From MMM: Chord Concordance between 14, 17, 19, 21, 23, and 24-EDO

I hope Carl and Steve will read this as well.

--- In MakeMicroMusic@yahoogroups.com, "lobawad" <lobawad@...> wrote:
> 24, 17, and 21 don't sound like 7/6 to me.
>
> I find that 17 and 21 sound like very dark minor thirds, not subminor thirds. They split
> the difference between 6/5 and 7/6.

How can that be? If that is indeed how we're hearing them, that means either a) 10:12:15 has a stronger field of attraction than 6:7:9, b) there's a chord between 6:7:9 and 10:12:15 which has a field of attraction in its own right, or c) a chord at a local maximum sounds better than a chord near a local minimum. The lowest triad between 6:7:9 and 10:12:15 is 16:19:24, followed by 22:26:33. Unless either of these is a local minimum, this looks like doom for triadic HE as a model of concordance.

Does anyone reading this have access to the triadic HE data that Steve calculated? If so, can we see how the following four triads compare in triadic HE?

0-257-686
0-285-686
0-261-678
0-250-700

> 24 offers a neutral interval, more related to a very large second I find. In more
> information-rich terms than those offered by the pseudo-scientific accuracy of using
> only cents, this proportion would be described as about 15/13.

And this would make the triads approximately 26:30:39. I know that Gene seems to prefer these over 14, as well. Over 19, even (which doesn't sound so hot). It's hard enough to swallow that a 7/6 that's 10 cents flat should sound worse than a decent 13/11, but that it should sound worse than a decent 15/13 as well definitely blows my mind.

> I think it's clear as to why close approximations to 7:6, such as that offered by 14,
> should look so good on paper and be so jarring when realized. If you check out the
> pattern of coincident partials in a pure 7:6, you'll find that it offers an interesting
> combination of coincidences within a pattern of strongly dissonant partial relationships, > tending to ride quite a bit in critical band areas. So, by getting very close to 7:6, you lose > the smoothing of the coincidences but keep the clanging of the dissonances. In
> combination with a doggy 9:7, 9:7 being an audible but not strong melding point in the > first place, it's not suprising (in retrospect) that 14 offers a a clangorous sonority here.

How can I look at the pattern of coincident partials? I'd like to see this.

Also, does this in any way explain why 23-EDO sounds better than 14-EDO, despite the fact that its 7/6 is almost as flat as 14's and its 9/7 and 3/2 are both even flatter?

> By the way, have you inspected the "consonant interval of Avicenna"? At 196:169(256.6 > cents) you can see that it is 14:13 stacked and pushing toward a "bad 7:6". But I find it to > be the earliest documented interval which qualifies in my opinion as consonant-by
> means of-asonance. I'm curious as to what you think of it.

I'll give it a try.

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> I hope Carl and Steve will read this as well.

On a related note, have you considered my request to use the 29edo version of your example?

Lobawad> "24, 17, and 21 don't sound like 7/6 to me.

> I find that 17 and 21 sound like very dark minor thirds, not subminor thirds. They split

> the difference between 6/5 and 7/6. "

    To note....I think the area around 13/11 is a relative dead-zone IE
one of the weakest dyadic areas there is far as having a unique
identity.  I could easily see it sucked into being a (although weak) 7/6
or a 6/5...especially considering 24 and 21TET both have a dyad in that
area closer the 6/5 than 7/6 (though in 17TET is seems a tad closer to
7/6) .

   Even then...the feel of the 17TET dyad getting pushed more toward 6/5
makes sense, since 6/5 has a stronger field of attraction.  I'm a
pretty strong advocate for areas too "weak" in tonal attraction to be
recognized by Harmonic Entropy, such as 11/9, 14/9 AKA 9/7 and even 22/15 AKA 15/11 inverted...but
the 13/11 to 19/16 area is too weak "even" for me.

Igs>>"this looks like doom for triadic HE as a model of concordance."

Here's something that could be related to 2HE and 3HE. Using sine wave tones the "strength" of a JI interval is, according to my formula:

(2 + 1/x + 1/y - y/x)/2

Note the above applies only when y/x<=0.9375 and x and y<256.

It seems to my ear that any *sine wave* dyad with a value less than 0.75 is no good.

I used the same formula to work out the strength value of the 6:7:9 sine wave triad and it has a value of 0.77 (greater than the cut-off point of 0.75 mentioned above). The maths say that the chord should sound good but if I listen carefully the 7/6 interval within the chord sounds bad and therefore the chord is bad. 7/6 on it's own has a value of 0.726 which is less than the cut-off point of 0.75. (Note that 7/6 is perfectly acceptable using complex tones with a 'regular' harmonic series, it only sounds bad using sine waves.) So it seems that even if the chord has an overall value equal to or greater than 0.75, if it contains at least one "bad" interval it should be illegal.

Next I worked out the strength value of a justly tuned sine wave minor chord:
10,15,20,24,30,40
and I got a value of -0.49. The maths say that the chord should be bad but it sounds okay (although definitely minor).
In this case, all the dyads within the chord are good (value>=0.75).

The point I am making is that it seems that as long as all the dyads in a chord are good, then the chord should be good.

With regard to HE, as long as all the dyads in a chord are "good" according to 2HE then there should be no need to work out triadic HE (3HE), or for that matter 4HE, 5HE, 6HE etc.

Although 3+HE would be useful for quantifying the "strengths" of chords with 3 or more notes.

John.

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> I hope Carl and Steve will read this as well.
>
> --- In MakeMicroMusic@yahoogroups.com, "lobawad" <lobawad@> wrote:
> > 24, 17, and 21 don't sound like 7/6 to me.
> >
> > I find that 17 and 21 sound like very dark minor thirds, not subminor thirds. They split
> > the difference between 6/5 and 7/6.
>
> How can that be? If that is indeed how we're hearing them, that means either a) 10:12:15 has a stronger field of attraction than 6:7:9, b) there's a chord between 6:7:9 and 10:12:15 which has a field of attraction in its own right, or c) a chord at a local maximum sounds better than a chord near a local minimum. The lowest triad between 6:7:9 and 10:12:15 is 16:19:24, followed by 22:26:33. Unless either of these is a local minimum, this looks like doom for triadic HE as a model of concordance.
>
> Does anyone reading this have access to the triadic HE data that Steve calculated? If so, can we see how the following four triads compare in triadic HE?
>
> 0-257-686
> 0-285-686
> 0-261-678
> 0-250-700
>
> > 24 offers a neutral interval, more related to a very large second I find. In more
> > information-rich terms than those offered by the pseudo-scientific accuracy of using
> > only cents, this proportion would be described as about 15/13.
>
> And this would make the triads approximately 26:30:39. I know that Gene seems to prefer these over 14, as well. Over 19, even (which doesn't sound so hot). It's hard enough to swallow that a 7/6 that's 10 cents flat should sound worse than a decent 13/11, but that it should sound worse than a decent 15/13 as well definitely blows my mind.
>
> > I think it's clear as to why close approximations to 7:6, such as that offered by 14,
> > should look so good on paper and be so jarring when realized. If you check out the
> > pattern of coincident partials in a pure 7:6, you'll find that it offers an interesting
> > combination of coincidences within a pattern of strongly dissonant partial relationships, > tending to ride quite a bit in critical band areas. So, by getting very close to 7:6, you lose > the smoothing of the coincidences but keep the clanging of the dissonances. In
> > combination with a doggy 9:7, 9:7 being an audible but not strong melding point in the > first place, it's not suprising (in retrospect) that 14 offers a a clangorous sonority here.
>
> How can I look at the pattern of coincident partials? I'd like to see this.
>
> Also, does this in any way explain why 23-EDO sounds better than 14-EDO, despite the fact that its 7/6 is almost as flat as 14's and its 9/7 and 3/2 are both even flatter?
>
> > By the way, have you inspected the "consonant interval of Avicenna"? At 196:169(256.6 > cents) you can see that it is 14:13 stacked and pushing toward a "bad 7:6". But I find it to > be the earliest documented interval which qualifies in my opinion as consonant-by
> > means of-asonance. I'm curious as to what you think of it.
>
> I'll give it a try.
>
> -Igs
>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> On a related note, have you considered my request to use the 29edo version of your
> example?

Oh, right, sorry. If you like it enough, then by all means go ahead, but I'm pretty sure I could do better in writing something specifically for the EDO.

-Igs

   I am working on an Adaptive JI algorithm which swaps semitones with 12/11 intervals whenever possible.  Quarter comma meantone does a great job, IMVHO, with eliminating the problem of the "evil" square root of 2 tritone in the diatonic scale under 12TET...but leaves a lot of very nasty sounding semitone clusters (21/20's) as a result.

    In some odd cases, of course, the swap will create an unbearable interval IE a 23/15 Wolf fifth and the program will knock it back to a 3/2 fifth.

    Here is the original scale used

1 9/8 6/5 4/3 7/5 3/2 8/5 5/3 9/5 2

...and here is the scale including notes substituted in when a cluster is found (notes used for clusters are in ()'s )...

1 9/8 6/5 (11/9) (9/7) 4/3 7/5 3/2 8/5 (18/11) 5/3 9/5 (11/6) 2

This works as the following.
   Say you have a clustered chord of 7/5 3/2 8/5.  The program will notice 3/2 and 8/5 are rather close and change the 8/5 to an 18/11.  Yes, of course, that causes commatic drift and introduces a few beating overtones...but fixes a lot of the heavy root-tone beating issues that cause musicians to often avoid very clustered chords (thus hopefully "opening" some new chords for many musicians).  And hopefully the tone shift required is not so startling that it causes any sort of overlapping identities/tonal-classes in the listener's mind.
  Now say you have a clustered chord of 7/5 3/2 8/5 5/3.  The program will stick with this chord as it will notice that moving 8/5 to 18/11 will put it too close to 5/3 (IE essentially in the same tonal class).

   Any ideas for how to improve this and/or things to watch out for?  

6:7:9 vs. 10:12:15?

   What keeps starting me in the face is that 6:7:9 has the dyad in it with the most root-tone critical band dissonance, 7/6 (vs. 6/5 in 10:12:15).  Plus the 6/5 has more dyadic HE attraction, plus 15:12 = 5/4...which is significantly more dyadic-ally stable than 9/7.

  I don't think triadic HE is "doomed"...but I think a triadic HE formula which does not BOTH take dyadic HE and root-tone critical band dissonance into account is grossly incomplete.  All three of these things (dyadic&triadic HE plus critical band dissonance) seem essential.

  What would be a hoot...I'd would REALLY like to see a weighted formula that uses dyadic and triadic HE and calculates HE using John's formula instead of Tenney Height.  John's formula, for the most part (unlike virtually every other dyadic formula I've seen) get dyadic dissonance, including critical band effects, right a vast majority of the time...no matter how "odd"/"non-standard" the way it comes up with its conclusions is.

--- "cityoftheasleep" <igliashon@...> wrote:

> Does anyone reading this have access to the triadic HE data
> that Steve calculated?

Filesection. My folder in the filesection.

-Carl

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> Oh, right, sorry. If you like it enough, then by all means go ahead, but I'm pretty sure I could do better in writing something specifically for the EDO.

Good idea!

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

> John's formula, for the most part (unlike virtually every other dyadic formula I've seen) get dyadic dissonance, including critical band effects, right a vast majority of the time...no matter how "odd"/"non-standard" the way it comes up with its conclusions is.

What's an example of where John's formula gets it right and TH doesn't?

Gene>"What's an example of where John's formula gets it right and TH doesn't?"

    12/7, 11/6...and a good few other.  In general Tenney Height unnecessarily punishes anything high limit enough to have a Tenney Height around to over 70 or so and often even some fractions a whole lot lower.

The fact the HE curve as based on Tenney Height has no dip around 11/9, 11/6, 15/8, and 9/5...for example, really makes me scratch my head...it's as if 11/6 and 15/8 are weak versions of 7/4?!...oh come on...

  According to Tenney Height, we might as well throw a huge amount of the ratios used in Mohajira and Middle Eastern scales in general out the window.  Tenney height seems fine until you get above 7-odd-limit or so...

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Gene>"What's an example of where John's formula gets it right and TH doesn't?"
>
>     12/7, 11/6...and a good few other.  In general Tenney Height unnecessarily punishes anything high limit enough to have a Tenney Height around to over 70 or so and often even some fractions a whole lot lower.

I didn't ask about harmonic entropy, I asked about John's formula versus Tenney height. What is an example where his formula gets it right and TH doesn't?

>   According to Tenney Height, we might as well throw a huge amount of the ratios used in Mohajira and Middle Eastern scales in general out the window.

Who says? Not Tenney Height, it ain't talkin'. Where do you get this stuff?

Damn, it's just the minima and maxima. Doesn't help me at all, because none of the triads I want to find the entropy for are in there. But thanks anyway.

-Igs

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- "cityoftheasleep" <igliashon@> wrote:
>
> > Does anyone reading this have access to the triadic HE data
> > that Steve calculated?
>
> Filesection. My folder in the filesection.
>
> -Carl
>

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

> Good idea!

Can you recommend a temperament that 29 does well, which won't require me to use more than 12 notes?

-Igs

On Tue, Mar 22, 2011 at 10:38 AM, cityoftheasleep
<igliashon@...> wrote:
>
> I hope Carl and Steve will read this as well.
>
> --- In MakeMicroMusic@yahoogroups.com, "lobawad" <lobawad@...> wrote:
> > 24, 17, and 21 don't sound like 7/6 to me.
> >
> > I find that 17 and 21 sound like very dark minor thirds, not subminor thirds. They split
> > the difference between 6/5 and 7/6.
>
> How can that be? If that is indeed how we're hearing them, that means either a) 10:12:15 has a stronger field of attraction than 6:7:9, b) there's a chord between 6:7:9 and 10:12:15 which has a field of attraction in its own right, or c) a chord at a local maximum sounds better than a chord near a local minimum. The lowest triad between 6:7:9 and 10:12:15 is 16:19:24, followed by 22:26:33. Unless either of these is a local minimum, this looks like doom for triadic HE as a model of concordance.

Because there's a difference in perception between 6:7:9 (resolved)
and 6:7:9 (unresolved). 6:7:9 (resolved) sounds like it's pointing to
1, and hence doesn't sound minor at all. 6:7:9 unresolved sounds like
it's pointing to 3. If you really wanted to test all of these triads,
and test the differences between them as "minor" triads, meaning in an
unresolved capacity, I'd say that you need to double that lowest voice
down an octave and compare the results via tetradic HE.

I'm not even sure that tetradic HE would do the trick well, as
tetradic HE wouldn't be factoring both 3:6:7:9 and 3:6:9/1:2:3 into
its probability calculation. This situation never came up with dyadic
HE, where there are no chord subsets.

-Mike

Gene and Michael,

I'll do the comparison test between my formula and Tenney Height myself. Remember that my formula applies to sine wave tones only but I used the formula in a program that works out the concordance values of intervals with complex tones (with a 'regular' harmonic series). So I'll compare the results of my program with TH.
If my understanding of TH is correct it's a "badness" measure and the formula is n*d. The higher the result the greater the badness, is this correct? So the "goodness" measure of TH would be the reciprocal of n*d which is 1/(n*d), is this correct?

Once I get these points clarified I'll perform the comparison test between my program and Tenney Height.

John.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, Michael <djtrancendance@> wrote:
> >
> > Gene>"What's an example of where John's formula gets it right and TH doesn't?"
> >
> >     12/7, 11/6...and a good few other.  In general Tenney Height unnecessarily punishes anything high limit enough to have a Tenney Height around to over 70 or so and often even some fractions a whole lot lower.
>
> I didn't ask about harmonic entropy, I asked about John's formula versus Tenney height. What is an example where his formula gets it right and TH doesn't?
>
> >   According to Tenney Height, we might as well throw a huge amount of the ratios used in Mohajira and Middle Eastern scales in general out the window.
>
> Who says? Not Tenney Height, it ain't talkin'. Where do you get this stuff?
>
>  
>

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
> Does anyone reading this have access to the triadic HE data that Steve calculated? If so, can we see how the following four triads compare in triadic HE?
>
> 0-257-686
> 0-285-686
> 0-261-678
> 0-250-700

Igs,
lobawad's comments and yours are both of great interest; I am still trying to fully understand them all. Meanwhile I can help with the numbers: I've listed:
* lower, upper, outer - intervals in cents
* lower, upper, outer - differences from the pure 6:7:9
* distance from 6:7:9 in triad space
* maximum absolute difference (of the three differences above)
* Triadic HE
* maximum probability for Tenney set member
* identity of that Tenney set member

As well as the four points you asked about, I included:
* the other ratios you mentioned
* the "best" approx to 6:7:9 in edos (9-14,17,19,22-24, 31)
* the point with max 3HE (35,35)
* the point with min 3HE (0,0)

l u o diff diff diff dist max 3HE maxprob Tenney point
256.00 430.00 686.00 -10.9 -5.1 -16.0 16.3 16.0 2.942 0.025 6 7 9
285.00 401.00 686.00 18.1 -34.1 -16.0 34.1 34.1 2.961 0.012 23 27 34
261.00 417.00 678.00 -5.9 -18.1 -24.0 25.0 24.0 2.968 0.016 19 22 28
250.00 450.00 700.00 -16.9 14.9 -2.0 18.5 16.9 2.942 0.020 6 7 9
266.87 435.08 701.96 0.0 0.0 0.0 0.0 0.0 2.887 0.062 6 7 9
315.64 386.31 701.96 48.8 -48.8 0.0 56.3 48.8 2.931 0.037 10 12 15
297.51 404.44 701.96 30.6 -30.6 0.0 35.4 30.6 2.946 0.023 16 19 24
289.21 412.75 701.96 22.3 -22.3 0.0 25.8 22.3 2.953 0.017 16 19 24
386.31 315.64 701.96 119.4 -119.4 0.0 137.9 119.4 2.809 0.090 4 5 6
266.67 400.00 666.67 -0.2 -35.1 -35.3 40.6 35.3 2.976 0.012 30 35 44 9edo
240.00 480.00 720.00 -26.9 44.9 18.0 45.2 44.9 2.960 0.013 27 31 41 10edo
218.18 436.36 654.55 -48.7 1.3 -47.4 55.5 48.7 2.962 0.017 15 17 22 11edo
300.00 400.00 700.00 33.1 -35.1 -2.0 39.4 35.1 2.941 0.022 16 19 24 12edo
276.92 461.54 738.46 10.1 26.5 36.5 37.7 36.5 2.959 0.018 17 20 26 13edo
257.14 428.57 685.71 -9.7 -6.5 -16.2 16.3 16.2 2.943 0.025 6 7 9 14edo
282.35 423.53 705.88 15.5 -11.6 3.9 16.1 15.5 2.937 0.026 6 7 9 17edo
252.63 442.11 694.74 -14.2 7.0 -7.2 14.2 14.2 2.930 0.031 6 7 9 19edo
272.73 436.36 709.09 5.9 1.3 7.1 7.6 7.1 2.900 0.051 6 7 9 22edo
260.87 417.39 678.26 -6.0 -17.7 -23.7 24.6 23.7 2.967 0.016 19 22 28 23edo
250.00 450.00 700.00 -16.9 14.9 -2.0 18.5 16.9 2.942 0.020 6 7 9 24edo
270.97 425.81 696.77 4.1 -9.3 -5.2 9.3 9.3 2.915 0.046 6 7 9 31edo
35.00 35.00 70.00 -231.9 -400.1 -632.0 639.4 632.0 3.036 0.009 47 48 49
0.00 0.00 0.00 -266.9 -435.1 -702.0 708.6 702.0 1.592 0.423 1 1 1

Steve M.

Gene>"I didn't ask about harmonic entropy, I asked about John's formula versus Tenney height. What is an example where his formula gets it right and
TH doesn't?"

Ok, easiest one I can think of is 11/6 vs. 9/7. Tenney Height says 9/7 is more consonant (54 vs. 66), but John's formula says 11/6 is more consonant (2.416 vs. 2116 where higher is better). I personally agree 11/6 wins here, no question.

Granted though, on second thought, both formulas seem to fail pretty badly in saying how much more "sour-sounding" things like 9 and 11-limit truly are... Actually, I think either formula fails pretty badly at that.

Note that 3/2 gets a rating of 32!!! according to John's formula and 6 according to Tenney height. According to such statistics...an 11/6 would be 7 times as dissonant as a 3/2 (by Tenney Height) and 13 times as dissonant by John's formula! Now if you scale them by the square root of the results...the scaling seems to make a lot more sense...but still seems way off to me.

Same goes for 15/8....both formulas rank it as sky high...but, in reality, it sounds like different than 11/6 to me far as dissonance. Not to mention 13/8, which sounding infinitely worse to me than 15/8...yet is ranked significantly better in both systems.
------

One way to solve this, I believe, is that, for ratios with a numerator or denominator 9 or over, do the following:
If a fraction has a numerator or denominator in it that can be divided by 3...divide one of the two (numerator of denominator) by 3 and take an average rating.

For example...15/8 when factoring out the 3 gives 5/3. Then 5/3 gets a Tenney Height of 15. Since the Tenney Height of 15/8 would be 120...take (120 + 15) / 2 = about 68...just a bit more than where 11/6 is in standard Tenney height (at 66).
Meanwhile for 11/9 you get 11/3 = 33 Tenney Height + 99 Tenney Height / 2 = about 66. 14/9 would give 14/3 with TH of 42 and averaged with the TH of 14/9 (126)...would give 84. 16/9 would get (48 + 144) / 2 = 96 (not great, but not off the charts sour). Meanwhile, 11/8 would get the usual TH value of 88 with 11/7 staying at 77. 13/9 would become 39 + 117 = 78.

Meanwhile fractions like 16/11 and 13/10 and 20/11, all of which I think sound much more dissonant than most others, would remain sky-high in this kind of modified Tenney Height as none of them are divisible by 3 in the numerator or denominator. I think to time of coming up with a version of Tenney Height that can reliably handle the 9 and 11-limit ratios (that often have a standard Tenney Height over 70) is LONG overdue.

As I said, my formula is *only* for sine wave tones, not regular complex tones.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Gene>"I didn't ask about harmonic entropy, I asked about John's formula versus Tenney height. What is an example where his formula gets it right and
> TH doesn't?"
>
>
> Ok, easiest one I can think of is 11/6 vs. 9/7. Tenney Height says 9/7 is more consonant (54 vs. 66), but John's formula says 11/6 is more consonant (2.416 vs. 2116 where higher is better). I personally agree 11/6 wins here, no question.
>
>
> Granted though, on second thought, both formulas seem to fail pretty badly in saying how much more "sour-sounding" things like 9 and 11-limit truly are... Actually, I think either formula fails pretty badly at that.
>
>
> Note that 3/2 gets a rating of 32!!! according to John's formula and 6 according to Tenney height. According to such statistics...an 11/6 would be 7 times as dissonant as a 3/2 (by Tenney Height) and 13 times as dissonant by John's formula! Now if you scale them by the square root of the results...the scaling seems to make a lot more sense...but still seems way off to me.
>
> Same goes for 15/8....both formulas rank it as sky high...but, in reality, it sounds like different than 11/6 to me far as dissonance. Not to mention 13/8, which sounding infinitely worse to me than 15/8...yet is ranked significantly better in both systems.
> ------
>
> One way to solve this, I believe, is that, for ratios with a numerator or denominator 9 or over, do the following:
> If a fraction has a numerator or denominator in it that can be divided by 3...divide one of the two (numerator of denominator) by 3 and take an average rating.
>
> For example...15/8 when factoring out the 3 gives 5/3. Then 5/3 gets a Tenney Height of 15. Since the Tenney Height of 15/8 would be 120...take (120 + 15) / 2 = about 68...just a bit more than where 11/6 is in standard Tenney height (at 66).
> Meanwhile for 11/9 you get 11/3 = 33 Tenney Height + 99 Tenney Height / 2 = about 66. 14/9 would give 14/3 with TH of 42 and averaged with the TH of 14/9 (126)...would give 84. 16/9 would get (48 + 144) / 2 = 96 (not great, but not off the charts sour). Meanwhile, 11/8 would get the usual TH value of 88 with 11/7 staying at 77. 13/9 would become 39 + 117 = 78.
>
> Meanwhile fractions like 16/11 and 13/10 and 20/11, all of which I think sound much more dissonant than most others, would remain sky-high in this kind of modified Tenney Height as none of them are divisible by 3 in the numerator or denominator. I think to time of coming up with a version of Tenney Height that can reliably handle the 9 and 11-limit ratios (that often have a standard Tenney Height over 70) is LONG overdue.
>

Thank you for this, Steve!

If I'm reading this right, the 3HE's are:
9-EDO: 2.976
14-EDO: 2.943
17-EDO: 2.937
19-EDO: 2.930
21-EDO: 2.961
22-EDO: 2.900
23-EDO: 2.967
24-EDO: 2.942

In order of lowest 3HE to highest, we have 22, 19, 17, 24, 14, 21, 23, 9.

I did a detailed comparative re-listening to the individual EDOs, and as a result, I would rank them from most concordant to least as follows:

24, 17, 22, 21, 19, 23, 9, 14

Granted, these were all played with sounds where the high partials are very audible; thus beating between them is more audible than it would be on, say, an acoustic guitar. None of these sounded very much like the Just version.

I'll put up an audio file of the different tunings of the chord arranged from most concordant to least (according to my own ears) and we'll see if everyone agrees that discordance increases from start to finish. If it's not just me, then this may be a significant issue for 3HE as a model of concordance.

-Igs

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@> wrote:
> > Does anyone reading this have access to the triadic HE data that Steve calculated? If so, can we see how the following four triads compare in triadic HE?
> >
> > 0-257-686
> > 0-285-686
> > 0-261-678
> > 0-250-700
>
> Igs,
> lobawad's comments and yours are both of great interest; I am still trying to fully understand them all. Meanwhile I can help with the numbers: I've listed:
> * lower, upper, outer - intervals in cents
> * lower, upper, outer - differences from the pure 6:7:9
> * distance from 6:7:9 in triad space
> * maximum absolute difference (of the three differences above)
> * Triadic HE
> * maximum probability for Tenney set member
> * identity of that Tenney set member
>
> As well as the four points you asked about, I included:
> * the other ratios you mentioned
> * the "best" approx to 6:7:9 in edos (9-14,17,19,22-24, 31)
> * the point with max 3HE (35,35)
> * the point with min 3HE (0,0)
>
> l u o diff diff diff dist max 3HE maxprob Tenney point
> 256.00 430.00 686.00 -10.9 -5.1 -16.0 16.3 16.0 2.942 0.025 6 7 9
> 285.00 401.00 686.00 18.1 -34.1 -16.0 34.1 34.1 2.961 0.012 23 27 34
> 261.00 417.00 678.00 -5.9 -18.1 -24.0 25.0 24.0 2.968 0.016 19 22 28
> 250.00 450.00 700.00 -16.9 14.9 -2.0 18.5 16.9 2.942 0.020 6 7 9
> 266.87 435.08 701.96 0.0 0.0 0.0 0.0 0.0 2.887 0.062 6 7 9
> 315.64 386.31 701.96 48.8 -48.8 0.0 56.3 48.8 2.931 0.037 10 12 15
> 297.51 404.44 701.96 30.6 -30.6 0.0 35.4 30.6 2.946 0.023 16 19 24
> 289.21 412.75 701.96 22.3 -22.3 0.0 25.8 22.3 2.953 0.017 16 19 24
> 386.31 315.64 701.96 119.4 -119.4 0.0 137.9 119.4 2.809 0.090 4 5 6
> 266.67 400.00 666.67 -0.2 -35.1 -35.3 40.6 35.3 2.976 0.012 30 35 44 9edo
> 240.00 480.00 720.00 -26.9 44.9 18.0 45.2 44.9 2.960 0.013 27 31 41 10edo
> 218.18 436.36 654.55 -48.7 1.3 -47.4 55.5 48.7 2.962 0.017 15 17 22 11edo
> 300.00 400.00 700.00 33.1 -35.1 -2.0 39.4 35.1 2.941 0.022 16 19 24 12edo
> 276.92 461.54 738.46 10.1 26.5 36.5 37.7 36.5 2.959 0.018 17 20 26 13edo
> 257.14 428.57 685.71 -9.7 -6.5 -16.2 16.3 16.2 2.943 0.025 6 7 9 14edo
> 282.35 423.53 705.88 15.5 -11.6 3.9 16.1 15.5 2.937 0.026 6 7 9 17edo
> 252.63 442.11 694.74 -14.2 7.0 -7.2 14.2 14.2 2.930 0.031 6 7 9 19edo
> 272.73 436.36 709.09 5.9 1.3 7.1 7.6 7.1 2.900 0.051 6 7 9 22edo
> 260.87 417.39 678.26 -6.0 -17.7 -23.7 24.6 23.7 2.967 0.016 19 22 28 23edo
> 250.00 450.00 700.00 -16.9 14.9 -2.0 18.5 16.9 2.942 0.020 6 7 9 24edo
> 270.97 425.81 696.77 4.1 -9.3 -5.2 9.3 9.3 2.915 0.046 6 7 9 31edo
> 35.00 35.00 70.00 -231.9 -400.1 -632.0 639.4 632.0 3.036 0.009 47 48 49
> 0.00 0.00 0.00 -266.9 -435.1 -702.0 708.6 702.0 1.592 0.423 1 1 1
>
>
> Steve M.
>

On Tue, Mar 22, 2011 at 7:09 PM, cityoftheasleep
<igliashon@...> wrote:
>
> I'll put up an audio file of the different tunings of the chord arranged from most concordant to least (according to my own ears) and we'll see if everyone agrees that discordance increases from start to finish. If it's not just me, then this may be a significant issue for 3HE as a model of concordance.

Here's an idea. Why don't you:

1) Post a complaint about triadic HE
2) Ignore the comments that address that complaint, which this time
are really just me parroting Paul, the guy who designed HE
3) Devise a test that purports to destroy 3HE

Sounds like a plan.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> Here's an idea. Why don't you:
>
> 1) Post a complaint about triadic HE
> 2) Ignore the comments that address that complaint, which this time
> are really just me parroting Paul, the guy who designed HE
> 3) Devise a test that purports to destroy 3HE

What in "Bob"'s name are you on about? You didn't address anything! Point me to where you address my actual "complaint", which was nothing more than pointing out that the ranking of some tempered versions of 6:7:9 by their 3HE scores does not match the ranking of them according to how concordant they actually sound. What difference does it make whether I voice the friggin' chords as tetrads with the 6 doubled an octave down as a 3, or with the actual fundamental as a 1? It doesn't change how the partials beat with each other. If 3HE is supposed to be a model for concordance, then oughtn't it darn well match up to *audible properties of chords*?

-Igs

On Tue, Mar 22, 2011 at 9:48 PM, cityoftheasleep
<igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > Here's an idea. Why don't you:
> >
> > 1) Post a complaint about triadic HE
> > 2) Ignore the comments that address that complaint, which this time
> > are really just me parroting Paul, the guy who designed HE
> > 3) Devise a test that purports to destroy 3HE
>
> What in "Bob"'s name are you on about? You didn't address anything! Point me to where you address my actual "complaint", which was nothing more than pointing out that the ranking of some tempered versions of 6:7:9 by their 3HE scores does not match the ranking of them according to how concordant they actually sound.

The point is that unless a 10:12:15 chord actually evokes a virtual 1,
that there's no point saying it's really being heard as 10:12:15. If
you load up a synth and set the timbre equal to three sine waves in a
perfect 10:12:15 ratio, chances are that you'll hear the whole
structure as a 5 with a slightly inharmonic, bell-like timbre. You
might be able to snap your brain into hearing it as a 1 if you really
try, but the 5 will probably dominate and I would imagine that it'll
dominate in a blind hearing test without any prior training.

So the point here, and this is straight from Paul, is that the
perception of a chord isn't just dependent on the f0's in the chord,
but on the f0's present in every subset of the chord as well. So you
have to take that into account.

In this case, the "minor" sound of 6:7:9, 10:12:15 and 16:19:24 has
"some mysterious thing to do" with its perception as 3/2 + crap. Just
because you don't like my specific way of explaining what this
mysterious thing is doesn't mean you should miss the forest for the
trees. You can do this listening test on your own and confirm that
it's true: 10:12:15, in a timbre, sounds more like a bell playing 5
than a sawtooth playing 1.

3HE is something that didn't exist until a few months ago, and these
drawbacks apply to it, since it doesn't actually look at all of the
subsets in the chord, just the chord as a whole. In this case it's
better than nothing, but it's still not the 3HE + 2HE of every dyad +
seeing how different f0's align with one another master solution that
has been proposed. Your comments are valid but they don't cast any
additional doubt on 3HE as a model because the whole thing is only
half done.

> What difference does it make whether I voice the friggin' chords as tetrads with the 6 doubled an octave down as a 3, or with the actual fundamental as a 1? It doesn't change how the partials beat with each other. If 3HE is supposed to be a model for concordance, then oughtn't it darn well match up to *audible properties of chords*?

What does partials beating with one another have to do with this? If
you play these chords with sine waves the same perception will result.
If by partials beating with one another you're talking about the
actual roots beating against one another, as in the "periodicity buzz"
sense, then there might be something there - I've been reading a lot
of research on that lately - but it has never been addressed how
periodicity buzz might contribute to f0 estimation here before.

But my point was that if you're hearing 6:7:9 as an actual 6:7:9, i.e.
you're hearing a C subminor triad as being the upper partials of an F,
then all of this doesn't apply, does it? It only sounds "minor" in the
sense that you know the notes are in a roughly minor proportion, but
it doesn't sound sad anymore. Nor does it sound rooted. So my point
was that if you put a 3 in there, that creates an extremely strong
1:2:3 structure at the 3:6:9, which as I and others have said is
important because you need to look at all of the subsets of the chord,
not just the chord itself.

I wouldn't expect that comparing different minor chords like this is
something that 3HE will excel at because if you aren't looking at the
HE of the subsets of the chord, then you aren't actually looking at
the things that cause minorness. 3HE might indirectly yield decent
results for this anyway by coincidence, but this is one of the known
limitations of just calculating the entropy of the triad without
looking at the constituent dyads. That doesn't mean that 3HE is
useless as a model of concordance.

-Mike

At least 2 things seem less than optimal about your formula, John.

1) It ranges negative, which means that neither an even root nor a log can
be taken.

2) You didn't try any other formulas. Gene pointed to what your formula
might do right, but there is not reason to think it is special in that. It
seems easy enough to modify TH in other ways. However, you were very clear
in explaining exactly what you did, which is good for understanding what was
tried.

>If my understanding of TH is correct it's a "badness" measure

I believe xenharmonic wiki states "inharmonicity". Badness is the wrong
construct; after all, many consider parallel perfects "bad". If we were to
say "badness", then other constructs would suit as well.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> So the point here, and this is straight from Paul, is that the
> perception of a chord isn't just dependent on the f0's in the chord,
> but on the f0's present in every subset of the chord as well. So you
> have to take that into account.

Who's talking about the perception of a chord? I'm talking about *concordance*. I don't give a damn what rational identity the brain is associating with these tempered chords, just how concordant they sound. I thought HE is supposed to model concordance?

> In this case, the "minor" sound of 6:7:9, 10:12:15 and 16:19:24 has
> "some mysterious thing to do" with its perception as 3/2 + crap. Just
> because you don't like my specific way of explaining what this
> mysterious thing is doesn't mean you should miss the forest for the
> trees. You can do this listening test on your own and confirm that
> it's true: 10:12:15, in a timbre, sounds more like a bell playing 5
> than a sawtooth playing 1.

Still not getting what this has to do with concordance.

> 3HE is something that didn't exist until a few months ago, and these
> drawbacks apply to it, since it doesn't actually look at all of the
> subsets in the chord, just the chord as a whole. In this case it's
> better than nothing, but it's still not the 3HE + 2HE of every dyad +
> seeing how different f0's align with one another master solution that
> has been proposed. Your comments are valid but they don't cast any
> additional doubt on 3HE as a model because the whole thing is only
> half done.

This is a completely different objection to my "complaint", and this one actually seems valid. If 3HE is still a work in progress, then there's no point in me complaining about it.

> What does partials beating with one another have to do with this?

This is about predicting the concordance of tempered chords. Concordance is just another word for timbral/harmonic fusion, and where there is audible beating, there is no fusion taking place.

The whole reason I'm going on and on about this is because 14-EDO's subminor triad sounds worse than 21-EDO's and even 23-EDO's, despite the fact that it is closer to 6:7:9 than the other two. I wanted to know if 3HE could explain why this is, and (in the state it currently exists), it can't. So far the only person who's given me any sort of an explanation is Lobawad, who has told me that looking at the coincidence of partials in a 6:7:9 triad explains why it should be more sensitive to a slight mistuning than to a gross mistuning.

That explanation actually makes sense to me, and maybe even also explains why I find 17-TET to be more concordant than 12-TET when it comes to playing major triads: harmonic identities associated with higher partials (like 7/6, 9/7, and even 5/4) could be more sensitive to a slight mistuning than a gross one because of critical band effects between the higher partials of a sound. If you actually listen to the sound example I posted, you can definitely hear beating between the higher partials. I may have done the math wrong, but assuming I haven't, the frequency difference between the higher partials of two tones is going to be larger than the frequency difference between the root tones.

For example, according to my calculations based on a fundamental of 400 Hz, there will be critical band dissonance between the 5th partial of the fundamental and the 4th partial of a note a 12-TET major 3rd above it, but there will NOT be critical band dissonance between the 5th partial of the fundamental and the 4th partial of a note a 17-TET supermajor 3rd above it. So regardless of the fact that 12-TET's 3rd is closer to a 5/4, 17's actually sounds better because it creates less audible beating in some registers.

To me, beating is the only factor of concordance that matters, because the only reason concordance is of interest to me in the first place is because it's supposed to tell me how high I can crank up the fuzz pedal on an interval and not have it go to s***.

And I have to say, recent events have made me entirely re-think my view of the importance of "approximation" in tempering. I used to think that just looking at absolute error from the nearest Just interval was enough, but now I see it's a lot more complicated than that. Absolute error is not a good indicator of critical band effects between (say) the first 16 partials. Neither, it seems, is harmonic entropy. In fact, it might be the case that looking at ETs in terms of JI approximation is ass-backwards, and I should just toss it out whole-hog, since it's (apparently) no predictor of how concordant that ET will sound?

> I wouldn't expect that comparing different minor chords like this is
> something that 3HE will excel at because if you aren't looking at the
> HE of the subsets of the chord, then you aren't actually looking at
> the things that cause minorness. 3HE might indirectly yield decent
> results for this anyway by coincidence, but this is one of the known
> limitations of just calculating the entropy of the triad without
> looking at the constituent dyads. That doesn't mean that 3HE is
> useless as a model of concordance.

Unless it can accurately predict the relative concordance of tempered chords, what is it useful for?

-Igs

    I noticed what seems to be an obvious trick that the diatonic scale uses.  The triads interlock in a Ls (major) and sL (minor) pattern...since all such triads interlock no wonder so many are available...

   Note I tried the same sort of thing with triads like 6:7:9 and 14:18:21...but very rarely did they chain back to the 2/1 octave and/or produce unique notes (IE chaining 6:7:9 and 14:18:21 seems to give both 8/7 and 7/6...forming a cluster and killing any chances of a strictly proper scale).   I even went out on a limb and started chaining things like 7:8:11 and 11:15:17 (since 11/8 and 15/8 are fairly close, as are 17/15 and 8/7)...but kept missing the octave.  
 
   Any ideas/tricks how to do this sort of triad chaining...without using meantone or a very similar tuning?

___

--- In tuning@yahoogroups.com, Daniel Nielsen <nielsed@...> wrote:
>
> At least 2 things seem less than optimal about your formula, John.
>
> 1) It ranges negative, which means that neither an even root nor a log can
> be taken.

It doesn't range negative, here it is again...

(2 + 1/x + 1/y - y/x)/2

where y/x <= 0.9375

The minimum result possible is +0.53125.

Chords with 3 or more notes may have a negative value but this is a different story.

>
> 2) You didn't try any other formulas.

As I outlined in my book I tried many various combinations of formulas eliminating them one by one until I got what I thought was the most likely formula. See my last post where I compare my Interval Calculator program (which uses the formula above) with Tenney Height. I found four inconsistencies (i.e. conflicting ratings of intervals) in the comparison.

Gene pointed to what your formula
> might do right, but there is not reason to think it is special in that. It
> seems easy enough to modify TH in other ways. However, you were very clear
> in explaining exactly what you did, which is good for understanding what was
> tried.
>
> >If my understanding of TH is correct it's a "badness" measure
>
> I believe xenharmonic wiki states "inharmonicity". Badness is the wrong
> construct; after all, many consider parallel perfects "bad". If we were to
> say "badness", then other constructs would suit as well.
>

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
> Thank you for this, Steve!
> If I'm reading this right ... In order of lowest 3HE to highest, we have 22, 19, 17, 24, 14, 21, 23, 9. ... I would rank them from most concordant to least as follows: 24, 17, 22, 21, 19, 23, 9, 14.

I think you are reading it right. re your original question about 6:7:9 vs 10:12:15 vs others nearby, this cross-section of the 3HE surface may (or may not) help:
/tuning/files/SteveMartin/h-o-702.png

(NB it is quite old and uses a lower Tenney limit than more recent calculations.)

Steve M.

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

>    Any ideas/tricks how to do this sort of triad chaining...without using meantone or a very similar tuning?

I've looked at that sort of thing, but I don't know what you mean by doing it without tempering.

Gene>"I've looked at that sort of thing, but I don't know what you mean by doing it without tempering."

    With tempering is fine.  I'm wondering if there are other triads that, within a 10 cent or so error margin, can be linked together similarly to form a 6-9 tone strictly proper scale.
  It looks to me like the Bohlen Pierce scale already partly does this with 3:5:7 chord, although only the 3 in the second chord overlaps with the 5 in the first chord (so there aren't as many triads available, apparently).

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
>     I noticed what seems to be an obvious trick that the diatonic scale uses.  The triads interlock in a Ls (major) and sL (minor) pattern...since all such triads interlock no wonder so many are available...
>
>    Note I tried the same sort of thing with triads like 6:7:9 and 14:18:21...but very rarely did they chain back to the 2/1 octave and/or produce unique notes (IE chaining 6:7:9 and 14:18:21 seems to give both 8/7 and 7/6...forming a cluster and killing any chances of a strictly proper scale).   I even went out on a limb and started chaining things like 7:8:11 and 11:15:17 (since 11/8 and 15/8 are fairly close, as are 17/15 and 8/7)...but kept missing the octave.  
>  
>    Any ideas/tricks how to do this sort of triad chaining...without using meantone or a very similar tuning?

/tuning-math/message/15501

Kalle

You're right about the not ranging negative part; I was thinking about the
formula without the recentering term. But why is it there and what does
0.53125 represent as a limit?

I'm not sure what you're asking me. All I know is that it seems that with sine waves, using my formula, any result 0.75 or higher sounds okay and any result below 0.75 sounds dissonant. 0.75 is midway between 0.5 and 1.0.

John.

--- In tuning@yahoogroups.com, Daniel Nielsen <nielsed@...> wrote:
>
> You're right about the not ranging negative part; I was thinking about the
> formula without the recentering term. But why is it there and what does
> 0.53125 represent as a limit?
>

On Wed, Mar 23, 2011 at 11:41 AM, Michael <djtrancendance@...> wrote:
>
>     I noticed what seems to be an obvious trick that the diatonic scale uses.  The triads interlock in a Ls (major) and sL (minor) pattern...since all such triads interlock no wonder so many are available...
>
>    Note I tried the same sort of thing with triads like 6:7:9 and 14:18:21...but very rarely did they chain back to the 2/1 octave and/or produce unique notes (IE chaining 6:7:9 and 14:18:21 seems to give both 8/7 and 7/6...forming a cluster and killing any chances of a strictly proper scale).   I even went out on a limb and started chaining things like 7:8:11 and 11:15:17 (since 11/8 and 15/8 are fairly close, as are 17/15 and 8/7)...but kept missing the octave.
>
>    Any ideas/tricks how to do this sort of triad chaining...without using meantone or a very similar tuning?

I'm not sure if this is exactly what you're talking about, but one of
the porcupine modes has two interlocked 8:10:11:12's a 6/5 apart.

How about mohajira with chained neutral triads?

-Mike

On Wed, Mar 23, 2011 at 11:30 AM, cityoftheasleep
<igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > So the point here, and this is straight from Paul, is that the
> > perception of a chord isn't just dependent on the f0's in the chord,
> > but on the f0's present in every subset of the chord as well. So you
> > have to take that into account.
>
> Who's talking about the perception of a chord? I'm talking about *concordance*.

...Is concordance not a percept of chords today?

> I don't give a damn what rational identity the brain is associating with these tempered chords, just how concordant they sound. I thought HE is supposed to model concordance?

When did I ever say anything about giving it a rational identity? If
anything it seemed like that's what you were doing, talking about the
field of attraction for 10:12:15 and 6:7:9 and so on.

> > 3HE is something that didn't exist until a few months ago, and these
> > drawbacks apply to it, since it doesn't actually look at all of the
> > subsets in the chord, just the chord as a whole. In this case it's
> > better than nothing, but it's still not the 3HE + 2HE of every dyad +
> > seeing how different f0's align with one another master solution that
> > has been proposed. Your comments are valid but they don't cast any
> > additional doubt on 3HE as a model because the whole thing is only
> > half done.
>
> This is a completely different objection to my "complaint", and this one actually seems valid. If 3HE is still a work in progress, then there's no point in me complaining about it.

It's not a completely different objection; the one was supposed to
lead into the other. Like, here's the psychoacoustic tonalness
concept, and here's what we know from it, and here's how it applies to
HE, and hence here's the big picture and why it applies to your
criticism of the model.

> > What does partials beating with one another have to do with this?
>
> This is about predicting the concordance of tempered chords. Concordance is just another word for timbral/harmonic fusion, and where there is audible beating, there is no fusion taking place.

Since when? There's audible beating on the fifths in 22 and 27 equal,
but they're still more concordant than sqrt(2). If I play a C-G dyad
and put a chorus pedal on it, the whole thing will "beat," and it'll
still sound concordant.

> The whole reason I'm going on and on about this is because 14-EDO's subminor triad sounds worse than 21-EDO's and even 23-EDO's, despite the fact that it is closer to 6:7:9 than the other two. I wanted to know if 3HE could explain why this is, and (in the state it currently exists), it can't. So far the only person who's given me any sort of an explanation is Lobawad, who has told me that looking at the coincidence of partials in a 6:7:9 triad explains why it should be more sensitive to a slight mistuning than to a gross mistuning.

Setharean dissonance is very important. Do you still feel that the
chords rank the same way if you use sine waves?

> That explanation actually makes sense to me, and maybe even also explains why I find 17-TET to be more concordant than 12-TET when it comes to playing major triads: harmonic identities associated with higher partials (like 7/6, 9/7, and even 5/4) could be more sensitive to a slight mistuning than a gross one because of critical band effects between the higher partials of a sound. If you actually listen to the sound example I posted, you can definitely hear beating between the higher partials. I may have done the math wrong, but assuming I haven't, the frequency difference between the higher partials of two tones is going to be larger than the frequency difference between the root tones.

Then maybe I misunderstood you; I think this is definitely true, but I
just think there are different things going on, all of which can cause
unpleasant "dissonance"-like sensations to arise. There's no point
comparing the two to see what makes better predictions, because there
are multiple things that can cause dissonance and both are valid in
different respects.

> For example, according to my calculations based on a fundamental of 400 Hz, there will be critical band dissonance between the 5th partial of the fundamental and the 4th partial of a note a 12-TET major 3rd above it, but there will NOT be critical band dissonance between the 5th partial of the fundamental and the 4th partial of a note a 17-TET supermajor 3rd above it. So regardless of the fact that 12-TET's 3rd is closer to a 5/4, 17's actually sounds better because it creates less audible beating in some registers.

Yes, but the further you get away from 5/4, the more it starts
sounding dissonant for other reasons. 450 cents, for instance, doesn't
have any beating at all, but you've said that you hate it. So both of
them play a factor.

Perhaps what would satisfy you is to multiply the two curves together
pointwise, such that you get a combined dissonance rating that factors
both of them into account. That way there's still a "bad spot" around
5/4 where critical band dissonance reaches a peak, but then once you
get out of that, although it may go back down for a little bit,
dissonance will then go back up as the major third goes into 450 cent
territory and beyond; and then back down once it gets into 4/3
territory and so on.

> To me, beating is the only factor of concordance that matters, because the only reason concordance is of interest to me in the first place is because it's supposed to tell me how high I can crank up the fuzz pedal on an interval and not have it go to s***.

That has far more to do with HE than it has to do with beating. I'll
spare you the mathematics involved - just minor chords and just major
chords both beat the same, but they end up sounding completely
different if you put distortion on them, whether you do it with sine
waves or not. The reason has to do with that distortion generates sum
and difference tones as well as perfectly harmonic combination tones -
so for a tempered interval these tones will end up falling very near
to one another and you'll get a huge complex of tones so closely
packed that the resulting beating clusterf*#@& will make you want to
die. This doesn't happen for JI intervals, where all of the resultant
tones overlap.

> And I have to say, recent events have made me entirely re-think my view of the importance of "approximation" in tempering. I used to think that just looking at absolute error from the nearest Just interval was enough, but now I see it's a lot more complicated than that. Absolute error is not a good indicator of critical band effects between (say) the first 16 partials. Neither, it seems, is harmonic entropy. In fact, it might be the case that looking at ETs in terms of JI approximation is ass-backwards, and I should just toss it out whole-hog, since it's (apparently) no predictor of how concordant that ET will sound?

I think that you can predict how concordant "certain" chords will
sound in an ET. For otonal chords, I recommend using some combination
of HE and Setharean dissonance, since that seems to be what you like
best. For utonal chords, you should maybe just stick with Setharean
dissonance for now, unless you want to join the crusade in hashing
some of the math out in looking at all of the subsets of the chord.
You can use HE for utonal chords as well, but as you can see you're
only getting half of the picture.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> Since when? There's audible beating on the fifths in 22 and 27 equal,
> but they're still more concordant than sqrt(2).

In some ways yes, in some ways no. The sqrt(2) might work better in death metal than the fifths of 27-ET.

> If I play a C-G dyad
> and put a chorus pedal on it, the whole thing will "beat," and it'll
> still sound concordant.

That seems like apples and oranges to me. When you have an out-of-tune interval (in the Setharesian sense), you can get a variety of different beat-frequencies happening simultaneously. I swear I can hear all sorts of different beat frequencies happening in the 14-EDO triad. But when you put a chorus effect on an in-tune interval, there's really only one smooth beat frequency.

> Setharean dissonance is very important. Do you still feel that the
> chords rank the same way if you use sine waves?

I don't care about sine waves. Why should I?

> Then maybe I misunderstood you; I think this is definitely true, but I
> just think there are different things going on, all of which can cause
> unpleasant "dissonance"-like sensations to arise. There's no point
> comparing the two to see what makes better predictions, because there
> are multiple things that can cause dissonance and both are valid in
> different respects.

Obviously there are different things going on! I've never denied that HE makes accurate predictions some times. But the fact that sometimes one model gets it right and sometimes the other model gets it right is messy. How are you supposed to know which to look at?

> Yes, but the further you get away from 5/4, the more it starts
> sounding dissonant for other reasons. 450 cents, for instance, doesn't
> have any beating at all, but you've said that you hate it. So both of
> them play a factor.

I hate it in a 10:13:15, but I like it in the utonal version of the chord. I like it melodically, too, because of its ambiguity.

> Perhaps what would satisfy you is to multiply the two curves together
> pointwise, such that you get a combined dissonance rating that factors
> both of them into account. That way there's still a "bad spot" around
> 5/4 where critical band dissonance reaches a peak, but then once you
> get out of that, although it may go back down for a little bit,
> dissonance will then go back up as the major third goes into 450 cent
> territory and beyond; and then back down once it gets into 4/3
> territory and so on.

I wonder how that would work...I have no idea how to do the math, but it would be interesting to see how it would rank the triads I've found problematic?

> That has far more to do with HE than it has to do with beating. I'll
> spare you the mathematics involved - just minor chords and just major
> chords both beat the same, but they end up sounding completely
> different if you put distortion on them, whether you do it with sine
> waves or not. The reason has to do with that distortion generates sum
> and difference tones as well as perfectly harmonic combination tones -
> so for a tempered interval these tones will end up falling very near
> to one another and you'll get a huge complex of tones so closely
> packed that the resulting beating clusterf*#@& will make you want to
> die. This doesn't happen for JI intervals, where all of the resultant
> tones overlap.

How does that have to do with HE, if it's the "beating cluster f***" that makes you "want to die"?

> I think that you can predict how concordant "certain" chords will
> sound in an ET. For otonal chords, I recommend using some combination
> of HE and Setharean dissonance, since that seems to be what you like
> best. For utonal chords, you should maybe just stick with Setharean
> dissonance for now, unless you want to join the crusade in hashing
> some of the math out in looking at all of the subsets of the chord.
> You can use HE for utonal chords as well, but as you can see you're
> only getting half of the picture.

"Some combination". What the hell does that *mean*? I don't see how that would be any more helpful than just *listening*.

-Igs

Awesome!   Thank you for the example, Kalle.

   So let's say I want 6:7:9 and 4:5:6 as the main chords....with 1/1 9/7 3/2 = 14:18:21 and 1.1 6.5 3.2 = 12:15:18 as the minor chords.  So here it looks like plain old 3/2 is the most common ratio.  

Starting with 6:7:9 I get

1/1
7/6
9/7
2/1

...adding 5ths I get......
1/1
7/6
3/2
7/4
27/14
2/1
...adding more.......
1/1
9/8
7/6
21/16 (about 4/3)
63/32 (about 13/9)
3/2
7/4
27/14 (no clue...terrible interval from the root)
2/1

Am I doing this correctly so far?

--- On Wed, 3/23/11, Kalle Aho <kalleaho@mappi.helsinki.fi> wrote:

From: Kalle Aho <kalleaho@...>
Subject: [tuning] Re: Interlocking 4:5:6 -> 10:12:15...can we use similar tricks outside meantone?
To: tuning@yahoogroups.com
Date: Wednesday, March 23, 2011, 3:25 PM

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

>

>     I noticed what seems to be an obvious trick that the diatonic scale uses.  The triads interlock in a Ls (major) and sL (minor) pattern...since all such triads interlock no wonder so many are available...

>

>    Note I tried the same sort of thing with triads like 6:7:9 and 14:18:21...but very rarely did they chain back to the 2/1 octave and/or produce unique notes (IE chaining 6:7:9 and 14:18:21 seems to give both 8/7 and 7/6...forming a cluster and killing any chances of a strictly proper scale).   I even went out on a limb and started chaining things like 7:8:11 and 11:15:17 (since 11/8 and 15/8 are fairly close, as are 17/15 and 8/7)...but kept missing the octave.  

>  

>    Any ideas/tricks how to do this sort of triad chaining...without using meantone or a very similar tuning?

/tuning-math/message/15501

Kalle

On Thu, Mar 24, 2011 at 3:37 AM, Michael <djtrancendance@...> wrote:
>
> Starting with 6:7:9 I get
>
> 1/1
> 7/6
> 9/7
> 2/1
>
> ...adding 5ths I get......
> 1/1
> 7/6
> 3/2
> 7/4
> 27/14
> 2/1
> ...adding more.......
> 1/1
> 9/8
> 7/6
> 21/16 (about 4/3)
> 63/32 (about 13/9)
> 3/2
> 7/4
> 27/14 (no clue...terrible interval from the root)
> 2/1
>
> Am I doing this correctly so far?

I'm totally lost here, but if you think that 21/16 is "about" 4/3 and
63/32 is "about" 13/9, you should consider tempering out the
difference between them and typing the resulting commas into Graham's
temperament finder.

If you're in the 13-limit, you'll need 4 commas to get you down to rank 2.

-Mike

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

> I'm totally lost here, but if you think that 21/16 is "about" 4/3 and
> 63/32 is "about" 13/9, you should consider tempering out the
> difference between them and typing the resulting commas into Graham's
> temperament finder.

I wouldn't recommend tempering out 567/416.

MikeB>"How about mohajira with chained neutral triads?"
  Good idea...used it tons of times but never looked at it that way.

I think Kalle hit the mark though...he gave a formula to make scales from interlocking chords....I'm just trying to understand it well....

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Awesome!   Thank you for the example, Kalle.
>
>    So let's say I want 6:7:9 and 4:5:6 as the main chords....with
1/1 9/7 3/2 = 14:18:21 and 1.1 6.5 3.2 = 12:15:18 as the minor
chords.  So here it looks like plain old 3/2 is the most common
ratio.  

I guess you mean 10:12:15, not 12:15:18?

>
> Starting with 6:7:9 I get
>
> 1/1
> 7/6
> 9/7
> 2/1

You should replace the 9/7 with a 3/2, don't you think?

> ...adding 5ths I get......
> 1/1
> 7/6
> 3/2
> 7/4
> 27/14
> 2/1
> ...adding more.......
> 1/1
> 9/8
> 7/6
> 21/16 (about 4/3)
> 63/32 (about 13/9)
> 3/2
> 7/4
> 27/14 (no clue...terrible interval from the root)
> 2/1
>
> Am I doing this correctly so far?

Well, I try: so, you want interlocking 6:7:9, 4:5:6, 14:18:21 and
10:12:15 chords. Note that there is no guarantee that there exists
such a scale but let's try to construct it.

Let's start with

1/1
5/4
3/2
2/1

All of the chords have a 2:3 frame so I think we should try to chain
chords at fifths. Let's put a 6:7:9 on top of that 4:5:6.

1/1
9/8
5/4
3/2
7/4
2/1

Next we must choose some of the chords to extend the scale from 9/8.
Let's try 6:7:9.

1/1
9/8
21/16
5/4
3/2
27/16
7/4
2/1

Now if you look at the interval matrix

1 2 3 4 5 6 7
1/1 : 9/8 5/4 21/16 3/2 27/16 7/4 2/1
9/8 : 10/9 7/6 4/3 3/2 14/9 16/9 2/1
5/4 : 21/20 6/5 27/20 7/5 8/5 9/5 2/1
21/16: 8/7 9/7 4/3 32/21 12/7 40/21 2/1
3/2 : 9/8 7/6 4/3 3/2 5/3 7/4 2/1
27/16: 28/27 32/27 4/3 40/27 14/9 16/9 2/1
7/4 : 8/7 9/7 10/7 3/2 12/7 27/14 2/1
2/1

you see that there are 4 of the chords we wanted and we could get
a further chord if we temper out either 63:64 or 80:81. If we temper
out both the differences between 7:9 and 4:5, and 6:7 and 5:6
disappear which is no good.

Tempering out 80:81 turns the 27:32:40 chord at 27/16 into another
10:12:15 chord. Tempering out 63:64 turns the 21:27:32 chord into
another 14:18:21 chord.

Here are the scales in .scl format:

!

7
!
193.43026
386.86052
461.29473
697.56439
890.99465
965.42887
1201.69852

and

!

7
!
217.95471
390.86635
489.88449
707.83920
925.79391
979.76897
1197.72368

Kalle

On Thu, Mar 24, 2011 at 3:49 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > I'm totally lost here, but if you think that 21/16 is "about" 4/3 and
> > 63/32 is "about" 13/9, you should consider tempering out the
> > difference between them and typing the resulting commas into Graham's
> > temperament finder.
>
> I wouldn't recommend tempering out 567/416.

Haha! Yeah, wait a second, 63/32? 13/9? I think you screwed something
up somewhere, Mike :)

-Mike

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@...> wrote:

> Here are the scales in .scl format:

Here's a scale for Mike to ponder: 5526526, or CDEE#GAA#C

The next is approximately 4324414.

Ig-dawg (if I may be permitted to use your formal title), how do the sonorities you posted rank as far as being accuate representations of ratios found within the traditionally accepted "realistic" range of about 16 partials?

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Wed, Mar 23, 2011 at 11:30 AM, cityoftheasleep
> <igliashon@...> wrote:
> >
> > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> > > So the point here, and this is straight from Paul, is that the
> > > perception of a chord isn't just dependent on the f0's in the chord,
> > > but on the f0's present in every subset of the chord as well. So you
> > > have to take that into account.
> >
> > Who's talking about the perception of a chord? I'm talking about *concordance*.
>
> ...Is concordance not a percept of chords today?
>
> > I don't give a damn what rational identity the brain is associating with these tempered chords, just how concordant they sound. I thought HE is supposed to model concordance?
>
> When did I ever say anything about giving it a rational identity? If
> anything it seemed like that's what you were doing, talking about the
> field of attraction for 10:12:15 and 6:7:9 and so on.
>
> > > 3HE is something that didn't exist until a few months ago, and these
> > > drawbacks apply to it, since it doesn't actually look at all of the
> > > subsets in the chord, just the chord as a whole. In this case it's
> > > better than nothing, but it's still not the 3HE + 2HE of every dyad +
> > > seeing how different f0's align with one another master solution that
> > > has been proposed. Your comments are valid but they don't cast any
> > > additional doubt on 3HE as a model because the whole thing is only
> > > half done.
> >
> > This is a completely different objection to my "complaint", and this one actually seems valid. If 3HE is still a work in progress, then there's no point in me complaining about it.
>
> It's not a completely different objection; the one was supposed to
> lead into the other. Like, here's the psychoacoustic tonalness
> concept, and here's what we know from it, and here's how it applies to
> HE, and hence here's the big picture and why it applies to your
> criticism of the model.
>
> > > What does partials beating with one another have to do with this?
> >
> > This is about predicting the concordance of tempered chords. Concordance is just another word for timbral/harmonic fusion, and where there is audible beating, there is no fusion taking place.
>
> Since when? There's audible beating on the fifths in 22 and 27 equal,
> but they're still more concordant than sqrt(2). If I play a C-G dyad
> and put a chorus pedal on it, the whole thing will "beat," and it'll
> still sound concordant.
>
> > The whole reason I'm going on and on about this is because 14-EDO's subminor triad sounds worse than 21-EDO's and even 23-EDO's, despite the fact that it is closer to 6:7:9 than the other two. I wanted to know if 3HE could explain why this is, and (in the state it currently exists), it can't. So far the only person who's given me any sort of an explanation is Lobawad, who has told me that looking at the coincidence of partials in a 6:7:9 triad explains why it should be more sensitive to a slight mistuning than to a gross mistuning.
>
> Setharean dissonance is very important. Do you still feel that the
> chords rank the same way if you use sine waves?
>
> > That explanation actually makes sense to me, and maybe even also explains why I find 17-TET to be more concordant than 12-TET when it comes to playing major triads: harmonic identities associated with higher partials (like 7/6, 9/7, and even 5/4) could be more sensitive to a slight mistuning than a gross one because of critical band effects between the higher partials of a sound. If you actually listen to the sound example I posted, you can definitely hear beating between the higher partials. I may have done the math wrong, but assuming I haven't, the frequency difference between the higher partials of two tones is going to be larger than the frequency difference between the root tones.
>
> Then maybe I misunderstood you; I think this is definitely true, but I
> just think there are different things going on, all of which can cause
> unpleasant "dissonance"-like sensations to arise. There's no point
> comparing the two to see what makes better predictions, because there
> are multiple things that can cause dissonance and both are valid in
> different respects.
>
> > For example, according to my calculations based on a fundamental of 400 Hz, there will be critical band dissonance between the 5th partial of the fundamental and the 4th partial of a note a 12-TET major 3rd above it, but there will NOT be critical band dissonance between the 5th partial of the fundamental and the 4th partial of a note a 17-TET supermajor 3rd above it. So regardless of the fact that 12-TET's 3rd is closer to a 5/4, 17's actually sounds better because it creates less audible beating in some registers.
>
> Yes, but the further you get away from 5/4, the more it starts
> sounding dissonant for other reasons. 450 cents, for instance, doesn't
> have any beating at all, but you've said that you hate it. So both of
> them play a factor.
>
> Perhaps what would satisfy you is to multiply the two curves together
> pointwise, such that you get a combined dissonance rating that factors
> both of them into account. That way there's still a "bad spot" around
> 5/4 where critical band dissonance reaches a peak, but then once you
> get out of that, although it may go back down for a little bit,
> dissonance will then go back up as the major third goes into 450 cent
> territory and beyond; and then back down once it gets into 4/3
> territory and so on.
>
> > To me, beating is the only factor of concordance that matters, because the only reason concordance is of interest to me in the first place is because it's supposed to tell me how high I can crank up the fuzz pedal on an interval and not have it go to s***.
>
> That has far more to do with HE than it has to do with beating. I'll
> spare you the mathematics involved - just minor chords and just major
> chords both beat the same, but they end up sounding completely
> different if you put distortion on them, whether you do it with sine
> waves or not. The reason has to do with that distortion generates sum
> and difference tones as well as perfectly harmonic combination tones -
> so for a tempered interval these tones will end up falling very near
> to one another and you'll get a huge complex of tones so closely
> packed that the resulting beating clusterf*#@& will make you want to
> die. This doesn't happen for JI intervals, where all of the resultant
> tones overlap.
>
> > And I have to say, recent events have made me entirely re-think my view of the importance of "approximation" in tempering. I used to think that just looking at absolute error from the nearest Just interval was enough, but now I see it's a lot more complicated than that. Absolute error is not a good indicator of critical band effects between (say) the first 16 partials. Neither, it seems, is harmonic entropy. In fact, it might be the case that looking at ETs in terms of JI approximation is ass-backwards, and I should just toss it out whole-hog, since it's (apparently) no predictor of how concordant that ET will sound?
>
> I think that you can predict how concordant "certain" chords will
> sound in an ET. For otonal chords, I recommend using some combination
> of HE and Setharean dissonance, since that seems to be what you like
> best. For utonal chords, you should maybe just stick with Setharean
> dissonance for now, unless you want to join the crusade in hashing
> some of the math out in looking at all of the subsets of the chord.
> You can use HE for utonal chords as well, but as you can see you're
> only getting half of the picture.
>
> -Mike
>