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Moving towards ET (Page 1)
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The Steel Guitar Forum
Pedal Steel Archive
Moving towards ET (Page 1)
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| This topic is 4 pages long: 1 2 3 4 This topic was originally posted in this forum: Pedal Steel |
| Author | Topic: Moving towards ET |
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John Steele Member Posts: 2469 |
posted 06 January 2001 12:37 PM
Preface: I can think of no more relevant discussion between players of the pedal steel than tuning concepts. If you are tired of hearing about it, spare us struggling newbies the moaning and gnashing of teeth. Hit the back button  And now, onward....3 and a half years I've been at it. Aside from one evening's brief flirtation with Equal Temperment (straight up on the tuner), I have basically been pursuing some comfort level with Just Intonation (tuning out the beats). |
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ajm Member Posts: 556 |
posted 06 January 2001 02:53 PM
This is kind of off the topic, but your statement about what Jay Dee said reminded me of another thing that happened at the convention this year. It was during Lloyd Green's set on Sunday. I don't remember what song it was, but somewhere along the line he kind of stumbled and muffed a note. If you weren't paying fairly close attention you'd have missed it. He kept right on going but cracked a smile and laughed to himself for a second. In a twisted sort of way, THAT gave me inspiration. |
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Lee Baucum Member Posts: 3201 |
posted 06 January 2001 03:28 PM
John - How ironic. I've been an ET tuner for about 6 or 8 years and have started moving towards JI.  There has to be a happy medium.
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Jeff Lampert Member Posts: 2636 |
posted 06 January 2001 03:33 PM
I don't think there is. |
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guitar543 Member Posts: 53 |
posted 06 January 2001 03:41 PM
John, The only approach I can find is to arrange songs in such a manner that the steel is not played against a keyboard or guitar that are also predominate in that part of the arrangement.Basically the steel is heard against bass and drums with very little volume from other instruments. JI will work better with this approach.Less clashing of tones. Just my .5 cent Mike |
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Steve Feldman Member Posts: 2983 |
posted 06 January 2001 04:28 PM
I feel like I'm sticking my neck out on this one, but here is a method that I've been using recently. It's a compromise between JT and ET, based on Ricky Davis' technique, drawing heavily from Buddy Emmons' harmonic tuning method (which he apparently no longer uses), and -b0b-'s Non-Critical Tuning (NCT) method that he proposed recently. This seems to work for me - at least better than anything else I tried. Comments very welcome. I use a tuner to establish three reference notes: The whole method is as follows: 1. Tune strings 3&6 (open, G#s) to –5 cents. Now, the 3rd,4th, 6th, & 8th strings are in tune, with 3 & 6 in tune with both pedals up and pedals down. From this point on, the method uses harmonics to tune out the beats. 4. Harmonic the 4th string @ 19th fret and the 5th string @ the 24th fret. Tune the 5th string tone to the 4th string tone. This gets the 5th string in tune with the E that you tuned with AB pedals down. Strings 1,2,5,7&10 are now in tune. From here, pedal tones and the open 9th string are tuned. 9. With A pedal down: Harmonic the 3rd string @ the 12th fret and the 5th string @ the 19th fret. Tune the 5th string (A pedal) tone to the 3rd string tone. [This message was edited by Steve Feldman on 06 January 2001 at 04:32 PM.] |
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Earnest Bovine Member Posts: 4687 |
posted 06 January 2001 06:26 PM
There are plenty of ways to compromise between JI and ET. For example, look at a chart such as Jeff Newman's which is basically a JI tuning. Then cut all the numbers in half, and you will be halfway between Jeff's JI and ET. |
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Lee Baucum Member Posts: 3201 |
posted 06 January 2001 06:50 PM
JI vs ET - Man attains the purposes of his being only when the two natures that are in him are in just equilibrium; and his life is a success only when it too is a harmony, and beautiful, like the great harmonies of God and the Universe. Lee, from South Texas |
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Bobby Lee Sysop Posts: 14849 |
posted 06 January 2001 07:32 PM
A few cents one way or the other doesn't make a whole lot of difference. That's my 2 cents worth. |
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Bob Hoffnar Member Posts: 4278 |
posted 07 January 2001 01:24 AM
Why not stop using a tuner ? Get an E note from somewhere and tune it till it sounds good. Record yourself with a band and adjust your tuning till it sounds right with the band. You might end up with ET or something else. It doesn't matter. What does matter is that you learn how to make your particular steel sound good with your particular hands. Whatever method you end up with you will most likely end up adjusting it forever anyway. You might look up the "Pythagorian Comma" . Then you can find out that its just plain impossible to tune anything ever. Careful not to end up in one of those JI societies. They make steel players look sane ! Bob [This message was edited by Bob Hoffnar on 07 January 2001 at 01:32 AM.] |
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Cliff Swanson Member Posts: 345 |
posted 07 January 2001 06:05 AM
John - I moved toward ET for precisely the same reason you stated. I listened to myself on tape with a band and even though my steel sounded "in tune" in the studio, things weren't sweet in the band mix. I decided to tune everything to 440 and persisted at it for about a month while I "got used to it" . Basically I didn't get used to it, so I decided to compromise by tuning my G#s to 438 and C#s to 436 (got a couple Hz cabinet drop with A&B down), F lever to 436 with A pedal down, and everything else to 440. I recently did a session and Voila! now things are sweet with the band AND I can listen to myself playing alone and not be driven to distraction. Best of both needs IMO. I'm sure there are subtle things I do unconsciously with the bar as well...listening while playing and finding the bar position that sounds right, as we all do.I think the key is to do what you've done, LISTEN critically and keep working. I completely agree with Bob Hoffnar that as we're being critical we compensate for things with our hand-ear connection. With that in mind, I also agree with b0b...a few cents either way on the tuner doesn't make much difference. Setting the tuning before the gig starts lays the foundation for what we'll then do with our hands. Cat |
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C Dixon Member Posts: 5912 |
posted 07 January 2001 06:34 AM
I admire those of you that can tune by ear. I will even say I envy you. I wish I had that ability. I DON'T!! Neve could tune a steel by ear so that when I played it, I felt it was in tune. With the advent of the tuner era, I am able to tune it electronically and it sounds great to my ears. I do have a chart and the 3rd's and 6th's are JI, while the 4th's and 5th's 7th's and 9th's are ET. What I end up with is NOT beatless!! Remember that ET, even on 4th's and 5th's have a very small amount of wobble. It was finding this that allowed me to finally be satisfied playing at home. Since I have not played in a band in almost 50 yrs, I have no problem with it. I am sure I would if I played in a band. God bless all of you, whether or not you have the "ear" for tuning, carl |
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Bobby Lee Sysop Posts: 14849 |
posted 07 January 2001 08:41 AM
Beatless just intonation sounds great with a ukelele and string bass. It's pretty rough with an electronic keyboard, though. In JI, thirds are tuned 386 cents wide instead of ET's 400 cents. In other words, if you play exactly the same C as the keyboard, your E will be 14 cents flat. Jeff Newman started this trend of tuning the E strings sharp and adjusting all of JI upwards by that amount. The idea is that the steel player can straddle ET with some notes sharp and other notes flat, and the listener's ear would average it out. It's actually a sound concept. You play the C 7 cents sharp of the keyboard, and the E is 7 cents flat. Some people can hear 7 cents as "out of tune", though. That's why Earnest says to take Jeff's numbers and cut them in half. There aren't too many people, even musicians, who hear a 3 or 4 cent difference as "out of tune". ------------------ |
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John Lacey Member Posts: 1843 |
posted 07 January 2001 09:58 AM
John, I have to agree with Bob Lee about the 2 or 3 cents thing. I've been tuning ET for the last year or so and it's still tough, especially live and especially alone. Until recently I used to flatten the 3rd. and 6th. a little bit but recently decided to go all of the way up. It really sits nice when I record to my Band in a Box tracks and most of the time with the live gigs, it's every once in a while, the sharpness grates on me. But I think it's a better compromise than the flatness I tried to live with for years. |
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John Kavanagh Member Posts: 378 |
posted 07 January 2001 10:45 AM
Let's bite the bullet. Give up those sweet thirds for the communal good, I say. (But then, I'm a Canadian.) If you want to get your ear bent about temperament, waylay a harpsichord player sometime. There are a MILLION compromises between JI and ET. It sounds like Steve Feldman is using one of these "Well-Tempered" compromises, and that kind of the solution is the norm in Baroque music. (I'd be happy if I could be consistently 100% in tune in any of them.) I'm a steel guitar dabbler, but my main instrument is the viola da gamba, a queer-looking seven-stringed monster from the dawn of time that is sort of a cross between a cello and a guitar. It has moveable frets and is tuned in fourths with one third like the guitar, so temperament is always an issue. At one point, I was so involved with temperaments that I installed extra frets on the neck so that I could have seventeen notes per octave, and that worked fine if I was playing music that stuck to four or five keys, and it did make my ears happy for awhile. However, I couldn't convince the rest of the world to tune to me. I wound up deciding that, since I usually played with a harpsichord, that it made sense to tune to the harpsichord - every open string with the keyboard instrument, and wiggle the frets around a little as necessary. When you're talking about melodic intervals - two notes played consecutively, or even about two different notes harmonizing in a chord, most people can hear an interval of two cents (the difference between a pure and an "equal tempered" fifth) in a lab setting, but in a musical setting, 3 or 4 cents difference is well within most people's toleration for pitch variance. It's probably less than the variation caused by ordinary vibrato. What really does hurt the ear is when you hear a mistuned unison - two notes played together that are almost, but not quite, the same. Even 3 or 4 cents can hurt.. If you played a just major chord and an ET major chord one after the other, they'll sound different for sure, but - especially because our ears are so used to the very sharp thirds of ET - they'll probably both be acceptable. With a little vibrato you won't notice the beats of the ET much. What is INTOLERABLE is to hear a JI and an ET chord at the same time, because the thirds (E in a C chord) will be 14 cents apart - 14% of a semitone, and your ears can really hear that. The only reason we don't run screaming when an ET keyboard or guitar and a JI steel play together is because the ET instruments don't sustain much, and the steel usually uses some vibrato, so the notes are moving in and out of tune in a way that's kind of interesting to the ear. Slight beating adds to the "fatness" of that sound, the way the double strings of a mandolin do, since they're usually beating a little bit too. The point I'm flailing around trying to make is that surely it's better to be comfortably and consistently in tune with the rest of the group than beautifully in tune with yourself, but hurting everybody's ears all the time? It's a little bloody-minded to be the only one who's right all the time. Doesn't it drive you crazy when people all tune up to their little electronic tuners and cheerfully ignore the fact that the tuners may not be in tune with each other, or that some poor soul playing harmonica or piano or any other fixed-pitch instrument is going to be out of tune with everybody all night? It's the same thing. What makes sense, if your goal is for the music to sound good, is for the person with the instrument that is the least pitch-flexible or the hardest to tune to get him- or her- self set up as best they can first, then everybody tune, and temper, to that instrument. The tuner is not in the band; you don't have to be in sync with the tuner, but you ARE going to play with that accordion all night, and if she's three cents above concert pitch, go with her, because she can't go with you. Likewise with temperament: if you have three ET instruments in the band, you'll all sound better if you ET your steel, won't you? I now play my gamba in a group with harpsichord, guitar, and fiddle. I have my frets set for equal temperament, not the norm for gamba players. The guitar is fretted ET, of course, so the harpsichord tunes ET, with an electronic tuner, and we all tune our open strings to the harpsichord. I think being better in tune with each other is worth the sacrifice. (If we had two fiddles, instead of fiddle and guitar, we might use a compromise temperament to sound better in the common keys.) In my "zoograss" stringband, we usually tune to a harmonica, and test out some chords in the keys we use most. If we use an electronic tuner, we all use the same one. It's a compromise, but I think the end result is improved. We Canadians are like that. |
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Steve Feldman Member Posts: 2983 |
posted 07 January 2001 11:21 AM
Like I said, for me, I've found a workable compromise. No beats (or almost imperceptible), the As are right on, the G#s are just flat of ET, and the Es are just sharp of ET. I like the idea of tuning one or two 'benchmark' notes via the tuner and then using harmonics the rest of the way. I can't stand those beats. Every divergence from ET seems to be within the range that the human ear will resolve correctly - or that you can't hear anyway - to borrow from -b0b-'s NCT thinking. Again, for me, this the best workaround I've tried yet. [This message was edited by Steve Feldman on 07 January 2001 at 11:21 AM.] |
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Jody Cameron Member Posts: 808 |
posted 07 January 2001 11:38 AM
I've been thinking about JI vs. ET a LOT lately. I've tried both, and eagerly read everything I can get my hands on about tuning theories. And after playing for 21 years with many musicians in all 50 states of the union and talking to all my friends and colleagues about it here's what I've concluded (at this point anyway): A steel guitar will never be perfectly "in tune". The very nature of the instrument (playing with a steel bar on raised strings, for starters) precludes any possibility of being "in tune" all the time. In fact, I dare say that this imperfection is part of what makes the sound of a steel what it is. I try to get the thing tuned as close as possible to what sounds in tune to my ears THEN I PLAY. But I noticed just last night that I am constantly and unconsiously adjusting my bar position throughout my performance to play in tune. I fretted about this for years and actually thought I was losing my mind at one point when I was on the road, because EVERYTHING sounded out of tune ALL THE TIME! I call this my "tuning crisis". Now I do the best I can and accept the limitations of the instrument and try to have fun again. Do the best you can and go on with learning to play. That's my $.02. |
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John Steele Member Posts: 2469 |
posted 07 January 2001 12:15 PM
Thanks alot, everyone, for your input. It seems every time this subject is brought up, there are new and helpful thoughts shared. The most frustrating thing about JI in a band to me, is that you sound out of tune, yet you can't tell exactly what isn't right...because the steel is in tune with itself. So it's hard to fix. Straining to hear and analyze it, I tend to minimize my vibrato... which just makes it worse. Lee, I have to take it that you have not been happy with ET (?) Care to elaborate ? b0b mentions, as many have, the difficulties of working with keyboards. I've never been faced with that, but why are they more difficult to deal with than guitars tuned ET? You people are a pretty smart bunch, thank you. Any other suggestions are welcome. [This message was edited by John Steele on 07 January 2001 at 12:20 PM.] |
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Lee Baucum Member Posts: 3201 |
posted 07 January 2001 01:24 PM
John - I'm not really unhappy with ET. The problem I have is that I rarely play with a band anymore. Most of my picking is done at home, where those sharp thirds really do stick out. When I do play with a band, ET works just fine. I just decided that I would try to experiment and find a happy medium. By the way, unless you have a guitar that has no cabinet drop, you can't tune everything "straight up". You have to allow for cabinet drop. I have always tuned my E's with the pedals down (even before Jeff Newman became an advocate). I then release the pedals and take a reading on the tuner. This is my reference point for all of the open strings, which will be a bit sharp. I tune everything to that reference point. Then I press the A and B pedals. That should bring the E's back to dead center on the tuner. Then I tune the A's and C#'s dead center. It sure is nice to have only two reference points to look for on a tuner when you are on the bandstand. Lee, from South Texas |
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Bobby Lee Sysop Posts: 14849 |
posted 07 January 2001 05:10 PM
Do you tune the F# strings with the pedals up or down? |
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Lee Baucum Member Posts: 3201 |
posted 07 January 2001 06:48 PM
I tune the F#'s with the pedals up. When I get through tuning, the F#'s do not produce any beats with the B's or the C#'s. That's how it works out on my Mullen and the same was true with my old push/pull Emmons. When I get through tuning, the only noticable beats are with the thirds. This drives me crazy when I'm playing by myself, but it seems to work out real well on the bandstand. Mikey Douchette turned me on to "straight up" tuning a long time ago. He got it from Buddy Emmons. Back when I played the Emmons is when I started tuning the E's with the pedals down. As you probably know, on a push/pull, you tune the pedals down at the key head and open strings are tuned with the allen wrench at the changer end. When I would get through tuning, the E's in the pedals up positions would be slightly sharp, on the tuner. Lee [This message was edited by Lee Baucum on 07 January 2001 at 06:52 PM.] |
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Larry Bell Member Posts: 4116 |
posted 08 January 2001 07:44 AM
You gonna play that thing or just TUNE it????  LTB |
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Gary Carriger Member Posts: 191 |
posted 08 January 2001 10:13 AM
Guys, if I never get anything else from the Forum, reading through this was worth it !! After 20+ years of playing (and tuning root notes to a tuner and ear tuning past that(is that ET?), it seems at sometimes that I had come to a dead end in gaining a complete handle on accurate tuning. Especially the comment about "tuning crisis"! Been there and done that! Working in South Texas all these years and usually always with fiddle, it has always been my intent to add chimes behind fiddle work (a B Wills legacy). If you've had that experience, you can understand one's self doubts in tuning expertise capabilities at times. But after reading through these comments and see JD's comment - it reassures me to know I'm obviously not alone. Hell, if playing steel was easy - then anyone could do it! Gary |
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David Pennybaker Member Posts: 1203 |
posted 08 January 2001 10:48 AM
I must be the only person alive who can't hear "beats". I can sorta hear it ONLY when one string is being tuned and being played with another fixed one. THEN (and only then), I can sorta hear a "sweet spot". If one string isn't being tuned, though, I can't tell. I don't know if this is a blessing or a curse! ------------------ |
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BDBassett Member Posts: 503 |
posted 08 January 2001 11:05 AM
I still claim that I get the best results by using a new set of strings, setting up at right angles to a well tuned piano and tuning each string and pedal accordingly. Of course that is not always available at your local Eagles hall but in the studio where it really counts, there is usually a decent piano standing by. Just my opinion. |
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Bobby Lee Sysop Posts: 14849 |
posted 08 January 2001 11:07 AM
Gary, it's very hard to get ET by ear. You have to carefully count the beats. Your ear wants to tune everything to just intonation (JI), because that's what's in tune according to the laws of physics. ET is the mathematical tuning compromise that makes the western 12 key music system possible. Our ears have been trained to accept/ignore ET's sharp major thirds by many years of exposure to pianos, organs, guitars, etc.. But all of the ET instruments in the world still can't change the laws of physics! ------------------ |
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Steve Schmidt Member Posts: 162 |
posted 08 January 2001 11:10 AM
For years I tuned to get the beats out, (JI). About 5 years ago I went to a studio and after listening to the result, I was really disappointed at how flat I was. I truly don't like the sound of ET when I practice at home, but ET sounds really in tune with the band. My solution, I have my playing out guitar tuned to ET and my at home guitar tuned to JI. Thats my compromise. |
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C Dixon Member Posts: 5912 |
posted 08 January 2001 12:26 PM
Steve, What you do is becoming more and more what I feel soo many are doing. When Buddy Emmons and Weldon Myrick both went to ET after 40+ years of JI, it was not made common knowledge. It still isn't. Not only that, but many have not a clue what we mean when we even use the terms. I once heard a very famous steel guitar player say, "I tune the beats out. And if the piano is out with me, it is because they have been tuning them WRONG all these years!!" Sounds good over a beer with your buddies and everyone laughs. But "tain't so!!" NO way could you ever tune all the notes on a piano so that no two would be beatless! You can prove that right on your E9th neck. Try for a million years and you will never tune your 4th, 5th 6th and 7th strings plus your A pedal down string to be be all beatless. NOT possible. Never has been. Never will be. Steel players who used the old Noel Boggs tuning: E found that out 40 years ago. It simply cannot be done. And the reason is in line with what b0bby Lee so poignantly pointed out, "it is the laws of physics" It would indeed be wonderful IF, every instrument could be tuned totally beatless. They can't. So we have had a dilemma ever since. And it will always be with us. The laws of physics dictate ET. The laws of our ears dictate JI. So what else is new in life? GOD bless you all, carl |
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Bob Hoffnar Member Posts: 4278 |
posted 08 January 2001 04:53 PM
quote: Actually the laws of physics dictate JI. The guy that just about invented physics based many of his studies on the relationship of musical overtones. Pragmatism and personal taste dictate whatever works or doesn't work. I don't know anybody that plays in tune that holds the bar straight all the time or uses the fret markers as anything more than vauge reference points anyway. Bob |
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C Dixon Member Posts: 5912 |
posted 08 January 2001 06:44 PM
Actually the laws of physics dictate ET. sorry carl |
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Bobby Lee Sysop Posts: 14849 |
posted 08 January 2001 08:15 PM
Two notes are in harmony when their frequencies are related by their harmonic sequence. That is physics, and that is the basis of Just Intonation (JI). As European music developed, the 12 key system evolved. Musical mathematicians devised a way to divide an octave into 12 equal steps. There is nothing in nature, nothing in the laws of physics, that dictates anything about 12 equally spaced frequencies per octave. When a string vibrates, it sets up a harmonic sequence of overtones. That's physics. If you tune your guitar to those overtones (which is hard to avoid if you tune carefully by ear), you have what is called "just intonation" or JI for short. An electronic tuner measures frequencies relative to a reference (usually A=440) multiplied by X times the 12th root of 2, where X is the number of half steps from the reference note. The number "the 12th root of 2" is not a part of the laws of physics. It does not occur in nature. It's an arbitrary invention of European musicians. So it's not the laws of physics that dictate equal temperament. It's the centuries-old rules of European music that dictate a tonal system with exactly 12 equally spaced notes per octave. Sorry, Carl, but I'm with Bob Hoffnar on this one. ------------------ |
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Bob Hoffnar Member Posts: 4278 |
posted 08 January 2001 10:51 PM
Yea, what he said. I'm not as look as I dumb ! |
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Earnest Bovine Member Posts: 4687 |
posted 08 January 2001 10:53 PM
quote: Actually it would be ... multiplied by {the twelfth root of 2} to the X power. So if s is the number of semitones from your note down to A 440, then your notes's frequency f is given by: |
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J D Sauser Member Posts: 1240 |
posted 09 January 2001 06:08 AM
I agree too, natural occurrence dictates the laws of physics and therefor the concept of JI: Play an open string tuned to whatever note relative to 440 (tuner needle on "0" or "12 o'clock"), then chime the 3rd harmonic with your finger on that string (fret # 4 or 9) and watch your tuner's needle take the dive... it's nature's way and therefor the basis to physics. When I started out on non-pedal steel, back in Europe, I had no guidance nor musical knowledge whatsoever. At first, I didn't even know what tunings there were and anything about the existence of ET or JI. So, I tuned to a tuner. It never sounded quite in tune to me, so I tried a number of tuners... Question: ------------------ |
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Steve Schmidt Member Posts: 162 |
posted 09 January 2001 06:19 AM
Another question: Is it possible that the scale of our instrument dictates the beatless intervals of JI. That may be worded fuzzy, but is it possible to find some scale (as opposed to 24 1/4" or 25") that would make beatless (JI) equal to ET. I have wondered this for years. |
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Bob Hoffnar Member Posts: 4278 |
posted 09 January 2001 07:50 AM
Steve, The string length has nothing to do with how Et and JI relate. They are 2 different systems of ratios. Like comparing miles to kilometers. The length of a road will not change how miles and kilometers relate to each other. JD, Anybody know how all those tuning programs they seem to run everything through in pop and country are set up ? Bob [This message was edited by Bob Hoffnar on 09 January 2001 at 08:03 AM.] |
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Bobby Lee Sysop Posts: 14849 |
posted 09 January 2001 11:05 AM
Earnest is right - my error. Duh. In JI, the "formula" for each interval is different. The frequencies are related by ratios of whole numbers. For example, to get the B, you multiply the E frequency by 3/2. See my page about Just Intonation of the E9th Pedal Steel. Harry Partch's "Genesis of a Music" has some very convenient tables that translate all of the JI intervals to cents. Here's an online version of one of them. |
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Pharaoh Member Posts: 211 |
posted 09 January 2001 12:24 PM
Bob Hoffnar - the "tuning programs," by which I guess you mean plugins like AutoTune and VoiceTweaker are pretty flexible on intonation. I know AutoTune can cope with a few methods of tempering scales, and am pretty sure it has JI capabilities too, after all it's just different numbers to the program. I've heard those plugins used ruthlessly when producers thought vocals weren't perfect. I think there's probably more expressive power in the human voice than anything, maybe even more than steel! And it breaks my heart to hear all the character being sapped out of great singers. On the other hand, it is a real big help sometimes when a singer is really out of tune! Ever put a steel thru one of those plugins? I'll have to try it tonight and hear how it sounds. Matt Farrow |
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BENGT ERLANDSEN unregistered Posts: 211 |
posted 09 January 2001 01:44 PM
Why the perfect 5th harmonic is not perfect at all. If the string is tuned to E and we divide it in three equal lengths you get a "perfect" B. Then we tune another string to that B note and find the F# harmonic. If we contine this and make "perfect" fifths until we get around to the A note and try to make the E harmonic you will be way out of tune with your original E note. So the perfect fifth should actually be a little flat. |
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C Dixon Member Posts: 5912 |
posted 09 January 2001 02:08 PM
Sorry folks, I shall not debate it after this post. Believe as you will, stand on what ever argument you wish. The "Law of physics is based in mathematics. Mathematics is at its very core. And the 12th root of the digit 2 just like any root of any number is a mathematical fact. The 12 note semitone system we call music in this country it based on the 12th root of the digit 2 simply because it is a fact that every 12 consecutive notes the starting frequency doubles or halves. This is fact. And while it may or may not occur in the plant world it is something our ears can hear. And since "nature" and our ears were both created by the same entity, it then does follow (in my mind) that it does in fact occur in nature in the broader sense. Again, I will NOT debate it again, nor listen to any rebuttal because it is moot in my mind. But from my years of academics, I must repeat that: The law of physics dictates ET. Finally this; Our ears do not like it. Our ears dictate that two frequencies in music be a harmonic of one another (JI), but the fact still remains that from a Physyics (mathematical) standpoint, ET is following the law of physics and JI is not. I agree to disagree. And on this one point vehemently so. God bless you all, carl |
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