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Oh and I also.....<insert ominous chord>.....spliced a bass string.

[*]Piano previously tuned (only three weeks ago) by an aural tuner who uses a 4ths & 5ths sequence, condemns the use of non-equal temperaments but offers Reverse Well as his own version of ET. Gets mad when you tell him it is not ET but Reverse Well instead and says he has never heard of "Reverse Well".

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Bill, just so I understand your posts better, could you explain what you mean by "Reverse Well"? Thanks!

What the HE** is

rEvErSe wElL...?

It's The Most Common Error In Tuning These Days.

Because so many people have asked me both in public and private mail, I will try to explain the meaning of the term, Reverse Well thoroughly. First of all, I think it can be considered jargon. I don't think the term can be found in any published book but may be found in articles and yes, I have used it extensively on the Piano Technician's list called Pianotech.

To me, the term is self explanatory. It describes a backwards version of a typical Well-Tempered Tuning (WT) [often called Well-Temperament]. (See my article on The True Meaning of Well Tempered Tuning). I did not invent the term but it may have been just something I heard another member of my Piano Technicians Guild (PTG) Chapter say to describe the kinds of errors we were noticing that other tuners were making with a disturbing, consistent and uncomfortable frequency. The term began to be repeated and since I have used it a lot, others have used it too both on this List and elsewhere. It has for me until recently always been meant to identify a very commonly made error in tuning.

I recall someone mentioning something about DeMorgan and Reverse Well but I had never read section 128 in Professor Owen Jorgensen's book, "Tuning" which describes this temperament. To my surprise, it describes a mildly unequal temperament that is exactly backwards of most other 19th Century temperaments which usually adhere to historical precedents.

To understand what Reverse Well is, you have to know and understand Andreas Werkmeister's Rules for Well Tempered Tuning. In these rules, it states that the major 3rds must progress in their beat speeds not chromatically as they do in Equal Temperament (ET) but in an alignment with the Cycle of 5ths. Thus, 3rds of a typical WT would sound uneven but in a specific way. (See my article on Anreas Werkmeister's Rules for Well Tempered Tuning).

A Reverse Well Temperament would be one in which this pattern of unevenness is more or less the opposite. Here is a general description of the way 3rds sound in most any mild WT and the way a piano erroneously tuned in Reverse Well would sound:

***EBVT/REVERSE WELL

F3-A3:slow/fast

F3#-A3#:fast/slow

G3-B3:slow/fast

G#3-C4:moderately fast/gentle or slow

A3-C#3:moderate/fast

A#3(Bb3-D4):gentle/very fast

B3-D#4:very fast/very slow

C4-E4:gentle/exremely fast

C#4(Db4):very fast/very slow


One of the reasons I felt free to create my own WT is that Professor Owen Jorgensen RPT, in his fine book called "Tuning" (see link to a website that has a review of Review of Professer Owen Jorgensen's Book, "Tuning"), documented that it was commonplace, historically to alter the usual practices of the time and place according to one's own taste... I found that no historically documented temperament suited me, so I created my own from the accumulated knowledge I had.

When trying to create your own WT, you need to be aware of what are called "imbalances". That is, you cannot have a 3rd from the top of the Cycle of 5ths beat faster than a 3rd from the bottom or vice-versa. That would break the rules and as such, is considered improper and contrary to the whole logic of the very reason to tune this way.

When trying to understand why so many tuners who believe only in tuning ET actually have as a result a backwards version of a WT, I have always thought it had to do with two things: the order in which notes are tuned and a tendency that the late John Travis RPT had noted in his book, "Let's Tune Up", the "tendency to err towards the just 5th". Mr. Travis believed that most tuners would try to tune the 5ths a little too close to pure rather than leaving them tempered by the right amount to have a true ET.

Most WT's have a pattern that starts with C, then to F, Bb, Eb, Ab and Db, all as pure or nearly pure 5ths. Then, the rest of the 5ths, mostly among the white keys would have to be tempered more than for ET in order for the temperament to work out. When an aural tuner uses a pattern that starts with either A or C and makes the error of tuning the first half of the 4ths and 5ths too purely, those among the white keys instead of the blacks, then the rest of them, those among the black keys, must be tempered a little more than would be proper for ET.

This would result in an uneven pattern of beating for the 3rds which is exactly opposite of the way they should be for a WT, hence the term, "Reverse Well". Here is what Jean Jousse said in 1832 about what I and other members of my Chapter had also noted:



ET...has the following disadvantages: ... it cannot be obtained in the strict sense as may be proved, not only mathematically, but also by daily experience; therefore the best equally tempered instruments are still unequally tempered, and, what is worse, oftentimes in [the] wrong places.

(see "Tuning", page 417).

Since 1980 or so, the Piano Technicians Guild (PTG) Standardized Tuning Exam has made a Quantum leap in many tuner's ability to tune a true ET. Dr. Albert Sanderson RPT not only invented the first Electronic Tuning Device (ETD) that could really compensate for Inharmonicity to produce a true ET on the modern piano, he also identified the 4:5 Ratio of Contiguous 3rds to the aural tuning profession. Since the time this Standardized Exam was implemented, more and more aural tuners have been able to produce a true ET because more of them know all of the interval checks, the 4:5 ratio being the most important of all.

Still, many tuners have not yet taken this Exam and have not learned these advanced techniques. An aural tuner who relies mostly on 4ths & 5ths and does not distinguish the fine gradations in the beating of 3rds & 6ths can easily produce a Reverse Well temperament and not even be aware of that fact. This may be particularly true if the error that John Travis identified is also made.

I have heard many tuners express the phrase, "temperament doesn't matter", that good unisons and octaves are more important. This may be true to a certain extent but what can happen if there is an accumulation of errors, that is, one error compounding another, the end result in the temperament can be a very significant effect that was not intended and not recognized for what it is but which does have an effect on the music and the way a person might play the piano. This effect might well be opposite of the kind of effect expected when tuning a true WT. Imagine the skewed perception of music a person might have who spent a whole career of study on a piano consistently tuned in Reverse Well!

There have also been other instances of Reverse Well, done erroneously that I cannot explain. It can even happen when using an ETD. I have done it myself. Making an error while programming an ETD can just as easily happen as when tuning aurally. It takes very little error to produce an audible inequality in ET. Any accumulation of errors will result in some kind of deviant pattern. It is unfortunate that when errors are made, they so often are made in opposition to what might be considered a beneficial alternative.

There have been a few people who have timidly confirmed that they have also witnessed Reverse Well Tunings. Many of these people would only dare to say it in private mail. I have found it virtually everywhere I have looked, from New York to Los Angeles, from Montreal to Mexico City and everywhere in between. I have always known and witnessed that the very discussion of temperament tends to bring out strongly held beliefs and emotions.

Technicians should not take the mindset however that "everything I always believed in is now invalid". There is simply more to the entire story, that's all. How else could there be such a blatant contradiction as that of the DeMorgan Temperament? I have long contended that the study of HT's is important for all tuners, even those who wish to only tune ET. It is so that the tuner can really recognize errors and their effects and so the Reverse Well error can be avoided.

Bill Bremmer, Registered Piano Technician

The True Meaning of Well Tempered Tuning

Based in part upon an article from Professor Owen Jorgensen RPT's publication, THE EQUAL-BEATING TEMPERAMENTS-A HANDBOOK FOR TUNING HARPSICHORDS AND FORTE PIANOS

Andreas Werkmeister (1645-1706) was an organist, composer and theorist highly respected by his peers. He was northern Europe's most influential writer and theorist of the 17th & 18th Centuries. According to Werkmeister, Well-Tempered Tuning (also commonly known as "Well-Temperament") (WT) is a philosophy based on the expected Key Color from the Cycle of 5ths.

In his time, chord structures with the fewest sharps and flats were used the most, so he reasoned that they should sound the best at the expense of those with many sharps and flats. He maintained, however that even the harshest chord structures must still be reasonable sounding enough to be acceptable in performance.

This means that modulation and transposition are unrestricted as opposed to earlier kinds of temperaments where chords in key signatures with many sharps and flats were thought to be unplayably out of tune.

In Werkmeister's systems, all 24 major and minor keys are musically acceptable but not the same as one another. Each one is different from the others and the distinctions are intended to be heard when modulating. There must be Key Color changes during modulation and transposition. (See my article about "Key Color"). Rather than all chords sounding alike except for pitch, they must differ.

In Werkmeister's time, composers would most often cling to the key signatures with the fewest sharps and flats even though they had become adept at modulation. The influence of the earlier and very restrictive Meantone Temperaments (MT)was still very strong. Music in the simple keys, those with fewer sharps and flats, simply sounded better to composers such as Mozart and Haydn.

Well-Tempered Tuning (WT) is NOT the same as Equal Temperament (ET)as is so commonly believed. By its very nature and design, ET lacks all of the qualities known as Key Color even though many people still insist they hear something that they would call "Key Color". (See my article which explains what Key Color is). Equal Temperament (ET) may not be as commonplace today as one might assume. It is very difficult to truly effect. Even the slightest of errors renders a temperament unequal. Perhaps this "Key Color" that many say they hear really is there but not intentionally and not by design. In fact, it can easily be exactly opposite of what Werkmeister intended and people in his time knew. (See my article called, "What the HE** is Reverse Well?").

The concept of Equal Temperament is as old as the ancient Chinese civilization, centuries before the establishment of the Common Era. 15th Century lutes were known to use it. It was tried but rejected many times throughout Western (occidental) music history because of its dissatisfying sound and lack of Key Color.

Even as late as 1879, William Pole wrote,

"The modern practice of tuning all organs to Equal Temperament (ET) has been a fearful detriment to their quality of tone. Under the old tuning, an organ made harmonious and attractive music, which was a pleasure to listen to. Now, the harsh thirds, applied to the whole instrument indiscriminately, give it a cacophonous and repulsive effect."

It was until well into the 20th Century when the teachings of the 19th Century scientist, mathematician and theorist (but NOT musician), Hermann Helmholtz and the well respected teacher of Piano Tuning & Technology (but again, NOT musician), William Braide White took preponderance over conventional wisdom that Equal Temperament (ET), as a concept began to be accepted. Unfortunately, neither Helmholtz nor White were able to provide the information and techniques necessary to truly effect ET for most practitioners. An error in Groves Dictionary of Music [that some say must have been intentional]is also thought to have contributed to the commonly believed misconception that Well-Tempered Tuning and Equal Temperament are synonymous. THIS IS NOT TRUE. The two terms are, in fact, mutually exclusive. Many history, culture and science books have repeated this error.

William Braide White contributed to this confusion by representing that there were only two possible tuning systems: Meantone and ET. While he "tipped his hat", so to speak to Meantone, saying that it was beautiful in its own way for early Music in his widely read book on Piano Technology, "Piano Tuning and Allied Arts", he deliberately ignored the infinite number of tuning systems found among the milder Meantones, the Modified Meantones (a separate class unto themselves)and the vast array of Well-Tempered Tunings at one's disposal to assert that only Equal Temperament (ET) would satisfy the demands of all music heard today.

White "proves" his case quite convincingly by omitting the now obvious fact that Meantone and ET are at complete opposite ends of an infinite spectrum. The unfortunate result is that nearly everyone believed in White's erroneous teachings and as a consequence, a large part of the art of tuning was nearly lost. It was only late in the 20th Century when through research, the true meaning of Well-Tempered Tuning and all of the possibilities it implies came to light.

Jorgensen says unashamedly that today's custom of performing J. S. Bach's Well-Tempered Clavier [Music]in ET is ridiculous because the [very] purpose of [those compositions] was to demonstrate the Key Color changes from one [kind of harmony] to the next in a Well-Tempered Tuning system. Yet, of all the many recordings by the many artists of this music on the modern piano, and even those on the Harpsichord, I know of not a single one made in a true Well-Tempered Tuning.

I did, however once read in the promotional writing for another such recording in the ubiquitous ET, the very same MISINFORMATION about how Bach had invented ET, wrote the Well-Tempered Clavier music to demonstrate how great it is and how it has been "...universally accepted ever since (sic)." I cancelled my subscription to that Record Club with a letter explaining exactly why but have only received offers to rejoin ever since, never hearing of a new recording on the modern piano of Bach's Music for the Well-Tempered Clavier in a true Well-Tempered Tuning.

The basic theory and practice of Werkmeister's ideas form the foundation for the creation of my new temperament for the modern piano, the Equal Beating Victorian Temperament (EBVT). There are other ideas included from the little known class of temperaments called the Modified Meantones and the unique effects of the principle known as Equal Beating but the EBVT adheres to Werkmeisters principles while at the same time being a tuning system acceptable and applicable to the modern piano and for virtually all of the music usually played upon it. Please see the separate page for Werkmeister's often referred to "Rules for Well Tempered Tuning".


Bill Bremmer, Registered Piano Technician

Andreas Werkmeister's Rules for Well-Tempered Tuning

From the publication by Professor Owen Jorgensen RPT, "Tuning the Historical TemperamentsBy Ear" (pages 246-7).

1. Each tonality or key center should sound differently and should have its own distinct color characteristics or "Affekt". This is in direct opposition to the ideals of Equal Temperament (ET) in which all tonalities [or keys]have the same identical and neutral color except for the increasing [(ascending)] or decreasing [(descending)] beat speeds of the Major thirds (M3rds) according to the position within the scale. The purpose of Well-Tempered Tuning is to allow harmonic color change through modulation and in a distinct alignment with the Cycle of 5ths].[.

2. The same as in ET, one must be able to modulate freely in all possible keys without experiencing any unacceptable, out-of-tune (or "Wolf") sounds. Thus, all 12 tones may be used enharmonically which is impossible in the class or restricted temperaments including Just Temperaments (JT) and Meantone Temperaments (MT).

3. During modulation through a series of 4ths or 5ths, the color changes of the "Affekts" must be consistent and even.

From a detailed study and evaluation of the beat speeds of Major 3rds and other qualities of all of the most prominent, historically documented Well-Tempered Tunings, (WT) the following rules for creating a WT are evident:

The Rules:

1. The 12 semitones of the octave must NOT be of the same, exact size.

2. No 5th, minor sixth (m6th) or minor third (m3rd) should be wider than just [(pure or beatless)].

3. No 4th, Major sixth (M6th) or Major third (M3rd) should be narrower than just [(pure or beatless)].

4. No Major or minor 3rd and no Major or minor 6th should be tempered from Just Intonation [(meaning pure or beatless)] by more than the value of one Syntonic Comma [(21.5 cents or a very rapid, borderline "sour" sounding beat)]. [The ET M3rds are all supposed to be 14 cents wide of Just Intonation].[( A moderate but fairly active sound)].

5. No 4th or 5th should be tempered more than half the value of the Syntonic Comma [(11 cents, which would be a moderately pulsing beat, quite noticeably more active than the nearly still adjusted for Inharmonicity) 5ths of ET. [(11 cent tempered 5ths are fairly rare and extreme. The typical WT tempered 5th averages 4 cents.)] [The ET 4ths & 5ths are all tempered by 2 cents, theoretically but are usually reduced to between 1.7 and 1.9 cents narrow.

Some Technicians even advocate an extremely stretched out version of ET where the 5ths are apparently pure or beatless but the 4ths beat noticeably and the M3rds are widened to about 16 cents, a very tart sounding interval. Tuning this way does produce a kind of clarity and melodic quality which may benefit many kinds of 19th Century Romantic styles as well as complex 20th Century styles but becomes fairly offensive to most 17th and 18th Century writing. Thus, it cannot be considered an "improvement" to ET or over any other style of temperament].

6. No Octave should be tempered from Just Intonation (JI). [Today, it is well known that with the Modern Piano, a Just Octave (truly pure or beatless) is not really possible. This is because of the important factor called "Inharmonicity", unknown in Werkmeister's time and not applicable to the organ or early keyboard instruments. See my article called, "How to Tune Tempered Octaves"].

7. No M3rd should be smaller or closer to JI than the 3rd, C-E.

8. The 3rd, F-A must be the same size or larger than C-E.

9. The 3rd, Bb-D must be the same size or larger than F-A and C-E.

10. The 3rd Eb-G must be the same size or larger than Bb-D.

11. The 3rd Ab-C must be the same size or larger than Eb-G.

12. The 3rd Db-F must be the same size or larger than Ab-C. Also, Db-F must be the same size or larger than B-D#. (Db-F may be the same size, smaller or larger than Gb-Bb. No other M3rds may be larger than Db-F and Gb-Bb.)

13. The 3rd Gb-Bb must be the same size or larger than Ab-C.

14. The 3rd B-D# must be the same size or smaller than F#-A#.

15. The 3rd E-G# must be the same size or smaller than B-D#.

16. The 3rd A-C# must be the same size or smaller than E-G#.

17. The 3rd D-F# must be the same size or smaller than A-C#. Also, D-F# must be larger than C-E.

18. The 3rd G-B must be the same size or smaller than D-F#.

19. The 3rd C-E must be the same size or smaller than G-B.

20. No Minor 3rd (m3rd) should be larger or closer to Just Intonation (JI)[pure or beatless]than the m3rds E-G or A-C. There may be necessary exceptions but it is definitely preferable for the m3rd AC to not be larger or closer to JI than E-G. To actually have E-G larger than A-C is considered a fine accomplishment.

21. The m3rd D-F must be the same size or smaller than A-C. ALso, D-F must be smaller than E-G.

22. The m3rd G-Bb must be the same size or smaller than D-F.

23. The m3rd C-Eb must be the same size or smaller than D-F.

24. The m3rd F-Ab must be the same size or smaller than C-Eb. Also, F-Ab must be the same size or smaller than Eb-Gb. (F-Ab may be the same size, smaller or larger than Bb-Db. No other m3rds may be smaller than F-Ab or Bb-Db.

25. The m3rd Bb-Db must be the same size or smaller than C-Eb.

26. The m3rd Bb-Db must be the same size or smaller than C-Eb.

27. The m3rd G#-B must be the same size or larger than D#-F#.

28. The m3rd C#-E must be the same size or larger than G#-B.

29. The m3rd F#-A must be the same size or larger than C#-E.

30. The m3rd B-D must be the same size or larger than F#-A.

31. The m3rd E-G must be the same size or larger than B-D.

32. The following tonalities [keys]should be more brilliant or richer [have faster beating Major or minor 3rds] than the same [keys] in ET:

F minor, Ab Major, Bb minor, Db Major, Eb minor, F# [or Gb]Major, G# minor, B Major.

33. The following tonalities [keys] should be less brilliant, plainer or milder [have slower beating Major or Minor 3rds] than the same [keys] in ET:

E minor, G Major, A minor, C Major, D minor, F Major.

34. The following tonalities [keys] are sometimes found to be quite similar to the same [keys] in ET:

C# minor, E Major, F# minor, A Major, B minor, D Major, G minor, Bb Major, C minor, Eb Major.

35. Breaking any of the above 34 rules ruins the evenness of Key Color progression and the position of C Major as the tonal or Key Color center. To do so may introduce unnecessary harshness [or produce the error known as Reverse Well, the opposite of the intended effect].

Bill Bremmer, Registered Piano Technician

From what I have read of Mr. Bremmer's discussion of "reverse well temperament," it would be a well temperament in which C is not the key which has fifths which are closest to being pure.

Bill, just so I understand your posts better, could you explain what you mean by "Reverse Well"? Thanks!

.....

Debate or discussion of how frequent or infrequent this kind of error is made and why it is made is a fair topic but if it is to be explored further, it would be better done in a thread on that topic, not in this one.