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Which 12-Tone Equal Temperament is the best / best compromise?

  • 1.  Which 12-Tone Equal Temperament is the best / best compromise?

    Posted 05-10-2019 11:58
    I have realised that 12-TET is the most practical compromise that is achievable after thoroughly experimenting with the mathematics of various tuning systems such as equal temperament, unequal temperament and just intonation.

    However, 12-TET is not one temperament because equal temperament is an open system. Any temperament can be classified as a 12-TET if it contains octaves equally divided into 12 semitones. Semitones can be as large or as small as one wants them to be.

    The most common forms of 12-TET that seem to be discussed on Pianotech are:

    • Pure 5th Equal Temperament.
    • Pure Octave Equal Temperament.
    • Pure 12th Equal Temperament.

    Therefore, I want to know if piano technicians have reached a consensus on which 12-TET is either the best or the best compromise and the rationale behind this consensus.


    I have developed an equal temperament myself that I have called RK 12-TET (Roshan Kakiya's 12-Tone Equal Temperament):

    https://my.ptg.org/communities/community-home/digestviewer/viewthread?GroupId=43&MessageKey=6e35edc2-b26b-4db2-a1af-238323302269&CommunityKey=6265a40b-9fd2-4152-a628-bd7c7d770cbf

    This is a unique way of distributing the Pythagorean comma. RK 12-TET lies exactly in the middle between Pure Octave 12-TET and Pure 5th 12-TET.


    Are any other forms of 12-TET being developed currently?

    ------------------------------
    Roshan Kakiya
    ------------------------------


  • 2.  RE: Which 12-Tone Equal Temperament is the best / best compromise?

    Posted 05-10-2019 12:21
     Roshan Kakiya wrote: "I want to know if piano technicians have reached a consensus on which 12-TET is either the best or the best compromise and the rationale behind this consensus."

    We are piano tuners. We tune; we don't much strive for consensus. That said pure 12th ET appears to be growing in acceptance, because of its balance between octaves and fifths, and its overall consonant effect.

    You should also be familiar with

    Pure 26th ET
    Pure 19th ET
    2-cent expanded octave ET (Rhodes Factory tuning from the early days of electronic tuning. Octave expanded by 2 cents instead of the fifth contracted by 2 cents.)

    There are others but the provenances are not well documented.







  • 3.  RE: Which 12-Tone Equal Temperament is the best / best compromise?

    Posted 05-10-2019 12:42
    The problem is what is one doing it for?

    All forms of expanded ET stretched consciously or not by reason of the string inharmonicity or by wanting purer fifths take the thirds further away from pure, without relief.

    Whether minor 1 or 2 cent deviations for 6 notes of the scale take the shimmer off the tuning is a matter of auditioning a real and harmonically rich piano tuned so.

    Whilst stretching to perfect 12ths might make the instrument consonant with itself it's actually consonant only with its inharmonicity, not the harmony. 

    It's for that reason that compressing the tuning to achieve some really perfect 3:2 5ths and getting a nice balance between some perfect 5ths and some perfect 3rds seems to me to have rather a musical advantage. The inharmonic stuff won't resonate and thereby becomes irrelevant.

    The piano is different to the organ in so far as organ tone is active where the relevant sound is what's pumped out whereas the piano is more subtle, more passive in so far as the sound is what is placed by one note and received by other strings, resonantly and then re-radiated. To get the resonance we have to look to the pure intervals. There may also be a difference of inharmonicity in strings in active or passive mode. When the string is hit, the soundboard becomes part of the vibration of the string so causing the node to be beyond the bridge for the fundamental note. But when it resonates, possibly the bridge is the node. Whilst the string might be quasi-lengthened with a metalength that exceeds agraffe to bridge in the fundamental mode, the harmonics will have less energy and so will not move the bridge with them, and higher frequency requiring the mass of the bridge to accelerate faster. So this might well be why we see the fundamental note flatter than the division of the aliquots, or rather the aliquots sharp with respect to the fundamental note.

    Best wishes

    David P

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    David Pinnegar, B.Sc., A.R.C.S.
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    +44 1342 850594





  • 4.  RE: Which 12-Tone Equal Temperament is the best / best compromise?

    Posted 05-10-2019 12:53
    David,

    Your idea of harmony is not the same as mine.

    Pure 12th ET is growing in acceptance.  It splits the difference between pure Octave ET and pure 5th ET.  It knits together large intervals on the piano.

    The idea that any ET which is more stretched than pure octave ET makes the M3rds more expanded and is therefore a negative departure from pure M3rds is true but irrelevant.  Who cares about pure M3rds?  Pure M3rds is not the definition of harmony.  Tempered M3rds are the pianist's vibrato.  They contribute to the sound of the piano that we have come to love.

    ------------------------------
    Carl Lieberman
    RPT
    Venice CA
    310-392-2771
    ------------------------------



  • 5.  RE: Which 12-Tone Equal Temperament is the best / best compromise?

    Posted 05-10-2019 13:11
    Carl - it's because I come as a horn player and many string players, brass players and singers come to music from a point of view of close harmony.

    Close harmony tends towards pure thirds.

    In the unequal temperaments some thirds are wide and you're saying that wide thirds don't cause a problem ;-)

    In the unequal temperaments many thirds are more pure so create more stillness in the music but we still get the sound you like in other keys and from other harmonics.

    The result of this is that we get some differences between the keys which were the reason why many classical composers used specific keys dedicated to different emotional landscapes. 

    The thirds all the same are like a landscape as flat as Holland but many people in England like going to the Lake District or in Europe the Alps or indeed climb Everest to see the view. The unequal temperaments make the soundscapes likewise not the same.

    I also come from harmony from organs. From organs, and if you listen carefully to the transient of a major triad on the piano you'll hear the same - a resultant beat note of the minor third two octaves below pure and the resultant beat note from the major third a quarter tone sharp, grating and beating.

    So to you the harmonious nature of the piano is the consonance of the inharmonious partials whilst to string and brass and choralists the harmony comes from relationships between the scale notes. It depends on whether one worships the piano or the music. Many musicians find the piano harsh and this is for the reason of the sharp thirds and constant beating throughout.

    The piano might be a universal medium of music making but the instrument's tuning doesn't have to be a monopoly on the instrument nor the music.

    Best wishes

    David P



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    David Pinnegar, B.Sc., A.R.C.S.
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    +44 1342 850594





  • 6.  RE: Which 12-Tone Equal Temperament is the best / best compromise?

    Posted 05-10-2019 12:56
    Roshan, 

    In my view, pure-12th ET works the best. Rather than holding the octave sacrosanct, and forcing all the compromise into narrowing the fifths, the octave is slightly wider than 12-tone ET, and the fifths are slightly less narrow, a win-win. Also, the piano sounds more coherent overall, and the stretch is easier to gauge. Further, it works well on a wide range of pianos with differing inharmonicity factors. In fact, the 12th into which you fit the 19-tone temperament is wider than 3:1 by, wait for it, the amount of inharmonicity present in  that interval {19*(19th root of 3.x), where x=inharmonicity}, thereby fitting the conceptual compromise to varied actual pianos. 

    Once I tried it, I have never looked back, and tune my own piano this way. My customers, mostly advanced musicians, while not being aware of these technical underpinnings, express enthusiasm and approval. 

    Mark Schecter, RPT
     | |   | | |   | |   | | | 






  • 7.  RE: Which 12-Tone Equal Temperament is the best / best compromise?

    Posted 05-10-2019 13:42
    Pure 12th Equal Temperament is an elegant solution:

    The Pure 12th's ratio is 3/1 which means that it is the 3rd harmonic of the Unison.

    The Pure Fifth is narrowed by 1.23 cents and the Pure Octave is widened by 1.23 cents.

    Has anyone noticed that Pure 12th Equal Temperament is a 12-TET that is based on a 12th?

    ------------------------------
    Roshan Kakiya
    ------------------------------



  • 8.  RE: Which 12-Tone Equal Temperament is the best / best compromise?

    Posted 05-10-2019 14:03
    Yes, the number 12 is overloaded in this context, but we trust readers to know it's 12 half-steps, versus the interval of a perfect 12th. Actually, 12-tone ET should perhaps be called 12-semitone ET, but that ship has sailed. 

    Mark Schecter
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  • 9.  RE: Which 12-Tone Equal Temperament is the best / best compromise?

    Posted 05-10-2019 14:04
    "Pure 12th Equal Temperament is a 12-TET that is based on a 12th"

    This is imprecise language that does not reflect the variable width of ET. "12-TET" is actually a meaningless term in a world with variable width ET. I prefer "19-tone to the Pure 12th ET". Or "7-tone to the pure 5th ET". And even "12-tone to the pure octave ET". Should specify the width.

    Sent from my iPad





  • 10.  RE: Which 12-Tone Equal Temperament is the best / best compromise?

    Posted 05-10-2019 14:22
    Kent,

    I'm not disagreeing here, but when you say "pure" octave, exactly what is pure? Is it a beatless 2:1 match, with inharmonicity included? Or is it something less real, like an exact 2:1 ratio of fundamental frequencies, which would not be audibly pure at any coincident partial? Since we're talking about language  ....

    Mark Schecter
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  • 11.  RE: Which 12-Tone Equal Temperament is the best / best compromise?

    Posted 05-10-2019 18:45
    Right. Theory and practical application are 2 different things. When I say 19-tone to the Pure 12th ET, I am referring to the zero-inharmonicity mathematical model.

    Applying the model to an inharmonic piano is a different subject in itself. But in order to apply a theoretical temperament to a real piano, we keep in mind the target beat rates, and try to find a best-fit compromise that retains the beat rate patterns of the model as closely as inharmonicity allows. Experience has shown over time that this strategy works and works well.

    Sent from my iPad




  • 12.  RE: Which 12-Tone Equal Temperament is the best / best compromise?

    Posted 05-10-2019 15:29
    But tuning "pure" 12ths on a piano means stretching not just by the temperament but by the inharmonicity added to them. So the thirds get stretched wider. You're tuning to the unmusicality of the instrument rather than the harmony of the music.

    Sorry to sound perverse and even pernickety perhaps but classical music is dying and hotel-foyer-piano tuning and piano brand worshippers are in a bubble which leaves good music sounding fuzzy and not engaging with the emotional communication so much as it can do when crisply harmonically related.

    Best wishes

    David P

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    David Pinnegar, B.Sc., A.R.C.S.
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    +44 1342 850594





  • 13.  RE: Which 12-Tone Equal Temperament is the best / best compromise?

    Posted 05-10-2019 17:08
    David, I think it can be said that anyone with a point of view is "in a bubble" or looking through a particular lens. The Classical music lens for example, or Jazz, solo performance or ensemble. There can be more than one coherent and logical approaches to tuning a piano as this thread articulates and various interpretations and methodologies to conform to the harmony of music, which is an extremely broad subject. If you prefer your tunings to be of a particular tint, that is fine, but it is just that, a particular tint.

    As to whether or not Classical music is dying, that is certainly arguable as it enjoys a listenership of 100's of millions, a much less rarified and parochial  audience than that of 1 or 200 years ago. The health of the Classical music genre, much of which doesn't even feature the piano, does not hinge upon the way pianos are tuned as much as you'd like to think and has little relevance in this technical forum.

    That thing about "hotel-foyer-piano tuning is pretty funny though considering how some of the great masters earned their rent money. And then again there's shopping mall tunings (right next to the waterfall)...

    ------------------------------
    Steven Rosenthal
    Honolulu HI
    808-521-7129
    ------------------------------



  • 14.  RE: Which 12-Tone Equal Temperament is the best / best compromise?

    Posted 05-10-2019 18:30
    Steven - across the pond in the UK the situation is really quite dire. If in the USA things are as better as you say they are then that's great but in the UK we can do better. People are thrashing about looking for the elixir and this is one of the reasons for the 432 quest, about which we probably share personal opinions. So for this reason it's worth seeing what other components of the equation can be tweaked.

    Mark - yes - from the perspective of someone who hasn't experienced the sort of thing that I've been able to achieve I'd be inclined to approach from the direction that you are. But the practice of unequal temperament that I've been advocating does actually enable all keys to be accessible and available and the major thirds in the worst keys being not greatly different to the major thirds in the stretched equal tunings.

    Whilst at first sight my placing a burden of causality on temperament rather than other contributing factors in performance might seem a little strained - but in fact when one changes the temperament even slightly the sensitive performer will respond in their performance so can be a causative connexion. In fact the greatest compliment that people can give me is that they can't hear the difference - that's great on the key usability front at least, but sensitive people do hear the difference and a better performance can be given thereby.

    Whilst sceptical and therefore thinking of the matter being not worth the bother really I recommend simply that you try it.

    Best wishes

    David P

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    David Pinnegar, B.Sc., A.R.C.S.
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    +44 1342 850594





  • 15.  RE: Which 12-Tone Equal Temperament is the best / best compromise?

    Posted 05-10-2019 19:00
    “Whilst sceptical and therefore thinking of the matter being not worth the bother really I recommend simply that you try it.”

    Likewise, sir. Likewise.

    Modern equal temperament is not the dismal picture that you depict. As I have said, there is a place in unequal piano temperaments for pure octave un-stretched tunings, but in modern equal temperament there is a place for carefully and consistently executed, tightly controlled levels of stretch, and it is a beautiful thing!

    Sent from my iPad




  • 16.  RE: Which 12-Tone Equal Temperament is the best / best compromise?

    Posted 05-11-2019 06:02
    Hmmm . . . Kent - this was actually the seduction brought into piano manufacturing and design in the 1870s or so - and why I disagree with Fred about his assertions of the universality of equal temperament 40 years before.

    Has anyone else noticed that instruments of the 1860s and before tend towards suppression of the 5th harmonic in the bass and bringing forward of the 3rd harmonic whereas post 1880 the instruments coming out of the newly industrial brands with full cast iron frames bring forward the 5th harmonic to an extent that's quite contrasting in comparison?

    It's this change of tonality of the instrument that exploited the "singing" of equal temperament, taken to the extreme by the 1899? concert Bechstein now in London where the strings for each note are a different length, the bridge being pinned evenly rather than in steps. The sound was seducing, in ET and as has progressed to stretched ET with pure 12ths or pure 5ths and it causes us to say "what a fantastic instrument!" rather than "how wonderfully this instrument expresses the music".

    So I agree with you that it can be a beautiful thing but it tunes the instrument for the instrument's sake rather than the music's sake.

    And there are methods of tuning which depart from the ET and from the stretch and achieve both objectives still.

    The result of the ET and stretched regime is that it gives a fuzziness to hotel-foyer pianos but a hardness to instruments of concert pretensions. That hardness might be described as a stressed sound, inducing stress. This is why the 432 movement thinks it has validity by making the pitch lower and so the sound less stressed. With stress in the sound and lack of variety, it's like skating around on ice or glass, and the performers go faster . . . because there is no reward for slowing down, for listening, for considering what beauty they are creating in their sound. The music becomes only a technical challenge, to play loud fast and with precision robotic accuracy to the timing and placement transfer from black notes on the page to fingers on keys. It's an entertainment without consideration of the sound. When the sustain pedal is used the sound is a jumble. So playing has become percussive rather than singing and without attention to phrasing of notes within phrases so that every note interrupts the music. https://www.youtube.com/watch?v=AF64GPfYw0Y is an example. Every note interrupts the previous note. The performances we hear now are staccato, "vertical" as a friend describes, rather than following the line of music and singing it. It's for this reason that I have the opinion that equal temperament and stretched ET might have sold pianos but it has killed the music. 

    This has gone to the extent in the UK that music is seen only as entertainment, optional, so justifying the cutting of budgets . . . and the killing of music other than the noise that is heard in nightclubs.

    So perhaps revisions in tuning and thinking about tuning might have some justification.

    Incidentally the Youtube recording above is an instrument I tuned and in my opinion the sound of the instrument is still beautiful and containing aspects of beauty that perhaps you might still be admiring were it tuned to perfect 12th ET. Or put it another way, has the beauty of the sound of this instrument suffered from my tuning rather than that of ET or p12ET? Likewise a https://www.youtube.com/watch?v=1z3o0x4dKJI although my tuning technique has developed since then. Is this instrument singing any less than you might expect?

    Best wishes

    David P





  • 17.  RE: Which 12-Tone Equal Temperament is the best / best compromise?

    Posted 05-11-2019 11:19
    David,

    If you have not yet tried it, I would like to suggest that you give Bill Bremmers Equal Beating Victorian temperament (EBVT) a significant test run.  You can download his instructions from his website or communicate directly with him.

    I have tried the Kellner, and though I hear (IOW I get it) what you are saying, I cannot yet wrap my head around the dissonant intervals...just too much for my ear. However, I am now thoroughly convinced (from applying it and hearing it, and listening to piano owners experiencing it) that the EBVT is an excellent (if not actually THE BEST) compromise between ET (of any form) and Kellner (or similar).  Now that I am applying it almost universally I fully understand what Bill has been saying for many years.

    It takes a while to get the hang of it (it does not follow ET rules) but it comes in time. You can also vary the strength of it (as it is in fact a true WT) to personal taste.

    I would be interested in your assessment as well as anyone else who is doing this.  On the whole I think it is superior to ET (this is just my personal opinion so don't anyone start throwing stones), but it took we quite a while to come to that conclusion, but primarily based on having the courage to actually do it.


    Pwg

    ------------------------------
    Peter Grey
    Stratham NH
    603-686-2395
    pianodoctor57@gmail.com
    ------------------------------



  • 18.  RE: Which 12-Tone Equal Temperament is the best / best compromise?

    Posted 05-11-2019 14:11
    Peter - it's really great to find that you've enjoyed experimenting. With regard to Kellner perhaps ensuring an unstretched execution of it is helpful but I'd comment that we might have a different perspective from the point of view of acutely listening to the tuning rather than a musician listening to the music. So the sharper intervals might not be objectionable and as they are only very marginally wider than stretched incarnations of ET it's actually the contrast with the really good near pure thirds that throws up how horrible the stretched (not much wider than equal) intervals really are. But in a context of sameness we don't notice. So I think perseverance may be rewarded.

    It's great to hear that Bill Bremmer's now being understood. I have experienced the EBVT which Michael Gamble demonstrated and - and this is a matter of pure taste rather than criticism, I found that the focus on beating took hold of my mind and drove me nuts. Each to their own. It's better than ET.

    As soon as we depart from ET we come on a journey. Welcome! It's exciting and we see new landscape, snow and flowers along the way.

    Best wishes

    David P

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    David Pinnegar, B.Sc., A.R.C.S.
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    +44 1342 850594





  • 19.  RE: Which 12-Tone Equal Temperament is the best / best compromise?

    Posted 05-10-2019 18:18
    Yes, David, but ...

    If one is wanting to align (inherently inharmonic) partials to achieve coherent resonant reinforcement, consider this: 1) the strength of partials is roughly inversely proportional to the partial number, i.e. the 3:2 beat of the P5th and the 3/1 beat of the P12th are considerably higher in amplitude than the 5:4 of the M3rd; 2) the absolute speed of any given M3rd varies depending on the inharmonicity of the particular piano's scale, and you are downgrading P12th-ET M3rds for being slightly faster without, in fairness, crediting it for eliminating beats in 5ths and 12ths; and 3) many musical forms/styles depend upon unlimited access to all keys, with no special benefits afforded to any individual key or group of keys, and with the expectation of equal usability in all keys, including for transcriptions or transpositions of original music into non-original keys; therefore in my view, your attributing the declining audience for classical music to what you consider the drawbacks of P8-ET or P12-ET to the slight differences between either of them and your proposed variation, to me is a bit persnickety, and frankly unpersuasive. 

    Or, to put it another way, I think the fact that you assign a heavier burden of causality to rather minute differences in tuning than I think is warranted, while leaving blameless many more obvious contributing factors in performance, suggests you are putting rather more focus upon the subject of our discussions than I think the general concert-going public ever has or ever will. I know of no way to test whether either your sense of it or mine is correct, but I know less than some cognitive neuroscience geeks who might weigh in. 

    No offense intended, of course. Jmho. 

    Mark Schecter, RPT
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  • 20.  RE: Which 12-Tone Equal Temperament is the best / best compromise?

    Posted 05-11-2019 11:50
    "...classical music is dying and hotel-foyer-piano tuning and piano brand worshippers are in a bubble which leaves good music sounding fuzzy and not engaging with the emotional communication so much as it can do when crisply harmonically related."

    Classical music is not dying, and even if it were, it would not be due to this or that tuning system. I am far from an expert on non-equal temperament tunings. But I believe that the specifics of temperament are far outweighed by the other aspects of musical performance, such as rubato, dynamic contrast and shading, tempo choices, and articulation. And even when all these things are "perfect," audiences know when a performance sounds canned or phoned-in. Compelling musical performance is a combination of all of the above, and audiences respond when they have witnessed the "total package."

    To insist otherwise is reductionism approaching the absurd. It would be like claiming that the best possible measure of a car is its performance from zero to 60, that horsepower is the sole measurement of greatness, and that car ownership among young people is declining because cars just aren't powerful enough.

    ------------------------------
    Scott Cole, RPT
    rvpianotuner.com
    Talent, OR
    (541-601-9033
    ------------------------------



  • 21.  RE: Which 12-Tone Equal Temperament is the best / best compromise?

    Posted 05-11-2019 15:24
    I think the best way of measuring the practicality of a tuning system is by its ability to make any music, that has been composed in the past, is being composed in the present and will be composed in the future, sound consistent overall.

    Equal temperament achieves the maximum amount of consistency that is achievable by making every semitone equal in terms of size. Therefore, equal temperament is the most practical tuning system.

    Equal temperament is, and always will be, the most practical tuning system for any music that has been composed in the past, is being composed in the present and will be composed in the future.

    ------------------------------
    Roshan Kakiya
    ------------------------------



  • 22.  RE: Which 12-Tone Equal Temperament is the best / best compromise?

    Posted 05-11-2019 16:12
    Practical?...yes,

    All purpose?...yes,

    Most beautiful?...sometimes.

    There is no question that ET has its place in musical history.  And, I think that pure 12th ET is the best form of ET.

    However, having had my ears attuned now to EBVT I have come to "tolerate" ET but not really like it anymore. I like SOME beating in my intervals (as a musician) but not gobs of it. This is probably why I am not totally drawn to Kellner since the virtually complete absence of beating (although nice) is just not my "cup of tea" (I prefer coffee anyway), and then the strong contrast in FBI is too much for me.  EBVT gives me (musically) just what I like...a slow down in the intervals I use mostly, and a little more contrast in lesser used but necessary intervals AND what appears to be an overall blossoming of the entire instrument.

    Best off all it is an all purpose temperament like ET but better (IMO).  One can play everything and not be bothered by anything (in general).

    Those reading this, please remember that I am speaking as a musician, not in technical terms here. But since in the end it is how the music sounds which is important, this is what I have come to prefer, and clients seem to agree.

    Equal is not ALWAYS better. Nothing in the universe is EQUALLY divided up (as far as I know)...it's all slightly unequal...interesting.

    Pwg

    ------------------------------
    Peter Grey
    Stratham NH
    603-686-2395
    pianodoctor57@gmail.com
    ------------------------------



  • 23.  RE: Which 12-Tone Equal Temperament is the best / best compromise?

    Posted 05-11-2019 19:01
    "Equal is not ALWAYS better. Nothing in the universe is EQUALLY divided up (as far as I know)...it's all slightly unequal...interesting."

    The Pythagorean comma is a fantastic example of how nature is slightly unequal.

    Equal temperament is the most practical tuning system. However, it is not necessarily the best because every tuning system has its own merits and its own flaws.

    ------------------------------
    Roshan Kakiya
    ------------------------------



  • 24.  RE: Which 12-Tone Equal Temperament is the best / best compromise?

    Posted 05-11-2019 19:44
    I agree, Scott.

    There are many aspects of musical performance of which the way in which instruments have been tuned is only one.

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    Roshan Kakiya
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  • 25.  RE: Which 12-Tone Equal Temperament is the best / best compromise?

    Posted 05-11-2019 20:37
    Scott - you're very lucky in the USA as all of you in the USA present a picture of a thriving classical musical scene. But in the UK and I believe other countries in Europe, but especially in the UK, that's not so. It is dying. Few schools have orchestras. Few children play the clarinet or the French horn. And so few play the piano that even where organs are maintained in churches there aren't even any pianists to play organs. The organ is dying and the piano is niche. Classical musical appreciation here is dismal.

    It can only be expressed by F minor in meantone  https://www.youtube.com/watch?v=rzrIWR3s84Q  and I don't recommend that recipe for pianos.

    Best wishes

    David P 

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    David Pinnegar, B.Sc., A.R.C.S.
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    +44 1342 850594