E9 Meantone, JI, ET, a book I read about temperament

[b0b]

Meantone is ill-suited for the piano because of the wolf tone. It makes certain keys unplayable. A pianist can't just move up or down a fret to get an in-tune position.

Ever try to play in C or F on the E9th, using only open strings? Same problem.

[Earnest Bovine]

Meantone is ill-suited for the piano because of the wolf tone. It makes certain keys unplayable.

[the following is not steel-guitar related]

They did sometimes tune organs and harpsichords in meantone in Bach's time. And they did avoid playing in some keys.

They also used other tunings back then. Some non-ET tunings allowed playing in all keys, but gave each key a different sound. Most scholars believe that Bach's Well-Tempered Klavier, with pieces in all 24 major and minor keys, was not written for ET.

[Charlie McDonald]

Meantone is ill-suited for the piano because of the wolf tone.

I guess that's why they call it mean.

Wolf tones are said to occur in pianos, but it seems to be an excuse for poor tuning.

I don't even want to know about well-tempering.

[Eric West]

A pianist can't just move up or down a fret to get an in-tune position.
-b0b-

And somehow 12,384 steel guitar players think they can...



EJL

[Charlie McDonald]

A pianist can't just move up or down a fret to get an in-tune position.
-b0b-

Yeah, I have a hard time agreeing with that too.

They use keys. Not to be flippant.
But that's why ET was invented, and only that.

[Eric West]

Just kidding mostly but it brings to mind a keyboard "player" that I know.

Big Dave.

He was good at the things he knew and he still is.

I know of others that do this too so it's not uncommon.

A Keyboard player in modern times, can change the key with a switch. nearly as fast as he can play. Sometimes between a verse and a chorus..

This said, you could have all "JI" tuned on your keyboard and ANY time you changed chords you could still have "JI".

Yeah..

Well that means at ANY and EVERY time you played ANY note, you have to know what key or chord you are "Justly Intoning" to.

Otherwise it's all just a bunch of what I maintain it is..

At LONG last, I find that to be the FATAL flaw in the PSG "Just Intonation Argument".

Could have saved typing a few hundred thousand words...



EJL

[David Doggett]

It's not just about different keys. On fixed-pitch instruments, JI has conflicts with chords even in one key. For example, in the key of C, if I, IV, V and VIm chords have good JI intervals, the IIm will be bad. If the D is tuned as the 5th of V, and the F is tuned a JI 5th below C, you cannot get anywhere close to a JI minor 3rd between the D and F for the Dm chord. If the G is +2.5 as the 5th of C, then the D will be +5.0 (to ET) as the 5th of G. The F will be -2.5 (a JI 5th below C). Therefore, the interval from D to F will be +2.5, very far from the +16 a JI minor 3rd wants. ET leaves all the 5th intervals at 0, a couple of cents flat, and it sounds like that is what mean tone does. Of course that doesn't solve the IIm problem either. Its minor 3rd interval will be 0, even further out of tune than with pure JI 5ths. Of course, this doesn't phase ET believers, because they simply redefine all intervals to be in tune when they are 0, whether they actually sound like it or not.

Steel guitar is unique in dealing with this problem. We just take the in-tune VIm chord, and move the bar up 5 frets or down 7 to get an in-tune IIm. That is what b0b meant by

A pianist can't just move up or down a fret to get an in-tune position.
-b0b-

We can do what pianos can't do.

Pedal steel can also get an in-tune IIm without moving the bar, by proper tuning of the C pedal and a compensator on the F# (which is the root of IIm in the key of E). I use another solution, which is to pull my E to F# on the C pedal, tuning that F# stop to sound good. See that's the beauty of having pedal and knee lever stops that can be tuned independent of how the open strings are tuned.

I'm not sure I really understand mean tone. But I have a problem with what seems to be the concept. We don't really have to worry about comas, because the open strings cover less than two octaves. So you will never work your way very far around the cycle of 5ths (or 3rds). You will never reach the coma. I suppose you could argue that you could play the cycle of 5ths as inversions within the two octave span. But as b0b points out, unless you have a truly chromatic tuning with all 12 tones on strings or changes, you will simply play the cycle of 5ths by moving the bar around. And in doing so you will compensate by ear and never get the coma.

The other thing I don't get is concentrating on the 5ths. Whether the 5ths are tuned a couple of cents one way or the other seems hardly an issue, when major and minor 3rds and 7ths are off by as much as 12-18 cents between ET and JI, and ET proponents swear that is just fine, and everybody should just quit quibling and get use to it. A very simple compromise is to tune 5ths straight up ET, and compromise 3rds half way, say around -7 for major 3rds and +8 for minor 3rds. And that's pretty easy to get on every chord in the harmonized scale with the extra strings and tunable pedal and knee stops of pedal steel. You will be more in tune with keyboards and ET guitars than they are within themselves. So is the arithmetic of mean tone really necessary and advantageous for steel?

[Earnest Bovine]

We don't really have to worry about comas, because the open strings cover less than two octaves.

Right conclusion; wrong reason.

So you will never work your way very far around the cycle of 5ths (or 3rds). You will never reach the coma.

THAT is the reason we don't encounter a coma or wolf interval using meantone. It is true even on keyboards instruments spanning several octaves.

[Earnest Bovine]

The other thing I don't get is concentrating on the 5ths. Whether the 5ths are tuned a couple of cents one way or the other seems hardly an issue,

The way you tune your fifths determines how all other intervals are tuned. If all your fifths are tuned ET, then your thirds (and every interval) will also be ET.

To take a simple example that we all know from E9 pedal steel:
There are four intervals of a fifth between E and G#:
E to B
B to F#
F# to C#
C# to G#
So if we tune each fifth 2 cents narrower than ET, then the major 3rd between E and G# will be 2*4 = 8 cents narrower than ET. That is the principle of meantone tuning.

I might use 2 cents, or 2.5 cents, or some other amount. If I use 2.5 cents, then the major thirds are 2.5 * 4 = 10 cents narrower than ET. That is a good compromise because fifths and 3rds both beat at about the same rate.

[b0b]

How do you calculate the "beat rate", Earnest?

[Earnest Bovine]

How do you calculate the "beat rate", Earnest?

The beat rate is the difference between 2 frequencies. If the two notes in an interval have partials (harmonics) that are close in frequency, you will hear a beat at the rate of the difference.

For example the A string on a 6-string guitar vibrates at a fundamental frequency of 110 Hz (2 octaves below A=440). At the same time it is also vibrating at 220, 330, 440, 550, 660, 770, etc. The closer you pick to the bridge, the more of the high partials you hear.

The Equal Tempered E note a twelfth above the open A string has a frequency of 329.63 Hz. So if you play the two together, you will hear a beat of 330 - 329.63 = 0.37 Hz. You can tune the beat out, slowing it down to 0, by raising the pitch of E up to 330 Hz. That is called Just Intonation.

Similarly the Equal tempered C# note two octaves and a major 3rd above that A string had a frequency of 554.37 Hz. When you play the A and C# together, you therefore hear a beat of 4.37 Hz. Notice this is a lot faster than the beating of ET fifths. That is why ET fifths sound more in tune than ET thirds. If you lower the pitch of C# from 554.37 to 550, the beat will slow down to 0. Again, that is Just Intonation.

[Eric West]

A very simple compromise is to tune 5ths straight up ET, and compromise 3rds half way, say around -7 for major 3rds and +8 for minor 3rds. DD

In one position? In two? In three? Using notes "out of position" in single note runs? In "accidental" substitutions?

And that's pretty easy to get on every chord in the harmonized scale with the extra strings and tunable pedal and knee stops of pedal steel.DD

Pretty easy to type maybe... I'm ballparking 1,384 changes..

And the gem...

You will be more in tune with keyboards and ET guitars than they are within themselves. DD

In other words.....

out of tune with them...

Since the Third Interval Throb or "Beat" I and others have come to expect and enjoy is 4.37hz, I'm wondering if any long ago or current tempos of songs were used to highlight this beating. Like 32.5, 65.5 or 131BPM or rough multiples of 4.37.

Interesting, since "Beats" and "Tempo" are integral parts of Music.

Probably moreso with the Fifths in early piano music. Maybe somebody can fill it in, or I'll do a search if I have time..

This being the FIRST time this has been mentioned in the beat-to-crap "JI" vs "In Tune" heap, it's worthy of at least some mention.



EJL

[Dave Mudgett]

Actually, the frequency of the signed envelope of the sum of two sinusoids of different frequencies is the difference frequency divided by two. The beating envelope can be seen mathematically from the sum-of-sinusoids formula - for example using cosines,

So the actual frequency of the signed envelope is the halved frequency difference, (f[sub]1[/sub] - f[sub]2[/sub])/2, term, which modulates the frequency average, (f[sub]1[/sub] + f[sub]2[/sub])/2, term.

But - if the two frequencies are close, the high-frequency term has many cycles between nulls in the envelope, which happen twice per envelope cycle - and the sign of the envelope is indecipherable to the ear. So, as Earnest says, we perceive a beat at twice the envelope frequency, or the difference frequency, f[sub]1[/sub] - f[sub]2[/sub].

This is an interesting discussion, but I've never tried it myself. I guess Paul Franklin and others here make such a good argument at a lot of levels (to me) for forcing oneself to learn to hear PSG tuning by ear that I just started doing it. But this sounds like a nice compromise that one can tailor to a large extent and sounds worth messing around with. It especially sounds interesting on guitar, which has begun to drive me nuts. JI is absolutely inappropriate there, but the ET compromise, in some situations especially with full triadic harmony, just sounds bad. Thanks.

[Earnest Bovine]

Actually, the frequency of the signed envelope of the sum of two sinusoids of different frequencies is the difference frequency divided by two. The beating envelope can be seen mathematically from the sum-of-sinusoids formula - for example using cosines,

Ooops, yes, half the difference.
There are plenty of nice illustrations on the wwweb. For example, here is an animation of two waves of slightly differing frequencies, and their sum. The big envelope around the sum is what we hear as a beat:
http://www.kettering.edu/~drussell/Demo ... ition.html
(Scroll down to the bottom of the page.)

...good argument at a lot of levels (to me) for forcing oneself to learn to hear PSG tuning by ear that I just started doing it.

Yes, I arrived at meantone first by ear and experiment.

[Dave Mudgett]

Actually, Earnest - as I said in my post, the perceived beat frequency is the difference frequency, as you said. It's only the "technical" signed envelope frequency that's half the difference frequency. You were exactly right on about this, and most people call the difference frequency the beat frequency. I was just explaining a bit further how this is arrived at.

[David Doggett]

To take a simple example that we all know from E9 pedal steel:
There are four intervals of a fifth between E and G#:
E to B
B to F#
F# to C#
C# to G#
So if we tune each fifth 2 cents narrower than ET, then the major 3rd between E and G# will be 2*4 = 8 cents narrower than ET. That is the principle of meantone tuning.

Okay, now I understand it. And I can see how this method would work well for a fixed pitch diatonic instrument, say a hammer dulcimer. But I don't find the reasoning compelling for pedal steel. The above notes are in different chords, which can be tuned independently on pedal steel. Because of this independence, it is possible to tune all the main chords found at the nut on the open strings and pedal and lever combinations much closer to JI than mean tone provides. By main chords, I mean E and B on the open strings, A (with the A and B pedal combination), C#m (the relative minor with the A pedal), C# (F lever and A pedal combination), F#m (IIm with the B and C pedals), G#m (the IIIm with the E lower lever).

So for example, the G# (string 6) would not need to be tuned as the 5th of C#. Rather it would be tuned directly as the 3rd of the E root. If the E was tuned to ET, the G# could be -14, a pure JI 3rd. Also, the C# on the A pedal stop would not be tuned as the 5th of F#. It would be tuned directly as the 3rd of the A chord with the A and B pedals down. The A note of the B pedal stop would be tuned a 5th below the straight up open E string, and so would be -2. Tuning the C# stop a pure JI 3rd to the A stop would make it -16. That C# stop on the A pedal has to do double duty as the root of the C#m chord (the relative minor). And it works perfectly for that. Since it is -16, the E string will be +16 relative to it, which is a perfect JI minor 3rd. And that C# stop at -16 will be a perfect JI 5th below the G# string, which was tuned -12 as the 3rd of the E chord. It really is amazing how perfect all the main chords work out. The E raise and lower stops are independent, and so can easily be tuned perfect JI for the C# and G#m chords. Likewise if you have a Bb change on string 5.

The one exception is the F# on string 7. If it is tuned as the 5th of the open string B chord, it will be +4. But for the root of the F#m chord (IIm) with the B and C pedals, it needs to be -16 relative to the straight up A on the B pedal stop. You can split the difference and make it -6. Or you can put a compensator for that 7th string on the C pedal. Or you can raise the 8th string E to F# on the C pedal (my preferred solution).

The reason all this works is because, every time you tune an independent stop, that potentially resets the cycle of 5ths, so no coma ever occurs. The same thing happens when you move the bar. The pedal steel guitar is really unique in this respect. The notes of the diatonic scale, and some of the chromatic scale, are set with independent strings and stops, so they don’t have to be connected by an unbroken cycle of 5ths.

The way you tune your fifths determines how all other intervals are tuned. If all your fifths are tuned ET, then your thirds (and every interval) will also be ET.

This does not hold true for pedal steel. As I showed above, you can get pure JI 5ths and 3rds. If you choose to set the 5ths in my examples above to ET (0 instead of +2), it would throw some of the 3rds off by only 2 cents, and other 3rds would be unaffected. Likewise, you could set the 5ths to ET and use “tampered” 3rds somewhere between JI and ET, say -8 for major 3rds, and +8 for minor 3rds. The 3rds are set by strings or stops independent of how the 5ths are set.

So it seems to me the whole basis of mean tone, an unbroken string of slightly compromised 5ths, is simply not necessary for pedal steel. It’s not a piano or a hammer dulcimer.

Now it is true, that by mixing and matching notes from all the above chords that one can get at the nut (or at any single fret), there can be intervals of 3rds, 5ths, or anything else, that do not fall out as good JI. That may be relevant to Ed Packard’s catalog of every interval available at a single fret; but it doesn’t seem relevant to me, because I would almost always get those intervals not by mixing and matching at the nut (or the one fret), but by moving the bar to a nearby fret and applying the appropriate pedal or lever combination for a chord with the desired interval. Slants would be used if there were no pedals or levers. I don’t generally play isolated intervals. I play chords. Even if I only pick two strings, I choose them from the appropriate chord at the appropriate fret. And this seems to work the same whether it is country, classical, or jazz. That’s simply the way most of us play the steel guitar I think.

On the other hand, I for one, don’t play anywhere near the level of Earnest or Buddy Emmons. Maybe they don’t choose their intervals from basic chords the way I do. Maybe they choose their intervals from something like Ed Packard’s catalog in their head. And for that they may require mean tone, or even complete ET. In that case I quote Emily Latella on the old Saturday Night Live, “Oh…well then…never mind.”

[Earnest Bovine]

... you can get pure JI 5ths and 3rds....
So it seems to me the whole basis of mean tone ... is simply not necessary for pedal steel.

Right; you can tune to Just Intonation. Many players prefer the sound of JI intervals and chords so strongly that they are willing to give up using many (or even most) combinations of notes in order to hear that pure sound. It looks like you are one of them.

[David Doggett]

Well, I'm keeping an open mind. ET 5ths are fine with me. And with my 3rds I err slightly towards ET, because they work better a little sharp than a little flat. And if I ever start running into intervals that are serious problems, I will reconsider the whole thing. But in terms of mean tone, the 5ths seem to be 4 cents flat of JI, which doesn't seem necessary and would be a little worrisome to me. And the 3rds don't seem better than I can get by simply tampering them a little.

[David Doggett]

Now, Eric, I’m glad to see you are recuperated enough to be up to your old tricks – mixing humor and obfuscation. First of all, all the cents counting in my above post is only for the purpose of this theoretical discussion. When you tune to JI by ear, you don’t have to count any cents, or use any complicated system or chart, or have a chromatic tuning meter. All you need is a single reference pitch and your ears. You just tune the 10 open strings and several pedal and lever stops (if necessary) to make a few basic chords sound good. It’s exactly the same number of strings and stops as you would tune straight up to your chromatic meter. I can tune JI by ear much faster than I can tune every string with a meter.

And it works out in all the positions (as I showed with some of the main ones in the above post), and also for single note scale runs, because all the chords and single notes come from the same one JI scale. To get another JI scale for another key, you just move the bar to another fret – you got 12 frets for 12 keys. No need for ET. It ain’t a piano.

And, assuming the conventional use that JI is theoretically “in tune,” and ET is an intentionally “out of tune” compromise, my example of a tampered tuning is more in tune to ET guitars and keyboards than they are within themselves, because the tampered tuning is never more than 8 cents from either pure JI or ET, either within itself or compared to another instrument; whereas, ET has 3rds that are 14-16 cents out of tune with roots WITHIN THE SAME INSTRUMENT. So as Jerry Byrd said, first you get the instrument in tune with itself. And we can do that, because it’s a steel guitar, not a piano.

I’m really not as religious about this as it might sound from some of the above. I would never presume to tell anyone they “should” tune JI, or that they “shouldn’t” tune straight up ET. All I’m saying is that you CAN tune a steel guitar pure JI if you want to, and play virtually any chord in any key. That is an inherent and unique capability of the instrument.

[Charlie McDonald]

There are two similar camps in piano tuning: tuners who use predominantly 5ths and 4ths, and those who use 3rds and 6ths. The former bunch are C tuners, using a C fork, and the latter are A tuners, who are more fanatical about a smooth progression of thirds, from about 7 bps to 12. sacrificing fifths.

I stress a strong fifth, with less emphasis on the 3rds. So my starting 5th (C-F on a piano) is a little wide of ET, and little shy of JI.

So meantone--a middle path--is great for me as it's a compromise between the tyrrany of JI and the tyrrany of ET.
It means that I can raise B to C# and E to F# and still have two useable fourths.
The beat rate is pretty much inconsequential. There's just a smoothness about it that is working for me, math aside.

I've only seen one depiction that works, a graph of the partials that indicates the geometry of sound, rather than the trigonometry. You can see it, but you can't calculate it. Pythagoreas couldn't get it to work out perfectly, and it never will.

In tune is only an illusion. To say ET is out of tune is just silly. But whatever gets me there on this crazy set of pedals, levers, cabinet drop, and wolves--well, it just enables me to play with more confidence when I know that the next note is going to work for my ear and I don't have to worry about its math.

[b0b]

There are two similar camps in piano tuning: tuners who use predominantly 5ths and 4ths, and those who use 3rds and 6ths. The former bunch are C tuners, using a C fork, and the latter are A tuners, who are more fanatical about a smooth progression of thirds, from about 7 bps to 12. sacrificing fifths.

Which of these is equal temperament?

Seriously, if piano tuners don't agree and guitar players are using staggered nuts and tweaking the length of each string at the bridge, the only real problem is the electronic keyboard. I have a Yamaha keyboard that measures straight up on the meter and it sounds pretty bad to me. Maybe not technically out of tune, but pushing it to the edge.

On the other hand, my A=442 marimba doesn't sound bad, even with A=440 guitars. It seems that timbre may be a big factor in whether an instrument sounds in tune or not.

[Charlie McDonald]

Which of these is equal temperament?

That is the question.

Obviously, C tuners are better.
But they're not looking for tuners with good taste....

[Brint Hannay]

DD, I like your term "tampered tuning"!

[Eric West]

vir·tu·al (vûrch-l)
adj.
1. Existing or resulting in essence or effect though not in actual fact, form, or name: the virtual extinction of the buffalo.

2. Existing in the mind, especially as a product of the imagination. -Online Websters-

That in the Foreground..

All I’m saying is that you CAN tune a steel guitar pure JI if you want to, and play virtually any chord in any key. That is an inherent and unique capability of the instrument. -DD-

You can "tune a guitar to 'pure JI'".

Saying NOTHING about the changes, and chordal positions.

Virtually?

Yes, according to my and the accepted definition of "Virtual", in The Imagination you probably can.

This is what I've pointed out tirelessy for years:

Just because you think something "ought to be", and you "want something to be" a certain way is one thing.

We all agree that Virtual Marxism, or Virtual Peace and Harmony are a "fact".

When the "Virtual Thing" comes out of the Imagination, however, it suffers somewhat, and immediately, or in the end, totally.

I, by the same token say empirically that Music Can Be Played in Equal Temprament, and Can Be Enjoyed. The PEdal Steel Guitar is perhaps the finest example of this.

BOTH Virtually AND Actually.

Your "JI/PSG Theorum" BARELY survives any but the most fertile Imaginations, and certainly not out of the "Virtual Realm". No more than the examples I cited...

That is the difference Dave.

Refer to the initial Definition.

At least Donny H says that he "feels" that "most pros tune and play 'J.I.'"

Virtually Yours,



FHLE.

[Eric West]

OK Kids..

I'll be off in the morning to a five day Camping Trip.

After a ROUGH month of recovery from my shoebox full of guts being taken out, I REALLY need it.

IN Lieu of an "Announcement", I'm just posting it here.

If not for the support, help, and prayers I've gotten from my friends, here in the "Virtual World", and in the "Real World", I can guarantee I wouldn't have made it. Simple.

Thanks, and I'll be back Tues or so..

...


To See if.... SOMEBODY can answer the question I raised about "Beats" in a "Third" or "Fifth" at their specific frequencies being incorporated or actually instead of being "dissonant", being supportive of music played at common, specific tempos.

The "Fifth Beat Freq" would have fit beautifully with the old Bach and Beethoven things, and I haven't done any research.

A 4/4 65bpm or 130bpm chord played in a section that had a 4.37hz beating third would sound as proper as I can imagine. Why not mathematical or fractional equations?

Has it been done?

It is that far off of consideration?

NOBODY has EVER mentioned it here to my knowledge.

Maybe that gives you an insight into my active thought process, and the things I'm actually willing to consider. A non-budging, stick in the mud, I'm not.

Thanks for any replies while I'm gone.

You're all some of my best friends, whether you know it or not..

Off camping..




EJL