RE: New tuning site at www.ppexpressivo.wyenet.co

>--- In tuning@y... <mailto:tuning@y...> , "jpehrson2" < jpehrson@r...
<mailto:jpehrson@r...> > wrote:
>> --- In tuning@y... <mailto:tuning@y...> , "paulerlich" < paul@s...
<mailto:paul@s...> > wrote:
>>
>> /tuning/topicId_35317.html#35435
</tuning/topicId_35317.html#35435>
>>
>>> --- In tuning@y... <mailto:tuning@y...> , "dkeenanuqnetau" <
d.keenan@u... <mailto:d.keenan@u...> > wrote:
>>>> --- In tuning@y... <mailto:tuning@y...> , "robert_wendell" <
rwendell@c... <mailto:rwendell@c...> > wrote:
>>>>> Thank you, Paul! Good points. I think we can generalize with a fair
>>>>> degree of safety (cringing defensively) that the octave is the most
>>>>> critical interval in terms of human tolerance for tuning error, the
>>>>> perfect fifth next, etc., right on up the harmonic series, but
>>>>> eliminating factors of two.
>>>>
>>>> Hi Bob, It's lucky you cringed defensively. :-)
>>>>
>>>> I feel that what you say holds until the complexity a*b of a ratio a:b

>>>> (in lowest terms) gets up around 35. Then I think tolerance for
>>>> mistuning (e.g. in cents) starts to decrease again, but for an
>>>> entirely different reason. A complex ratio doesn't become much more
>>>> dissonant with mistuning like a simple ratio does, since it is already
>>>> fairly dissonant, but it loses its justness, or its identity _as_ that
>>>> ratio, with a smaller mistuning the more complex the ratio, until
>>>> finally up around a complexity of 100 to 120 (depending on the timbre
>>>> and lots of other things) it _has_ no identity distinct from its
²>>>> nearby mistunings and its justness cannot be found.
>>>>
>>>> The above assumes we're talking about bare dyads. If the more complex
>>>> ratios are embedded in larger otonal chords then I think the tolerance
>>>> stays about the same as we keep going up.
>>>>
>>>> Just my humble opinion.
>>>
>>> i think dave keenan is exactly right. there's no substitute for
>>> actually *listening* to lots of just and mistuned ratios, and dave
>>> has obviously done a great deal of this.
>>
>>
>> ***This is terribly interesting, but I'm only *half* getting it.
>>>
>> Bob Wendell is saying that the tolerance for error increases as we go
>> further up the harmonic series. Dave Keenan says this is true, but
>> after a certain a*b (num*denom) then again we need to be more
>> sensitive to error again, yes?
>
>right, between a*b=35 and a*b=100, or higher.
>
>>
>> But the reasoning is the "justness" of the larger ratio which becomes
>> rather an "academic" distinction, yes, since we're in the realm
>> of "rational" intonation rather than truly "just" intonation.
>
>no, that comes in with even more complex ratios.

Hello, I am just an amateur musician interrested in physics of music.

Back to the basic:

When a fifth (3/2F0) is out of tune of dF with a fundamental (F0), the
beating harmonics are around 3F0, 6F0 and so on, with corresponding beat
frequencies of 3*dF, 6*dF etc.

When a major third (5/4F0) is out of tune, the beating harmonics
are around, 5F0, 10F0 and so on with corresponding beat
frequencies of 5*dF, 10*dF etc.

When a minor third (6/5F0) is out of tune, the beating harmonics
are around, 6F0, 12F0 and so on with corresponding beat frequencies
of 6*dF, 10*dF etc.

So, for a given mistuning value of dF, the first beat if faster for
minor third than for major third and faster for major third than for
a fifth. Thats for the hard facts. But how annoying it sound depends
of other (partly subjective) factors.

For most acoustic sources, there is an overall enveloppe damping
that is typically between 3 and 6 dB per octave. So typically,for
a given value of dF the beating frequency of the third shall be faster,
but also less intense (from -6 to -12dB) than the beating of the fifth.
The accoustic damping may compensate partly for the increase of speed.
This compensation is more important for instrument with great damping
(such as modern grand piano) than for instrument like harpsichord
having a a flatter spectrum with much more intense high wavenumber
harmonics.

In my opinion that may explain why ET is much more tolerable on a
piano that on an harpsichord.

Further, there is a threshold in frequency beyond which a beat is
not perceived as a beat but as a "roughness". Figures: 4Hz out of
tune for A4 at 440 Hz represents around 15 cents (this is approximately
how much out of tune is ET thirds). This gives a 4*5=20 (maj. 3rd)
or 4*6=24 (min.third) beats per second for a third; quite rough but
for a fifth that whould give an annoying (nearly intolerable) 12 beats
with an even greater intensity.

For ratio between greater numbers (let say 17/19), the beating harmonics
are farther away from the fondamental, much more rapid, less intense
and thus less likely to be perceived. Further, most ears are not used
to the smooth sound of a just 17/19 :-).

François Laferrière