Yahoo Tuning Groups Ultimate Backup tuning Re: Here is the Excel worksheet I have worked with

Hello Bradley, in between the lines:

(And I'm forwarding this to the tuning list for its relevance to the
whole issue of your recipe of Lehman-Bach Temperament versus my recipe
for UWT nr.3)

✩ ✩ ✩
www.ozanyarman.com

On May 22, 2010, at 12:45 AM, Brad Lehman wrote:

> Hi Oz,
>
> Among other problems, you're profoundly misreading my instructions
> vis-a-vis what the word "same" means. It's not "same beat
> frequency." It's "same tempered quality in the interval"; not at all
> equivalent concepts here.
>

In your recipe, taking as reference the 440 Hz diapason for A, I read
tenor A to middle E slightly flat in the same quality as those other
fifths F-C and G-D. You have given instructions to compare beat ratios
against temporary notes and that is how the calculations are done to
find G, C and D. You have specifically written that middle C-E is to
beat 4.5 times per second. Do you not understand the implications of
what you have written? The fifths that one arrives at per your
instructions are:

F-C = 694.498 cents

G-D = 700.473 cents

A-E = 698.016 cents.

What room is there for complaint here? Apparently you are doing
something wrong, unless I made an error somewhere, and it appears I
have not.

> C#-G# and Ab-Eb aren't supposed to have identical beat frequencies.> They are supposed to have the same tempering, 1/12 PC each. That
> "same" tempering, in this instance, means that Ab-Eb should beat 3/2
> as fast as C#-G#, because it's up a 5th.
>

Following this information, one derives of course:

1.5 * (4Eb - 3x) = (2x - 3C#)

1.5 * (1,247.2888888 - 3x) = (2x - 833.4140625)

giving us 416.053445492307 Hz for G# in the middle octave, which
yields a 699.23750842516474 cent fifth between C#-G# compared to what
I found via equalized beat frequencies: 699.60008746885019; and
700.7458558855475 cents fifth between Ab-Eb compared to what I found
in the same fashion: 700.38327684186286.

Unless you have the extraordinary ability to differentiate 0.36 cents
difference in each case acoustically, let alone in a musical pace,
your complaint is totally mute.

> Similarly, your explanation of Bb here makes some wrong assumptions
> on what's supposed to be "same".
>

Humour me, what do you mean by: "Lower Bb slightly so it has a slow
beat from Eb like that of C#-G# and Ab-Eb."

Am I to be accused of misrepresentation when the differences you
suggest are so miniscule when I'm following your recipe to the letter
based on your reliance on beat frequencies and ratios?

> I still have no clue where you got 877, 550, 825, etc. What is 877
> for F? Did you just move it up and down until it gave a desired
> beat rate against some A? Which A?
>

My dear colleague, I have made it very clear that I took 440 Hz for A
and 220 Hz for tenor A as per your instructions. Follow the math, you
can't go wrong there. F-A in the tenor region beats exactly 6 times
per second as you wanted if you assign it the value 877/550. The berth
from A down to F is 877/1100 if you trust your recipe.

> Why reduce ANY of this to ratios?

To demonstrate how some claims do not measure up in reality?

> Accurate comma-splitting geometrically has nothing to do with ratios!

They do, when you pour a recipe in to the oven of mathematical
verification.

> As you surely know, it's not possible to get there by that method,
> even if you'd pick million-digit integers or longer.
>

The whole purpose was to see, if your recipe was as trustworthy as
mine, or vice versa, your heavy reliance on your ears and experience
had any reflections in the real world of verification by number
crunching.

> If you'll compare your "Cents fr A" column E against my "Cents"
> column at
> http://www-personal.umich.edu/~bpl/larips/bach-beats-440.gif
> you'll see that ALL of your numbers are wrong by various amounts,
> except of course the 0.00 for A!
>

If my numbers are wrong, then so is your recipe Brad. I suggest you
quintuple and hextuple the effort to check your calculations.

> Why not tune all this on a real harpsichord (which is the point),
> instead of trying to come up with numbers for everything?
>

How are we to rely on any of this if the basic verification method
shows that what you describe as the recipe for setting up your
temperament correctly is incorrect?

It has been observed that you do not correctly identify the beat
frequency difference between tenor C to E and middle C to E.

> I appreciate the attempt to try to reverse-engineer a set of ratio
> approximations from the instructions, but I feel it misses the point
> of this temperament!
>

And I feel you are being a bit biased and unfair when it comes to
evaluating UWT nr.3, both mathematically and acoustically, when it is
evident that you have not correctly set it up on your harpsichord and
that my recipe, be it digital or analog, is more trustworthy even then
compared to yours.

> Brad
>
>

Cordially,
Oz.

> On 5/21/2010 5:25 PM, Ozan Yarman wrote:
>> Oh, those ratios in B & D are the end results after I followed the
>> recipe.
>>
>> The temporary ratios were:
>>
>> C at (877 / 550) x (3 /4) = 2631/2200 (310 cents)
>> D at 4/3 (498 cents)
>> Bb at 877/825 (106 cents)
>>
>> Temporary tenor D and C were pitted against G to acquire the beat
>> proportion 3/2. G was calculated to be 1961/1100
>>
>> A and C were pitted against D above middle C to acquire the same
>> brat. D
>> was calculated to be 78440000000000/58716923076923.
>>
>> G# was tuned so that the beating was the same for both C#-G# in the
>> middle octave and Ab-Eb across middle C, with the result that the
>> ratio
>> of G# came out to be 1479754324905530/782298000000000.
>>
>> Bb was lowered so that the same beat frequency as above (1.13 Hz)
>> could
>> be found between Eb-Bb. Bb was calculated to be
>> 546273819459862/514506666666663
>>
>> Cordially,
>> Oz.
>>
>>
>>
>> ✩ ✩ ✩
>> www.ozanyarman.com
>>
>> On May 21, 2010, at 11:20 PM, Brad Lehman wrote:
>>
>>> Where does the recipe get entered into this? Columns B & D? Where do
>>> all those integers come from?
>>>
>>> Brad
>>>
>>>
>>> On 5/21/2010 4:08 PM, Ozan Yarman wrote:
>>>> I usurped George Secor's worksheet for the purpose, but you can
>>>> check
>>>> the calculations for yourself.
>>>>
>>>> Oz.
>>>>
>>>> ✩ ✩ ✩
>>>> www.ozanyarman.com
>>
>>
>>
>>

🔗bplehman27 <bpl@...>

Dear Oz,

This has clearly become a pointless discussion.

For one thing, it's wrong to copy somebody's private e-mail onto a discussion list, and hack it apart in response.

For another, you've cited here the snapshot that gives all the CORRECT cent values and A=440 beat rates of my temperament, to 2 decimal places:
http://www-personal.umich.edu/~bpl/larips/bach-beats-440.gif
...but, you've disregarded it, and have chosen instead to go with your own calculations that arise from one mistaken assumption after another.

That table of mine is to compare against the set of my instructions here:
http://www-personal.umich.edu/~bpl/larips/practical.html
You are reverse-engineering those instructions into a spreadsheet of your own, USING RATIOS that mis-represent the nature of this temperament (and the nature of the instructions). It's a comma-splitting temperament. It would therefore deal in IRRATIONAL numbers (the twelfth root of the Pythagorean Comma), if any numbers were actually needed in the process of setting this up. They're not. Just a simple listening process and MUSICAL skills, knowing what to listen for, and how to compare the qualities of intervals until they're "the same" as one another.

Take a closer look. My instructions say that they are "Practical temperament instructions by ear". That they are. By ear, to the best accuracy that anyone can reasonably hope for when working directly at a real harpsichord. The recipe is not a pretext for calculations. It is a straightforward set of practical instructions to do a physical task, to reasonably deliverable accuracy. I take the two-decimal accuracy of the beat rates from that snapshot (and two decimals are more than enough for anybody, already!)...and I round all of them off to a single decimal, for the by-ear instructions. Have you got that point? That's why the instructions say "3 beats per second" for the F-A (it's rounded off to 3.0 from 2.98), and "4.5 per second" for C-E (it's rounded off from 4.46).

1/10th of a beat per second, in the middle octaves of a keyboard, listening to 5ths and 3rds? That's plenty, in a task that takes only two to three minutes (setting the temperament bearings for all twelve pitch-classes), working by ear and having the necessary physical skills of adjusting tuning pins with the right tools. 1/100th of a beat per second would be ludicrous, and that's why I round off my instructions to 1/10th, which itself is overkill.

To be able to say that I'm allegedly wrong, here below you've produced 12-digit and 14-digit decimals. Come on! In trying to demonstrate that my practical instructions are somehow faulty, you have only demonstrated that you aren't reading them properly, and that you are not even approaching them within the medium for which they're intended.

Because you can't deal with the rounding-off of two decimal places to one, in interpreting my instructions at the very few places where they mention beat rates as specific constants, you misinterpret everything else from there forward. You come up with three differently-sized 5ths in F-C, G-D, and A-E, even though I have clearly stated that they are to be geometrically identical: with 1/6 PC of tempering, each.

You also take the F-A to be a rational distance, 877/1100, which is (as I noted above) impossible. To get it right, IF CALCULATION WERE EVEN IMPORTANT, you'd have to do your calculations from irrational numbers...and you're obviously not prepared to do that with your spreadsheet. Your input of the temperament is a set of integers, making ratios. There is no way that that approach is ever going to hit the spot, in assessing a temperament (or a set of instructions) based on comma-splitting.

More importantly: to follow my instructions "to the letter", as you are evidently fond of saying, you'd have to be sitting there actually tuning a harpsichord by ear, not playing with spreadsheets and producing 14-digit decimals.

Brad Lehman

--- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>
> Hello Bradley, in between the lines:
>
> (And I'm forwarding this to the tuning list for its relevance to the
> whole issue of your recipe of Lehman-Bach Temperament versus my recipe
> for UWT nr.3)
>
> â© â© â©
> www.ozanyarman.com
>
> On May 22, 2010, at 12:45 AM, Brad Lehman wrote:
>
> > Hi Oz,
> >
> > Among other problems, you're profoundly misreading my instructions
> > vis-a-vis what the word "same" means. It's not "same beat
> > frequency." It's "same tempered quality in the interval"; not at all
> > equivalent concepts here.
> >
>
>
> In your recipe, taking as reference the 440 Hz diapason for A, I read
> tenor A to middle E slightly flat in the same quality as those other
> fifths F-C and G-D. You have given instructions to compare beat ratios
> against temporary notes and that is how the calculations are done to
> find G, C and D. You have specifically written that middle C-E is to
> beat 4.5 times per second. Do you not understand the implications of
> what you have written? The fifths that one arrives at per your
> instructions are:
>
> F-C = 694.498 cents
>
> G-D = 700.473 cents
>
> A-E = 698.016 cents.
>
> What room is there for complaint here? Apparently you are doing
> something wrong, unless I made an error somewhere, and it appears I
> have not.
>
>
> > C#-G# and Ab-Eb aren't supposed to have identical beat frequencies.
> > They are supposed to have the same tempering, 1/12 PC each. That
> > "same" tempering, in this instance, means that Ab-Eb should beat 3/2
> > as fast as C#-G#, because it's up a 5th.
> >
>
>
> Following this information, one derives of course:
>
> 1.5 * (4Eb - 3x) = (2x - 3C#)
>
> 1.5 * (1,247.2888888 - 3x) = (2x - 833.4140625)
>
> giving us 416.053445492307 Hz for G# in the middle octave, which
> yields a 699.23750842516474 cent fifth between C#-G# compared to what
> I found via equalized beat frequencies: 699.60008746885019; and
> 700.7458558855475 cents fifth between Ab-Eb compared to what I found
> in the same fashion: 700.38327684186286.
>
> Unless you have the extraordinary ability to differentiate 0.36 cents
> difference in each case acoustically, let alone in a musical pace,
> your complaint is totally mute.
>
>
>
> > Similarly, your explanation of Bb here makes some wrong assumptions
> > on what's supposed to be "same".
> >
>
>
> Humour me, what do you mean by: "Lower Bb slightly so it has a slow
> beat from Eb like that of C#-G# and Ab-Eb."
>
> Am I to be accused of misrepresentation when the differences you
> suggest are so miniscule when I'm following your recipe to the letter
> based on your reliance on beat frequencies and ratios?
>
>
> > I still have no clue where you got 877, 550, 825, etc. What is 877
> > for F? Did you just move it up and down until it gave a desired
> > beat rate against some A? Which A?
> >
>
>
> My dear colleague, I have made it very clear that I took 440 Hz for A
> and 220 Hz for tenor A as per your instructions. Follow the math, you
> can't go wrong there. F-A in the tenor region beats exactly 6 times
> per second as you wanted if you assign it the value 877/550. The berth
> from A down to F is 877/1100 if you trust your recipe.
>
>
> > Why reduce ANY of this to ratios?
>
>
> To demonstrate how some claims do not measure up in reality?
>
>
> > Accurate comma-splitting geometrically has nothing to do with ratios!
>
>
> They do, when you pour a recipe in to the oven of mathematical
> verification.
>
>
> > As you surely know, it's not possible to get there by that method,
> > even if you'd pick million-digit integers or longer.
> >
>
>
> The whole purpose was to see, if your recipe was as trustworthy as
> mine, or vice versa, your heavy reliance on your ears and experience
> had any reflections in the real world of verification by number
> crunching.
>
>
> > If you'll compare your "Cents fr A" column E against my "Cents"
> > column at
> > http://www-personal.umich.edu/~bpl/larips/bach-beats-440.gif
> > you'll see that ALL of your numbers are wrong by various amounts,
> > except of course the 0.00 for A!
> >
>
>
> If my numbers are wrong, then so is your recipe Brad. I suggest you
> quintuple and hextuple the effort to check your calculations.
>
>
> > Why not tune all this on a real harpsichord (which is the point),
> > instead of trying to come up with numbers for everything?
> >
>
>
> How are we to rely on any of this if the basic verification method
> shows that what you describe as the recipe for setting up your
> temperament correctly is incorrect?
>
> It has been observed that you do not correctly identify the beat
> frequency difference between tenor C to E and middle C to E.
>
>
> > I appreciate the attempt to try to reverse-engineer a set of ratio
> > approximations from the instructions, but I feel it misses the point
> > of this temperament!
> >
>
>
> And I feel you are being a bit biased and unfair when it comes to
> evaluating UWT nr.3, both mathematically and acoustically, when it is
> evident that you have not correctly set it up on your harpsichord and
> that my recipe, be it digital or analog, is more trustworthy even then
> compared to yours.
>
>
> > Brad
> >
> >
>
>
> Cordially,
> Oz.
>
>
> > On 5/21/2010 5:25 PM, Ozan Yarman wrote:
> >> Oh, those ratios in B & D are the end results after I followed the
> >> recipe.
> >>
> >> The temporary ratios were:
> >>
> >> C at (877 / 550) x (3 /4) = 2631/2200 (310 cents)
> >> D at 4/3 (498 cents)
> >> Bb at 877/825 (106 cents)
> >>
> >> Temporary tenor D and C were pitted against G to acquire the beat
> >> proportion 3/2. G was calculated to be 1961/1100
> >>
> >> A and C were pitted against D above middle C to acquire the same
> >> brat. D
> >> was calculated to be 78440000000000/58716923076923.
> >>
> >> G# was tuned so that the beating was the same for both C#-G# in the
> >> middle octave and Ab-Eb across middle C, with the result that the
> >> ratio
> >> of G# came out to be 1479754324905530/782298000000000.
> >>
> >> Bb was lowered so that the same beat frequency as above (1.13 Hz)
> >> could
> >> be found between Eb-Bb. Bb was calculated to be
> >> 546273819459862/514506666666663
> >>
> >> Cordially,
> >> Oz.
> >>
> >>
> >>
> >> â© â© â©
> >> www.ozanyarman.com
> >>
> >> On May 21, 2010, at 11:20 PM, Brad Lehman wrote:
> >>
> >>> Where does the recipe get entered into this? Columns B & D? Where do
> >>> all those integers come from?
> >>>
> >>> Brad
> >>>
> >>>
> >>> On 5/21/2010 4:08 PM, Ozan Yarman wrote:
> >>>> I usurped George Secor's worksheet for the purpose, but you can
> >>>> check
> >>>> the calculations for yourself.
> >>>>
> >>>> Oz.
> >>>>
> >>>> â© â© â©
> >>>> www.ozanyarman.com
> >>
> >>
> >>
> >>
>

🔗Ozan Yarman <ozanyarman@...>

My dear colleague, Bradley,

✩ ✩ ✩
www.ozanyarman.com

On May 22, 2010, at 3:24 AM, bplehman27 wrote:

>
>
>
>
>
> Dear Oz,
>
> This has clearly become a pointless discussion.
>

Possibly. Because I see that you have no intention of yielding to the
facts:

1. The recipe that you gave as the course to follow "for an
experienced harpsichord tuner to set up an entire harpsichord in 15
minutes to your tuning", DOES NOT produce the temperament which you
claim to set by ear using fifths tempered by GEOMETRIC FRACTIONS of a
rational interval. Every interval save G is skewed 3 to 3.6 cents down
from your values (even taking into consideration your protestations
against my formulations). On the one hand, you are being extremely
liberal in setting up your temperament, on the other, you throw about
criticisms against my UWT major thirds and fifths which differ from
yours by even smaller increments (that is, if you actually follow the
recipe right and cross-check by a tuning device for human errors).

2. The feature that you boast for your temperament being absent in
mine, that is, the regularity of fifth temperings, does not arise from
your recipe. Where does it come from? It comes from your Excel
spreadsheet and the Scala file. How can anyone tune irrational
integers exactly the same size by ear? By following your recipe. Maybe
you possess an extra-ordinary capacity to disregard the recipe and
tune everything right to the nth decimal, I don't know. But is your
Excel number-crunching of the irrational figures of a 1/6 comma Well-
Temperament to take precedence over another Excel number crunching
that yields a balanced, ear-tunable Well-Temperament with proportional
beat rates that can be set up EXACTLY according to my instructions?

3. You OBVIOUSLY listen to, count and compare beats just the same in
your recipe when setting up your temperament "in 15 minutes" compared
to my recipe for UWT nr.3 (where the second is the unit time or not).
That there is nothing to suggest your approach is analog versus myalleged "digital". That if my approach has CONSTANTS, then your
approach are not BEREFT of them.

> For one thing, it's wrong to copy somebody's private e-mail onto a
> discussion list, and hack it apart in response.
>

I saw no harm in doing so, for the messages contained nothing
"private" as I could tell. The technical relevance of the contents to
the matter at hand took precedence in my view. Sorry if I caused you
any alarm.

> For another, you've cited here the snapshot that gives all the
> CORRECT cent values and A=440 beat rates of my temperament, to 2
> decimal places:
> http://www-personal.umich.edu/~bpl/larips/bach-beats-440.gif
> ...but, you've disregarded it, and have chosen instead to go with
> your own calculations that arise from one mistaken assumption after
> another.
>

My dear colleague. I have done no mistaken assumptions. I have taken
your recipe to tune the harpsichord in 15 minutes at face value. If
those instructions mean anything and have any relevance to your
temperament, as you clearly insinuate is the case, then what argument
can you have against the results that show such gross deviations in
the cycle and intended pitches? It's either:

1. You are not correctly describing what you are doing, or,
2. You are not adhering to what you describe, or, in the remotest
possibility (yet!),
3. I have made some truly terrible error that affects the results up
to the point that your recipe yields your temperament to the nth
remainder for every irrational number you hold dear.

> That table of mine is to compare against the set of my instructions
> here:
> http://www-personal.umich.edu/~bpl/larips/practical.html
> You are reverse-engineering those instructions into a spreadsheet of
> your own, USING RATIOS that mis-represent the nature of this
> temperament (and the nature of the instructions). It's a comma-
> splitting temperament. It would therefore deal in IRRATIONAL
> numbers (the twelfth root of the Pythagorean Comma), if any numbers
> were actually needed in the process of setting this up. They're
> not. Just a simple listening process and MUSICAL skills, knowing
> what to listen for, and how to compare the qualities of intervals> until they're "the same" as one another.
>

The nature of the instructions you have given yield the ratios I
provided, not these in your worksheet which you take as reference for
your instructions:

C 1/1
C# 8926735/8435236
D 2907629/2596260
D# 8495647/7152031
E 6837827/5451757
F 12177389/9112438
F# 8926735/6326427
G 3285409/2195225
G# 4613353/2909514
A 2299792/1372105
A# 1419143/797367
B 15193459/8075767
C 2/1

Do these ratios misrepresent the nature of your paper-calculated
temperament when they are accurate beyond the ability of human hearing
to discern?

So are my ratios accurate beyond the ability of human discernment in
representing the outcome of what happens if we trust your recipe and
follow "what to listen for" and make intervals "the same quality as
one another".

If you don't like what you see, why would you not re-check your
methods? It is mighty clear that you either tolerate a 694.5 cent
fifth in your tuning while bombarding me for having specified a 695
cent fifth in E-B and F-C (where they are supposed to be tuned 698.5
and 700 cents respectively), or do not follow the instructions.

Being compromised in such a fashion, you cannot throw about criticisms
against UWT nr.3, most of which are demonstrably baseless.

> Take a closer look. My instructions say that they are "Practical
> temperament instructions by ear". That they are. By ear, to the
> best accuracy that anyone can reasonably hope for when working
> directly at a real harpsichord.

3.5 cents error on every tone except G? And you still claim the
superiority of regular sized geometrically tempered fifths?

The discrepancy between your claims and the reality are enormously
glaring my dear colleague.

And I'm sure, if you work at the numbers a bit more, you can arrive at
even closer approximations to the desired numbers in your Excel
worksheet.

> The recipe is not a pretext for calculations. It is a
> straightforward set of practical instructions to do a physical task,
> to reasonably deliverable accuracy.

Well!

> I take the two-decimal accuracy of the beat rates from that
> snapshot (and two decimals are more than enough for anybody,
> already!)...and I round all of them off to a single decimal, for the
> by-ear instructions. Have you got that point? That's why the
> instructions say "3 beats per second" for the F-A (it's rounded off
> to 3.0 from 2.98), and "4.5 per second" for C-E (it's rounded off
> from 4.46).

All of which end up yielding a 694.5 cent fifth right there between F-
C. How you get rid of that fifth which is arrived at by methods that
are "enough for anybody", I don't know.

>
> 1/10th of a beat per second, in the middle octaves of a keyboard,> listening to 5ths and 3rds? That's plenty, in a task that takes
> only two to three minutes (setting the temperament bearings for all
> twelve pitch-classes), working by ear and having the necessary
> physical skills of adjusting tuning pins with the right tools.
> 1/100th of a beat per second would be ludicrous, and that's why I
> round off my instructions to 1/10th, which itself is overkill.
>

Overkill enough for 3 to 3.6 skew on every pitch except G and that
monsterous 694.5 cent fifth?

Well!

> To be able to say that I'm allegedly wrong,

Oh believe me, there is nothing allegedly wrong about the end result
of your tuning instructions which you just admitted above is: "the
best accuracy that anyone can reasonably hope for when working
directly at a real harpsichord".

> here below you've produced 12-digit and 14-digit decimals. Come
> on! In trying to demonstrate that my practical instructions are
> somehow faulty, you have only demonstrated that you aren't reading
> them properly, and that you are not even approaching them within the
> medium for which they're intended.
>

Is this a joke? Your instructions do not result in the tuning you say
it does my colleague. Please take a step back and consider your
position here.

> Because you can't deal with the rounding-off of two decimal places
> to one, in interpreting my instructions at the very few places where
> they mention beat rates as specific constants, you misinterpret> everything else from there forward.

Me? Me the one misinterpreting your instructions? I followed them just
as you described them. Them being, in your own words, "the best
accuracy that anyone can reasonably hope for when working directly at
a real harpsichord".

If you have a recipe that helps you set it up RIGHTLY this time,
please share it with us.

> You come up with three differently-sized 5ths in F-C, G-D, and A-E,
> even though I have clearly stated that they are to be geometrically
> identical: with 1/6 PC of tempering, each.
>

Yes yes yes, you always state the reality of the figures of your Excel
worksheet. Wonderful, tremendous... But how do you go on about
IMPLEMENTING them, if not via your recipe or some other recipe?

Pray tell us if you indeed have a method that lets you tune your
Harpsichord to the Temperament of your desire aside from the recipe
which you say is "the best accuracy that anyone can reasonably hope
for when working directly at a real harpsichord".

> You also take the F-A to be a rational distance, 877/1100, which is
> (as I noted above) impossible.

You start by calibrating A to 220 Hz, want the 4:5 to beat 3 times per
second wide, and you can arrive at something that represents F-A spoton other than 1100/877?

I'm shocked.

What does the division 220 Hz / 175.4 Hz give you my dear colleague?

> To get it right, IF CALCULATION WERE EVEN IMPORTANT, you'd have to
> do your calculations from irrational numbers...

Well!

> and you're obviously not prepared to do that with your spreadsheet.
> Your input of the temperament is a set of integers, making ratios.

And there is some way to calculate the interval between two
frequencies by any other method NOT involving ratios?

If you are vexed by rationals, it is the easiest thing to substitute
all the ratios in the worksheet by integers to the nth decimal, if
that be your desire! But the change would still be fathoms beyond the
human ability to discern.

> There is no way that that approach is ever going to hit the spot,
> in assessing a temperament (or a set of instructions) based on comma-
> splitting.
>

Sure, whatever.

> More importantly: to follow my instructions "to the letter", as you
> are evidently fond of saying, you'd have to be sitting there
> actually tuning a harpsichord by ear, not playing with spreadsheets
> and producing 14-digit decimals.
>

I would, if the recipe actually yielded the Lehman-Bach Temperament
results satisfactorily as the tuner-musician says it does!

>
> Brad Lehman

Cordially,
Dr. Oz.

🔗genewardsmith <genewardsmith@...>

🔗Ozan Yarman <ozanyarman@...>

🔗genewardsmith <genewardsmith@...>

🔗bplehman27 <bpl@...>

--- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:

> In your recipe, taking as reference the 440 Hz diapason for A, I read
> tenor A to middle E slightly flat in the same quality as those other
> fifths F-C and G-D. You have given instructions to compare beat ratios
> against temporary notes and that is how the calculations are done to
> find G, C and D. You have specifically written that middle C-E is to
> beat 4.5 times per second. Do you not understand the implications of
> what you have written? The fifths that one arrives at per your
> instructions are:
>
> F-C = 694.498 cents
>
> G-D = 700.473 cents
>
> A-E = 698.016 cents.
>
> What room is there for complaint here? Apparently you are doing
> something wrong, unless I made an error somewhere, and it appears I
> have not.

Hi Oz, I've had opportunity this morning to take a closer look in the spreadsheet you sent me yesterday, plus the e-mailed notes about how you derived things.

First, I'll humor your use of F-A of 877/1100, because even with the use of that approximation, you ought to get better results than you actually did.... I still can't see all the processes you did, because you didn't show them, but here's how I would do it:

Temporary C a pure 4th below that F: 877/1100 x 3/4 = 2631/4400

Temporary D a pure 5th below the A: 2/3

Next step is to find the G that's a 4th above that D, and a 5th above that C, such that it's at the GEOMETRIC mean, and such that the D-G 4th beats triplets against the C-G 5th's duplets.

The pure G from that D would be 2/3 x 4/3 = 8/9

The pure G from that C would be 2631/4400 x 3/2 = 7893/8800

The mean spot between those two phantom G candidates, geometrically, is sqrt(8/9 x 7893/8800) = .8929012976+

That's the G we get by following my instructions, tuning by ear and seeking the spot where we get triplets beating against duplets from the two temporary notes.

The G *you* got (somehow??) was "1961/1100", which when adjusted to the proper octave at 1961/2200 is .8913636363+

Your notes don't say anything about moving the C (per my instructions) to a proper spot, after using the temporary one. Again, I can't see how you derived C in its final position with your ratios, but clearly you've ended up with C at a wrong spot vis-a-vis A. Yours is at 302.27 cents, and it ought to be 305.87; you're off by more than 3 cents!

From your wrong G, you somehow came up with a D that you say is "calculated to be 78440000000000/58716923076923". I can't follow any of your arithmetic here, because you didn't show it, but clearly you're already accumulating so many errors recursively that it's pointless to continue. Your D ends up at 501.3758 cents from A, somehow, while it ought to be at 501.96. That's not as bad as the errors in the C and the G, but anyway, it's off. They're all off. Somehow, you accidentally found ways to diverge from the instructions, while trying to test those instructions.

More importantly, I can't see that you have used square roots anywhere, which is the proper procedure to place a note at a mean point relative to two other notes. Without square roots, and therefore some irrational numbers occasionally, you're not going to get there.

My "triplets against duplets" beating procedure, comparing successive 5ths, works whenever we're making 5ths the same size GEOMETRICALLY. All the 5ths have some constant amount taken off them (which happens to be a fraction of the PC in mine, but it could be any constant), and the frequencies of both notes are 3/2 higher each time we move up(**), so the beat rates also increase as 3/2, "triplets against duplets". If we are looking at (for example) the open-string notes of violins and violas, and we have regularly tempered 5ths everywhere, and the A-E is beating 3/sec, the D-A will be 2/sec, the G-D is 4/3 per sec, and the C-G is 8/9 per sec.

(**) "3/2 higher" shortened by some small tempered amount, but close enough to 3/2 for all practical purposes on acoustic instruments, when making this comparison....

To calculate all those notes with their REGULAR GEOMETRIC tempering, if they actually have to be calculated (as opposed to tuning them directly by ear, which is much simpler!), we have to use square roots....not ratios.

The MUSICAL skill here, as opposed to the mathematical skill, is to recognize all those identically-tempered 5ths by their quality, directly: not having to count the beats that prove they're spot-on. With practice (and I've had 27 years of doing this), it's easy to get two neighboring 5ths set up with similar quality, without any conscious counting of beats. And so, as I said several days ago, that's how I tune: and that's how I write tuning instructions for readers who have that similarly practiced musical skill. That "triplets against duplets" check is a useful test for the similarity of the intervals, but the similarity is already directly perceptible (close enough) without it.

People who prefer mathematically-based instructions based on beat-counting are free to formulate their own sequence of setting the notes. All the correct values, to two decimal places, are shown at my page:
http://www-personal.umich.edu/~bpl/larips/math.html

Brad Lehman

🔗martinsj013 <martinsj@...>

🔗Ozan Yarman <ozanyarman@...>

My dear colleague Dr. Lehman,

I'm utterly astonished by what I perceive as the latest attempt on
your part to distort data to deliver yourself from the grip of facts.
You clearly do not wish to comprehend what the calculations you
venture to abuse signify: They are the direct pitch results between
220 Hz A and 440 Hz A after following your recipe to the letter (with
so tight a precision, if you will, beyond the discernment of the human
ear) found at:
http://www-personal.umich.edu/~bpl/larips/practical.html

They are, of course, NOT meant to JUSTIFY your hearsay ratios of 1/6
PC tempered fifths which you CLAIM to IMPLEMENT on a real harpsichord
and which DO NOT ARISE if one takes the route described in your
recipe, among the "...recipes on my http://www.larips.com site are
*by* me, describing the way I set all those various temperaments by
ear" scrutinized vis-a-vis my calculations.

I shall soon demonstrate how my numbers yield the results that TRULY
CORRESPOND to your instructions, and how your numbers are just wishful
fabrications that have nothing to do with your recipe.

You can protest that I am reverse-engineering your recipe all you
like. The data show that, if you rely on that recipe of yours, you
WILL NOT get the temperament you hold dear to your heart my colleague.

Let us remember the whole purpose of my delving into this dialague
with you here. You formulated a bunch of protestations against UWT nr.
3 to nullify the proportional beat ratios feature and forward the
presumptive precedence of your Bach Temperament or some other no-
brainer temperament above it based on what proved to be spurious
illations.

Before we come to all that, let me shed light on my calculations on
your recipe step by step:

1. Tune tenor A to 220 Hz. Ok, no-brainer.

2. Tenor F down from this A beating thrice per second, a wide major
third.

(220 * 4 - 3) / 5 = 175.4 Hz for tenor F.

yielding exactly the interval 1100/877 at 392.226 cents

3. Temporary tenor C a pure fourth down therefrom. No-brainer:

(175.4 * (3/4)) = 131.55 Hz for tenor C. Ratio from A 1/1 is 2631/2200.

4. Temporary tenor D a pure fifth down from tenor A. No-brainer again:

(220 * 2/3) = 146 2/3 Hz for tenor D. Ratio from A 1/1 is 4/3.

5. Tenor G will be acquired in such a fashion that D-G will beat 1.5
times as fast as C-G as you say. Tricky math, but easy with the formula:

1.5 * (3x - 4D) = -(2x - 3C)

1.5 * (3x - (4*(146+(2/3)))) = -(2x - (3*131.55))

1.5 * (3x - (1760/3)) = 394.65 - 2x

Variable x is calculated as 196.1 Hz for G.

1.63333333333333 beats per second D-G VS -2.45 beats per second C-G.
Exactly 3/2 times absolutely.

And, surprise surprise, what do we find as the ratio? Why, that makes
a 1961/1100 from 1/1 A, the ratio for G you wondered where was comingfrom.

6. Middle C as the result of G-C beating 1.5 times as fast as F-C as
aforetime:

1.5 * (3x - 4G) = -(2x - 3F)

1.5 * (3x - (4*196.1)) = -(2x - (3*175.4))

1.5 * (3x - 784.4) = 526.2 - 2x

Variable x is calculated as 261.9692307692308 Hz for C.

1.5076923076924 beats per second G-C VS -2.2615384615384 beats per
second F-C. Exactly 3/2 times absolutely.

And, what do we get? Why, could it be middle C at its proper spot
with the ratio 877/550 * 2619692307692308/3508000000000000 =
654923076923077/550000000000000 from A 1/1? Which is simplified to
256860830769231/215710000000000 (302.273 cents) with the sole
difference being the raising of the last digit in the last decimal of
the relative frequency by 1? (1.190769230769231 VERSUS
1.190769230769232)

Figures.

7. Next order of business, finding D in the same fashion from G and A.
Again our number-cruncher equation:

1.5 * (3x - 4A) = -(2x - 3G)

1.5 * (3x - (4*220)) = -(2x - (3*196.1))

1.5 * (3x - 880) = 588.3 - 2x

Variable x is calculated as 293.5846153846154 Hz for D.

0.7538461538462 beats per second A-D VS -1.1307692307692 beats per
second G-D, once more exactly 3/2 times absolutely.

The ratio of D being 1961/1100 * 2935846153846154/3922000000000000 =
1467923076923077/1100000000000000 from A 1/1, yielding 499.53 cents.
Ok, I admit I made the horrible mistake of 1.8 cents by saying it was78440000000000/58716923076923 that yields 501.37580. A slip from
losing my sanity after all that number-crunching. Even worse for you,
because the previous value was almost identical with your calculated
interval on paper. So, the 700.5 cent fifth moved from G-D to D-A in
the cycle. "Big fat hairy deal". You still don't get a 1/6 PC tempered
fifth where you seek.

8. The most complicated part comes now. Complicated because it's
impossible to ascertain what piece of information to rely on to set
the "golden mean" (another monicker for brats you shun?) for WTC 1 C
Major Prelude, since taking one refutes the other. We want E to be
lowered so that A-E becomes the "same quality" as the previous fifths
we worked out so far. What to do, what to do? You specifically
instruct us that C-E is to beat 4.5 times per second wide. What do we
get?

(261.9692307692308 x 5 + 4.5) / 4 = 328.5865384615385 Hz for tenor E.

What of the A-E fifth? Ay Ay ay, 694.524 cents. Surely that can't be
right.

Let's then take the other path: A-E fifth beating exactly 1/2 that of
F-A, sharing the same A. Easy as pie. That means 1.5 times per second.
Hence the formula:

2x - 660 = 1.5

x = 330.75 Hz.

Ah, an even worse A-E fifth at 705.885 cents. No can do!

Let's try another path: You say A-E beats 1.5 times the D-A fifth in
the tenor. Hence the equation:

(2x - (220 * 3)) = 1.5*((220*2) - (146.7923076923077 * 3))

= 329.7173076923077 Hz.

Close, but no cigar. 700.471 cents between A-E this time.

Let's try this: C-E 1.5 times the beat of F-A in the tenor.

(4x - (261.9692307692308 * 5)) = 4.5

Yielding 328.5865384615385 Hz, and a glaring 694.5 cent fifth again!

Nothing works, so I have to figure out how to comply with this demand
of yours: "fuss with both these E's until all four of the following
points are true", a task, which is impossible from where we stand. Not
even taking 1100/877 over to C in compliance with "These two major
thirds (F-A, C-E) should have exactly the same character as one
another, although the beat rate is different" solves the 694.5 cent
narrow fifth problem on A-E. All that remains is trying to keep A-E
"same quality as these other fifths", D-A, F-C, G-D. The best I can
do, is to keep A-E the same size as G-D and C-G. I chose -1.5 beats
per second for the job. Criticize me all you like! The fifth size in
the end is what you desire: 698.016 cents versus your 698.045 cents.
Thus the ratio for E becomes 1317/880 (You ought to thank me for
saving your neck right at this step!).

9. Tuning all fifths pure from E up to C# and from F up to Eb. Piece
of cake. Now we are calibrating G# from pure to low, so that C#-G# in
the middle octave has the "same quality" as Ab-Eb a perfect fourth
below. Tricky! Since you want to have a very slow beat from each. What
the heck, it has been shown that the difference in between
distributing the beating uniformly between them matters only as much
as 0.36 cents. But just to humour you, let's consider that C#-G# is to
beat 1.5 times that of Ab-Eb. You said the latter was up a fifth. What
was the original equation?

1.5 * (4Eb - 3x) = (2x - 3C#)

1.5 * (1,247.2888888 - 3x) = (2x - 833.4140625)

And what was the differences from my uniform beating scheme? Your
implied C#-G# at 699.238 cents VS my C#-G# at 699.6 cents & your
implied Ab-Eb 700.746 cents VS my Ab-Eb at 700.383 cents!

The ratio you imply for G# is 3508/2475 *
4160534454923077/3118222222222222 =
663416130357734278/350799999999999975 at 1103.118 cents compared to my
1103.481 cents, boiling down to a 0.36 cents difference!

Finally, let's take the case of Eb-Bb, where Bb is to be lowered
slightly to achieve the "same quality" as the two previous fifths. Do
you have any proposition where it will result in drastic modifications
compared to my 699.857 cent fifth between Eb-Bb?

And you still did not reveal how 0.36 cents is important to you!

______________________________________________

Here is the grand total result of your recipe with that 1.8 cent error
at D "corrected":

0: 1/1 C Dbb unison, perfect prime
1: 101.608 cents C# Db
2: 197.255 cents D Ebb
3: 301.592 cents D# Eb
4: 395.743 cents E Fb
5: 505.502 cents F Gbb
6: 599.653 cents F# Gb
7: 698.631 cents G Abb
8: 801.208 cents G# Ab
9: 897.727 cents A Bbb
10: 1001.449 cents A# Bb
11: 1097.698 cents B Cb
12: 1200.000 cents C Dbb

Cycle of fifths:

|
0: 0.000 cents 0.000 0 0
commas C
7: 698.631 cents -3.324
-102 G
2: 698.624 cents -6.655
-204 D
9: 700.473 cents -8.138 -250 A
4: 698.016 cents -12.077-371 E
11: 701.955 cents -12.077
-371 B
6: 701.955 cents -12.077
-371 F#
1: 701.955 cents -12.077
-371 C#
8: 699.600 cents -14.432
-443 G#
3: 700.383 cents -16.003
-491 Eb
10: 699.857 cents -18.101
-556 Bb
5: 704.053 cents -16.003
-491 F
12: 694.498 cents -23.460 -720 -1 Pyth.
commas C
Average absolute difference: 12.8686 cents
Root mean square difference: 14.4709 cents
Maximum absolute difference: 23.4600 cents
Maximum formal fifth difference: 7.4567 cents

Errors from the values of your temperament consigned to paper:

1: 1: -3.563 cents -3.563280 0.5704 Hertz, 34.2269
cycles/min.
2: 2: -1.165 cents -1.164790 0.1972 Hertz, 11.8321
cycles/min.
3: 3: -3.547 cents -3.546650 0.6373 Hertz, 38.2390
cycles/min.
4: 4: -3.563 cents -3.563280 0.6761 Hertz, 40.5653
cycles/min.
5: 5: -3.547 cents -3.546650 0.7170 Hertz, 43.0188
cycles/min.
6: 6: -3.563 cents -3.563280 0.7606 Hertz, 45.6359
cycles/min.
7: 7: -0.586 cents -0.585590 0.1325 Hertz, 7.9479
cycles/min.
8: 8: -3.163 cents -3.163370 0.7587 Hertz, 45.5217
cycles/min.
9: 9: -3.592 cents -3.592380 0.9109 Hertz, 54.6525
cycles/min.
10: 10: -3.404 cents -3.403790 0.9164 Hertz, 54.9836
cycles/min.
11: 11: -3.563 cents -3.563280 1.0141 Hertz, 60.8479
cycles/min.
12: 12: 1/1 0.000000 0.0000 Hertz, 0.0000
cycles/min.
Mode: 1 1 1 1 1 1 1 1 1 1 1 1 Twelve-tone Chromatic
Total absolute difference : 33.2563 cents
Average absolute difference: 2.7714 cents
Root mean square difference: 3.0570 cents
Highest absolute difference: 3.5924 cents
Number of notes different: 11

And you still have to account for that 694.5 cent fifth at F-C all
while you were so haughtily bombarding UWT nr. 3 fifths that you
apparently mistuned on your harpsichord.

_________________________________________________________________

First you claimed that your temperament has regular fifths, and UWT nr.
3 does not, when it became clear that you cannot produce a recipe that
yields the correct fifths you obtained on paper exactly everywhere,
and that that set of instructions is, in your own words, "the best
accuracy that anyone can reasonably hope for when working directly at
a real harpsichord". I have naturally come to the understanding that
this is the best you can do to implement a semblance of your Bach
tuning on a real harpscihord by ear, which is a DISASTER really. All
pitches save G and D are 3.5 cents down on the average! Have you not
yourself admitted: "And, all these "procured" recipes on my http://www.larips.com
site are *by* me, describing the way I set all those various
temperaments by ear."? The recipe for Lehman-Bach Temperament does not
hold as you claim if you actually follow the steps you use. I stand
ready to be corrected if you have some other method of setting thosefifths tempered exactly by said geometrical fractions of a rational
commatic interval to the nth precision you demand that I deliver in my
calculations.

You alleged that your temperament is easier to set up than mine simply
because you say your methods are "analog" versus my "digital". It has
been shown that you yourself rely upon counting beats, listening to
beats and comparing beats in the only recipe you have for setting your
tuning on an acoustical instrument. You too take the 440 Hz=A
diapason, you too count the seconds (rounded down or not), you
moreover rationalize the beat count of one interval to another, etc...
That's all very well and "analog" while my instructions for tuning by
counting the beats of fifths here and there and cross-checking with
the beat count of major thirds in seconds (or some other unit)
"digital"?

You always say things about where intervals are ought to be, and how
to take the square roots and geometric fractions (as if your wrist
commands the tuning wrench by them!) but you failed with your recipe
and refuse to recognize the end results. Your failure to set up UWT nr.
3 on an harpsichord using instructions that promise to EXACTLY match
the end result is a catastrophic development that I fear incapacitates
your criticisms and evaluations of UWT nr.3 and the nature of
proportional beat ratios.

Now is it me who: "accidentally found ways to diverge from the
instructions, while trying to test those instructions"? Check the
math, do stop making groundless assumptions and please desist from
throwing about broad comments on proportional beat rates that
apparently have not yet presented themselves to you on your
harpsichord my dear colleague.

I implore you to utilize some electronic aid to check your results
next time so that it is not just some fancy tuning you are evaluating
that does not represent the descriptions!

Cordially,
Dr. Oz.

✩ ✩ ✩
www.ozanyarman.com

On May 22, 2010, at 5:19 PM, bplehman27 wrote:

>
>
>
>
> --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>
>> In your recipe, taking as reference the 440 Hz diapason for A, I read
>> tenor A to middle E slightly flat in the same quality as those other
>> fifths F-C and G-D. You have given instructions to compare beat
>> ratios
>> against temporary notes and that is how the calculations are done to
>> find G, C and D. You have specifically written that middle C-E is to
>> beat 4.5 times per second. Do you not understand the implications of
>> what you have written? The fifths that one arrives at per your
>> instructions are:
>>
>> F-C = 694.498 cents
>>
>> G-D = 700.473 cents
>>
>> A-E = 698.016 cents.
>>
>> What room is there for complaint here? Apparently you are doing
>> something wrong, unless I made an error somewhere, and it appears I
>> have not.
>
> Hi Oz, I've had opportunity this morning to take a closer look in
> the spreadsheet you sent me yesterday, plus the e-mailed notes about
> how you derived things.
>
> First, I'll humor your use of F-A of 877/1100, because even with the
> use of that approximation, you ought to get better results than you
> actually did.... I still can't see all the processes you did,
> because you didn't show them, but here's how I would do it:
>
> Temporary C a pure 4th below that F: 877/1100 x 3/4 = 2631/4400
>
> Temporary D a pure 5th below the A: 2/3
>
> Next step is to find the G that's a 4th above that D, and a 5th
> above that C, such that it's at the GEOMETRIC mean, and such that
> the D-G 4th beats triplets against the C-G 5th's duplets.
>
> The pure G from that D would be 2/3 x 4/3 = 8/9
>
> The pure G from that C would be 2631/4400 x 3/2 = 7893/8800
>
> The mean spot between those two phantom G candidates, geometrically,
> is sqrt(8/9 x 7893/8800) = .8929012976+
>
> That's the G we get by following my instructions, tuning by ear and
> seeking the spot where we get triplets beating against duplets from
> the two temporary notes.
>
> The G *you* got (somehow??) was "1961/1100", which when adjusted to
> the proper octave at 1961/2200 is .8913636363+
>
> Your notes don't say anything about moving the C (per my
> instructions) to a proper spot, after using the temporary one.
> Again, I can't see how you derived C in its final position with your
> ratios, but clearly you've ended up with C at a wrong spot vis-a-vis
> A. Yours is at 302.27 cents, and it ought to be 305.87; you're off
> by more than 3 cents!
>
> From your wrong G, you somehow came up with a D that you say is
> "calculated to be 78440000000000/58716923076923". I can't follow
> any of your arithmetic here, because you didn't show it, but clearly
> you're already accumulating so many errors recursively that it's
> pointless to continue. Your D ends up at 501.3758 cents from A,
> somehow, while it ought to be at 501.96. That's not as bad as the
> errors in the C and the G, but anyway, it's off. They're all off.
> Somehow, you accidentally found ways to diverge from the
> instructions, while trying to test those instructions.
>
> More importantly, I can't see that you have used square roots
> anywhere, which is the proper procedure to place a note at a mean
> point relative to two other notes. Without square roots, and
> therefore some irrational numbers occasionally, you're not going to
> get there.
>
> My "triplets against duplets" beating procedure, comparing
> successive 5ths, works whenever we're making 5ths the same size
> GEOMETRICALLY. All the 5ths have some constant amount taken off
> them (which happens to be a fraction of the PC in mine, but it could
> be any constant), and the frequencies of both notes are 3/2 higher
> each time we move up(**), so the beat rates also increase as 3/2,
> "triplets against duplets". If we are looking at (for example) the
> open-string notes of violins and violas, and we have regularly
> tempered 5ths everywhere, and the A-E is beating 3/sec, the D-A will
> be 2/sec, the G-D is 4/3 per sec, and the C-G is 8/9 per sec.
>
> (**) "3/2 higher" shortened by some small tempered amount, but close
> enough to 3/2 for all practical purposes on acoustic instruments,
> when making this comparison....
>
> To calculate all those notes with their REGULAR GEOMETRIC tempering,
> if they actually have to be calculated (as opposed to tuning them
> directly by ear, which is much simpler!), we have to use square
> roots....not ratios.
>
> The MUSICAL skill here, as opposed to the mathematical skill, is to
> recognize all those identically-tempered 5ths by their quality,
> directly: not having to count the beats that prove they're spot-on.
> With practice (and I've had 27 years of doing this), it's easy to
> get two neighboring 5ths set up with similar quality, without any
> conscious counting of beats. And so, as I said several days ago,
> that's how I tune: and that's how I write tuning instructions for
> readers who have that similarly practiced musical skill. That
> "triplets against duplets" check is a useful test for the similarity
> of the intervals, but the similarity is already directly perceptible
> (close enough) without it.
>
> People who prefer mathematically-based instructions based on beat-
> counting are free to formulate their own sequence of setting the
> notes. All the correct values, to two decimal places, are shown at
> my page:
> http://www-personal.umich.edu/~bpl/larips/math.html
>
>
> Brad Lehman
>

🔗Ozan Yarman <ozanyarman@...>

🔗genewardsmith <genewardsmith@...>

🔗Mike Battaglia <battaglia01@...>

🔗martinsj013 <martinsj@...>

🔗Ozan Yarman <ozanyarman@...>

Yes, my mistake. The results are modified thus:

G= (3x - (4*(146+(2/3)))) = 1.5 * -(2x - (3*131.55))

G= 196.44 + 25/90000 (the octave is 392.88055555555556 Hz)

C= (3x - (4*196.4402777777778)) = 1.5 * -(2x - (3*175.4))

C= 262.5101851851852 Hz

D= (3x - (4*220)) = 1.5 * -(2x - (3*196.4402777777778))

D= 293.996875 Hz.

E try1= (293.996875 x 5 + 4.5) / 4 = 329.2627314814815 Hz (A-E is
698.083 cents)

E try2= (2x - (220 * 3)) = 1.5*((220*2) - (146.9984375 * 3)) =
329.253515625 Hz (A-E is 698.0344 cents, good enough!)

B= 329.253515625 * (3/4) = 246.94013671875 Hz

F#= 246.94013671875 * (3/2) = 370.410205078125 Hz

C#= 370.410205078125 * (3/4) = 277.80765380859375 Hz

temp. Bb= 350.8 * (4/3) = 467.7 + 1/30 Hz.

Eb= (467.7 + 1/30) * (2/3) = 311.8 + 2/90 Hz.

G#= ((4*(311.8 + (2/90))) - 3x) = 1.5 * (2x - (3 *
277.80765380859375)) = 416.2372218379268 Hz

Bb= (4x - (3*(311.8 + (2/90)))) = ((2 * (311.8 + 2/90)) - (3*
208.1186109189634)) = 233.5998960494995 Hz.

F was = 350.8 Hz (beating six times wide against A=440 Hz.)

All the frequencies from tenor A to A =

220.0
233.5998960494995
246.94013671875
262.5101851851852
277.80765380859375
293.996875
311.822222222222222
329.253515625
350.8
370.410205078125
392.88055555555556
416.2372218379268
440.0

Cents from C:

0: 1/1 C Dbb unison, perfect prime
1: 98.056 cents C# Db
2: 196.113 cents D Ebb
3: 298.020 cents D# Eb
4: 392.191 cents E Fb
5: 501.930 cents F Gbb
6: 596.101 cents F# Gb
7: 698.061 cents G Abb
8: 798.039 cents G# Ab
9: 894.156 cents A Bbb
10: 997.999 cents A# Bb
11: 1094.146 cents B Cb
12: 1200.000 cents C Dbb

Cycle:

0: 0.000 cents 0.000 0 0
commas C
7: 698.061 cents -3.894
-120 G
2: 698.052 cents -7.797
-239 D
9: 698.043 cents -11.709
-359 A
4: 698.034 cents -15.629
-480 E
11: 701.955 cents -15.629
-480 B
6: 701.955 cents -15.629
-480 F#
1: 701.955 cents -15.629
-480 C#
8: 699.984 cents -17.601
-540 G#
3: 699.981 cents -19.575
-601 Eb
10: 699.979 cents -21.551
-661 Bb
5: 703.931 cents -19.575
-601 F
12: 698.070 cents -23.460 -720 -1 Pyth.commas C
Average absolute difference: 15.6399 cents
Root mean square difference: 17.2708 cents
Maximum absolute difference: 23.4600 cents
Maximum formal fifth difference: 3.9206 cents

My apologies to my colleague Brad for this glaring slip. His
calculations are very much right and I was wrong.

Still, Bradley should labour to set up UWT nr.3 rightly on his
harpsichord according to my instructions if there is no error with the
recipe that I gave, which I will check again.

Cordially,
Oz.

✩ ✩ ✩
www.ozanyarman.com

On May 23, 2010, at 11:17 AM, martinsj013 wrote:

> --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>> My dear colleague Dr. Lehman,
>> ...
>> 5. Tenor G will be acquired in such a fashion that D-G will beat 1.5
>> times as fast as C-G as you say. Tricky math, but easy with the
>> formula:
>> 1.5 * (3x - 4D) = -(2x - 3C)
>> 1.5 * (3x - (4*(146+(2/3)))) = -(2x - (3*131.55))
>> 1.5 * (3x - (1760/3)) = 394.65 - 2x
>> Variable x is calculated as 196.1 Hz for G.
>> 1.63333333333333 beats per second D-G VS -2.45 beats per second C-G.
>> Exactly 3/2 times absolutely.
>
> Hi Oz,
> I think it is supposed to be the other way around - D-G beats faster
> than C-G - that is what I did (I mean on paper of course!) and I get
> G 196.44 Hz, D-G beats at 2.65 (i.e. wide) and C-G at -1.77 (i.e.
> narrow). If correct, this will have knock-on effects for the rest
> of your calculation.
>
> Kind Regards,
> Steve M.
>
>> Cordially,
>> Dr. Oz.
>
>
>
>
>
> ------------------------------------
>
> You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
> tuning-subscribe@yahoogroups.com - join the tuning group.
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>
>
>

🔗Ozan Yarman <ozanyarman@...>

🔗bplehman27 <bpl@...>

🔗Ozan Yarman <ozanyarman@...>

Yes O Magister, I bow down humbly to your superior knowledge and
experience and avow to better my presumptuous foolhardy ways.

Me clumsy silly absent-minded "mathematician" appear to have made the
horrible error of swapping the beat number with the second numbers in
my temperament recipe in a moment of carelessness! AIE. Ok, I will
curl up and die now.

Here is the corrected sequence:

______________________________________

Dear Brad, here is a recipe for UWT nr.3:

Make sure you memorize the ticking rhythm of a metronome at 60 beats
per minute. Take a tuning fork or tuner device to calibrate A4 to 440
Hz.

Then, tune the fifth D-A to yield a narrow fifth beating 2 times per
sec.

Next, G'-D (G in the lower octave) again to a narrow fifth 2 times per
sec.

Then, C'-G' (still in the lower octave) to 2 beats per 3 seconds.

Find the pure octaves of C' and G', which are C, c, and G.

Also tune D' from D.

From C, go down a fifth to F', again 2 beats per 3 seconds.

Find the pure octave of F', which is F.

From F, down to Bb', yielding once again 2 beats per 3 seconds.

Check Bb'-D, should make 6 beats per second. If it's hard to count
this way, go an octave down from Bb' and count 3 beats per second
against D.

Tune the octave of Bb' which is Bb.

___________________________

Stop right there. Back to A.

Go down an octave, find A'.

A'-E is pure

E-B is a narrow fifth beating 2 times per sec.

B-f# (upper octave) is a narrow fifth beating once per sec.

Find the octave of f#, which is F#.

Check D' against F# (or D against f#). Should beat 5 times per second.

F#-c# is a narrow fifth beating 2 times every 3 seconds.

Find the octave of c#, which is C#.

C#-G# is pure.

Go down an octave from G# to G#'

G#'-Eb is pure too.

Lastly, Eb-Bb should sound beatless and pure.

________________________

My deepest humblest apologies my dear colleague, for being so blinded
with my excessive trust in numbers and having lost track of the
importance of what they are meant to serve. I hope that you will be
magnanimous and forgive my scatter-brained mistakes, and even perhaps
devote yet again a portion of your precious time to see, if this time,
the corrected recipe produces a change in the beat patterns.

Most cordially,
Dr. Oz.

✩ ✩ ✩
www.ozanyarman.com

On May 23, 2010, at 3:27 PM, bplehman27 wrote:

>
>
> --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>>
>> Oh, and thank you Steve, for pointing that glaring error out! Much
>> obliged.
>>
>> To err is human, I suppose.
>
> So, Oz, are we all settled, now? That I'm not ludicrously off-base,
> as you spent several days alleging vehemently?
>
> I did your "UWT #3" again on the harpsichord yesterday, this time > taking your E-B of 2/sec in a different octave, and it worked out a
> little better (at least in that one interval). I spent almost an
> hour playing Couperin and some other Bach with it: a handful of
> movements from the French Suites. As I said before, it works
> decently, but I still don't like the way it gives a too-high E# and
> A# for their musical contexts. It's nearly-equal enough that
> nothing is terrible, and the enharmonic swaps work OK except for
> maybe the E#/F. The E# came up quite a bit in the Couperin Ordre
> that's in A major and A minor. I still don't hear anything either
> perceptible or musically meaningful in the putative "proportional
> beating", even though I know cognitively that it's there (because
> the recipe put it there). As I said before, Your Mileage May Vary.
>
> Brad Lehman
>
>

🔗a_sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:

> The nature of the instructions you have given yield the ratios I
> provided, not these in your worksheet which you take as reference
> for your instructions:
>
> C 1/1
> C# 8926735/8435236
> D 2907629/2596260
> D# 8495647/7152031
> E 6837827/5451757
> F 12177389/9112438
> F# 8926735/6326427
> G 3285409/2195225
> G# 4613353/2909514
> A 2299792/1372105
> A# 1419143/797367
> B 15193459/8075767
> C 2/1
>
Many thanks Oz,
for clearing what Brad's instruction's really do mean:

!Lehman_Yarman.scl
Bradley Lehman's ratios, compiled from 'instructions' by Ozan Yarman
12
!
8926735/8435236 ! 1200*ln(8926735/8435236)/ln(2) = ~98.0449991...C
2907629/2596260 ! 1200*ln(2907629/2596260)/ln(2) = ~196.089998...C
8495647/7152031 ! 1200*ln(8495647/7152031)/ln(2) = ~298.044999...C
6837827/5451757 ! 1200*ln(6837827/5451757)/ln(2) = ~392.179997...C
12177389/9112438 ! 1200*ln(12177389/9112438)/ln(2)= ~501.955001...C
8926735/6326427 ! 1200*ln(8926735/6326427)/ln(2) = ~596.089998...C
3285409/2195225 ! 1200*ln(3285409/2195225)/ln(2) = ~698.044999...C
4613353/2909514 ! 1200*ln(4613353/2909514)/ln(2) = ~798.044999...C
2299792/1372105 ! 1200*ln(2299792/1372105)/ln(2) = ~894.134997...C
1419143/797367 ! 1200*ln(1419143/797367 )/ln(2) = ~998.044999...C
15193459/8075767 ! 1200*ln(15193459/8075767)/ln(2)=~1094.135......C
2/1
!
![eof]

Just remember
http://www-personal.umich.edu/~bpl/larips/cpeb.html
Quote:
"
According to his son C.P.E. Bach, J.S. Bach always insisted on tuning his instruments himself, and it never took him more than 15 minutes.

As CPE wrote to Forkel for the biography of JSB,
"The exact tuning of his instruments as well as of the whole orchestra had his greatest attention. No one could tune and quill his instruments to please him. He did everything himself." (1774, New Bach Reader, #394)

Forkel then presented it thus:
"Nobody could install the quill-plectrums of his harpsichord to his satisfaction; he always did it himself. He also tuned both his harpsichord and his clavichord himself, and was so practised in the operation that it never cost him above a quarter of an hour. But then, when he played from his fancy, all the 24 keys were in his power; he did with them what he pleased. He connected the most remote as easily and as naturally together as the nearest; the hearer believed he had only modulated within the compass of a single key. He knew nothing of harshness in modulation; even his transitions in the chromatic style were as soft and flowing as if he had wholly confined himself to the diatonic scale. His Chromatic Fantasy, which is now published, may prove what I here state. All his extempore fantasies are said to have been of a similar description, but frequently even much more free, brilliant, and expressive." (1802, New Bach Reader, p436. English translation by Kollmann, 1820.)
"

It is well known, that Bach loved jokes about "number-symbolism"
http://www.bach-cantatas.com/Topics/Numbers.htm
Quote:
"
In the Symbolum Nicenum of the B-minor Mass (BWV 232), consider the two movements, "Credo in unum Deum" and "Patrem omnipotentem"
The total number of measures/bars for these two movements: 129
Apply gematria: C(3)+R(7)+E(5)+D(4)+O(14) = 43
The number 129 comes from (43 + 43 + 43) X 3 3 = Triune God
The word 'Credo' is sung 7 X 7 = 49 times..."

Appearently an (more or less) intended reference to Werckmeister's
initial monochord stringlength 49:= 7*7 in his:

http://en.wikipedia.org/wiki/14_(number)

"A number 'encoded' in much of the music of Johann Sebastian Bach.[citation needed] Bach may have considered this number a sort of signature, since given A = 1, B = 2, C = 3, etc., then B + A + C + H = 14. (See also 41)"

http://www2.nau.edu/tas3/wtc/ii14s.pdf
http://jan.ucc.nau.edu/tas3/wtc/i04s.pdf

http://www.ntnu.no/gemini/2000-06e/32-34.htm

http://books.google.de/books?id=xdKJl5Ja7igC&pg=PA196&lpg=PA196&dq=bach++number-symbolism+14+41&source=bl&ots=hyD_SoXeIC&sig=W5zt_MFAIidqZDwThOQX_iENzRg&hl=de&ei=rRH8S524Hc6TOKXb9PYB&sa=X&oi=book_result&ct=result&resnum=6&ved=0CDoQ6AEwBQ#v=onepage&q=bach%20%20number-symbolism%2014%2041&f=false

Especially for Brad:
http://www.harpsichord.org.uk/EH/Vol2/No2/bachmath.pdf

http://www.flagmusic.com/work.php?r=BWV_1087

http://www.glenwilson.eu/article1.html
"
The gematric explanation for the surpising length of this excursion would be that it contains 41 notes, a number also often thought to be a signature (J9+ S18 +BACH14), and the reverse of 14. [22].
"

http://www.vanrecital.com/events/concert_notes.cfm?concertid=220&type=program

http://www.biography.com/articles/Johann-Sebastian-Bach-9194289&part=9?print
"
Symbolism

A repertoire of melody types existed, for example, that was generated by an explicit âdoctrine of figuresâ that created musical equivalents for the figures of speech in the art of rhetoric. Closely related to these âfiguresâ are such examples of pictorial symbolism in which the composer writes, say, a rising scale to match words that speak of rising from the dead or a descending chromatic scale (depicting a howl of pain) to sorrowful words. Pictorial symbolism of this kind occurs only in connection with wordsâ"in vocal music and in chorale preludes, where the words of the chorale are in the listener's mind. There is no point in looking for resurrection motifs in The Well-Tempered Clavier. Pictorialism, even when not codified into a doctrine, seems to be a fundamental musical instinct and essentially an expressive device. It can, however, become more abstract, as in the case of number symbolism, a phenomenon observed too often in the works of Bach to be dismissed out of hand.

Number symbolism is sometimes pictorial; in the St. Matthew Passion it is reasonable that the question âLord, is it I?â should be asked 11 times, once by each of the faithful disciples. But the deliberate search for such symbolism in Bach's music can be taken too far. Almost any number may be called âsymbolicâ (3, 6, 7, 10, 11, 12, 14, and 41 are only a few examples); any multiple of such a number is itself symbolic; and the number of sharps in a key signature, notes in a melody, measures in a piece, and so on may all be considered significant. As a result, it is easy to find symbolic numbers anywhere, but ridiculous to suppose that such discoveries invariably have a meaning.
"

But nevertheless, even when takeing JSB's
funny jokes about number-symbolism apart aside:

Quest:
However, in heaven, whoever is able to explain,
why Bach should had chosen just the above given
compliecated ratios, not mention of tuning them
properly within
"less than an quarter hour" on his harpsichord?

The above allegation about Bach's 'authentic-tuning' rings hollow!
And I do agree with Lindley & Ortgies,
that such broade noncredible claims have to be;
condemmend as untrustworthy.
http://sites.google.com/site/iboortgies/errataandcorrigendatolindleyortgies%3A%22bac
"
In it we discuss some of the historically wild and methodologically wrong speculations published last year (2005) by Bradley Lehman in his article in Early Music (Note 1), where he claimed to have discovered "Bach's temperament.â
The Lehman temperament is of modern design. It is, like the one designed by the late Herbert Anton Kellner...."

http://em.oxfordjournals.org/cgi/content/full/34/4/613
Quote:
"A significant degree of aesthetic and historical consensus is indicated when the wife of a man who is passionately committed to a certain aesthetic point agrees with him.
Caveat lector." [Footnote 19]
[19] Dr Lehman sets out additional arguments (of similar value) in his EM article and at www.larips.com and other websites....

So far about the specific opinion
of two leading scholars, that are
well-reckognized experts in the
topic about Baroque tunings.

But at least, however, one can say:
that nobody else had previously ever tried before
to foist upon J.S.Bach such large number-proportions as
"intended-original-tuning"

I.m.h.o: Outrageous hair-raising!

Yours-Sincerely
A.S.

🔗George Sanders <georgesanders11111@...>

Andreas,

Before everyone starts going off the deep end with Bach and number symbolism, I ould advocate a look at Ruth Tatlow's

Bach and the Riddle of the Number
Alphabet (

David Yearsley's Bach and the Meanings of Counterpoint

also has some sane reflections on the issue). There's a tradition in Germany, much of which stems from one of the main mid-20th century editors of the Bach complete edition, that assumes that one can count the number of notes in any Bach theme and derive a theological message from it, that every instrumental work has a theological message, and so forth. It's very difficult to challenge these assertions, because you basically have to have an encyclopedic knowledge of Bach's works to do so. Tatlow is probably the best de-mythologizer of all of these tendencies. She doesn't deny that certain numbers keep coming up, but she doubts the significance of the number alphabet that's commonly claimed to be the one Bach is using (owing to its anachronism) and doubts claims about mystical proportioning, etc. (Golden Section, etc.; again, this is anachronistic). She believes the mysteries of recurring numbers have a more prosaic origin: Bach needed to
calculate approximate lengths of movements in order to fit pieces into the intended time slot in the service; for the instrumental pieces, she thinks he aimed for whole-number proportions in the "good construction" tradition of Quintillian, I believe (I saw a lecture she gave at the national Society of Music Theory conference two years ago).

I found a fascinating article of hers online a couple of years ago that might still be available: The Use and Abuse of Fibonacci Numbers and the Golden Section in Musicology Today.

best

Franklin

--- On Tue, 5/25/10, a_sparschuh <a_sparschuh@...> wrote:

From: a_sparschuh <a_sparschuh@...>
Subject: [tuning] The real Lehman tuning, Re: Here is the Excel worksheet I have worked with
To: tuning@yahoogroups.com
Date: Tuesday, May 25, 2010, 7:16 PM

--- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:

> The nature of the instructions you have given yield the ratios I

> provided, not these in your worksheet which you take as reference

> for your instructions:

> C 1/1

> C# 8926735/8435236

> D 2907629/2596260

> D# 8495647/7152031

> E 6837827/5451757

> F 12177389/9112438

> F# 8926735/6326427

> G 3285409/2195225

> G# 4613353/2909514

> A 2299792/1372105

> A# 1419143/797367

> B 15193459/8075767

> C 2/1

Many thanks Oz,

for clearing what Brad's instruction's really do mean:

!Lehman_Yarman.scl

Bradley Lehman's ratios, compiled from 'instructions' by Ozan Yarman

8926735/8435236 ! 1200*ln(8926735/8435236)/ln(2) = ~98.0449991...C

2907629/2596260 ! 1200*ln(2907629/2596260)/ln(2) = ~196.089998...C

8495647/7152031 ! 1200*ln(8495647/7152031)/ln(2) = ~298.044999...C

6837827/5451757 ! 1200*ln(6837827/5451757)/ln(2) = ~392.179997...C

12177389/9112438 ! 1200*ln(12177389/9112438)/ln(2)= ~501.955001...C

8926735/6326427 ! 1200*ln(8926735/6326427)/ln(2) = ~596.089998...C

3285409/2195225 ! 1200*ln(3285409/2195225)/ln(2) = ~698.044999...C

4613353/2909514 ! 1200*ln(4613353/2909514)/ln(2) = ~798.044999...C

2299792/1372105 ! 1200*ln(2299792/1372105)/ln(2) = ~894.134997...C

1419143/797367 ! 1200*ln(1419143/797367 )/ln(2) = ~998.044999...C

15193459/8075767 ! 1200*ln(15193459/8075767)/ln(2)=~1094.135......C

2/1

![eof]

Just remember

http://www-personal.umich.edu/~bpl/larips/cpeb.html

Quote:

According to his son C.P.E. Bach, J.S. Bach always insisted on tuning his instruments himself, and it never took him more than 15 minutes.

As CPE wrote to Forkel for the biography of JSB,

"The exact tuning of his instruments as well as of the whole orchestra had his greatest attention. No one could tune and quill his instruments to please him. He did everything himself." (1774, New Bach Reader, #394)

Forkel then presented it thus:

"Nobody could install the quill-plectrums of his harpsichord to his satisfaction; he always did it himself. He also tuned both his harpsichord and his clavichord himself, and was so practised in the operation that it never cost him above a quarter of an hour. But then, when he played from his fancy, all the 24 keys were in his power; he did with them what he pleased. He connected the most remote as easily and as naturally together as the nearest; the hearer believed he had only modulated within the compass of a single key. He knew nothing of harshness in modulation; even his transitions in the chromatic style were as soft and flowing as if he had wholly confined himself to the diatonic scale. His Chromatic Fantasy, which is now published, may prove what I here state. All his extempore fantasies are said to have been of a similar description, but frequently even much more free, brilliant, and expressive." (1802, New Bach Reader, p436. English translation
by Kollmann, 1820.)

It is well known, that Bach loved jokes about "number-symbolism"

http://www.bach-cantatas.com/Topics/Numbers.htm

Quote:

In the Symbolum Nicenum of the B-minor Mass (BWV 232), consider the two movements, "Credo in unum Deum" and "Patrem omnipotentem"

The total number of measures/bars for these two movements: 129

Apply gematria: C(3)+R(7)+E(5)+D(4)+O(14) = 43

The number 129 comes from (43 + 43 + 43) X 3 3 = Triune God

The word 'Credo' is sung 7 X 7 = 49 times..."

Appearently an (more or less) intended reference to Werckmeister's

initial monochord stringlength 49:= 7*7 in his:

http://en.wikipedia.org/wiki/14_(number)

http://www2.nau.edu/tas3/wtc/ii14s.pdf

http://jan.ucc.nau.edu/tas3/wtc/i04s.pdf

http://www.ntnu.no/gemini/2000-06e/32-34.htm

Especially for Brad:

http://www.harpsichord.org.uk/EH/Vol2/No2/bachmath.pdf

http://www.flagmusic.com/work.php?r=BWV_1087

http://www.glenwilson.eu/article1.html

The gematric explanation for the surpising length of this excursion would be that it contains 41 notes, a number also often thought to be a signature (J9+ S18 +BACH14), and the reverse of 14. [22].

http://www.vanrecital.com/events/concert_notes.cfm?concertid=220&type=program

http://www.biography.com/articles/Johann-Sebastian-Bach-9194289&part=9?print

Symbolism

A repertoire of melody types existed, for example, that was generated by an explicit â€œdoctrine of figuresâ€� that created musical equivalents for the figures of speech in the art of rhetoric. Closely related to these â€œfiguresâ€� are such examples of pictorial symbolism in which the composer writes, say, a rising scale to match words that speak of rising from the dead or a descending chromatic scale (depicting a howl of pain) to sorrowful words. Pictorial symbolism of this kind occurs only in connection with wordsâ€"in vocal music and in chorale preludes, where the words of the chorale are in the listener's mind. There is no point in looking for resurrection motifs in The Well-Tempered Clavier. Pictorialism, even when not codified into a doctrine, seems to be a fundamental musical instinct and essentially an expressive device. It can, however, become more abstract, as in the case of number symbolism, a phenomenon observed too
often in the works of Bach to be dismissed out of hand.

Number symbolism is sometimes pictorial; in the St. Matthew Passion it is reasonable that the question â€œLord, is it I?â€� should be asked 11 times, once by each of the faithful disciples. But the deliberate search for such symbolism in Bach's music can be taken too far. Almost any number may be called â€œsymbolicâ€� (3, 6, 7, 10, 11, 12, 14, and 41 are only a few examples); any multiple of such a number is itself symbolic; and the number of sharps in a key signature, notes in a melody, measures in a piece, and so on may all be considered significant. As a result, it is easy to find symbolic numbers anywhere, but ridiculous to suppose that such discoveries invariably have a meaning.

But nevertheless, even when takeing JSB's

funny jokes about number-symbolism apart aside:

Quest:

However, in heaven, whoever is able to explain,

why Bach should had chosen just the above given

compliecated ratios, not mention of tuning them

properly within

"less than an quarter hour" on his harpsichord?

The above allegation about Bach's 'authentic-tuning' rings hollow!

And I do agree with Lindley & Ortgies,

that such broade noncredible claims have to be;

condemmend as untrustworthy.

http://sites.google.com/site/iboortgies/errataandcorrigendatolindleyortgies%3A%22bac

In it we discuss some of the historically wild and methodologically wrong speculations published last year (2005) by Bradley Lehman in his article in Early Music (Note 1), where he claimed to have discovered "Bach's temperament.â€�

The Lehman temperament is of modern design. It is, like the one designed by the late Herbert Anton Kellner...."

http://em.oxfordjournals.org/cgi/content/full/34/4/613

Quote:

"A significant degree of aesthetic and historical consensus is indicated when the wife of a man who is passionately committed to a certain aesthetic point agrees with him.

Caveat lector." [Footnote 19]

[19] Dr Lehman sets out additional arguments (of similar value) in his EM article and at www.larips.com and other websites....

So far about the specific opinion

of two leading scholars, that are

well-reckognized experts in the

topic about Baroque tunings.

But at least, however, one can say:

that nobody else had previously ever tried before

to foist upon J.S.Bach such large number-proportions as

"intended-original-tuning"

I.m.h.o: Outrageous hair-raising!

Yours-Sincerely

A.S.

🔗a_sparschuh <a_sparschuh@...>

🔗George Sanders <georgesanders11111@...>

🔗Andy <a_sparschuh@...>

--- In tuning@yahoogroups.com, George Sanders <georgesanders11111@...> wrote:
>
>
> ...Smend is the father of the modern numerological tradition...

Dear Franklin,
are you aware about the preceeding 'grand-fahter' of that somehow questionable business:

http://de.wikipedia.org/wiki/Arnold_Schering
http://en.wikipedia.org/wiki/Arnold_Schering

http://www.kirchenlexikon.de/s/s1/schering_a.shtml
Quote:
"...
SCHERING, Arnold, Musikforscher, * 2.4. 1877 in Breslau als Sohn eines Verlegers, + 7.3. 1941 in Berlin. S. besuchte die Kreuz- und die Annenschule in Dresden, und nahm Violinunterricht. Bereits in früher Jugend erhielt er reiche Eindrücke durch das Dresdner Musikleben. Seit 1896 studierte er an der Hochschule für Musik in Berlin (J. Joachim, Violine, R. Succo, Komposition). Ab 1898 studierte er Musikwissenschaft, Literaturgeschichte und Philosophie in Berlin (O. Fleischer, C. Stumpf), München und Leipzig (H. Kretzschmar). Seine Diss. (1902) über die »Geschichte des Instrumentalkonzerts bis A. Vivaldi« wurde in erweiterter Form als »Geschichte des Instrumentalkonzerts bis auf die Gegenwart« 1905 in Leipzig gedruckt. Bereits während seines Studiums war er als Musikkritiker für Leipziger Zeitungen tätig. 1903-1905 war S. Herausgeber der Neuen Zeitschrift für Musik, 1904-39 Herausgeber des Bach-Jahrbuches. 1907 habilitierte er sich in Leipzig mit dem Thema »Die Anfänge des Oratoriums«, die 1911 erweitert als »Geschichte des Oratoriums« gedruckt wurde. Seit 1909 hatte er eine Lehrauftrag für Musikgeschichte am Leipziger Konservatorium. 1915 wurde er zum außerordentlichen Professor für Geschichte und Ästhetik der Musik an der Universität Leipzig ernannt. 1920 wurde er in der Nachfolge von H. Abert zum Ordinarius in Halle berufen und übernahm 1927 den Vorsitz der Händelgesellschaft. 1928 folgte er H. Abert auf das Ordinariat für Musikgeschichte an der Berliner Universität, wo er bis zu seinem Tod blieb. Er war Vorsitzender der Kommission der DDT und Präsident der Gesellschaft für Musikwissenschaft. Als gleichermaßen praktisch ausgebildeter Musiker legte er großen Wert auf die Verbindung von Musikgeschichte und Praxis, was sich z.B. in seiner Tätigkeit als Leiter der Collegia musica in Leipzig, Halle und Berlin zeigte. S. war einer der bedeutendsten deutschen Musikforscher der ersten Hälfte des 20. Jahrhunderts. Schwerpunkte seiner Arbeit waren die Bach-Forschung (Textbehandlung, Symbolik), Aufführungspraxis alter Musik (Besetzung, Figurenlehre) und Beethoven (musikalische Symbolik, Ästhetik)...." &ct..pp...

tr:
"
SCHERING, Arnold, music researcher, * 2.4. 1877 in Breslau, the son of a publisher, + 7.3. 1941 in Berlin. S. went to the cross and the Anne School in Dresden, and took violin lessons. Already in early youth, he received many impressions through the musical life of Dresden. Since 1896, he studied at the Hochschule für Musik in Berlin (J. Joachim, violin, R. Succo, composition). From 1898 he studied musicology, literature, history and philosophy in Berlin (O. Fleischer, C. Stumpf), Munich and Leipzig (H. Kretzschmar). His Diss (1902) about the "history of the concerto by Vivaldi," was in expanded form as a "history of the concerto to be printed on the presence of" 1905 in Leipzig. During his studies he worked as a music critic for newspapers in Leipzig. P. 1903-1905 was editor of the Neue Zeitschrift für Musik, 1904-39 editor of the Bach-Jahrbuch. In 1907 he qualified as in Leipzig, with the theme "The beginnings" of the oratorio, which expanded in 1911 as "History of the Oratorio" was printed. Since 1909, he was a lecturer in music history at the Leipzig Conservatory. In 1915 he was appointed extraordinary professor of history and aesthetics of music at the University of Leipzig. In 1920 he was appointed to the succession of H. Abert Hall to full professor in 1927 and took over the chairmanship of the Handel Society. In 1928, he was followed by H. Abert in the chair of music history at the University of Berlin, where he remained until his death. He was chairman of the Commission of DDT and President of the Society for Musicology. As a trained musician, he practically equally attached great importance to the combination of music history and practice, which, for example, showed in his work as head of the Collegia musica in Leipzig, Halle and Berlin. S. was one of the most important German musicologist of the first half of the 20th Century. The focus of his work were the Bach-research (text treatment, symbolism), Performance Practice of Early Music (occupation, character education) and Beethoven (musical symbolism, aesthetics )...." & ct .. pp ...

Here comes an small sample of his style in gothic facsimile,
inklusvie an apt translation to English by a good friend:

http://www.bach-cantatas.com/Articles/ScheringFistulanten.pdf

> One thing that's illuminating about Tatlow is that I think she's
> demonstrated how much of the fascination with certain types of
> numerology such as the Golden Section actually stems from the 19th > century, starting in the 1830's in Germany, when this term was
> invented (although the proportion was known earlier,

...which insustainable claims
comes up against my own extreme skepticism...

http://en.wikipedia.org/wiki/Golden_ratio
see there chapter: Architecture:
"Some scholars deny that the Greeks had any aesthetic association with golden ratio....And Keith Devlin says, "Certainly, the oft repeated assertion that the Parthenon in Athens is based on the golden ratio is not supported by actual measurements. In fact, the entire story about the Greeks and golden ratio seems to be without foundation. The one thing we know for sure is that Euclid, in his famous textbook Elements, written around 300 BC, showed how to calculate its value."[21] Near-contemporary sources like Vitruvius exclusively discuss proportions that can be expressed in whole numbers, i.e. commensurate as opposed to irrational proportions.

For instance:
Take just the there said so often-cited example of the:
All I can detect in the
http://en.wikipedia.org/wiki/Parthenon
's 'floor-plan'
http://en.wikipedia.org/wiki/File:Parthenon-top-view.svg

All I can only detect in the are the Pythagorean-theorem
a^2 + b^2 = c^2 rectangular triangles of:

1.) Relicts of the inner ancient coulumnade of marble pillars:
5^2 + 12^2 = 13^2 of the
http://en.wikipedia.org/wiki/Older_Parthenon

2.) Later added coulmnade of the outer pillars with proportion:
8^2 + 15^2 = 17^2

> according to
>Tatlow it was not applied in paintings, music, or architecture,
> where integer proportions--numbers of bars, etc.--were used).
> It reached a peak in mystical circles at the turn into the 20th
> century, receded, then popped up in academic circles after WWII.

By far, most of that pamphletes must be considered as nonsenseial:
http://en.wikipedia.org/wiki/Pseudoscience
alike the so called "Rosetta-Stone - discoveries"
in the bold-face-lied "Da-Vinci-Code" manner of a fair-swindle
that appears now and then even in that group here
ang got again and again perpetuaded.

> Tatlow's talk a couple years ago was fantastic, because she had
> discovered that the bar-number proportions within every set of
> pieces by Bach that was published or prepared for publication were > in simple integer proportions--3:2, 4:3, etc.

Please attend,
that statement in not quite correct,
because:
Her work depends by far mostly on the earlier research results of
my old friend Ulrich Siegele:

http://de.wikipedia.org/wiki/Violinkonzerte_%28Bach%29#cite_ref-1
Footnote:
"# b Ulrich Siegele: Kompositionsweise und Bearbeitungstechnik in der Instrumentalmusik Joh. Seb. Bachs, 1956, ISBN 3-7751-0117-9

'The compsing and editing technique in Bach's instrumental music.'
Footnote #4:
# b Ulrich Siegele: Proportionierung als kompositorisches Arbeitsinstrument in Konzerten J. S. Bachs in: Martin Geck (Her.): Bachs Orchesterwerke, Bericht über das 1. Dortmunder Bach-Symposion 1996. Witten 1997, ISBN 3-932676-04-1

'Proportions as a compositional working-tool in Bach's concertos.'

http://de.wikipedia.org/wiki/3._Brandenburgisches_Konzert_%28Bach%29

"Her research in this area is especially credible owing to her "success in debunking a lot of numerological claims."

" But she believes that most of this had to do with a tradition of "proper construction" coming out of classical architectural models "(you're right, it was Vitruvius she mentioned,

Absolutely correct observed:
as for instance some other Pythagorean-Triple propo:

29^2 = 21^2 + 20^2 for famous J+S+B or S+D+G hypothenouse 29.

and also too:

41^2 = 40^2 + 9^2 for the corresponding hypothenouse 41 respectively.

Probably JSB overtook that technique already in his youth from
when he probably detected just that same proportions in the ancient
compositions of:
http://en.wikipedia.org/wiki/Ars_nova
http://en.wikipedia.org/wiki/Notre_Dame_school

and also in that old tradition especially preferable:

http://en.wikipedia.org/wiki/Josquin_des_Prez

> not Quintillian...thanks for catching this).

> Obviously there are traditions where number symbolism is crucial
> to the understanding of certain artworks.

only the discovery and the deeper understanding of the above said proportions and other inherent structures,
enables me now to conduct J.S.Bach's conduct more freely
completely without having to look into note score text,
which I surely would have consult much more often
without the knowledge what will happen within the next few bars,
when using that as an memory-aid that was inherent built into
the composition by the composer himself,
alike a well written mathemtical proof:

Carl Friedrich Gauss used to say:
"
A finished perfect building one can no longer watch behind
its completion, after the scaffolding skeleton got removed,
under whose help it once had been built
by the architekt's blueprint.

Additional further literature:
For a deeper study of that technique I reccomend to consult:
http://en.wikipedia.org/wiki/Johann_Kirnberger
's
"Die Kunst des reinen Satzes in der Musik (The Art of Strict Composition in Music, 1774, 1779)

http://www.amazon.com/Strict-Musical-Composition-Johann-Kirnberger/dp/0300024835

have a lot of fun in yours own studies of JSB's scores
bye
A.S.

🔗Andy <a_sparschuh@...>

--- In tuning@yahoogroups.com, George Sanders <georgesanders11111@...> wrote:
>...demonstrated how much of the fascination with certain types of
> numerology such as the Golden Section actually stems from the 19th > century, starting in the 1830's in Germany, when this term was
> invented (although the proportion was known earlier, according to
> Tatlow it was not applied in paintings, music, or architecture,
> where integer proportions--numbers of bars, etc.--were used).

Dear Franklin,
here some more recent crude balnoey stuff to yours amusement

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibInArt.html
"
...However, the "most pleasing shape" idea is open to criticism. The golden section as a concept was studied by the Greek geometers several hundred years before Christ, as mentioned on earlier pages at this site, But the concept of it as a pleasing or beautiful shape only originated in the late 1800's and does not seem to have any written texts (ancient Greek, Egyptian or Babylonian) as supporting hard evidence. ...
...So this page has lots of speculative material on it...
...George Markowsky's Misconceptions about the Golden ratio
in The College Mathematics Journal Vol 23, January 1992, pages 2-19 is an important article that points out the weaknesses in parts of "the golden-section is the most pleasing shape" theory.
This is readable and well presented. Perhaps too many people just take the (unsupportable?) remarks of others and incorporate them in their works? You may or may not agree with all that Markowsky says, but this is a good article which tries to debunk a simplistic and unscientific "cult" status being attached to Phi, seeing it where it really is not!

http://www.atrise.com/golden-section/
http://en.wikipedia.org/wiki/Golden_ratio
Disputed sightings
Examples of disputed observations of the golden ratio include the following:
....
# In 2003 Weiss and Weiss came on a background of psychometric data and theoretical considerations to the conclusion that the golden ratio underlies the clock cycle of brain waves.[76] In 2008 this was empirically confirmed by a group of neurobiologists.[77]
# In 2010 the journal Science reported that the golden ratio is present at the atomic scale in the magnetic resonance of spins in cobalt niobate atoms.

http://www.geom.uiuc.edu/~demo5337/s97b/art.htm
http://www.goldennumber.net/classic/goldsect.htm

> Obviously there are traditions where number symbolism is crucial to > the understanding of certain artworks...

Here you mentioned justly Dante,
Even Kurt Vonnegut refered to that ancient pratice in his:
http://www.goodreads.com/book/show/4982.The_Sirens_of_Titan
My preferred quote:
"
But mankind wasn't always so lucky. Less than a century ago men and women did not have easy access to the puzzle boxes within them.

**They could not name even one of the fifty-three portals to the soul.***

Gimcrack religions were big business. Mankind, ignorant of the truths that lie within every human being, looked outwardpushed ever outward. What mankind hoped to learn in its outward push was who was actually in charge of all creation, and what all creation was all about. Mankind flung its advance agents ever outward, ever outward. Eventually it flung them out into space, into the colorless, tasteless, weightless sea of outwardness without end.
"

Probably here, K.Vonnegut, refers to the alleged ancient claims in:
http://en.wikipedia.org/wiki/Gospel_of_the_Hebrews
Clemens_of_Alexandria 6235 Fol.56R
§27: "In that two towns (namely Behthsaida and Chorazin)
happened (due to influence of Jesus) eactly 53 wonderous miracles."

Here some later comments about such miracoulos events:
http://newtheologicalmovement.blogspot.com/2010/04/disciples-caught-153-fish.html
Quote:
"
St. Augustine says: The catch of fish tells us of the salvation of men, but man cannot be saved without keeping the 10 commandments. But, on account of the fall, man cannot even keep the commandments without the help of grace and the 7 gifts of the Holy Spirit. Moreover, the number 7 signifies holiness, since God blessed the 7th day and made it holy (Gen 2:3). But 10 plus 7 equals 17, and if all the numbers from 1 to 17 are added together (1+2+3+17), they equal 153. Hence, the 153 fish signify that all the elect are to be saved by the gift of grace (7) and the following of the commandments (10).
Or rather: St. Augustine notes that there were 7 disciples in the boat (Peter, Thomas, Nathanael, the sons of Zebedee, James and John, and two other disciples), who had all been filled with the 7 gifts of the Holy Spirit. 7 times 7 equals 49. 49 plus 1 makes the perfection of 50. Now, 3 is the number of the Trinity and also of our faith (which is founded on the Trinity); but 50 times 3 (for our faith) is 150, plus 3 (for the Trinity) is 153. Hence, the 153 fish signify the fullness of the Church (7), filled with the Holy Spirit (7), perfected (50) in her faith (3) in the most holy Trinity (3).
St. Gregory the Great says: 10 and 7 are perfect numbers, which added together make 17. This, times 3, for the perfection of faith in the Trinity, makes 51. This, times 3 again, makes 153.
St. Cyril breaks 153 into 100 (the great number of gentiles to be saved), plus 50 (the smaller number of Jews to be saved), plus 3 (the Trinity who saves all). Others follow St. Cyril, but modify this as follows: 100 (the multitude of married lay faithful in the Church), plus 50 (the many faithful who commit themselves later in life to continence either living as widows or living with their spouse in a brother-sister relationship), plus 3 (the precious few who commit their whole lives to celibacy as virgins) equals 153 (the whole Church taken together as a single body).
"

Just those ennumerations corresponds exactly with the
arithmetically calculations of
bars in Friedrich-Smend's works
and also with A.Werckmeister considerations about 'holy' concepts
of how to divide the Commata into subparts, see also:
http://tokenofperdition.com/numbers/nu153.php
"
Gematria
Adding the numerical value of the Greek word Maria, 152, with the "alpha" of the God, 1, we obtain 153, which makes say to Peignot that God needed Mary to do useful work the salvation."

Therefore I do suppose, that an somehow harsh sounding

153/152 = (17*3^2) / (19*2^3) = ~11.35...Cents almost but < PC^(1/2)

tempered 5th: D-A
should express maybe the pain of
mother-mary(19) about the crucification of her son(17).
But already Bach's uncles and others used that codification by numbers
in theirs sacral-compositions. Hende I assume that the crude deviation of 152/153 was not only tolerated but even sought intensionally. In the same sense one has need to overcome 152/153 in
order to get rid of the PC, when doing so useful work in the
solution of subdividing the comma.

Appearently Ruth-Tatlow comletely overvied such aspcts
in hers considerations about medivial numerological nomenclauture,
especially in religious contexts.
Then in the middle age,
it was common sense among sacal-music composer
to express theris humility for the 'holy'-family of the bible,
as devotional aspect when dividing the score by
the corresponding numbers in the coeval standard meaning.
Obviously J.B.Bach secretly put in [1722] his own crown on this old tradition.

Hence without beeing aware of that decisive background:
No wonder that she had gained mostly barely negative results.
I am surprised that she had not noticed this aspect and so
she had missed that decisve aspect

In Baroque times, even scientists alike,
Isaac Newton refers to that 53 in his famous tuning-scheme:

http://mto.societymusictheory.org/issues/mto.93.0.3/mto.93.0.3.lindley7.gif

http://www.sciencedirect.com/science?_ob=MiamiCaptionURL&_method=retrieve&_udi=B6WG9-4NS2GPC-2&_image=fig004&_ba=4&_user=2717328&_rdoc=1&_fmt=full&_orig=search&_cdi=6817&view=c&_acct=C000056831&_version=1&_urlVersion=0&_userid=2717328&md5=ecae25c10d0d7cfb603c20e7a93bf911

References, Manuscripts:

Cambridge, University Library, Add MS. 4000 Cambridge, University Library, Add MS. 4000, ff. 137r143v: Isaac Newton, essay "Of Musick".

Cambridge, University Library, Add MS. 4000 Cambridge, University Library, Add MS. 4000, ff. 104r113v and Add MS. 3958 (B), f. 31r: Isaac Newton, musical calculations.

so far ours circle over the millenias
closes back from the old Greek
Philolaos 53-tone division of the octave, theirs
distortion and misconceptions among the church fathers,
their usage by A.Werckmeister, J.G.Walther and J.S.Bach,
and finally theirs clarification by I.Newton and F.Smend,
in ours hitherto poor understanding of the inerhernt processes
within J.S.Bach masterpieces.

Conlusion:
F.Smend opened us the doors for an deeper understanding
of Bach's gothic concepts and usage
about numerologically symbolic meanings.

bye
Andy