From 'septenarius' to an analog 'quaternarius'-interperatation,was Re: Bach's
- --- In firstname.lastname@example.org, "Tom Dent" <stringph@...> wrote:
>in using algebraically irrational numbers
> --- In email@example.com, Afmmjr@ wrote:
>>took up theoretically from exactly where W. had left off.while W stayed alwas reamaining within the rational fraction concept,
which was citizied by N as 'inferior' versus his own
abstract calculations using Simon Stevins ET invention,
that W refused in reference to the Pythagorean concept of
of all possible intervalls to the unit 1.
N. abonded that traditional demand, still indispensable for W.
henc I do agree, that:
> > Werckmeister III is different systems.against N's mathematically more advanced
way of computation in his 'sectio-canonis'.
> Each key is intervallicallyyielding just
> different from another.
that W considered still as essential.
N. firstly advocated ET without that feature,
needed time depart from ET
for accepting the traditional view also for himself too.
> Some keys embody Pythagorean tuning.That's right observed,
W. intended fully aware just that sound in the rare keys
for a change in the 'variationibus' when modulating.
>Appearently W. designed that nice effect consciously wanted
> Hum, I don't hear F# major and C# major having much intervallic
> difference myself. Rather too much Pythagoras.
especially for fast melodic transitions,
alike violinist change to pythagorean when playing runs rapidly.
>Simply wrong, because:
>> W's later turn towards ET
He never refers to that new mathematically concept, not even
squareroots do appear nowhere in any of his writings,
not to mention logarithms.
Alike later Kirnberger W. never cared about that
modern "mathematical-stuff", as JSB
inbeween them also too refused that.
>>and/or dilutedW. considered that outdated practice simply as: 'wrong'.
>> meantones may have acknowledged this.
JSB refused Silbermann's 'barbaric 3rds!'
Who needs even today an much over-broadly based
wolf 5th of ~704Cents?
inept placed inbetween Bb and F unduly, absolute needless, even
faulty labeled as an alleged theoretically "diminished 6th"?
in 55ET, 'a tuning that had never existed' in the Baroque era.
>Long ago disproved early 20th century scholary nonsene!
>> Werckmeister VI, a basically near ET tuning found with
>> measurements like one/seventh of a Pythagorean comma.
> Oh, please! Werckmeister VI is based on (complicated) j u s tCave!
> intervals, it is nothing to do with divisions of a comma, the text >of
> Musicalische Temperatur explicitly says as much. Unfortunately many
> 20th century 'historians' starting with Dupont & Barbour have > >utterly
> misread the text and imagine (simply because of the number 7 popping
> up) that it has something to do with 1/7 comma.
That both time-honored authors should be touched only with croucher
tools due to careless deformation of historical source texts: All in
all: Obsolete out-dated faulty below todays scientific standards.
> Well, that isn't even a good approximation.but even worser:
Theirs deceptive description misleads astray
about W's concept behind his plain rational number arithmetics.
> Actually the 'septenarius' is better approximated by fifth-commaThere's no need for improving him:
Why using barely approximations instead
staying in his own original values
given concrete in absolute monochord-stringlengths:
196 C 1
186 C# 98/93
176 D 49/44
165 Eb 196/165
156 E 49/39
147 F 4/3
139 F# 196/139
131 G 196/131
124 G# 49/31
117 A 196/117
110 Bb 98/55
104 B 49/26
98 C' 2
obtained from tempered 5hs-circle
196 C (_393_)392;196 start
131 G 393/3:= 131(132,66,33)
176 D (_351_)352;176;88,44,22,11:= 33/3
156 E 78,_39_:= 117/3
104 B (_417_)416,208;104;52,26,13:= 39/3
139 F# (_279_)278;139:= 417/3
186 C# 93:= 279/3
124 G# (_495_)496,248;124;62,31:= 93/3
110 Bb (_441_)440,220;110;55:= 165/3 not Scheibler's pitch: 440cps
_147_F := 441/3
196 C 98,49:= 147/3 returned to begin
so that seven 5ths are tempered by the pure rational fractions:
Analog it's also possible to fit the corresponding four 5ths of his
#3(1691) the 'quaternarius' according in the same manner,
My actual interpretation sounds:
C 6560/6561 G 204/205 D 152/153 A>E>B 512/513 F#>C#>G#>Eb>Bb>F>C
expanded in absolute frequencies:
273.375 C ((17))2187:= 3^7
410 G (17*3=51,102,204)205;410;820,1640,3280,6560(6561:= 3^8)
306 D (19,38,76,152)153:= 17*9
456 A 57:= 19*3
342 E 171:= 19*9
256.5 B (1,...,512)513:= 19*27
384 F# 3
288 C# 9
432 G# 27
324 Eb 81
486 Bb 243
364.5 F 729:= 3^6
273.375 C 2187:= 3^7
that's in ascending pitch order
273.375 C 1 middle C
288 C# 256/243
306 D 272/243
324 Eb 32/27
342 E 304/243
364.5 F 4/3
384 F# 1024/729
410 G 3280/2187 coeval Cammer-tone ~410cps
432 G# 128/81
456 A 1216/729 coeval Choir-tone ~456cps
486 Bb 152/81
512.5 B 16/9
546.75 C' 2
Further refinement in
> Oh well. There's only so long I can go on about the same stuff - theThe original sources are still worth to study and recheck again and
> sheer multiplicity of historical existence, compared to the scarcity
> of historical sources;
again, against meanwhile unsustainable claims and questionable
alike the historical W3 would consist in the later (20th-century)
foisted modern PC^(1/4) of ~ 6Cents variant or even worser 12-ET.
> the need (nevertheless) to actually read >thoseIn order to get at least partially rid of historically obsolete
> sources and see what they do or don't say.
ballast, that meanwhile has got mouldy.
- Good points, Tom. I especially agree with you regarding the need for
objective criteria, and like you, decry the 'personalized' language
BTW, you picked up an important point regarding the general ignorance
of statistics---some of the arguments about the virtues of a given
temperament based on what occurs in Bach's music remind me of the
inane logic that occurs around 'The Bible Code'---you know, you can
find such and such a message if you read the characters in
such-and-such an ordering.
--- In firstname.lastname@example.org, "Tom Dent" <stringph@...> wrote:
> --- In email@example.com, Brad Lehman <bpl@> wrote:
> > (...) Lehman thinks that 'flat' keys should be tuned purer than
> > > keys, other things being equal; Lindley thinks the reverse.
> > >
> > > - Which key should be the most 'far out' in tuning? Lehman
> thinks E
> > > major/C# minor, Lindley thinks C# major/F minor.
> > 'Scuse me, but where did I ever say such a thing about C# minor, let
> > alone think it? This is news to me.
> I am writing in a STRICTLY objective and impartial manner here. You
> favour a tuning in which the intervals of the chord of C# minor are
> the farthest from pure 5-limit intervals of any minor chord. Lindley
> favours a tuning in which the chord of F minor is the farthest from
> 5-limit just intonation.
> That is the simple and logical meaning of 'far out', in context.
> > I would also disagree with the assertion: "Lehman thinks that 'flat'
> > keys should be tuned purer than 'sharp' keys, [other things being
> equal]." I don't think that.
> Again, this is a simple and objective use of the word 'pure'. And you
> *do* favour a tuning in which F-A is purer than G-B, Bb-D is purer
> than D-F#, Eb-G is purer than A-C#, and so on. I think it is an
> entirely reasonable summary that, in any objective sense, the tuning
> you think is correct makes flat keys purer than sharp ones.
> > I
> > do think that the flat keys and sharp keys make a comparably "pure"
> > *musical effect* when played in compositions by Bach.
> This is a subjective opinion, which depends on particular
> circumstances and the choice of repertoire. We can't meaningfully
> discuss tuning at all if everyone reserves the right to redefine the
> meaning of 'purity' however they want, according to what they feel
> when playing their favourite pieces on their own instrument(s) at
> home. 'Purity' of intervals and chords is a word which *does* have an
> objective meaning in discussing tuning.
> With your definition, every sentence becomes subjective, relative, and
> arguable. Total confusion. I don't see the advantage if we replace the
> objective meaning of 'purity' by 'Lehman-purity' which is based
> entirely on Bradley Lehman's internal perceptions. Well, I see the
> advantage to one person, but not to the rest of the world. This is a
> solipsist's definition.
> Not to mention that 'Lehman-purity' back in 2003 encompassed a Bach
> tuning which was basically 1/5-comma meantone, with Eb modified
> partway towards D#. That version of 'Lehman-purity', of course, was
> wrong ... but the present-day version of 'Lehman-purity' *must*
> represent a historical truth about Bach!!
> If you want to talk about what you feel, by all means do so, but don't
> give it the label of a fact.
> > hearing it daily as I practice Bach repertoire on my
> > harpsichords.
> Fine, but every time I tried to practice the A major English Suite on
> mine, I got the opposite impression about the 'purity', or otherwise,
> of the tuning. De gustibus, I hope.
> > N.B. This is not "purity" in the shallow sense of counting up
> > simple-ratio intervals;
> I see. Simple, objective, easily-understood definitions of intervallic
> purity are now 'shallow', and must be replaced by personal, subjective
> and endlessly arguable non-meanings. Words must be made ambiguous,
> confusing and endlessly manipulable. Cui bono?
> > but rather, the sense that each scale (both
> > melodically and harmonically) makes a crisp, clear, interesting
> > with integrity, and no rough spots anywhere.
> Sounds like a vodka advertisement. Who chose that list of adjectives?
> Why 'crisp', but not 'mellow' or 'crunchy' or 'smooth'?
> And how can we tell when a scale has 'integrity both melodically and
> harmonically'? Otherwise it's meaningless.
> Again, I disagree about the presence of 'rough spots'. So it seems
> 'Lehman-purity' and 'Dent-purity' mean two quite different things,
> which I would call an entirely natural and healthy state of affairs,
> and tells us nothing about 'Bach-purity'.
> > ... convincing focus to the music. In my opinion, of course!
> But this is just what is pointless for anyone except you to write
> about, because none of us has any business to summarize or represent
> your internal perceptions. (Let alone Bach's...)
> > But, I also believe that this is the same type of
> > "purely played" effect CPE Bach was writing about
> CPE Bach on tuning is so vague that you can believe almost anything
> about it, with or without the aid of subjectively redefining the
> meaning of words and thinking about malt whiskies. But why would your
> belief be more accurate than anyone else's?
> And what would CPE have been thinking when he endorsed Barthold
> Fritz's little book about how to tune clavichords (approximately) in
> > "This
> > fugue (unusually) has real entrances in _six_ different keys,
> > all the following diminished 4ths: B#-E, Fx-B, E#-A, Cx-F#, D#-G and
> > A#-D. Those are all the six positions in Bach's temperament where
> > interval is narrower than or equal to its size in equal temperament."
> > (May 2005 _Early Music_, top right of page 223)
> EVERY prelude and fugue has some peculiarities in its use of
> intervals. With sufficient diligence one can ferret out
> 'coincidences', or curious facts, that might be presented as having
> some significance in relation to a particular temperament. What this
> can never do is show that one temperament was used by the composer in
> preference to another. (You could also argue that it makes the music
> sound good, but it might be that a slightly different temperament made
> it sound even better...)
> And WHATEVER temperament you choose, there will be some 'interesting
> coincidences', which you can pick out and dazzle your readers with -
> at least, for readers having a rather loose grasp on statistics.
> This is no more evidence of anything than finding (say) the numbers 14
> and 41 everywhere in Bach.
> > I believe that Bach *knew* that they're the six smallest
> of course, knowing what fifths he had tuned between what notes on any
> given day, it would be obvious with or without listening... but who's
> to say if he always tuned the same fifths??
> > and used this as an
> > unusual compositional feature to help generate this piece of music.
> Well, I myself disbelieve! I disbelieve every single assertion from
> 2005 of Bach using specialized or esoteric connections between one
> particular temperament and his compositions! I believe that you could
> search for 'unusual compositional features' in relation to a dozen
> different irregular circulating temperaments, and turn up equally
> convincing 'evidence' for each of them!
> Where does that then leave us? Does anyone with a sufficiently strong
> belief get to publish his or her personal opinions about 'purity' in
> an Oxford Journal?