Interval Calculator version 6.0

To all,

I have uploaded the new version of the calculator to the JohnOSullivan folder in the "Files" section. It hasn't been *fully* tested but the anomalies that turned up before have disappeared.

Why do we need a calculator? Consider the 15/8 interval. Is it good or bad? I imagine if everyone on this list was asked some would say good and some would say bad. I couldn't be sure, hence I wrote the calculator.

With this latest version the algorithm I used was pretty much the same. I made a few programming errors in the last version. Gene asked for a formula. Here it is...

1/x + 1/y - diss (x should be less than or equal to y)
If x/y<=0.9375 then 'diss' = x/y
If X/y> 0.9375 then 'diss' = (1 - x/y)*15

I used this formula, pairing the first 1024 of the elements/harmonics of both notes in an interval with each other and 'weighting' each calculation according to the strength value of the quieter element in each pair.

So far I cannot 100% guarantee that the program is accurate but I suspect that my methods are going in the right direction. Test it yourself and let me know what you think of it.

John.

Actually (John), I'd say 15/8 is actually bad "considering it's position" of being so far apart from the root tone IE it's good in general but bad compared to many other notes toward the higher end octave interval...it sounds to me like a bit of a stretched soprano tone with the vocalist straining a bit to "belch" it out.

The very nearby 19/10 in fact sounds considerably more relaxed to me despite barely being any further away from the root. The 11/6 interval actually sounds as good to me as the 15/8 just...different; gives more of an slightly strained "alto" feel. Virtually anything between that and 8/5 loosens any tension in that alto feel as it approaches 8/5 (to me between 8/5 and 11/6 is a relative "sweet spot" that I'm guessing can't simply be summarized by any formula involving fractions).

I've also noticed...if you play 11/6 and then 15/8 then the 15/8 sounds "off" and if you do vice-versa the 11/6 sounds off...it doesn't seem to be a good idea to rate intervals by testing them directly after each other (I'm guessing your mind tries to find a "key" and treats the first note you play as the "tonic or dominant" tone). In this way...the idea of a calculator does make sense to give you a general feel of the "curve" of consonance. But if you want to explain the kinks in the curve, such as the one above...aesthetics do IMVHO play a large role and you have to decide say, on the average, where people's definition of tonal ranges with a certain feel begin and end in addition to crunching numbers intelligently to really explain how people see dyadic relationships.

I will definitely give your calculator a shot, though. :-)

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:

> 1/x + 1/y - diss (x should be less than or equal to y)
> If x/y<=0.9375 then 'diss' = x/y
> If X/y> 0.9375 then 'diss' = (1 - x/y)*15

You said you'd changed things, so I hoped it was to something lesss as hoc. This formula is compeltely unmotivated and extremely bizarre, so it's kind of hard tp work up any enthusiasm for analyzing anything in terms of it.

If you leave off the 'diss', then you are using
1/x+1/y, with a reciprocal of xy/(x+y). This can be compared to the often-used Tenney figure, which is simply xy. Tenney rates 5/3 and 15 as equally good, whereas you, without the 'diss', rate 15 as better. If you use the 'diss' the advantage to 15 increases even further, which seems counterproductive. It seems to me you'd be better off without it, and you've prodived no reason for including it, nor justification for its bizarre definition.

Michael,

I think it's crucial to use the right 'voice' on your keyboard when testing intervals. For me the best voice is the Church Organ (voice #22 on my Yamaha keyboard).

I tested the 19/10 interval (the calculator gives it -3.18) and I didn't like it when I played it. Maybe you meant to give a different example. You said that that 11/6 interval sounds good to you. The calculator concurs with this and gives it a value of 1.187.

Try testing the 10/9 interval using a church organ voice and 'stare' at it. To my ear the beating is obvious and unpleasant. In other words, IMVHO, 9/8 is the narrowest 'legal' interval.

John.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Actually (John), I'd say 15/8 is actually bad "considering it's position" of being so far apart from the root tone IE it's good in general but bad compared to many other notes toward the higher end octave interval...it sounds to me like a bit of a stretched soprano tone with the vocalist straining a bit to "belch" it out.
>
> The very nearby 19/10 in fact sounds considerably more relaxed to me despite barely being any further away from the root. The 11/6 interval actually sounds as good to me as the 15/8 just...different; gives more of an slightly strained "alto" feel. Virtually anything between that and 8/5 loosens any tension in that alto feel as it approaches 8/5 (to me between 8/5 and 11/6 is a relative "sweet spot" that I'm guessing can't simply be summarized by any formula involving fractions).
>
> I've also noticed...if you play 11/6 and then 15/8 then the 15/8 sounds "off" and if you do vice-versa the 11/6 sounds off...it doesn't seem to be a good idea to rate intervals by testing them directly after each other (I'm guessing your mind tries to find a "key" and treats the first note you play as the "tonic or dominant" tone). In this way...the idea of a calculator does make sense to give you a general feel of the "curve" of consonance. But if you want to explain the kinks in the curve, such as the one above...aesthetics do IMVHO play a large role and you have to decide say, on the average, where people's definition of tonal ranges with a certain feel begin and end in addition to crunching numbers intelligently to really explain how people see dyadic relationships.
>
> I will definitely give your calculator a shot, though. :-)
>

Gene,

You said: "This formula is compeltely unmotivated and extremely bizarre, so it's kind of hard tp work up any enthusiasm for analyzing anything in terms of it. "
I have spent 15 years working on and testing my formula, hardly ad hoc. As I said several times before the formula is explained in chapters 4 and 6 of my book which can be found in the JohnOSullivan folder in the "Files" section.

John.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "john777music" <jfos777@> wrote:
>
> > 1/x + 1/y - diss (x should be less than or equal to y)
> > If x/y<=0.9375 then 'diss' = x/y
> > If X/y> 0.9375 then 'diss' = (1 - x/y)*15
>
> You said you'd changed things, so I hoped it was to something lesss as hoc. This formula is compeltely unmotivated and extremely bizarre, so it's kind of hard tp work up any enthusiasm for analyzing anything in terms of it.
>
> If you leave off the 'diss', then you are using
> 1/x+1/y, with a reciprocal of xy/(x+y). This can be compared to the often-used Tenney figure, which is simply xy. Tenney rates 5/3 and 15 as equally good, whereas you, without the 'diss', rate 15 as better. If you use the 'diss' the advantage to 15 increases even further, which seems counterproductive. It seems to me you'd be better off without it, and you've prodived no reason for including it, nor justification for its bizarre definition.
>

John>"I tested the 19/10 interval (the calculator gives it -3.18) and I didn't like it when I played it."
Could well be. But, just to make sure you aren't operating on a timbre-related slant, it might makes sense to compare all intervals with the following
A) Guitar for even overtones matching (for example, Diatonic JI intervals excel at this)
B) Flute for odd overtone matching (BP intervals excel at this)
C) Sine for root tone matching (anything fairly well spaced and with fair or better periodicity should work here)
If, on the average, 15/8 does better than 19/10 at that I think it's fair to say 15/8 is better...but if it, say, simply fairs well with a guitar but fairs badly with the other two it's a problem.
Another take...if it does well at C) but not A or B chances are it would be a good scale to use with custom-made Setharesian timbres only (since you probably wouldn't want to be stuck with only sine waves as instruments).

A side note: organs can yield odd results as, say, the odd and even overtone-volume-controlling "drawbars" can be adjusted in random ways, and the it can slant any number of ways between focusing on "even" or "odd" timbre...plus the overtones don't oscillated/change in amplitude giving a very flat/non-moving sound that doesn't realistically match many other types of instrument. And a piano has an untypical amount of neither even nor odd overtones which some across as "inharmonic" and also a bad way way to test a scale for use with something other than just a piano.
Thus, I would highly recommend testing with a guitar over an organ sound, piano sound, or other type of "neither even nor odd" sound.

>"Maybe you meant to give a different example.
You said that that 11/6 interval sounds good to you. The calculator
concurs with this and gives it a value of 1.187."
Well,
1) What does that calculator give for 19/10?
2) How does 19/10 do on said-above tests...and does it match with the rating in #1?

Also, does anyone else have a special opinion about there intervals?

Do you have a reference for these matching timbres?

On Thu, Apr 29, 2010 at 3:31 PM, Michael <djtrancendance@...> wrote:
  But, just to make sure you aren't operating on a timbre-related
slant, it might makes sense to compare all intervals with the
following
A) Guitar for even overtones matching (for example, Diatonic JI
intervals excel at this)
B) Flute for odd overtone matching (BP intervals excel at this)
C) Sine for root tone matching (anything fairly well spaced and with
fair or better periodicity should work here)

Chris>" Do you have a reference for these matching timbres?"
http://en.wikipedia.org/wiki/Timbre
...for example...
"William Sethares (2004) wrote that just intonation and the western equal tempered scale are related to the harmonic spectra/timbre of many western instruments in an analogous way that the inharmonic
timbre of the Thai renat (a xylophone-like instrument)
is related to the seven-tone near-equal temperament in which they are
tuned. Similarly, the inharmonic spectra of Balinese metallophones combined with harmonic instruments such as the stringed rebab or
the voice, are related to the five-note near-equal tempered slendro scale commonly found in Indonesian gamelan music."

also look at...
http://www.stanford.edu/~hoffert/projects/music/figure3.jpg
Here you see a similar to flute wind type instrument producing much stronger odd than even harmonics plus a guitar where harmonic #2 (even) is the strongest overtone by a considerable amount and the rest of the overtones are about the same (thus it leads toward even harmonics...although it isn't as leaned toward even harmonics as a flute is to odd ones).

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
>
> Gene,
>
> You said: "This formula is compeltely unmotivated and extremely bizarre, so it's kind of hard tp work up any enthusiasm for analyzing anything in terms of it. "
> I have spent 15 years working on and testing my formula, hardly ad hoc. As I said several times before the formula is explained in chapters 4 and 6 of my book which can be found in the JohnOSullivan folder in the "Files" section.

I really don't see how an explanation which makes much sense is even possible. From what you've said, you've fiddled with it to get it to produce results you like with the range of intervals you look at, but even so why do you *want* 5/3 to be rated worse than 15? Moreover, your cited formula, which is asymmetic in terms of numerator and denomiator, does not fit the results of your calculator.

> John.
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> >
> >
> >
> > --- In tuning@yahoogroups.com, "john777music" <jfos777@> wrote:
> >
> > > 1/x + 1/y - diss (x should be less than or equal to y)
> > > If x/y<=0.9375 then 'diss' = x/y
> > > If X/y> 0.9375 then 'diss' = (1 - x/y)*15
> >
> > You said you'd changed things, so I hoped it was to something lesss as hoc. This formula is compeltely unmotivated and extremely bizarre, so it's kind of hard tp work up any enthusiasm for analyzing anything in terms of it.
> >
> > If you leave off the 'diss', then you are using
> > 1/x+1/y, with a reciprocal of xy/(x+y). This can be compared to the often-used Tenney figure, which is simply xy. Tenney rates 5/3 and 15 as equally good, whereas you, without the 'diss', rate 15 as better. If you use the 'diss' the advantage to 15 increases even further, which seems counterproductive. It seems to me you'd be better off without it, and you've prodived no reason for including it, nor justification for its bizarre definition.
> >
>

The graph from Stanford is excellent - thanks!

On Thu, Apr 29, 2010 at 4:11 PM, Michael <djtrancendance@...> wrote:
>
>
>
> Chris>"  Do you have a reference for these matching timbres?"
> http://en.wikipedia.org/wiki/Timbre
> ...for example...

>
> also look at...
> http://www.stanford.edu/~hoffert/projects/music/figure3.jpg
>   Here you see a similar to flute wind type instrument producing much stronger odd than even harmonics plus a guitar where harmonic #2 (even) is the strongest overtone by a considerable amount and the rest of the overtones are about the same (thus it leads toward even harmonics...although it isn't as leaned toward even harmonics as a flute is to odd ones).
oogroups.com - turn off mail from the group.

Maybe instead of using the "right" voice, you should use voices that you might actually to make music with? Unless you're going to write music exclusively with that church-organ voice for the rest of your life...

-Igs

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
>
> Michael,
>
> I think it's crucial to use the right 'voice' on your keyboard when testing intervals. For me the best voice is the Church Organ (voice #22 on my Yamaha keyboard).
>
> I tested the 19/10 interval (the calculator gives it -3.18) and I didn't like it when I played it. Maybe you meant to give a different example. You said that that 11/6 interval sounds good to you. The calculator concurs with this and gives it a value of 1.187.
>
> Try testing the 10/9 interval using a church organ voice and 'stare' at it. To my ear the beating is obvious and unpleasant. In other words, IMVHO, 9/8 is the narrowest 'legal' interval.
>
> John.
>
> --- In tuning@yahoogroups.com, Michael <djtrancendance@> wrote:
> >
> > Actually (John), I'd say 15/8 is actually bad "considering it's position" of being so far apart from the root tone IE it's good in general but bad compared to many other notes toward the higher end octave interval...it sounds to me like a bit of a stretched soprano tone with the vocalist straining a bit to "belch" it out.
> >
> > The very nearby 19/10 in fact sounds considerably more relaxed to me despite barely being any further away from the root. The 11/6 interval actually sounds as good to me as the 15/8 just...different; gives more of an slightly strained "alto" feel. Virtually anything between that and 8/5 loosens any tension in that alto feel as it approaches 8/5 (to me between 8/5 and 11/6 is a relative "sweet spot" that I'm guessing can't simply be summarized by any formula involving fractions).
> >
> > I've also noticed...if you play 11/6 and then 15/8 then the 15/8 sounds "off" and if you do vice-versa the 11/6 sounds off...it doesn't seem to be a good idea to rate intervals by testing them directly after each other (I'm guessing your mind tries to find a "key" and treats the first note you play as the "tonic or dominant" tone). In this way...the idea of a calculator does make sense to give you a general feel of the "curve" of consonance. But if you want to explain the kinks in the curve, such as the one above...aesthetics do IMVHO play a large role and you have to decide say, on the average, where people's definition of tonal ranges with a certain feel begin and end in addition to crunching numbers intelligently to really explain how people see dyadic relationships.
> >
> > I will definitely give your calculator a shot, though. :-)
> >
>

>"Maybe instead of using the "right" voice, you should use voices that you might actually to make music with? Unless you're going to write music
exclusively with that church-organ voice for the rest of your life..."
Exactly. The key to testing, IMVHO, is to find a few instruments that cover most of the typical harmonic structures you want to use. For example
1) Clarinet (fairly low stronger odd harmonics like many wind instruments)
2) Guitar (fairly low stronger even harmonics like many stringed instruments)
3) Trumpet (higher even and odd harmonics like many brass instruments)
4) Sine wave (Setharesian "spectral aligned" instruments...if you want to focus on strictly electronic music and particularly music made with the his TransFormSynth softsynth)

And here (again) isa graph of the overtone curves of those three: http://www.stanford.edu/~hoffert/projects/music/figure3.jpg
In addition you may want to try
5) Xylophone and/or Bell (many in-harmonic partials)

I have found the sine wave gives you a decent idea of how well a scale will match timbres "on the average". So it might be a good idea to make sure your intervals sound quite good with sine waves before bothering to test them with other instruments as, in general, consonance can only go down with non-sine timbres. Again I don't recommend saw waves as the abundance of high harmonics is very untypical of most instruments...I'd consider brass a such much realistic option for testing high harmonic timbre "matching" for your scales.

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> Maybe instead of using the "right" voice, you should use voices that you might actually to make music with? Unless you're going to write music exclusively with that church-organ voice for the rest of your life...

Exactly.

I'm wondering, what with all the "good vs. bad" intervals and a reliance on Church Organs, if this isn't the Pope posting as an alt with a very discernible agenda...