introduction and a question about Viennese temperaments early 1800's

dear all, greetings!
I recently joined the club. I have what could be called a varied background. Presently I am self-employed as a harpsichord technician, I do all kinds of repairs and upgrades, in and out of the workshop, and am based in Amsterdam. I have been active as a singer as well, in old modal styles particularly, medieval chant from the manuscripts and Indian classical chant, most of all Dhrupad of the Dagar-family tradition. Acoustics, resonance, harmonic control, tuning tanpuras in function of a raga and playing sarangi and tuning its 38 sympathetic resonance strings have had my deep interest since the late 1980's. Fixing, and tuning harpsichords according to various historic temperaments is also quite engrossing.
As I see more dealings with pianofortes in the future, I'm looking around for specific info on Viennese temperaments as used in the first half of the 19th century. So far, did not get further than Owen Jorgenson's Big Red Book, who has some discriptions and values to offer. However, I noticed that mr. Jorgenson notates his centsvalues in up to five decimals behind the comma. The largest value in cents for Hummel for instance (cts deviation from 12TET) is 0,53 cts and the smallest 0,02 cts. This all gives me a dreary feeling and somehow raises doubts. Who in his right mind would think it is relevant to work with such abstract numbers? I confess I have not read all mr Jorgenson has to say, but can anyone chime in on this if they have worked with the Big Red Book in real life? Or knows of other sources?

​I knew Mr. Owen Jorgenson, he took pride in his work and was very
meticulous in his work. Do not let the precision of his math through
you. He had one of the best TI calculators at the time he wrote the
book. Even his hand writing look like some of the whiting of a devout
monk, particularly the way he made his T's

Another good book that you may want to read is by J. Murray Barbour,
Tuning and Temperament. Barbour started out as a mathematician, but he
carried his math only to the nearest ¢. (so that should make you happy).

I asked Jorgenson why Barbour had said some of what he said about ET in his
book. It has been over 40 years cense I read the book so do not rember
the comment now. Jorgenson was a piano technician at Michigan
State University at time that Barbour taught there. When Barbour wrote his
book he had never heard an music other than 12 ET and did not think that it
mattered how the different tuning made music sound. Jorgenson shows
Barbour that there was a difference. If Barbour had known he likely
would have written his book differently.

ƒg

On Sat, Jan 17, 2015 at 5:02 PM, martinspaink@... [TUNING] <
TUNING@yahoogroups.com> wrote:

>
>
> dear all, greetings!
>
> I recently joined the club. I have what could be called a varied
> background. Presently I am self-employed as a harpsichord technician, I do
> all kinds of repairs and upgrades, in and out of the workshop, and am based
> in Amsterdam. I have been active as a singer as well, in old modal styles
> particularly, medieval chant from the manuscripts and Indian classical
> chant, most of all Dhrupad of the Dagar-family tradition. Acoustics,
> resonance, harmonic control, tuning tanpuras in function of a raga and
> playing sarangi and tuning its 38 sympathetic resonance strings have had my
> deep interest since the late 1980's. Fixing, and tuning harpsichords
> according to various historic temperaments is also quite engrossing.
>
> As I see more dealings with pianofortes in the future, I'm looking around
> for specific info on Viennese temperaments as used in the first half of the
> 19th century. So far, did not get further than Owen Jorgenson's Big Red
> Book, who has some discriptions and values to offer. However, I noticed
> that mr. Jorgenson notates his centsvalues in up to five decimals behind
> the comma. The largest value in cents for Hummel for instance (cts
> deviation from 12TET) is 0,53 cts and the smallest 0,02 cts. This all gives
> me a dreary feeling and somehow raises doubts. Who in his right mind would
> think it is relevant to work with such abstract numbers? I confess I have
> not read all mr Jorgenson has to say, but can anyone chime in on this if
> they have worked with the Big Red Book in real life? Or knows of other
> sources?
>
>
>
>

Hey Martin,

Jorgenson's Big Red Tome is very popular in the states, and although he seems to have been a very nice fellow, his scholarship was second-rate at best. He didn't speak/read German, so he more or less totally ignores the vast body of historical literature on the subject. He also had some really strange ideas, like the idea that before modern methods of beat counting people didn't know that a specific interval tempered by a specific amount beats faster or slower when it is higher or lower on the keyboard. This is utter nonsense; not only is it impossible to believe that people wouldn't have noticed (for example, fifths transposed by an octave, or Pythagorean thirds higher or lower), but Werckmeister specifically described this change in beat speed. Neither Jorgenson nor Barbour are considered to be trustworthy sources anymore by those well-schooled in the original literature, both are considered to be more or less fatally flawed, so I would take anything found in either of them with many many grains of salt.

I have read all the historical literature in German I could find over the last 4 years or so, I'm currently up to 137 works from 1511 to 1881. I know of no original source - not one - in German which indicates that there was any conscious use of anything other than equal temperament in any German-speaking country after about 1800. One may argue as to how accurate the realization of ET may have been, but there is no evidence about that, either.

Ciao,

Paul

Hi Paul, I was hoping my question would reach your august attention! So basically, my misgivings about Jorgenson's data with 5 decimal points behind the comma where he gives cents values are confirmed by what you have researched. Any advice then on what temperaments are feasible for 1800-1830? Jorgenson mentions Prinz, Hummel, Tuner's guide, but none of these are eligible due to lack of precise data? Still, exact 12 TET was not commonly used before late 1850's ? thanks for your feedback!martin

There is no reason to exclude "exact" ET from the picture. Of course, there is no such thing as "exact", not even mathematically, since the ratios are irrational, and no matter how many decimals you use, you can always add one more. Tuning an instrument is similar; no matter how "perfect" you think it is tuned, you can always go one step finer in what you accept as error. The whole question is pointless, as is Jorgenson's assertion that it was impossible.

From Neidhardt (1706) on the tool of an accurately-marked monochord was available, and from Sorge (1744) on there was a method which could well achieve a functionally "perfect" ET... it merely depended upon the skill of the tuner. I would suspect that by 1800, any good tuner who really wanted ET was capable of doing it to a degree that no one could detect any inequality. So just tune ET and be done with it. All this crap about Victorian temperaments is just so much crap.

Once again, Jorgenson did not read German, and his ignorance of the literature really shows.

Ciao,

P

The only thing i would like to add is very small numbers have a habit of building up into bigger ones, so no one has failed by being too accurate, while the assumption of close enough have lead to some real problems
,',',',Kraig Grady,',',',
'''''''North/Western Hemisphere:
North American Embassy of Anaphoria island
'''''''South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria
',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Paul

The intervals of any near-equal division would presumably demonstrate strictly monotonic beating when chromatically ascending and descending – indeed, in the absence of counting, that would be the primary qualitative check when tuning some flavour of ET by ear. Accordingly, I’m curious as to when this notable characteristic was first mentioned in the German tuning literature.

Kind regards

Charles

From: TUNING@yahoogroups.com [mailto:TUNING@yahoogroups.com]
Sent: 19 January 2015 16:12
To: TUNING@yahoogroups.com
Subject: [TUNING] Re: introduction and a question about Viennese temperaments early 1800's

There is no reason to exclude "exact" ET from the picture. Of course, there is no such thing as "exact", not even mathematically, since the ratios are irrational, and no matter how many decimals you use, you can always add one more. Tuning an instrument is similar; no matter how "perfect" you think it is tuned, you can always go one step finer in what you accept as error. The whole question is pointless, as is Jorgenson's assertion that it was impossible.

From Neidhardt (1706) on the tool of an accurately-marked monochord was available, and from Sorge (1744) on there was a method which could well achieve a functionally "perfect" ET... it merely depended upon the skill of the tuner. I would suspect that by 1800, any good tuner who really wanted ET was capable of doing it to a degree that no one could detect any inequality. So just tune ET and be done with it. All this crap about Victorian temperaments is just so much crap.

Once again, Jorgenson did not read German, and his ignorance of the literature really shows.

Ciao,

Hi Charles,

I have always been of the opinion that Jorgenson’s assertion that tuners in ancient times did not know that indentically-tempered intervals beat faster and slower when they are higher or lower is not credible because the most basic experiences of tuning a keyboard teach you that it is so. The lessons lie in two phenomena:

1. Octave transposition of a tempered interval
2. Unintentional constant ”tempering” in the form of the Pythagorean ditone. Everyone knew that the ditone was of a constant size, and that it represented a constant deviation from a 5/4 third. And yet, if one has tuned Pythagorean by using only pure fifths, the ditones beat faster of slower in proportion to their relative height on the keyboard.

So it is essentially impossible to tune a keyboard instrument and NOT notice that equally-sized impure intervals beat at different rates.

The answer to the question of who actually first spoke this is enormously complicated by the modern misunderstanding of the word “schweben/Das Schweben/Die Schwebung(en)”, which is ubiquitously taken to mean “to beat/the beating”. This is a post-Helmholtz definition, and figuring out what it meant before the middle of the 19th century is complex, due in no large part to the vast range of connotations of the word in normal German usage over the ages (read the entry in Grimm’s Deutsches Wörterbuch - 8 pages long!). My exhaustive survey of the historical literature (135+ books from 1511 to c.1880 and about 30 historical reference works) has demonstrated that most often it refers to an aberration or an offset from the pure interval or the correct position of a note. Sometimes it seems to refer to the acoustic quality of a tempered interval without referring directly to the beating itself; and occasionally it does indeed seem to refer directly to beating, but usually when they wanted to do that, they used the words such as Tremorem, Zittern, Bebung, etc. Looking at isolated quotes one can quite easily misapply the “beating” definition, and sometimes it even seems like “beating” is the most obvious. But one is easily tricked, and it really takes an in-depth familiarity with the large body of literature in order to figure out what is meant.

All that said, in regards to your question, while it is obvious that people were well aware of the fact that tempered intervals beat, nobody used this to consciously control the amount of tempering. In fact, the first mention I am aware of in German of the unequal beating of tempered intervals of constant size is used precisely to the opposite effect, i.e. to emphasize that it is difficult to temper consistently precisely because beats rates are not constant. This is in Werckmeister’s tempering instructions in the appendix of his 1698 Die Nothwendigsten Anmerckungen und Regeln wie der Bassus Continuus oder General-Bass wol könne tractiret werden, starting on page 61 (my translation):

“Some experience is required in this [tempering], for even when one already knows that one note of a consonance should hover [schweben] by one-, two-, or three-quarters of a comma against the other, our poor sense of hearing cannot judge accurately if the hovering [Schwebenungen] is too great or too small [NB - size, not speed!!], or if they beat [schlagen] too slowly or too rapidly. There is also a big difference when two small or two large pipes are tuned with one another. For example, I have to tune an octave, such as c and c', 4 foot and 2 foot, or c' and c", 2 foot and 1 foot. If c-c' were a comma too small, or impure, and if c' and c" were to have the same distance between them and also be a comma too small or impure, the c and c' between the 4 and 2 foot would tremble [tremuliren], or hover [schweben], once again more slowly [i.e. half as much] than the c' and c", and so on. The fifths, however, will tremble [tremuliren] faster or slower according to their proportion of 2/3 when the deeper and higher pipes, or larger and smaller, are sounded or tuned against one another. That's why it is difficult to determine with complete precision these small differences with the sense of hearing alone.”

So Werckmeister not only knew that octave transposition doubled or halved the beat rate, but he also obviously understood that when a tempered interval is transposed by ANY interval, the change in beat speed is equal to the ratio of the interval of transposition.

In spite of this, German temperament authors curiously failed to take advantage of it until Scheibler. For instance, when Sorge’s “student” poses the question to the “master” of how he can know if he has tempered his fifths by the right amount or not, Sorge says one uses the Scharfe (biting, cutting, piercing, harsh quality) of the major thirds as the final arbiter. His basic method is to set C-E-G#/Ab-C all “equally harsh” and then temper the four fifths within each third equally. As to how to do that, he doesn’t really say. Only in the 1758 Zuverlässige Anweisung, Claviere und Orgeln behörig zu temperiren und zu stimmen, as part of his severe critique of Fritz’s unreliable method, does give a somewhat accurate though not complete description of the relative beat speeds of the four fifths within the tempered third C-E (§39).

Ciao,

P

Paul

You wrote ‘All that said, in regards to your question, while it is obvious that people were well aware of the fact that tempered intervals beat, nobody used this to consciously control the amount of tempering.’

As you likely know, around 1700 Roger North was an English civil servant responsible for overseeing church organ installation contracts, with the actual work typically performed by foreign builders such as Bernhardt Schmidt. Not being a member of an organ guild, North was free to report what he saw and heard. Here are a some relevant quotes from his writings:

‘If the notes are remote from tune, the disorder will be chattering, or beating very distinguishable and fast, and as it comes nearer, grow slower, and near the truth, fall into a waiving or rowling manner, which proceeding, will quite die in the accord. The artists who deal much in tuning, will by the manner of the beats, judge in what distance the notes are from accord...’

‘...therefore the use is to tune the fifth perfect, and just take it off that perfection, leaving it so as the defect may not readily be perceived. Some observe the beats or wallows of the sound, and leave them to go on in time as slow quavers.’

‘And in that manner these mechanics, injudicious of true Music, shall tune more exactly than the nicest musician.’

‘But I am not in a capacity of subtilizing on this subject, which is peculiar to the artists I mentioned, (and who are not willing to communicate the secrets of their art).’

So apparently beats were used by organ builders back in 1700.

You wrote ‘So Werckmeister not only knew that octave transposition doubled or halved the beat rate, but he also obviously understood that when a tempered interval is transposed by ANY interval, the change in beat speed is equal to the ratio of the interval of transposition.’

There is some extrapolation here. Werckmeister’s assertion regarding 2/3 will trivially tell us that the pure fifths in Pythagorean tuning beat at the same rate, i.e. not at all, and as it stands, the principle cannot be used to tune ET, since no fifths have the needed ratio of 2/3.

You wrote ‘His basic method is to set C-E-G#/Ab-C all “equally harsh” and then temper the four fifths within each third equally.’

As I recall, another of Sorge’s suggestions was to tune in Pythagorean manner to prepare the sensitive ear of his readers for the harsh world of ET. But surely given few, if any, pieces at the time ended on Ab-C, wouldn’t a sensible musician naturally temper C-E to be less harsh than than Ab-C?

Would it be correct to conclude from your exhaustive survey of the German historical literature, that when tuning ET, there’s nowhere mention of a check for accelerating beats of thirds and fifths while proceeding up the keyboard?

Kind regards

Charles

Charles,

Yes, of course I am aware of North’s description. There’s also Beeckman’s. But neither of these indicate that there was any use of proportional beat rates to control for consistent tempering of a series (if not all) of fifths, which is what I thought you were asking about. Also, neither of them were writing in German, which is what I thought you were asking about, the German tempering tradition.

To the contrary, I am not aware of any German-language tuning instruction which describes in any specific manner the use of beating to temper. Sorge says that one can test the tempering of a fifth or third by listening to the sound of the tempered third against the pure third that “sings along” in the sound of a low tone two octaves below the root note of the close interval, or in the case of the fifth, one octave down. But even under repeated questioning of the master by the student in his several Socratic dialogues, there is no mention of beat counting nor acceleration/deceleration when playing chromatic ascending/descending intervals. We should not forget that Werckmeister, Neidhardt and Sorge were all licensed organ inspectors, and Sorge claimed to have personally “liberated” a number of organs from the limited meantone tuning. So if German organ builders were using proportional changes in beat rates as a tempering tool, it is extremely odd that nobody mentions it.

Furthermore, I haven’t the foggiest notion how you arrive at the conclusion you do about what Werckmeister is saying in regards to the fifths. i.e.. that it has something to do with pure fifths. I think you completely misunderstand the passage, perhaps due to the English translation. It is always tricky to adapt German syntax to English, especially Werckmeister’s German, which is rather quirky and takes some getting used to. It is quite clear in the original German what he means. He constructs the model of a chain of two successive octaves, c1-c2-c3, both of which are tempered by a comma, and says that c1-c2 beats half the rate of c2-3. Then he shifts the model to fifths, saying that in a similar series of successive identically tempered fifths - for example f-c1-g1-d2 (not his example but it is clear he means something like this) - the higher fifth beats 3/2 times the middle fifth and the lower 2/3 times the middle. There is nothing whatsoever to indicate he is talking about purity, the entire paragraph deals explicitly with the challenge of controlling the degree of tempering by ear.

I cannot recall any passage in Sorge where he recommends tuning Pythagorean in order to accustom one’s ear to the sound of an ET third, but perhaps I missed it. If you could direct me to the work/page number, I’d be much obliged.

As to whether or not anyone specifically directs the tuner to check for or acceleration of beating in thirds or fifths in order to tune ET, I can’t recall any such passage. The closest one comes to it is Sorge’s rather nit-picky little finger-wagging pamphlet aimed at Schröter’s moncohord numbers for ET, rather appropriately titled “Gründliche Untersuchung, ob die im 3. Th. des 3. Bds. der Mizlerischen musik. Bibliothek S. 457 et 580 befindliche Schröterische Clavier-Temperaturen für gleichschwebend passiren könen oder nicht” (1754). Here, he gives lists of the “Schwebungen” of all twelve fifths arranged chromatically: F-C, F#-C#, G-D, G#-D#, etc. The figures in question are not beats, but rather the absolute aberrations in monochord units from the lengths which would give a pure fifth for each case, i.e. the degree of tempering in absolute, not proportional, amounts. He says that “anybody that has any understanding of such things” can easily see that his correct Schwebungen values follow a log progression while those of Schröter do not. It may seem like a stretch to expect someone to simply look at a series of numbers and judge whether or not they increase of decrease by a log progression, but it’s not so strange for organ builders, who spend there whole lives looking at just such series of pipe lengths. Personally, as someone who has spent his whole life looking at series of string lengths which follow log progressions, I have a pretty good feel for it and can indeed just see it by looking at a list of numbers. If you have a good feeling for the multifaceted meaning of schweben-derived terminology (which one can only acquire a feeling for by reading all the literature), it would be but a very small leap - if a leap at all - to understand that this also meant a smooth progression in beat rates, were one so inclined to listen for them.

Ciao,

P

Paul

I’m rather late responding to your interesting comments from last month (been busy). I’ll certainly provide you with the reference to preparing the ear for ET using Pythagorean tuning when I come across it again.

You wrote: ‘As to whether or not anyone specifically directs the tuner to check for or acceleration of beating in thirds or fifths in order to tune ET, I can’t recall any such passage.’

Interesting, don’t you think? How about meantone, I wonder?

You wrote: ‘Then he shifts the model to fifths, saying that in a similar series of successive identically tempered fifths - for example f-c1-g1-d2 (not his example but it is clear he means something like this) - the higher fifth beats 3/2 times the middle fifth and the lower 2/3 times the middle.’

In your example above I don’t see how the beat ratio of ‘successive identically tempered fifths’ can be 2/3 or 3/2. On the one hand if the fifths were impure it wouldn’t be 2/3, while if the fifths were pure it would be a singularity.

You wrote: “Gründliche Untersuchung, ob die im 3. Th. des 3. Bds. der Mizlerischen musik. Bibliothek S. 457 et 580 befindliche Schröterische Clavier-Temperaturen für gleichschwebend passiren könen oder nicht” (1754)

Got to love that cute title!

Kind regards

Charles

Charles, don't be ridiculous! Of course, if the fifths are tempered, the ratio of the beating will not be 3:2... not PRECISELY, but this is what is commonly referred to as "a difference without a distinction". Take a meantone fifth, for example, which has a ratio of 1,49534878122121. Now consider such a fifth which is beating at precisely 2 beats/second (this is essentially the beat rate of the fifth on tenor Bb at c=246,9, actual beat rate is 2,05). The fifth upon the upper note of this fifth, F-C, will beat at a rate of 2,9906976 beats/second. Now, do you really wish to imply that anyone can hear the difference between this and a true 3 b/s?

Please, let's keep the discussion grounded in reality!

Paul

Paul

Are you suggesting Werckmeister timed beats? Let’s not forget he thought in theological ways, and for him ratios and proportions were conceived in that manner. ‘God is the octave’ as he put it – hence the importance of the number 2; while the number 3 presumably represented the Trinity. Hence ratios such as 2:3 were divinely ordained for him. The anachronistic beat rates you provide concerning the ratio 1 : 1 + 4/10 + 9/100 + 5/1000 etc. go beyond what Werckmeister actually stated, which was only for the ratio 2:3. Try to engage with Werckmeister’s reality to better understand him.

Kind regards

Charles

From: TUNING@yahoogroups.com [mailto:TUNING@yahoogroups.com]
Sent: 17 February 2015 13:30
To: TUNING@yahoogroups.com
Subject: RE: [TUNING] Re: introduction and a question about Viennese temperaments early 1800's

Charles, don't be ridiculous! Of course, if the fifths are tempered, the ratio of the beating will not be 3:2... not PRECISELY, but this is what is commonly referred to as "a difference without a distinction". Take a meantone fifth, for example, which has a ratio of 1,49534878122121. Now consider such a fifth which is beating at precisely 2 beats/second (this is essentially the beat rate of the fifth on tenor Bb at c=246,9, actual beat rate is 2,05). The fifth upon the upper note of this fifth, F-C, will beat at a rate of 2,9906976 beats/second. Now, do you really wish to imply that anyone can hear the difference between this and a true 3 b/s?

Please, let's keep the discussion grounded in reality!

Paul

You're mixing apples and oranges. Werckmeister said that successive fifths beat at a ratio of 3:2. That is what he said, literally. You said they wouldn't if they were tempered. I simply countered that although you are technically correct, there is no perceivable difference and therefore Werckmeister was correct in his assertion. It is obvious that Werckmeister knew that a 1/4 comma fifth ≠ 3:2, but whether or not he knew the exact ratio or even cared is not relevant. What IS relevant is that the validity of the rule of thumb that two successive fifths of ANY reasonable degree of tempering will manifest a beat speed of 3:2 upper fifth/lower fifth for all practical puproses. All this religious mumbo-jumbo has nothing to do with that validity.

Please, try to keep the discussion grounded in reality.

Additionally, I'm not sure you have actually read the quote in question. It's in Die Nothwendigsten Anmerckungen und Regeln wie der Bassus Continuus oder General-Bass wol könne tractiret werden on page 63. He first talks about what happens with octaves which are tempered by a comma, so where's your magic perfect Sky God there? Then he says that fifths will "tremulieren" by there ratio of 2:3. So if these are Holy Trinity fifths, how come they are trembling? Maybe they are Catholic fifths. Or even worse, Jewish fifths! ;-)

Please, Charles, read the passage and figure it out before confusing the issue with Werckmeister's speculative theological musings in his Musicalische Paradoxal-Discourse.

Ciao,

P

The point is, Charles, these guys were both practical folk looking for real solutions to real problems, and on occasion, capable of the most extreme theological and/or philosophical flights of fancy. You should not confuse the one with the other. Sorge, for example, was so enamored of the "charming little game of Nature" (as he called it) that one could hear in a single tone the root, third and fifth of a major harmony, that he found it to be such compelling evidence of the "proof" of the Holy Trinity that it should be sufficient to convince even the "unbelieving Jew" (which is what I was referring to in my last post). Nonetheless, this Holy Harmonic Perfection did not stop him from condemning the use of pure major thirds, comparing them to stagnant water which becomes putrefied, or to a sort of musical doldrums where the ship of harmonic motion becomes trapped on the becalmed seas, whereas the "Schwebungen" of tempered thirds (as long as it is not too severe) drives the Goode Shippe Harmony happily onward to port.

Paul

You wrote: ‘I simply countered that although you are technically correct, there is no perceivable difference...’

Let’s assume for arguments sake that the 2:3 rule-of-thumb ratio was not arrived at a priori from metaphysics, but was rather an empirical discovery. Then wouldn’t the discoverer have first needed to relate together the beat ratios of fifths in order to arrive at 2/3 rather than some other fraction?

How widely disseminated was that discovery? If Werckmeister’s remark was overlooked or considered uninteresting, it presumably had little impact on the writings and tunings of other theorists. Conversely, if common knowledge, wouldn’t it show up in discussions of regular tunings?

Kind regards

Charles

Yes, Charles, you cut to the very heart of the Grand Mystery. I am convinced that they knew about proportional beat rates empirically, and not a priori, and that it was indeed wide-spread knowledge. I base this belief on the fact that there are only two possibilities; 1) a person tuning a keyboard instrument is completely unaware of beating, i.e. he can't hear any beating no matter how fast or slow, and therefore he can no more be under a delusion that he must use equal-beating than he can be aware that he must use proportional beating; 2) he can hear of beating, and in that case, he would have to be pretty dense (or deaf) NOT to notice that any beating interval will beat twice as fast at the octave, or twice as slow an octave down. Once you cop to that, it is but a small step to notice the same at the fifths, or the third, knowing their natural ratios (and once again, the minor deviation for the tempered versions do not cause beats rates to change so much that one would notice the difference between the actual ratio of tempered intervals and the theoretical natural ratio). This is why I believe that all this equal-beating nonsense is just that, a bunch of modern nonsense. As I said before, I think it was started by Owen Jorgenson, and before him, it simply does not appear in any literature, neither modern nor historical.

That said, it remains a great mystery as to why they did not use the knowledge of proportional beating to control the size of intervals - something which seems so mind-bogglingly obvious to us that we cannot imagine how anyone could fail to do so... but evidently it was NOT so obvious to them. Bear in mind that Werckmeister speaks of proportional beating as a reason why it is DIFFICULT to know what you are doing by ear, exactly the opposite of how it seems to us. But there it is.

The one thing we CANNOT do is assume that because they didn't use beat rates in tuning-by-ear recipes they were so out-to-lunch that they didn't notice the larger correlation between pitch and beat speed, and therefore they suffered under the delusion that identically-tempered fifths beat at identical rates. That idea is simply logically self-contradictory, because at some point you will get to the great leap where the beat rate doubles all of a sudden. For example, if you set your bearings by tempering the fifths on all the notes between (tenor) e and (middle) d#1, as did Fritz, you are going to get a sudden doubling if you compare the fifth d#1-g#1 to e1-b1, of a sudden halving comparing e-b to d#-g#. Yet one would know that by tuning upwards and downwards by perfectly pure octaves outward from the notes within the beating octave, if you think that all fifths within the bearing octave are identically tempered, it would follow logically that all fifths would be tempered by the same amount. So wouldn't that sudden doubling or halving cause you to realize that the whole idea was utter bollocks?

Once again, let me emphasize that not only is the idea itself bonkers, but there isn't one single bit of historical evidence to indicate anyone ever thought this way. I keep asking modern supporters of this theory for such, and nobody ever seems capable of providing anything. So why should we believe it? It is the intellectual equivalent of the "God of the Gaps" argument for the existence of a supreme being. Because you don't understand something, you simply stick some explanation into the gap, without any logic or evidence.