# Newbe question: pitch and octave frequency ratio?

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Pitch math ... can somebody explain what that: "an octave ... which
itself is a frequency ratio of 2:1" means?

This is an excerpt from the book: "Audio Explained" by Michael
Talbot-Smith:

------
Pitch ... the frequency ratio between two adjacent semitones is
approximately 6%. In scientific terms the exact number is ¹²V2 (the
twelfth root of 2).

[V = suppose to the root char, cannot find the key combination for it
on the windows system.]

The reasoning behind this is that there are 12 equal semitone
‘intervals’ in an octave, which itself is a frequency ratio of 2:1.
Each step must therefore be ¹²V2.
------

I understand that a octave have 12 ’intervals‘ as well as how you do
the math (twelfth root of 2) but not why an octave: "is itself a
frequency ratio of 2:1".

Any help will be most appreciated, thanks
Mike
Anonymous

ras wrote:
> I understand that a octave have 12 'intervals' as well as how you do
> the math (twelfth root of 2) but not why an octave: "is itself a
> frequency ratio of 2:1".
>
> Any help will be most appreciated, thanks
> Mike

And octave below A 440 is 220, and an octave higher is 880. The octave
above and below those is 110 and 1660Hz..

Not sure if that relationship is naturally musical, or whether it is a thing
learned by us.

geoff

Quoth Geoff Wood ...

Aha, it is because it doubles the frequency.
Thank you kindly for helping,
mike

>ras wrote:
>> I understand that a octave have 12 'intervals' as well as how you do
>> the math (twelfth root of 2) but not why an octave: "is itself a
>> frequency ratio of 2:1".
>>
>> Any help will be most appreciated, thanks
>> Mike
>
>And octave below A 440 is 220, and an octave higher is 880. The octave
>above and below those is 110 and 1660Hz..
>
>Not sure if that relationship is naturally musical, or whether it is a thing
>learned by us.
>
>
>geoff
>
Related resources
Anonymous

"Geoff Wood" <geoff@paf.co.nz-nospam> wrote in message
news:BlkEc.5435\$NA1.501831@news02.tsnz.net...
| ras wrote:
| > I understand that a octave have 12 'intervals' as well as how
you do
| > the math (twelfth root of 2) but not why an octave: "is
itself a
| > frequency ratio of 2:1".
| >
| > Any help will be most appreciated, thanks
| > Mike
|
| And octave below A 440 is 220, and an octave higher is 880.
The octave
| above and below those is 110 and 1660Hz..
|
| Not sure if that relationship is naturally musical, or whether
it is a thing
| learned by us.
|
|
| geoff
|
|

It is indeed naturally musical. The ratio 2:1 is quite simple,
no? Let's say that we hear two notes simultaneously. If we hear
A above middle C (440) and A an octave above that (880), this
ratio sounds calm and peaceful. However, if we hear a whole
series of a few hundred different notes, all accompanied by
another synchronized series an octave above that, we might become
irritated. But play both notes one after the other and we can
experience the sequence as expressive. The psychology of music is
indeed complex.

Many harpsichordists like to couple up the mechanism so that
they're constantly playing simultaneous octaves, even one octave
above and one below, so that three pitches are sounding. It makes
the instrument louder, an advantage for a naturally quiet
instrument when playing a concerto with an orchestra. However, I
find such coupling obnoxious: to me, it ruins the musicality --
the sound is bright, but it's also really annoying. I'd prefer a
proper Baroque situation: one string at a time or two strings in
unison (1:1). This is an authentic 18th Century configuration. To
bring the ensemble into balance, just make the orchestra really
small. We lose majesty and brightness; we gain subtlety,
excitement, and warmth. Classical music becomes more like jazz.
Wow: which would you choose?

As these ratios become more and more complex (7:1, 23:1), we
perceive them as being increasingly dissonant, interesting,
raucous, wonderful, exciting, horrible (take your pick). The
history of music is a continuous series of one generation's
dissonance becoming the consonance of the next generation. Each
generation bends the envelope of the last one. And so the
artist's pallette progresses with more colors.

In practice, musical ratios are not as simple as I've presented
above. I've given you the general, theoretical ideal. After a
certain point in musical history, many of these ratios don't work
out quite "right." Bach wrote The Well-Tempered Clavier in an
attempt to promote a modern tuning system in which the ratios
were stretched very slightly. Our modern "equal temperament"
gives all musical keys the same characteristics as one another,
allowing a profound, innovative composer such as Bach the ability
to modulate from one key to another almost without limits. In the
world since the Baroque period, musical physics has been bent to
suit the art -- the ratios are subtly altered. In fact, piano
tuners have been tuning the entire instrument slightly wide,
obtaining a brightness of sound as a result (which drives
woodwind players insane when they try to match the pitches of
that piano). And Charlie Parker blew some very complicated music,
making use of it: believe it! The piano tuner is much more of a
trickster than you thought, huh?

However, the sound of an un-tempered organ or harpsichord is
subtly and fascinatingly different. Every key has its own
coloration, its own character. A concerto in D major is happy,
for example; in the old world, it was a happy key. We're not used
to this in modern times. That's why authenticity in performing
baroque music can really matter. I'm going to stop dead in my
tracks right now because this subject is just so intricate. It's
worthy of a book.

So, yes, an octave is 2:1.

Richard
Anonymous

"Richard Steinfeld" <rgsteinBUTREMOVETHIS@sonic.net> wrote in message news:<ifpEc.19635\$Fo4.260403@typhoon.sonic.net>...
>
> Many harpsichordists like to couple up the mechanism so that
> they're constantly playing simultaneous octaves, even one octave
> above and one below, so that three pitches are sounding.

Having built 11 harpsichords, all based on historical models, having
examined maybe 30 others from antiquity, and many dozens more modern
instruments, I must say that have NEVER EVER seen an instrument that
can be "coupled up" as you describe.

The largest common instruments, from then and now, are two manual
instruments with three sets of strings. The lower manual plays two
strings, on sounding at unisin, the other sounding at an octave,
while the upper manual plays the second unison choir. The ONLY
"coupling up" such instruments are capable of is coupling the
uppoer manual to the lower. This does not couple up or down an
octave, but rather at unison, so you have the capability of playing
the 2 unison choirs together, or adding the octave choir to the
unison.

The occasional and, in fact, rare example of instruments based on the
large Hass model will have two more choirs on the lower manual, tuned
and octave down and two octaves up, but such instrument are, by far,
exceedingly rare exceptions to the rule.

But, once again, there simply is no such thing in harpsichords, modern
or ancient, of "coupling up" by octaves.

> It makes
> the instrument louder, an advantage for a naturally quiet
> instrument when playing a concerto with an orchestra. However, I
> find such coupling obnoxious: to me, it ruins the musicality --
> the sound is bright, but it's also really annoying. I'd prefer a
> proper Baroque situation: one string at a time or two strings in
> unison (1:1). This is an authentic 18th Century configuration.

Sorry, but you're ignoring the entire class of authentic Baroque
instruments that include the octave (4') choir. Included in that
class are most double manual instruments of the era, from France,
Flanders, Germany and England.

> To
> bring the ensemble into balance, just make the orchestra really
> small. We lose majesty and brightness; we gain subtlety,
> excitement, and warmth. Classical music becomes more like jazz.
> Wow: which would you choose?

I would choose authenticity to the original than something, uhm,
"jazzed up." :-)

> As these ratios become more and more complex (7:1, 23:1), we
> perceive them as being increasingly dissonant, interesting,
> raucous, wonderful, exciting, horrible (take your pick).

Uh no, not exactly. There are entire sets of small whole-number
ratios that are quite consonant. You chose ratios that are, in fact
consonant. The latter example is, well, wierd in that it is so widely
separated as to be heard as two distinct tones.

> In practice, musical ratios are not as simple as I've presented
> above.

That much I'll certainly agree on.

> I've given you the general, theoretical ideal. After a
> certain point in musical history, many of these ratios don't work
> out quite "right." Bach wrote The Well-Tempered Clavier in an
> attempt to promote a modern tuning system in which the ratios
> were stretched very slightly. Our modern "equal temperament"
> gives all musical keys the same characteristics as one another,
> allowing a profound, innovative composer such as Bach the ability
> to modulate from one key to another almost without limits.

Please, I REALLY hope you are not equating "well-temperement" with
"equal-temperment." They are most decidely NOT the same thing, as
bach and many before and since nknew full well.

> However, the sound of an un-tempered organ or harpsichord is
> subtly and fascinatingly different.

I should say, never, ever having heard and "untempered" keyboard
instrument. That would suggest Pythogrean intonation, something
that is physically impossible on a fretted or keyed instrument
with a finite number of freat or keys.

I suspect you mean "just intoned" or "well-tempered" or other
tuning schemes which are not "equally tempered." Yes, such, on the
appropriate instrument playing the appropriate repertoire can be
most enjoyable.

> I'm going to stop dead in my
> tracks right now because this subject is just so intricate. It's
> worthy of a book.

Several of which I have on the shelf beside me as we speak, from
a reprint of Pietro Aron to Barbour's eanalysis of this and other
just intonations, to Helmholtz's Sensation of Tone, discussing
aural consonance and sissonance and the role of hole-number ratios
to Fespermann Equal-Beating Temperments and many others.

> So, yes, an octave is 2:1.

Yes, it is, but you utterly failed to answer the question as to why.

Here's a moe general explanation as to why small, simple whole-number
ratios, such as 2:1 (the octave), 3:2 (the pure 5th), 4:3 (the fourth)
5:4 (a justly intoned major third), and so on, sound pleasing.

It has to do with the concept of "coincident partials". Except in the
case of pure sine waves, musical tones have overtones or "partials"
that are whole number ratios of the original frequency. For example,
play the middle A note on a piano, organ or harpsichord, look at it
with a high-resolution spectrum analyzer, and not only will you find
fundamental tone at 440 Hz (or, 415 if you have it tuned, like my
harpsichords, about a semitone low), but you'll find prominent tones
at whole-number multiples of that, 2 times, 3 times, 4 times and so on,
at frequencies of 880, 1320, 1760 Hz and so on. This is true of each
and every one of the notes you play. Indeed, on some instruments, such
as the harpsichord, MOST of the energy is in these uppper partials
or harmonics.

Now, play the octave above A440, together with A440. See what happens?
the 2nd harmonic of A440 is "coincident" with the fundamental of the
note 1 octave up (or double the frequency), at 880:

A440 * 2 = 880
A880 * 1 = 880
--------------
Difference 0

These two notes, assuming the instrument is tuned properly, coincide
and reinforce on another. But, let's take the upper A sightly off-tune.

A440 * 2 = 880 Hz
A881 * 1 = 881
--------------
Deff: 1 Hz

That difference will be heard in a wavering of the note at a rate of 1
second. tune it further away, and the wavering gets faster, until about
the time when the difference is 6-10 Hz, when it stops being perceived
as a waver and starts being perceived as "fuzz" make it far enough away,
the difference tone perceived sounds quite discordant.

Let's look at another example. As I mentioned, an interval of a pure
fifth has the ratio of 3:2. IN our example, the fifth above A is E. It
has a frequency 3:2 times that of the fundamenatl, or:

440 * 3/2 = 660 Hz

Now, watch what happens. Take the THRID harmonic af the A and the SECOND
hamonic of the E:

A440 * 3 = 1320 Hz
E660 * 2 = 1320 Hz

Voila! coincident partials! And the same rule applies. As long as they
are close in frequency, they sound "in tune" because of the lack of
wavering.

You can extend this to more complex rations, like, as mentioned,
4:3 (4th), 5:4 (major 3rd), and so forth. The problem is that,
generally, the amount of energy diminshes as the harmonic number
increases and the efect of out-of-tuneness is less apparent. This
is one reason why music with strange interbvals may sound fine on
a piano, but intlerable on a harpsichord: there's FAR more energy
in these upper partials in a harpshord then a piano.

Of ALL these intervals, those that have small whole-number ratios,
the simplest and most obvious is the unison 1:1. Next is the octave
(2:1).

Now, the next question is, why is it called "an octave" if it's
ratio is 2:1? The word derives form the Latin for "eight."
Anonymous

"Dick Pierce" <dpierce@cartchunk.org> wrote in message
| "Richard Steinfeld" <rgsteinBUTREMOVETHIS@sonic.net> wrote in
message news:<ifpEc.19635\$Fo4.260403@typhoon.sonic.net>...
| >
| > Many harpsichordists like to couple up the mechanism so that
| > they're constantly playing simultaneous octaves, even one
octave
| > above and one below, so that three pitches are sounding.
|
| Having built 11 harpsichords, all based on historical models,
having
| examined maybe 30 others from antiquity, and many dozens more
modern
| instruments, I must say that have NEVER EVER seen an instrument
that
| can be "coupled up" as you describe.
|

I was the leading harpsichord tuner in New York for a few years;
you've heard my work. Think of your garden-variety two-manual
instrument with two sets of 8' strings, one set of 4' strings,
one set of 16 foot. I worked on them fairly often, especially
Neuperts. Or, briefly, the high-tension aviation-age Pleyels
beloved by Wanda Landowska and Rafael Puyana.

In the recording world, in the typical studio situation, the
emphasis is on practicality rather than historical accuracy.
Although I set up instruments for people like Fernando Valente,
Sylvia Kind, the NY Philharmonic, Hague Philharmonic, chamber
music recordings, etc., etc., most of my bread-and-butter was
with Zuckermann Harpsichords and especially Caroll Instrument
Rental Service. And you wanna know what most of the recordings
were that used harpsichords? They were TV commercials and top-40

So, yes, I came away from all that with respect for both
historical accuracy and respect for modern materials and designs.
On one hand, there was the day that I struggled with an authentic
with files, glue, and shims; mechanisms made with real boar's
hair springs. I played with a harpsichordist who owned a
Christopher Bannister instrument -- the builder shot crows in New
Jersey for their authentic quills! With this type of madness,
one can gain an appreciation for plastic jack guides, uniform
plastic quills, aluminum mechanicals, etc.: they're reliable.
When one is playing rapid repeated notes in Scarlatti, does one
get better musical service from natural frayed quills that'll
hang on the strings or from self-lubricating hard plastic? Will
the player's artistry be better served by having to spend hours
every week futzing with the harpsichord or from practicing the
music?

The entire world of historic performance always involves
myself, "If Bach came here today and heard our modern
instruments, what would he have preferred? What would have been
the best tools to serve his music?"

| The largest common instruments, from then and now, are two
manual
| instruments with three sets of strings. The lower manual plays
two
| strings, on sounding at unisin, the other sounding at an
octave,
| while the upper manual plays the second unison choir. The ONLY
| "coupling up" such instruments are capable of is coupling the
| uppoer manual to the lower. This does not couple up or down an
| octave, but rather at unison, so you have the capability of
playing
| the 2 unison choirs together, or adding the octave choir to the
| unison.
|

I'd actually prefer such an instrument. I've been out of this for
a long time, so things change. The larger Neuperts I recall were:
top manual -- 8' + 4', lower manual -- 8' + 16'
For the bystander, let me explain that Neupert was (is?) the
foremost manufacturer of mass-produced harpsichords. These were
based upon certain historical models, but were not authentic
reproductions. The firm also made custom instruments to
historical configurations. These custom instruments, however,
still made use of modern mechanism materials.

| The occasional and, in fact, rare example of instruments based
on the
| large Hass model will have two more choirs on the lower manual,
tuned
| and octave down and two octaves up, but such instrument are, by
far,
| exceedingly rare exceptions to the rule.
|

Yes. I've never seen such an instrument (2 octaves up). I've
worked on such rarities as pedal harpsichords and
electonically-amplified clavichords, but not that configuration.

| But, once again, there simply is no such thing in harpsichords,
modern
| or ancient, of "coupling up" by octaves.
|

I think we're splitting hairs. Yes, of course, the coupling is
between the two keyboards.

| > It makes
| > the instrument louder, an advantage for a naturally quiet
| > instrument when playing a concerto with an orchestra.
However, I
| > find such coupling obnoxious: to me, it ruins the
musicality --
| > the sound is bright, but it's also really annoying. I'd
prefer a
| > proper Baroque situation: one string at a time or two strings
in
| > unison (1:1). This is an authentic 18th Century
configuration.
|
| Sorry, but you're ignoring the entire class of authentic
Baroque
| instruments that include the octave (4') choir. Included in
that
| class are most double manual instruments of the era, from
France,
| Flanders, Germany and England.
|

Now that we're talking shop, let me continue. What bugs me is the
harpsichordist who just couples up the instrument and plays
flat-out that way from beginning to the end of the piece --
there's no contrast in the sound. A fine artist of the time, I'd
hope, would engage the 4' for occasional color or emphasis. Since
the harpsichord cannot gradate the intensity of the pluck, I see
one of the tools in the harpsichordist's bag of tricks being to
convey the -impression- of dynamic changes through the art of
phrasing, subtle (really subtle) metric changes, etc.

| > To
| > bring the ensemble into balance, just make the orchestra
really
| > small. We lose majesty and brightness; we gain subtlety,
| > excitement, and warmth. Classical music becomes more like
jazz.
| > Wow: which would you choose?
|
| I would choose authenticity to the original than something,
uhm,
| "jazzed up." :-)
|

I don't mean "jazzed up," but rather, let's say, playfulness,
interplay, in the spirit of the music of the times. I think that
figured bass was a roadmap for sensible improvisation (not
Charlie Parker on the keyboard).

| > As these ratios become more and more complex (7:1, 23:1), we
| > perceive them as being increasingly dissonant, interesting,
| > raucous, wonderful, exciting, horrible (take your pick).
|
| Uh no, not exactly. There are entire sets of small whole-number
| ratios that are quite consonant. You chose ratios that are, in
fact
| consonant. The latter example is, well, wierd in that it is so
widely
| separated as to be heard as two distinct tones.
|
| > In practice, musical ratios are not as simple as I've
presented
| > above.
|
| That much I'll certainly agree on.
|
| > I've given you the general, theoretical ideal. After a
| > certain point in musical history, many of these ratios don't
work
| > out quite "right." Bach wrote The Well-Tempered Clavier in an
| > attempt to promote a modern tuning system in which the ratios
| > were stretched very slightly. Our modern "equal temperament"
| > gives all musical keys the same characteristics as one
another,
| > allowing a profound, innovative composer such as Bach the
ability
| > to modulate from one key to another almost without limits.
|
| Please, I REALLY hope you are not equating "well-temperement"
with
| "equal-temperment." They are most decidely NOT the same thing,
as
| bach and many before and since nknew full well.
|

| > However, the sound of an un-tempered organ or harpsichord is
| > subtly and fascinatingly different.
|
| I should say, never, ever having heard and "untempered"
keyboard
| instrument. That would suggest Pythogrean intonation, something
| that is physically impossible on a fretted or keyed instrument
| with a finite number of freat or keys.
|
| I suspect you mean "just intoned" or "well-tempered" or other
| tuning schemes which are not "equally tempered." Yes, such, on
the
| appropriate instrument playing the appropriate repertoire can
be
| most enjoyable.
|

I'm probably talking about "just intonation."

| > I'm going to stop dead in my
| > tracks right now because this subject is just so intricate.
It's
| > worthy of a book.
|
| Several of which I have on the shelf beside me as we speak,
from
| a reprint of Pietro Aron to Barbour's eanalysis of this and
other
| just intonations, to Helmholtz's Sensation of Tone, discussing
| aural consonance and sissonance and the role of hole-number
ratios
| to Fespermann Equal-Beating Temperments and many others.
|
| > So, yes, an octave is 2:1.
|
| Yes, it is, but you utterly failed to answer the question as to
why.
|

????????????????

| Here's a moe general explanation as to why small, simple
whole-number
| ratios, such as 2:1 (the octave), 3:2 (the pure 5th), 4:3 (the
fourth)
| 5:4 (a justly intoned major third), and so on, sound pleasing.
|
| It has to do with the concept of "coincident partials". Except
in the
| case of pure sine waves, musical tones have overtones or
"partials"
| that are whole number ratios of the original frequency. For
example,
| play the middle A note on a piano, organ or harpsichord, look
at it
| with a high-resolution spectrum analyzer, and not only will you
find
| fundamental tone at 440 Hz (or, 415 if you have it tuned, like
my
| harpsichords, about a semitone low), but you'll find prominent
tones
| at whole-number multiples of that, 2 times, 3 times, 4 times
and so on,
| at frequencies of 880, 1320, 1760 Hz and so on. This is true of
each
| and every one of the notes you play. Indeed, on some
instruments, such
| as the harpsichord, MOST of the energy is in these uppper
partials
| or harmonics.
|
| Now, play the octave above A440, together with A440. See what
happens?
| the 2nd harmonic of A440 is "coincident" with the fundamental
of the
| note 1 octave up (or double the frequency), at 880:
|
| A440 * 2 = 880
| A880 * 1 = 880
| --------------
| Difference 0
|
| These two notes, assuming the instrument is tuned properly,
coincide
| and reinforce on another. But, let's take the upper A sightly
off-tune.
| Let's make it 881 instead:
|
| A440 * 2 = 880 Hz
| A881 * 1 = 881
| --------------
| Deff: 1 Hz
|
| That difference will be heard in a wavering of the note at a
rate of 1
| second. tune it further away, and the wavering gets faster,
| the time when the difference is 6-10 Hz, when it stops being
perceived
| as a waver and starts being perceived as "fuzz" make it far
enough away,
| the difference tone perceived sounds quite discordant.
|
| Let's look at another example. As I mentioned, an interval of a
pure
| fifth has the ratio of 3:2. IN our example, the fifth above A
is E. It
| has a frequency 3:2 times that of the fundamenatl, or:
|
| 440 * 3/2 = 660 Hz
|
| Now, watch what happens. Take the THRID harmonic af the A and
the SECOND
| hamonic of the E:
|
| A440 * 3 = 1320 Hz
| E660 * 2 = 1320 Hz
|
| Voila! coincident partials! And the same rule applies. As long
as they
| are close in frequency, they sound "in tune" because of the
lack of
| wavering.
|
| You can extend this to more complex rations, like, as
mentioned,
| 4:3 (4th), 5:4 (major 3rd), and so forth. The problem is that,
| generally, the amount of energy diminshes as the harmonic
number
| increases and the efect of out-of-tuneness is less apparent.
This
| is one reason why music with strange interbvals may sound fine
on
| a piano, but intlerable on a harpsichord: there's FAR more
energy
| in these upper partials in a harpshord then a piano.
|
| Of ALL these intervals, those that have small whole-number
ratios,
| the simplest and most obvious is the unison 1:1. Next is the
octave
| (2:1).
|
| Now, the next question is, why is it called "an octave" if it's
| ratio is 2:1? The word derives form the Latin for "eight."

Yup. I did forget to mention that. Your explanation is great! I
didn't want to go down that path since I was trying to stay
somewhat within the OP's range of experience. I think that I was
afraid that if I started on the "8" explanation, with white keys
as a visual picture, the explanation would have gotten very far
afield. So, I didn't.

Good to meet you.

Richard
Anonymous

"Richard Steinfeld" <rgsteinBUTREMOVETHIS@sonic.net> wrote in message
news:ifpEc.19635\$Fo4.260403@typhoon.sonic.net...
>
> "Geoff Wood" <geoff@paf.co.nz-nospam> wrote in message
> news:BlkEc.5435\$NA1.501831@news02.tsnz.net...
> | ras wrote:
> | > I understand that a octave have 12 'intervals' as well as how
> you do
> | > the math (twelfth root of 2) but not why an octave: "is
> itself a
> | > frequency ratio of 2:1".
> | >
> | > Any help will be most appreciated, thanks
> | > Mike
> |
> | And octave below A 440 is 220, and an octave higher is 880.
> The octave
> | above and below those is 110 and 1660Hz..

Well--1760Hz--but who's counting.

Norm Strong
Anonymous

Hang on a sec, one octave up being double the requency is fine. But this is
only valid in base 10 number systems. Is there something inherently more
natural about base 10 rather than say, base 7 ?

If humans had 6 fingers would music be different ?

geoff
Anonymous

"Geoff Wood" <geoff@paf.co.nz-nospam> wrote in message
news:NcFEc.5577\$NA1.518368@news02.tsnz.net...
>
> Hang on a sec, one octave up being double the requency is fine. But this
is
> only valid in base 10 number systems. Is there something inherently more
> natural about base 10 rather than say, base 7 ?

It's valid in any base numbering system. Nature doesn't care what numbering
system we humans use to measure it. Computers use base 2, but synthesizers
like Microsoft GS Wavetable SW Synth work just fine.

> If humans had 6 fingers would music be different ?

Musical instruments would certainly be different. Music would be different
because of that. Music theory would probaby be about the same, because the
physics of sound would be the same.

> geoff
Anonymous

"Geoff Wood" <geoff@paf.co.nz-nospam> wrote in message
news:NcFEc.5577\$NA1.518368@news02.tsnz.net...
|
| Hang on a sec, one octave up being double the requency is fine.
But this is
| only valid in base 10 number systems. Is there something
inherently more
| natural about base 10 rather than say, base 7 ?
|
| If humans had 6 fingers would music be different ?
|
|

Think of what Theloneus Monk would have turned out with two more
fingers. Crazy, man.

Richard
Anonymous

"Dick Pierce" wrote ...
> These instruments should suffer the fate that befell the
> instruments in the Palace of Versais in the early 19th
> century: they should be broken down for firewood (that
> woudln't work with a Pleyel: these ARE pianos, complete
> with enormous cast-iron frames and behemoth piano cases).

Or those plexiglass and aluminum(?) things I saw in a couple
of pop groups in the 70s.

> There are a number of well-temperments, but, in general,
> they sacrifice a little bit on the purity of the fifths and gain
> an enormous advantage in better consonance of the thirds.

A friend of mine just sent me a CD he engineered of the new
Pasi dual-temperment organ at St. Cecilia's in Omaha...
http://www.tcvomaha.com/ArchivedIssues/Sept19-2003/SEPT...
http://www.pasiorgans.com/opus14.html
Anonymous

Dick Pierce wrote:

<snip>

Richard,

I've been following this thread a bit and found it very
interesting. I must admit that I learned quite a bit from

Just out of curiosity:

Do you happen to know what type of harpsichord Keith Jarret
plays "Bach: The Goldberg Variations, ECM" on and how it is
tempered? How do you like his play? It sure is completely
different from Glenn Gould's on piano...

Cheers,

Franco
Anonymous

On Thu, 1 Jul 2004 08:16:34 +1200, "Geoff Wood"
<geoff@paf.co.nz-nospam> wrote:

>
>Hang on a sec, one octave up being double the requency is fine. But this is
>only valid in base 10 number systems. Is there something inherently more
>natural about base 10 rather than say, base 7 ?
>
>If humans had 6 fingers would music be different ?

Double is double, however you represent the numbers.
Anonymous

"Franco Del Principe" <franco.delprincipe@lis.ch> wrote in
message news:2kl846F3nu74U1@uni-berlin.de...
| Dick Pierce wrote:
|
| <snip>
|
| Richard,
|
| I've been following this thread a bit and found it very
| interesting. I must admit that I learned quite a bit from
|

Thank you.
Dick Pierce, too, has been providing a different perspective, I
feel that it's as valid as mine, although we're coming from
completely different vantage points. I do, though, agree with his
assessments of the older factory instruments -- someone had to
come first in the 20th Century. After studying performance
practice at two conservatories, and especially having been both a
performer and an instrument maintainer (and an audio person,
too), my view is that there is no one right way to build
instruments and no single right way to reproduce an "authentic"
performance. For example, if given our modern plastics and
metalurgy, what kinds of harpsichords would the builders of the

I'm a woodwind player. I have four plastic recorders built by
three different builders. Two of them were molded of phenolic and
finished by hand; the other two are stright out of their
injection molds. They are all excellent; listening to me play,
you'd think that the instruments were made of wood. Their molds

There are scholars who can point the way from limited historic
examples; we do enlightened groping, and hope that in the process
that we can share a profound musical experience with our
listeners. There are performers whose renditions make sublime
musical sense to me; others who make sound supported by
scholarship that's uninspired, repetitive, predictable, and
boring. These days, musicians (I hope) render
"historically-informed" performances, not "historically dictated"
versions.

Until Mr. Pierce replied to my post, I'd forgotten about how the
harpsichord world has denizens who are fiercely loyal to one or
another builder or school of construction. each convinced that he
knows the correct answers and that everyone else is wrong. I was
familiar with the excellent work of the Boston builders
(including the incredible woodwind artisan Friedrich von Heune).
However, being based in New York, I only saw the Hubbard and Dowd
harpsichords that Dick mentions occasionally. My experience
included a few brands other than the ones I mentioned. Most of
those were really unsuited to everyday use, since they were
constantly falling apart.

| Just out of curiosity:
|
| Do you happen to know what type of harpsichord Keith Jarret
| plays "Bach: The Goldberg Variations, ECM" on and how it is
| tempered?

I'm sorry; haven't heard the recording. One thing that I must
mention is that, like it or not, the equal-temperment system is
the modern standard. Musicians are used to it, and when playing
in an ensemble of diverse instruments, all players will attempt
to adhere to a common standard. In other words, chances are that
Jarret's instrument was tuned in modern fashion.

How do you like his play? It sure is completely
| different from Glenn Gould's on piano...
|

I haven't heard it, Franco.
Since I sometimes worked at the Columbia 30th Street studio, I
was able to get a few looks at the two pianos that Gould kept
there. They were custom Steinways that were modified to his
requirements. The patented Steinway piano action resists the
player; working against that resistance gives the performer the
ability to achieve more subtle gradations than with other (not
all) instruments. Gould went one step further; his actions were
raised closer to the strings. This reduced the maximum loudness
of the instrument, but permitted even further subtlety in the
piano-to-medium range. In order to play loud on his pianos, Gould
had to pound like hell. That's how Gould wanted it. He was able
to work with those actions to produce Bach fugues in which the
listener can pick out the different voices and follow them with
the mind, the voices having different colorations and emphasis.

By comparison, most other concert pianists can't even comprehend
the lower end of the piano's dynamic range; their playing is
almost all Forte through Fortissimo, with occasional quiet
moments to set off the bombast. It hurts my ears as well as my
sensibility.

Of course, the piano had just begun to exist in Bach's waning
years. So Gould's renditions on the piano are not authentic. I
ask myself, though, if Gould's performances were true to the
spirit of the compositions. I ask myself, "If J.S. Bach was
sitting in the control room at the 30th Street studio while Gould
was being taped, would he be picking the performance apart
angrily or would Bach been sitting there digging the music, head
bobbing, shouting 'Yeah, Yeah!,'" like the tuned-in listeners at
a bebop gig. And in my mind's eye, I suspect the second vision.

When Gould was good, he was very, very good. When he was bad, he
ones. I have a promotional interview with his producer in which
he said that he renounced the renditons of his early records (the
reasons have been repeated and are well-known). I do enjoy the
good ones very, very much. Gould had an innate understanding of
the need for keeping steady tempos, which convey the hypnotic
quality that's built into so much of Bach's work. His recordings
were played and edited to build the listener's experience from
beginning to end of each movement; we're carried upon a journey.
Most concert pianists and many modern "historically-accurate"
conductors are deaf to this -- they speed up and slow down
constantly, putting their own "emotions" into it and wrenching
the listener away from what the composer wrote. In doing a
rhythmic analysis of Bach's work while in college, I was
impressed with the with the subtle and patient way that Bach
introduces changes into his lines. The music takes its time (you
may notice that), builds and morphs, gradually. It takes a
special performer to bring this out; one dedicated to the spirit
of the music and not just his own ego.

One great thing about music is that there's so much to it that
there's always something else to learn. I will never know all the
answers, and that's just fine with me!

Richard
Anonymous

an octive higher than a freq is 2x the freq
ie a= 440 next one up= 880 and so on
news:9v44e01idksuhjs4mkg1edv9rhiv2mqgma@4ax.com...
>
> Pitch math ... can somebody explain what that: "an octave ... which
> itself is a frequency ratio of 2:1" means?
>
> This is an excerpt from the book: "Audio Explained" by Michael
> Talbot-Smith:
>
> ------
> Pitch ... the frequency ratio between two adjacent semitones is
> approximately 6%. In scientific terms the exact number is ¹²V2 (the
> twelfth root of 2).
>
> [V = suppose to the root char, cannot find the key combination for it
> on the windows system.]
>
> The reasoning behind this is that there are 12 equal semitone
> 'intervals' in an octave, which itself is a frequency ratio of 2:1.
> Each step must therefore be ¹²V2.
> ------
>
> I understand that a octave have 12 'intervals' as well as how you do
> the math (twelfth root of 2) but not why an octave: "is itself a
> frequency ratio of 2:1".
>
> Any help will be most appreciated, thanks
> Mike
>
Anonymous

Mike wrote
>>
>> I understand that a octave have 12 'intervals' as well as how you do
>> the math (twelfth root of 2) but not why an octave: "is itself a
>> frequency ratio of 2:1".

Well, you have it right there: Multiplying the 12 intervals each a
ratio 2^(1/12) apart gives (2^(1/12))^12 = 2! Voila!

Or did I miss you misunderstanding?

Per.

>>
>> Any help will be most appreciated, thanks
>> Mike
>>
>
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