Yahoo Tuning Groups Ultimate Backup tuning Difference tones versus harmonic beating

Steve wrote :

> > > (Jacques) :
> > > What's the beating of such a A:E dyad for A = 440 hz ?
> > > (3 * A) - (2 * E) =
> > > (3 * 440 hz) - (2 * 658.219542747 hz) = 3.5609145 hz
> > > It's a beating frequency, not a difference tone.
> > > The difference tone of this dyad is B - A =
> > > 658.219542747 hz - 440 hz = 218.219542747 hz
> > (Gene): Other people use the words differently:
> > http://en.wikipedia.org/wiki/Combination_tone
> > I'd be interested to know what's considered most acceptable.
>
> Although a comparative noob, I must say that my understanding of > the terms is
> exactly as Jacques says.
>
> BTW the Wiki page isn't clear to me - it says there are three related
> psychoacoustic phenomena but doesn't clearly say what they are, > instead giving
> an example and relating it to missing fundamental.
>
> Steve M.

I agree. It clearly mixes up "missing fundamental" and difference tones from the beginning.
It doesn't mention nor explains either 1st order /2nd order / 3rd order difference tones.
It does'nt mentions double-ocarinas, which provide the most efficient initiation ;)
What's cool is that it has a link to a page on a "Difference Tone Training" method, very complete, but using speakers (and repeating the usual old belief that difference tones occur at fairly high volumes, which is wrong).
Then it says something surprising, that Max Mathews and John Pierce wrote that "Heinz Bohlen proposed what is now known as the B-P scale on the basis of combination tones". But what combination tones and on what basis ? that's interesting but rather strange because it is easy to demonstrate that the BP intervals, wether in addition or soustraction, introduces octaves (but why not, they would be the "missing octaves" then).
It has, on the other hand, much more complete information on "Binaural beats" :
"For a time it was thought that the inner ear was solely responsible whenever a sum or difference tone was heard. However, experiments show evidence that even when using headphones providing a single pure tone to each ear separately, listeners may still hear a difference tone. Since the peculiar, non-linear physics of the ear doesn't come into play in this case, it is thought that this must be a separate, neural phenomenon."

About difference tones/harmonic beating :
We can say that the (harmonic) beating of a tempered dyad is the 1st order difference tone of their concordant harmonics.
But another difference is in the range : a beat, passed 20hz becomes a tone ; and a difference tone below 20hz becomes an infra-sound and a beat.
So again my subsidiary question :
We have a meantone with a fifth ratio of 1.4959535062432 (697.2784049 c.).
What's the frequency of the beat you hear in a major third tuned in 1/1 = 440hz ?
And more important, how you find it ?
Nobody can answer ?
- - - - - - - - - -
Jacques

🔗cameron <misterbobro@...>

🔗jacques.dudon <fotosonix@...>

🔗cameron <misterbobro@...>

Just got back on the Internet the other day, travelling! I have a show tonight, feeling kind of nervous- great location here in Tomar, Portugal, but heavily muffled acoustics, I far prefer an acoustic place and no microphone.

About the ongoing discussion about difference tones, I find that moderate volumes and nice acoustics are the best conditions for hearing and feeling these things. Well, those are the best conditions for pretty much anything in music anyway.

Anyway beating occurs most prominently at those places where the partials would coincide were the interval to be perfectly Just.

-Cameron Bobro

--- In tuning@yahoogroups.com, "jacques.dudon" <fotosonix@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> >
> >
> >
> > --- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@> wrote:
> >
> > > So again my subsidiary question :
> > > We have a meantone with a fifth ratio of 1.4959535062432
> > > (697.2784049 c.).
> > > What's the frequency of the beat you hear in a major third tuned in
> > > 1/1 = 440 hz ?
> > > And more important, how you find it ?
> > > Nobody can answer ?
> > > - - - - - - - - - -
> > > Jacques
> > >
> >
> > In a major third diad, the fourth partial of the higher tone will beat against the the fifth partial of the lower. So, .8902 Hz.
> >
> > Correct me if I'm wrong. :-)
>
> Hi Cameron !
> It's been sometime since we heard about you, where have you been ?
> .8902 Hz., mmmh ?... everybody agrees ?
>

🔗jacques.dudon <fotosonix@...>

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
> Just got back on the Internet the other day, travelling! I have a show tonight, feeling kind of nervous- great location here in Tomar, Portugal, but heavily muffled acoustics, I far prefer an acoustic place and no microphone.

All the best and good luck with the sound ! you better get ready then !

> About the ongoing discussion about difference tones, I find that moderate volumes and nice acoustics are the best conditions for hearing and feeling these things. Well, those are the best conditions for pretty much anything in music anyway.

That's so true !

> Anyway beating occurs most prominently at those places where the partials would coincide were the interval to be perfectly Just.
>
> -Cameron Bobro

Can't say better ! it's exactly how it works.
- - - - - - -
Jacques
>
> --- In tuning@yahoogroups.com, "jacques.dudon" <fotosonix@> wrote:
> >
> >
> >
> > --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> > >
> > >
> > >
> > > --- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@> wrote:
> > >
> > > > So again my subsidiary question :
> > > > We have a meantone with a fifth ratio of 1.4959535062432
> > > > (697.2784049 c.).
> > > > What's the frequency of the beat you hear in a major third tuned in
> > > > 1/1 = 440 hz ?
> > > > And more important, how you find it ?
> > > > Nobody can answer ?
> > > > - - - - - - - - - -
> > > > Jacques
> > > >
> > >
> > > In a major third diad, the fourth partial of the higher tone will beat against the the fifth partial of the lower. So, .8902 Hz.
> > >
> > > Correct me if I'm wrong. :-)
> >
> > Hi Cameron !
> > It's been sometime since we heard about you, where have you been ?
> > .8902 Hz., mmmh ?... everybody agrees ?
> >
>

🔗genewardsmith <genewardsmith@...>

🔗martinsj013 <martinsj@...>

🔗jacques.dudon <fotosonix@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, "jacques.dudon" <fotosonix@> wrote:
> > > (Cameron) :
> > > In a major third diad, the fourth partial of the higher tone will beat against the the fifth partial of the lower. So, .8902 Hz.
> > >
> > > Correct me if I'm wrong. :-)
> >
> > Hi Cameron !
> > It's been sometime since we heard about you, where have you been ?
> > .8902 Hz., mmmh ?... everybody agrees ?
>
> The higher note is 550.89... Hz, and the difference between four times this and five times 440 is 3.56.. Hz.

OK you guys, you are all good acousticians but Gene beats you on the maths. Cameron has a excuse because his was in Portugal just before his concert (Wow, he is singing now !) and he didn't read well that 1/1 was 440 hz, not 100 hz (but why choosing 440 hz ?).

Now remember, the fifth A:E of the same meantone was beating at
(3 * 440 hz) - (2 * 658.219542747 hz) = 3.5609145 hz =
same frequency ! = this is what we mean by "equal-beating" (here between the fifth and the major third beatings, but it could be other intervals).
So now you can check it with a plain sawtooth waveform and a multitrack, because harmonic beatings ARE completely audible (and you can also see them on the signal), and experience how it feels with the triad A : C# : E (you can also imagine it would be simpler to tune if you were an harpsichord tuner).

OK, one more thing and I have finished : calculate the difference tone between 2A (A at the higher octave), and C# :
880 - 550.8902286265 = 329.1097713735 hz = E/2 (E at the lower octave)
And hear it, if you can rather with sinus sounds. So this is a simple example (Skisni meantone) of how the two phenomenas, difference tones and harmonic beating, while operating in very different domains of audition, are in fact, in recurrent sequences completely connected.
What's the significance of that ? If you can hear it, it is significant. But if YOU can't hear it, it does not mean it's not significant. Not all singers hear their harmonics, but all of them produce harmonics with their voices.
- - - - - - - -
Jacques

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, "jacques.dudon" <fotosonix@...> wrote:

> Now remember, the fifth A:E of the same meantone was beating at
> (3 * 440 hz) - (2 * 658.219542747 hz) = 3.5609145 hz =
> same frequency ! = this is what we mean by "equal-beating" (here between the fifth and the major third beatings, but it could be other intervals).

If f is the fifth and t the major third of a major triad in close root position, then (6t-5f)/(4t-5) is what I've saddled with the moniker "brat", from Beat RATio. The above is then the meantone tuning with a brat of 4, one of the "magic" figures which give interesting relationships between the beating of the major third, minor third, and fifth. Since in fully regular meantone we have the relationship t = f^4/4, we can substitute into the brat and find a brat of 4 corresponds to a fifth of f^4 + 2f - 4 = 0, so we get a fifth which is the unique positive real root of the above equation, an algebraic integer in case anyone cares, but not the conjugate with largest absolute value. Someone may recall who first suggested this as a meantone tuning; I've discussed it along with some others here.

This sort of thing is not, of course, confined to meantone.

> So now you can check it with a plain sawtooth waveform and a multitrack, because harmonic beatings ARE completely audible (and you can also see them on the signal), and experience how it feels with the triad A : C# : E (you can also imagine it would be simpler to tune if you were an harpsichord tuner).

They may be audible, but they don't exactly leap out and bite you on the leg. Even so, there's something to be said for using them as tunings; one thing simply being why not?

🔗Ozan Yarman <ozanyarman@...>

The equation:

> f^4 + 2f - 4 = 0

yields either -1.6243 or 1.144 as real solutions, none of which are
the desired fifths for a brat of 4 in a tempered 4:5:6.

Oz.

✩ ✩ ✩
www.ozanyarman.com

On May 6, 2010, at 4:23 AM, genewardsmith wrote:

>
>
>
>
>
> --- In tuning@yahoogroups.com, "jacques.dudon" <fotosonix@...> wrote:
>
>> Now remember, the fifth A:E of the same meantone was beating at
>> (3 * 440 hz) - (2 * 658.219542747 hz) = 3.5609145 hz =
>> same frequency ! = this is what we mean by "equal-beating" (here
>> between the fifth and the major third beatings, but it could be
>> other intervals).
>
> If f is the fifth and t the major third of a major triad in close
> root position, then (6t-5f)/(4t-5) is what I've saddled with the
> moniker "brat", from Beat RATio. The above is then the meantone
> tuning with a brat of 4, one of the "magic" figures which give
> interesting relationships between the beating of the major third,
> minor third, and fifth. Since in fully regular meantone we have the
> relationship t = f^4/4, we can substitute into the brat and find a
> brat of 4 corresponds to a fifth of f^4 + 2f - 4 = 0, so we get a
> fifth which is the unique positive real root of the above equation,
> an algebraic integer in case anyone cares, but not the conjugate
> with largest absolute value. Someone may recall who first suggested
> this as a meantone tuning; I've discussed it along with some others
> here.
>
> This sort of thing is not, of course, confined to meantone.
>
>> So now you can check it with a plain sawtooth waveform and a
>> multitrack, because harmonic beatings ARE completely audible (and
>> you can also see them on the signal), and experience how it feels
>> with the triad A : C# : E (you can also imagine it would be simpler
>> to tune if you were an harpsichord tuner).
>
> They may be audible, but they don't exactly leap out and bite you on
> the leg. Even so, there's something to be said for using them as
> tunings; one thing simply being why not?
>
>
>
>
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🔗genewardsmith <genewardsmith@...>

🔗Ozan Yarman <ozanyarman@...>

Yes, that's correct now. The desired result is: 1.495953506243232, or
697.2784049059542 cents, which is practically a 5/23 synt. comma
Baroque fifth.

Oz.

✩ ✩ ✩
www.ozanyarman.com

On May 6, 2010, at 5:14 AM, genewardsmith wrote:

>
>
> --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>>
>> The equation:
>>
>>> f^4 + 2f - 4 = 0
>>
>>
>> yields either -1.6243 or 1.144 as real solutions, none of which are
>> the desired fifths for a brat of 4 in a tempered 4:5:6.
>
> Sorry, Oz and all: that should be
>
> f^4 + 2f - 8 = 0.
>
>
>
> ------------------------------------
>
> You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - leave the group.
> tuning-nomail@yahoogroups.com - turn off mail from the group.
> tuning-digest@yahoogroups.com - set group to send daily digests.
> tuning-normal@...m - set group to send individual emails.
> tuning-help@yahoogroups.com - receive general help information.
> Yahoo! Groups Links
>
>
>

🔗Carl Lumma <carl@...>

Hi Jacques,

> About difference tones/harmonic beating :
> We can say that the (harmonic) beating of a tempered dyad is
> the 1st order difference tone of their concordant harmonics.

I'm not sure what you mean here, but beating and combination
tones are different phenomena.

> But another difference is in the range : a beat, passed 20hz
> becomes a tone ; and a difference tone below 20hz becomes an
> infra-sound and a beat.

Beats above 20Hz do NOT become tones. A pair of sine tones
close together in frequency will sound like a single beating
pitch . As the tones diverge, the beating becomes roughness.
If they diverge farther and cross the limit of frequency
discrimination they will be heard as two rough pitches, and
when they are no longer in the same critical bandwidth they
will be heard as two clear pitches. There may or may not be
a variety of combination tones present, depending on the
amplitudes and positions of the tone sources.

> We have a meantone with a fifth ratio of 1.4959535062432
> (697.2784049 c.).
> What's the frequency of the beat you hear in a major third
> tuned in 1/1 = 440hz ?
> And more important, how you find it ?
> Nobody can answer ?

You haven't told us about the timbre. But for typical timbres
a good rule of thumb is that the most prominent beats will be
at mN - nM Hz, where in this case

m = 4, n = 5, M = 440, N = 550.890228626445

so the beat rate should be about 3.5 Hz. Why do you ask?

-Carl

🔗cameron <misterbobro@...>

Hahaha! A year or two ago here at the list I corrected someone for making the exact same mistake I made yesterday, calculating the beat rate at the octave reduction of the ratio whereas of course it's happening at the ratio of the harmonic partials in question themselves. (5:1):(4:1) not 5:4 so to speak.

I've got a lot of observations and experiences about differential coherence and proportional beat rates- they're very concrete positive things in my experience as well- but I don't have time until next week.

BTW last night's Byzantine Cadillac show was simultaneously the best and the worst we've done, to such a humorous extreme that it worked well as a total performance, good audience response.

take care Cameron Bobro

--- In tuning@yahoogroups.com, "jacques.dudon" <fotosonix@...> wrote:
>
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@> wrote:
> >
> > --- In tuning@yahoogroups.com, "jacques.dudon" <fotosonix@> wrote:
> > > > (Cameron) :
> > > > In a major third diad, the fourth partial of the higher tone will beat against the the fifth partial of the lower. So, .8902 Hz.
> > > >
> > > > Correct me if I'm wrong. :-)
> > >
> > > Hi Cameron !
> > > It's been sometime since we heard about you, where have you been ?
> > > .8902 Hz., mmmh ?... everybody agrees ?
> >
> > The higher note is 550.89... Hz, and the difference between four times this and five times 440 is 3.56.. Hz.
>
> OK you guys, you are all good acousticians but Gene beats you on the maths. Cameron has a excuse because his was in Portugal just before his concert (Wow, he is singing now !) and he didn't read well that 1/1 was 440 hz, not 100 hz (but why choosing 440 hz ?).
>
> Now remember, the fifth A:E of the same meantone was beating at
> (3 * 440 hz) - (2 * 658.219542747 hz) = 3.5609145 hz =
> same frequency ! = this is what we mean by "equal-beating" (here between the fifth and the major third beatings, but it could be other intervals).
> So now you can check it with a plain sawtooth waveform and a multitrack, because harmonic beatings ARE completely audible (and you can also see them on the signal), and experience how it feels with the triad A : C# : E (you can also imagine it would be simpler to tune if you were an harpsichord tuner).
>
> OK, one more thing and I have finished : calculate the difference tone between 2A (A at the higher octave), and C# :
> 880 - 550.8902286265 = 329.1097713735 hz = E/2 (E at the lower octave)
> And hear it, if you can rather with sinus sounds. So this is a simple example (Skisni meantone) of how the two phenomenas, difference tones and harmonic beating, while operating in very different domains of audition, are in fact, in recurrent sequences completely connected.
> What's the significance of that ? If you can hear it, it is significant. But if YOU can't hear it, it does not mean it's not significant. Not all singers hear their harmonics, but all of them produce harmonics with their voices.
> - - - - - - - -
> Jacques
>

🔗jacques.dudon <fotosonix@...>

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
> Hahaha! A year or two ago here at the list I corrected someone for making the exact same mistake I made yesterday, calculating the beat rate at the octave reduction of the ratio whereas of course it's happening at the ratio of the harmonic partials in question themselves. (5:1):(4:1) not 5:4 so to speak.

Oh yes ? I thought you calculated it from 100 hz ! Steve had the good answer then. Sorry Steve ! You deserve the first prize too !

> I've got a lot of observations and experiences about differential coherence and proportional beat rates- they're very concrete positive things in my experience as well- but I don't have time until next week.

Oh yes, the differentials will wait.

> BTW last night's Byzantine Cadillac show was simultaneously the best and the worst we've done, to such a humorous extreme that it worked well as a total performance, good audience response.
>
> take care Cameron Bobro

Take care,
- - - - - - - -
Jacques

🔗jacques.dudon <fotosonix@...>

🔗cameron <misterbobro@...>

🔗Jacques Dudon <fotosonix@...>

> --- In tuning@yahoogroups.com, Gene wrote:
>
> > (Jacques) : Now remember, the fifth A:E of the same meantone was > beating at
> > (3 * 440 hz) - (2 * 658.219542747 hz) = 3.5609145 hz =
> > same frequency ! = this is what we mean by "equal-beating" (here > between the fifth and the major third beatings, but it could be > other intervals).
>
> If f is the fifth and t the major third of a major triad in close > root position, then (6t-5f)/(4t-5) is what I've saddled with the > moniker "brat", from Beat RATio. The above is then the meantone > tuning with a brat of 4, one of the "magic" figures which give > interesting relationships between the beating of the major third, > minor third, and fifth. Since in fully regular meantone we have the > relationship t = f^4/4, we can substitute into the brat and find a > brat of 4 corresponds to a fifth of f^4 + 2f - 4 = 0, so we get a > fifth which is the unique positive real root of the above equation, > an algebraic integer in case anyone cares, but not the conjugate > with largest absolute value. Someone may recall who first suggested > this as a meantone tuning; I've discussed it along with some others > here.
>
> This sort of thing is not, of course, confined to meantone.

I didn't use this "brat" parameter to find my equal-beating sequences, which arrive not surprisingly to the same results.
But it looks like one among other pertinent ways to analyse linear temperaments and I would like to integrate it to my data.
I have no claim for the paternity here of x^4 = 8 - 2x and if one wants to do such investigation I would advise to have a look into Jorgensen's works, who himself says these (equal-beating meantone tunings) were even used by more ancients.
I don't have his (expensive) books and if one has some information I would be interested to know about his discoveries.
I took the initiative to name all these recurrent sequences, because having a huge number of them you need to find them by several ways and naming them helps the communication. But this is open and for various reasons these names have to be retuned from time to time, it's a question you might be familiar with, I guess when working with regular temperaments.

> > (Jacques) : So now you can check it with a plain sawtooth > waveform and a multitrack, because harmonic beatings ARE completely > audible (and you can also see them on the signal), and experience > how it feels with the triad A : C# : E (you can also imagine it > would be simpler to tune if you were an harpsichord tuner).
>
> They may be audible, but they don't exactly leap out and bite you > on the leg.

I am not sure if when you come closer to the wolves they can't byte you real hard ! :D

> Even so, there's something to be said for using them as tunings; > one thing simply being why not?

Working for years with a good diversity of them now what I can say is that I find more and more strong differences between them in their colors, vibrations, resonances of all kind.
- - - - - - - -
Jacques

🔗gdsecor <gdsecor@...>

I haven't had much free time to read this list lately (much less all the messages in just this one thread), so I've only scanned the message list & read a few messages here & there. I noticed that this subject is rather closely related to a recent announcement I made, directed especially to Gene & Oz, who just happen to be participants in this current thread.

I've pasted here two of the messages that I read, with my comments:

Message #88742:
--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, "jacques.dudon" <fotosonix@> wrote:
>
> > Now remember, the fifth A:E of the same meantone was beating at
> > (3 * 440 hz) - (2 * 658.219542747 hz) = 3.5609145 hz =
> > same frequency ! = this is what we mean by "equal-beating" (here between the fifth and the major third beatings, but it could be other intervals).
>
> If f is the fifth and t the major third of a major triad in close root position, then (6t-5f)/(4t-5) is what I've saddled with the moniker "brat", from Beat RATio. The above is then the meantone tuning with a brat of 4, one of the "magic" figures which give interesting relationships between the beating of the major third, minor third, and fifth. Since in fully regular meantone we have the relationship t = f^4/4, we can substitute into the brat and find a brat of 4 corresponds to a fifth of f^4 + 2f - 4 = 0, so we get a fifth which is the unique positive real root of the above equation, an algebraic integer in case anyone cares, but not the conjugate with largest absolute value. Someone may recall who first suggested this as a meantone tuning; I've discussed it along with some others here.
>
> This sort of thing is not, of course, confined to meantone.
>
> > So now you can check it with a plain sawtooth waveform and a multitrack, because harmonic beatings ARE completely audible (and you can also see them on the signal), and experience how it feels with the triad A : C# : E (you can also imagine it would be simpler to tune if you were an harpsichord tuner).
>
> They may be audible, but they don't exactly leap out and bite you on the leg. Even so, there's something to be said for using them as tunings; one thing simply being why not?

I've also observed that the brat number for a tempered root-position major triad can be calculated by dividing the beat rate of the minor 3rd by the beat rate of the major 3rd, and I've found the same calculation to be useful for minor triad brats. Gene, were you also intending brats to apply to minor triads?

Message #88748:
--- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>
> On May 6, 2010, at 5:14 AM, genewardsmith wrote:
>
> > --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@> wrote:
> >>
> >> The equation:
> >>
> >>> f^4 + 2f - 4 = 0
> >>
> >> yields either -1.6243 or 1.144 as real solutions, none of which are
> >> the desired fifths for a brat of 4 in a tempered 4:5:6.
> >
> > Sorry, Oz and all: that should be
> >
> > f^4 + 2f - 8 = 0.
> >
> Yes, that's correct now. The desired result is: 1.495953506243232, or
> 697.2784049059542 cents, which is practically a 5/23 synt. comma
> Baroque fifth.
>
> Oz.

Something I (and others) have previously noticed is that, although a chain of fifths of this size (an irrational ratio) produces major triads with equal beat rates for the fifth and major 3rd (and 1:4 for the fifth and minor 3rd), the beat rates in the minor triads will be close to simple ratios, although not exact. In order to get exact proportional beating for both major & minor triads, you need a rational temperament, for example, the following tones in a chain of 5ths,
F = 631/472, C=1/1, G=353/236, D=66/59, A=395/236, E=591/472, B=221/118,
will have all 3 major triads with brats = 4, exactly. Although the brats for the 3 minor triads will also be exact, they will all be different (2.6 for Dm, 2.5 for Am, and 8/3 for Em).

Now, as to how this ties in with my recent announcement about how I had significantly improved my 12-tone temperament (extra)ordinaire, I am now both delighted & embarrassed to say that I've found a way to improve the 12-TX even more -- delighted that it came about as a result of reading this thread: C & G major triads already had brats = 4, so I tried changing F to have the same brat (4) and Bb to make a just 5th with F (brat = 1.5), and embarrassed that this has happened so soon after what I thought would be the "final" version.

The "latest & greatest" 12-TX is now:

! Secor5_23TX.scl
!
George Secor's synchronous 5/23-comma temperament extraordinaire
12
!
62/59
66/59
70/59
591/472
631/472
331/236
353/236
745/472
395/236
631/354
221/118
2/1

This change makes a noticeable improvement in both the F & Bb major triads, especially F, and there are now 6 major triads with the major 3rd tempered less than 8 cents.

Oz, I was a bit reluctant to try this, because the 3rds in the F# & C# major triads are a couple of cents wider than before, but once I did a listening test comparing these with the previous version, I found that I couldn't hear any difference. Since you're so enthused about "ordinaire" temperaments, could you try this new tuning and let me know whether those two triads are acceptable.

--George

🔗genewardsmith <genewardsmith@...>

🔗Carl Lumma <carl@...>

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

> Sort of, as they have an approximate relationship to the major
> triads. It might give an added argument for using recurrences
> rather than irrational generators, depending on how the details
> work out. The same might possibly turn out to be true for higher
> limit ratios.

So if you really want to waste your time, you can take up my
VRWT challenge, which is to find a WT with all 5ths >= 696c
and <= 3/2, and all major 3rds <= 404c, such that the count of
19-limit consonances is > 68.

RWTs have good brats by implication.

Here's what I think is the most most equal of the VRWTs with
exactly 68 19-limit consonances (19-odd-limit intervals).

! 12_lumma_vrwt.scl
!
[2 3 17 19] well temperament.
12
!
19/18
323/288
19/16
64/51
4/3
24/17
3/2
19/12
57/34
16/9
32/17
2/1
!

George poo-pooed it as unmusical, but I meticulously tuned my
piano to it and I didn't have any such complaint. However, I
also didn't notice any great advantage to the rational intervals.
The 17/12s maybe. Not the 24/19s.

-Carl

🔗Jacques Dudon <fotosonix@...>

George Secor wrote :

> I've also observed that the brat number for a tempered root-> position major triad can be calculated by dividing the beat rate of > the minor 3rd by the beat rate of the major 3rd, and I've found the > same calculation to be useful for minor triad brats. Gene, were you > also intending brats to apply to minor triads?

Eq-b minor triads do not have whole numbers or simple rational beat ratios, according to the "brat" definition.
That's a limitation of the brat parameter, if used alone.
There are alternative beat ratios that would give whole numbers or simple rational ratios for Eq-b minor triads :
- major 6th between C and A beat / major 3rd beat
- major 6th between G and E beat / minor 3rd beat
- minor 3rd beat / major 3rd between G and B (or b/x...)
- same as minor 3rd beat between lower A and C / major 3rd between C and E

Therefore I would define a complementary beat ratio for minor triads, and the simplest would be :
major 6th beat / major 3rd beat = (3s - 5/(4t - 5)
or (3x^3 - 10) / (x^4 - 5), where x = the fifth generator.
It finds a simple relation with the min3rd / maj3rd "brat", as its value is simply :
2b/x

> .../...
> Something I (and others) have previously noticed is that, although > a chain of fifths of this size (an irrational ratio) produces major > triads with equal beat rates for the fifth and major 3rd (and 1:4 > for the fifth and minor 3rd), the beat rates in the minor triads > will be close to simple ratios, although not exact. In order to get > exact proportional beating for both major & minor triads, you need > a rational temperament, for example, the following tones in a chain > of 5ths,
> F = 631/472, C=1/1, G=353/236, D=66/59, A=395/236, E=591/472, > B=221/118,
> will have all 3 major triads with brats = 4, exactly. Although the > brats for the 3 minor triads will also be exact, they will all be > different (2.6 for Dm, 2.5 for Am, and 8/3 for Em).

Among many other advantages, that I already detailed, harmonic (or rational) temperaments have this one, to be able to combine both major and minor eq-b in a number of cases.

I'll take an example with Deolia = 1.4962797197298
Its "major brat" is 2.9925594393851
but its "minor brat", as I just defined it, is 4.
It can be combined with Olonie = 1.49627602525755362
whose major brat is 3.
The following series :
2412 : 3609 : 5400 : 8080 : 12090 : 27067 : 40500 : 60600 : 90676
satisfies both of their eq-b characteristics (with nano-roundings of less than one unit)
and it is impossible to say which one it belongs to.

Of course shorter series will be able to combine even more distant generators - this is one exemple with my "guhzeng.scl" (Ethno collection/EA1), that I resume here to 5 tones :

Guzheng cithara optimal tuning in F, triple eq-b of minor and major triads
5
!
428/383
512/383
1147/766
641/383
2/1
!
! Skisni, Deolia and Hummy -c and eq-b recurrent sequences, Dudon 2006
! Triple equal-beating of minor and major triads and more :
! 4A - 5F = 3F - 4C = 6C - 4G = 6D -4A = (3/2)A - (5/2)C = 3D - (5/2)F

- - - - - - -
Jacques

🔗jacques.dudon <fotosonix@...>

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
I wrote :

> its value is simply : 2b/x

... my excuses, its value is b/x.
(x = fifth ratio)

That should be the best relevant beat ratio for minor triads.
and this was correct :
mb = major 6th beat / major 3rd beat = (3s - 5)/(4t - 5)
ex. (3A - 5C) / (4E - 5C)
or (3x^3 - 10) / (2x^4 - 10), where x = the fifth generator (but for meantone only).
So the relation of this "minor brat" to the brat, is even simpler :

mb = b/(fifth ratio)

examples :

Deolia 1,4962797197298 mb = 2
Michemine 1,4946848271484 mb = -1
Sireine 1,4907403889753 mb = 1/2

Therefore, temperaments giving synchronous-beating renditions of minor or major triads can be resumed to a simple ratio, either for their major, or their minor brat (mb).

Does that make sense ?

- - - - - - -
Jacques

🔗a_sparschuh@...

--- In tuning@yahoogroups.com mailto:tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
> ...temperaments giving synchronous-beating renditions of triads can be resumed to a simple ratio,..
> Does that make sense ?

...at least on my own piano, when tuned in an tempered cycle of a dozen 5ths modulo 19 octaves:

F: (C/3 = 21.9 43.8 <) 43.9 [< 44 88 = A/5 := 440Hz/5 against an standard normal tuning-fork]
C: (G/3 = 4.1 8.2 16.4 32.8 65.6 <) 65.7 [< 65.8 32.9 = E/5]
G: 12.3
D: 2.3 4.6 9.2 18.4 36.8 (< 36.9 = G*3)
A: (6.9 13.8 27.6 >) 27.5 55 110 220 [439 F*5 <] 440Hz
E: 164.5 329 (< 330 A*3) [657 C*5 <] 658 (<660 A*3)
B : 246.7 493.4 (< 493.5 987 E*3)
F#: [D*5 23 46 <] 46.25 92.5 [D*5 184 <] 185 370 740 (< 740.1 B*3)
C#: 138.75 := F#*3 277.5 [A*5 = 550 <] 555
G#: 52.03 104.06 208.12 416.24 (< 416.25 C#*3
Eb: 156.09 := G#*3
Bb: 468.27 := Eb*3 = G#*9
F: [440Hz/5 = A/5 = 11 22 44 >] 43.9 87.8 175.6 351.2 702.4 1404.8 (<1404.81 =Bb*3=Eb*9 =G#*27)

That procedure yields in ascending chromatic order the following absolute pitches:

c' : 262.84 middle-C4 = one-lined c'
c#: 277.5
d' : 294.4
eb: 312.18
e' : 329
f' : 351.8
f#: 370
g' : 393.6
g#: 416.24
a' : 440 Hz, see: http://en.wikipedia.org/wiki/A440_%28pitch_standard%29
bb: 468.27
b' : 493.4
c": 525.68 tenor- or treble-C5 = two-lined c"

Tune that frequencies at least ten times better than the accuracy of the ISO norm specification:
http://www.iso.org/iso/iso_catalogue/catalogue_tc/catalogue_detail.htm?csnumber=3601
Quotation
"Abstract
Specifies the frequency for the note A in the treble stave and shall be 440 Hz. Tuning and retuning shall be effected by instruments producing it within an accuracy of 0,5 Hz."

Hence:
That coarse offical Normalization allows an deviation of about ~2 cents depart from the exact value.

Or simply let do the job by 'Sacla' on a MIDI device:

! SpaEquBeatFAC_CEG.scl
Sparschuh's 'Equal-Beating' major triads F~A~C & C~E~G well-temperament (2014)
12
! Middle C4 = 262.84 Hz or one-lined c'
27750/26284 ! C#
29440/26284 ! D
31218/26284 ! Eb
32900/26284 ! E = (5/4)(658/657 +~2.633...cents)
35180/26284 ! F = (4/3)(439/438 +~3.948...cents)
37000/26284 ! F#
39360/26284 ! G = (3/2)(656/657 -~2.637...cents)
41624/26284 ! G#
44000//26284 ! A = F*(5/4)(440/439 +~3.939...cents) @ atleast 440.00Hz (+-~0.05Hz ~0.2 cents)
46827/26284 ! Bb
49340/26284 ! B
2/1 ! C' = F*(3/2)*(438/439 -~3.948...cents)
!
! [eof]

Have a lot of fun with the above tuning on yours pianos, organs or other instruments.
bye
Andy

🔗a_sparschuh@...

sorry Manuel,
the absolute-pitch of the middle C4 is 262.4Hz := 65.7*4
instead of the preliminary wrong value of 262.84,
also F4 should be corrected from the false 351.8 downwrds to 351.2 := 43.9*4
hence the scala file needs some little updates:

! SpaEquBeatFAC_CEG.scl
Sparschuh's 'Equal-Beating' major triads F~A~C & C~E~G well-temperament (2014)
12
! Middle C4 = 262.80 Hz or one-lined c'
27750/26280 ! C#
29440/26280 ! D
31218/26280 ! Eb
32900/26280 ! E = (5/4)[658/657 +~2.633...cents]
35120/26280 ! F = (4/3)(439/438 +~3.948...cents)
37000/26280 ! F#
39360/26280 ! G = (3/2)(656/657 -~2.637...cents)
41624/26280 ! G#
44000/26280 ! A = F*(5/4)[440/439 +~3.939...cents] @ a'= 440.00Hz
46827/26280 ! Bb
49340/26280 ! B
2/1 ! C' = F*(3/2)*(438/439 -~3.948...cents)
!
! [eof]

Here attend the intended deviations from JI inside the two major triads F~A~C & C~E~G.
The round brackets indicate the abberations of the 5ths from 3/2,
and the squared brackets the departure of the corresponding 3rds from 5/4.

Now i hope the revision of the Scals- file should sound proper.
bye
Andy