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I am a musician. To me, equal temperment is nothing more than a pure octave (Hz X 2 = Octave) with 11 subdivisions in between. It is purely mathmatical. It does not sound very good in reality. That may be ET to you, fine. But don't expect your understanding to hold true as a general and universal definition! I am also a musician. And I submit to you that if you want to discuss temperament, especially on a piano forum, you widen your horizons, and possibly correct your (mis)understanding. For starters (but by no means the only source), you might go to Wikipedia: An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. No mention of any interval size there. Simply a geometric progression of frequencies - that's it. In fact, there are infinitely many equal temperaments, of which your understanding (octave = 2x Hz / x Hz, with 12 semitones, i.e. semitone ratio equals 12th root of 2, often called 12 tone equal temperament, 12-TET) is only one. To my knowledge, ET is neither restricted to the use of the octave (2x/x) as defining interval, nor does the defining interval have to be divided into 12 equal parts. Some other ETs that have pretty similar semitones to 12-TET, are 19th root of 3 (pure twelfths spanning 19 equal semitones) and 7th root of 3/2 (pure fifths spanning 7 equal semitones, Pythagorean tuning). But then there are also those ETs that have much larger or smaller (semi)tones, e.g. 7-TET (7 equal tones in a 2x/x octave) or 19-TET (19 equal semitones in a 2x/x octave). In fact, there is a whole continuum of equal temperaments, as shown further down that Wikipedia page: http://en.wikipedia.org/wiki/File:Syntonic_Tuning_Continuum.jpg And all of these equal temperaments have not even taken inharmonicity into account! The fact that octaves are stretched on a piano, in order to obtain a better matching of partials, doesn't make the temperament any less equal. There is no discrepancy between the maths, the physics and the musicality of a piano. Just my grad - umm, schisma - umm, 2 cents, give or take...
Last edited by Mark R.; 06/05/12 06:55 AM. Reason: added 12-TET as a term, also "to my knowledge"
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I am a musician. To me, equal temperment is nothing more than a pure octave (Hz X 2 = Octave) with 11 subdivisions in between. It is purely mathmatical. It does not sound very good in reality. That may be ET to you, fine. But don't expect your understanding to hold true as a general and universal definition! I am also a musician. And I submit to you that if you want to discuss temperament, especially on a piano forum, you widen your horizons, and possibly correct your (mis)understanding. For starters (but by no means the only source), you might go to Wikipedia: An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. No mention of any interval size there. Simply a geometric progression of frequencies - that's it. In fact, there are infinitely many equal temperaments, of which your understanding (octave = 2x Hz / x Hz, with 12 semitones, i.e. semitone ratio equals 12th root of 2, often called 12 tone equal temperament, 12-TET) is only one. To my knowledge, ET is neither restricted to the use of the octave (2x/x) as defining interval, nor does the defining interval have to be divided into 12 equal parts. Some other ETs that have pretty similar semitones to 12-TET, are 19th root of 3 (pure twelfths spanning 19 equal semitones) and 7th root of 3/2 (pure fifths spanning 7 equal semitones, Pythagorean tuning). But then there are also those ETs that have much larger or smaller (semi)tones, e.g. 7-TET (7 equal tones in a 2x/x octave) or 19-TET (19 equal semitones in a 2x/x octave). In fact, there is a whole continuum of equal temperaments, as shown further down that Wikipedia page: http://en.wikipedia.org/wiki/File:Syntonic_Tuning_Continuum.jpg And all of these equal temperaments have not even taken inharmonicity into account! The fact that octaves are stretched on a piano, in order to obtain a better matching of partials, doesn't make the temperament any less equal. There is no discrepancy between the maths, the physics and the musicality of a piano. Just my grad - umm, schisma - umm, 2 cents, give or take... Wikipedia? Seriously? The encyclopedia that ANYONE can edit and every other "fact" has *citation needed* appended to it, meaning there is nothing backing it up? Right. Wikipedia is a place where good questions get bad answers. It is not reliable. At the college where my daughter teaches, it's not even allowable as a reference source for that reason. Btw, his definition of ET is correct. It divides the octave into 12 equal semitones. Without it, F# and Gb are no longer the same note.
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Picture a ruler with all of those equally spaced markings...
Now picture that same ruler printed on a stretchy piece of rubber. Pull on the ends (change the octave width) and all of the markings are still equally spaced...
Ron Koval
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Loren,
Firstly, I did say: for starters, and by no means the only source.
Secondly, Marty's definition included very pertinently the octave as a doubling of frequency. If you call his definition correct, how can you even say you're tuning a piano to ET if A4 = 440Hz but ... A5 is not tuned to 880Hz? ... A6 is not tuned to 1760Hz? ... nor A3 to 220? ... nor A2 to 110?
By his definition, your octaves aren't even octaves, so your temperament can't be ET. So what temperament are you tuning then?
Last edited by Mark R.; 06/05/12 07:24 AM. Reason: cross-posted with Ron, hence addressed this post to Loren, also corrected A1 to A2
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If A4 is 440, A5 is going to be 880 + some, due to stretch. 880.9, for instance. But the tuning is still based on a temperament octave that is divided into 12 equal semitones. As Ron said, stretch the rubber ruler, and the markings are still evenly spaced.
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You're mixing stretch due to inharmonicity with temperament, which are two different things. The tuning is still based on a system of 12 equal semitones. If, on a piano, you tuned pure mathematical octaves, you'd wind up with a poorly tuned piano. The treble would be flat and the bass would be sharp.
edit: This holds true no matter what temperament the tuning is based on.
Last edited by Loren D; 06/05/12 07:40 AM.
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Loren,
If we're both in agreement with Ron, why are you raking me over the coals?
One of the stretches of Ron's rubber ruler will correspond exactly to 19 equal markings in a P12. Is that ET or not? Certainly Bernhard Stopper calls it an ET - I just checked on his website. Bernhard sent me his temperament, and it is indeed based on a P12, not a P8.
Now, if Marty is indeed correct, as you say he is, then I'm afraid Mr Stopper is wrong in calling this an ET.
Autodidact interested in piano technology. 1970 44" Ibach, daily music maker. 1977 "Ortega" 8' + 8' harpsichord (Rainer Schütze, Heidelberg)
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Picture a ruler with all of those equally spaced markings...
Now picture that same ruler printed on a stretchy piece of rubber. Pull on the ends (change the octave width) and all of the markings are still equally spaced...
Ron Koval That is the point and you have just illustrated what I am saying. You are attempting to change what an octave is and inserting semitones between an interval which is not the original octave. It still forms a geometric relationship, but it is no longer based on the starting defined Hz. So, all you have done is to raise the pitch of the stretchy rubber. If the fundimental pitch of your hypothetical octave was C, you might now have an octave with the fundimental of C#. (or D, or D# ...) Even though, visually, it appears to be stretching an octave, it is merely raising the pitch and the semitones in between.
Marty in Minnesota
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Who's raking you over the coals? I responded to your assertion that If A4 is 440 and A5 is not 880, it's not equal temperament. If I misread or misunderstood, sorry!
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I am a musician. To me, equal temperment is nothing more than a pure octave (Hz X 2 = Octave) with 11 subdivisions in between. It is purely mathmatical. It does not sound very good in reality. That may be ET to you, fine. But don't expect your understanding to hold true as a general and universal definition! I am also a musician. And I submit to you that if you want to discuss temperament, especially on a piano forum, you widen your horizons, and possibly correct your (mis)understanding. For starters (but by no means the only source), you might go to Wikipedia: An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. No mention of any interval size there. Simply a geometric progression of frequencies - that's it. In fact, there are infinitely many equal temperaments, of which your understanding (octave = 2x Hz / x Hz, with 12 semitones, i.e. semitone ratio equals 12th root of 2, often called 12 tone equal temperament, 12-TET) is only one. To my knowledge, ET is neither restricted to the use of the octave (2x/x) as defining interval, nor does the defining interval have to be divided into 12 equal parts. Some other ETs that have pretty similar semitones to 12-TET, are 19th root of 3 (pure twelfths spanning 19 equal semitones) and 7th root of 3/2 (pure fifths spanning 7 equal semitones, Pythagorean tuning). But then there are also those ETs that have much larger or smaller (semi)tones, e.g. 7-TET (7 equal tones in a 2x/x octave) or 19-TET (19 equal semitones in a 2x/x octave). In fact, there is a whole continuum of equal temperaments, as shown further down that Wikipedia page: http://en.wikipedia.org/wiki/File:Syntonic_Tuning_Continuum.jpg And all of these equal temperaments have not even taken inharmonicity into account! The fact that octaves are stretched on a piano, in order to obtain a better matching of partials, doesn't make the temperament any less equal. There is no discrepancy between the maths, the physics and the musicality of a piano. Just my grad - umm, schisma - umm, 2 cents, give or take... Wikipedia? Seriously? The encyclopedia that ANYONE can edit and every other "fact" has *citation needed* appended to it, meaning there is nothing backing it up? Right. Wikipedia is a place where good questions get bad answers. It is not reliable. At the college where my daughter teaches, it's not even allowable as a reference source for that reason. Btw, his definition of ET is correct. It divides the octave into 12 equal semitones. Without it, F# and Gb are no longer the same note. Wikipedia is collaborative and may chnge definitions dpending of the day. But what is learneef in some colleages is certainly no more scientific truth so your argument is irrelevant
Professional of the profession. Foo Foo specialist I wish to add some kind and sensitive phrase but nothing comes to mind.!
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I am a musician. To me, equal temperment is nothing more than a pure octave (Hz X 2 = Octave) with 11 subdivisions in between. It is purely mathmatical. It does not sound very good in reality. That may be ET to you, fine. But don't expect your understanding to hold true as a general and universal definition! I am also a musician. And I submit to you that if you want to discuss temperament, especially on a piano forum, you widen your horizons, and possibly correct your (mis)understanding. For starters (but by no means the only source), you might go to Wikipedia: An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. No mention of any interval size there. Simply a geometric progression of frequencies - that's it. In fact, there are infinitely many equal temperaments, of which your understanding (octave = 2x Hz / x Hz, with 12 semitones, i.e. semitone ratio equals 12th root of 2, often called 12 tone equal temperament, 12-TET) is only one. To my knowledge, ET is neither restricted to the use of the octave (2x/x) as defining interval, nor does the defining interval have to be divided into 12 equal parts. Some other ETs that have pretty similar semitones to 12-TET, are 19th root of 3 (pure twelfths spanning 19 equal semitones) and 7th root of 3/2 (pure fifths spanning 7 equal semitones, Pythagorean tuning). But then there are also those ETs that have much larger or smaller (semi)tones, e.g. 7-TET (7 equal tones in a 2x/x octave) or 19-TET (19 equal semitones in a 2x/x octave). In fact, there is a whole continuum of equal temperaments, as shown further down that Wikipedia page: http://en.wikipedia.org/wiki/File:Syntonic_Tuning_Continuum.jpg And all of these equal temperaments have not even taken inharmonicity into account! The fact that octaves are stretched on a piano, in order to obtain a better matching of partials, doesn't make the temperament any less equal. There is no discrepancy between the maths, the physics and the musicality of a piano. Just my grad - umm, schisma - umm, 2 cents, give or take... Wikipedia? Seriously? The encyclopedia that ANYONE can edit and every other "fact" has *citation needed* appended to it, meaning there is nothing backing it up? Right. Wikipedia is a place where good questions get bad answers. It is not reliable. At the college where my daughter teaches, it's not even allowable as a reference source for that reason. Btw, his definition of ET is correct. It divides the octave into 12 equal semitones. Without it, F# and Gb are no longer the same note. Wikipedia is collaborative and may chnge definitions dpending of the day. But what is learneef in some colleages is certainly no more scientific truth so your argument is irrelevant Not so. In a college, someone doesn't just walk in off the street and starting teaching a class. Colleges use text books where information is sourced and referenced. Does scientific understanding and knowledge change? Yes, of course. But it's a result of greater understanding, not because someone who doesn't know what he's talking about decided to present information as fact.
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*citation needed* means just that. It's not backed up or referenced. If you feel comfortable with that, go for it!
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Gentlemen,
This thread has now come to the premiss of what I have been stating throughout this thread.
Equal Temperament is, in fact, not equal. If an ocatve is no longer a doubling of Hz, it is no longer an octave. The definitation of "octave" has been changed in your understanding.
May I propose that what you refer to as ET should actually be called AET.
Please proceed with tuning pianos to Adjusted Equal Temperament. Or if you prefer, call it Stretched Equal Temperament. SET would be better, anyway.
Marty in Minnesota
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Marty, we are talking about the properties of the acoustic string and its associated anomalies (inharmonicity) which require that stretch.
Tune an electronic organ, and the stretch goes away, making mathematically correct octaves possible. Both are equal temperament. One has partials that run sharp of the fundamental, while the other doesn't.
Call it what you want, but the temperament octave on which the whole tuning is based is a system where 12 semitones are equally spaced.
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Who's raking you over the coals? I responded to your assertion that If A4 is 440 and A5 is not 880, it's not equal temperament. If I misread or misunderstood, sorry! Well, your reaction to my earlier post, while ignoring completely what I'd written, contained essentially two parts: 1) A long part, saying that Wikipedia is #$%^&*! 2) A short part, saying that Marty's definition of ET is correct. (Do you still think so?) The 220/440/880 post was not my assertion, but based on his definition of an octave. I'm still wondering: is P12 = 19 equal semitones (Stopper-Stimmung) an ET, or isn't it? By Marty's definition, with which you agreed earlier, it can't be. But by Ron's rubber ruler analogy, with which you apparently also agree, it is! According to Bernhard himself, it's certainly an ET. So what will it be? You said that I'm mixing iH-related stretch with temperament - as though the two could really be separate on a piano. But can they ever? I've never heard of a (piano) temperament with 2:1 temperament octave. To my knowledge, it is already stretched at least to 4:2 to allow for iH. In Bernhard's case, the amount of stretch corresponds to P12 = 19 equal semitones. And that's all I tried to say in my Wiki-containing post: I think it's pointless and frankly wrong to speak of a strict 2:1 octave when defining ET for the piano. One of the things I like about Bernhard's approach is that he applies only this one yardstick, perfect 12ths, from A0 to C8. The temperament octave is no different from any other on the piano. In a certain sense, this is "more equal" than 4:2 in the temperament, 6:3 in the bass, "mindless" in the treble and possibly 2:1 in the high treble.
Last edited by Mark R.; 06/05/12 09:08 AM. Reason: changed C88 to C8
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Alfredo, perhaps you are unfamiliar with "the blues"???
Ron Koval Ron, can you expand on that? What is it that you mean to say? The Blues is an American musical style that comes from pain and suffering... It is common to use notes not available on the keyboard - forcing pianist to play two chromatic notes next to each other to approximate the effect.. I was responding to your statement about tonal temperaments being "out of tune", which (outside of unisons and octaves) is a cultural and learned phenomenon. While many tuners would like to think that equal temperament is universal, it is actually a very small subset in the music world; it only really exists on keyboard instruments, as well as some fretted instruments - and then only if the tuning follows very strict parameters that many ET technicians have admitted not really following... The idea of the Blues brings up a concept that often gets lost in these discussions. The idea of "sounding better" or "sounding worse" shouldn't really be the focus. I like to think of it as offering a range, or expanding the palette available to the composer. One of the early demonstrations I went to (before I was interested in anything other than ET) included an old piece of music that was written after the death of the composer's wife. Played in ET, it was sad - kindof "I hit my toe on the dresser and it hurts pretty bad". When it was played in the other temperament (don't know what) The pain and anguish made possible by the tuning made me squirm in my seat. I wish I knew the composer and piece now! Ron Koval
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Wikipedia IS *$^%&$. I stand by that. Stretch vs. temperament. Yes, they are two different things. As I said, tune an organ, skip the stretch. No need to stretch since you are not dealing with the inharmonicity inherent to a vibrating string. Again, using the electronic organ, there is your 2:1 octave, working well within equal temperament. Inharmonicity is something piano tuners must deal with, but it is still separate from temperament. Stretch or not (piano or organ), the octave is still divided into 12 equal semitones. Stretching causes the tones to get wider as we progress up and down, but relative to each other, they are still equally spaced.
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Gentlemen, I don't understand why the topic of octave stretch, be it natural or artificial, raises questions about temperament. You are discussing apples and oranges. The temperament refers to what is happening within the octave (generally), the rest is simply an expansion of that to the extents of the keyboard. For the purposes of tuning, some folks expand the temperament octave slightly, but ending up 35 cents sharp of theoretical on the highest note is not tempering, its stretching.
From a dictionary.... equal temperament....the system commonly used in keyboard instruments, giving a scale based on an octave divided into twelve exactly equal semitones
Piano Technician George Brown College /85 Niagara Region
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Loren,
If we're both in agreement with Ron, why are you raking me over the coals? [...]
I think we reached a point in these threads a while ago where just about everybody is ready to rake anybody over the coals. It's like putting a whole bunch of grown cats in a small room, and even cats that would otherwise get along fine are hissing and spitting at each other. If I could bring a bit of Pianist Corner into the Tuner/Tech forum today, it would be to say, "WHAT AN AWESOME THREAD!"
I may not be fast, but at least I'm slow.
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Marty, we are talking about the properties of the acoustic string and its associated anomalies (inharmonicity) which require that stretch.
Tune an electronic organ, and the stretch goes away, making mathematically correct octaves possible. Both are equal temperament. One has partials that run sharp of the fundamental, while the other doesn't.
Call it what you want, but the temperament octave on which the whole tuning is based is a system where 12 semitones are equally spaced. I couldn't agree more and I understand all that has been stated in this thread. However, please go back to the very beginning of this thread. I prefer to have all three of my pianos tuned to a temperament other than ET. Are the octaves stretched? Certainly. Are the intervals in between based on an absolute mathematical progression? No, they are not. The derivation, measured in cents, is miniscule when compared to absolute ET. The result, perceived by me, is the generation of tonal color. My musical ear wants to hear the differentation between keys. I prefer to perceive a tonal difference between D-Maj and D#-Maj. If needed or desired, I can always transpose a composition. The new key will then have a different tonal color than the original. That is the point. There is much discussion of ET as a theoretical concept which has, through evolution, become the norm. In practice, it really doesn't exist with anything other than fixed tone generators. It is the inharmonicity, overtones, partials, which bring life to any given pitch. Single pitch, and not an interval. It is the interaction of intervals to then form our concept of intonation. From there, it becomes layer upon layer as more notes are added. The concept holds true for any scale, diatonic or not. It all boils down to what we percieve aurally. I prefer a temperament choice which emphasizes key color rather than trying to negate it. Some temperaments suit a given piano better than others. That is where the tuning skill (art) of a gifted tuner comes into play.
Marty in Minnesota
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