back this up:
Play for instance C Eb G Bb D F A C
The triads are C Eb G, Eb G Bb, G Bb D, Bb D F, D F A, F A C
It's mathematically impossible to have every triad in this chord as either
1/1 5/4 3/2 or 1/1 6/5 3/2.

There are MANY such chords which are impossible to have every tris 1/1 5/4
3/2 or 1/1 6/5 3/2.
As a consequence there are many chord progressions which are impossible to
have every triad 1/1 5/4 3/2 or 1/1 6/5 3/2.
Some people say this is proof that JI doesn't work.
I say it is proof that not every triad is 1/1 5/4 3/2 or 1/1 6/5 3/2, and
the only "problem" of JI is lack of human insight.
(don't take this as me saying that I have all the answers which I don't
offcourse)

-Marcel

🔗Petr Parízek <p.parizek@...>

Marcel wrote:

> A fifth of 40/27 from the fundemental bass is indeed unusable.

The only situation where 40/27 might be acceptable could be possibly sine waves. Anything else will make beats like hell. 40/27 is so close to 3/2 that the beats will always be clearly audible.

> Have you tried playing my 2 examples?

I have. And I’ve done lots of other experiments as well. And I have particular reasons to say these progressions will always sound out of tune. If you look for „parizek_syndiat.scl“ in Manuel’s scale archive, you can try out lots of that stuff yourself with an ordinary 12-tone keyboard.

> I've researches the minor third above the fundamental bass a lot
> and I'm quite sure it is never 6/5.

If you’re so courageous that you claim a minor third not to be 6/5, then you can’t claim this to be the „model“ version of 5-limit JI -- for one very simple reason. You’ve destroyed the „senario“ of 1:2:3:4:5:6, which Zarlino had used as the model for his „Armonia perfetta“. And the reason why he had used the „senario“ is also very simple -- it’s the most acoustically synchronous triad ever possible. And if you don’t want to use the senario as the starting point, then you simply can’t claim your version to be the „correct“ version of 5-limit JI, because 5-limit JI comes right from that. And for that matter, why don’t you use minor triads of 16:19:24 if you don’t want to use 10:12:15? You’re completely free to use higher primes than 5 and I don’t see any reason why 1215/1024 should be better than 19/16.

> Also please try playing my second example and play 27/10
> instead of 10935/4096. I hope your ears will object as much to this 27/10
> as mine do.

I have to dissappoint you -- they don’t. It doesn’t sound synchronous enough to me. And the beating in the E-A fourth of 27/20 is so strong that it’s just inomittable. If you think it isnt‘, then show me your own audio rendering.

Petr

🔗clumma <carl@...>

🔗Marcel de Velde <m.develde@...>

🔗Aaron Johnson <aaron@...>

On Wed, Sep 16, 2009 at 1:30 PM, clumma <carl@...> wrote:

> --- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:
>
> > It does make a 5-limit triad.
> > The triad is 1/1 5/4 27/16
>
> This is a 27-limit triad.
> It's incorrect to apply prime limits to chords.
>
>
Not that I agree with Marcel, but according to whom is it incorrect to say
'5-limit triad'. If someone said to me '5-limit triad' I would know exactly
what was meant, wouldn't you?

>
>
>
>
> ------------------------------------
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Aaron Krister Johnson
http://www.akjmusic.com
http://www.untwelve.org

🔗Marcel de Velde <m.develde@...>

Hi Petr,

The only situation where 40/27 might be acceptable could be possibly sine
> waves. Anything else will make beats like hell. 40/27 is so close to 3/2
> that the beats will always be clearly audible.
>

I think there's a difference between beeting of an interval with the
fundamental bass and beating of an interval with another interval that is
not the fundamental bass.
In this case the 40/27 is not beating against the fundamental bass (the
chord is 1/1 5/4 27/16) and I do not find it unacceptable.

> > Have you tried playing my 2 examples?
>
> I have. And I’ve done lots of other experiments as well. And I have
> particular reasons to say these progressions will always sound out of tune.
> If you look for „parizek_syndiat.scl“ in Manuel’s scale archive, you can try
> out lots of that stuff yourself with an ordinary 12-tone keyboard.
>
> > I've researches the minor third above the fundamental bass a lot
> > and I'm quite sure it is never 6/5.
>
> If you’re so courageous that you claim a minor third not to be 6/5, then
> you can’t claim this to be the „model“ version of 5-limit JI -- for one very
> simple reason. You’ve destroyed the „senario“ of 1:2:3:4:5:6, which Zarlino
> had used as the model for his „Armonia perfetta“. And the reason why he had
> used the „senario“ is also very simple -- it’s the most acoustically
> synchronous triad ever possible. And if you don’t want to use the senario as
> the starting point, then you simply can’t claim your version to be the
> „correct“ version of 5-limit JI, because 5-limit JI comes right from that.
> And for that matter, why don’t you use minor triads of 16:19:24 if you don’t
> want to use 10:12:15? You’re completely free to use higher primes than 5 and
> I don’t see any reason why 1215/1024 should be better than 19/16.
>
>
Well I think Zarlino was wrong to use the 6/5 from the fundamental bass.
Wasn't it Zarlino btw who also first brought up the fundamental bass?
And didn't Rameau shortly after corrected Zarlino in many things regarding
this fundamental bass and it's use?
I don't see it as courageous to claim minor third not to be 6/5.
6/5 has been debated many times before. 1/1 6/5 3/2 doesn't even point to
the fundamental bass of 1/1, it points to 5/6.
Furthermore the power of the octave is unlimited. 1215/1024 may seem high in
it's natural occurance when you see it as 1215/1 but since one can transpose
by octaves as much as you like it's not a crazy interval.
It's 243/128 down from 5/4, 135/128 up from 9/8. Fits perfectly in a
chromatic scale etc etc.
And it works mathematically nice with other intervals in chord progressions.
I beleive correct 5-limit to be a long chain of major triads connected by
thirds (and never 25/16 etc).
This ensures maximum consonance.

Btw I'm not sure what your ideas are about JI problems.
The chord I gave you that's impossible to make with major and minor triads
of 1/1 6/5 3/2 and 1/1 5/4 3/2, and the resulting proof that not all chord
progressions can be played with only 1/1 6/5 3/2 and 1/1 5/4 3/2.
Are you of the idea that JI is imperfect and that these chord progressions
can't be played properly in JI, Or are you of the idea that this is indeed
proof that 1/1 5/4 3/2 and 1/1 6/5 3/2 are not the only correct tunings for
the major and minor intervals? (or are you undecided? :)

> Also please try playing my second example and play 27/10
> > instead of 10935/4096. I hope your ears will object as much to this 27/10
> > as mine do.
>
> I have to dissappoint you -- they don’t. It doesn’t sound synchronous
> enough to me. And the beating in the E-A fourth of 27/20 is so strong that
> it’s just inomittable. If you think it isnt‘, then show me your own audio
> rendering.
>
I will make a midi rendering soon.Bussy now with another piece I have to
finish first.
Too bad Michaels midi generator doesn't work for me (yet), it would've been
perfect for this.

-Marcel

🔗Marcel de Velde <m.develde@...>

🔗Charles Lucy <lucy@...>

It's really interesting to watch "you guys" discussing the limitations of JI, and the problems involved in attempts to produce sonorous triads etc. using JI.

You have covered criticisms of JI far beyond what I considered before, many years ago, arrogantly "dumping" JI as a practical way to produce harmonious music.

I appreciate your endeavours, thank you.

On 16 Sep 2009, at 20:47, Marcel de Velde wrote:

>
> 6/5 has been debated many times before. 1/1 6/5 3/2 doesn't even > point to the fundamental bass of 1/1, it points to 5/6.
>
> Ugh sorry that should've been it points to 4/5, or actually 1/5.
> (though I do think it has a use too as 1/1 5/4 5/3 with fundamental > bass of 1/1 where 5/3 is a note foreign to the fundamental bass > (sory don't know a better desciption))
>
> -Marcel
>
>
Charles Lucy
lucy@...

- Promoting global harmony through LucyTuning -

for information on LucyTuning go to:
http://www.lucytune.com

For LucyTuned Lullabies go to:
http://www.lullabies.co.uk

🔗Carl <carl@...>

🔗clumma <carl@...>

--- In tuning@yahoogroups.com, Aaron Johnson <aaron@...> wrote:

> > > It does make a 5-limit triad.
> > > The triad is 1/1 5/4 27/16
> >
> > This is a 27-limit triad.
> > It's incorrect to apply prime limits to chords.
>
> Not that I agree with Marcel, but according to whom is it
> incorrect to say '5-limit triad'. If someone said to me
> '5-limit triad' I would know exactly what was meant, wouldn't
> you?

You *would* (or wouldn't?) know what was meant? Try this
entry if you meant wouldn't:
http://en.wikipedia.org/wiki/Limit_(music)

When discussing the consonance of chords, odd limit is
always meant, by any theorist of any repute. You might
catch Harrison or Partch in the occasional vague paragraph,
but generally they get it right. Prime limit is actually
quite rare in the literature -- even Lou Harrison et al,
who invented it, don't use it for chords. That's because
it trivially leads to absurd results.

-Carl

🔗Marcel de Velde <m.develde@...>

>
> If we're talking about the consonance of chords, prime
> limit does NOT matter. This has been established so many
> times on this list, and so many people like you have been
> forced to eat crow over it (including myself circa 1998),
> we won't be discussing it again now. Try the search
> function.
>

Aah no the last thing I want is another big discussion over such a thing.
I'll take your word for it.
However, consonance in this way is a very abstract meaning.
And I currently have the belief that in common practice music any chord
other than prime 5-limit will be out of tune, no matter it's consonance.
Out of tune and consonance are somewhat linked in my mind, but I understand
this is not how consonance is described on this list.
One other thing I belief is that for instance 1/1 5/4 27/16 has more
"consonance potential" than for instance 1/1 5/4 13/8.
In other words, it plays more nicely with other intervals. 1/1 5/4 27/16 has
in my mind many other notes that can be played and even though they're not
played they do matter :)

-Marcel

🔗Marcel de Velde <m.develde@...>

🔗Daniel Forro <dan.for@...>

🔗Carl <carl@...>

--- In tuning@yahoogroups.com, Marcel de Velde <m.develde@...> wrote:

> And I currently have the belief that in common practice music
> any chord other than prime 5-limit will be out of tune, no
> matter it's consonance.

What is the evidence for such a belief? Isn't the fact
that millions consider meantone to be in tune counterevidence?

> Out of tune and consonance are somewhat linked in my mind,
> but I understand this is not how consonance is described on
> this list.

There's psychoacoustic consonance and musical consonance,
the latter being only partially related to the former.
Both have been discussed on this list, but usually the former
is meant if the author doesn't specify.

> One other thing I belief is that for instance 1/1 5/4 27/16
> has more "consonance potential" than for instance 1/1 5/4 13/8.
> In other words, it plays more nicely with other intervals.
> 1/1 5/4 27/16 has in my mind many other notes that can be
> played and even though they're not played they do matter :)

That sounds more like musical consonance. That is indeed
the realm for which prime limits were designed, as mentioned
in the wikipedia entry.

-Carl

🔗Carl <carl@...>

🔗Marcel de Velde <m.develde@...>

>
> Results like 1/1-5/4-40/27 being as consonant as 1/1-5/4-3/2.

Ah that is an absurd result indeed :)
But this leads me to think that there is an error in the logic/thinking used
that lead to such a result, not nessecarily an error of starting with prime
limits for chords.

Btw since you are close to a human library of this tuning list, I'd love to
ask you the following question :)
Has it been discussed / researched before a just intonation minor third that
is close to the equal tempered minor third? (as in 32/27 or 1215/1024 or
19/16)
I'm only aware that I think it was Mike who used 19/16 for the minor third
from the fundamental bass.
As I said before, the minor third of 1/1 6/5 3/2 where 1/1 is the
fundamental bass sounds wrong to my ears.
This is not a short burst of insanity I have :)
I've researched / played it a lot over a longer period of time. Tried it in
different settings/chord progressions etc.
My ears simply will not accept 6/5 here, it sounds out of tune and too wide
to me. (though for instance 1/1 5/4 5/3 is fine with me)
I find it strange that I have not heard more people say this. Only Michael
shortly agreed with me when I pointed it out to him with the Beethoven piece
as n example, and Mike with 19/16.

-Marcel

🔗Aaron Johnson <aaron@...>

In my case, it's a matter of context: I use JI as well as temperaments. The
right tool for the job.

There are some things JI does phenomenally well that no other tuning can
touch. I't hard for me to describe, but there's a mysterious energy there.

I still fully enjoy all the different meantone varieties like 31 and 19, the
spuper Pythagorean tunings like 17 and 22, and the 'out there' stuff:
PHI-based tunings, ethnic tunings, 15-equal, 23-equal, non-octaves.

But man, especially on real instruments, or decent samples, JI is just
great.

I just DON'T UNDERSTAND at all the back and forth bickering about tuning
systems. I cannot relate to being in one camp or another about all this
stuff.

Best,
Aaron.

On Wed, Sep 16, 2009 at 4:35 PM, Charles Lucy <lucy@...> wrote:

>
>
> It's really interesting to watch "you guys" discussing the limitations of
> JI, and the problems involved in attempts to produce sonorous triads etc.
> using JI.
> You have covered criticisms of JI far beyond what I considered before, many
> years ago, arrogantly "dumping" JI as a practical way to produce harmonious
> music.
>
> I appreciate your endeavours, thank you.
>
>
>
> On 16 Sep 2009, at 20:47, Marcel de Velde wrote:
>
>
> 6/5 has been debated many times before. 1/1 6/5 3/2 doesn't even point to
>> the fundamental bass of 1/1, it points to 5/6.
>
>
> Ugh sorry that should've been it points to 4/5, or actually 1/5.
> (though I do think it has a use too as 1/1 5/4 5/3 with fundamental bass of
> 1/1 where 5/3 is a note foreign to the fundamental bass (sory don't know a
> better desciption))
>
> -Marcel
>
>
> Charles Lucy
> lucy@...
>
> - Promoting global harmony through LucyTuning -
>
> for information on LucyTuning go to:
> http://www.lucytune.com
>
> For LucyTuned Lullabies go to:
> http://www.lullabies.co.uk
>
>
>
>
>
>
>

Aaron Krister Johnson
http://www.akjmusic.com
http://www.untwelve.org

🔗Daniel Forro <dan.for@...>

🔗Carl <carl@...>

🔗Mike Battaglia <battaglia01@...>

> Ah that is an absurd result indeed :)
> But this leads me to think that there is an error in the logic/thinking used that lead to such a result, not nessecarily an error of starting with prime limits for chords.
> Btw since you are close to a human library of this tuning list, I'd love to ask you the following question :)
> Has it been discussed / researched before a just intonation minor third that is close to the equal tempered minor third? (as in 32/27 or 1215/1024 or 19/16)
> I'm only aware that I think it was Mike who used 19/16 for the minor third from the fundamental bass.

I'm not sure if you mean me or Michael Sheiman here. As far as I'm
concerned, 19/16 and 6/5 have differences, but the highly fragile
psychoacoustic nature of all of this means that one can often function
as the other from a perceptual standpoint. Hearing 19/16 as a direct
consonance is somewhat difficult for me unless there are other notes
in the chord to reinforce it as such (the key one being 5/4 and
sometimes 7/4). Something like C-E-Bb-Eb is the proverbial example of
this - what's the most perceptually coherent way to map this chord
back into JI?

Then again, perhaps it might be more perceptually intuitive to feel
the Eb out as 7/3. I suppose it would hinge on whether you tend to
hear the Bb-Eb as being a really strong 4/3 or not. I'm not really an
expert on determining which of the zillions of intonations of "minor
third" is the one I'm hearing at any particular moment. I think you
will agree that the Eb in C-E-Bb-Eb has a completely different kind of
flavor than a straight C-Eb-G minor chord though, and I expect this to
be because the two cases lead to them being perceived as two different
harmonic intervals. Just my humble opinion.

> As I said before, the minor third of 1/1 6/5 3/2 where 1/1 is the fundamental bass sounds wrong to my ears.
> This is not a short burst of insanity I have :)
> I've researched / played it a lot over a longer period of time. Tried it in different settings/chord progressions etc.
> My ears simply will not accept 6/5 here, it sounds out of tune and too wide to me. (though for instance 1/1 5/4 5/3 is fine with me)
> I find it strange that I have not heard more people say this. Only Michael shortly agreed with me when I pointed it out to him with the Beethoven piece as n example, and Mike with 19/16.

I do agree with this in certain circumstances - especially when the
chord is in a tenor or baritone register. The lower the minor chord,
the more excessively wide that minor third starts to sound to me. That
being said, the heightened dissonance of the relatively wide minor
third in a 10:12:15 chord actually adds something to the already
dissonant enough minor triad as far as I'm concerned -- it's a
beautiful sound to my ears, and sometimes it adds a poignance to the
minor chord that just isn't there with 0-300-700 cents.

As I hear it, replacing 10:12:15 with 16:19:24 in these lower
registers usually serves to increase critical band smearing effects a
bit, but with the tradeoff of an increase in the underlying harmonic
consonance of the whole triad. It might be because the fundamentals of
most of the dyads in 10:12:15, as well as the triad as a whole, are at
a pretty steep angle to the root that we'd all like to hear the minor
chord as having. This phenomenon seems to be noticeably more annoying
as the chord gets lower and lower, and does not happen with 16:19:24.

That being said, intonating it as 6:5 and simply placing it in a
higher register makes for an extremely "bright" minor chord, and I do
tend to like that sound an awful lot.

-Mike

🔗Mike Battaglia <battaglia01@...>

> Marcel wrote:
> > Btw since you are close to a human library of this tuning list,
> > I'd love to ask you the following question :)

LOL, ain't he though?

> It has been discussed. Another candidate is 13/11.
> I happen to think that 16:19:24 is probably the most stable
> minor triad, even though 10:12:15 can beat less with many
> timbres. I believe Kraig Grady, Paul Erlich, and Joseph
> Pehrson have agreed with that, among others, but apologies
> if I misremember. Certainly, you are not alone in rejecting
> 10:12:15.
>
> -Carl

From everything I've seen involving Paul's harmonic entropy data with
triads, 10:12:15 was more consonant than 16:19:24 and 6:7:9. That is
to say, it had a larger area on the Voronoi cell distribution posted
around here somewhere (it might have been on the Harmonic Entropy
list, I'm too tired to dig around now). I don't remember how 6:7:9
fared up to 16:19:24 -- I think 6:7:9 was a bit larger.

I've been communicating with Paul recently and he mentioned that those
graphs were in some way incomplete or an intermediate step to the
final calculations -- I'm not sure how exactly, so perhaps I'm
misinterpreting the graphs somewhat here. I assume that they just
haven't been blurred in accordance with that s parameter yet, although
I don't really know for sure.

-Mike

🔗Mike Battaglia <battaglia01@...>

> Just a simple example to back this up:
> Play for instance C Eb G Bb D F A C
> The triads are C Eb G, Eb G Bb, G Bb D, Bb D F, D F A, F A C
> It's mathematically impossible to have every triad in this chord as either 1/1 5/4 3/2 or 1/1 6/5 3/2.
> There are MANY such chords which are impossible to have every tris 1/1 5/4 3/2 or 1/1 6/5 3/2.
> As a consequence there are many chord progressions which are impossible to have every triad 1/1 5/4 3/2 or 1/1 6/5 3/2.
> Some people say this is proof that JI doesn't work.
> I say it is proof that not every triad is 1/1 5/4 3/2 or 1/1 6/5 3/2, and the only "problem" of JI is lack of human insight.
> (don't take this as me saying that I have all the answers which I don't offcourse)
> -Marcel

Sonically, I actually prefer having that C on top as 81/20 for that
chord, which technically would solve all of those mathematical
problems anyway. Just keep stacking those 6/5's and 5/4's and that's
what you get. It always amazes me how dissonant the raw 81/20 sounds
by itself, and how amazingly consonant it sounds when you stack the
rest of the 6/5's and 5/4's in there. It really doesn't sound like
81/20 at all.

Or, maybe it finally DOES sound like 81/20 instead of a screwed up
4/1. It certainly does sound like it fits better as an "outward" chord
extension of the Cm13 sonority you're building up below it, rather
than as a benign octave doubling of the root two octaves below. The
former is what the 81/20 would be, in actuality.

-Mike

🔗Mike Battaglia <battaglia01@...>

> > What do you mean by that using prime limit for chords leads
> > to absurd results? Which absurd results? I'm truly curious.
>
> Results like 1/1-5/4-40/27 being as consonant as 1/1-5/4-3/2.
>
> -Carl

This is no doubt absurd, but is it not just as absurd to say that
8:10:12:15 is less consonant than 8:10:12:13, just because of the 15
being involved? Certainly doesn't sound that way to me. Probably
because the 8:15 dyad is the only really dissonant thing in the
original, compared to 8:13, 10:13, 12:13 in the second one... I
suppose this is mixing the odd-limit and prime-limit approach in a
way.

Don't take this as another argument over the definition of consonance.
I've always had this qualm about the use of odd-limit to describe the
consonance of chords. Maybe I just don't understand how the concept is
supposed to be used. Then again, the limiting of psychoacoustic
"consonance" to a single quantitative dimension (even ignoring
critical band effects) seems to be an increasingly restrictive
paradigm for me these days, so maybe I'm just drifting away from the
fold.

-Mike

🔗Petr Parízek <p.parizek@...>

Marcel wrote:

> In this case the 40/27 is not beating against the fundamental bass
> (the chord is 1/1 5/4 27/16) and I do not find it unacceptable.

No matter if you want to concentrate your mind more on beats against the fundamental bass or beats against any other pitch, the mistuned fifth or fourth will always be heard and I don’t know about any means to convince your brain not to interpret it as being mistuned.

> Well I think Zarlino was wrong to use the 6/5 from the fundamental bass.

The reason he had done that was not just that „he liked it that way“ but because he knew that chords like 4:5:6 or 3:4:5 are actually parts of the harmonic series and that’s why he was using them as the „model“ triads. Whatever you may object to this, you can try to explore the properties of such a sound with a sound editor and you’ll find that no other triads are acoustically as synchronous and periodic as these. That’s also why they are essentially the easiest to recognize just by ear.

> Wasn't it Zarlino btw who also first brought up the fundamental bass?
> And didn't Rameau shortly after corrected Zarlino in many things
> regarding this fundamental bass and it's use?

I would rather say that Rameau wanted to correct the people who misinterpreted Zarlino’s words.

> I beleive correct 5-limit to be a long chain of major triads connected by thirds
> (and never 25/16 etc).

Maybe you’re rediscovering Eivint Groven’s ideas but this is a completely different topic and should not be confused with the original concept of pure 5-limit JI. IIRC, Groven’s first idea was the fact that a chain of 8 pure fourths gets only about 2 cents lower than 10/1. Then he devised a tuning in which each fourth is widened by ~0.25 cents to make 8 fourths arrive exactly at 10/1.

> Are you of the idea that JI is imperfect and that these chord progressions
> can't be played properly in JI, Or are you of the idea that this is indeed proof that
> 1/1 5/4 3/2 and 1/1 6/5 3/2 are not the only correct tunings for the major
> and minor intervals? (or are you undecided? :)

First of all, JI is not imperfect since JI intervals can be maximally „in tune“. Then, it depends on what you call JI. Don’t forget that 5-limit JI is not the same as 19-limit JI. But if you say that 4:5:6 is a wrong choice for a major triad, then I have to strongly disagree. Try playing two tones of a triad and singing the third tone. I would be surprised if you even then kept favoring some other than the 5-limit triads as the most easily recognizable ones.

> I will make a midi rendering soon.

Okay, listen to this first: www.sendspace.com/file/4uoaai

Petr

🔗Carl <carl@...>

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

> I've been communicating with Paul recently and he mentioned that
> those graphs were in some way incomplete or an intermediate step
> to the final calculations -- I'm not sure how exactly, so perhaps
> I'm misinterpreting the graphs somewhat here. I assume that they
> just haven't been blurred in accordance with that s parameter
> yet, although I don't really know for sure.
>
> -Mike

The cell areas are preliminary to harmonic entropy, but are
still meaningful in their own right. Just as in the dyadic
case, s characterizes a Gaussian that is partitioned by the
voronoi cells, and the entropy of the partition is the
harmonic entropy. Among simple JI triads, the areas alone
are an accurate indicator of relative consonance. The
entropy lets you handle tempered chords and complex JI chords
also. The minor triad may already be complex enough to
warrant its use.

-Carl

🔗Carl <carl@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > > What do you mean by that using prime limit for chords leads
> > > to absurd results? Which absurd results? I'm truly curious.
> >
> > Results like 1/1-5/4-40/27 being as consonant as 1/1-5/4-3/2.
> >
> > -Carl
>
> This is no doubt absurd, but is it not just as absurd to say
> that 8:10:12:15 is less consonant than 8:10:12:13, just because
> of the 15 being involved?

A few points:

1. These two chords have less contrast than the two I gave.

2. I don't claim that odd limit is free of problems.
Just that it is better than prime limit for ranking the
consonance of chords, and that it's about the best that can
be done when octave equivalence is assumed.

3. 12:13 is a smaller interval than occurs any in the first
chord, so critical band roughness may be a factor. No
back-of-the-envelope JI formula can deal with roughness.

4. If you consider the mean odd limit of the dyads in the
chords (rather than the max), the former chord indeed comes
out as being more consonant, since the 15 will simplify with
10 and 12.

> Certainly doesn't sound that way to me. Probably
> because the 8:15 dyad is the only really dissonant thing in the
> original, compared to 8:13, 10:13, 12:13 in the second one...

Yes.

> I suppose this is mixing the odd-limit and prime-limit approach
> in a way.

Not really, because prime limit runs into problems even for
bare dyads. Like 21:16 is as consonant as 3:2. It's more down
to a mean vs. max approach.

> I've always had this qualm about the use of odd-limit to
> describe the consonance of chords. Maybe I just don't understand
> how the concept is supposed to be used.

It's just the simplest consonance formula there is after you
impose octave equivalence. Integer limit is advocated by some
when octave equivalence is not assumed. It's just dead simple,
and it works fairly well.

> Then again, the limiting of psychoacoustic "consonance" to
> a single quantitative dimension (even ignoring critical band
> effects) seems to be an increasingly restrictive paradigm
> for me these days, so maybe I'm just drifting away from the
> fold.

Maybe it's like IQ. There's a lot more to intelligence than
a single factor. But, a single factor *does* capture quite a
lot (more than some would like to admit).

-Carl

🔗Marcel de Velde <m.develde@...>

Hi Petr,

> > In this case the 40/27 is not beating against the fundamental bass
> > (the chord is 1/1 5/4 27/16) and I do not find it unacceptable.
>
> No matter if you want to concentrate your mind more on beats against the
> fundamental bass or beats against any other pitch, the mistuned fifth or
> fourth will always be heard and I don’t know about any means to convince
> your brain not to interpret it as being mistuned.bas
>

I disagree here. I have several examples where my brain defenately expects
27/16 from the fundamental bass and 5/3 sounds very out of tune.
This is with 5/4 also from the fundamental bass making 1/1 5/4 27/16

> Well I think Zarlino was wrong to use the 6/5 from the fundamental bass.
>
> The reason he had done that was not just that „he liked it that way“ but
> because he knew that chords like 4:5:6 or 3:4:5 are actually parts of the
> harmonic series and that’s why he was using them as the „model“ triads.
> Whatever you may object to this, you can try to explore the properties of
> such a sound with a sound editor and you’ll find that no other triads are
> acoustically as synchronous and periodic as these. That’s also why they are
> essentially the easiest to recognize just by ear.
>

Agreed it's the shortest periodic minor triad. But I don't think it's
allways about that.

> I beleive correct 5-limit to be a long chain of major triads connected by
> thirds
> > (and never 25/16 etc).
>
> Maybe you’re rediscovering Eivint Groven’s ideas but this is a completely
> different topic and should not be confused with the original concept of pure
> 5-limit JI. IIRC, Groven’s first idea was the fact that a chain of 8 pure
> fourths gets only about 2 cents lower than 10/1. Then he devised a tuning in
> which each fourth is widened by ~0.25 cents to make 8 fourths arrive exactly
> at 10/1.
>

No I'm not into tempering :)
This tempers out the Schisma.
Giving 53tet?
It would make life very easy I agree, yet it fails in perfection and
explaining music fully.

> > Are you of the idea that JI is imperfect and that these chord
> progressions
> > can't be played properly in JI, Or are you of the idea that this is
> indeed proof that
> > 1/1 5/4 3/2 and 1/1 6/5 3/2 are not the only correct tunings for the
> major
> > and minor intervals? (or are you undecided? :)
>
> First of all, JI is not imperfect since JI intervals can be maximally „in
> tune“. Then, it depends on what you call JI. Don’t forget that 5-limit JI is
> not the same as 19-limit JI. But if you say that 4:5:6 is a wrong choice for
> a major triad, then I have to strongly disagree. Try playing two tones of a
> triad and singing the third tone. I would be surprised if you even then kept
> favoring some other than the 5-limit triads as the most easily recognizable
> ones.
>

Well but still you can't get around the fact that 1/1 6/5 3/2 for every
minor triad simply doesn't work and never will work.
So if you truly belief that this is the only way to tune every minor triad
then JI has a big problem that can't be solved.

> > I will make a midi rendering soon.
>
> Okay, listen to this first: www.sendspace.com/file/4uoaai
>
> Petr
>
Thanks a lot for the rendering!
However I do not think it is a correct rendering of my first example.
My first example clearly states the fundamental bass of the D F A chord to
be F, making 1/1 5/4 27/16.
You've used D a lot as the fundamental bass for D F A in which case you
should have used my second example and D F A should be 1/1 1215/1024 3/2.
(or 1/1 32/27 3/2 or 1/1 19/16 3/2, whatever your prefered poison for the
minor third from the fundamental bass other than the 6/5)
With D as the fundamental bass you've created a mistuned fifth of 40/27 from
the fundamental bass which is agree is not right.

-Marcel

🔗Petr Parízek <p.parizek@...>

Marcel wrote:

> Agreed it's the shortest periodic minor triad. But I don't think it's allways about that.

If you want to use different interval models for major chords than for minor chords, you have to apply a completely different concept than what is considered to be the „pure“ version of 5-limit JI. If we really were to apply a different concept, then we should agree on what are the starting chords and what we do or don’t consider a mistuned chord. If we really go for totally different minor chords than simple inversions of the major chords, then I could say, for example, that the most synchronous major chord could be 6:7:9. Even though it may sound terribly out of tune for traditional tonal music, 6:7:9 is easily recognizable by our ears. If I want the lowest tone to be an octave equivalent of the fundamental frequency, I may go for 16:19:24, which I can, though with some higher effort, also learn to recognize by ear. But I don’t see any reason for minor thirds like 1215/1024, which are so complex that I‘ll probably never hear their proper size and I’ll most likely happily mistune them by 6 or 7 cents if I was trying to find them by ear. And isn’t JI primarily about easily recognizable intervals?

> No I'm not into tempering :)

I was not pointing primarily to the schismatic temperament but rather to the tuning where one in eight fifths is a schisma smaller than the seven others in the chain.

> Well but still you can't get around the fact that 1/1 6/5 3/2
> for every minor triad simply doesn't work and never will work.

But then you’re using your specific modified version of 5-limit JI which can’t be called „the correct version“. You’re trying to change the definition of something that has already been given a quite clear definition in the past. I’ll give you an anologous example. Suppose that mathematicians had stated that you couldn’t get square roots of negative numbers. Then one day, I would come and say: „You’re doing square roots and powers in the wrong way, you should multiply the square root of the absolute value by the sign and then the square root of -36 comes out as -6 and also -6 squared comes out as -36“. And some mathematicians might, quite understandably, respond by saying: „But the definition of a square root is such and such; what you’re doing is not in accordance with this definition, which means that you’re using a modified version of a square root“. And this in turn means that I simply couldn’t call my method of squaring the „correct method“ because the general approach for squares says that -6 squared is (-6)^2, not -(6^2). And what I’m trying to explain here all the time is that you’re simply using your own modification of 5-limit JI which, in some ways, is not in accordance with the original concept of pure 5-limit JI, and therefore can’t be called „the correct version of 5-limit JI“.

> However I do not think it is a correct rendering of my first example.

If you were listening carefully enough, you would have noted that I was alternating between two different versions of the progression. One of them is „1/1_3/2_2/1_5/2, 5/6_5/3_2/1_5/2, 2/3_4/3_5/3_9/4, 3/4_3/2_15/8_9/4“. The other is „1/1_2/1_5/2_3/1, 1/1_2/1_5/2_27/8, 9/8_9/4_27/10_27/8, 3/2_15/8_9/4_3/1“. Each of them contains at least one wide fourth of 27/20, which sounds just out of tune because it’s still close enough to be interpreted as a mistuned 4/3.

Petr

🔗Marcel de Velde <m.develde@...>

Hi Petr,

If you want to use different interval models for major chords than for minor
> chords, you have to apply a completely different concept than what is
> considered to be the „pure“ version of 5-limit JI.
>

Yes I think a different concept is needed than this "pure" classical 5-limit
JI since pure classical 5-limit JI doesn't work, it's impossible
mathematically.

If we really were to apply a different concept, then we should agree on what
> are the starting chords and what we do or don’t consider a mistuned chord.
> If we really go for totally different minor chords than simple inversions of
> the major chords, then I could say, for example, that the most synchronous
> major chord could be 6:7:9. Even though it may sound terribly out of tune
> for traditional tonal music, 6:7:9 is easily recognizable by our ears. If I
> want the lowest tone to be an octave equivalent of the fundamental
> frequency, I may go for 16:19:24, which I can, though with some higher
> effort, also learn to recognize by ear. But I don’t see any reason for minor
> thirds like 1215/1024, which are so complex that I‘ll probably never hear
> their proper size and I’ll most likely happily mistune them by 6 or 7 cents
> if I was trying to find them by ear. And isn’t JI primarily about easily
> recognizable intervals?
>

I may agree with you on the 1215/1024, I've allready changed my mind again
to 1/1 32/27 3/2 as the minor chord in major mode :)
But I don't think JI is about chords, I think it's mainly an interlocking
system of melodies, and with melodies you could conceivably create something
like a 1215/1024 interval that is recognisable.

> No I'm not into tempering :)
>
> I was not pointing primarily to the schismatic temperament but rather to
> the tuning where one in eight fifths is a schisma smaller than the seven
> others in the chain.
>
> > Well but still you can't get around the fact that 1/1 6/5 3/2
> > for every minor triad simply doesn't work and never will work.
>
> But then you’re using your specific modified version of 5-limit JI which
> can’t be called „the correct version“. You’re trying to change the
> definition of something that has already been given a quite clear definition
> in the past. I’ll give you an anologous example. Suppose that mathematicians
> had stated that you couldn’t get square roots of negative numbers. Then one
> day, I would come and say: „You’re doing square roots and powers in the
> wrong way, you should multiply the square root of the absolute value by the
> sign and then the square root of -36 comes out as -6 and also -6 squared
> comes out as -36“. And some mathematicians might, quite understandably,
> respond by saying: „But the definition of a square root is such and such;
> what you’re doing is not in accordance with this definition, which means
> that you’re using a modified version of a square root“. And this in turn
> means that I simply couldn’t call my method of squaring the „correct method“
> because the general approach for squares says that -6 squared is (-6)^2, not
> -(6^2). And what I’m trying to explain here all the time is that you’re
> simply using your own modification of 5-limit JI which, in some ways, is not
> in accordance with the original concept of pure 5-limit JI, and
> therefore can’t be called „the correct version of 5-limit JI“.
>

Ok then I'll give it a new name.
DeVelde-JI till I or someone else comes up with a better one :)
Pure 5-limit JI doesn't work, proved that a few messages ago allready and
I'm not interested in something that's proven wrong for music.

> > However I do not think it is a correct rendering of my first example.
>
> If you were listening carefully enough, you would have noted that I was
> alternating between two different versions of the progression. One of them
> is „1/1_3/2_2/1_5/2, 5/6_5/3_2/1_5/2, 2/3_4/3_5/3_9/4, 3/4_3/2_15/8_9/4“.
> The other is „1/1_2/1_5/2_3/1, 1/1_2/1_5/2_27/8, 9/8_9/4_27/10_27/8,
> 3/2_15/8_9/4_3/1“. Each of them contains at least one wide fourth of 27/20,
> which sounds just out of tune because it’s still close enough to be
> interpreted as a mistuned 4/3.
>
> Petr
>

Aah ok.But then why did you say it was my example?
Because it is not.
In my example I clearly gave the fundamental bass in the 2 examples and said
it was about the fundamental bass.
All I heard in your example was out of tune things and I listened for the
bass and heard it was not my example.

-Marcel

🔗Marcel de Velde <m.develde@...>

🔗Petr Parízek <p.parizek@...>

Marcel wrote:

> But I don't think JI is about chords, I think it's mainly an interlocking systém
> of melodies, and with melodies you could conceivably create
> something like a 1215/1024 interval that is recognisable.

Again, depends on what you call JI. This concept of melodic lines may be quite nicely applicable to 3-limit JI. And it even was applied like this quite often in the past. But the primary change brought by 5-limit intervals in the 16th century was the concept of chords as autonomous elements usable in music which have a completely different meaning than melodic lines. It’s quite probable that this entire systém and classification of chords would never have arrived if 5-limit intervals hadn’t come into use.

> In my example I clearly gave the fundamental bass in the 2 examples and said
> it was about the fundamental bass.

Okay, I’ve redone them with the sole exception that I’ve replaced the 15/4 with 15/8, hope you don’t mind. Again, the examples are alternating one after the other. Here’s the link: www.sendspace.com/file/2mi997

Petr

new (old) music

🔗Aaron Johnson <aaron@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Aaron Johnson <aaron@...>

🔗Petr Parízek <p.parizek@...>

🔗Petr Parízek <p.parizek@...>

🔗Aaron Johnson <aaron@...>

🔗Ozan Yarman <ozanyarman@...>

🔗Petr Pařízek <p.parizek@...>

🔗Aaron Johnson <aaron@...>

🔗Petr Pařízek <p.parizek@...>

🔗Aaron Johnson <aaron@...>

🔗Petr Pařízek <p.parizek@...>

🔗Aaron Johnson <aaron@...>

🔗Aaron Johnson <aaron@...>

🔗Petr Pařízek <p.parizek@...>

🔗Aaron Johnson <aaron@...>

🔗Petr Pařízek <p.parizek@...>

🔗Marcel de Velde <m.develde@...>

🔗Aaron Johnson <aaron@...>

🔗Aaron Johnson <aaron@...>

🔗Aaron Johnson <aaron@...>

🔗Petr Parízek <p.parizek@...>

🔗Petr Parízek <p.parizek@...>

🔗Marcel de Velde <m.develde@...>

🔗Marcel de Velde <m.develde@...>

🔗Marcel de Velde <m.develde@...>

🔗Petr Parízek <p.parizek@...>

🔗clumma <carl@...>

🔗Marcel de Velde <m.develde@...>

🔗Aaron Johnson <aaron@...>

🔗Marcel de Velde <m.develde@...>

🔗Marcel de Velde <m.develde@...>

🔗Charles Lucy <lucy@...>

🔗Carl <carl@...>

🔗clumma <carl@...>

🔗Marcel de Velde <m.develde@...>

🔗Marcel de Velde <m.develde@...>

🔗Daniel Forro <dan.for@...>

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🔗Carl <carl@...>

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🔗Mike Battaglia <battaglia01@...>

🔗Mike Battaglia <battaglia01@...>

🔗Mike Battaglia <battaglia01@...>

🔗Petr Parízek <p.parizek@...>

🔗Carl <carl@...>

🔗Carl <carl@...>

🔗Marcel de Velde <m.develde@...>

🔗Petr Parízek <p.parizek@...>

🔗Marcel de Velde <m.develde@...>

🔗Marcel de Velde <m.develde@...>

🔗Petr Parízek <p.parizek@...>