Interval Calculator vs Tenney Height

Here are the results I got comparing my Interval Calculator program against Tenney Height. I'm assuming that the "goodness" value of Tenney Height is is 1/(n*d), is this correct?
The program uses the (2 + 1/x + 1/y - y/x)/2 formula (note y/x<=0.9375) which applies to sine wave tones.

Interval--IC------TH
3/2 32.229537 0.166667
4/3 19.114845 0.083333
5/3 16.668730 0.066667
5/4 12.124845 0.05
7/4 9.428482 0.035714
6/5 7.811470 0.033333
7/5 6.661762 0.028571
8/5 5.809393 0.025
9/5 5.196255 0.022222!!!
7/6 4.804254 0.023810!!!
8/7 2.613538 0.017857
11/6 2.416868 0.015152!!!
9/7 2.116143 0.015873!!!
10/7 1.579299 0.014286
11/7 1.204370 0.012987!!!
9/8 0.920317 0.013889!!!
12/7 0.870487 0.011905
13/7 0.425974 0.010989!!!
11/8 0.195058 0.011364!!!

My program says 9/5 is better than 7/6, Tenney Height says the opposite. The program says 11/6 is better than 9/7, TH says the opposite. The program says 11/7 is better than 9/8, TH says the opposite. The program says 13/7 is better than 11/8, TH says the opposite. The last example should be the clearest.

I haven't done any listening tests on these yet (it's late where I am) so I'll have more on this tomorrow.

John.

I did the listening tests for my Interval Calculator program (which uses my formula for sine waves) against Tenney Height. Note that Tenney Height does not consider the harmonic series of the notes and my calculator program does. My program says 9/5 is better than 7/6, Tenney Height says the opposite. The program says 11/6 is better than 9/7, TH says the opposite. The program says 11/7 is better than 9/8, TH says the opposite. The program says 13/7 is better than 11/8, TH says the opposite.

With the 9/5 vs 7/6, 11/6 vs 9/7, and 13/7 vs 11/8 pairs it was very obvious that the predictions of my calculator seemed correct and TH seemed wrong. With the 11/7 vs 9/8 pair the winner wasn't *very* obvious but seemed to me to lean more towards my program than TH. You might like to test these yourself.

John.

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
>
> Here are the results I got comparing my Interval Calculator program against Tenney Height. I'm assuming that the "goodness" value of Tenney Height is is 1/(n*d), is this correct?
> The program uses the (2 + 1/x + 1/y - y/x)/2 formula (note y/x<=0.9375) which applies to sine wave tones.
>
>
> Interval--IC------TH
> 3/2 32.229537 0.166667
> 4/3 19.114845 0.083333
> 5/3 16.668730 0.066667
> 5/4 12.124845 0.05
> 7/4 9.428482 0.035714
> 6/5 7.811470 0.033333
> 7/5 6.661762 0.028571
> 8/5 5.809393 0.025
> 9/5 5.196255 0.022222!!!
> 7/6 4.804254 0.023810!!!
> 8/7 2.613538 0.017857
> 11/6 2.416868 0.015152!!!
> 9/7 2.116143 0.015873!!!
> 10/7 1.579299 0.014286
> 11/7 1.204370 0.012987!!!
> 9/8 0.920317 0.013889!!!
> 12/7 0.870487 0.011905
> 13/7 0.425974 0.010989!!!
> 11/8 0.195058 0.011364!!!
>
> My program says 9/5 is better than 7/6, Tenney Height says the opposite. The program says 11/6 is better than 9/7, TH says the opposite. The program says 11/7 is better than 9/8, TH says the opposite. The program says 13/7 is better than 11/8, TH says the opposite. The last example should be the clearest.
>
> I haven't done any listening tests on these yet (it's late where I am) so I'll have more on this tomorrow.
>
> John.
>

Gene,

I uploaded four mp3s to the JohnOSullivan folder in the Files section that compare intervals which my interval calculator program and Tenney Height disagree on.

In all four examples the interval with the higher note should (if I'm right) sound sweeter than the interval with the lower note.

In each mp3 the interval with the higher note sounds smoother to me and the interval with the lower note sound harsher.

I should point out that in the past I have listened for either 'strength' or 'sweetness' or 'resolution' and found in many cases I could not decide which interval I thought sounded better. In Ralph Denyer's "The Guitar Handbook" he says that the octave (2/1) somehow sounds 'similar' to the tonic (1/1). Based on this idea I have found that listening for 'similarity' works best and easily. For example take 1/1 paired with 11/6 and 1/1 paired with 9/7. It seems to me that the 11/6 sounds more 'similar' to 1/1 than the 9/7.

What is your opinion after listening?

BTW, prime limits are not a factor in my interval calculator program but I do not *ignore* them. I used the prime limit idea when I was choosing notes for my Blue Just Tuning scale.

John.

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
>
> Gene,
>
> I uploaded four mp3s to the JohnOSullivan folder in the Files section that compare intervals which my interval calculator program and Tenney Height disagree on.

Thanks!

> What is your opinion after listening?

(1) 9/7 is better than 11/6

(2) 9/8 is better than 11/7

(3) 13/7 is better than 11/8

(4) 7/6 is better than 9/5.

Hmmm, I will concede (as I have before) that my calculator program might be wrong. Still, to your ears 13/7 is better than 11/8. If you are right then Tenney Height must be wrong as it predicts that 13/7 should be worse than 11/8.

Food for thought indeed.

John.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "john777music" <jfos777@> wrote:
> >
> > Gene,
> >
> > I uploaded four mp3s to the JohnOSullivan folder in the Files section that compare intervals which my interval calculator program and Tenney Height disagree on.
>
> Thanks!
>
> > What is your opinion after listening?
>
> (1) 9/7 is better than 11/6
>
> (2) 9/8 is better than 11/7
>
> (3) 13/7 is better than 11/8
>
> (4) 7/6 is better than 9/5.
>

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
>
> Hmmm, I will concede (as I have before) that my calculator program might be wrong. Still, to your ears 13/7 is better than 11/8. If you are right then Tenney Height must be wrong as it predicts that 13/7 should be worse than 11/8.
>
> Food for thought indeed.

It could be because of the proximity of 11/8 to 4/3, which is a very strong interval.

I doubt it, 11/8 is 551.3 cents and 4/3 is 498.0 cents, a difference of 53.3 cents. A significant difference in my opinion.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "john777music" <jfos777@> wrote:
> >
> > Hmmm, I will concede (as I have before) that my calculator program might be wrong. Still, to your ears 13/7 is better than 11/8. If you are right then Tenney Height must be wrong as it predicts that 13/7 should be worse than 11/8.
> >
> > Food for thought indeed.
>
> It could be because of the proximity of 11/8 to 4/3, which is a very strong interval.
>

John>"Still, to your ears 13/7 is better than 11/8. If you are right then
Tenney Height must be wrong as it predicts that 13/7 should be worse
than 11/8. Also note...the octave inverse of 11/8 is "

   Indeed.  I think that's a quite obvious example of your theory being better for certain things...I hear it that way as well.  Plus its inverse is 16/11...which I'm pretty sure you'll agree with me is horrid sounding, even relative to intervals around it like 13/7.

   Here's a side question for everyone...how would you rank all of the 11-odd-limit intervals, from best to worst?   Better yet, if in interval is very close to a lower limit interval, explain how much you think it sounds like said lower-limit interval.

   I think that would definitely be food for thought as I think both Tenney Height and John's calculator fall quite flat for 11-limit.  Perhaps after we get a good handle what the results are (on average)...we can hypothesize how to push past the ability of things like Tenney Height with either new theory or modifying existing ones (IE John's and Tenney Height). 

   For the record, I think the rule "is the numerator or denominator divisible by 3?" says a lot about what makes certain 11-limit intervals better than others...if it's divisible, changes are it's noticeably more stable sounding than intervals of similar Tenney Height around it (IE 15/11.vs. 16/11 vs. 18/11).

>"It could be because of the proximity of 11/8 to 4/3, which is a very strong interval."

   Funny, because, to me the difference between 11/8 and 4/3 is like night and day and seems to point out one of the huge weaknesses of HE and also Tenney Height (IE they fall to pieces at 11-limit or higher).
   In fact, the furthest point from 4/3 and 7/5 (the lowest limit nearby intervals) is a tad over 15/11...and to my ears...15/11 has its own field of attraction, although a very small one, which is just far enough away to escape the pulls of 4/3 and 7/5 (to note, the field of attraction seems to be stronger above than below 15/11...so indeed 4/3 seems to have a slight pull on 15/11 over 7/5).    

   Another example, between 6/5 and 5/4 you get 11/9...so is 11/9 a warped 6/5 or a warped 5/4?...to my ears, it's clearly its own identity and reaches up to about 8 cents over 11/9 and a couple of cents below it (IE it's not conveniently symmetrical).

  And how about 7/4 vs. 9/5 vs. 11/6 vs. 15/8?  Sure, 7/4 seems the strongest, but I dare anyone to tell me they hear, say, 15/8 or even 9/5 as a "weak 7/4".

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> >"It could be because of the proximity of 11/8 to 4/3, which is a very strong interval."
>
>    Funny, because, to me the difference between 11/8 and 4/3 is like night and day

This is obvious, as otherwise I would have rated it better than 13/7.

> ...and seems to point out one of the huge weaknesses of HE and also Tenney Height (IE they fall to pieces at 11-limit or higher).

No more than O'Sullivan's measure does. Anyway, we need better data than we have to draw very much in the nature of conclusions.

I think we need to know what people mean by "better". For I can agree with either side here, depending on whether I choose "fused, as an isolated physical identity" or "pretty-sounding dyad" as my definition of "better".

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, Michael <djtrancendance@> wrote:
> >
> > >"It could be because of the proximity of 11/8 to 4/3, which is a very strong interval."
> >
> >    Funny, because, to me the difference between 11/8 and 4/3 is like night and day
>
> This is obvious, as otherwise I would have rated it better than 13/7.
>
> > ...and seems to point out one of the huge weaknesses of HE and also Tenney Height (IE they fall to pieces at 11-limit or higher).
>
> No more than O'Sullivan's measure does. Anyway, we need better data than we have to draw very much in the nature of conclusions.
>

Yes I know...there's a song (yes, VOCAL song) of the same name. 

   The idea here is to get something with as many Just triads per note as the diatonic scale and I dare anyone on this list to get more triads per note in a 7-tone scale without using the Diatonic scale!  :-D
----------------------------------
   As for how to get the scale...I'm using the 18th harmonic (hence the scale name)...take 18-36th harmonic fractions (factors of 18 used = 2,3,6,9) and make them into a scale.  I got this idea from the fact that diatonic JI can be summarized between the 24th and 48th harmonic (factors of 24 used = 2,3,4,8,9).  So yes, diatonic has more factors...but 18 is a close second (note 12 has 2,3,4...16 has 2,4,8...10 has 1,2,5...20 has 1,2,4,5...so 20 might not be a bad alternative).

  For comparison, the "18 (till I die)" scale is
18/18 20/18 21/18 24/18 28/18 30/20 32/18 36/18 AKA
18:20:21:24:28:30:32:36

 While diatonic JI is
---------9/8-------5/4------4/3---------3/2----------5/3------15/8
24/24 27/24  30/24   32/24    36/24      40/24    45/24-------2/1

The reduced version of the "18 till I die" scale is simply
>>>>>>>>1/1 10/9 7/6 4/3 14/9 5/3 16/9 2/1<<<<<<<<<<<<<<

Triads/tetrads possible (from different root notes...derived from Scala) include
3:4:5 and 4:5:7 (chained)
6:7:8 and 7:8:9 (chained)
5:6:8 and 6:8:9 and 8:9:15 and 9:15:21 (all chained!)
10:12:15 and 12:15:16 and 15:16:18 and 16:18:21 (all chained!)
7:8:10 and 8:10:12
4:5:6 and 5:6:7
12:15:16 and 15:16:18 and 16:18:21 and 18:20:21
5:6:7 and 6:7:8
4:5:6 and 5:6:9 and 3:5:7
8:9:10 and 9:10:12 and 10:12:15 and 12:15:21

--- In tuning@yahoogroups.com, "lobawad" <lobawad@...> wrote:
>
> I think we need to know what people mean by "better". For I can agree with either side here, depending on whether I choose "fused, as an isolated physical identity" or "pretty-sounding dyad" as my definition of "better".

It would be interesting to see you rate each pair both ways.

--- In tuning@yahoogroups.com, "lobawad" <lobawad@...> wrote:
>
> I think we need to know what people mean by "better". For I can agree with either side here, depending on whether I choose "fused, as an isolated physical identity" or "pretty-sounding dyad" as my definition of "better".

Good point. For anyone who is interested here's a story. I performed a large number of listening tests to discover which of these four formulas was the most likely to be correct:
1/(x*y), 1(x+y), 1/x + 1/y and 1/x.
First I listened for 'strength' but with many intervals I couldn't decide which ones were better. Next I listened for 'sweetness' and I still couldn't be sure. Next I listened for 'resolution' and I was still flummoxed. Then I remembered reading in Ralph Denyer's "The Guitar Handbook" that the octave (2/1) sounds *similar* to the tonic (1/1), they could almost be treated as being the same note. Based on this I started listening for *similarity* and, Bingo, the notes that sounded more *similar* to the tonic were clearly and easily identified.

So all of my listening tests from then on are based on listening for *similarity*. The question is: does *similarity* automatically imply *sweetness*? It certainly does for intervals with low numbers (e.g. 2/1 and 3/2) but maybe it doesn't for more complex intervals. If it doesn't then much of work was for naught. If it does then my work should carry some weight.

Here are four pairs of intervals and how I rate them...

(i) 9/5 is better than 7/6
(ii) 11/6 is better than 9/7
(iii)11/7 is better than 9/8
(iv) 13/7 is better than 11/8

Tenney Height predicts that I'm wrong and the opposite is true.

In the JohnOSullivan folder in the Files section there are four MP3s that demonstrate each of the four pairs listed above. In each MP3 I predict that the higher note paired with 1/1 should sound more similar to 1/1 than the lower note paired with 1/1. Anyone want to give these a listen?

I think that the question above (does 'similarity' imply sweetness?) is important because when listening for *similarity*, listening tests seem to be much easier and clearer (for me at least) than listening for strength, sweetness or resolution.

John.

>
>
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> >
> >
> >
> > --- In tuning@yahoogroups.com, Michael <djtrancendance@> wrote:
> > >
> > > >"It could be because of the proximity of 11/8 to 4/3, which is a very strong interval."
> > >
> > >    Funny, because, to me the difference between 11/8 and 4/3 is like night and day
> >
> > This is obvious, as otherwise I would have rated it better than 13/7.
> >
> > > ...and seems to point out one of the huge weaknesses of HE and also Tenney Height (IE they fall to pieces at 11-limit or higher).
> >
> > No more than O'Sullivan's measure does. Anyway, we need better data than we have to draw very much in the nature of conclusions.
> >
>

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
>
> --- In tuning@yahoogroups.com, "lobawad" <lobawad@> wrote:
> >
> > I think we need to know what people mean by "better". For I can agree with either side here, depending on whether I choose "fused, as an isolated physical identity" or "pretty-sounding dyad" as my definition of "better".
>
> Good point. For anyone who is interested here's a story. I performed a large number of listening tests to discover which of these four formulas was the most likely to be correct:
> 1/(x*y), 1(x+y), 1/x + 1/y and 1/x.
> First I listened for 'strength' but with many intervals I couldn't decide which ones were better. Next I listened for 'sweetness' and I still couldn't be sure. Next I listened for 'resolution' and I was still flummoxed. Then I remembered reading in Ralph Denyer's "The Guitar Handbook" that the octave (2/1) sounds *similar* to the tonic (1/1), they could almost be treated as being the same note. Based on this I started listening for *similarity* and, Bingo, the notes that sounded more *similar* to the tonic were clearly and easily identified.
>
> So all of my listening tests from then on are based on listening for *similarity*. The question is: does *similarity* automatically imply *sweetness*? It certainly does for intervals with low numbers (e.g. 2/1 and 3/2) but maybe it doesn't for more complex intervals. If it doesn't then much of work was for naught. If it does then my work should carry some weight.
>
> Here are four pairs of intervals and how I rate them...
>
> (i) 9/5 is better than 7/6
> (ii) 11/6 is better than 9/7
> (iii)11/7 is better than 9/8
> (iv) 13/7 is better than 11/8
>
> Tenney Height predicts that I'm wrong and the opposite is true.
>

Except for iii which is kind of a tie for me, I'd rate the same as you but for NOT being "similar", rather, as sweeter sounding dyads. For "fused together", I'd rate, in this case, the same as Tenney height does.

This is all in isolation of course, context can change things greatly.