--- In firstname.lastname@example.org
, "Brad Lehman" <bpl@...> wrote:
simply short all nominators over denominators by the factor 10
in order to meet scala the convention of integral fractions:
SC=80:81 inbetween F~C~G~D~A~E & schisma in E~B_F#_C#_G#~Eb~Bb~F
2790/2643 ! 279.0C#4 / 264.3C4
2960/2643 ! 296.0D_4 / 264.3C4
6275/4926 ! 627.5Eb4 / 528.6C5
6615/4926 ! 661.5E_4 / 264.3C4 = (5:4)(882:881) ~+1.964 Cents sharp
3524/2643 ! 352.4F_4 / 264.3C4 = (4:3)(882:881) ~+1.964 Cents sharp
3720/2643 ! 372.0F#4 / 264.3C4
3960/2643 ! 396.0G_4 / 264.3C4 = (3:2)(880:881) ~-1.966 Cents flat
4185/2643 ! 418.5G#4 / 264.3C4
4425/2643 ! 442.5A_4 / 264.3C4 440Hz(+2.5Hz = 150 MetronomeBeats/min)
4705/2643 ! 470.5Bb4 / 264.3C4
4960/2643 ! 496.0B_4 / 264.3C4
That results in epimoric beating lowered 5ths, all amounts given
in rational, ~Cents~ & ~TUs~ for the Syntonic-comma in F~C~G~D~A~E
F 881:882 C 880:881 G 296:297 D 295:296 A 294:295 E = product 80:81
F ~-1.963 C ~-1.966 G ~-5.839 D ~-5.859 A ~-5.879 E = sum ~-21.506C
F ~-60.28 C ~-60.34 G ~-179.2 D ~-179.8 A ~-180.4 E = sum ~660.04TUs
and the schisma = 2^15/5/3^8 = 32768:32805 ~-1.954Cents ~-59.96TUs
E 3968:3969 B_F#_C#_G# 2510:2511 Eb 3764:3765 Bb 4704:4705 F
E ~-0.43624 B_F#_C#_G# ~-0.68959 Eb ~-0.45988 Bb ~-0.36799 F
E ~-13.3886 B_F#_C#_G# ~-21.1641 Eb ~-14.1141 Bb ~-11.2940 F
> With only an A=440 tuning fork in one hand, a harpsichord tuning lever
> in the other hand, and absolutely NO electronic devices of any kind:
> how exactly should one proceed to get all twelve of your notes
> correctly tuned onto a harpsichord, using this scheme?
The wanted precision of accessible accuracy
depends on several factors:
1. Quality of tuneability for the instrument alike
deviations due to inhamonicty of the strings?
2. Counting beats barely by own heart-pulse of
under aid of an clock or even better an adjustable Metronome?
3. Tuner is rested/relaxed or fatigued/exhausted or
may be even incompetent?
> And with no
> way of measuring integer frequencies, either, or knowing when they've
> been achieved precisely?
In yours personal "squiggle" impreciseness
of barely PC^(1/12)= 60TUs
probable the following rouding would be sufficient
for yours personal taste?:
F -1 C -1 G -3 D -3 A -3 E for approximation of about an ~SC
F -60 C -60 G -180 D -180 A -180 E for exaclty 660TUs~=~SC
E -0.25 B_F#_C#_G# -0.25 Eb -0.25 Bb -0.25 F in PC^(1/12) units
E -~15~ B_F#_C#_G# -~15~ Eb -~15~ Bb -~15~ F with sum=60TUs=~schisma
> To what precision are errors acceptable? And why?
That approximation in yours personal style
-if you would achive 15TUs = PC^(1/48) = ~0.5 Cents
would deviate maximal even less than 6 TUs = PC^(1/120) ~0.2 Cents.
> Does one first have
> to agree with your goal of proportional beating, and your constraint
> of integer frequencies?
Never at all, due to the possibilty to translate into
your's terms within an error of less than ~1/5 Cents.
> All this stuff just looks like
> nearly-meaningless tables of numerals to me, sorry;
How about that "squiggle"-type notation with 4 different grades of
5ths with the cycle:
with extended legend that has additional an dot "." for PC(-1/48)
' = PC(-1/12)
''' = PC^(-1/4) as in W#3 on the average C'''G'''D'''A_E_B'''F#_..._C
. = PC^(-1/48) that 1/4 of yours usual unit '
_ = an JI 5th of exactly 3/2=1.5
If you don't accept PC^(1/48) = ~schisma^(1/4) as smallest unit
then try intead that modifiaction
without any subschismatic refinements:
> the only way I
> know to assess its quality is to see if it agrees with *your own*
I.m.h.o. ~0.2 Cents on the average in precision will suffice enough.
> which doesn't tell us one way or another about the usefulness
> for anything else *but* your own goal of proportional beating (or
> whatever it is).
Meanwhile I'm more tolerant in that aspect:
Never mind if you persist in inprecise tuning-methods
without counting beats exactly.
> If I'd somehow take the time and get this temperament set up on my
Simply try it out!
> within some acceptable error tolerance but without using
> any electronic devices:
Surely with yours ability and daily practice
in hearing you should achieve at least for:
an exactness of
even less than about PC(-1/24)accuracy
or equivalent ~1 Cents precision an the average.
> how would the resulting temperament sound in
> playing (say) some late Couperin?
Works fine, due to the pronouced Baroque key-characteristics.
> What does it do for the music,
> harmonically and melodically?
C-major deviates the least from JI.
> That's the kind of thing I personally
> care about: a temperament that sounds great in the music, and that
> be done entirely by ear in less than 10 minutes without having to
> calculate (or even refer to) a page of numbers.
even my modern 3-fold stringed piano with 88keys over 7 octaves
needs only a few minutes retuning when the weather changes.
> And, what happens if I'd want to start on A=430 or something else
Transposing is no problem for scala.
442.5 / 430 = 177/172 = 35.4/34.4 = ~1.02906977... or ~2.9 %
(1 200 * ln(430 / 442.5)) / ln(2) = -49.6089545...Cents
that's about an half of an 12-EDO semitone = 50Cents lower than 442.5.
Where's the problem there,
except that for A4=430Hz the string tension becomes to loose in an
modern standard piano?
> (maybe not having anything to do with integers!), or on some C?
> it all need to be recalculated?
Scala does that job quite well.
> Help out the practical musicians who
> just want to listen to the sounds of intervallic relationships,
Please let me know your's opinion
after you have tryed out some of the above versions.
I do agree with you, that without:
the corresponding acustical ratios, there is almost
understood in tempering key-instruments in an properly way.